http://www.vliz.be/v/index.php?title=Speciaal:NieuwePaginas&vliz=1&feed=atomKust Wiki - Nieuwe pagina's [nl]2017-10-17T05:22:21ZUit Kust WikiMediaWiki 1.27.2http://www.vliz.be/wiki/Wave_ripple_formationWave ripple formation2017-07-20T15:33:13Z<p>Dronkers J: </p>
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<div>==Introduction==<br />
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[[Image: WaveRippleFormationFig0.jpg|thumb|350px|right|Figure 1. Ripples observed at Sea Rim State Park, along the coast of east Texas close to the border with Louisiana (courtesy by Zoltan Sylvester).]]<br />
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When a sandy seabed is subject to wave action and the wave orbital motion is strong enough to move sand grains, ripples often appear. The ripples induced by wave action are called “wave ripples”; their characteristics being different from those of the ripples generated by steady flows.<br />
The most striking difference between wave ripple fields and current ripple fields is the regularity of the former. Indeed, regular long-crested wave ripple fields are often observed on tidal beaches from which the sea has withdrawn at low water (see figure 1).<br />
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The characteristics of wave ripples are described in the accompanying article [[Wave ripples]], which also explains their crucial importance for sand transport in the coastal zone (see also the articles [[Sand transport]] and [[Sediment transport formulas for the coastal environment]]).<br />
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The formation of wave ripple fields, characterized by well-defined wavelengths, is initiated by the interaction of seabed disturbances with the wave orbital motion. This article deals with the underlying processes giving rise to different wave ripple patterns.<br />
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==The mechanism of ripple formation==<br />
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Wave ripples form because the interaction of the oscillatory flow, induced by surface waves close to the bottom, with a bottom undulation of small amplitude generates steady streamings which consist of recirculating cells (see the flow visualizations of Kaneko and Honji<ref> Kaneko A. and Honji H. (1979.) Double structures of steady streaming in the oscillatory viscous flow over a wavy wall. J. Fluid Mech. 93, 727-736.</ref>). Indeed the presence of the bottom waviness induces periodic spatial variations of the streamwise oscillatory velocity component. Hence the nonlinear self-interaction of the velocity field, that is due to the convective acceleration, generates time-independent terms into momentum equation which can be balanced only by the presence of a steady velocity component. <br />
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[[Image: WaveRippleFormationFig1.jpg|thumb|400px|right|Figure 2. Steady streaming generated by an oscillatory flow over a wavy wall for <math>R_\delta=0.1</math>, <math>2 \pi \delta/\lambda=0.15</math> being the dimensionless wavenumber of the bottom waviness (adapted from Blondeaux<ref name=B90></ref>). In the figure <math>x</math> and <math>y</math> are dimensioness spatial coordinates scaled with <math>\delta</math>.]]<br />
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The spatial distribution of this steady streaming depends on the parameters of the problem, namely, 1) the ratio <math>\eta/\lambda</math> between the height <math>\eta</math> and the length <math>\lambda</math> of the bottom waviness, 2) the ratio <math>U_0/(\omega\lambda)</math> between the amplitude <math>U_0/\omega</math> of the fluid displacement oscillations and <math>\lambda</math> (<math>U_0</math> and <math>\omega</math> denote the amplitude and angular frequency of the velocity oscillation of the fluid particles), 3) the ratio between <math>U_0/\omega</math> and the conventional thickness <math>\delta=\sqrt{2 \nu/\omega}</math> of the viscous bottom boundary layer (the reader should notice that <math>U_0/(\omega \delta)=U_0\delta/(2\nu)=R_\delta/2</math>, <math>R_\delta</math> being the Reynolds number characteristic of the bottom boundary layer). As these parameters are varied, different balances take place into momentum equation among the local acceleration, the convective acceleration, the pressure gradient and the viscous terms. A discussion of the relative importance of these terms and of the different approches which are used to determine the flow field as function of the values of the parameters (see among others Lyne <ref>Lyne W.H. (1971). Unsteady viscous flow over a wavy wall. J. Fluid Mech. 50, 33-48.</ref>, Sleath <ref name=S76>Sleath J.F.A. (1976). On rolling-grain ripples. J. Hydraul. Res. 14, 69-81.</ref>, Blondeaux <ref name=B90>Blondeaux P. (1990). Sand ripples under sea waves. Part 1. Ripple formation. J. Fluid Mech. 218, 1-17.</ref>, Vittori <ref name=V89>Vittori G. (1989). Non-linear viscous oscillatory flow over a small amplitude wavy wall J. Hydraulic Research 27 (2), 267-280.</ref>) can be found in Hara and Mei <ref name=HM>HaraT. and Mei C.C. (1990). Oscillating flows over periodic ripples. J. Fluid Mech. 211,<br />
183-209.</ref>.<br />
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Figure 2 shows an example of the steady recirculating cells determined on the basis of Blondeaux's <ref name=B90></ref> analysis. When the steady velocity component close to the bed is directed from the troughs towards the crests of the bottom undulation and is strong enough to drag the sediments, the sediments tend to move from the troughs towards the crests. The tendency of sediments to pile up near the crests is opposed by the component of the gravity force acting in the down-slope direction. It follows that the growth of the amplitude of the bottom waviness is controlled by a balance between these two effects. If gravity prevails over drag, the amplitude decays, otherwise it grows leading to the appearance of ripples. Moreover, Sleath <ref name=S76></ref> argued that the effect is stronger for ripple wavelengths of the same order of magnitude as the amplitude of the fluid displacement oscillations since in this case particle settling locations will tend, after several cycles, to the nearest ripple crest.<br />
Once formed, ripples do not continue to grow indefinitely because the steady streaming is modified by nonlinear effects and, as the ripples get steeper, an equilibrium configuration is attained.<br />
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==Models of ripple formation==<br />
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Idealized models allow to predict the conditions leading to the appearance of ripples and some of their geometrical characteristics. Even though they are not straightforward to be used, it is worthwhile to introduce them and briefly summarize the results they provide.<br />
First of all, they show that the ripple wavelength and height do not depend on just one dimensionless parameter (this finding is also supported by some laboratory observations) and they suggest the most appropriate parameters to be used to improve the empirical predictors described in article [[Wave ripples]].<br />
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===Two-dimensional ripples===<br />
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The formation of two-dimensional ripples can be investigated by determining the time development of a two-dimensional random perturbation of the sea bottom characterized by a small amplitude and subject to a harmonic oscillatory flow (see [[Stability models]]). The assumption of a small amplitude of the perturbation allows the hydrodynamic and morphodynamic problems to be linearized and solved by considering the time development of a generic sinusoidal component of the bottom waviness characterized by a wavenumber <math>\alpha=2 \pi /\lambda</math> with <math>\lambda</math> of order <math>\delta</math>.<br />
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<math>y= \eta (x,t) = A(t) e^{i \alpha x} + c.c. . \qquad(1)</math><br />
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To define the bottom profile (1), a Cartesian coordinate system is introduced with the origin located on the average bottom, the <math>y</math>-axis vertical and pointing upwards and the <math>x</math>-axis aligned in the direction of the fluid oscillations.<br />
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The hydrodynamic problem (the determination of the oscillatory flow over a wavy bed) was first solved by Sleath <ref name=S76></ref>, who considered moderate values of the Reynolds number such that the flow regime is laminar and both small and large values of the ratio between the ripple wavelength and the fluid displacement oscillations. Later Blondeaux <ref name=B90></ref> solved the problem for fluid displacement oscillations of the same order of magnitude as the wavelength of the bottom waviness.<br />
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The time development of the bottom waviness can be evaluated by considering sediment continuity (Exner) equation, after the introduction of an appropriate sediment transport predictor which relates the sediment flux to the bottom shear stress.<br />
The linearized sediment continuity equation leads to <br />
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<math>\frac{d A}{dt}= \gamma (t) A(t) , \qquad(2) </math><br />
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where <math>\gamma</math> is a complex quantity (<math>\gamma=\gamma_r + i \gamma_i</math>) (see Blondeaux<ref name=B90></ref>).<br />
The time average <math>\overline \gamma_r</math> of <math>\gamma_r</math> describes the growth/decay of the bottom waviness, depending on its positive/negative value and is named 'growth rate', while the time average <math>\overline \gamma_i</math> of <math>\gamma_i</math> is related to the migration speed of the bottom forms. The periodic parts of <math>\gamma</math>, characterized by a vanishing time average (namely <math>\gamma_r-\overline \gamma_r</math> and <math>\gamma_i-\overline \gamma_i</math>), turn out to be small and describe the vertical and horizontal oscillations of the bottom profile, around its average position, which take place during the wave cycle.<br />
If the small oscillations of the bottom profile around its averaged position are neglected, the bottom profile turns out to be described by<br />
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<math>\eta( x, t) = A_0 \exp \left[ \overline \gamma_r \tau \right] \exp \left[ i \alpha \left( x + \frac{ \overline \gamma_i}{\alpha} \tau \right) \right] + c.c. , \qquad (3) </math><br />
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where <math>\tau=Q t</math> is the morphodynamic temporal scale and the value of <math>Q</math> depends on the formula used to quantify the sediment transport rate.<br />
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The analysis of Blondeaux <ref name=B90></ref> shows that, at the leading order of approximation, <math>\overline \gamma_i</math> vanishes and the bottom forms do not migrate, because of the symmetry of the problem. On the other hand the growth rate <math>\overline \gamma_r</math> of the bottom perturbation depends on the dimensionless parameters <math>R_\delta=U_0 \delta/\nu, R_p=\sqrt{\left( \rho_s/\rho-1\right) g d^3}/\nu, \psi_d=U_0^2/\left( \left( \rho_s/\rho-1\right) g d \right), s = \rho_s/\rho</math> or their combinations, where <math>\rho_s</math> and <math>\rho</math> are the densities of the sediment particles and the fluid, respectively, <math>g</math> is gravity acceleration and <math>d</math> is the grain size.<br />
When <math>\overline{\gamma_r}</math> is plotted as function of <math>\alpha</math> for assigned values of the relative density <math>s</math> and the flow and sediment Reynolds numbers, denoted by <math>R_\delta</math> and <math>R_p</math> respectively, the results of Blondeaux<ref name=B90></ref> show that a critical value <math>\psi_{d,crit}</math> of <math>\psi_d</math> exists such that for values of <math>\psi_d</math> larger than the critical value, perturbation components characterized by wavenumbers falling within a restricted range have a positive growth rate and grow exponentially in time. Increasing values of <math>\psi_d</math> lead to an increase of the range of unstable wavenumbers while decreasing values of <math>\psi_d</math> make the unstable wavenumbers to collapse around a single value of the wavenumber (<math>\alpha_{crit}</math>) named 'critical wavenumber'.<br />
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[[Image: WaveRippleFormationFig2.jpg|thumb|400px|right|Figure 3. Region of existence of ripples and flat beds in the <math>(\psi_d,R_\delta)</math>-plane. Experimental observations for <math>5<R_d<15</math> and <math>s=2.65, \mu=0.15, n=0.4</math> (adapted from Blondeaux<ref name=B90></ref>).]]<br />
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Figure 3 shows the regions of existence of ripples (<math>\psi_d > \psi_{d,crit}</math>) and flat beds (<math>\psi_d < \psi_{d,crit}</math>) in the <math>(R_\delta,\psi_d)</math>-plane, as predicted by the stability analysis, along with the experimental observations of Blondeaux et al.<ref name=B88>Blondeaux P., Sleath J.F.A. and Vittori G. (1988). Experimental data on sand ripples in an oscillatory flow. Rep. 01/88. Inst. of Hydraulics, University of Genoa.</ref> who observed ripple formation using an oscillating tray apparatus. <br />
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[[Image: WaveRippleFormationFig3.jpg|thumb|400px|right|Figure 4. Critical value <math>\alpha_{crit}</math> of <math>\alpha</math> plotted versus the flow Reynolds number <math>R_\delta</math> for <math>s=2.65, \mu=0.15, n=0.4</math> and different values of <math>R_d=R_p\sqrt{\psi_d}</math>.]]<br />
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As pointed out in the accompanying article [[Wave ripples]], the dynamics of sediment grains and bed forms in oscillating trays is different from that forced by oscillatory flows in water tunnels or under sea waves. Indeed, although the fluid velocity distribution relative to axes fixed in the bed is the same, the forces acting on a sediment particle are not the same. This is because the force <math>\rho U_0 \omega V</math> on a particle of volume <math>V</math> due to the pressure gradient in an oscillatory flow is not equal to the inertia force <math>\rho_s U_0 \omega V</math> on a similar particle in an oscillating tray. On the other hand, the force on the particle due to the shear stress is the same in both cases. However, the measurements of Zala Flores and Sleath <ref name=ZS>Zala Flores N. and Sleath J.F.A. (1998). Mobile layer in oscillatory sheet flow. J. Geophys. Res. 103 (N. C6) 12783-12793.</ref> show that the movement of sediment grains is dominated by the shear stress and inertia plays a minor role when the parameter <math>\frac{U_0 \omega}{(s - 1) g} < 0.1</math> . The reader can easily verify that the data described in the following satisfy this condition. In figure 3, the theoretical values are obtained for <math>s=2.65</math> and different values of <math>R_p</math> such that <math>R_d=U_0d/\nu=R_p \sqrt{\psi_d}=10</math>, while the experimental observations are characterized by values of <math>R_d</math> falling in the range <math>(5,15)</math>.<br />
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Notwithstanding the quantitative differences between the theoretical results and the experimental data, the stability analysis appears to provide a reliable description of the process which leads to ripple formation. Moreover, the model shows that coarser sediments, which are characterized by larger values of <math>R_p</math>, give rise to longer ripples. This theoretical finding is in agreement with laboratory measurements as shown in figure 4, where the critical values of <math>\alpha_{crit}</math> are plotted versus the Reynolds number <math>R_\delta</math> of the bottom boundary layer for different values of the sediment Reynolds number <math>R_d=R_p \sqrt{\psi_d}</math>.<br />
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The comparison between the theoretical values and the experimental measurements shows that the analysis underpredicts the observed wavelengths but a qualitative agreement with the laboratory measurements can be observed. Moreover, figure 3 shows that the ripple wavelength can not be predicted on the basis of just one parameter since both <math>R_\delta</math> and <math>R_d</math> affect the length of the bedforms.<br />
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==Rolling-grain ripples and vortex ripples==<br />
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[[Image: WaveRippleFormationFig4.jpg|thumb|350px|left|Figure 5. Visualization of the sediments picked-up from the bed and carried into suspension by the vortices shed at the crests of large amplitude ripples in an oscillatory flow (courtesy of Dr. Megale).]]<br />
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A linear approach, which considers ripples characterized by small (strictly infinitesimal) amplitudes, cannot follow the time development of the bottom forms for long times and determine their equilibrium amplitude. The prediction of the equilibrium amplitude of ripples is quite important since for large amplitudes the flow separates from the crests of the ripples and vortices are shed which modify the mechanism of sediment transport (see figure 5).<br />
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A model to determine the temporal growth of ripples for long times and to predict their equilibrium amplitude was developed by Vittori and Blondeaux<ref name=VB90>Vittori G., Blondeaux P. (1990). Sand ripples under sea waves. Part 2. Finite amplitude development. J. Fluid Mech. 218, 19-39.</ref> taking into account nonlinear effets but assuming that they are weak, i.e. when the amplitude of the ripples is moderate and the bottom boundary layer does not separate from the ripple crests. <br />
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[[Image: WaveRippleFormationFig4a.jpg|thumb|450px|left|Figure 6a. Rolling-grain ripples (courtesy of John F.A. Sleath).]]<br />
[[Image: WaveRippleFormationFig4b.jpg|thumb|450px|right|Figure 6b. Vortex ripples (courtesy of John F.A. Sleath).]]<br />
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Vittori and Blondeaux<ref name=VB90></ref> showed that, for fixed values of <math>s</math> and <math>R_p</math>, the plane <math>(R_\delta,\psi_d)</math> can be divided into three regions.<br />
In the first region, defined by values of <math>\psi_d</math> smaller than <math>\psi_{d,crit}</math>, the flat bottom is stable and ripple do not appear. For values of <math>\psi_d</math> larger than <math>\psi_{d,crit}</math> the linear analysis of Blondeaux<ref name=B90></ref> predicts the formation of ripples. In this region, Vittori and Blondeaux<ref name=VB90></ref> identified two sub-regions. In one sub-region the ratio between the predicted height and length of the ripples at equilibrium is smaller than about <math>0.1</math> and the analysis, according with the criterion suggested by Sleath<ref name=S84>Sleath J.F.A. (1984) Sea bed mechanics. John Wiley and Sons.</ref>, predicts the appearance of rolling grain ripples as equilibrium bedforms (see figure 6a). In the other sub-region, the analysis shows that no equilibrium is possible assuming that nonlinear effects are weak. In this case the amplitude of the bottom forms grows and nonlinear effects become increasingly more important till the boundary layer separates from the ripple crests and vortex ripples are generated (see figure 6b).<br />
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A comparison between theoretical predictions and the experimental observations of Blondeaux et al. <ref name=B88></ref> and Horikawa and Watanabe<ref name=HW></ref> is shown in figure 7.<br />
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[[Image: WaveRippleFormationFig5ab.jpg|thumb|800px|center|Figure 7. Regions in the <math>(R_\delta,\psi_d)</math>-plane where a flat bed, rolling-grain ripples, and vortex ripples are expected to appear according to Vittori and Blondeaux<ref name=VB90></ref>. Comparison between the theoretical predictions and the laboratory observations. Left panel: <math>R_d=R_p\sqrt{\psi_d}=10, \beta=0.15, s=2.65</math> and experimental data by Blondeaux et al. <ref name=B88></ref> (<math>5<R_d<10</math>); Right panel: <math>R_d=R_p\sqrt{\psi_d}=40, \beta=0.15, s=2.65</math> and experimental data by Horikawa and Watanabe <ref name=HW>Horikawa K. and Watanabe, A. (1968). Laboratory study on oscillatory boundary layer flow. Proc. 11th Coastal Eng. Conf., 467-486.</ref> (<math>30<R_d<50</math>).]]<br />
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===Brick-pattern ripples===<br />
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[[Image: WaveRippleFormationFig6a.jpg|thumb|350px|right|Figure 8. Brick-pattern ripples (courtesy of John F.A. Sleath).]]<br />
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Vittori and Blondeaux<ref name=VB92>Vittori G., Blondeaux P. (1992). Sand ripples under sea waves. Part 3. Brick-pattern ripple formation. J. Fluid Mech. 239, 23-45.</ref> developed an idealized model for the formation of brick-pattern ripples (see figure 8) by considering the time development of a sandy bottom subject to an oscillatory flow when three-dimensional initial disturbances of the bottom profile are present. The analysis shows that brick-pattern ripples can be originated by the simultaneous growth of two-dimensional and three-dimensional components of the initial disturbance, which interact with a mechanism similar to that described by Craik<ref name=C>Craik A.D.D. (1971). Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393-413.</ref> in a different context. The results allow to identify the regions in the parameter space where brick-pattern ripples appear.<br />
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In summary, on the basis of the analyses of Blondeaux<ref name=B90></ref>, Vittori and Blondeaux<ref name=VB90></ref> and Vittori and Blondeaux<ref name=VB92></ref> the plane <math>(R_\delta, \psi_d)</math> (for fixed values of <math>s</math> and <math>R_d</math>) can be divided into four different regions where the stability analyses predict i) stable flat bed, ii) rolling-grain ripples, iii) vortex ripples, iv) brick-patterns ripples. Figure 9 shows the different regions in the <math>(R_\delta,\psi_d)</math>-plane for fixed values of <math>s</math> and <math>R_d</math> along with the experimental results of Sleath and Ellis<ref name=SE>Sleath J.F.A. and Ellis A.C. (1978). Ripple geometry in oscillatory flow. Univ. Cambridge Dept. Engr. Rept. A/Hydraulics TR2, 15 pp.</ref> and Horikawa and Watanabe<ref name=HW></ref>. <br />
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[[Image: WaveRippleFormationFig7.jpg|thumb|400px|right|Figure 9. Limiting curves dividing the <math>(R_\delta, \psi_d)</math>-plane in regions where a flat bed, rolling grain ripples, two-dimensional vortex ripples, brick-pattern ripples are expected to form (<math>R_d=R_p\sqrt{\psi_d}=40, s=2.65, \beta= 0.15</math>). Also shown are experimental data by Sleath and Ellis<ref name=SE></ref> and Horikawa and Watanabe<ref name=HW></ref> for <math>35 < R_\delta < 45</math> (white points = rolling grain ripples, black points = vortex ripples, triangles = brick pattern ripples). Adapted from Vittori and Blondeaux<ref name=VB92></ref>.]]<br />
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===Tile ripples===<br />
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Field and experimental observations show the existence of hexagonal ripples, named tile ripples. Roos and Blondeaux<ref name=RB>Roos P.C., Blondeaux P. (2001). Sand ripples under sea waves. Part 4. Tile ripple formation. J. Fluid Mech. 447, 227-246.</ref> developed a model similar to that of Vittori and Blondeaux<ref name=VB92></ref> but they considered the forcing flow generated by the simultaneous presence of two surface waves characterized by the same angular frequency but different amplitudes and directions of propagation, such as the wave field which can be observed when a wave is partially reflected by a coastal structure. In this case, close to the bottom, the irrotational flow is not unidirectional but characterized by an elliptical behaviour.<br />
The model of Roos and Blondeaux <ref name=RB></ref> predicts the conditions for the formation of tile ripples.<br />
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==Models of ripple migration due to waves only==<br />
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When a steady current is superimposed to surface waves, the so-called wave-current ripples are observed, the characteristics of which are somewhat between those of wave ripples, previously described, and those of current ripples. Since the flow, generated close to the bottom by sea waves, is characterized by an oscillatory velocity component and by a steady velocity component originated by nonlinear effects <ref>Longuet-Higgins M.S. (1953). Mass transport in water waves. Philos. Trans. R. Soc. 345. 535-581.</ref>, the ripples generated by sea waves of large amplitude have geometrical and kinematical characteristics which are similar to those of the ripples generated by the interaction between waves and currents.<br />
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[[Image: WaveRippleFormationFig9.jpg|thumb|500px|right|Figure 10. Equilibrium profile of the ripples predicted by the stability analysis of Blondeaux et al. <ref name=B16>Blondeaux P., Foti E., Vittori G. (2015). A theoretical model of asymmetry wave ripples. Phil. Trans. R. Soc. A 373 (2033). pii: 20140112. doi: 10.1098/rsta.2014.0112.</ref> for <math>R_\delta=15, R_d= 25, \frac{\sqrt(\psi_d-\sqrt{\psi_{dc}}}{\sqrt{\psi_{dc}}} = 0.1, 2 \pi \delta/\lambda= 0.364, \delta/L = 0.004</math> (continuous line, considering <math>O(\delta/L)</math>-effects; broken line, neglecting <math>O(\delta/L)</math>-effects). The ratio between the ripple height <math>\eta</math> and the ripple wavelength <math>\lambda</math> is abount <math>0.067</math>.]]<br />
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The distinguishing geometric characteristic of the ripples generated by large amplitude waves<br />
is the asymmetry of their profile (see figure 10).<br />
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Moreover, the presence of a steady streaming induces the migration of the bottom forms. Blondeaux et al. <ref name=B20>Blondeaux P., Foti E., Vittori G. (2000). Migrating sea ripples. European Journal of Mechanics - B/Fluids 19 (2), 285-301.</ref> investigated whether the steady velocity component has a destabilizing or a stabilizing effect on the formation of ripples. Moreover, they determined the migration speed of the ripples.<br />
This information is of practical interest since it is common practice to evaluate the average sediment transport rate from measurements of ripple migration assuming that the sediment transport rate is related to the migration speed times the height of the ripples <ref name=FD>Fredsøe J. and Deigaard R. (1992). Mechanics of coastal sediment transport. Advances Series on Ocean Engineering 3 World Scientific.</ref>.<br />
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When nonlinear effects due to large amplitude sea waves are taken into account to determine the forcing flow, the value of <math>\gamma</math> turns out to have both a real <math>\gamma_r</math> part and imaginary <math>\gamma_i</math> part. The value of <math>\overline \gamma_r</math> differs from that determined by Blondeaux<ref name=B90></ref> because of terms of order <math>a/L</math> and the migration speed of the bottom forms is proportional to the steepness of the surface wave (<math>a</math> and <math>L</math> denote the amplitude and the length of the sea waves). A discussion of the results of the linear analysis can be found in Blondeaux et al. <ref name=B20></ref>, where the interested reader can also find an exhaustive discussion of the influence of second order terms in the wave slope on the stability of the sea bottom.<br />
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Figure 11 shows a comparison between the migration speed predicted by the theoretical analysis<br />
and that measured by Blondeaux et al. <ref name=B20></ref> during laboratory experiments.<br />
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[[Image: WaveRippleFormationFig10.jpg|thumb|450px|left|Figure 11. Theoretical and experimental values of the dimensioless migration speed of ripples plotted versus <math>R_\delta</math>. The theoretical values are obtained by evaluating the sediment transport rate by means of Hallermeier's formula <ref>Hallermeier R.J. (1982). Oscillatory bedload transport: data review and simple formulation. Continental Shelf Res. 1, 159-190.</ref> and the migration speed is scaled introducing the morphodynamic time scale <math>\tau</math> <ref name=B20></ref>. The experimental measurements are for <math>5<R_d<10</math> (white circles), <math>10<R_d < 15</math> (black circles), <math>15 <R_d < 20</math> (white triangles), <math>20 < R_d< 25</math> (black triangles), <math>25 < R_d < 30 </math> (while diamond).]]<br />
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[[Image: WaveRippleFormationFig11.jpg|thumb|450px|right|Figure 12. Theoretical value of the symmetry index plotted versus <math>U_s/U_0</math>, where the value of <math>U_s/U_0</math> for a monochromatic wave is assumed to be <math>3 \pi \delta R_\delta/(4L) </math>. <math>R_d = 25</math> and <math>R_\delta = 15,30</math> are values characteristic of Blondeaux et al.'s<ref></ref> (2000) experiments. The experimental data of Inman<ref name=I></ref>, Tanner<ref name=T></ref>, Blondeaux et al. <ref name=B20></ref> are also shown. The values of <math>\psi_d</math> are related to <math>F_d</math> by <math>\psi_d=F_d^2</math>.]]<br />
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The analysis of Blondeaux et al. <ref name=B20></ref> neglects the nonlinear effects related to the amplitude of the bottom forms. Hence, the bottom profile turns out to be sinusoidal and its amplitude can not be determined. To describe the process, which shapes the bottom profile giving rise to bottom forms characterized by crests sharper than the troughs and up-current slopes smaller than the down-current slopes, Blondeaux et al. <ref name=B16></ref> considered the interaction of the bottom perturbation with itself.<br />
<br />
Figure 10 shows the equilibrium profile of a ripple obtained by means of the theory, for fixed values of the parameters along with that obtained neglecting <math>O(a/L)</math> effects. The characteristic profile of ripples affected by a steady streaming can be recognized from the lee side of the ripple being steeper than the stoss side. The steady velocity component in figure 10 is from the right to the left. In this case the ratio <math>\lambda_2/\lambda_1</math> turns out to be <math>1.81</math>. The experimental data by Inman<ref name=I>Inman D.L. (1957). Tech. Mem. U.S. Beach Erosion N. 100.</ref>, Tanner<ref name=T>Tanner W.F. (1971). Numerical estimates of ancient waves, water depth and fetch. Sedimentology 16, 71-88.</ref>, who used sand, and those described in Blondeaux et al. <ref name=B20></ref>, who used high density plastic particles, show that the symmetry index increases as the ratio between the steady velocity component and the amplitude of the oscillating one increases but a limiting value exists. The same dependence is observed in the theoretical results presented in figure 12. Interestingly, for strong steady currents, the different curves tend to converge towards a common value that appears to be the maximum value of the symmetry index and ranges between <math>2</math> and <math>3</math>.<br />
A quantitative comparison between experimental observations and theoretical findings can be made looking at table 1, which shows the values of <math>\lambda_2/\lambda_1</math> detected during experiments nr. 39, 40, 42 of Blondeaux et al. <ref name=B20></ref> along with the theoretical values computed for the same parameters of the problem.<br />
<br />
{| style="border-collapse:collapse;background:ivory;" cellpadding=5px align=center width=50%<br />
|+ Table 1. Experimental and theoretical values of <math>\lambda_2/\lambda_1</math> for the experiments of Blondeaux et al. <ref name=B20></ref> characterized by the presence of rolling grain ripples. <br />
|- style="font-weight:bold; text-align:center; background:lightgrey" <br />
! width=10% style=" border:1px solid gray;"| exp. nr.<br />
! width=10% style=" border:1px solid gray;"| <math>(\lambda_2/\lambda_1)_{exp.}</math><br />
! width=10% style=" border:1px solid gray;"| <math>(\lambda_2/\lambda_1)_{theor.}</math><br />
|- <br />
| style="border:1px solid gray;"| <br />
| style="border:1px solid gray;"|<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| 39<br />
| style="border:1px solid gray;"| 1.19<br />
| style="border:1px solid gray;"| 1.17<br />
|- <br />
| style="border:1px solid gray;"| 40<br />
| style="border:1px solid gray;"| 1.23<br />
| style="border:1px solid gray;"| 1.17<br />
|- <br />
| style="border:1px solid gray;"| 42<br />
| style="border:1px solid gray;"| 1.30<br />
| style="border:1px solid gray;"| 1.20<br />
|}<br />
<br />
<br />
For experiments nr. 30, 35, 36, 41, the theory predicts the formation of vortex ripples and hence the symmetry index can not be computed.<br />
<br />
<br />
==Ripple formation: the turbulent boundary layer case==<br />
<br />
[[Image: WaveRippleFormationFig12.jpg|thumb|500px|right|Figure 13. Ratio between the amplitude <math>U_0/\omega</math> of the fluid displacement oscillations and the wavelength <math>\lambda</math> of the ripples plotted versus the parameter <math>\rho d/(\rho_s-\rho) g T^2</math>. The continuous lines are the theoretical predictions of the linear stability anaysis of Foti and Blondeaux<ref name=FB></ref> developed assuming the flow regime to be turbulent. The experimental data of Manohar<ref>Manohar M. (1955) 'Mechanics of bottom sediment movement due to wave action', Tech. Memo. 75, 121 pp., U.S. Army Corps of Eng., Beach Erosion Board, Washington, D.C.</ref> (points) and Sleath<ref name=S76></ref> (open circles) are taken from the book of Sleath<ref></ref> (adapted from Foti and Blondeaux<ref name=FB></ref>).]]<br />
<br />
The analyses previously summarized explain the appearance of ripples and predict their characteristics at incipient formation but for moderate values of the Reynolds number, such that the flow regime is laminar. For field conditions, the Reynolds number <math>R_\delta</math> is often larger than its critical value and the flow regime is turbulent. The model of Blondeaux (1990) was extended to the turbulent regime by Foti and Blondeaux<ref name=FB>Foti E., Blondeaux P. (1995). Sea ripple formation: the turbulent boundary layer case. Coastal Eng., 25 (3-4), 227-236.</ref> who considered the Reynolds averaged momentum equations and used a simple turbulence model. They evaluated, in particular, the Reynolds stresses by using a constant turbulent eddy viscosity <math>\nu_T</math>. In order to obtain a reasonable velocity profile, as suggested by Sleath<ref name=S91>Sleath J.F.A. (1991). Velocities and shear stresses in wave-current flows. J. Geophys. Res. 96(C8), 15237-15244.</ref> and by Engelund and Fredsøe<ref name=EF>Engelund F. and Fredsøe J. (1982). Sediment ripples and dunes. Ann. Rev. Fluid Mech. 14, 13-37.</ref> in another context, they further replaced the no-slip condition at the bottom by a partial slip condition.<br />
Figure 13 shows a comparison between laboratory data and the theoretical predictions made by means of this model.<br />
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<br />
==Vortex ripples==<br />
<br />
[[Image: WaveRippleFormationFig13.jpg|thumb|500px|right|Figure 14. Vorticity contours of the oscillatory flow over a wavy wall <math>R_\delta=50, \eta/\lambda=0.15, U_0/(\omega \lambda)=0.75</math> (The vorticity isolines are <math>0.15 U_0/\omega</math> apart, thick lines = clockwise vorticity, thin lines = counter-clockwise vorticity). Panel a) <math>\omega t = \pi/2</math>; Panel b) <math>\omega t = \pi</math>; Panel c) <math>\omega t =5 \pi/4</math>; Panel d) <math>\omega t = 3 \pi/2</math>; Panel e) <math>\omega t = 2 \pi</math>. Adapted from Blondeaux and Vittori<ref name=BV91></ref>.]]<br />
<br />
The models based on linear and weakly nonlinear stability analyses are no longer valid when the parameters of the problem are far from the critical conditions.<br />
In this case, nonlinear effects become relevant and a perturbation approach can be used no longer. Under these circumstances, only numerical simulations of momentum and Exner equations can be used to determine the fluid flow and the time development of the bottom forms.<br />
<br />
The oscillatory flow over vortex ripples was first determined by the numerical integration of momentum equation by Shum<ref>Shum K.T. (1988) A numerical study of vortex dynamics over rigid ripples. PhD Thesis M.I.T. Dep. of Civil Engineering, Cambridge, Mass.</ref> and Blondeaux and Vittori<ref name=BV91>Blondeaux P., Vittori G. (1991). Vorticity dynamics in an oscillatory flow over a rippled bed. J. Fluid. Mech. 226, 257-289.</ref>.<br />
Figure 14 shows the spanwise component of vorticity over two-dimensional vortex ripples<br />
at different phases from the beginning of the numerical simulation, as computed by the latter authors. When the fluid moves from the left to the right, clockwise vorticity is generated along the bottom profile and tends to roll up and to generate a well defined vortex (figure 14a,b).<br />
<br />
As the external fluid velocity reverses its direction, the clockwise vortex is no longer fed but it is convected from the right to the left by the external flow (figure 14c,d). Simultaneously, counter-clockwise vorticity is shed from the crest and the phenomenon repeats similarly during the following half cycle (figure 14e).<br />
Of course, the size, the strength and the number of vortex structures generated by the oscillatory flow over a rippled bed depends on the parameters of the problem. In particular, these first simultations considered moderate values of the Reynolds number such that the flow regime can be assumed to be laminar.<br />
<br />
[[Image: WaveRippleFormationFig14.jpg|thumb|500px|right|Figure 15. Time evolution of the crests of the ripples generated by the growth of a sandy Gaussian hump which interacts with an oscillatory flow characterized by <math>U_0=0.26</math> m/s, <math>T=6</math> s, <math>d=0.3</math> mm (adapted from Marieu et al. <ref name=M></ref>.]]<br />
<br />
Nowadays, the power of computers is such as to allow the evaluation of the turbulent flow field<br />
by means of Direct Numerical Simulations (DNS) of Navier-Stokes and continuity equations or using Large Eddy Simulations (LES) (Scandura et al. <ref>Scandura P., Vittori G. and Blondeaux P. (2000). Three-dimensional oscillatory flow over steep ripples. J. Fluid Mech. 412, 355-378.</ref>, Barr and Slinn<ref>Barr B. and Slinn, D. (2004). Numerical simulation of turbulent, oscillatory flow over<br />
sand ripples. J. Geophys. Res., 109 (C9), 1-19.</ref>, Zedler and Street<ref>Zedler E.A. and Street R.L. (2006). Sediment transport over ripples in oscillatory flow. J. Hydraul. Eng. A.S.C.E. 132 (2), 1-14</ref>, Chalmoukis and Dimas<ref>Chalmoukis I.A. and Dimas A.A. (2016). Numerical simulation of oscillatory flow over 3-D vortex ripples using the Immersed Boundary Method. 26th Int. Ocean and Polar Eng. Conf., 26 June-2 July, Rhodes, Greece. ISOPE-I-16-549.</ref>, Leftheriotis and Dimas<ref>Leftheriotis G. and Dimas A. (2016). Large Eddy Simulation of oscillatory flow and mor-<br />
phodynamics over ripples. Proc. 35th Conference on Coastal Engineering, Antalya, Turkey,<br />
2016.</ref>). However, for practical applications, it is convenient to compute the turbulent oscillatory flow over vortex ripples by using the Reynolds averaged equations and a turbulence model.<br />
<br />
Andersen<ref>Andersen K.H. (1999). The dynamics of ripples beneath surface waves and topics in shell models of turbulence. PhD thesis, Det Naturvidens-kabelige Fakultet Københavns Universitet.</ref> was one of the first to couple the evaluation of the turbulent flow by means of a RANS approach with the evaluation of the time development of the bottom profile computed by means of Exner equation. More recently, Marieu et al. <ref name=M>Marieu V., Bonneton P., Foster D.L. and Ardhuin F. (2008). Modeling of vortex ripple morphodynamics. J. Geophys. Res. 113 (C09007), 1-15.</ref> used a similar approach to simulate ripple growth from a quasi-flat bed. Turbulence closure was achieved by means of the model of Wilcox<ref>Wilcox D.C. (1988). Re-assessment of the scale-determining equation for advanced turbulence models. AIAA Journal 26 (11), 1299-1310.</ref> while the sediment transport rate took into account both the bed load contribution and the suspended load contribution. Moreover, the morphology module simulated the avalanches that take place at the crests of the ripples when the steepness of the profile becomes too large. Figure 15 shows the time development of the position of the crests of the ripples generated by the interaction of an oscillatory flow with a Gaussian hump located at the middle of the computational domain, It clearly appears that further ripples are generated around the initial hump. Later these ripples are characterized by a complex nonlinear dynamics; they migrate, split, merge, annihiliate and they eventually attain an equilibrium configuration.<br />
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==Related articles==<br />
<br />
[[Wave ripples]]<br />
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[[Stability models]]<br />
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[[Bedforms and roughness]]<br />
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[[Sand transport]]<br />
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[[Sediment transport formulas for the coastal environment]]<br />
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==References==<br />
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<references/><br />
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{{2Authors<br />
|AuthorID1=14087<br />
|AuthorFullName1= Paolo Blondeaux<br />
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|AuthorID2=14090 <br />
|AuthorFullName2= Giovanna Vittori<br />
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}}<br />
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[[Category:Land and ocean interactions]]<br />
[[Category:Geomorphological processes and natural coastal features]]<br />
[[Category:Coastal processes, interactions and resources]]<br />
[[Category:Coastal and marine natural environment]]</div>Dronkers Jhttp://www.vliz.be/wiki/Shallow-water_wave_theoryShallow-water wave theory2017-07-03T18:34:51Z<p>Dronkers J: Created page with "== Introduction== This article explains some theories of periodic progressive waves and their interaction with shorelines and coastal structures. The first section provides a..."</p>
<hr />
<div>== Introduction==<br />
<br />
This article explains some theories of periodic progressive waves and their interaction with shorelines and coastal structures. The first section provides a descriptive overview of the generation of wind waves, their characteristics, the processes which control their movement and transformation. The following sections describe some aspects of wave theory of particular application in coastal engineering. Some results are quoted without derivation, as the derivations are often long and complex. The interested reader should consult the references provided for further details.<br />
<br />
It should be noted that this article has been abstracted from the text book “Coastal Engineering: Processes, Theory and Design Practice” 2nd edn (2012) and 3rd edn (in press)<ref> Reeve, D., Chadwick, A. J., Fleming, C. (2012). Coastal Engineering: Processes, Theory and Design Practice (2nd ed) E & FN Spon.</ref>, with the permission of Spon Press.<br />
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==Wave generation==<br />
<br />
[[Image: ChadwickFig1.jpg|thumb|400px|left|Figure 1. Wave generation and dispersion.]]<br />
<br />
Ocean waves are mainly generated by the action of wind on water. The waves are formed initially by a complex process of resonance and shearing action, in which waves of differing wave height, length, period are produced and travel in various directions. Once formed, ocean waves can travel for vast distances, spreading in area and reducing in height, but maintaining wavelength and period as shown in Figure 1. <br />
<br />
In the storm zone generation area high frequency wave energy (e.g. waves with small period) is both dissipated and transferred to lower frequencies. Waves of differing frequencies travel at different speeds and therefore outside the storm generation area the sea state is modified as the various frequency components separate. The low frequency waves travel more quickly than the high frequency waves resulting in a swell sea condition as opposed to a storm sea condition. This process is known as dispersion. Thus wind waves may be characterised as irregular, short crested and steep containing a large range of frequencies and directions. On the other hand swell waves may be characterised as fairly regular, long crested and not very steep containing a small range of low frequencies and directions.<br />
<br />
[[Image: ChadwickFig2a.jpg|thumb|445px|left|Figure 2. Wave transformations at Bigbury bay, Devon, England. Photograph courtesy of Dr. S. M. White.]]<br />
[[Image: ChadwickFig2b.jpg|thumb|445px|right|Figure 3. Wave transformation, main concepts.]]<br />
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As waves approach a shoreline, their height and wavelength are altered by the processes of refraction and shoaling before breaking on the shore. Once waves have broken, they enter what is termed the surf zone. Here some of the most complex transformation and attenuation processes occur, including generation of cross and longshore currents, a set-up of the mean water level and vigorous sediment transport of beach material. Some of these processes are evident in Figure 2.<br />
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Where coastal structures are present, either on the shoreline or in the nearshore zone, waves may also be diffracted and reflected resulting in additional complexities in the wave motion. Figure 3 shows a simplified concept of the main wave transformation and attenuation processes which must be considered by coastal engineers in designing coastal defence schemes.<br />
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Additionally, the existence of wave groups are of considerable significance as they have been shown to be responsible for the structural failure of some maritime structures designed using the traditional approach. The existence of wave groups also generates secondary wave forms of much lower frequency and amplitude called bound longwaves (see [[Infragravity waves]]). Inside the surf zone these waves become separated from the 'short' waves and have been shown to have a major influence on sediment transport and beach morphology producing long and cross shore variations in the surf zone wave field.<br />
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== Small amplitude wave theory==<br />
<br />
The earliest mathematical description of periodic progressive waves is that attributed to Airy in 1845. Airy wave theory is strictly only applicable to conditions in which the wave height is small compared to the wavelength and the water depth. It is commonly referred to as linear or first order wave theory, because of the simplifying assumptions made in its derivation.<br />
<br />
===Derivation of the Airy Wave equations===<br />
<br />
[[Image: ChadwickFig3.jpg|thumb|450px|right|Figure 4. Definition sketch for a sinusoidal wave.]]<br />
<br />
The Airy wave was derived using the concepts of two-dimensional ideal fluid flow. This is a reasonable starting point for ocean waves, which are not greatly influenced by viscosity, surface tension or turbulence.<br />
<br />
Figure 4 shows a sinusoidal wave of wavelength <math>L</math>, height <math>H</math> and period <math>T</math>, propagating on water with undisturbed depth <math>h</math>. The variation of surface elevation with time, from the still water level, is denoted by <math>\eta</math> (referred to as excursion) and given by<br />
<br />
<math> \eta =\frac{H}{2} \cos \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\} , \qquad(1)</math><br />
<br />
where <math>x</math> is the distance measured along the horizontal axis and <math>t</math> is time. The wave celerity, the speed <math>c</math> at which the wave moves in the <math>x</math>-direction, is given by <math> c=L/T </math>.<br />
<br />
Equation (1) represents the surface solution to the Airy wave equations. The derivation of the Airy wave equations starts from the Laplace equation for irrotational flow of an ideal fluid. The Laplace equation is simply an expression of the continuity equation applied to a flow net and is given by<br />
<br />
<math>\frac{\partial u}{\partial x} +\frac{\partial w}{\partial z} =0=\frac{\partial ^{2} \phi }{\partial x^{2} } +\frac{\partial ^{2} \phi }{\partial z^{2} } , </math> <br />
<br />
where <math>u</math> is the velocity in the <math>x</math> direction <math>w</math> is the velocity in the <math>z</math> direction <math>\phi</math> is the velocity potential and<br />
<br />
<math>u=\frac{\partial \phi }{\partial x} , \quad w=\frac{\partial \phi }{\partial z}.</math><br />
<br />
A solution for <math>\phi</math> is sought which satisfies the Laplace equation throughout the body of the flow. Additionally this solution must satisfy the boundary conditions at the bed and on the surface. At the bed assumed horizontal, the vertical velocity w must be zero. At the surface, any particle on the surface must remain on the surface, hence<br />
<br />
<math>w=\frac{\partial \eta }{\partial t} +u\frac{\partial \eta }{\partial x}, \quad z=\eta</math><br />
<br />
and the (unsteady) Bernoulli's energy equation must be satisfied,<br />
<br />
<math>\frac{p}{\rho } +\frac{1}{2} \left(u^{2} +w^{2} \right)+g\eta +\frac{\partial \phi }{\partial t} =C(t), \quad z=\eta . </math><br />
<br />
Making the assumptions that <math>H \ll L</math> and <math>H \ll h</math> results in the linearised boundary conditions (in which the smaller, higher order and product terms are neglected). The resulting kinematic and dynamic boundary equations are then applied at the still water level, given by,<br />
<br />
<math>w=\frac{\partial \eta }{\partial t}, \quad g\eta +\frac{\partial \phi }{\partial t} =0 , <br />
\quad z=0 .</math><br />
<br />
The resulting solution for <math>\phi</math> is given by<br />
<br />
<math>\phi =-gH\left[\frac{T}{4\pi } \right]\frac{\cosh \left\{\left(\frac{2\pi }{L} \right)\left(h+z\right)\right\}}{\cosh \left\{\left(\frac{2\pi }{L} \right)h\right\}} \sin \left(\frac{2\pi x}{L} -\frac{2\pi t}{T} \right)</math><br />
<br />
Substituting this solution for <math>\phi</math> into the two linearised surface boundary conditions yields the surface profile given in Equation (1) and the wave celerity <math>c</math> given by<br />
<br />
<math>c = (gT/2 \pi) \tanh(2 \pi h/L) = (g / \omega) \tanh (kh) , \qquad (2) </math><br />
<br />
where the wave number is <math>k = 2\pi / L</math> and the wave angular frequency is <math>\omega=2 \pi / T</math>. <br />
<br />
Substituting for <math>c</math> from Equation (2) gives<br />
<br />
<math>\omega ^{2} =gk\tanh \left(kh\right) . \qquad (3) </math> <br />
<br />
Equation (3) is known as the wave dispersion equation. <br />
<br />
<br />
===Numerical Solution of the Wave Dispersion Equation===<br />
<br />
In order to solve this problem from first principles it is first necessary to solve the wave dispersion equation for <math>k=2 \pi / L</math> in any depth <math>h</math>. This may be done by substituting in Equation (3) successive estimates of <math>L</math> starting with an initial estimate <math>L=L_0</math> into the <math>\tan (kh)</math> term. A more efficient technique is described by Goda <ref name=G>Goda, Y., 2000. Random Seas and Design of Maritime Structures, Advanced Series on Ocean Engineering, Vol. 15, World Scientific. </ref>, based on Newton's method, given by <br />
<br />
<math>x_{2} =x_{1} -\frac{ x_1 - D \coth x_1 }{1+D ({\coth} ^2 x_1 - 1)} , </math><br />
<br />
where <math>x=kh, \; D=k_0 h=2 \pi h/ L_0</math> and the best estimate for the initial value is <math>x_1=D , \; D \ge 1 ; \quad x_1=D^{1/2}, \; D < 1 . </math> <br />
This provides an absolute error of less than 0.05 <math>\% </math> after three iterations.<br />
<br />
A direct solution was derived by Hunt <ref>Hunt, J. N., 1979. Direction solution of wave dispersion equation. Journal of Waterway, Port, Coastal, and Ocean Engineering (ASCF), 105 (WW4), 457-459. </ref>, given by<br />
<br />
<math>\frac{gh}{c^{2} } =D+\frac{1}{1+0.6522D+0.4622D^{2} +0.0864D^{4} +0.0675D^{5}} , </math><br />
<br />
where <math>D = k_0 h</math> , which is accurate to 0.1 <math>\%</math> for <math>0<D<\infty</math> .<br />
<br />
<br />
===Water particle velocities, accelerations and paths===<br />
<br />
The equations for the horizontal, <math>u</math>, and vertical, <math>w</math>, velocities of a particle at a mean depth <math>-z</math> below the still water level may be determined from <br />
<math>\partial \phi / \partial x</math> and <math>\partial \phi / \partial z</math> respectively. The corresponding local accelerations, <math>a_x</math> and <math>a_z</math>, can then be found from <math>\partial u / \partial t</math> and <math>\partial w / \partial t </math>. <br />
<br />
[[Image: ChadwickFig4.jpg|thumb|400px|right|Figure 5. Particle displacements for deep and transitional waves.]]<br />
<br />
Finally the horizontal,<math>\zeta </math>, and vertical, <math>\xi</math>, displacements can be derived by integrating the respective velocities over a wave period. The resulting equations are given by<br />
<br />
<math>\zeta =-\frac{H}{2} \left[\frac{\cosh \left\{k\left(z+h\right)\right\}}{\sinh kh} \right]\sin \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\}, \qquad (4a) </math><br />
<br />
<math>u=\frac{\pi H}{T} \left[\frac{\cosh \left\{k\left(z+h\right)\right\}}{\sinh kh} \right]\cos \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\} , \qquad (4b) </math><br />
<br />
<math>a_{x} =\frac{2\pi ^{2} H}{T^{2} } \left[\frac{\cosh \left\{k(z+h)\right\}}{\sinh kh} \right]\sin \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\} , \qquad (4c) </math><br />
<br />
<math>\xi =\frac{H}{2} \left[\frac{\sinh \left\{k\left(z+h\right)\right\}}{\sinh kh} \right]\cos \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\} , \qquad (5a) </math><br />
<br />
<math>w=\frac{\pi H}{T} \left[\frac{\sinh \left\{k\left(z+h\right)\right\}}{\sinh kh} \right]\sin \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\}, \qquad(5b) </math><br />
<br />
<math>a_{z} =\frac{\begin{array}{l} {} \\ {-2\pi ^{2} H} \end{array}}{T^{2} } \left[\frac{\sinh \left\{k\left(z+h\right)\right\}}{\sinh kh} \right]\cos \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\} . \qquad (5c) </math><br />
<br />
All the equations have three components. The first is a magnitude term, the second describes the variation with depth and is a function of relative depth and the third is a cyclic term containing the phase information. Equations (4a) and (5a) describe an ellipse, which is the path line of a particle according to linear theory. Equations (4b,c) and (5b,c) give the corresponding velocity and accelerations of the particle as it travels along its path. The vertical and horizontal excursions decrease with depth, the velocities are 90<math>^{\circ}</math> out of phase with their respective displacements and the accelerations are 180<math>^{\circ}</math> out of phase with the displacements. These equations are illustrated graphically in Figure 5.<br />
<br />
Readers who wish to see a full derivation of the Airy wave equations are referred to Sorensen <ref name=So>Sorensen, R.M., 1993. Basic Wave Mechanics for Coastal and Ocean Engineers, John Wiley & Sons, New York. </ref> and Dean and Dalrymple <ref name=DD>Dean, R.G. & Dalrymple, R.A., 1991. Water wave mechanics for engineers and scientists, Advanced Series on Ocean Engineering, Vol. 2, World Scientific, Singapore. </ref>, in the first instance, for their clarity and engineering approach.<br />
<br />
<br />
===Pressure variation induced by wave motion===<br />
<br />
The equation for pressure variation under a wave is derived by substituting the expression for velocity potential into the unsteady Bernoulli equation and equating the energy at the surface with the energy at any depth. After linearising the resulting equation by assuming that the velocities are small, the equation for pressure results, given by<br />
<br />
<math>p=-\rho gz+\rho g\frac{H}{2} \cos (kx-\omega t)\frac{\cosh \left\{k(h+z)\right\}}{\cosh kh}=-\rho gz+\rho g\eta K_{p} (z) , \quad z=0 , </math><br />
<br />
valid at or below the still water level, where <math>K_p (z) </math> is known as the pressure attenuation factor, given by<br />
<br />
<math>K_{p} (z)=\frac{\cosh \left\{k(h+z)\right\}}{\cosh kh} </math>.<br />
<br />
The pressure attenuation factor is unity at the still water level, reducing to zero at the deep water limit (ie <math>h/L \geq 0.5</math>). At any depth (<math>-z</math>) under a wave crest, the pressure is a maximum and comprises the static pressure, <math>p_0=-\rho g z</math>, plus the dynamic pressure, <math>\rho gH K_{p} (z) /2 </math>. The reason why it is a maximum under a wave crest is because it is at this location that the vertical particle accelerations are at a maximum and are negative. The converse applies under a wave trough.<br />
<br />
Pressure sensors located on the seabed can therefore be used to measure the wave height, provided they are located in the transitional water depth region. The wave height can be calculated from the pressure variation by calculating <math>K_p (z) </math> and subtracting the hydrostatic pressure (mean value of recorded pressure). This requires the solution of the wave dispersion equation for the wavelength in the particular depth, knowing the wave period. This is easily done for a simple wave train of constant period. However, in a real sea comprising a mixture of wave heights and periods, it is first necessary to determine each wave period present (by applying Fourier analysis techniques). Also, given that the pressure sensor will be located in a particular depth, it will not detect any waves whose period is small enough for them to be deep-water waves in that depth.<br />
<br />
<br />
===The influence of water depth on wave characteristics===<br />
<br />
'''Deep water'''<br />
<br />
The particle displacement Equations (4a) and (5a) describe circular patterns of motion in (so-called) deep water. At a depth (<math>-z=L/2</math>), the diameter is only 4<math>\%</math> of the surface value and this value of depth is normally taken as the lower limit of deep water waves. Such waves are unaffected by depth, and have little or no influence on the seabed.<br />
<br />
For <math>h / L\ge 0.5, \quad \tanh (kh) \approx 1</math>. Hence Equation (3a) reduces to<br />
<br />
<math> c_{0} = gT / 2 \pi = (g L_0 / 2 \pi )^{1/2} , \qquad (6) </math> <br />
<br />
where the subscript 0 refers to deep water. Thus, the deep water wave celerity and wavelength are determined solely by the wave period.<br />
<br />
'''Shallow Water''' <br />
<br />
For <math>h/L \le 0.04, \quad \tanh (kh) \approx 2\pi h /L </math>. This is normally taken as the upper limit for shallow water waves. Hence Equation (3b) reduces to<br />
<math>c=ghT/L</math> and substituting this into Equation (2) gives <math>c=\sqrt{gh} </math>.<br />
Thus, the shallow water wave celerity is determined by depth, and not by wave period. Hence shallow water waves are not frequency dispersive whereas deep-water waves are.<br />
<br />
'''Transitional Water'''<br />
<br />
This is the zone between deep water and shallow water, i.e. <math>0.5 >h/L>0.04 </math>. In this zone <math>\tanh(kh)< 1</math>, hence<br />
<br />
<math>c=\frac{gT}{2\pi } \tanh \left(kh\right)=c_{0} \tanh \left(kh\right)<c_{0} </math>.<br />
<br />
This has important consequences, exhibited in the phenomena of refraction and shoaling. In addition, the particle displacement equations show that, at the sea bed, vertical components are suppressed so only horizontal displacements now take place (see Figure 5). This has important implications regarding sediment transport.<br />
<br />
<br />
=== Group velocity and energy propagation===<br />
<br />
The energy contained within a wave is the sum of the potential, kinetic and surface tension energies of all the particles within a wavelength and it is quoted as the total energy per unit area of the sea surface. For Airy waves, the potential (<math>Ep</math>) and kinetic (<math>EK</math>) energies are equal and <math>EP = EK =\rho gH^2 L/16</math>. Hence, the energy (<math>E</math>) per unit area of ocean is (ignoring surface tension energy which is negligible for ocean waves)<br />
<br />
<math>E=\rho gH^{2} /8 . </math><br />
<br />
This is a considerable amount of energy. For example, a (Beaufort) Force 8 gale blowing for 24 h will produce a wave height in excess of 5 m, giving a wave energy exceeding 30 kJ/m2.<br />
<br />
<br />
One might expect that wave power (or the rate of transmission of wave energy) would be equal to wave energy times the wave celerity. This is incorrect, and the derivation of the equation for wave power leads to an interesting result which is of considerable importance. Wave energy is transmitted by individual particles which possess potential, kinetic and pressure energy. Summing these energies and multiplying by the particle velocity in the <math>x</math>-direction for all particles in the wave gives the rate of transmission of wave energy or wave power (<math>P</math>), and leads to the result (for an Airy wave)<br />
<br />
<math>P=\frac{\rho gH^{2} }{8} \frac{c}{2} \left(1+\frac{2kh}{\sinh 2kh} \right) = E c_{g} , \qquad (8) </math><br />
<br />
where <math>c_g</math> is the group wave celerity, given by<br />
<br />
<math> c_{g} =\frac{c}{2} \left(1+\frac{2kh}{\sinh 2kh} \right) . \qquad (9) </math>.<br />
<br />
In deep water (<math>h/L>0.5</math>) the group wave velocity <math>c_g = c/2</math>, and in shallow water <math>c_g = c</math>. Hence, in deep water wave energy is transmitted forward at only half the wave celerity. <br />
<br />
<br />
===Radiation Stress (Momentum Flux)===<br />
<br />
The radiation stress is defined as the excess flow of momentum due to the presence of waves (with units of force/unit length). It arises from the orbital motion of individual water particles in the waves. These particle motions produce a net force in the direction of propagation (<math>S_{XX}</math>) and a net force at right angles to the direction of propagation (<math>S_{YY}</math>). The original theory was developed by Longuet-Higgins and Stewart <ref>Longuet-Higgins, M.S. & Stewart, R.W., 1964. Radiation stresses in water waves: a physical discussion, with applications, Deep Sea Res., 11, 529-562.</ref>. Its application to longshore currents was subsequently developed by Longuet-Higgins <ref name=LH>Longuet-Higgins, M.S., 1970. Longshore currents generated by obliquely incident sea waves. Journal of Geophysical Research, 75, 6778-6789. </ref>. The interested reader is strongly recommended to refer to these papers that are both scientifically elegant and presented in a readable style. Further details may also be found in Horikawa <ref name=H>Horikawa, K., 1978. Coastal Engineering, University of Tokyo Press, Tokyo. </ref> and Komar <ref name=K>Komar, P.D., 1976. Beach Processes and Sedimentation, Prentice-Hall, Englewood Cliffs, NJ. </ref>. Here only a summary of the main results is presented.<br />
<br />
The radiation stresses were derived from the linear wave theory equations by integrating the dynamic pressure over the total depth under a wave and over a wave period, and subtracting from this the integral static pressure below the still water depth. Thus, using the notation of Figure 4, <br />
<br />
<math>S_{XX} =\overline{\int _{-h}^{\eta } (p+\rho u^{2} )dz } - \int _{-h}^{0} p_0 dz , \qquad<br />
S_{YY} =\overline{\int _{-h}^{\eta }(p+\rho v^{2} )dz }-\int _{-h}^{0}p_0 dz , </math><br />
<br />
where <math>v</math> is the horizontal component of orbital velocity in the <math>y</math>-direction and <math>p_0</math> the hydrostatic pressure. The first integral is the mean value of the integrand over a wave period where <math>u</math> is the horizontal component of orbital velocity in the <math>x</math> direction. After considerable manipulation it may be shown that<br />
<br />
<math>S_{XX} =E \left(\frac{2kh}{\sinh 2kh} +\frac{1}{2} \right) . \qquad(10) </math> <br />
<br />
For waves travelling in the <math>X</math>-direction (<math>v = 0</math>) <br />
<br />
<math> S_{YY} = E \left(\frac{kh}{\sinh 2kh} \right) . \qquad (11)</math><br />
<br />
In deep water <math>S_{XX} =\frac{1}{2}E , \quad S_{YY} =0</math> ; <br />
in shallow water <math>S_{XX} =\frac{3}{2} E , \quad S_{YY} =\frac{1}{2} E .</math><br />
Thus both <math>S_{XX}</math> and <math>S_{YY} </math> increase in reducing water depths.<br />
<br />
<br />
==Wave transformation and attenuation processes==<br />
<br />
As waves approach a shoreline, they enter the transitional depth region in which the wave motions are affected by the seabed. These effects include reduction of the wave celerity and wavelength, and thus alteration of the direction of the wave crests (refraction) and wave height (shoaling) with wave energy dissipated by seabed friction and finally breaking.<br />
<br />
<br />
===Refraction===<br />
<br />
[[Image: ChadwickFig5.jpg|thumb|300px|right|Figure 6. Wave refraction.]]<br />
<br />
Wave celerity and wavelength are related through Equations (2, 3a) to wave period (which is the only parameter which remains constant for an individual wave train):<br />
<br />
<math>c/c_0=\tanh(kh) =L/L_0</math> .<br />
<br />
To find the wave celerity and wavelength at any depth h, these two equations must be solved simultaneously. The solution is always such that <math>c<c_o</math> and <math>L<L_0</math> for <math>h<h_0</math> (where the subscript o refers to deep water conditions).<br />
<br />
Consider a deep-water wave approaching the transitional depth limit (<math>h/L_0 =0.5</math>), as shown in Figure 6. A wave travelling from A to B (in deep water) traverses a distance <math>L_0</math> in one wave period <math>T</math>. However, the wave travelling from C to D traverses a smaller distance, L, in the same time, as it is in the transitional depth region. Hence, the new wave front is now BD, which has rotated with respect to AC.<br />
<br />
Letting the angle <math>\alpha</math> represent the angle of the wave front to the depth contour, then<br />
<br />
<math>\sin \alpha = L/BC</math> and <math>\sin \alpha_0=L_0/BC</math>. Hence <br />
<br />
<math>\frac{\sin \alpha }{\sin \alpha _{0} } =\frac{L}{L_{0} } =\frac{c}{c_{0} } =\tanh \left(kh\right) . \qquad (12) </math><br />
<br />
[[Image: ChadwickFig6.jpg|thumb|400px|right|Figure 7. Variations of wave celerity and angle with depth.]]<br />
<br />
As <math>c < c_0</math> then <math>\alpha < \alpha_0</math>, which implies that as a wave approaches a shoreline from an oblique angle the wave fronts tend to align themselves with the underwater contours. Figure 7 shows the variation of <math>c/c_0</math> with <math>h/L_0</math>, and <math>\alpha / \alpha_0</math> with <math>h/L_0</math> (the later specifically for the case of parallel contours). It should be noted that <math>L_0</math> is used in preference to <math>L</math> as the former is a fixed quantity.<br />
<br />
In the case of non-parallel contours, individual wave rays (i.e. the orthogonals to the wave fronts) must be traced. Figure 7 can still be used to find <math>\alpha</math> at each contour if <math>\alpha_0</math> is taken as the angle (say <math>\alpha_1</math>) at one contour and <math>\alpha</math> is taken as the new angle (say <math>\alpha_2</math>) to the next contour. The wave ray is usually taken to change direction midway between contours. This procedure may be carried out by hand using tables or figures <ref name=S>Silvester, R., 1974. Coastal Engineering, Vols 1 and 2, Elsevier, Oxford. </ref> or by computer as described later in this section.<br />
<br />
Equation (12) is also known as Snell’s law, according to which <math> \sin \alpha / c</math> = constant along a wave ray. Multiplication with the radial frequency <math>\omega</math> shows that similar constancy holds for the component of the wave vector <math>\vec k </math> parallel to the depth contour (the wave vector <math>\vec k </math> follows the wave propagation direction and its length equals the wave number <math>k=\omega /c</math>). <br />
<br />
It can be shown that Snell’s law can also be expressed as<br />
<br />
<math> \vec \nabla \times \vec k |_{vertical component} \equiv \partial (k \sin \alpha) / \partial x - \partial (k \cos \alpha) / \partial y =0 . \qquad (13) </math> <br />
<br />
The proof that this Equation is equivalent to (12) is given in Dean & Dalrymple <ref name=DD></ref>.<br />
<br />
The wave energy conservation equation reads <br />
<br />
<math> \partial (E c_g \cos \alpha) / \partial x + \partial (E c_g \sin \alpha) / \partial y = - \epsilon_d , \qquad (14) </math> <br />
<br />
where <math> \epsilon_d</math> represents energy losses (due to seabed friction, cf Equation (14)). Koutitas <ref>Koutitas, G.K., 1988. Mathematical Models in Coastal Engineering, Pentech Press, London. </ref> gives a worked example of a numerical solution to Equations (13) and (14).<br />
<br />
<br />
===Shoaling===<br />
<br />
[[Image: ChadwickFig7.jpg|thumb|400px| right |Figure 8. Variation of the shoaling coefficient with depth.]]<br />
<br />
Consider first a wave front travelling parallel to the seabed contours (ie no refraction is taking place). Making the assumption that wave energy is transmitted shorewards without loss due to bed friction or turbulence, then from Equation (8), <br />
<br />
<math> \frac{P}{P_{0} } =1=\frac{Ec_{g} }{E_{0} c_{g_{0} } }. </math><br />
<br />
Substituting Equation (7) gives <br />
<br />
<math>\frac{P}{P_{0} } =1=\left(\frac{H}{H_{0} } \right)^{2} \frac{c_{g} }{c_{g_{0} } } ,\quad <br />
\frac{H}{H_{0} } =\left(\frac{c_{g_{0} } }{c_{g} } \right)^{1/ 2}=K_{S} , </math><br />
<br />
where <math>K_S</math> is the shoaling coefficient.<br />
<br />
The shoaling coefficient can be evaluated from the equation for the group wave celerity, Equation (9), <br />
<br />
<math>K_S =\left( \frac{c_{g_0} }{c_g} \right)^ {1/2}=\left[ (c/c_0 ) ( 1+\frac{2kh}{\sinh 2kh} ) \right]^ {-1/2} . \qquad (15) </math><br />
<br />
The variation of <math>K_S</math> with <math>d/L_0</math> is shown in Figure 8.<br />
<br />
<br />
===Combined refraction and shoaling===<br />
<br />
Consider next a wave front travelling obliquely to the seabed contours as shown in Figure 9. In this case, as the wave rays bend, they may converge or diverge as they travel shoreward. At the contour <math>h/L_0 = 0.5, \quad BC=b_0 / \cos \alpha _0 =b / \cos \alpha . </math> Thus<br />
<br />
<math>b/b_0 = \cos \alpha / \cos \alpha _0 . </math><br />
<br />
[[Image: ChadwickFig8.jpg|thumb|300px| right |Figure 9. Divergence of wave rays over parallel contours.]]<br />
<br />
Again, assuming that the power transmitted between any two wave rays is constant (i.e. conservation of wave energy flux), then<br />
<br />
<math>\frac{P}{P_0} = 1 =\frac{E b c_g}{E_0 b_0 c_{g_0 }} = \left( \frac{H}{H_0} \right)^2 \frac{\cos \alpha }{\cos \alpha _0 } \frac{c_g }{c_{g_0 }} . </math><br />
<br />
Hence<br />
<br />
<math>H/H_0=K_R K_S , \qquad (16) </math><br />
<br />
where <math>K_{R} =\left( \cos \alpha _{0} /\cos \alpha \right)^{1/2}</math> is called the refraction coefficient.<br />
<br />
For the case of parallel contours, <math>K_R</math> can be found using Figure 9. In the more general case, <math>K_R</math> can be found from the refraction diagram directly by measuring <math>b</math> and <math>b_0</math>.<br />
<br />
As the refracted waves enter the shallow water region, they break before reaching the shoreline. The foregoing analysis is not strictly applicable to this region, because the wave fronts steepen and are no longer described by the Airy waveform. However, it is common practice to apply refraction analysis up to the so-called breaker line. This is justified on the grounds that the inherent inaccuracies are small compared with the initial predictions for deep-water waves, and are within acceptable engineering tolerances. To find the breaker line, it is necessary to estimate the wave height as the wave progresses inshore and to compare this with the estimated breaking wave height at any particular depth. As a general guideline, waves will break when<br />
<br />
<math> h_{b} =1.28H_{b} , \qquad (17)</math><br />
<br />
where the subscript b refers to the point of breaking. The subject of wave breaking is of considerable interest both theoretically and practically. <br />
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<br />
===Bathymetry effects on Refraction===<br />
<br />
In general, the seabed contours are not straight and parallel, but are curved. This results in some significant refraction effects. Within a bay, refraction will generally spread the wave rays over a larger region, resulting in a reduction of the wave heights. Conversely, at headlands the wave rays will converge, resulting in larger wave heights. Over offshore shoals the waves may be focused, resulting in a small region where the wave heights are much larger. If the focusing is so strong that the wave rays are predicted to cross, then the wave heights become so large as to induce wave breaking.<br />
<br />
<br />
===Shoaling and refraction of directional wave spectra===<br />
<br />
So far, the discussion of shoaling and refraction has been restricted to considering waves of single period, height and direction ( a monochromatic wave ). However, a real sea state is more realistically represented as being composed of a large number of components of differing periods, heights and directions (known as the directional spectrum). Therefore, in determining an inshore sea state due account should be taken of the offshore directional spectrum.<br />
<br />
This can be achieved in a relatively straightforward way, provided the principle of linear superposition can be applied. This implies that non-linear processes such as seabed friction and higher-order wave theories are excluded. The principle of the method is to carry out a refraction and shoaling analysis for every individual component wave frequency and direction and then to sum the resultant inshore energies at the new inshore directions at each frequency and hence assemble an inshore directional spectrum.<br />
<br />
[[Image: ChadwickFig9a.jpg|thumb|350px| left | Figure 10. Some of Goda’s results for the diffraction coefficient <math>K_R</math> as a function of relative depth <math>h/L_0 </math> for a typical wind wave state and different predominant wave incident angles <math>\alpha_0 </math> at deep water. Adapted from Goda <ref name=G></ref>.]]<br />
<br />
[[Image: ChadwickFig9b.jpg|thumb|350px| right | Figure 11. Comparison of results for the diffraction coefficient <math>K_R</math> between a monochromatic wave and Goda’s result for a directional spectrum of a typical wind wave state and predominant wave incident angle <math>\alpha_0 =30^{\circ}</math>. Adapted from Goda <ref name=G></ref>.]]<br />
<br />
[[Image: ChadwickFig10NEW.jpg|thumb|350px| left | Figure 12. Some of Goda’s results for the predominant wave direction at a range of relative depths for a typical wind wave state and different predominant wave incident angles <math>\alpha_0 </math> at deep water. Adapted from Goda <ref name=G></ref>.]]<br />
<br />
<br />
Goda <ref name=G></ref> presents a set of design charts for the effective refraction coefficient (<math>K_R </math>) and predominant wave direction (<math>\alpha_0 </math>) over parallel contours for a range of relative depths using the Betchneider-Mitsuyasu frequency spectrum and Mitsuyasu spreading function, which facilitate the ready application of the method described above. Figure 10 illustrates some of Goda’s results for <math>K_R </math> as a function of relative depth for a typical wind wave state. Figure 11 shows a comparison of results for <math>K_R </math> between a monochromatic wave and Goda’s result for a directional spectrum of a typical wind wave state. Figure 12 shows some of Goda’s results for the principal wave direction at a range of relative depths for a typical wind wave state.<br />
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<br />
===Seabed Friction===<br />
<br />
In the foregoing analysis of refraction and shoaling it was assumed that there was no loss of energy as the waves were transmitted inshore. In reality, waves in transitional and shallow water depths will be attenuated by wave energy dissipation through seabed friction. Such energy losses can be estimated, using linear wave theory, in an analogous way to pipe and open channel flow frictional relationships. In contrast to the velocity profile in a steady current, the frictional effects under wave action produce an oscillatory wave boundary layer which is very small (a few millimetres or centimetres). In consequence, the velocity gradient is much larger than in an equivalent uniform current that in turn implies that the wave friction factor will be many times larger.<br />
<br />
Firstly, the mean seabed shear stress (<math>\tau_b</math> ) may be found using<br />
<br />
<math>\tau _{b} =\frac{1}{2} f_{w} \rho u_{m}^{2} , \qquad (18) </math><br />
<br />
where <math>f_w</math> is the wave friction factor and <math>u_m</math> is the maximum near bed orbital velocity; <math>f_w</math> is a function of a local Reynolds' number (<math>Re_w</math>) defined in terms of <math>u_m</math> (for velocity) and either <math>a_b</math>, wave amplitude at the bed or the seabed grain size <math>k_s</math> (for the characteristic length). A diagram relating <math>f_w</math> to <math>Re_w</math> for various ratios of <math>a_b/k_s</math>, due to Jonsson, is given in Dyer <ref> Dyer, K.D., 1986. Coastal and Estuarine Sediment Dynamics, Wiley, Chichester </ref>. This diagram is analogous to the Moody diagram for pipe friction factor (<math>\lambda</math>). Values of <math>f_w</math> may take values in the range <math>\; 0.5 \times 10^{-3} - 5 \;</math>. Hardisty<ref name=Ha> Hardisty, J., 1990. Beaches Form and Process, Unwin Hyman, London </ref> summarizes field measurements of <math>f_w</math> (from Sleath) and notes that a typical field value is about 0.1. Soulsby <ref> Soulsby, R.L., 1997. Dynamics of Marine Sands. Thomas Telford, London</ref> provides details of several equations which may be used to calculate the wave friction factor. For rough turbulent flow in the wave boundary layer, he derived a new formula which best fitted the available data, given by<br />
<br />
<math>f_w=0.237 r^{-0.52}, \quad r=A/k_s, \quad A=u_m T/2 \pi . \qquad (19) </math><br />
<br />
Using linear wave theory <math>u_m</math> is given by <math>u_{m} =\pi H / (T\sinh kh) .</math><br />
<br />
The rate of energy dissipation may then be found by combining the expression for <math>\tau_b</math> with linear wave theory to obtain<br />
<br />
<math>\frac{dH}{dx} =-\frac{4f_{w} k^{2} H^{2} }{3\pi \sinh (kh)(\sinh (2kh)+2kh)} . \qquad (20) </math><br />
<br />
The wave height attenuation due to seabed friction is of course a function of the distance travelled by the wave as well as the depth, wavelength and wave height. Thus the total loss of wave height (<math>\Delta H_f</math>) due to friction may be found by integrating over the path of the wave ray.<br />
<br />
BS6349 <ref name=BS>BSI, BS6349, 1984. Maritime Structures, British Standards Institute, London, UK</ref> presents a chart from which a wave height reduction factor may be obtained. Except for large waves in shallow water, seabed friction is of relatively little significance. Hence, for the design of maritime structures in depths of 10 m or more, seabed friction is often ignored. However, in determining the wave climate along the shore, seabed friction is now normally included in numerical models, although an appropriate value for the wave friction factor remains uncertain and is subject to change with wave induced bed forms. Furthermore, wave energy losses due to other physical processes such as breaking can be more significant.<br />
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<br />
===Wave-Current Interaction===<br />
<br />
So far, consideration of wave properties has been limited to the case of waves generated and travelling on quiescent water. In general, however, ocean waves are normally travelling on currents generated by tides and other means. These currents will also, in general, vary in both space and time. Hence two distinct cases need to be considered here. The first is that of waves travelling on a current and the second when waves generated in quiescent water encounter a current (or travel over a varying current field).<br />
<br />
For waves travelling on a current, two frames of reference need to be considered. The first is a moving or relative frame of reference, travelling at the current speed. In this frame of reference, all the wave equations derived so far still apply. The second frame of reference is the stationary or absolute frame. The concept which provides the key to understanding this situation is that the wavelength is the same in both frames of reference. This is because the wavelength in the relative frame is determined by the dispersion equation and this wave is simply moved at a different speed in the absolute frame. In consequence, the absolute and relative wave periods are different.<br />
<br />
Consider the case of a current with magnitude (<math>u</math>) following a wave with wave celerity (<math>c</math>), the wave speed with respect to the seabed (<math>c_a</math>) becomes <math>c + u</math>. As the wavelength is the same in both reference frames, the absolute wave period will be less than the relative wave period. Consequently, if waves on a current are measured at a fixed location (eg in the absolute frame), then it is the absolute period (<math>T_a</math>) which is measured. The current magnitude must, therefore, also be known in order to determine the wavelength. From the dispersion Equation (3) it follows that <br />
<br />
<math>c_a=L/T_a=\left(\frac{gL}{2\pi } \tanh \frac{2\pi h}{L} \right)^ {1 2} +u . \qquad (21)</math><br />
<br />
This equation thus provides an implicit solution for the wavelength in the presence of a current when the absolute wave period has been measured.<br />
<br />
[[Image: ChadwickFig11.jpg|thumb|400px| right |Figure 13. Deep water wave refraction by a current.]]<br />
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Conversely, when waves travelling in quiescent water encounter a current, changes in wave height and wavelength will occur. This is because as waves travel from one region to the other requires that the absolute wave period remains constant for waves to be conserved. Consider the case of an opposing current, the wave speed relative to the seabed is reduced and therefore the wavelength will also decrease. Thus wave height and steepness will increase. In the limit the waves will break when they reach limiting steepness. In addition, as wave energy is transmitted at the group wave speed, waves cannot penetrate a current whose magnitude equals or exceeds the group wave speed and thus wave breaking and diffraction will occur under these circumstances. Such conditions can occur in the entrance channels to estuaries when strong ebb tides are running, creating a region of high, steep and breaking waves.<br />
<br />
Another example of wave-current interaction is that of current refraction. This occurs when a wave obliquely crosses from a region of still water to a region in which a current exits or in a changing current field. The simplest case is illustrated in Figure 13 showing deep-water wave refraction by a current.<br />
<br />
In an analogous manner to refraction caused by depth changes, Jonsson showed that in the case of current refraction<br />
<br />
<math>\sin \alpha _{c} =\frac{\sin \alpha }{\left(1-\frac{u}{c} \sin \alpha \right)^{2} } . \qquad (22) </math><br />
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The wave height is also affected and will decrease if the wave orthogonals diverge (as shown) or increase if the wave orthogonals converge. For further details of wave-current interactions, the reader is referred to Hedges <ref>Hedges, T.S., 1987. Combinations of waves and currents: an introduction. Proc. Inst. Civ. Engrs, Part 1,1987, June, 567-585. </ref> in the first instance.<br />
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<br />
===Wave Reflection===<br />
<br />
Waves normally incident on solid vertical boundaries (e.g. harbour walls and sea walls) are reflected such that the reflected wave has the same phase but opposite direction and substantially the same amplitude as the incident wave. This fulfils the necessary boundary condition that the horizontal velocity is always zero. The resulting wave pattern set up is called a standing wave, as shown in Figure 14. Reflection can also occur when waves enter a harbour or estuary. This can lead to `resonance' where the waves are amplified.<br />
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The equation of the standing wave (subscript s) may be found by adding the two waveforms of the incident (subscript i) and reflected (subscript r) waves. Thus,<br />
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<math>\eta _{s} =\eta _{i} +\eta _{r} , \qquad \eta _{i} =\frac{H_{i} }{2} \cos \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\}, \qquad \eta _{r} =\frac{H_{r} }{2} \cos \left\{2\pi \left(\frac{x}{L} +\frac{t}{T} \right)\right\} , \qquad (23) </math><br />
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Taking <math>H_r=H_i =H_s / 2</math>, then<br />
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<math> \eta_s=H_s \cos (2 \pi x/L) \cos (2 \pi t/T) . \qquad (24) </math><br />
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[[Image: ChadwickFig12a.jpg|thumb|445px|left|Figure 14. Standing Waves, idealised.]]<br />
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At the nodal points there is no vertical movement with time. By contrast, at the antinodes, crests and troughs appear alternately. For the case of large waves in shallow water and if the reflected wave has a similar amplitude to the incident wave, then the advancing and receding crests collide in a spectacular manner, forming a plume known as a clapotis (see Figure 15). This is commonly observed at sea walls. Standing waves can cause considerable damage to maritime structures, and bring about substantial erosion.<br />
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[[Image: ChadwickFig12b.jpg|thumb|445px|right|Figure 15. Standing Waves, observed clapotis.]]<br />
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'''Clapotis Gaufre'''<br />
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[[Image: ChadwickFig13.jpg|thumb|400px|left|Figure 16. Plan view of oblique wave reflection.]]<br />
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When the incident wave is at an angle <math>\alpha</math> to the normal from a vertical boundary, then the reflected wave will be in a direction <math>\alpha</math> on the opposite side of the normal. This is illustrated in Figures 16 and 17. The resulting wave motion (the clapotis gaufre) is complex, but essentially consists of a diamond pattern of island crests which move parallel to the boundary. It is sometimes referred to as a short crested system. The crests form at the intersection of the incident and reflected wave fronts. The resulting particle displacements are also complex, but include the generation of a pattern of moving vortices. A detailed description of these motions may be found in Silvester <ref name=S></ref>. The consequences of this in terms of sediment transport may be severe. Very substantial erosion and longshore transport may take place. Considering that oblique wave attack to sea walls is the norm rather than the exception, the existence of the clapotis gaufre has a profound influence on the long term stability and effectiveness of coastal defence works. This does not seem to have been fully understood in traditional designs of sea walls, with the result that collapsed sea walls and eroded coastlines have occurred.<br />
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<br />
'''Mach Stem'''<br />
<br />
[[Image: ChadwickFig14.jpg|thumb|400px| right |Figure 17. Wave impact and reflection during a storm.]]<br />
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When periodic or solitary waves approach a steep barrier at an oblique angle, the amplitude of the wave against the barrier may be magnified by a phenomenon known as the Mach Stem. The crest immediately adjacent to the wall alters its alignment to create a wave travelling along the face of the wall with increased crest height and this is the Mach Stem wave, illustrated in Figure 17. This reflection phenomenon was first discovered in aerodynamics, but is equally applicable to water waves for which it can begin to occur when the angle of obliquity to the wall becomes less than about 45 degrees. The height of the crest gives rise to a velocity that is equivalent to the component of the incident wave's celerity in the direction of the alignment of the wall. Since the waves do not strike the wall due to a growing slipstream zone, reflection is much reduced until for angles of obliquity less than about 20 degrees reflection becomes non-existent.<br />
<br />
Miles <ref>Miles J., 1980, Solitary Waves. Annual Review of Fluid Mechanics}, 12, 11-43, January. </ref> demonstrated theoretically that the Mach Stem wave could be amplified by as much as four times the incoming waves, two time greater than a linear superposition of incident and reflected wave. However, Melville <ref>Melville,W.K.,1980. On the Mach Relexion of a Solitary Wave. Journal of Fluid Mechanics}, 98, 285-297. </ref> was unable to reproduce such large amplification factors in the laboratory. More recently Yoon and Liu <ref>Yoon, S.B. and Liu, P.L.-F., 1989. Stem Waves along Breakwater. Journal of Waterway, Port, Coast and Ocean Engineering (ASCE), 115, 635-648.</ref> employed parabolic approximations to study the stem waves induced by an oblique cnoidal wave train in front of a vertical barrier and Honda and Mase <ref>Honda, K. and Mase, H., 2007. Application of non-linear frequency-domain wave model to mach stem evolution and wave transformation on reef. Proceedings of the 5th Coastal Structures International Conference, ASCE, Venice, Italy. </ref> have applies a non-linear frequency-domain wave model to mach stem evolution and wave transformation on a reef structure.<br />
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In practical design terms it is important to recognise the potential for Mach stem waves to exist as the enhanced wave height and high velocities running along a breakwater can result in increased overtopping, armour instability, toe scour problems and beach erosion at the root. <br />
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===Wave reflection coefficients===<br />
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Defining a reflection coefficient <math>K_r = H_r /H_i</math> then typical values are as follows <br />
<br />
<br />
{|<br />
|- style="font-weight:bold; text-align:center; background:lightgrey" <br />
! width=25% style=" border:1px solid gray;"| Reflection barrier <br />
! width=15% style=" border:1px solid gray;"| <math>K_r</math><br />
|- <br />
| style="border:1px solid gray;"| <br />
| style="border:1px solid gray;"|<br />
|- <br />
| style="border:1px solid gray;"| Concrete sea walls <br />
| style="border:1px solid gray;"| 0.7 to 1.0<br />
|- <br />
| style="border:1px solid gray;"| Rock breakwaters <br />
| style="border:1px solid gray;"| 0.4 to 0.7<br />
|- <br />
| style="border:1px solid gray;"| Beaches <br />
| style="border:1px solid gray;"| 0.05 to 0.2<br />
|}<br />
<br />
It should be noted that the reflected wave energy is equal to <math>K_r^2</math> as energy is proportional to <math>H^2</math>.<br />
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<br />
'''Predictive Equations for Wave Reflection from Rock Slopes'''<br />
<br />
The Rock Manual (CIRIA/CUR manual <ref>CIRIA and CUR, 1991. Manual on the Use of Rock in Coastal and Shoreline Engineering. </ref> and CIRIA/CUR/CETMEF 2007<ref> CIRIA/CUR/CETMEF, 2007. The Rock Manual. The Use of Rock in Hydraulic Engin¬eering (2nd edition), C683, CIRIA, London</ref>) gives an excellent summary of the development of wave reflection equations based on laboratory data of reflection from rock breakwaters. This work clearly demonstrates that rock slopes considerably reduce reflection compared to smooth impermeable slopes. Based on this data, the best fit equation was found to be<br />
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<math>K_{r} =0.125\xi _{p}^{0.7} </math>, <br />
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where <math>\xi _{p} </math> is the Iribarren Number = <math>\tan \beta (H/L_p)^{-1/2} </math> and the subscript <math>p</math> refers to peak frequency.<br />
<br />
Davidson et al <ref>Davidson, M.A., Bird, P.A.D., Bullock, G.N. and Huntley, D.A., 1996. A new non-dimensional number for the analysis of wave reflection from rubble mound breakwaters. Coastal Engineering, 28, p93—120. </ref> subsequently carried out an extensive field measurement programme of wave reflection at prototype scale at the Elmer breakwaters (Sussex, UK) and after subsequent analysis proposed a new predictive scheme. A new dimensionless reflection parameter was proposed given by<br />
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<math> R=\frac{d_{t} L_0^{2} }{H_{i} D^{2} \cot \beta } , \qquad (25)</math><br />
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where <math>d_t</math> (m) is water depth at the toe of structure, <math>L_0</math> is deep water wavelength at peak frequency, <math>H_i</math> is significant incident wave height, <math>D</math> is characteristic diameter of rock armour and <math>\tan \beta</math> is the structure gradient. <math>R</math> was found to be a better parameter than <math>\xi</math> in predicting wave reflection. The reflection coefficient is then given by<br />
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<math> K_r = 0.151 R^{0.11} \qquad (26) </math> <br />
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or, alternatively<br />
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<math> K_{r} =\frac{0.635 R^{0.5} }{41.2+R^{0.5} } . \qquad (27)</math><br />
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<br />
===Wave reflection due to refraction===<br />
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Wave reflection due to refraction alone can also occur due to very rapid changes in the seabed. In particular, when waves approach a deep dredged channel with the direction or propagation at a sufficiently acute angle to the dredged side slope and there is a sufficiently large change in water depth which, in turn, results in a large and rapid change in wave speed, the wave may reflect off the side of the channel. An analogous example of this phenomenon is the internal reflection of light rays in a glass prism due to changes in wave speed between the glass (shallow water) and air (deep water), the essential difference being that, as wave speed is a function of water depth, it is not a constant on the wave approach or on the channel side slope. This is a very real phenomenon and, if not recognised, can result in wave energy inadvertently being reflected into a port area. The converse also applies as this process can also be used to advantage to reflect wave energy away from a harbour entrance. It should also be appreciated that longer period waves will also be more susceptible to this phenomenon due to the their relatively greater speed in deeper water. When it comes to wave modelling described in Section 3.9 it follows that any numerical grid used in a wave model must be fine enough to capture the detail of the dredged channel in order to properly reproduce this effect.<br />
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===Wave Diffraction===<br />
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[[Image: ChadwickFig15a.jpg|thumb|300px| right |Figure 18. Idealised wave diffraction around an impermeable breakwater.]]<br />
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This is the process whereby waves bend round obstructions by radiation of the wave energy. Figure 18 shows an oblique wave train incident on the tip of a breakwater. There are three distinct regions:<br />
<br />
# the shadow region in which diffraction takes place;<br />
# the short crested region in which incident and reflected waves form a clapotis gaufre;<br />
# an undisturbed region of incident waves.<br />
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In region (1) the waves diffract with the wave fronts forming circular arcs centred on the point of the breakwater. When the waves diffract, the wave heights diminish as the energy of the incident wave spreads over the region. The real situation is, however, more complicated than that presented in Figure 18. The reflected waves in region (2) will diffract into region (3) and hence extend the short crested system into region (3).<br />
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'''Mathematical Formulation of Wave Diffraction'''<br />
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Mathematical solutions for wave diffraction have been developed for the case of constant water depth using linear wave theory. The basic differential equation for wave diffraction is known as the Helmholtz equation. This can be derived from the Laplace equation (refer to “Section Small amplitude wave theory”) in three dimensions<br />
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<math>\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}+\frac{\partial^2 \phi}{\partial z^2}=0</math>.<br />
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Now, let<br />
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<math>\phi(x,y,z) = Z(z) F(x,y) e^{i \omega t}</math>,<br />
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(i.e. <math>\phi</math> is a function of depth and horizontal co-ordinates and is periodic and <math>i</math> is the imaginary number ).<br />
For uniform depth an expression for <math>Z(z)</math> satisfying the no flow bottom boundary condition is<br />
<math>Z(z)=\cosh (k(h+z))</math>.<br />
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Substituting for <math>\phi</math> and <math>Z</math> in the Laplace equation leads (after further manipulation) to the Helmholtz equation<br />
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<math>\frac{\partial^2 F}{\partial x^2}+\frac{\partial^2 F}{\partial y^2}+k^2 F(x,y)=0</math>.<br />
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'''Solutions to the Helmholtz Equation'''<br />
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A solution to the Helmholtz equation was first found by Sommerfeld in 1896 who applied it to the diffraction of light (details may be found in Dean & Dalrymple (1991)). Somewhat later, Penney & Price (1952) showed that the same solution applied to water waves and presented solutions for incident waves from different directions passing a semi-infinite barrier and for normally incident waves passing through a barrier gap <math>b</math>. For the case of normal incidence on a semi-infinite barrier, it may be noted that, for a monochromatic wave, the diffraction coefficient <math>K_d</math> (ratio of incident and diffracted wave height) is approximately 0.5 at the edge of the shadow region and that <math>K_d</math> exceeds 1.0 in the 'undisturbed' region due to diffraction of the reflected waves caused by the (perfectly) reflecting barrier. Their solution for the case of a barrier gap is essentially the superposition of the results from two mirror image semi-infinite barriers. <br />
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Their diagrams apply for a range of gap width to wavelength (<math>b/L</math>) from 1 to 5. When <math>b/L</math> exceeds 5 the diffraction patterns from each barrier do not overlap and hence the semi-infinite barrier solution applies. For <math>b/L</math> less than one the gap acts as a point source and wave energy is radiated as if it were coming from a single point at the centre of the gap. It is important to note here that these diagrams should '''not be used for design'''. This is because of the importance of considering directional wave spectra. <br />
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[[Image: ChadwickFig15e.jpg|thumb|300px| left |Figure 19. Diffraction (ratio of diffracted wave height and incident wave height) of a normally incident directional random sea state for a semi-infinite barrier. Adapted from Goda <ref name=G></ref>.]]<br />
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[[Image: ChadwickFig15f.jpg|thumb|300px| right | Figure 20. Diffraction (ratio of diffracted wave height and incident wave height) of a normally incident directional random sea state for a breakwater gap width of <math>b=L</math>. Adapted from Goda <ref name=G></ref>.]]<br />
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Goda (2000) pioneered the use of wave directional spectra in the determination of wave diffraction. In a similar manner to the technique described in the section on refraction and shoaling of directional spectra, he calculated an effective diffraction coefficient from the superposition of diffraction of all the component wave directions and frequencies present in a typical wind wave state and hence constructed a new set of diffraction diagrams.<br />
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These diagrams show that the diffraction of a directional random sea state differs quite markedly from the case of a monochromatic sea. At the edge of the shadow zone for a semi-infinite barrier <math>K_d</math> is approximately 0.7 (cf. <math>K_d</math>= 0.5 for a monochromatic wave), and waves of greater height penetrate the shadow zone at equivalent points. This is illustrated in Figure 19. For the case of a barrier gap (width <math>b</math>), the wave height variations are smoothed compared with the monochromatic case, with smaller heights in the area of direct penetration and larger heights in the shadow regions, as shown in Figure 20. <br />
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<br />
===Combined refraction and diffraction===<br />
<br />
Refraction and diffraction often occur together. For example, the use of a wave ray model over irregular bathymetry may produce a caustic (i.e. a region where wave rays cross). Here diffraction will occur spreading wave energy away from regions of large wave heights. Another example is around offshore breakwaters; here diffraction is often predominant close to the structure with refraction becoming more important further away from the structure. A solution to the Laplace equation over irregular bathymetry is required, which allows diffraction as well as refraction. Such a solution was first derived in 1972 by Berkhoff<ref>Berkhoff, J.C.W., 1972. Computation of combined refraction-diffraction, Proc. 13th International Conference on Coastal Engineering, Lisbon, 55-69. </ref>. This is generally known as the mild slope equation because the solution is restricted to slowly bathymetry that varies slowly relative to the wavelength.<br />
It may be written as<br />
<br />
<math> \frac{\partial }{\partial x} \left(cC_{g} \frac{\partial \phi }{\partial x} \right)+\frac{\partial }{\partial y} \left(cC_{g} \frac{\partial \phi }{\partial y} \right)+\omega ^{2} \frac{C_{g} }{c} \phi =0<br />
, \qquad (28) </math><br />
<br />
where <math>\phi (x,y)</math> is a complex wave potential function. The solution of this equation is highly complex and beyond the scope of this text. However, the interested reader is directed to Dingemans <ref>Dingemans, M.W., 1997. Water Wave Propagation over Uneven Bottoms. Advanced Series on Ocean Engineering, Vol 13. World Scientific, London.</ref> for a review of the subject. One of the more recent developments in solving the mild slope equation is that due to Li <ref>Li, B., 1994. A generalised conjugate gradient model for the mild-slope equation, Coastal Engineering, 23, 215-225. </ref>. This version of the mild slope equation allows the simultaneous solution of refraction, diffraction and reflection. <br />
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[[Image: ChadwickFig15b.jpg|thumb|452px|left|Figure 21. Photograph of real wave diffraction at the Elmer breakwater scheme, Sussex, England.]]<br />
[[Image: ChadwickFig15c.jpg|thumb|448px|right|Figure 22. Physical model study of (21) in the UK Coastal Research Facility at HR Wallingford.]]<br />
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[[Image: ChadwickFig15d.jpg|thumb|400px|left|Figure 23. Aerial photograph of wave diffraction at the Happisburgh to Winterton scheme, Norfolk, England (courtesy of Mike Page).]]<br />
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It has also been the subject of a field validation study. Initial results may be found in Ilic and Chadwick <ref>Ilic, S. & Chadwick, A.J., 1995. Evaluation and validation of the mild slope evolution equation model for combined refraction—diffraction using field data, Coastal Dynamics 95, Gdansk, Poland, pp. 149-160. </ref>. They tested this model at the site of the Elmer offshore breakwater scheme (shown in Figure 21) where refraction and reflection are the main processes seaward of the breakwaters with diffraction and refraction taking place shoreward of the breakwaters, and in a physical model (shown in Figure 22). Figure 23 illustrates wave diffraction at the Happisburgh to Winterton scheme, Norfolk, England. <br />
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==Finite amplitude waves==<br />
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It has already been noted that the Airy wave equations only strictly apply to waves of relatively small height in comparison to their wavelength and water depth. For steep waves and shallow water waves the profile becomes asymmetric with high crests and shallow troughs. For such waves, celerity and wavelength are affected by wave height and are better described by other wave theories. To categorise finite amplitude waves, three parameters are required. These are the wave height (<math>H</math>), the water depth (<math>h</math>) and wavelength (<math>L</math>). Using these parameters various non-dimensional parameters can be defined, namely relative depth (<math>h/L</math>), wave steepness (<math>H/L</math>) and wave height to water depth ratio (<math>H/h</math>). Another useful non-dimensional parameter is the Ursell number (<math>U_r=H L^{2} /h^{3}</math>), first introduced in 1953.<br />
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The first finite amplitude wave theory was developed by Stokes in 1847. It is applicable to steep waves in deep and transitional water depths. Following Stokes, Korteweg and de Vries developed a shallow water finite amplitude wave theory in 1895. They termed this Cnoidal theory, analogous to the sinusiodal Airy wave theory. Both of these theories relax the assumptions made in Airy theory which, as previously described, linearises the kinematic and dynamic surface boundary conditions. In Stokes' wave theory <math>H/L</math> is assumed small and <math>h/L</math> is allowed to assume a wide range of values. The kinematic free surface boundary condition is then expressed as a power series in terms of <math>H/L</math>, and solutions up to and including the nth order of this power series are sought. Stokes derived the second order solution. In Cnoidal theory, <math>H/h</math> is assumed small and <math>U_r</math> of the order of unity. Korteweg and de Vries derived a first order solution. Much more recently (1960's to 1980's), these two theories have been extended to higher orders (third and fifth). The mathematics is complex and subsequently other researchers developed new methods whereby solutions could be obtained to any arbitrary order, by numerical solution.<br />
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Stokes' solution for the surface profile is given by:<br />
<br />
<math>\eta =\frac{H}{2} \cos \left\{2\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\}+\frac{\pi H}{8} \left(\frac{H}{L} \right)\frac{\cosh \left\{kh(2+\cosh 2kh)\right\}}{\sinh ^{3} kh} \cos \left\{4\pi \left(\frac{x}{L} -\frac{t}{T} \right)\right\} . \qquad (29)</math><br />
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This equation differs from the linear solution by the addition of the second order term. Its frequency is twice that of the first order term, which therefore increases the crest height, decreases the trough depth and thus increases the wave steepness. To second order, the wave celerity remains the same as linear theory. However, to third order the wave celerity increases with wave steepness and is approximately 20<math>\% </math> higher than given by linear theory in deep water at the limiting steepness (1/7).<br />
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[[Image: ChadwickFig16.jpg|thumb|400px| right |Figure 24. Approximate regions of validity of analytical wave theories.]]<br />
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A full mathematical description of all these theories is beyond the scope of this book and the reader is referred to Dean and Dalrymple <ref name=DD></ref> and Sorensen <ref name=So></ref> for further details. However, it is useful here to provide some information on the circumstances under which these finite amplitude wave theories can be applied. Figure 24, taken from Hedges <ref>Hedges, T.S., 1995. Regions of validity of analytical wave theories. Proc. Inst. Civ. Eng., Wat., Marit., & Energy, 112, June, 111-114. </ref>, provides useful guidance. It may be noted that the range of validity of linear theory is reassuringly wide, covering all of the transitional water depths for most wave steepnesses encountered in practice. For engineering design purposes, the main implication of using linear theory outside its range of validity is that wave celerity and wavelength are not strictly correct, leading to (some) inaccuracies in refraction and shoaling analysis. In addition, the presence of asymmetrical wave forms will produce harmonics in the Fourier analysis of recorded wave traces which could be incorrectly interpreted as free waves of higher frequency.<br />
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==Wave forces==<br />
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Wave forces on coastal structures are highly variable and depend on both the wave conditions and the type of structure being considered. Three cases of wave conditions need to be considered, comprising unbroken, breaking and broken waves. Coastal structures may also be considered as belonging to one of three types, vertical walls (e.g. sea walls, caisson breakwaters), rubble mound structures (e.g. rock breakwaters, concrete armoured breakwaters) and individual piles (e.g. for jetty construction). Here consideration is limited to outlining some of the concepts and mentioning some of the design equations that have been developed. <br />
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<br />
===Vertical Walls===<br />
<br />
The forces exerted on a vertical wall by wave action can be considered to be composed of three parts: the static pressure forces, the dynamic pressure forces and the impulsive forces. When the structure is placed such that the incident waves are unbroken, then a standing wave will exist seaward of the wall and only the static and dynamic forces will exist. These can be readily determined from linear wave theory. As a standing wave comprises two superposed progressive waves, travelling in opposite directions, the resulting equation for pressure under a standing wave is of the same form as that for a progressive wave. The standing wave height must be used in the equation, rather than the incident wave height. However, more commonly the structure will need to resist the forces produced by breaking or broken waves. The most widely used formulae for estimating the quasi static pulsating forces for either broken or unbroken waves is due to Goda <ref> Goda, Y., 1974. New wave pressure formulae for composite breakwaters, Proc. 14th Int. Conf on Coastal Eng., ASCE, New York </ref><ref name=G></ref>.<br />
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Additionally, very high localised impulsive forces may also arise due to breaking waves. These trap pockets of air, which are rapidly compressed, resulting in highly variable impulse forces (between 10 to 50 times higher than the pulsating forces). The study of this phenomenon is an ongoing area of research and currently there are no widely accepted formulae for the prediction of such forces (see Cuomo et al <ref>Cuomo, G., Allsop, W., Bruce, T. & Pearson, J. (2010). Breaking wave loads at vertical seawalls and breakwaters. Coastal Engineering 57, 424-439</ref> for recent results). Impact pressure forces are of very short duration (of the order of tenths of a second) and consequently typically affect the dynamic response of the structure rather than its static equilibrium.<br />
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===Rubble Mound===<br />
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In the case of rubble mound structures, the waves will generally break on the structure and their energy is partly dissipated by turbulence and friction, with the remaining energy being reflected and possibly transmitted. Many breakwaters are constructed using large blocks of rock (the 'armour units') placed randomly over suitable filter layers. More recently, rock has been replaced by numerous shapes of massive concrete blocks (for example, dolos, tetrapod and cob). The necessary size of the armour units depends on several interrelated factors (wave height, amour unit type and density, structure slope and permeability). Traditionally, the Hudson formula has been used. This was derived from an analysis of a comprehensive series of physical model tests on breakwaters with relatively permeable cores and using regular waves. More recently (1985-1993) these equations have been superseded by Van der Meer's equations for rock breakwaters. These equations were also developed from an extensive series of physical model tests. In these tests random waves were used and the influence of wave period and number of storm waves were also considered. A new damage criterion and a notional core permeability factor were developed. The equations are for use where the structure is placed in deep water with the waves either breaking on the structure or causing surging.<br />
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===Vertical Piles===<br />
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Finally for the case of unbroken wave forces on piles, the Morrison equation <ref>Morrison, J.R., Johnson, J.W., O'Brien, M.P. and Schaaf, S.A., 1950. The forces exerted by surface waves on piles, Petroleum Transactions, American Institute of Mining Engineers, Vol 189, 145-154. </ref> is an option that is used for design. This equation presumes that there are two forces acting. These are a drag force (<math>F_D</math>) induced by flow separation around the pile and an inertia force (<math>F_I</math>) due to the flow acceleration. For the case of a vertical pile, only the horizontal velocities (<math>u</math>) and accelerations (<math>a_x</math>) need be considered (see Equations 4b, 4c). The drag and inertia forces per unit length of pile of diameter <math>D</math> are given by <br />
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<math>F_D=C_D \rho D u|u| </math>, where <math>C_D</math> is a drag coefficient, and <br />
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<math> F_I= \rho C_M (\pi D^2 / 4) a_x </math>, where <math>C_M</math> is an inertia coefficient.<br />
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The total ‘in-line’ foce is given by <math>F=F_D+F_I . \qquad (30)</math><br />
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Morrison's equation is derived from a combination of theoretical considerations and empirical evidence, not from first principles. The equation does not include lift and slam forces and is most appropriately applied to slender circular piles or pipes subject to unbroken waves. Considering linear waves the velocity <math>u</math> and corresponding component of acceleration are given by Equation 4b and 4c respectively. The result is<br />
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<math>F=\frac{C_D \rho D A^2 \omega^2}{8} \cos(kx- \omega t) |\cos(kx- \omega t)| + \frac{C_M \rho \pi D^2 A \omega^2}{8} \sin (kx - \omega t) . \qquad (31)</math><br />
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Both these forces reduce with increasing depth and are 90<math>^{\circ}</math> out of phase. The total force acting on a vertical pile must be found as their sum, integrated over the length of the pile. Typical values of <math>C_D</math> and <math>C_M</math> for cylinders are 1 and 2 respectively. The number A/D has special significance and is known as the Keulegan-Carpenter number. Accurate values of <math>C_D</math> and <math>C_M</math> are difficult to establish from field measurements, but recommended values have been published (see the Shore Protection Manual <ref>Shore Protection Manual, 1984. USACE, Coastal Engineering Research Centre, Waterways Experimental Station, Vicksburg, USA</ref>and BS6349 <ref name=BS></ref>). Both numbers are functions of Reynolds' number <math>Re</math> and Keulegan-Carpenter number. Results from many laboratory and field experiments have been compiled by the US Army Coastal Engineering Centre who have recommended design values for shown in the tables below.<br />
<br />
{|<br />
|- style="font-weight:bold; text-align:center; background:lightgrey" <br />
! width=40% style=" border:1px solid gray;"| Reynolds number <br />
! width=30% style=" border:1px solid gray;"| <math>C_M</math><br />
|- <br />
| style="border:1px solid gray;"| <br />
| style="border:1px solid gray;"|<br />
|- <br />
| style="border:1px solid gray;"| <math>Re < 2.5 \times 10^5</math><br />
| style="border:1px solid gray;"| <math>2.0</math><br />
|- <br />
| style="border:1px solid gray;"| <math>2.5 \times 10^5 < Re < 5 \times 10^5 </math> <br />
| style="border:1px solid gray;"| <math>2.5 - 2 \times 10^{-6} Re</math> <br />
|- <br />
| style="border:1px solid gray;"| <math>Re > 5 \times 10^5</math><br />
| style="border:1px solid gray;"| <math>1.5</math><br />
|}<br />
<br />
<br />
{|<br />
|- style="font-weight:bold; text-align:center; background:lightgrey" <br />
! width=40% style=" border:1px solid gray;"| Reynolds number <br />
! width=30% style=" border:1px solid gray;"| <math>C_D</math><br />
|- <br />
| style="border:1px solid gray;"| <br />
| style="border:1px solid gray;"|<br />
|- <br />
| style="border:1px solid gray;"| <math>Re < 10^5</math><br />
| style="border:1px solid gray;"| <math>1.2</math><br />
|- <br />
| style="border:1px solid gray;"| <math> 10^5 < Re < 4 \times 10^5 </math> <br />
| style="border:1px solid gray;"| <math>0.6 - 1.2</math> <br />
|- <br />
| style="border:1px solid gray;"| <math>Re > 4 \times 10^5</math><br />
| style="border:1px solid gray;"| <math>0.6 - 0.7</math><br />
|}<br />
<br />
If these tables are used then the Reynolds number must be calculated using the maximum velocity associated with the wave.<br />
<br />
<br />
==Surf Zone Processes==<br />
<br />
===A General Description of the Surf Zone===<br />
<br />
[[Image: ChadwickFig17a.jpg|thumb|400px|left|Figure 25. The surf zone, conceptual.]]<br />
<br />
For simplicity, consider the case of a coast with the seabed and beach consisting of sand. The bed slope will usually be fairly shallow (say 0.01 <math> < \beta < </math> 0.03). Waves will therefore tend to start to break at some distance offshore of the beach or shoreline (i.e. the beach contour line which corresponds to the Still Water Level, see Figure 25). At this initial break point the wave will be of height <math>H_b</math> and at angle <math>\alpha_b</math> to the beach line. The region between this initial point and the beach is known as the surf zone. In this region, the height of an individual wave is largely controlled by the water depth. The wave height will progressively attenuate as it advances towards the beach, and the characteristic foam or surf formation will be visible on the wave front (see Figure 26 for a real example). <br />
<br />
[[Image: ChadwickFig17b.jpg|thumb|490px|right|Figure 26. A real surf zone at Hope Cove, Devon, England.]]<br />
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The mechanics of this progressive breaking are very complex. A brief summary is as follows:<br />
*Turbulence and aeration are produced.<br />
* Significant rates of change are induced in the momentum of the elements of fluid which constitute the wave. This produces a momentum force which may be resolved into two components (Figure 25). The component which lies parallel to the shoreline is the cause of a corresponding 'longshore current'. The component which is perpendicular to the shoreline produces an increase in the depth of water above the Still Water Level, and this is usually called the 'set up'.<br />
* Energy is lost due to bed friction and due to the production of turbulence. The frictional losses are produced both by the oscillatory motion at the seabed due to the wave and by the unidirectional motion of the longshore current. The two motions are not completely independent, and their interaction has significant effects on the bed friction.<br />
<br />
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<br />
===Wave Breaking===<br />
<br />
There are two criteria which determine when a wave will break. The first is a limit to wave steepness and the second is a limit on the wave height to water depth ratio. Theoretical limits have been derived from solitary wave theory, which is a single wave with a crest and no trough. Such a wave was first observed by Russell in 1840, being produced by a barge on the Forth and Clyde canal. The two criteria are given by:<br />
# Steepness <math>H/L < 1/7</math>. This normally limits the height of deep water waves.<br />
# Ratio of height to depth: the breaking index <math>\gamma = H/h = 0.78</math>. <br />
In practice <math>\gamma</math> can vary from about 0.4 to 1.2 depending on beach slope and breaker type.<br />
<br />
Goda <ref name=G></ref> provides a design diagram for the limiting breaker height of regular waves, which is based on a compilation of a number of laboratory results. He also presents an equation, which is an approximation to the design diagram, given by:<br />
<br />
<math>\frac{H_{b} }{L_{o} } =0.17\left\{1-\exp \left[-\frac{1.5\pi h}{L_{o} } \left(1+15\tan ^{4/3} \beta \right)\right]\right\}. \qquad (32)</math><br />
<br />
where <math>\tan \beta</math> is the beach slope, <math>H_b</math> the wave height at breaking and <math>L_0</math> the deep-water wavelength. For the case of random Goda <ref name=G></ref> also presents an equation set to predict the wave heights within the surf zone, based on a compilation of field, laboratory and theoretical results. <br />
<br />
<br />
===Breaker Types===<br />
<br />
[[Image: ChadwickFig18a.jpg|thumb|400px|left|Figure 27. Principal types of breaking waves.]]<br />
<br />
Breaking waves may be classified as one of three types as shown in Figure 27. The type can be approximately determined by the value of the surf similarity parameter (or Iribarren Number) <math>\xi _{b} =\tan \beta /\sqrt{H_{b} /L_{b} }, </math> where <math>L_b</math> is the wavelength at breaking.<br />
<br />
Spilling breakers (Figure 28) occur when <math> \xi _{b} < </math>0.4, plunging breakers (Figure 29) when 0.4<math>\le \xi _{b} \le </math>2.0 and surging breakers when <math>\xi _{b} > </math>2.0.<br />
<br />
Battjes <ref>Battjes, J. A., 1968. Refraction of water waves. J. Waterways and Harbours Div. ASCE, WW4, 437-457. </ref> found from real data that for 0.05 <math> <\xi_0 < </math>2 (subscript <math>o</math> indicates deep water)<br />
<br />
<math>\gamma \approx \xi _{0}^{0.17} +0.08 . \qquad (33)</math> <br />
<br />
Further details may be found in Horikawa <ref name=H></ref> and Fredsoe and Deigaard <ref>Fredsoe, J. & Deigaard, R., 1992. Mechanics of Coastal Sediment Transport, Advanced Series on Ocean Engineering 3, World Scientific, Singapore. </ref>.<br />
<br />
[[Image: ChadwickFig18b.jpg|thumb|452px|left|Figure 28. Example of a spilling breaker.]]<br />
[[Image: ChadwickFig18c.jpg|thumb|452px|right|Figure 29. Example of a plunging breaker.]]<br />
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===Wave set-down and set-up===<br />
<br />
In the case of shore-perpendicular wave incidence, the onshore momentum flux (i.e. radiation stress) <math>S_{XX}</math> , defined in Section Radiation Stress Theory, must be balanced by an equal and opposite force for equilibrium. This manifests itself as a slope in the mean still water level (given by <math>d \eta / dx</math>).<br />
<br />
[[Image: ChadwickFig19.jpg|thumb|400px|right|Figure 30. Diagram for derivation of wave setdown / setup.]]<br />
<br />
Consider the control volume shown in Figure 30 in which a set up <math>\overline{\eta }</math> on the still water level exists induced by wave action. The forces acting are the pressure forces <math>F_p</math>, the reaction force on the bottom <math>R_x</math> and the radiation stresses (all forces are wave period averaged). For equilibrium the net force in the <math>x</math>-direction is zero. Hence<br />
<br />
<math>(F_{p_{1} } -F_{p_{2} } )+(S_{XX_{1} } -S_{XX_{2} } )-R_{x} =0 , \qquad (34)</math><br />
<br />
where <math>F_{p_{2} } =F_{p_{1} } +\frac{dF_{p} }{dx} \delta x , \qquad S_{XX_{2} } =S_{XX_{1} } +\frac{dS_{XX} }{dx} \delta x .</math> <br />
<br />
As <math>F_{p} =\frac{1}{2} \rho g(h+\overline{\eta })^{2} </math> (i.e. the hydrostatic pressure force), then <br />
<br />
<math>\frac{dF_{p} }{dx} =\frac{1}{2} \rho g\frac{d}{dx} (h+\overline{\eta })^{2} =\rho g(h+\overline{\eta })\left(\frac{dh}{dx} +\frac{d\overline{\eta }}{dx} \right) \qquad (35) </math><br />
<br />
and as <math>R_x</math> for a mildly sloping bottom is due to bottom pressure,<br />
<br />
<math>R_{x} =\overline{p}\delta h=\overline{p}\frac{dh}{dx} \delta x=\rho g(h+\overline{\eta })\frac{dh}{dx} \delta x . \qquad (36)</math><br />
<br />
After substitution of Equations (35, 36) into Equation (34) the final result is <br />
<br />
<math> \frac{dS_{XX} }{dx} +\rho g(h+\overline{\eta })\frac{d\overline{\eta }}{dx} =0 , \qquad (37)</math><br />
<br />
where <math>\overline{\eta}</math> is the difference between the still water level and the mean water level in the presence of waves.<br />
<br />
Outside the breaker zone Equation (37) may be integrated to obtain<br />
<br />
<math> \overline{\eta _{d} }=-\frac{1}{8} \frac{kH_{b}^{2} }{\sinh (2kh)} . \qquad (38) </math><br />
<br />
This is referred to as the ''set-down'' (<math>\overline{\eta _{d} }</math>) and demonstrates that the mean water level decreases in shallower water. Inside the breaker zone the momentum flux rapidly reduces as the wave height decreases. This causes a ''set-up'' (<math>\overline{\eta _{u}} </math>) of the mean still water level. Making the assumption that inside the surf zone the broken wave height is controlled by depth such that<br />
<br />
<math>H=\gamma (\overline{\eta }+h) , \qquad (39) </math><br />
<br />
where <math>\gamma \approx </math>0.8 , then combining Equations (10, 37) and (39) leads to the result<br />
<br />
<math>\frac{d\overline{\eta }}{dx} =\frac{1}{1+\frac{8}{3\gamma ^2 } } \tan \beta , \qquad (40) </math><br />
<br />
Thus for a uniform beach slope it may be shown that<br />
<br />
<math> \overline{\eta _{u} }=\frac{3 \gamma^2}{8} (h_{b} -h)+\overline{\eta _{d_{b} } } , \qquad (41)<br />
</math><br />
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demonstrating that inside the surf zone there is a rapid increase in the mean water level.<br />
Thus it may be appreciated that set-down is quite small and the set-up much larger. In general wave set-down is less than 5<math>\%</math> of the breaking depth and wave set-up is about 20-30<math>\%</math> of the breaking depth. It may also be noted that for a real sea, composed of varying wave heights and periods, the wave set-up will vary along a shoreline at any moment. This can produce the phenomenon referred to as surf beats (see [[Infragravity waves]]). Wave set-up also contributes to the overtopping of sea defence structures, during storm conditions, and may thus be a contributory factor in coastal flooding.<br />
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<br />
===Radiation Stress Components for Oblique Waves===<br />
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[[Image: ChadwickFig20NEW.jpg|thumb|300px| right |Figure 31. Relationships between principal axes and shoreline axes. The <math>X</math>-axis follows the wave propagation direction; the <math>y</math>-axis is parallel to the breaker line.]]<br />
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The radiation stresses <math>S_{XX}</math>, <math>S_{YY}</math> are, in fact, principal stresses. Utilising the theory of principal stresses, shear stresses will also act on any plane at an angle to the principal axes. This is illustrated in Figure 31 for the case of oblique wave incidence to a coastline. The wave incidence angle <math>\alpha</math> is usually taken equal to the wave incidence angle at the breaker line, which is defined as the depth contour where waves start breaking according to the breaker criterion <math>H=\gamma h</math>.<br />
<br />
The relationships between the principal radiation stresses and the direct and shear components in the <math>x,y</math> directions are<br />
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<math>S_{xx} =S_{XX} \cos ^{2} \alpha +S_{YY} \sin ^{2} \alpha =\frac{1}{2} E\left[\left(1+G\right)\cos ^{2} \alpha +G\right] , </math><br />
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<math>S_{yy} =S_{XX} \sin ^{2} \alpha +S_{YY} \cos ^{2} \alpha =\frac{1}{2} E\left[\left(1+G\right)\sin ^{2} \alpha +G\right] , </math><br />
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<math>S_{xy} =S_{XX} \sin \alpha \cos \alpha -S_{YY} \sin \alpha \cos \alpha =\frac{1}{2} E\left[\left(1+G\right)\sin \alpha \cos \alpha \right] , \qquad (42)</math><br />
<br />
where <math>G=2kh/\sinh (2kh) . </math> The right-member expressions follow from equations (10, 11). <br />
<br />
<br />
===Longshore Currents===<br />
<br />
Radiation stress theory has been successfully used to explain the presence of longshore currents. The original theory is eloquently explained by Longuet-Higgins <ref name=LH></ref>. Subsequently Komar <ref name=K></ref>, as a result of his own theoretical and field investigations, developed the theory further and presented revised equations. All of the foregoing is succinctly summarised in Hardisty <ref name=Ha></ref>. Here a summary of the main principles is given together with a statement of the main equations.<br />
<br />
An expression for the mean wave period averaged longshore velocity (<math>\overline{\nu _{l} }</math>) was derived from the following considerations. Firstly outside the surf zone the energy flux towards the coast (<math>P_x</math>) of a wave travelling at an oblique angle (<math>\alpha </math>) is constant and given by (see Equation (8))<br />
<br />
<math>P_{x} =Ec_{g} \cos \alpha . \qquad (43) </math><br />
<br />
Secondly, the radiation stress (<math>S_{xy}</math>) which constitutes the flux of <math>y</math>-momentum parallel to the shoreline across a plane <math>x</math> = constant is given by<br />
<br />
<math>S_{xy} = \frac{1 }{2} E (1+G) \cos \alpha \sin \alpha =E\left(\frac{c_{g} }{c} \right)\cos \alpha \sin \alpha . \qquad (44) </math><br />
<br />
Hence, combining Equations (43, 44), <math>S_{xy} =P_{x} c / \sin \alpha </math> outside the surf zone. Because of Snell’s law <math>\sin \alpha / c</math>= constant, <math>S_{xy}</math> is also constant. However, inside the surf zone this is no longer the case as wave energy flux is rapidly dissipated. The net thrust (<math>F_y</math>) per unit area exerted by the waves is given by<br />
<br />
<math>F_{y} =\frac{-\partial S_{xy} }{\partial x} . \qquad (45) </math><br />
<br />
Substituting for <math>S_{xy}</math> from Equation (44) and taking conditions at the wave break point (at which <math>c_{g} =c=\sqrt{gh_{b} } , \; H_{b} /h_{b} =\gamma , \; u_{m} =(\gamma /2) \sqrt{gh_{b} } </math>). <br />
<br />
Longuet-Higgins <ref name=LH></ref> derived an expression for <math>F_y</math> given by<br />
<br />
<math>F_{y} =\frac{5}{4} \rho u_{mb}^{2} \tan \beta \sin \alpha . \qquad (46) </math><br />
<br />
Finally, by assuming that this thrust was balanced by frictional resistance in the longshore (<math>y</math>) -direction he derived an expression for the mean longshore velocity <math>\overline{\nu _{l} }</math>, given by<br />
<br />
<math> \overline{\nu _{l} }=\frac{5\pi }{8C} u_{mb} \tan \beta \sin \alpha, \qquad (47) </math><br />
<br />
where <math>C</math> is a friction coefficient.<br />
<br />
Subsequently Komar <ref name=K></ref> found from an analysis of field data that <math>\tan \beta /C </math> was effectively constant and he therefore proposed a modified formula given by<br />
<br />
<math> \overline{\nu _{l} }=2.7u_{mb} \sin \alpha \cos \alpha, \qquad (48) </math><br />
<br />
in which the <math>\cos \alpha </math> term has been added to cater for larger angles of incidence (Longuet-Higgins <ref name=LH></ref> assumed <math>\alpha</math> small and therefore <math>\cos \alpha \to 1</math>).<br />
<br />
The distribution of longshore currents within the surf zone was also studied by both Longuet-Higgins and Komar. The distribution depends upon the assumptions made concerning the horizontal eddy coefficient, which has the effect of transferring horizontal momentum across the surf zone. Komar <ref name=K></ref> presents a set of equations to predict the distribution.<br />
<br />
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<br />
==Further reading==<br />
<br />
*Reeve, D., Chadwick, A. J., Fleming, C. (2012). Coastal Engineering: Processes, Theory and Design Practice (2nd ed) E & FN Spon.<br />
*Open University, 1989. Waves, Tides and Shallow Water Processes, Pergamon Press, Oxford.<br />
*Horikawa, K. (ed.), 1988. Nearshore Dynamics and Coastal Processes, Theory Measurement and Predictive Models, University of Tokyo Press, Tokyo.<br />
<br />
<br />
<br />
==Related articles==<br />
<br />
[[Waves]]<br />
<br />
[[Infragravity waves]]<br />
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<br />
==References==<br />
<br />
<references/><br />
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{{author<br />
|AuthorID=11189<br />
|AuthorFullName= Andrew Chadwick<br />
|AuthorName= Andrew Chadwick }}<br />
<br />
[[Category:Land and ocean interactions]]<br />
[[Category:Hydrodynamics]]<br />
[[Category:Coastal processes, interactions and resources]]<br />
[[Category:Coastal and marine natural environment]]</div>Dronkers Jhttp://www.vliz.be/wiki/Wave_ripplesWave ripples2017-07-03T12:23:22Z<p>Dronkers J: </p>
<hr />
<div>==Introduction==<br />
<br />
Sea waves shape the bottom and generate different morphological patterns,<br />
which are characterized by a wide range of length scales. The ripples are<br />
the smallest bedforms but, notwithstanding their relatively small size,<br />
they play a prominent role in many transport processes. Indeed,<br />
usually, the flow separates at their crests and vortices are<br />
generated which increase momentum transfer, sediment transport and, in<br />
general, mixing phenomena.<br />
<br />
Even though the ripples generated by sea waves (wave ripples) appear to be<br />
similar to the ripples generated by steady currents/slowly varying tidal<br />
currents, they have different characteristics since they are the result of<br />
a different mechanism of formation. What follows concerns wave ripples, which hereinafter<br />
are simply named 'ripples'. The ripples generated by steady currents<br />
are not considered in the present article.<br />
<br />
<br />
==Ripple geometry==<br />
<br />
[[Image: BlondeauxFig1.jpg|thumb|400px|right|Figure 1. Ripple marks in the Mediterranean Sea. Courtesy of José B. Ruiz.]]<br />
<br />
The geometry of the most common ripples is almost two-dimensional and similar<br />
to that of a wave with a crest and a trough (see figure 1). However, the wavelength <math>\lambda</math> of the<br />
ripples is of the order of magnitude of the amplitude of the fluid displacement<br />
oscillations close to the sea bottom and it turns out to be of <math>O </math> (10 cm) , while the height of the ripples is of a few centimetres.<br />
<br />
The estimate of the geometrical characteristics of the ripples is important for many reasons. First,<br />
ripple presence enhances the suspended sediment transport. Indeed,<br />
when the steepness of the ripples becomes larger than approximately 0.1 <ref name=S>Sleath J.F.A. (1984) Sea bed mechanics. John Wiley and Sons.</ref>, the oscillatory bottom<br />
boundary layer separates from the crest of the bedforms and a vortex is generated, twice per<br />
wave cycle, on the lee side of the ripple by the roll-up of the<br />
free shear layer shed from the ripple crest. This vortex picks-up the sediments<br />
from the bottom and at flow reversal, when it is convected in the opposite direction by the free stream, it ejects the sediments far from the bottom. <br />
Later, the vortex decays and the sediments settle down.<br />
Moreover, because of the vortices they shed, the ripples significantly increase the bottom friction and the dissipation of energy.<br />
Finally, ripples may act as a source of nutrients. In fact, for high values of the bottom shear stress,<br />
ripples are washed out and the nutrients are released into the water column at a rate which depends on the time development of their geometrical characteristics.<br />
<br />
Empirical formulae to predict ripple wavelength and height are used for practical purposes.<br />
The physical quantities which affect the ripple geomety can be assumed to be i) the<br />
density <math>\rho</math> of the water, ii) the kinematic viscosity of the water <math>\nu</math>,<br />
iii) the period <math>T</math> of the velocity oscillations induced close to the sea bed by the surface waves or, alternatively, the angular frequency <math>\omega=2 \pi /T</math>,<br />
iv) the amplitude <math>U_0</math> of the velocity oscillations (<math>U_0=a\omega/\sinh (kh)</math>, <math>k=2 \pi/L</math> being the wavenumber of the surface wave, <math>a</math> being its amplitude and <math>h</math> the local water depth),<br />
v) the sediment size <math>d</math>, vi) the density <math>\rho_s</math><br />
of the sediment and vii) gravity acceleration <math>g</math>.<br />
By applying dimensional arguments, the wavelength and height of the ripples turn<br />
out to depend on four dimensionless parameters, i.e. the relative<br />
density <math>s=\rho_s / \rho</math>, the ratio <math> d /\delta</math> between<br />
the grain size and the viscous length <math>\delta=\sqrt{2 \nu / \omega}</math>,<br />
a sediment Reynolds number <math>R_p = \sqrt{ (s-1)g d^3} / \nu</math> and a flow Reynolds<br />
number <math>R_{\delta} = U_0 \delta / \nu</math>. Of course, these parameters<br />
can be replaced by their combinations and in the literature<br />
it is common to encounter other parameters such as the mobility number <math>\psi= U_0^2 / ((s-1)gd)</math>, the Reynolds number of the sediment defined by <math>R_d = U_0 d / \nu</math> and the flow Reynolds number <math>Re = U_0^2 / (\nu \omega) = R^2_\delta / 2</math>.<br />
<br />
Both a simple dimensional analysis and idealized models, based on linear stability analyses (see the<br />
article [[Wave ripple formation]]), show that the geometrical characteristics of ripples can not<br />
be predicted from the knowledge of just one dimensionless parameter. However,<br />
the empirical formulae which can be found in the literature use one<br />
parameter for simplicity. Hence, the first question to be<br />
addressed is: which is the independent parameter that mainly controls<br />
ripple geometry ?<br />
<br />
The plethora of predictors using different parameters, along with<br />
the significant differences among the predictions they provide, suggest that the problem<br />
of predicting ripple characteristics is far to be definitively solved.<br />
An exhaustive description of all the predictors and the<br />
advantages/disadvantages of each of them is beyond the aim of the present article.<br />
The paper of Nelson et al. <ref name=N>Nelson T.R., Voulgaris G. and Traykovski P. (2013). Predicting wave-induced ripple equilibrium geometry. J. Geophys. Res.: Ocean 118, 3202-3220. </ref> describes and discusses some of the predictors commonly used.<br />
In the following, to give an idea of these predictors and of their performances, we describe only a few of them.<br />
<br />
<br />
<br />
===Ripple wavelength===<br />
<br />
Laboratory and field data indicate that the wavelengths of the ripples<br />
generated by regular surface waves are somewhat different from those generated<br />
by irregular waves.<br />
However, Soulsby and Whitehouse <ref name=SW>Soulsby R.R. & Whitehouse R.J.S. (2005) Prediction of Ripple Properties in Shelf Seas. Mark 2 Predictor for Time Evolution Final Technical Report TR 154. HR Wallingford Limited. </ref> and Nelson et al. <ref name=N></ref> proposed a single predictor to be used under both regular and irregular waves and they related<br />
the ratio between the ripple wavelength <math>\lambda</math> and the amplitude<br />
<math>U_0/\omega</math> of the fluid displacement oscillations close to the bottom<br />
to the parameter <math> U_0 / ( \omega d)</math>:<br />
<br />
<math>\frac{\lambda}{U_0/\omega}=\frac{1}{ a_1 + b_1 \frac{U_0}{\omega d}<br />
\left[1-e^{-\left( c_1 \frac{U_0}{\omega d} \right)^{d_1}}\right]} , \qquad(1) </math><br />
<br />
where the constants suggested by <ref name=S></ref> are<br />
<br />
<math>a_1=1, \ \ \ \ \ b_1= 1.87 \times 10^{-3}, \ \ \ \ \ c_1=2.0 \times 10^{-4}, \ \ \ \ \ d_1=1.5 ,<br />
\qquad (2) </math><br />
<br />
while the constants suggested by <ref name=N></ref> are<br />
<br />
<math>a_1=0.72, \ \ \ \ \ b_1= 2.00 \times 10^{-3}, \ \ \ \ \ c_1=1.57 \times 10^{-4}, \ \ \ \ \ d_1=1.15 \qquad (3) </math><br />
<br />
[[Image: BlondeauxFig2.jpg|thumb|350px|right|Figure 2. Comparison between predicted (lines) and observed (points) ripple wavelengths. The experimental data are some of those collected by Nelson et al. <ref name=N></ref>.]]<br />
<br />
Figure 2 shows measured ripple wavelengths along with the values provided by equation (1)<br />
with the values of the constants suggested by both Nelson et al. <ref name=N></ref> and Soulsby and Whitehouse <ref name=SW></ref>.<br />
<br />
The blue points of figure 2 are extracted from the database collected by <ref name=N></ref> for regular waves<br />
or oscillatory flows. Of course when data for irregular waves (red points) are added by considering the flow generated by the significant wave (red points),<br />
the scatter of the data slightly increases.<br />
The experimental data obtained using<br />
an oscillating tray are discarded because Miller and Komar <ref>Miller M.C. and Komar P.D. (1980). A field investigation of the relationship between ripple spacing and near-bottom water motions. J. Sed. Petrology 50, 183-191. </ref> concluded that the results of oscillating bed experiments are<br />
different from water tunnel, wave channel and field results.<br />
Moreover, some of the data of <ref name=N></ref> are not plotted in figure 2 because they refer to field data or data obtained in large flume facility where the average characteristics of the surface waves change over time. Even though Nelson et al. <ref name=N></ref> introduced heuristic criteria to consider only ripples which attained a morphological equilibrium, it might be that some of the data refer to relic ripples the geometry of which does not depend on the actual characteristics of the surface waves.<br />
<br />
[[Image: BlondeauxFig3.jpg|thumb|350px|right|Figure 3. Comparison between predicted (lines) and observed (points) ripple wavelengths. The experimental data are some of those collected by Nelson et al. <ref name=N></ref>.]]<br />
<br />
[[Image: BlondeauxFig4.jpg|thumb|350px|right|Figure 4. Comparison between predicted (lines) and observed (points) ripple wavelengths. The experimental data are some of those collected by Nelson et al. <ref name=N></ref>.]]<br />
<br />
On the other hand, Inman <ref name=I>Inman D.L. (1957). Tech. Mem. U.S. Beach Erosion N. 100. </ref> suggested that, for an assigned sediment size, the ripple wavelength depends on<br />
<math>2 U_0 / \omega</math>, i.e. twice the amplitude of the fluid displacement oscillations close to the bottom.<br />
In particular Inman <ref name=I></ref>, by analysing experimental data, showed that the ripple<br />
wavelength is directly proportional to <math>2 U_0 / \omega</math> up to a critical value whereupon the crest-to-crest distance becomes inversely proportional<br />
to <math>2 U_0 / \omega</math> and finally it attains a constant value.<br />
Figure 3 shows that the values of <math>\lambda/d</math> plotted versus <math>(2 U_0/\omega)/d</math>, for the data already considered in figure 2.<br />
The laboratory measurements dominate the left hand side of the plot while the field measurements dominate the right hand side.<br />
Later, Clifton <ref>Clifton H.E. (1976). Wave-formed sedimentary structures - A conceptual model. In Davis and Ethington ed: Beach and nearshore sedimentation. SEPM special publication 24, 126-148</ref> classified ripples as orbital, anorbital and suborbital ripples. Orbital ripples are<br />
characterized by a wavelength proportional to the amplitude of the fluid displacement oscillations<br />
<br />
<math>\lambda \simeq 0.65 \frac{2 U_0}{\omega} . \qquad (4) </math><br />
<br />
Anorbital ripples appear for large values of <math>2 U_0/(\omega d)</math> and their wavelength is almost independent of <math>2 U_0 / \omega</math><br />
and ranges between <math>400 d</math> and <math>600 d</math>, even though the measurements show a large scatter which does not allow to obtain a more precise value.<br />
The critical point is to predict what type (orbital/anorbital) of ripple appears for given<br />
hydrodynamic and morphodynamic parameters. Moreover, suborbital ripples exist too, which have a wavelength which depends on both<br />
<math>2 U_0/ \omega</math> and the grain size <math>d</math>.<br />
This problem is not present if formula (1) is used to predict ripple wavelength,<br />
however also figure 2 shows that a significant number of observations (values of <math>U_0/(\omega d)</math> in the range<br />
<math>(10^3,10^4)</math>) deviate from the trend predicted by relationship equation (1)<br />
<br />
Nielsen <ref name=N81>Nielsen P. (1981). Dynamics and geometry of wave generated ripples. J. Geophys Res. 86 (C7), 6467-6472. </ref> proposed to predict the spacing of the ripples as function of the sediment mobility number<br />
<br />
<math>\psi=\frac{U_0^2}{\left( s-1 \right) g d} \qquad (5) </math><br />
<br />
and he suggested<br />
<br />
<math>\frac{\lambda}{U_0/\omega}= \exp\left(\frac{693-0.37 \ln^8\psi}{1000 +0.75 \ln^7 \psi}\right) \qquad(6) </math><br />
<br />
for ripples observed in the field and<br />
<br />
<math>\frac{\lambda}{U_0/\omega}= 2.2 - 0.345 \psi^{0.34} \qquad (7) </math><br />
<br />
for the ripples generated by a regular oscillatory flow. The values of the ripple wavelength predicted by means of (6) and (7) are plotted as function of the mobility number in figure 4, along with the experimental data.<br />
<br />
<br />
<br />
===Ripple height===<br />
<br />
[[Image: BlondeauxFig5.jpg|thumb|400px|right|Figure 5. Comparison between predicted (line) and observed (points) ripple steepnesses. The experimental data are some of those collected by Nelson et al. <ref name=N></ref>.]]<br />
<br />
Once ripple wavelength is estimated, the height <math>\eta</math> can be obtained from the prediction of ripple steepness.<br />
Soulsby and Whitehouse <ref name=SW></ref> suggest to predict the ripple steepness <math>\eta/\lambda</math> as function of the parameter <math>U_0/(\omega d)</math><br />
<br />
<math>\frac{\eta}{\lambda}= 0.15 \left[ 1 - \exp\left[ -( \frac{5.0 \times 10^{3} } {\left(U_0/\omega d\right)} )^{3.5} \right] \right] . \qquad (8) </math><br />
<br />
On the other hand, Nelson et al. <ref name=N></ref> suggest to predict the ripple steepness as function of the ripple wavelength according to the formula<br />
<br />
<math>\frac{\eta}{\lambda}=0.12 \lambda^{\left(-0.056 \right)} , \qquad (9) </math><br />
<br />
where <math>\lambda</math> should be in metres. The predictor equation (9) has the disadvantage of predicting the ratio <math>\eta/\lambda</math><br />
as function of a dimensional quantity. Moreover, to predict the ripple steepness, it is necessary either to know the wavelength or to<br />
predict the value of <math>\lambda</math> by using equations (1) and (3).<br />
Figure 5 shows a comparison between the results provided by equation (8) and the experimental measurements for regular and<br />
and irregular waves.<br />
<br />
<br />
<br />
===Ripple symmetry index===<br />
<br />
[[Image: BlondeauxFig6.jpg|thumb|400px|left|Figure 6. Ripple symmetry index plotted versus the ratio between the mass-transport velocity and the maximum near-bed orbital velocity.<br />
The experimental data are those described in Allen <ref name=A>Allen J.R.L. (1984). Sedimentary structures, their character and physical basis. Elsevier.</ref> and in Blondeaux et al. <ref name=B></ref>.]]<br />
<br />
Even though the profile of the ripples is almost symmetric with respect to their<br />
crest, a small degree of asymmetry is invariably generated by the steady<br />
streaming, which is present under a propagating wave because of nonlinear<br />
effects <ref name=L>Longuet-Higgins M.S. (1953). Mass transport in water waves. Philos. Trans. R. Soc. 345, 535-581. </ref>, and by the difference between the forward<br />
fluid velocity, which takes place under the crests of the surface wave,<br />
and the backward velocity under the troughs. Hence, a symmetry index of the ripple can be defined as the ratio between the length <math>l_2</math> of the<br />
gentle (up-current) side to the length <math>l_1</math> of the steep (down-current)<br />
side of the bottom forms. Figure 6 shows the values of<br />
<math>\; (l_2 / l_1)-1 \;</math> plotted versus the strength of the steady streaming for the experimental data of Inman <ref name=I></ref>, Tanner <ref>Tanner W.F. (1971). Numerical estimates of ancient waves, water depth and fetch. Sedimentology 16, 71-88. </ref> and Blondeaux et al. <ref name=B>Blondeaux P., Foti E., Vittori G. (2000). Migrating sea ripples. European Journal of Mechanics - B/Fluids 19 (2), 285-301. </ref>.<br />
In figure 6, the value of the steady velocity component <math>U_s</math> is estimated by means of<br />
the theory of Longuet-Higgins <ref name=L></ref> (<math>U_s = 3\pi a^2\omega /[2L\sinh^2(2\pi h/L)]</math>).<br />
Hence, the abscissa of figure 6 is equal to <math>3\pi a /[2L\sinh(2\pi h/L)]</math>.<br />
As expected, the results plotted in figure 6<br />
indicate that ripples tend to become more asymmetric as the mass transport velocity increases.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
===Three-dimensional ripples===<br />
<br />
Often, the ripple profile is two-dimensional but depending on sediment and flow characteristics,<br />
other ripple shapes are observed. For example, figure 7a shows the brick-pattern ripples<br />
observed by Sleath <ref name=S></ref> during a laboratory experiment. Brick pattern ripples have the crests perpendicular to the direction of the fluid oscillations, as two-dimensional ripples. However, the crests are joined by equally spaced bridges of smaller amplitude which are parallel to the direction of fluid oscillations and shifted by half a wavelength between adjacent sequences. It follows that the overall bottom topography resembles a wall made by bricks.<br />
<br />
[[Image: BlondeauxFig7ab.jpg|thumb|700px|center|Left: 7a. Brick-pattern ripples observed during a laboratory experiment. Centre: 7b. Three-dimensional vortex ripples observed during a laboratory experiment (7a and 7b courtesy of John F.A. Sleath).]]<br />
<br />
Other three-dimensional ripples do exist and photos can be found in the books of<br />
Sleath <ref name=S></ref> and Allen <ref name=A></ref>.<br />
<br />
<br />
==Friction factor for ripples==<br />
<br />
In large scale hydrodynamic problems, it is not possible to compute the flow field with the spatial resolution required to compute the flow around each ripple. Hence, these small scale bedforms are usually modelled as a roughness of the bed of appropriate size. Experimental measurements indicate that the size of the roughness is related to the ripple height. Van Rijn <ref name=R>van Rijn L.C. (1993). Principles of sediment transport in rivers, estuaries and coastal seas. Aqua Publications, Amsterdam, the Netherlands.</ref> suggests that the roughness size ranges between one and three ripples heights. These values are supported by the data shown in figure 3.6.7 of Nielsen's book <ref name =N92>Nielsen P. (1992). Coastal bottom boundary layers and sediment transport. World Scientific</ref>. However, experimental measurements carried out for low flow intensities indicates that also the shape of the ripples, and in particular their steepness <math>\eta/\lambda</math>, affects the equivalent roughness size.<br />
In the literature, it is suggested that the equivalent roughness size <math>k_s</math> can be given values close to <math>c_1 \eta^2/\lambda</math>,<br />
where <math>c_1</math> is a constant. Nielsen <ref name=N92></ref> proposed <math>c_1=8</math> while Grant and Madsen <ref>Grant W.D. and Madsen O.S. (1982). Movable bed roughness in unsteady oscillatory flow. J. Geophys Res. 87, 469-481. </ref> and Li et al. <ref>Li M.Z., Wright L.D. and Amos C.L. (1996). Predicting ripple roughness and sand resuspension under combined flows in a shoreface environment. Marine Geology 130, 139-161. </ref><br />
proposed <math>c_1=28</math> and Van Rijn <ref name=R></ref> suggested <math>c_1 =20</math>, even though in a previous work Van Rijn <ref>van Rijn L.C. (1984). Sediment Transport, Part III: Bed forms and alluvial roughness. J. Hydraul. Eng., ASCE, 110, 1733-1754. </ref> assumed<br />
<math>k_s=1.1 \eta \left(1-e^{-25 \eta/\lambda} \right)</math><br />
<br />
For large flow intensities, when the coherent vortices shed by the ripples pick-up a lot of sediments from the bed and put<br />
them into suspension, Nielsen <ref name=N92></ref> suggests to estimate the equivalent roughness by means of<br />
<br />
<math> k_s= 8 \frac{\eta^2}{\lambda}+170 d \sqrt{\theta -0.05} , </math><br />
<br />
where <math>\theta=\tau / ((\rho_s-\rho) g d)</math> is the Shields parameter evaluated by using the skin friction <math>\tau</math>.<br />
<br />
<br />
<br />
==Grain-sorting over ripples==<br />
<br />
[[Image: BlondeauxFig10.jpg|thumb|500px|left|Figure 8. Grain sorting over ripples. The yellow sediment grains are coarser than the red sediments and pile up at the crests of the ripples which<br />
are generated in a U-tube by an oscillatory flow. The figure shows a top<br />
view of the bottom at two different phases of the cycle<br />
(adapted from Foti and Blondeaux<ref name=F>Foti E. and Blondeaux P. (1995). Sea ripple formation: the heterogeneous sediment case. Coastal Eng., 25(3-4), 237-253. </ref>).]]<br />
<br />
<br />
In the field, the sediment is often a mixture of particles having different<br />
sizes and ripples give rise to sorting phenomena.<br />
Foti and Blondeaux <ref name=F></ref> made laboratory experiments with a sediment mixture characterized by a bimodal<br />
grain size distribution and observed that the coarser fraction oscillated around the crests of the bedforms while the fine<br />
fraction tended to move towards the troughs (see figure 8).<br />
Moreover, they found that the sorting phenomena<br />
affect the dynamics of the ripples. Indeed, the ripples generated by a well sorted sediment turn out to be shorter<br />
than those generated by a poorly sorted sediment.<br />
<br />
<br />
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<br />
<br />
==Related articles==<br />
<br />
[[Wave ripple formation]]<br />
<br />
[[Bedforms and roughness]]<br />
<br />
[[Sand transport]]<br />
<br />
[[Sediment transport formulas for the coastal environment]]<br />
<br />
<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
<br />
<br />
{{2Authors<br />
|AuthorID1=14087<br />
|AuthorFullName1= Paolo Blondeaux<br />
<br />
|AuthorID2=14090 <br />
|AuthorFullName2= Giovanna Vittori<br />
<br />
}}<br />
<br />
[[Category:Land and ocean interactions]]<br />
[[Category:Geomorphological processes and natural coastal features]]<br />
[[Category:Coastal processes, interactions and resources]]<br />
[[Category:Coastal and marine natural environment]]</div>Dronkers Jhttp://www.vliz.be/wiki/Swash_zone_dynamicsSwash zone dynamics2017-06-01T08:39:51Z<p>Dronkers J: </p>
<hr />
<div>==Introduction==<br />
<br />
[[Image: BaldockFig1.jpg|thumb|440px|left|Figure 1. Definition sketch for the nearshore littoral zone (swash zone width exaggerated). After <ref> Elfrink, B. and T. Baldock (2002). Hydrodynamics and sediment transport in the swash zone: a review and perspectives. Coastal Engineering 45: 149-167</ref>.]]<br />
<br />
The swash zone forms the land-ocean boundary at the landward edge of the surf zone, where waves runup the beach face (figures 1, 2). It is perhaps the region of the ocean most actively used by recreational beach users and, being very visible, is the region of the littoral zone most associated with beach erosion and the impacts of climate change. The landward edge of the swash zone is highly variable in terms of geomorphology, and may terminate in dunes, cliffs, marshes, ephemeral estuaries and a wide variety of sand, gravel, rock or coral barriers. This influences the exchange of sediment between the land and ocean, which ultimately forms the coastline.<br />
<br />
In terms of coastal processes and coastal protection, a large part of the littoral sediment transport occurs in the swash zone, both cross-shore and longshore, which influences beach morphology, and beach erosion and beach recovery during and after storms. Wave runup is an important factor in the design of coastal protection and also generates hazards for beach users, and is the dominant process leading to the erosion of coastal dunes. Swash hydrodynamics also influence the ecology of the intertidal zone and groundwater levels in sub-aerial littoral beaches and low lying islands, which is often critical for freshwater water supply on islands and atolls <ref>Nielsen P., 1999. Groundwater dynamics and salinity in coastal barriers. J. Coastal Res., 15: 732-740. </ref>. <br />
<br />
<br />
==Characteristics of the swash zone==<br />
<br />
<br />
<br />
[[Image: BaldockFig2a.jpg|thumb|445px|left|Figure 2a. Seven Mile Beach, NSW, Australia, a dissipative beach. Photo shows conditions after a swash rundown, with only small bores reaching the swash zone. Photo: Dr Hannah Power, University of Newcastle, NSW, Australia.]]<br />
<br />
[[Image: BaldockFig2b.jpg|thumb|445px|right|Figure 2b. Avoca Beach, NSW, Australia, a reflective beach. Photo shows the inner surf zone and a bore reaching the swash zone in the background and a swash uprush reaching the top of a beach berm in the foreground.]]<br />
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The hydrodynamic processes in the swash zone are characterised by very different types of fluid flow compared to the open ocean and surf zone, as illustrated in figure 2, where the strongly orbital motion of waves is transformed into flow parallel to the bed (velocity <math>V</math>), usually in thin shallow sheets (thickness <math>d</math>). In terms of fluid mechanics, the key difference is the occurrence of supercritical flow in the swash zone, where the Froude number,<math>V/ \sqrt{gd}</math>, is greater than 1, which has important implications for the nature of the flow. Other important distinctions are that friction becomes more important in controlling aspects of the shallow flow in the swash zone than in the surf zone, and that turbulence and sediment transport in the swash zone is generated locally in the swash zone and advected into the swash zone from the surf zone. A key feature of both the hydrodynamics and sediment dynamics in the swash zone is intermittency, where the extent and degree of inundation of the swash zone varies over different timescales, from orders of seconds and tidal periods to years and decades. This is a challenge for coastal scientists, both in terms of measurement and modelling of the physical processes. For the purposes of this summary, the swash zone will be considered as the region of the beach face exposed to the atmosphere over wind, swell and infragravity wave durations, i.e., seconds to a few minutes.<br />
<br />
The characteristics of the swash zone hydrodynamics and sediment transport are governed by the inner surf zone and the underlying beach, with feedback of course between the morphology and hydrodynamic processes. The beach slope is a controlling parameter <ref>Guza, R. T., Thornton, E. B. and Holman, R. A., 1984. Swash on steep and shallow beaches. Proc. 19th Int. Conf. Coastal Eng., ASCE, 708-723. </ref>. On dissipative beaches, with wide surf zones, most of the wind wave and swell energy is dissipated seaward of the swash zone. Therefore, swash processes are dominated by those due to long, or infragravity, waves, which are frequently non-breaking standing waves (figure 2a). On intermediate and reflective beaches, short wave energy reaches the beach face in the form of bores or shore-breaks, which collapse at the beach, initiating a runup motion characterised by a thin sheet of water with a rapidly propagating wave tip which is analogous to a dam-break flow over a dry bed (figure 2b). This sheet of water is slowed by gravity and friction until the flow reverses and forms another shallow flow seaward, the backwash. On coarse grained sand and gravel beaches a significant volume of the uprush and some of the backwash may percolate into the beach, reducing the volume of water in the surface backwash flow. These two distinct types of swash zone make modelling hydrodynamic processes difficult, since parametric models rely on similarity of processes, and therefore phase resolving models of the whole surf zone, or at least the inner surf zone, are required if the details of the hydrodynamics are required. Fortunately some processes are modelled very well by parametric models, perhaps more accurately than phase-resolving models, particularly wave runup.<br />
<br />
<br />
==Wave runup and overtopping==<br />
<br />
[[Image: BaldockFig3NEW2.jpg|thumb|500px|left|Figure 3. Typical pattern of bore-driven swash oscillations (vertical component). Data show shoreline elevation versus time at Avoca Beach, NSW. Red and green squares indicate maxima and minima of individual swash events. Data courtesy of Dr Michael Hughes, NSW Office of Environment and Heritage.]]<br />
<br />
Wave runup is perhaps the most important aspect of swash zone flows. While the motion of the water volume as a whole may be considered as runup, conventionally wave runup refers to the landward limit of the swash motion on the beach face, usually defined vertically above the ocean level. The runup and rundown of the shoreline is referred to as the swash excursion or oscillation. A typical plot of the shoreline motion is shown in figure 3.<br />
<br />
Remarkably, the maximum runup is still most reliably described by a simple empirical parametric formula, which is perhaps one of the oldest regularly in use, proposed by Hunt (1959). Despite numerous variations, the underlying scaling still holds over a very large range of wave conditions, both in the laboratory and field.<br />
The scaling for runup proposed by Hunt <ref>Hunt, I.A., 1959. Design of sea-walls and breakwaters. Transactions of the American Society of Civil Engineers, 126: 542-570. </ref> is:<br />
<br />
<math>R_{max}=\xi H_0 = \tan \beta \sqrt{H_0 L_0} , \qquad (1)</math><br />
<br />
where <math>\xi</math> is the surf similarity parameter, or Iribarren number (<math>\xi =\tan\beta / \sqrt{H_0/L_0})</math> and <math>H_0, L_0</math> and <math>\beta</math> are offshore wave height, wave length and the swash zone beach face slope, respectively. This is strictly only applicable for monochromatic waves and the maximum runup. However, the same formulation has been widely used to describe random wave runup, using appropriate statistical parameters to describe the wave conditions. This parameter is usually the runup elevation exceeded by 2% of the waves, <math>R_{2\%}</math>. The most widely used derivative of Hunt's formula is perhaps due to Stockdon et al. <ref name=S> Stockdon, H. F., Holman, R. A., Howd, P. A. & Sallenger JR, A. H. 2006. Empirical parameterization of setup, swash, and runup. Coastal Engineering, 53, 573-588.</ref>, which also allows for a contribution from infragravity waves:<br />
<br />
<math> R_{2/%} = 1.1 \; \left( 0.5 \sqrt{H_s L_p \; (0.563 \beta^2 + 0.004)} +0.35 \; \beta \sqrt{H_s L_p} \right) , \qquad \xi_p \ge 0.3 , \qquad (2)</math><br />
<br />
<math> R_{2/%}= 0.43 \; \beta \sqrt{H_s L_p} , \qquad \xi_p < 0.3 , \qquad (3)</math><br />
<br />
where the term <math>0.35 \; \beta \sqrt{H_s L_p}</math> represents the setup, <math>H_s</math> is the deep water significant wave height, and <math>L_p</math> is the wavelength corresponding to the deep water peak wave period, <math>T_p</math>. Atkinson et al. <ref>Atkinson, A.L., Power, H.E., Moura, T., Hammond, T., Callaghan, D.P. and Baldock, T.E., 2017. Assessment of runup predictions by empirical models on non-truncated beaches on the south-east Australian coast. Coastal Engineering, 119: 15-31. </ref> compared a number of recent runup formulations to measurements from a range of beaches and concluded that the models generally predict runup with errors of order <math>\pm 25 \%</math>.<br />
<br />
[[Image: BaldockFig4.jpg|thumb|500px|left|Figure 4. Swash energy spectra from (a) reflective, (b) intermediate, and (c) dissipative beach-states. The grey lines show individual spectra, the coloured lines show the average spectrum for the beach-state, the black line represents an <math>f^{-4}</math> energy roll-off and the vertical dashed line demarcates the short-wave and long-wave frequency bands. From Hughes et al. <ref>Hughes, M. G., T. Aagaard, T. E. Baldock and H. E. Power, 2014. Spectral signatures for swash on reflective, intermediate and dissipative beaches. Marine Geology 355: 88-97. </ref>, with permission.]]<br />
<br />
Swash-swash interactions occur through the overtaking of a swash uprush by the following bore or during the collision of the backwash flow with the next uprush. The magnitude, or vertical excursion, of the swash oscillations, from rundown position to runup, is strongly influenced by interaction between wave uprush and backwash, with the period of the incident waves also controlling the period of the swash oscillations at swell and wind wave frequencies <ref>Holman, R. A. 1986. Extreme value statistics for wave runup on a natural beach. Coastal Engineering, 9: 527-544. </ref>. Hence, given a finite time for the uprush and backwash to occur, there is a finite magnitude for a swash oscillation at a given frequency on a given beach slope if the motion is solely controlled by gravity. This leads to swash saturation, where an increase in incident wave height does not increase the magnitude of the swash oscillations. This can be parameterised for individual events, or through a spectral representation.<br />
<br />
For non-breaking monochromatic waves, this limit is given by Miché <ref>Miché, R., 1951. Le pouvoir réfléchissant des ouvrages maritimes exposés à l'action de la houle. Ann. Ponts et Chaussees, 121: 285-319. </ref>:<br />
<br />
<math>\varepsilon_s=\frac{a_s \omega^2}{g \beta^2} = 1 , \qquad (4)</math><br />
<br />
where <math>a_s</math> is the vertical amplitude of the shoreline motion, <math>\omega</math> is the angular wave frequency (<math>=2 \pi f</math> - where <math>f</math> is the wave frequency), <math>g</math> the gravitational acceleration and <math>\beta</math> the beach slope. This assumes a saturated surf zone and, based on the limiting amplitude for monochromatic unbroken standing waves, <math>\varepsilon \approx 1</math>. For swash initiated by breaking wave bores, Baldock and Holmes <ref name=B>Baldock T. E. and Holmes P., 1999. Simulation and prediction of swash oscillations on a steep beach. Coastal Engineering, 36: 219-242. </ref> derived a theoretical value <math>\varepsilon \approx 2.5</math>, where saturation is controlled by swash-swash interaction as opposed to surf zone saturation. Spectra of the shoreline oscillations also indicate saturation at higher frequencies, with a typical roll-off in the energy density that is proportional to <math>f^{-4}</math>, which is also evident from equation 4. Huntley et al. <ref>Huntley, D. A., R. T. Guza and A. J. Bowen, 1977. Universal form for shoreline runup spectra. Journal of Geophysical Research-Oceans and Atmospheres 82: 2577-2581. </ref> proposed a uniform spectral form for saturated swash spectra, but variations occur due to different surf zone conditions (figure 4).<br />
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[[Image: BaldockFig5.jpg|thumb|300px|right|Figure 5. Washover fan deposited on Ocracoke Island, North Carolina, during Hurricane Isabel, September 2003. Source: Adapted from Donnelly, Kraus, and Larson (2006) <ref>Donnelly, C., N. Kraus and M. Larson, 2006. State of knowledge on measurement and modeling of coastal overwash. Journal of Coastal Research 22: 965-991. </ref>. Reproduced with permission of the Coastal Education and Research Foundation, Inc.]]<br />
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When the runup exceeds the elevation of the crest of a structure, beach berm or dunes, wave overtopping or wave overwash occurs. This process is very important in building beach berms higher, in association with the spring-neap tidal cycle, but also leads to coastal flooding and inundation of the backshore region. The geomorphology of barrier islands and gravel barriers is strongly dependent on swash overtopping, and breaching of these systems by landward transport of sediment during the overtopping can lead to rapid and potentially catastrophic failure of protective coastal barriers (figure 5). The response of coastlines to sea level rise is also be influenced by swash overtopping and sediment overwash, which increases recession in comparison to the classical Bruun Rule <ref>Rosati, J. D., R. G. Dean and T. L. Walton, 2013. The modified Bruun Rule extended for landward transport. Marine Geology 340: 71-81. </ref>. A combination of parametric modelling and numerical techniques is required to model these scenario <ref>Roelvink, D., Reniers, A., Van Dongeren, A.P., de Vries, J.V.T., McCall, R. and Lescinski, J., 2009. Modelling storm impacts on beaches, dunes and barrier islands. Coastal engineering, 56: 1133-1152. </ref><br />
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==Swash zone hydrodynamics==<br />
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[[Image: BaldockFig6.jpg|thumb|500px|left|Figure 6. Forward (solid) and backward (dashed) characteristic curves (a), and contours of flow velocity (b), surface elevation (c) and depth (d) for swash initiated by a near uniform bore in the non-dimensional coordinates of Peregrine and Williams <ref name=P></ref>. Dotted lines in panel (a) show locus of u=c (critical flow) for uprush and backwash. From <ref name=GB></ref>, with permission.]]<br />
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Swash zone flows have several features that differ from those in the surf zone, but this is dependent on the dominant wave conditions in the inner surf zone as noted above. Many key characteristics again depend on the Iribarren number <ref> Roos, A. and Battjes, J.A., 1977. Characteristics of flow in runup of periodic waves. Proc. 15th International Conference on Coastal Engineering, ASCE, pp. 781-795</ref>. For infragravity standing long waves, the variation in flow depth and flow velocity at a point is relatively symmetrical during uprush and backwash. Short wave bores generate more asymmetrical flows, with the maximum velocity occurring at the start of inundation, with a rapid rise to the maximum depth, with an almost linear deceleration to flow reversal, and a correspondingly similar uniform acceleration in the backwash, at least until the flow becomes very shallow, when friction retards the flow significantly. A key aspect of these flows is diverging flow, which means the swash lens thins rapidly.<br />
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Runup and backwash velocities in the field reach 2-5 m/s, which are generally larger than those in the surf zone. The runup durations are typically shorter than the backwash duration, and the backwash depths are shallower than during the uprush, and therefore the velocity moments tend to be skewed offshore, which has important implications for the sediment dynamics <ref>Raubenheimer, B., R. T. Guza, S. Elgar and N. Kobayashi, 1995. Swash on a gently sloping beach. Journal of Geophysical Research-Oceans 100(C5): 8751-8760. </ref>. The asymmetry is however affected by the mass and momentum advected into the swash zone, which depends on the flow in the inner surf zone. Self-similar solutions for different boundary conditions are presented by Guard and Baldock <ref name=GB>Guard, P. A. and T. E. Baldock, 2007. The influence of seaward boundary conditions on swash zone hydrodynamics. Coastal Engineering 54: 321-331. </ref>, following the work of Peregrine and Williams <ref name=P>Peregrine, D. H. and S. M. Williams, 2001. Swash overtopping a truncated plane beach. Journal of Fluid Mechanics 440: 391-399. </ref>, figure 6. These indicate the fundamental nature of the hydrodynamics, which comprise of a near parabolic motion of the shoreline (due to gravity being the dominant process) and a saw-tooth shaped variation in velocity with time, which decreases at a near linear rate from the peak velocity, which occurs as the shoreline passes a given location. The water surface slope dips seaward for nearly the whole swash cycle, i.e. the total fluid acceleration is offshore throughout the swash cycle.<br />
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This results in a key difference between the surf zone and swash zone bed boundary conditions, namely that there is generally little phase lag between velocity and the bed shear stress in the swash zone, i.e. maxima in bed shear stress occur close to the instants of maxima in velocity. Close to the time of flow reversal, the flow near the bed does however reverse prior to the flow higher in the water column, due to the adverse pressure gradient during the uprush <ref>Kikkert, G. A., T. O'Donoghue, D. Pokrajac and N. Dodd, 2012. Experimental study of bore-driven swash hydrodynamics on impermeable rough slopes. Coastal Engineering 60: 149-166. </ref>. The boundary layer is thinnest at the seaward edge of the swash zone during uprush, and grows following the flow up the beach. The boundary layer largely vanishes at flow reversal, and again grows from the bed as the flow recedes. Accounting for such processes in sediment transport models remains to be tackled.<br />
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[[Image: BaldockFig7NEW.jpg|thumb|400px|right|Figure 7. Contours of surface elevation with two different resistance coefficients, R=0 (blue lines) and R=0.01 (red dashed lines) in the non-dimensional coordinates of Peregrine and Williams <ref name=P></ref>. From Deng et al. <ref>Deng, X., H. Liu, Z. Jiang and T. E. Baldock, 2016. Swash flow properties with bottom resistance based on the method of characteristics. Coastal Engineering 114: 25-34</ref>, with permission.]]<br />
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There are several sources of turbulence in the swash zone. In the runup, turbulence is advected from the inner surf zone, which combines with further generation of turbulence at the bed. The boundary layer is evolving, generally increasing in thickness and may become depth limited. During the backwash turbulence generation occurs mainly at the bed, with swash-swash interactions generating further turbulence as the next wave arrives. Overall, the high turbulence near the bed leads to high bed shear stresses and the potential for high concentrations of suspended sediment transport <ref>Puleo, J. A., R. A. Beach, R. A. Holman and J. S. Allen, 2000. Swash zone sediment suspension and transport and the importance of bore-generated turbulence. Journal of Geophysical Research-Oceans 105(C7): 17021-17044. </ref>.<br />
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Friction plays a large role in controlling the shoreline motion, with friction factors based on conventional fluid mechanics principles typically in the range <math>0.02<f<0.1</math>. Friction effects are strongest when the flow is shallowest, at the swash tip, reducing runup excursions, and late in the backwash, so the shoreline recedes more slowly. Simple models, comprising of a ballistic motion plus friction, describe the shoreline motion reasonably well, although details are missing <ref>Hughes M. G., 1995. Friction factors for wave uprush. J. Coastal Res., 13: 1089-1098</ref><ref name=PH>Puleo, J. A. and K. T. Holland, 2001. Estimating swash zone friction coefficients on a sandy beach. Coastal Engineering 43: 25-40</ref>. Inclusion of friction effects in the internal flow requires numerical modelling at present, or the use of integral models which can avoid the uncertainty in the treatment of friction at the shoreline <ref>Archetti, R. and M. Brocchini, 2002. An integral swash zone model with friction: an experimental and numerical investigation. Coastal Engineering 45: 89-110. </ref>. However, recent results suggest that the effects of friction on the internal flow are small compared to the effects at the swash tip. For example, figure 7 shows a method of characteristics solution for the swash flow with and without friction. The numerical results indicate that the two solutions are similar when water is present, which is interesting given the significant effect of friction on the location of the shoreline. The reason is the supercritical nature of the flow, which is a particular feature of the swash zone. This means that the large change in the shoreline position due to friction does not significantly affect the flow seaward of the shoreline until the flow becomes subcritical, which does not occur until late in the uprush. Similarly, the supercritical nature of the backwash flow means that the changes to the shoreline position further landward cannot significantly affect flow further seaward. Thus, the supercritical nature of the swash flow means that the significant changes in shoreline position do not significantly affect the flow in the interior of the swash lens.<br />
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==Sediment transport mechanics==<br />
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[[Image: BaldockFig8.jpg|thumb|400px|left|Figure 8. A turbulent bore containing entrained suspended sediment just prior to reaching the swash zone. The sediment is then advected into the swash zone during the runup. The end of a supercritical backwash flow is visible in the right of the image. The posts are 1m apart and the orange stringlines are horizontal. Source: Adapted from Hughes, Aagaard, and Baldock (2007) <ref name=H></ref>. Reproduced with permission of the Coastal Education and Research Foundation, Inc.]]<br />
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Cross-shore sediment transport in the swash zone generally occurs as a combination of bed load under sheet flow conditions, with a flat bed, plus an additional component of suspended load, generated locally and advected into the swash zone by surf zone bores. For the bed load, the Meyer-Peter and Muller <ref> Meyer-Peter, E. and Müller, R., 1948. Formulas for bed-load transport. Proc. IAHR, Stockholm </ref> formulation, or derivatives, generally perform well with calibration, i.e. determining the transport coefficient and friction factor remain problematic. In these models, the transport is typically a function of the velocity cubed. The relative balance between bed load, which is generated locally, and suspended load depends on the sediment grain size, and also on the quantity of sediment advected into the swash zone from the inner surf zone (figure 8). This can be considerable, and affects the distribution of suspended load across the swash zone. While the basic sediment transport equations still apply, model-data comparisons are lacking, particularly close to the bed where suspended sediment concentrations are largest.<br />
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[[Image: BaldockFig9.jpg|thumb|350px|right|Figure 9. Measured normalized suspended sediment concentration <math>c </math> indicated by colours mapped onto the normalised <math>x_* - t_*</math> plane representing the swash excursion and duration. Colour bar indicates normalised concentration from high (hot) to low (cold). Source: Adapted from Hughes, Aagaard, and Baldock (2007) <ref name=H></ref>. Reproduced with permission of the Coastal Education and Research Foundation, Inc.]]<br />
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In simple <math>u^3</math> type sediment transport models the asymmetry of the flow tends to export sediment from the swash zone. In reality, this is balanced by the suspended sediment imported and advected into the swash zone from the inner surf <ref name=H>Hughes, M. G., T. Aagaard and T. E. Baldock, 2007. Suspended sediment in the swash zone: Heuristic analysis of spatial and temporal variations in concentration. Journal of Coastal Research 23: 1345-1354. </ref>, the distribution of which is illustrated in figure 9. Measurement of suspended sediment in the backwash remains a challenge, and is difficult to separate from bed-load. The quantity of suspended load entering the swash zone affects the net deposition pattern, and hence zones of erosion or accretion, as illustrated by Pritchard and Hogg <ref>Pritchard, D. and A. J. Hogg, 2005. On the transport of suspended sediment by a swash event on a plane beach. Coastal Engineering 52: 1-23</ref>. Data is still lacking in the field to reliably quantify this process, which is complicated by sediment suspended by swash-swash interactions and the influence of turbulence in the inner surf zone, which is significant <ref>Butt, T., P. Russell, J. Puleo, J. Miles and G. Masselink, 2004. The influence of bore turbulence on sediment transport in the swash and inner surf zones. Continental Shelf Research 24(7-8): 757-771. </ref>. The infiltration and exfiltration of water into the beach during swash flows is an important contributor to groundwater processes, particularly on coarse grain beaches, and also influences sediment transport <ref>Turner, I.L. and G. Masselink, 1998. Swash infiltration-exfiltration and sediment transport. Journal of Geophysical Research, 103(C13): 30,813-30,824</ref>.<br />
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The mechanics of longshore sediment transport in the swash zone are well-understood in principle, but data and modelling is limited, despite the clear presence of a significant contribution to the total longshore transport in the littoral zone. In the swash zone the flow direction during both runup and rundown has a longshore component under oblique waves (the usual case). The boundary between the surf and swash zone is also an active region of longshore transport, particularly on steep cobble or coral beaches, where sediment may also move offshore into the inner surf zone, then alongshore, and then back onshore as the wave conditions or tide change <ref>Kench, P. S., E. Beetham, C. Bosserelle, J. Kruger, S. M. L. Pohler, G. Coco and E. J. Ryan, 2017. Nearshore hydrodynamics, beach face cobble transport and morphodynamics on a Pacific atoll motu. Marine Geology 389: 17-31. </ref>. While the relative importance of longshore transport in the swash zone compared to the surf zone is greater during milder wave conditions than during storms, longshore sediment transport in the swash zone may account for up to 50% of the total longshore transport <ref>Kamphuis J. W., 1991. Alongshore sediment transport rate distribution. Coastal Sediments '91 Conference, ASCE, 170-183. </ref>. Longshore transport in the swash zone is also relatively more important on steep beaches and where small oblique waves break frequently at the shoreline, e.g., in estuary mouths and landward of lagoons behind fringing and barrier reefs.<br />
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==Morphodynamics==<br />
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The evolution of the beach face in the swash zone is controlled by the sediment fluxes across the boundaries, both longshore and cross-shore. Thus, the morphology is influenced by the presence of dunes, cliffs and hard structures at the landward extent of the beach, by conditions in the inner surf one, notably bars, rip cells, beach steps, and by lateral boundaries such as groynes, breakwaters and estuary mouths. While the general processes are again well understood, and similar to those applied for littoral transport in the surf zone, the details of dune erosion, barrier degradation and progradation and the short and long term balance of sediment deposition and erosion remain a challenge <ref> Masselink, G. and J. A. Puleo (2006). Swash-zone morphodynamics. Continental Shelf Research 26(5): 661-680</ref>. For example, rates of deposition and erosion vary significantly on wave-by-wave time scales, wave group time-scales, and tidal time-scales, plus seasonal and annual changes due to variations in wave and wind climate. Further, while the active swash zone is usually relatively plane (excluding dunes), feedback between morphology and hydrodynamics does lead to more complex morphology such as [[Beach Cusps]]. Varying tides also lead to the formation of beach ridges and berms at different elevations, which complicate the overall topography. Dunes act as significant sources and sinks of sediment that control swash zone morphodynamics and that of the whole beach, and are essential in developing the sediment budget for the whole beach. A further aspect is the influence of vegetation, which can provide important stabilising mechanisms for dunes. The upper beach and swash zone is where the impacts of sea level rise will be most visible, with loss of the upper beach if the coastline cannot recede landward.<br />
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==Measurement techniques==<br />
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[[Image: BaldockFig10.jpg|thumb|500px|left|Figure 10. A LIDAR system mounted above the swash zone, together with ultrasonic distance point sensors. Photo: Dr Chris Blenkinsopp, University of Bath, UK.]]<br />
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Research on swash zone processes is both assisted and hindered by the nature of the swash. The beach face is generally accessible for deployment of instruments, particularly with the aid of large tides, it is close to the shore and associated infrastructure, and much wave energy has been dissipated in the surf zone. However, flows are very shallow and also intermittent, making measurements over the depth difficult and time-averaging complex. In terms of in-situ instruments, the shoreline motion can be measured by runup wires or point instruments such as pressure sensors or ultrasonic sensors, flow velocity with electromagnetic or acoustic instruments, and morphology changes with standard survey techniques or ultrasonic sensors. To tackle larger scales and to avoid in-situ measurements and to sample over longer time-scales, remote sensing by video has adopted techniques first developed for the surf zone by Aagaard and Holm <ref>Aagaard, T. and J. Holm, 1989. Digitization of wave runup using video records. Journal of Coastal Research 5: 547-551. </ref> to determine friction factors <ref name=PH></ref>, runup <ref name=S></ref> and internal kinematics <ref>Power, H. E., R. A. Holman and T. E. Baldock, 2011. Swash zone boundary conditions derived from optical remote sensing of swash zone flow patterns. Journal of Geophysical Research, Oceans 116, C06007, doi:10.1029/2010JC006724</ref>. Shore-mounted LIDAR is becoming a promising tool for high resolution and long term monitoring <ref>Blenkinsopp, C. E., M. A. Mole, I. L. Turner and W. L. Peirson, 2010. Measurements of the time-varying free-surface profile across the swash zone obtained using an industrial LIDAR. Coastal Engineering 57(11-12): 1059-1065. </ref>, figure 10. However, both these techniques suffer from loss of resolution in the thinning backwash, and do not provide flow velocity with any degree of reliability. Sediment transport measurements using sediment traps <ref>Horn, D. P. and T. Mason, 1994. Swash zone sediment transport modes. Marine Geology 120(3-4): 309-325. </ref><ref> Masselink, G. and M. Hughes (1998). Field investigation of sediment transport in the swash zone. Continental Shelf Research 18(10): 1179-1199</ref> still provide the most reliable measure of total load transport rates in the field, supplemented by sediment transport rates derived from morphological measurements through sediment continuity, which can be particularly useful during overwash events. Both techniques require predominantly cross-shore transport.<br />
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==Related articles==<br />
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[[Beach Cusps]]<br />
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[[Infragravity waves]]<br />
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[[Gravel Beaches]]<br />
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[[Shallow-water wave theory]]<br />
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==References==<br />
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{{author<br />
|AuthorID=34603<br />
|AuthorFullName=Tom Baldock<br />
|AuthorName= Tom Baldock}}<br />
[[Category:Coastal processes, interactions and resources]] <br />
[[Category:Geomorphological processes and natural coastal features]] <br />
[[Category:Hydrodynamics]] <br />
[[Category:Land and ocean interactions]]<br />
[[Category:Hydrological processes and water]]</div>Dronkers Jhttp://www.vliz.be/wiki/Rhythmic_shoreline_featuresRhythmic shoreline features2017-05-22T14:57:42Z<p>Dronkers J: </p>
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<div>==Introduction==<br />
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[[Image: RSFFFig1.jpg|thumb|500px|left|Figure 1. Left: Rhythmic shoreline features forced by a breakwater system. Right: Picture taken at the Arcachon bay (courtesy of Isabel Casanovas) showing ripple marks and shoreline undulations (<math>L \sim </math> 1-2 m) at low tide. On the right there is a wooden board. Existing studies demonstrate that the ripple marks are self-organized, emerging from an instability of the flat bed under the coupling of water and sand. The origin of the shoreline undulations is unknown but they are probably self-organized too, because there are no nearby features at these particular scales that could force them.]]<br />
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Sandy shorelines are rarely strait but they commonly exhibit undulations or cuspate shapes that are most often irregular. Sometimes, however, the shoreline position is nearly or roughly periodic in space along the shore with a wavelength <math>L</math> or, at least, there exists the suggestion of a recurrent alongshore spacing <math>L</math>. We then refer to these undulations or cuspate features as ''rhythmic shoreline features''.<br />
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The spacing <math>L</math> is sometimes dictated by external constraints like human-made structures (e.g., groins or breakwaters, see Figure 1a) or the inherited geology (e.g., sea bed large-scale morphology like drowned canyons). In other cases, a pre-existing hydrodynamic template can imprint its spacing <math>L</math> on the developing morphology. This is the case of bars created by standing waves. We refer to these two types as ''forced features''.<br />
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Intriguingly, rhythmic shoreline features can also exist without any forcing by external constraints or by a pre-existing template in the hydrodynamic forcing at the length scale <math>L</math> (see Figure 1b). We refer to them as ''self-organized features'' <ref> Coco, G. and A.B. Murray (2007), Patterns in the sand: From forcing templates to self-organization, Geomorphology, 91, 271-290</ref>. We are here concerned with this class and we will hereinafter refer to them unless stated otherwise. They can be very striking, there are many types and their length scale <math>L</math> can span several orders of magnitude, roughly from 1 m to 100 km (see Figure 2). It is hard to think that they simply come out of random processes and their simplicity or their ordered complexity strongly suggest that they are the result of collective processes at the length scale <math>L</math> involving waves, currents, tides and sand. It has been found that they emerge out of positive feedbacks between the hydrodynamics and the morphology, and their spacing <math>L</math> is<br />
internally generated as the length scale that makes the feedback most efficient.<br />
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[[Image: RSFFFig2NEW.jpg|thumb|800px|center|Figure 2. Rhythmic shoreline features at various lengthscales. a) Beach cusps, <math>L\approx 2 </math>m (Matingarahi, NZ. Source: courtesy of Prof. A.D. Short); b) up: megacups associated to transverse bars, <math>L \approx </math>20 m (Ebro delta, Catalonia (Spain), image taken by the authors); b) down: large beach cusps <math>L \approx </math> 60 m (Angola. Source: Google Earth, image from Digital-Globe); c) up: megacusps associated to a crescentic bar, <math>L \approx </math> 300 m (Saint Cyprien, France. Source: Google Earth, image from Data SIO, NOAA, U.S. Navy, NGA, GEBCO); c) down: megacusps associated to transverse bars (Lighthouse Beach, NSW, Australia. Source: courtesy of Prof. A.D. Short), <math>L \approx </math>500 m; d) up: rhythmic spits, <math>L \approx</math> 7 km (Gulf of Amur. Source: Google Earth, image from TerraMetrics); d) down: shoreline sand waves, <math>L \approx</math> 5 km (Namibia. Source: Google Earth, image from Digital Globe); e) rhythmic spits <math>L \approx</math> 50-80 km (Azov sea. Source: Google Earth, image from Data SIO, NOAA, U.S. Navy, NGA, GEBCO).]]<br />
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Rhythmic shoreline features are interesting by themselves in the context of coastal geomorphology. But they are also a visible footprint of important physical mechanisms associated to the coupling between hydrodynamics and morphology so that their study offers a way of getting insight into this coupling. From an engineering point of view, the dynamics of shoreline undulations leads to the existence of erosional hotspots or zones of increased coastal vulnerability.<br />
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Rhythmic shoreline features can be classified either according to their alongshore lengthscale <math>L</math> or according to the processes involved in their formation. According to the latter, we can distinguish three types: a) ''beach cusps'', which are associated to swash zone processes, b) ''megacusps'', which are associated to surf-zone processes and c) ''large scale shoreline features'', which are associated to processes at a length scale which is larger than the surf zone width, <math>X_b</math>, so that <math>L \gg X_b</math>. The length scale typically increases in the order a, b, c. However, depending mainly on the wave energy level, their lengthscale can overlap (see Figure 2). For instance, megacups or large scale shoreline features on low energy beaches can be smaller than cusps or megacusps on open ocean beaches, respectively. We will here deal with types b) and c), since beach cusps are specifically treated in another article ([[Beach Cusps]] by G.Coco).<br />
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==Megacusps and rhythmic surf zone bars==<br />
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[[Image: RSFFFig3NEW.jpg|thumb|500px|left|Figure 3. Images from the Castelldefels beach video station (Catalonia, Spain) showing surf zone bar morphology at different times. a) shore parallel straight bar, b) incipient crescentic bar with relatively subtle undulations, c) large amplitude crescentic bar with crescents developped into transverse bars (TBR) attaching the shoreline and determining megacusps, d) complex morphology encompassing a crescentic bar, rip channels, megacups and small transverse medium-energy finger bars.]]<br />
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The surf zone of sandy beaches commonly features sand deposits or bars separated by bed depressions or troughs.<br />
The associated bathymetry is often complex and can be rhythmic in the alongshore direction. The rhythmic bar systems can influence the shoreline and imprint their wavelength <math>L</math> on it. As shown in Figure 2c, the ''megacups'' are the resulting undulations or cuspate features on the shoreline. Bar systems can be very complex and sometimes it is difficult to assign any known type to the observed morphology (see Figure 3). But rhythmic bars are in principle classified into two types: crescentic bars and transverse bar systems.<br />
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===Crescentic bars===<br />
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A crescentic bar consists of an alongshore sequence of shallower and deeper sections alternating<br />
shoreward and seaward (respectively) of a line parallel to the shore in such a way that the bar shape is undulating in plan view (<ref name=ER>van Enckevort, I. M. G., B. G. Ruessink, G. Coco, K. Suzuki, I. L. Turner, N. G. Plant, and R. A. Holman (2004), Observations of nearshore crescentic sandbars, J. Geophys. Res., 109 (C06028), doi:10.1029/2003JC002214</ref><ref name=CB>Castelle, B., P. Bonneton, H. Dupuis, and N. Senechal (2007), Double bar beach dynamics on the high-energy meso-macrotidal french aquitanian coast: A review, Mar. Geol., 245, 141-159</ref> <ref name=R4>Ribas, F., A. Falques, H. E. de Swart, N. Dodd, R. Garnier, and D. Calvete (2015), Understanding coastal morphodynamic patterns from depth-averaged sediment concentration, Rev. Geophys., 53, doi:10.1002/2014RG000457</ref> and references therein; also see Figures 2c and 3b, c, d ). In some cases the undulation is quite subtle, the bar being almost straight (Figure 3b), but occasionally, it features pronounced crescent moons with the horns pointing shoreward and the bays (deeps) located seaward (Figures 2c, 3c).<br />
The deeper sections are called rip channels because strong seaward directed currents called rip currents <ref name=M1>MacMahan, J. H., E. B. Thornton, and A. J. H. M. Reniers (2006), Rip current review, Coastal Eng., 53, 191-208</ref> <ref name=DM>Dalrymple, R. A., J. H. MacMahan, A. J. H. M. Reniers, and V. Nelko (2011), Rip currents, Annu. Rev. Fluid Mech., 43, 551-581</ref> are concentrated there. This is why they are sometimes called ''rip channel systems''. Note, however, that rip channels, i.e., cross-shore-oriented channels in the surf zone where rip currents concentrate, can also be observed associated to transverse bar systems without the presence of crescentic bars (see next section).<br />
Crescentic bars have been reported on microtidal to mesotidal sandy beaches at different scales with a mean alongshore spacing, <math>L</math>, ranging from tens of meters up to 2-3 km.<br />
They can influence the shoreline to form megacusps in two ways: directly, the horns of the crescentic bars connecting to the shoreline and thus forming the megacusps at the attachment points (Figure 3c) or indirectly, the bar and the shoreline being separated by a trough (Figure 3d). In the latter case the undulations in the crescentic bar and in the shoreline are still linked and roughly share the same alongshore spacing <math>L</math>. This can happen because of the alongshore rhythmic topographic control exerted by the bars on the waves and by the induced circulation cells. Usually, the undulations are either in phase (i.e., seaward/shoreward displacements in phase) or out of phase (i.e., shoreward displacements of the bar in front of the seaward displacements of the shoreline and vice versa).<br />
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===Transverse bar systems===<br />
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Transverse bars extend perpendicularly to the coast or with an oblique orientation. No matter their angle with the shore normal being small or large. Shepard <ref name=S>Shepard, F. P. (1952), Revised nomenclature for depositional coastal features, Bull. Am. Assoc. Petrol. Geol., 36 (10), 1902-1912</ref> introduced the term transverse bars to distinguish them from the shore-parallel bars. They usually occur in patches of a few of them up to tens, they are separated by troughs and they display a rhythmic pattern along the shore. Significant rip currents sometimes concentrate at the troughs that can then be considered rip channels <ref name=T>Thornton, E. B., J. MacMahan, and A. H. Sallenger (2007), Rip currents, mega-cusps, and eroding dunes, Marine Geology, 240, 151 167</ref>. The alongshore spacing, <math>L</math>, is defined as the distance between successive bar crests. They are typically attached to the shore and the shoreline attachments determine the megacusps. In the presence of an alongshore current, they migrate down-drift with migration rates up to <math>40</math> m/d <ref name=R1>Ribas, F., and A. Kroon (2007), Characteristics and dynamics of surfzone transverse finger bars, J. Geophys. Res., 112 (F03028), doi:10.1029/2006JF000685</ref><ref name=P>Pellon, E., R. Garnier, and R. Medina (2014), Intertidal finger bars at El Puntal, Bay of Santander, Spain: observation and forcing analysis, Earth Surface Dynamics, 2, 349-361</ref>. Oblique bars in this case can have its distal end shifted down-current or up-current with respect to the shore attachment and are then called down-current oriented or up-current oriented, respectively. They sometimes show an asymmetry of their cross-section (the down-current flank being steeper than the up-current flank <ref name=P></ref>. Many types of transverse bars (in their characteristics and origin) have been reported in the literature and we here state a tentative classification based on that provided by <ref name=P></ref>. However, we group two of the types in that paper in only one class because we think they do not essentially differ.<br />
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'''TBR Bars'''<br />
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They are associated to the Transverse Bar and Rip (TBR) state in the standard beach state classification <ref name=W>Wright, L. D., and A. D. Short (1984), Morphodynamic variability of surf zones and beaches: A synthesis, Mar. Geol., 56, 93-118</ref> (see Figures 2c and 2c). They are typically wide and short-crested and their origin is the merging of a crescentic bar into the beach as has been described previously. Therefore, their spacing <math>L</math> is the spacing of the pre-existing crescentic bar. As in the case of crescentic bars, TBR bars also show strong and narrow rip currents flowing seaward in the troughs and wider and weaker onshore flows over the crests. They can be either shore-normal or down-current oriented if the wave incidence is oblique.<br />
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'''Medium Energy Finger Bars'''<br />
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They have been observed in open microtidal beaches under medium-energy conditions <ref name=KH>Konicki, K. M., and R. A. Holman (2000), The statistics and kinematics of transverse bars on an open coast, Mar. Geol., 169, 69-101</ref><ref name=R1> </ref> and they always coexist with shore-parallel (or crescentic) bars. They are thin and elongated (hence finger bars) in contrast with the wider and shorter TBR bars. They are ephemeral (residence time from 1 day to 1 month), attached to the low-tide shoreline or, occasionally, to the shore-parallel bar. They are linked to the presence of alongshore wave driven current and they are up-current oriented. Their spacing is in the range <math>L \approx</math> 15 -200 m. It seems that some medium energy finger bars show up in Figure 3d.<br />
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'''Long Finger Bars'''<br />
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[[Image: RSFFFig4.jpg|thumb|400px|right|Figure 4. Long transverse finger bars at Horn Island, Mississippi, USA. Geographical coordinates: <math>30^{\circ} 44' 44''</math> N, <math>88^{\circ} 41' 17''</math> W. Wavelength <math>L \approx</math> 100-150 m (Source: Google Earth).]]<br />
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They are persistent features in low to medium energy beaches without shore-parallel bars and whose foreshore is a very flat terrace <ref name=E>Evans, O. F. (1938), The classication and origin of beach cusps, J. Geology, 46, 615-627</ref> <ref name=N>Niederoda, A. W., and W. F. Tanner (1970), Preliminary study on transverse bars, Mar. Geol., 9, 41-62</ref><ref name=F89>Falques, A. (1989), Formacion de topografia ritmica en el Delta del Ebro, Revista de Geofisica, 45 (2), 143-156</ref><ref name=B>Bruner, K. R., and R. A. Smosna (1989), The movement and stabilization of beach sand on transverse bars, Assateague Island, Virginia, J. Coastal Res., 5 (3), 593-601</ref><ref name=G>Gelfenbaum, G., and G. R. Brooks (2003), The morphology and migration of transverse bars off the west-central Florida coast, Mar. Geol., 200, 273-289</ref>. They are characterized by long crests which are typically larger than the alongshore spacing which may vary in the range <math> L \approx</math> 10 -500 m. The incident wave focusing by the bars caused by topographic refraction seems to be an essential process to them. Although they are most often observed on microtidal beaches, they may also exist on meso and macrotidal coasts <ref name=L>Levoy, F., E. J. Anthony, O. Monfort, N. Robin, and P. Bretel (2013), Formation and migration of transverse bars along a tidal sandy coast deduced from multi-temporal lidar datasets, Mar. Geol., 342, 39-52</ref><ref name=P></ref>. We group in this class both the 'large-scale finger bars' and the 'low-energy finger bars' types of <ref name=P></ref>. Figures 2b and 4 show some examples.<br />
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==Large-scale shoreline features==<br />
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[[Image: RSFFFig5.jpg|thumb|400px|left|Figure 5. Shoreline sand waves at the Gulf of Finland, Russia. Geographical coordinates: <math>59^{\circ} 57' 22''</math> N, <math>29^{\circ} 33' 53''</math> E. Wavelength increasing down-drift from <math>L \approx</math> 200 m to 1100 m (Source: Google Earth, image from TerraMetrics and GeoEye).]]<br />
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This type is difficult to define and probably includes a number of distinct classes, which are very different in morphology, size and involved physical processes. We define them here as being associated to processes at a length scale larger than the surf zone width, <math>L \gg X_b</math>.<br />
They include shoreline undulations called shoreline sand waves, cuspate forelands and sandy spits <ref name=AMA>Ashton, A., A. B. Murray, and O. Arnault (2001), Formation of coastline features by large-scale instabilities induced by high-angle waves, Nature, 414, 296-300</ref> (see Figures 2d, e and 5). On low energy shores where <math>X_b</math> is very small, <math>L</math> can be <math>\approx</math> O(100 m), in the same range than megacusps on open ocean beaches. However, large-scale shoreline features can also be very large, up to tens of km, on these ocean shores. An important characteristic is that they are typically linked to similar undulations in the bathymetric contours beyond the surf zone into the shoaling zone, sometimes up to considerable depths <ref name=K3>Kaergaard, K., J. Fredsoe, and S. B. Knudsen (2012), Coastline undulations on the West Coast of Denmark: Offshore extent, relation to breaker bars and transported sediment volume, Coastal Eng., 60, 109-122</ref>. Because of the large length and time scales involved in their dynamics, it is often difficult to ascertain whether they are self-organized or they are forced by offshore bathymetric anomalies or by geological constraints. When there is a dominant littoral drift they tend to migrate down-drift.<br />
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==Feedback mechanisms==<br />
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We here describe the main feedback mechanisms that have been identified as potential drivers for self-organized shoreline features.<br />
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===Surf zone: bedsurf mechanism===<br />
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[[Image: RSFFFig6.JPG|thumb|300px|left|Figure 6. Bed-surf feedback mechanism based on the rip-current circulation in the surf zone. The yellow colour indicates emerged or submerged shallower areas, blue colour indicates submerged deeper areas. The lower/higher density of brown spots indicate the lower/higher concentration of suspended sediment. The blue lines with arrows indicate the depth averaged currents.]]<br />
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Let us assume a shore-parallel bar with alternating shallows and deeps or channels on its crest (see Figure 6). The breaking is more intense over the shallows so that there is more wave-induced sea level setup shoreward of them. This creates currents flowing alongshore from the areas shoreward of the shallows to the areas shoreward of the channels. In this way, a rip current circulation flowing seaward at the channels and shoreward at the shallows is originated. Then, let us consider the sediment transported in suspension by the current. Since the larger waves break more offshore and the wave height is gradually reduced onshore, the maximum suspended concentration (i.e., sediment load induced by wave breaking) will therefore occur offshore. Then, the current flowing shoreward at the shoals will carry a lot of sediment into places where the wave agitation is weak and the equilibrium suspended sediment concentration is low. Therefore, the sediment will settle down and sediment deposition will occur at the shoals. On the contrary, the water flowing seaward along the rip channels from the inner surf zone, where the sediment load is weak, will carry small sediment concentrations to places where the wave agitation is high allowing larger concentrations. Therefore, sediment will be picked-up and the channel will be eroded. As a result, the circulation induced by the morphology will bring sediment from the channels into the shoals, i.e., will reinforce<br />
the morphology so that a positive feedback will occur. This is the so-called ''bed-surf mechanism''. It was first described in these terms by Falques et al. <ref name=F0>Falques, A., G. Coco, and D. A. Huntley (2000), A mechanism for the generation of wave-driven rhythmic patterns in the surf zone, J. Geophys. Res., 105 (C10), 24,071-24,088</ref> but was earlier implicitly included in the modelling studies by Hino <ref name=H>Hino, M. (1974), Theory on formation of rip-current and cuspidal coast, in Coastal Eng. 1974, pp. 901-919, Am. Soc. of Civ. Eng. </ref> and Deigaard et al. <ref name=DD>Deigaard, R., N. Droenen, J. Fredsoe, J. H. Jensen, and M. P. Joergensen (1999), A morphological stability analysis for a long straight barred coast, Coastal Eng., 36 (3), 171-195</ref>. The latter paper showed that the bedsurf mechanism can explain the formation of a crescentic bar out of a shore-parallel bar. Numerous subsequent modelling studies have confirmed it in different conditions and with different models <ref name=RR>Reniers, A. J. H. M., J. A. Roelvink, and E. B. Thornton (2004), Morphodynamic modeling of an embayed beach under wave group forcing, J. Geophys. Res., 109 (C01030), doi:10.1029/2002JC001586</ref> <ref name=R>Ranasinghe, R., G. Symonds, K. Black, and R. Holman (2004), Morphodynamics of intermediate beaches: A video imaging and numerical modelling study, Coastal Eng., 51, 629-655</ref><ref name=CD>Calvete, D., N. Dodd, A. Falques, and S. M. van Leeuwen (2005), Morphological development of rip channel systems: Normal and near normal wave incidence, J. Geophys. Res., 110 (C10006),Doi:10.1029/2004JC002803</ref> <ref name=G8>Garnier, R., D. Calvete, A. Falques, and N. Dodd (2008), Modelling the formation and the long-term behavior of rip channel systems from the deformation of a longshore bar, J. Geophys. Res., 113 (C07053), doi:10.1029/2007JC004632</ref> <ref name=SR>Smit, M., A. Reniers, B. Ruessink, and J. Roelvink (2008), The morphological response of a nearshore double sandbar system to constant wave forcing, Coastal Eng., 55, 761-770</ref> <ref name=O>Orzech, M. D., A. J. H. M. Reniers, E. B. Thornton, and J. H. MacMahan (2011), Mega-cusps on rip channel bathymetry: Observations and modeling, Coastal Eng., 58, 890907</ref> <ref name=CC>Castelle, B., and G. Coco (2012), The morphodynamics of rip channels on embayed beaches, Cont. Shelf Res., 43, 10-23</ref>}. It has been tested in a wave-tank experiment by Michallet et al. <ref name=MC>Michallet, H., B. Castelle, E. Barth elemy, C. Berni, and P. Bonneton (2013), Physical modeling of three-dimensional intermediate beach morphodynamics, J. Geophys. Res., 118, 1-15, doi:10.1002/jgrf.20078</ref>.<br />
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One could argue that in order that the positive feedback occurs, the rhythmic morphology should be already there at the beginning so that the question of where it comes from would remain. This issue is solved by the instability theory which can be sketched as follows. The initial configuration can be assumed as the superposition of a featureless reference morphology (alongshore uniform) plus a perturbation. Then, the perturbation can be expanded as a superposition of normal modes encompassing all alongshore wavelengths. If the amplitude of the perturbation is small enough each mode will evolve separately due to linearity of the governing equations.<br />
Thus the feedback can occur for each mode. Depending on their shape and wavelength, some modes will lead to a negative feedback and others to a positive feedback. The most efficient in causing positive feedback will grow faster in time and will eventually dominate the dynamics imprinting its shape and its wavelength, <math>L</math>, on the emerging morphology. The fundamentals of instability theory to explain self-organized coastal features is presented in a comprehensive manner in <ref name=D>Dronkers, J. (2016), Dynamics of Coastal Systems second edition, World Scientific Publ. Co., Singapore</ref> and in [[Stability models]]. A more detailed formulation can be found in <ref name=DB>Dodd, N., P. Blondeaux, D. Calvete, H. E. de Swart, A. Falques, S. J. M. H. Hulscher, G. Rozynski, and G. Vittori (2003), The use of stability methods in understanding the morphodynamical behavior of coastal systems, J. Coastal Res., 19 (4), 849-865</ref> and an application example can be seen in <ref name=F0></ref>.<br />
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The bedsurf mechanism can also work in a similar way in absence of shore-parallel bars and it can induce the formation of transverse bars <ref name=G6>Garnier, R., D. Calvete, A. Falques, and M. Caballeria (2006), Generation and nonlinear evolution of shore-oblique/transverse sand bars, J. Fluid Mech., 567, 327-360</ref>. In this case the circulation consists of shoreward flow over the bars and seaward flow at the rip channels which are the troughs in between the bars. This type of coupled morphology and circulation in absence of any shore-parallel bar has been observed, for example, in Monterey Bay, CA (US) <ref name=T></ref> <ref name=M2>MacMahan, J. H., E. B. Thornton, A. J. H. M. Reniers, T. P. Stanton, and G. Symonds (2008), Low-energy rip currents associated with small bathymetric variations, Marine Geology, 255, 156-164</ref>.<br />
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===Surf zone: bedflow mechanism===<br />
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[[Image: RSFFFig7.JPG|thumb|500px|left|Figure 7. Bed-flow feedback mechanism associated to the wave driven longshore current in the surf zone. It can be based either on the shoreward deflection of the current by down-current oriented bars (a) or on the seaward deflection of the current by up-current oriented bars (b). Colours, brown spots, blue lines and arrows have the same meaning as in Figure 6.]]<br />
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It is well known that a current flowing over a sandy bed can de-stabilize the flat bed and originate a variety of bedforms as ripples, dunes, antidunes, sand waves, bars, sand ridges, etc. Falques et al. <ref name=F96>Falques, A., A. Montoto, and V. Iranzo (1996), Bedflow instability of the longshore current, Cont. Shelf Res., 16 (15), 1927-1964</ref> examined how the wave-driven longshore current in case of oblique wave incidence could generate rhythmic bars in the surf zone<br />
following the analogy with alternate bars in rivers. Following this analogy, the gradients in wave breaking caused by the rhythmic bars (the essential driver of bed-surf mechanism) were ignored and the counterpart of river alternate bars in the nearshore was found. The corresponding feedback mechanism was called ''bedflow mechanism''. In contrast with the bedsurf mechanism, where the differential breaking is essential but there is no longshore current, the current is essential for the bedflow mechanism but differential breaking is ignored. While bedsurf mechanism alone can be realistic in the surf zone (in case of normal wave incidence), bedflow mechanism is not because the current is generated by oblique wave incidence in which case the differential breaking is also present. Still, the bedflow mechanism has been conceptually important since it points to the feedbacks essentially associated to the longshore current in the surf zone.<br />
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In order to analyze the possible feedbacks driven by the longshore current in a more realistic way,<br />
the models describing the bedsurf mechanism have been extended to account for the differential breaking, therefore mixing bedflow and bedsurf mechanisms <ref name=R2>Ribas, F., A. Falques, and A. Montoto (2003), Nearshore oblique sand bars, J. Geophys. Res., 108 (C43119), doi:10.1029/2001JC000985</ref> <ref name=G6> </ref> <ref name=R3>Ribas, F., H. E. de Swart, D. Calvete, and A. Falques (2012), Modeling and analyzing observed transverse sand bars in the surf zone, J. Geophys. Res., 117 (F02013), doi:10.1029/2011JF002158</ref>}. It has been found that there are indeed two possible feedbacks between oblique bars and the wave-driven longshore current (see Figure 7).<br />
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In case of down-current oriented bars it is found that the current is deflected onshore over the bars and offshore at the troughs as a result of mass conservation, current inertia, friction and differential breaking (so with much more complexity than in case of bedsurf mechanism alone). Then, if we assume that the sediment concentration is maximum at the breaking line and decreases onshore, by the same reason explained in the previous section, there will be sediment deposition over the bars and erosion at the channels, hence a positive feedback (Figure 7a).<br />
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For up-current oriented bars the current is deflected in the opposite way, offshore over the bars, onshore at the channels. Therefore, if there is a maximum in concentration at the breaking line, the feedback is negative. Only if there is a peak of suspended concentration near the shoreline and a decreasing concentration in the inner surf zone, the feedback can be positive in the inner surf zone<br />
(Figure 7b). This is possible thanks to the turbulent bores propagating onshore from the breaking line and has been proved as a possible origin of the medium energy finger bars <ref name=R3></ref>.<br />
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===Surf-shoaling zones: wave energy mechanism===<br />
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[[Image: RSFFFig8.JPG|thumb|500px|left|Figure 8. Feedback mechanisms between shoreline undulations and waves involving both the surf and the shoaling zones. Wave incidence from the left. The blue lines with arrows indicate the wave rays. The brown arrows indicate the larger or smaller total sediment transport rate, <math>Q</math>. A crest (C) is shown along with the two nearest embayments (E). The areas where sediment excess (deposition) or deficit (erosion) occur are indicated. The wave-energy mechanism which is based on the differences in wave crest stretching between up-drift and down-drift of C is shown. The larger/smaller wave height, <math>H_b</math>, causing the gradients in <math>Q</math>, is indicated.]]<br />
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The length scales at which bedsurf and bedflow mechanisms operate and rhythmic surf zone bars and megacusps emerge are comparable to the surf zone width, <math>X_b</math>, so typically tens to hundreds of m. The time scales involved are of hours to days. Therefore, if we are interested in the behavior of a sandy coast at time scales of years and length scales <math>\gg X_b</math>, i.e., km's, it is reasonable to rely on averages where the internal dynamics of the surf zone is filtered out. In case of oblique wave incidence this can be done by considering the total cross-shore integrated alongshore sediment transport rate, <math>Q</math> (m<math>^3</math>/s) and looking at its alongshore gradients. Where there is convergence of <math>Q</math> there is overall sand accumulation and the shoreline progrades and where there is divergence of <math>Q</math> there is overall sand deficit and the shoreline retreats. The sediment transport rate <math>Q</math> can be evaluated with semi-empirical formulae in terms of the wave characteristics at breaking. Let us assume oblique wave incidence on an undulating coastline with an angle <math>\theta_b</math> at breaking with respect to the mean shoreline trend (absolute incidence angle). If <math>\phi</math> is the local orientation of the shoreline with respect to its mean trend, the wave incidence angle with respect to the local shore normal is <math>\alpha_b = \theta_b - \phi</math> (relative incidence angle). Then, if <math>H_b</math> is the wave height at breaking, <math>Q</math> can typically be cast into<br />
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<math> Q = f(H_b) \Psi(\alpha_b) , \qquad (1)</math><br />
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where both <math>f,\Psi</math> are increasing functions (in case of <math>\Psi</math> it is increasing only for <math>\theta_b < 45^{\circ}</math>, but due to wave refraction <math>\theta_b</math> is rarely <math>> 45^{\circ}</math>). A well-known example is the CERC formula <ref name=KM>Komar, P. D., and R. A. Holman (1986), Coastal processes and the development of shoreline erosion, Annual Review Earth and Planetary Sciences, 14, 237-265</ref>.<br />
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Having all this in mind, we first examine how large scale (<math>L \gg X_b</math>) shoreline undulations cause gradients in <math>H_b</math> that cause, in turn, gradients in <math>Q</math>. Because of topographic wave refraction the wave crests stretch so that the wave energy undergoes dispersion and wave height tends to decrease (Figure 8). This effect is more pronounced as more intense is the wave ray bending. A shoreline undulation is commonly linked to similar undulations in the bathymetric contours up to certain depth, <math>D_c</math>. This affects wave refraction so that the ray bending will be more pronounced at the down-drift sides of the headlands of the undulation than at the up-drift sides. This will induce less wave height at the down-drift sides than at the up-drift with the result that <math>Q</math> will decrease moving from up-drift to down-drift making the headlands prograde (Figure 8). Similarly, divergence of <math>Q</math> will occur at the embayments so that the shoreline will retreat there. Thus, a positive feedback is originated. We call it ''wave-energy mechanism'' <ref name=F17>Falques, A., F. Ribas, D. Idier, and J. Arriaga (2017), Formation mechanisms for self-organized km-scale shoreline sand waves, J. Geophys. Res., 10.1002/2016JF003964, in press</ref>. It is important to stress that the feedback is essentially based on the alteration of wave refraction by the perturbed depth contours before wave breaking. Thus, it takes place at the surf and shoaling zones together. Consequently, to sustain the feedback, the bathymetric undulations must follow the shoreline undulations: moving offshore when the shoreline progrades and onshore when it retreats. This process is not simultaneous to the morphological changes driven by <math>Q</math>, which take place at the surf zone or near. It is accomplished gradually by the wave driven cross-shore transport and both, longshore transport and cross-shore transport work together only on average at large time scales. Most of the modelling studies that have explored this mechanism have considered an instantaneous link between shoreline and bathymetry on the basis of the large time scales <ref name=AMA></ref> <ref name=F5>Falques, A., and D. Calvete (2005), Large scale dynamics of sandy coastlines. Diffusivity and instability, J. Geophys. Res., 110 (C03007), doi:10.1029/2004JC002587</ref> <ref name=K1>Kaergaard, K., and J. Fredsoe (2013a), Numerical modeling of shoreline undulations part 1: Constant wave climate, Coastal Eng., 75, 64-76</ref> <ref name=K2>Kaergaard, K., and J. Fredsoe (2013b), Numerical modeling of shoreline undulations part 2: Varying wave climate and comparison with observations, Coastal Eng., 75, 77-90</ref>, but a study where this assumption was lifted out showed that the mechanism does not depend essentially on it <ref name=B1>van den Berg, N., A. Falques, and F. Ribas (2012), Modelling large scale shoreline sand waves under oblique wave incidence, J. Geophys. Res., 117 (F03019), doi:10.1029/2011JF002177</ref>. The role of wave-energy mechanism as a potential driver of large scale shoreline features will be discussed in the last section.<br />
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We must also mention that the distribution of wave energy along a shoreline undulation is also affected by wave focusing by the headlands and de-focusing by the embayments. This can alter and inhibit the wave-energy mechanism as it tends to move the maximum in wave energy down-drift, near the headland. In fact, it becomes always dominant for small enough wavelengths and controls the characteristic wavelength of the emerging shoreline features <ref name=B2>van den Berg, N., A. Falques, F. Ribas, and M. Caballeria (2014), On the wavelength of self-organized shoreline sand waves, J. Geophys. Res. Earth Surf., 119, 665-681, doi:10.1002/2013JF002751</ref>.<br />
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===Surf-shoaling zones: wave angle mechanism===<br />
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[[Image: RSFFFig9.jpg|thumb|500px|left|Figure 9. Feedback mechanisms between shoreline undulations and waves involving both the surf and the shoaling zones. Left: Definition of absolute and relative wave incidence angles, <math>\theta</math> and <math>\alpha=\theta - \phi</math>, respectively. The straight brown line represents the local shoreline orientation. Right: The wave-angle mechanism which is based on the differences in relative wave-angle between up-drift and down-drift of C is shown. The larger/smaller relative wave angle, <math>\alpha_b = \theta_b - \phi</math>, causing the gradients in <math>Q</math>, is indicated. As in Figure 8, the wave incidence is from the left and the symbols have the same meaning. An example where the wave-angle mechanism induces a positive feedback is presented.]]<br />
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Let us now examine how a large scale undulating coastline causes gradients in <math>Q</math> through the gradients in relative wave angle, <math>\alpha_b = \theta_b - \phi </math>. We first consider the gradients in <math>\theta_b</math> (ignoring for a while the gradients in <math>\phi</math>). As has been explained in the previous section, wave refraction is more intense at the down-drift sides of headlands than at the up-drift sides. Thus, <math>\theta_b</math> is smaller at the down-drift side and so does <math>Q</math>. Therefore, this effect makes <math>Q</math> decrease moving from the up-drift to the down-drift sides with the result that it causes sediment accumulation and progradation of the headlands. Just the contrary happens at the embayments, so that the shoreline tends to retreat. Thus, the gradients in absolute wave angle, <math>\theta_b</math>, always induce a positive feedback between waves and morphology.<br />
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However, the local shoreline angle, <math>\phi</math>, counteracts this positive feedback. Indeed, the maximum in <math>\phi</math> is located up-drift of each headland and the minimum (maximum magnitude with minus sign) is located down-drift of the headland. This tends to locate the minimum in relative angle, <math>\alpha_b = \theta_b - \phi </math>, at the up-drift side of the headland and the maximum at the down-drift side, contrarily to the effect of <math>\theta_b</math>. Therefore, the gradients in <math>\phi</math> counteract the positive feedback originated by the gradients in <math>\theta_b</math>. We call the interplay of both effects ''wave-angle mechanism'' <ref name=F17></ref>. Which effect is dominant, i.e., whether the wave-angle mechanism induces a positive or a negative feedback depends on a number of factors: the cross-shore mean bathymetric profile, the depth of closure, <math>D_c</math> and the wave conditions (angle, height and period) <ref name=F17></ref>. It also depends on the shape of the bathymetric perturbation associated to the shoreline undulations, so that a positive feedback requires bathymetric undulations with larger amplitude than the shoreline undulations <ref name=I17>Idier, D., A. Falques, J. Rohmer, and J. Arriaga (2017), Self-organized kilometre-scale shoreline sandwave generation: sensitivity to model and physical parameters, J. Geo-phys. Res., submitted</ref>. Figure 9 illustrates a case where the feedback resulting from the gradients in <math>\alpha_b</math> is positive.<br />
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===Instabilities driven by the litoral drift===<br />
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As it has been seen in the previous sections, the gradients in <math>Q</math> are governed by the wave-energy and wave-angle mechanisms. For large enough wavelength, <math>L</math>, the wave-energy always induces a positive feedback. However, the feedback induced by the wave-angle can be either positive or negative. The most common situation is the latter one and, in this case, it turns out that the resulting feedback is negative for <math>\theta < \theta_c</math> and positive for <math>\theta > \theta_c</math>, where the wave angles are evaluated at <math>D_c</math>. The threshold angle, <math>\theta_c</math>, ranges between <math>40^{\circ}</math> and <math>90^{\circ}</math>. This instability is called ''High-angle wave instability'' (HAWI) and it has been hypothesized that it is the origin of a number of large scale features like<br />
shoreline sand waves, cuspate forelands and sandy spits <ref name=AMA></ref> <ref name=AM>Ashton, A., and A. B. Murray (2006), High-angle wave instability and emergent shoreline shapes: 2. Wave climate analysis and comparisons to nature, J.Geophys.Res., 111, F04,012,doi:10.1029/2005JF000,423</ref> <ref name=F6>Falques, A. (2006), Wave driven alongshore sediment transport and stability of the Dutch coastline, Coastal Eng., 53, 243-254</ref> <ref name=MF>Medellin, G., A. Falques, R. Medina, and M. Gonz alez (2009), Sand waves on a low-energy beach at 'El Puntal' spit, Spain: Linear Stability Analysis, J. Geophys. Res., 114 (C03022), doi:10.1029/2007JC004426</ref> <ref name=AML>Ashton, A. D., A. B. Murray, R. Littlewood, D. A. Lewis, and P. Hong (2009), Fetch-limited self-organization of elongate water bodies, Geology, 37, 187-190</ref> <ref name=K1></ref> <ref name=I14>Idier, D., and A. Falques (2014), How kilometric sandy shoreline undulations correlate with wave and morphology characteristics: preliminary analysis on the Atlantic coast of Africa, Advances in Geosciences, 39, 55-60, doi:10.5194/adgeo-39-55-2014</ref>.<br />
In case that the bathymetric undulations have larger amplitude than the shoreline undulations, for low gradient shorefaces and large enough <math>D_c</math>, the wave-angle mechanism can induce a very weak negative feedback or even a positive one. In this case, the critical angle <math>\theta_c</math> can be very low or even <math>0</math>, which means that the shoreline can be unstable for very low wave angles. This situation was called ''Low-angle wave instability'' (LAWI) by Idier et al <ref name=I11>Idier, D., A. Falques, B. G. Ruessink, and R. Garnier (2011), Shoreline instability under low-angle wave incidence, J. Geophys. Res., 116 (F04031), doi:10.1029/2010JF001894</ref> and has been extensively studied by Falques <ref name=F17></ref>, where it is shown that the shoreline sand waves observed at Holmslands Tange (west Danish coast) might be jointly driven by both wave-energy and wave-angle mechanisms.<br />
<br />
<br />
==Acknowledgments==<br />
<br />
This article has been written within the project CTM2015-66225-C2-1-P, which is funded by the Spanish Government and cofounded by the E.U. (FEDER). Useful advice and comments by Prof. J. Dronkers as well as pictures provided by Prof. A.D. Short are gratefully acknowledged.<br />
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==Related articles==<br />
<br />
[[Stability models]]<br />
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[[Beach Cusps]]<br />
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[[Sand transport]]<br />
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[[Sediment transport formulas for the coastal environment]]<br />
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[[Swash zone dynamics]]<br />
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==References==<br />
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<references/><br />
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{{3Authors<br />
|AuthorID1=12926<br />
|AuthorFullName1= Albert Falques<br />
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|AuthorID2=13662 <br />
|AuthorFullName2= Francesca Ribas<br />
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|AuthorID3=13734 <br />
|AuthorFullName3= Daniel Calvete<br />
}}<br />
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[[Category:Land and ocean interactions]]<br />
[[Category:Geomorphological processes and natural coastal features]]<br />
[[Category:Coastal processes, interactions and resources]]<br />
[[Category:Coastal and marine natural environment]]</div>Dronkers Jhttp://www.vliz.be/wiki/Siltation_in_harbors_and_fairwaysSiltation in harbors and fairways2017-05-04T15:58:59Z<p>Dronkers J: </p>
<hr />
<div>==Introduction==<br />
This article addresses the siltation in semi-enclosed harbor basins and fairways in open water with sediments from the surrounding waters. In its most basic form, siltation occurs when the sediment transport capacity is locally exceeded by the supply of sediment. <br />
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==Siltation in semi-enclosed harbor basins==<br />
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For harbor basins, siltation with fine (cohesive) sediments is in general more problematic than siltation with coarser sediments (sand), as the siltation rates with fines are often larger, and at times, these sediments are contaminated as well. Thus we focus on siltation with fine sediments, which are generally carried by the flow in suspension (see [[Dynamics of mud transport]]).<br />
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The harbor basin and the ambient water system exchange sediment-laden water by the three mechanisms described below. Because the basin is semi-enclosed, no net exchange of water occurs (over a long-enough period). However, there is a gross exchange of water – sediment-rich water from the ambient system is exchanged with sediment-poor water from the basin itself. In zero-order approach we assume that the sediment flux may be treated as the product of water flux and SPM (suspended particulate matter) concentration, corrected for a trapping efficiency (the fraction of sediment that enters the harbor and deposits). <br />
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A harbor basin can be situated along a river, a lake, a tidal river, a coast, an estuary etc. We always assume that the water body in front of the harbor entrance flows with a characteristic velocity U. In a tidal river and along a coast, tides play a major role, whereas in estuaries, and along some coasts, density currents are also important.<br />
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[[Image: HanFig1.jpg|thumb|400px|right|Figure 1: Visualization of exchange processes between harbor basin and surrounding (courtesy Vanlede and Dujardin<ref> Vanlede, J. and A. Dujardin, 2014. A geometric method to study water and sediment exchange in tidal harbors. Ocean Dynamics 64:1631–1641 DOI 10.1007/s10236-014-0767-9</ref> and de Boer and Winterwerp<ref> De Boer, W.P. and J.C. Winterwerp, 2016. The role of fresh water discharge on siltation rates in harbor basins. Proceedings, PIANC-COPEDEC IX, Rio de Janeiro, Brazil </ref>.]]<br />
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The three relevant mechanisms exchanging sediment-laden water between the ambient water body and the harbor basin are (see also Fig. 1):<br />
# Horizontal exchange: large scale circulations in the harbor’s mouth driven by flow separation, entrainment and stagnation effects at the downstream side of the basin’s entrance. This mechanism always plays a role in flowing ambient water. Net exchange off water is always zero.<br />
# Tidal filling: during rising tide, sediment-laden water flows into the basin, while during falling water, the same amount of water flows out of the basin, containing less sediment, though. Over a tidal period, no net amount of water is exchanged. This mechanism plays a role in tidal rivers and along open coasts.<br />
# Density currents: gradients in salinity induce density currents with a near-bed current (which contains the majority of the sediment) against the direction of that gradient. Salinity-induced density currents play a role in estuaries and coastal systems with salinity gradients. Eysink (1989) emphasized the importance of this mechanism. Density currents can also be induced by gradients in SPM and temperature.<br />
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Note that these mechanisms interact. For instance, during tidal filling, the wake induced by flow separation in the harbor’s mouth is deflected into the basin, and horizontal exchange thus becomes less effective in exchanging water between the basin and the surrounding water body.<br />
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[[Image: HanFig2.jpg|thumb|350px|right|Figure 2: Timing of water exchange processes over a tidal period, assuming that salinity and velocity are in phase.]]<br />
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These three mechanisms are illustrated below for an estuary, where salinity variations are more or less in phase with ebbing and flooding of the tide. In general, water level and tidal velocity are out of phase, and in the following we assume a phase difference of <math>45^{\circ}</math>. During flood, flow separation occurs at the down-estuary side of the harbor basin, and the wake is deflected into the basin during rising tide, and out of the basin during falling tide. Thus, relatively, little water is exchanged during falling water and ebb, which is therefore neglected in this zero-order estimation. Tidal filling is in phase with rising and falling water. The salinity in the estuary in front of the harbor basin is more or less in phase with flood and ebb – for a harbor basin situated down-estuary (close to the sea) salinity in the estuary is larger than in the basin during flood, while the opposite is true during ebb. Hence, a near-bed sediment-rich density current flows into the basin during flood, while, owing to siltation in the basin, a near-bed sediment-poor density current flows out of the basin during ebb. This phasing is sketched in Fig. 2, showing that sediment import into a harbor basin is unevenly distributed over a tidal period.<br />
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[[Image: HanFig3.jpg|thumb|400px|left|Figure 3: Zero-order mass balance for harbor siltation, where <math>F_s</math> = siltation rate, and <math>\alpha</math> an efficiency parameter (here the trapping efficiency) – in the following this efficiency is accounted for through the settling velocity of the sediment in the basin (which thus becomes the effective settling velocity).]]<br />
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Eysink <ref name=E> Eysink, W.D. (1989). “Sedimentation in harbor basins – small density differences may cause serious effects”, Proceedings of the 9th International Harbor Congress, Antwerp, Belgium, June 1988; also: Delft Hydraulics, Publication No 417</ref> was the first to quantify the siltation rate in harbor basins, using a zero-order assessment (see also <ref> Van Rijn, L.C., 2005. Principles of sedimentation and erosion engineering in rivers, estuaries and coastal seas. Aqua Publications, The Netherlands </ref><ref> Winterwerp, J.C. and W.G.M. van Kesteren, 2004. Introduction to the physics of cohesive sediments in the marine environment, Elsevier, Developments in Sedimentology, 56</ref>). The basis for this assessment is the zero-order sediment balance, sketched in Fig. 3, where it is assumed that the SPM-values in the harbor basin <math>c_e</math> (and thus the siltation rate) are proportional to the ambient SPM-value <math>c_a</math>. The equilibrium solution to this simple differential equation is given in equation (1):<br />
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<math>c_e=\frac{<Q>}{<Q>+\alpha S W_s} c_a , \qquad (1) </math> <br />
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in which <math>S</math> = projected basin’s surface. The exchange flow <math>Q</math> is determined by the three processes described above:<br />
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<math>\frac{1}{T} \int_0^T Q dt \equiv <Q>=<Q_t>+<Q_e>+<Q_d> , \qquad(2) </math><br />
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where <math><Q_t></math> is the gross water exchange by tidal filling (angular brackets implies averaging over tidal period), <math><Q_e></math> is the gross water exchange by horizontal circulation (entrainment), and <math><Q_d></math> is the gross water exchange by density currents.<br />
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Eysink <ref name=E></ref> proposes a number of coefficients to quantify these gross water exchange rates:<br />
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<math><Q_t>=\frac{V_t}{T}, \quad <Q_e>=f_e A U - f_{e,t}<Q_t> , \quad <Q_d>=f_d A \sqrt{ \frac{\Delta \rho_s g h_0}{\rho} } - f_{d,t} <Q_t> , \qquad(3)</math><br />
<br />
where <math>V_t</math>= tidal volume of harbor basin, <math>T</math>= tidal period, <math>A</math>= cross section harbor entrance, <math>U</math>= characteristic velocity along the harbor entrance, <math>\Delta \rho_s</math> = characteristic salinity-induced density difference across the harbor entrance, <math>h_0</math> is local water depth, the coefficients <math>f_{e,t}, f_{d,t}</math> reflect a reduction in exchange efficiency during rising tide, <math>\alpha=1-u_h^2/u_{cr}^2</math>, <math>u_h</math> is a characteristic velocity in the harbor basin, and <math>u_{cr}</math> is a threshold velocity below which sediment can permanently settle of the basin’s bed. <br />
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The following empirical coefficients were proposed by Eysink:<br />
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[[Image: HanFig4.jpg|thumb|250px|left|Figure 4: Empirical coefficients for harbor siltation.]]<br />
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[[Image: HanFig5.jpg|thumb|250px|right|Figure 5: Cartoon of shipping-induced siltation, especially important in lakes and canals.]]<br />
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Next to these three exchange mechanisms, also shipping itself can induce sediment import into harbor basins, in particular in lakes and canals, where the above-mentioned mechanisms are small. Fig. 5 sketches this process, for which however no general quantification exists.<br />
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==Siltation in fairways==<br />
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Siltation in fairways in open water can occur with fine (silt) and coarse (sand) sediment, which may be transported as suspended load and/or as bed load. Fairways in rivers and estuaries (“open water”) behave differently. In the latter case full morphodynamic analyses are required for assessing these siltation rates, as the navigation channel and ambient water system can interact strongly. This is for instance the case in the Western Scheldt where fairway deepening influences the morphodynamic development of this estuary<ref> Jeuken, M.C.J.L. and Z.B. Wang. Impact of dredging and dumping on the stability of ebb–flood channel systems. Coastal Engineering 57, 553–566</ref> .<br />
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[[Image: HanFig6.jpg|thumb|250px|left|Figure 6: Refraction of current oblique to navigation channel.]]<br />
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Fig. 6 depicts the refraction of a current, oblique to a navigation channel, over that channel. To understand this picture, the scales of the system have to be considered. A cross-current experiences a sudden increase in water depth (the width of the channel is small compared to the system’s dimensions), thus the flow decelerates locally, as the water flux does not change. However, along the channel, the channel dimensions are much larger than the channel width. Given a constant pressure gradient along the channel, the larger depth within the channel reduces the effective hydraulic drag accelerating the current. Thus, depending on the angle of incidence, the flow accelerates of decelerates. A semi-quantitative analysis is presented below.<br />
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The flow velocity in the channel as a function of relative channel depth and angle of incidence (with <math>C</math>= Chézy coefficient) can be derived from simple geometric arguments:<br />
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<math>\frac{U_{y,1}}{U_{y,0}}=\frac{h_0}{h_1} ,\quad \frac{U_{x,1}}{U_{x,0}}=\frac{C_1}{ C_0} \left( \frac{h_1}{h_0} \right)^{1/2} , \qquad(4)</math><br />
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where <math>C \approx C_1 \approx C_2</math> and <br />
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<math>U_1=U_0 \frac{h_0}{h_1} [ \sin^2 \alpha_0 + \left ( \frac{h_1}{h_0} \right )^3 \cos^2 \alpha_0 ]^{1/2} , \quad \tan \alpha_1=\left ( \frac{h_0}{h_1} \right )^{3/2} \tan \alpha_0. </math> <br />
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The subscripts <math>x, y</math> indicate the along-channel and cross-channel projections of the velocity, respectively. <br />
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[[Image: HanFig7NEW.jpg|thumb|350px|left|Figure 7: Ratio flow velocity inside and outside channel according to equation 4 (black curves) and the three-dimensional approach by Jensen et al. (red curves)<ref>Jensen, J.H., Madsen, E.Ø. and Fredsøe, J., 1999, “Oblique flow over dredged channels – I: Flow description”, ASCE, Journal of Hydraulic Engineering, Vol 125, No 11, pp 1181-1189</ref><ref>Jensen, J.H., Madsen, E.Ø. and Fredsøe, J., 1999, “Oblique flow over dredged channels – II: Sediment transport and morphology”, ASCE, Journal of Hydraulic Engineering, Vol 125, No 11, pp 1190-1198</ref>.]]<br />
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The ratio between the flow velocities inside and outside the navigation channel are plotted in Fig. 7 as a function of the angle of incidence of the ambient flow. Note that up to quite large angles, the flow velocity within the channel is larger than outside the channel. A larger flow velocity implies a larger transport capacity. The implications for channel self-cleansing are discussed below. <br />
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[[Image: HanFig8.jpg|thumb|300px|right|Figure 8: Cartoon of extra water to be attracted.]]<br />
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A larger flow velocity through a larger cross section implies a larger specific discharge, and the extra water has to come from the channel’s surrounding waters. However, this extra water carries also extra sediment. For simplicity, only along-channel flow is discussed, as illustrated in Fig. 8. Continuity and substitution from equation (4) yields a relation for an “effective channel width” <math>b_0</math> representing the area from which the “extra” water and sediment is attracted,<br />
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<math>\frac{b_0}{b_1}=\frac{U_1 h_1}{U_0 h_0}=\frac{C_1}{C_0} \left ( \frac{h_1}{h_0} \right )^{3/2} . \qquad(5)</math><br />
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Ignoring settling and erosion time lag effects, the equilibrium transport <math>T_e</math> is represented by a power of the local flow velocity: <math>T_e \propto u_*^n</math>, where <math>u_*</math> = shear velocity and in which <math>n</math> = 3 represents bed load and <math>n</math> = 4 – 6 represents suspended load (see e.g. [[Sediment transport formulas for the coastal environment]]). Substitution from equations (4) and (5) yields a relation for the sediment transport capacity inside the channel in relation to the sediment transport capacity outside the channel:<br />
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<math>\frac{T_{e,1}}{T_{e,0}}=\frac{b_1}{b_0} \left ( \frac{u_{*,1}}{u_{*,0}} \right )^n=\frac{b_1}{b_0} \left ( \frac{U_1 C_0}{U_0 C_1} \right )^n=\frac{b_1}{b_0} \left ( \frac{h_1}{h_0} \right )^{n/2}= \frac{C_0}{C_1} \left ( \frac{h_1}{h_0} \right )^{(n-3)/2} . \qquad(6)</math><br />
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As <math>C_1</math> is (slightly) larger than <math>C_0</math>, the sediment transport capacity within the channel is smaller than outside in case of bed load (<math>n</math> = 3). Thus the channel will silt up. However, for suspended load, equation (6), when <math>n</math> = 4 – 6, equation (6) predicts that the fairway is self-cleansing, and possibly even eroding. Advanced three-dimensional numerical sediment transport models yield a similar prediction, at least qualitatively. This result conflicts with observed fairway sedimentation, however.<br />
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This paradox illustrates that some important phenomena are missing in the above analysis. This holds in particular for the omission of wave effects. Waves stir up and can transport large amounts of sediments in the shallows surrounding the fairway, whereas wave activity on the channel’s bed is small. Linear wave theory gives a first-order estimate of the ratio of maximum wave-induced bed shear stresses:<br />
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<math>\frac{\hat \tau_{b,c}}{\hat \tau_{b,\infty}}= \left ( \frac{u_{b,c}}{u_{b,\infty}} \right )^2=\left ( \frac{\sinh kh_{\infty}}{\sinh kh_c} \right )^2 \approx \left ( \frac{h_{\infty}}{h_c} \right )^2 , \qquad(7) </math><br />
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where the subscript <math>b</math> indicates the near-bed value. The most right member follows from the inequality <math>k=2 \pi / \lambda << 1 /h</math> (wavelength <math>\lambda >> h </math>).<br />
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[[Image: HanFig9new.jpg|thumb|300px|right|Figure 9: Ratio of maximum wave-induced bed shear stresses on the channel bed and in the surrounding waters as a function of relative channel depth.]]<br />
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This ratio is plotted in Fig. 9, showing that bed shear stresses on the channel bed decrease rapidly with channel depth. Moreover, wave-induced shear stresses are often larger than flow-(tide) induced stresses. Hence, ignoring effects of waves on the siltation rates in navigation channels may give an entirely wrong picture of channel’s maintenance needs. <br />
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Other effects, amongst which estuarine circulation, may also affect channel siltation. However, these are not accounted for in the present zero-order assessment<br />
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This analysis yields two important observations:<br />
# Channel siltation may vary strongly over the year, in particular if the local climate is characterized by seasonal variations in wave conditions,<br />
# Channel orientation may strongly influence siltation rates, thus maintenance costs.<br />
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Literature provides some simple engineering rules for assessing channel siltation <math>F_c</math> with channel width B:<br />
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* Mayor-Mortensen-Fredsøe<ref> Mayor-Mora, R., P. Mortensen and J. Fredsoe, 1976. Sedimentation studies on the Niger River Delta. 15th ICCE, Honolulu, Hawaii </ref>:<br />
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<math>F_c = \frac{T_0}{\cos(\alpha_0-\alpha_1)} \left [ 1 - \exp \left ( - \frac{A_m h_0 B}{h_1 \sin \alpha_1 \cos(\alpha_0-\alpha_1)} \right ) \right ] -T_1 \left [ 1- \exp \left (- \frac{A_m B}{\sin \alpha_1} \right ) \right ] \sin \alpha_1 , \qquad(8) </math><br />
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with <math>A_m=W_s^2/ \epsilon_1 U_1 , \quad \epsilon_1=0.085 h_1 u_{*,1} </math>.<br />
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* Bijker<ref>Bijker, E., 1980. Sedimentation in channels and trenches. 17th ICCE, Sydney, Australia</ref> :<br />
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<math>F_c = \left ( \frac{T_0}{\cos(\alpha_0-\alpha_1)}-T_1\right ) \left[ 1- \exp \left (- \frac{A_b B}{\sin \alpha_1} \right ) \right ] \sin \alpha_1 , \qquad(9) </math><br />
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with <math>A_b=\frac{F_B W_s}{h_1 U_1}, \quad F_B=\frac{a_1}{a_2-a_3}</math> and<br />
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<math>a_1=\frac{u_{*,1}}{u_{*,0}-u_{*,1}}, \quad a_2=\frac{W_s}{0.085 u_{*,1}} \left [1- \exp \left (-\frac{W_s}{0.085 u_{*,0}} \right ) \right ], \quad a_3=\left ( \frac{u_{*,1}}{u_{*,0}} \right )^3 \frac{W_s}{0.085 u_{*,0}} \left [ 1- \exp \left (-\frac{W_s}{0.085 u_{*,1}} \right ) \right ]</math>.<br />
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* Eysink & Vermaas<ref> Eysink, W. and H. Vermaas, 1983. Computational method to estimate the sedimentation in dredged channels and harbor basins in estuarine environments. Int. Conf. on Coastal and Ports Engineering in Developing Countries, Colombo, Sri Lanka; also: Delft Hydraulics, Publication No 307</ref>:<br />
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<math>F_c = \left ( \frac{T_0}{\cos(\alpha_0-\alpha_1)} - T_1 \right ) \left [ 1- \exp \left ( - \frac{A_{ev} B}{h_1 \sin \alpha_1} \right ) \right ] \sin \alpha_1 , \qquad(9) </math> <br />
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with <math>A_{ev} = 0.03 \frac{W_s}{u_{*,1}} \left ( 1+\frac{2W_s}{u_{*,1}} \right ) \left [ 1+4.1 \left ( \frac{k_s}{h_1} \right )^{0.25} \right ]</math>.<br />
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Note that from our analysis above, the shear velocity has to be corrected for the effects of waves, otherwise misleading results will be obtained. <br />
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* The fourth model is by Allersma, known as the “volume-of-cut” method, but which has never been published:<br />
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<math>h_{T*}=h_0-(h_0-h_e) \left [ 1- \exp \left (-\frac{v T_*}{h_0} \right ) \right ]</math>.<br />
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This simple model is particularly useful in case of deepening an existing channel – data on previous dredging volumes can be used for calibration of the parameter <math>v</math>. <math>T_*</math> is an arbitrary time scale (one year, five years, ..), <math>h_0</math> the initial channel depth, and <math>h_e</math> the channel’s equilibrium depth (which may be the local water depth in open water). <br />
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<br />
==List of symbols==<br />
<br />
{|<br />
|- style="font-weight:bold; text-align:center; background:lightgrey" <br />
! width=20% style=" border:1px solid gray;"| parameter<br />
! width=50% style=" border:1px solid gray;"| definition<br />
! width=10% style=" border:1px solid gray;"| unit<br />
|- <br />
| style="border:1px solid gray;"| <br />
| style="border:1px solid gray;"|<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>A</math><br />
| style="border:1px solid gray;"| cross-section harbor mouth<br />
| style="border:1px solid gray;"| <math>m^2</math><br />
|- <br />
| style="border:1px solid gray;"| <math>A_b, A_{ev}, A_m</math><br />
| style="border:1px solid gray;"| empirical coefficients<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>b</math><br />
| style="border:1px solid gray;"| channel width<br />
| style="border:1px solid gray;"| <math>m</math><br />
|- <br />
| style="border:1px solid gray;"| <math>C</math><br />
| style="border:1px solid gray;"| Chezy coefficient<br />
| style="border:1px solid gray;"| <math>m^{1/2}/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>c_a</math><br />
| style="border:1px solid gray;"| ambient suspended particulate matter (SPM) concentration<br />
| style="border:1px solid gray;"| <math>kg/m^3</math><br />
|- <br />
| style="border:1px solid gray;"| <math>c_e</math><br />
| style="border:1px solid gray;"| equilibrium concentration in harbor<br />
| style="border:1px solid gray;"| <math>kg/m^3</math><br />
|- <br />
| style="border:1px solid gray;"| <math>c_h</math><br />
| style="border:1px solid gray;"| SPM concentration in harbor<br />
| style="border:1px solid gray;"| <math>kg/m^3</math><br />
|- <br />
| style="border:1px solid gray;"| <math>F_s</math><br />
| style="border:1px solid gray;"| siltation rate in harbor<br />
| style="border:1px solid gray;"| <math>kg/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>F_h</math><br />
| style="border:1px solid gray;"| siltation rate in channel<br />
| style="border:1px solid gray;"| <math>kg/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>F_B</math><br />
| style="border:1px solid gray;"| empirical coefficient<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>f_d, f_e, f_{d,t}, f_{e,t}</math><br />
| style="border:1px solid gray;"| empirical coefficients<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>h</math><br />
| style="border:1px solid gray;"| depth<br />
| style="border:1px solid gray;"| <math>m</math><br />
|- <br />
| style="border:1px solid gray;"| <math>k_s</math><br />
| style="border:1px solid gray;"| Nikuradse roughness height<br />
| style="border:1px solid gray;"| <math>m</math><br />
|- <br />
| style="border:1px solid gray;"| <math>n</math><br />
| style="border:1px solid gray;"| power in transport formula<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>Q</math><br />
| style="border:1px solid gray;"| exchange flow rate<br />
| style="border:1px solid gray;"| <math>m^3/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>Q_d, Q_e, Q_t</math><br />
| style="border:1px solid gray;"| same; density-, entrainment-, tide-induced<br />
| style="border:1px solid gray;"| <math>m^3/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>S</math><br />
| style="border:1px solid gray;"| harbor projected surface<br />
| style="border:1px solid gray;"| <math>m^2</math><br />
|- <br />
| style="border:1px solid gray;"| <math>T</math><br />
| style="border:1px solid gray;"| tidal period<br />
| style="border:1px solid gray;"| <math>s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>T_*</math><br />
| style="border:1px solid gray;"| characteristic time scale<br />
| style="border:1px solid gray;"| <math>s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>T_e</math><br />
| style="border:1px solid gray;"| equilibrium sediment transport<br />
| style="border:1px solid gray;"| <math>kg/m</math><br />
|- <br />
| style="border:1px solid gray;"| <math>T_0 (T_1)</math><br />
| style="border:1px solid gray;"| sediment transport outside (inside) channel<br />
| style="border:1px solid gray;"| <math>kg/m</math><br />
|- <br />
| style="border:1px solid gray;"| <math>U</math><br />
| style="border:1px solid gray;"| characteristic velocity<br />
| style="border:1px solid gray;"| <math>m/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>u_*</math><br />
| style="border:1px solid gray;"| shear velocity<br />
| style="border:1px solid gray;"| <math>m/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>u_{cr}</math><br />
| style="border:1px solid gray;"| critical velocity<br />
| style="border:1px solid gray;"| <math>m/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>u_h</math><br />
| style="border:1px solid gray;"| characteristic velocity in harbor basin<br />
| style="border:1px solid gray;"| <math>m/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>V</math><br />
| style="border:1px solid gray;"| harbor volume<br />
| style="border:1px solid gray;"| <math>m^3</math><br />
|- <br />
| style="border:1px solid gray;"| <math>W_s</math><br />
| style="border:1px solid gray;"| settling velocity<br />
| style="border:1px solid gray;"| <math>m/s</math><br />
|- <br />
| style="border:1px solid gray;"| <math>x</math><br />
| style="border:1px solid gray;"| coordinate along channel<br />
| style="border:1px solid gray;"| <math>m</math><br />
|- <br />
| style="border:1px solid gray;"| <math>y</math><br />
| style="border:1px solid gray;"| coordinate perpendicular to channel<br />
| style="border:1px solid gray;"| <math>m</math><br />
|- <br />
| style="border:1px solid gray;"| <math>\alpha</math><br />
| style="border:1px solid gray;"| efficiency parameter for harbor siltation<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>\alpha_0</math><br />
| style="border:1px solid gray;"| flow angle ambient current<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>\alpha_1</math><br />
| style="border:1px solid gray;"| flow angle within channel<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>\Delta \rho_s</math><br />
| style="border:1px solid gray;"| salinity-induced density difference<br />
| style="border:1px solid gray;"| <math>kg/m^3</math><br />
|- <br />
| style="border:1px solid gray;"| <math>\epsilon_1</math><br />
| style="border:1px solid gray;"| empirical coefficient<br />
| style="border:1px solid gray;"| <br />
|- <br />
| style="border:1px solid gray;"| <math>v</math><br />
| style="border:1px solid gray;"| empirical coefficient<br />
| style="border:1px solid gray;"| <math>m/s</math><br />
|}<br />
<br />
<br />
==Related articles==<br />
<br />
[[Dynamics of mud transport]]<br />
<br />
[[Sediment deposition and erosion processes]]<br />
<br />
[[Sediment transport formulas for the coastal environment]]<br />
<br />
[[Sand transport]]<br />
<br />
[[Manual Sediment Transport Measurements in Rivers, Estuaries and Coastal Seas]]<br />
<br />
<br />
==Further reading==<br />
<br />
PIANC, 2008. Minimising harbour siltation, Pianc. Brussels, Belgium.<br />
<br />
http://www.leovanrijn-sediment.com/papers/Harboursiltation2012.pdf<br />
<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
<br />
{{author<br />
|AuthorID=15072<br />
|AuthorFullName=Johan Winterwerp<br />
|AuthorName=Johan Winterwerp}}<br />
<br />
[[Category:Land and ocean interactions]]<br />
[[Category:Geomorphological processes and natural coastal features]]<br />
[[Category:Coastal processes, interactions and resources]]<br />
[[Category:Coastal and marine natural environment]]</div>Dronkers J