Case studies: Long term predictions for estuaries

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This case-study illustrates how the formulation and validation of a detailed numerical model can utilise observational data ranging from process measurements, extended observations to permanent monitoring. Likewise, it shows how the Theoretical Frameworks are used to interpret ensemble sensitivity simulations.


Tidal ranges in the Mersey vary from 4 to 10 m over the extremes from neap to spring. The estuary has been widely studied because of its vital role in shipping. The ‘Narrows’ at the mouth of the 45 km-long estuary is approximately 1.5 km wide with a mean depth (below chart datum) of 15 m (Figure 8.1), and tidal currents through this section can exceed 2 m s−1. Further upstream in the inner estuary basin, the width can be as much as 5 km, and extensive areas are exposed at low water. Freshwater flow into the estuary, Q, varies from 25 to 300 m3 s−1 with a mean ‘flow ratio’ (Q × 12.42 hr)/Volume between high and low water) of approximately 0.01. Flow ratios of less than 0.1 usually indicate well-mixed conditions, though in certain sections during part of the tidal cycle, the Mersey is only partially mixed.

 Figure. 8.1 Liverpool Bay and the Mersey Estuary location map, showing the 1992 transect line, positions P2, P3, and tide gauges marked with dots. Depths (1997 bathymetry) are in metres below ordnance datum Newlyn (ODN). Chart datum is approximately the lowest astronomical tide level, and is 4.93 m below ODN.

 Table 8.1 Comparison of Lagrangian model results with observations. Values in the Narrows. Position on the transect lines Fig. 8.1: (1) 280 m from Wirral. (2) 290 m from Liverpool shore. aLane (2004), bPrandle et al. (1990), cThomas et al. (2002). *90% of sediments have concentration less than this value

Figure 8.2 Observed Sediment Concentrations in The Narrows: (a) surface, (b) mid-depth 1986; ( c) 280m from Wirral shore and (d) 290m from Liverpool shore 1992.

Suspended sediments and net deposition

Figure 8.2 shows observed suspended sediment time series from locations in the Narrows recorded in 1986 and 1992; Table 8.1 summarises these results. The 1986 observations included five simultaneous moorings across the Narrows, providing estimates of net spring and neap tidal fluxes of sediments.Prandle et al. (1990) analysed four sets of observa�tions of SPM indicating tidally averaged cross-sectional mean concentrations varying as a function of tidal amplitudes, ζ, as follows: 32 mg l−1 for ζ = 2.6 m, 100 mg l−1 for ζ = 3.1 m, 200 mg l−1 for ζ = 3.6 m and 213 mg l−1 for ζ = 4.0 m. These values correspond to a tidal flux (on ebb or flood) of 40000 t on a mean tide, reducing to as little as 2500 t at neap and increasing by up to 200000 t on springs – in reasonable agreement with earlier estimates of Price and Kendrick (1963).

Hutchinson and Prandle (1994) used contaminant sequences in sediment cores ( analogous to tree-ringing) to estimate net accre�tion rates in the adjacent and similarly sized Dee Estuary, These amounted to 0.3 Mt a−1 between 1970 and 1990 and 0.6 Mt a−1 between 1950 and 1970. Hill et al. (2003) derived settling velocities, ws, of 0.0035 m s−1 for spring tidal conditions and 0.008 m s−1 for neaps. Noting that particle diameter d (mm) ~ 1000 ws½ (m s−1), these correspond to d = 59 and 89 mm, respectively.


Data are available from surveys carried out by the Mersey Docks and Harbour Company in 1906, 1936, 1956, 1977 and 1997. Differences in volume within the Narrows are of the order of a few percent from one data set to the next. The largest changes appear in the inter-tidal regions of the inner estuary basin, particularly from Hale and Stanlow to Runcorn where the low water channel positions change readily, and differences between successive surveys exceed 10%. The overall pattern is for the estuary volume to decrease by about 60 Mm3 or 8% between 1906 and 1977, despite sea level rise averaging 1.23 mm per year during the past century (Woodworth et al. 1999). After this period, there is a small increase of 10 Mm3.


Here we illustrate the capabilities and limitations of a 3-D Eulerian Hydrodynamic model coupled with a Lagrangian sediment module (Lane and Prandle, 2006) to quantify impacts on the estuarine sediment regime and indicate the rate and nature of bathymetric evolution. Particular emphasis is on quantifying the variations in sediment concentrations and fluxes in sensitivity tests of: bed roughness, eddy viscosity, sediment supply (particle sizes 10 to 100 μm), salinity intrusion and 2-D versus 3-D formulations of the hydrodynamic model. The model was not intended to reproduce bed-load transport associated with coarser sediments.

Recognising the limited capabilities to monitor the often extremely heterogeneous suspended particulate matter, a wide range of observational data was used for assessing model performance. These include: suspended concentrations (axial profiles of mean and ‘90th percentile’), tidal and residual fluxes at cross-sections, estuary-wide net suspension and deposition on spring and neap tides, surficial sediment distributions and sequences of bathymetric evolution.

The Eulerian hydrodynamic model provides velocities, elevations, and diffusivity coefficients for the Lagrangian ‘random-walk’ particle model in which up to a million particles represent the sediment movements. It includes a wetting-drying scheme to account for the extensive inter-tidal areas. Forcing involved specifying tidal elevation constituents at the seaward limit in the Mersey Narrows, and river flow at the head. The model uses a 120-metre rectangular grid horizontally and a 10-level sigma-coordinate scheme in the vertical. Calibration of the model (Lane, 2004) involved simulating effects of ‘perturbations’ (based on varying the mean sea level, bed friction coefficient, vertical eddy viscosity and the river flow) and finding the optimum combination to minimise differences from observed tidal elevation constituents. The model indicated that the estuary (particularly in the inner basin) is most sensitive to changes in bathymetries and bed friction coefficients. River flow only has an appreciable effect for discharges significantly higher than those usually encountered.

Lagrangian, random-walk particle module for non-cohesive sediment

Random-walk particle models replicate solutions of the Eulerian advection-diffusion equation by calculating, for successive time steps, the height above the bed, z and horizontal location of each particle following: a vertical advective movement −wsΔt (downwards) a diffusive displacement l (up or down), horizontal advection.

The displacement length l = Ö(2 Kz Δt) , (Fischer et al., 1979), withKz approximated by k Û D (Prandle, 1982). Contacts with the surface and bed during this diffusion step are reflected elastically.Deposition occurs when the particle reaches the bed calculated in a discrete advective settlement step −wsΔt . New particles are released into suspension by accumulation of the erosion potential.

A simple algorithm for the erosion source was adopted

ER = g r k U P (8.1)

where k is the bed friction coefficient, ρ is water density and a value of P = 2 was assumed. Having specified P, all subsequent calculations of concentration, flux and sedimentation rates are linearly proportional to the coefficient γ. A value of γ = 0.0001 m−1 s was found to produce suspended sediment concentrations comparable with those in Figure 8.3. The corresponding values of tidal and residual cross-sectional fluxes were also in reasonable agreement with observed values shown in Table 8.1.

Sensitivity Tests

Full details of the sensitivity tests are shown by Lane and Prandle (2006), these are summarised in Table 8.2 and 8.3. Figure 8.3 shows cross-sectional mean suspended sediment concentration , at successive locations landwards from the mouth, over two spring-neap cycles commencing from the initial introduction of sediments . The examples chosen are for sediment fall velocities, ws of 0.005 m s−1 (coarse sediment d = 70 μm, black lines) and 0.0005 m s−1 (finer sediment, d = 22 μm, grey lines) respectively.

 Figure 8.3 Suspended sediment concentrations at 12 positions along the Mersey (1 the mouth, 12 the head). Grey lines settling velocity Ws = 0.0005 m s-1; black lines Ws = 0.005 m s-1 .

Starting with no sediment in the estuary, all particles are introduced at the seaward boundary of the model using the erosion formula (8.1). An unlimited supply is assumed together with zero axial concentration gradient (dC/dx = 0) for inflow conditions. To reflect the effect of changing distributions of surficial sediments on the bed friction coefficient, this was specified as 0.0158 ws¼.

For ws = 0.0005 m s−1 (grey lines), the suspended sediment time series change from predominantly semi-diurnal (linked to advection) at the mouth to quarter-diurnal (linked to localised resuspension) further upstream. Even close to the mouth, a significant quarter-diurnal component is generated at spring tides. Close to the mouth, peak concentrations occur some three tidal cycles after maximum spring tides and at up to seven cycles later further upstream.

 Table 8.2

Sensitivity of modelled sediments

Particle diameters d= 10 to 100 m. for Ws = 10 -6 d2 m s-1.

For the coarser sediment, ws = 0.005 m s−1 (black lines), Figure 8.3 shows much reduced concentrations largely confined to the seaward region, although the slower ‘adjustment’ rate suggests that a longer simulation is required to introduce the coarser sediments further upstream. The time series is predominantly quarter-diurnal and peak concentrations coincide with peak tides; the sediments have a much shorter half-life in suspension as described in Chapter 5.

Figure 8.4(a) shows corresponding time-series of cumulative inflow and outflow of sediments across the mouth of the estuary model. Differences between inflow and outflow, in Figure 8.4(b), indicate net suspension (high frequency) and net deposition (low-frequency). For ws = 0.0005 m s−1, the mean tidal exchange of sediments is around 110000 t per tide, of which approximately 6% is retained amounting to 7000 t per tide. For ws = 0.005 m s−1, the mean exchange is 22000 tonnes of which approximately 12% is retained or about 3000 t per tide.

Sensitivity to sediment size

 Figure 8.4 (a) Cumulative inflow and outflow at the mouth of the Mersey (b) Net suspension (high frequency) and deposition (low frequency).

For a more extensive quantitative evaluation of the model, single neap-spring tidal cycle simulations were used. Results are summarised in Table 8.2 for particle diameters d from 10 to 100 μm.

The model reveals that mean suspended sediment concentrations vary approximately with d−2. Equation (7.29a) indicates variability ranging from d0 to d−4 for finer to coarser sediments. The extent of landward intrusion increases progressively for finer sediments. A minimum capture rate of 2.8% occurs for d = 30 μm with a corresponding deposition rate of 1 Mt per year. While capture rates increase progressively with increasing sediment size (above d = 30 μm), corresponding decreases in concentration yield a maximum deposition at 50 μm of 2 Mt per year. This maximum is close to the preponderance of sediments with ws = 0.003 m s−1 (d = 54 μm) found by Hill et al. (2003). In Chapter 7, it was shown that the size of suspended sediments corresponding to ‘equilibrium’ conditions of zero net deposition or erosion is in the range 20 to 50 µm. Net sedimentation remains surprisingly constant, between 1 and 2 Mt per year, throughout the range of d = 30 to 100 μm. This sedimentation rate is in close agreement with observational evidence (Table 8.1).

Sensitivity to model parameters

The model’s responses to the following parameters were quantified: vertical structure of currents, eddy diffusivity and salinity, as well as the bed friction coefficient and sediment supply. Table 8.3 shows, for ws = 0.0005 m s−1 (d = 22 μm), the sensitivity to Run Numbers: No:

  1. vertical current shear, i.e., a 2-D hydrodynamic model.
  2. Depth-varying eddy diffusivity with depth-mean value Kz at the bed, 1.33 Kz at z = 0.33 and 0 at the surface, Kz(z) = Kz (−3z2 + 2z + 1).
  3. A time varying value of Kz(t), with a quarter-diurnal variation of amplitude 0.25 Kz producing a peak value one hour after peak currents.
  4. Mean salinity-driven residual current profile (4.15); Uz = g Sx D3 / E {−0.1667 z3 + 0.2687 z2 − 0.0373 − 0.0293}, where the salinity gradient Sx was specified over a 40 km axial length and eddy viscosity E = Kz.
  5. Bed friction coefficient halved, k = 0.5 × 0.0158 ws¼.
  6. Bed friction coefficient doubled, k= 2.0 × 0.0158 ws¼.
  7. Erosion rate at mouth 0.5 γ, i.e., halving the rate of supply of marine sediments.
  8. BASE-LINE simulation

While the calculated values of sediment concentration and net fluxes varied widely and irregularly for varying sediment sizes, the net deposition remained much more constant. The acute and complex sensitivity to bed roughness and related levels of eddy diffusivity and viscosity is evident from Table 8.3. This acute sensitivity to bed-roughness and sediment supply leads to concern that migration of new flora and fauna might lead to ‘modal shifts’ with potentially catastrophic consequences. To comprehend these sensitivities, we can approximate, from Prandle (2004), the following dependencies on the friction factor ‘k’:

  • tidal velocity amplitude U ~ k−½,
  • sediment concentration C ~ k ½,
  • tidal sediment flux UC ~ k0,
  • residual sediment flux <UC> ~ UC cos θ ~ k ½,

where θ is the phase lag of tidal elevation relative to currents and residual sediment flux corresponds to net upstream deposition. These theoretical results are consistent with the increases in concentration and residual fluxes for larger values of k shown by the model for both sediment types.

 Table 8.3 Sensitivity of modelled sediments, for Ws = 0.0005 m s -1 ( d = 22 m)

By introducing all sediments at the open boundary, the sensitivity to changes in marine supply, Run (7), is immediately evident. Thus a 50% reduction in supply at the mouth reduced concentrations by nearly a half, capture rates reduced to a factor of 0.6, and deposition to one quarter. Since many estuaries will have marine supplies substantially below the maximum ‘carrying capacity’ assumed here, we anticipate typical capture rates and net sedimentation to be much less than those shown in Table 8.3. However, this sensitivity does highlight the potential for accelerated deposition rates in many estuaries if the marine supply increases (e.g., by dredging disposal or sea-bed disturbance in the offshore approaches). Using the computed patterns of bed ‘sorting’ (i.e., varying axial distributions of deposited sediments), each sediment size can be compared with distributions of surficial sediments to indicate the nature and quantity of the marine source.


A century of bathymetric surveys indicate a net loss of estuarine volume of about 0.1%, or 1 million cubic metres, per year. Similar results are found in many of the large estuaries of NW Europe. In contrast, sea level rise of 1.2 mm a−1 represents only a 0.02% annual increase. This relative stability persists in a highly dynamic regime with suspended sediment concentrations exceeding 2000 mg 1−1 and spring tide fluxes of order 200000 t. Detailed analyses of the bathymetric sequences indicate most significant changes occur in the upper estuary and in inter-tidal zones. A long period, up to 63 years, of tidal elevation records (in the lower estuary) shows almost no changes to the predominant M2 and S2 constituents.

A 3-D Eulerian fine-resolution hydrodynamic model coupled with a Lagrangian, random-walk sediment module was used to show how the dominant fluxes involve fine (silt) sediments on spring tides. Model estimates of net imports of sediments agree with observed ranges for sediments of diameter of approximately 50 mm. and both dredging records and in situ observations indicate that sediments of this kind predominate. The model showed little influence of river flow, saline intrusion or channel deepening on the sediment regime. Conversely, the net fluxes were sensitive to both the bed friction coefficient and the phase lag q of elevation relative to velocity. Upper-bound rates of infill of up to 10 Mt a−1 are indicated by the model, comparable with annual dredging rates of up to 5 Mt. The limited mobility of coarse sediments was contrasted with the near-continuously suspended nature of the finest clay. A sensible match between the net sedimentation rates indicated by the model and the net observed deposition rate was found to occur for silty sand corresponding directly with evidence from dredging records and from direct sampling. While the model indicated sedimentation rates might increase significantly for much finer particles, this is likely to be restricted by the limited availability of such material in the adjacent coastal zone. The present approach can be readily extended to study changes in biological mediation of bottom sediments, impacts of waves, consolidation, and the interactions between mixed sediments.

The main author of this article is Prandle, David
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Citation: Prandle, David (2008): Case studies: Long term predictions for estuaries. Available from [accessed on 23-08-2017]