|Mixing and flushing of tidal embayments in the western Dutch Wadden Sea, Part II: Analysis of mixing processes|
Zimmerman, J.T.F. (1976). Mixing and flushing of tidal embayments in the western Dutch Wadden Sea, Part II: Analysis of mixing processes. Neth. J. Sea Res. 10(4): 397-439
In: Netherlands Journal of Sea Research. Netherlands Institute for Sea Research (NIOZ): Groningen; Den Burg. ISSN 0077-7579, more
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A theory is developed describing the diffusive effect of a two-dimensional quasi-random Eulerian time-independent velocity field superimposed upon a tidal current. It is shown that the superposition gives rise to Lagrangian motions which are partly random functions of time. The application of TAYLOR'S (1921) classical theory of single particle diffusion appears to be an appropriate means for the treatment of diffusion caused by these random motions. The theory suggests the introduction of the concept of "tidal random walk" by which is meant that part of the displacement over half a tidal period which arises from the presence of an erratic residual current field. It is taken into account that the random displacements during successive ebb and flood periods are correlated. The theory leads to diffusion coefficients for the longitudinal direction (the direction of the tidal current) and the lateral direction (perpendicular to it). It is shown that in the case of an isotropic random residual current field the dispersion is anisotropic, being largest in the direction of the tidal current. If the time-independent residual velocity field is assumed to consist of a random superposition of characteristic eddies of a given strength and with given dimensions, the diffusion coefficients can be alternatively expressed in the form of: K = g(λ2)ζ(υ,λ) or of K = g(λ2) ζ(υ,λ) where Uo and lo are the amplitudies of the tidal velocity and the tidal displacement, is the mean-square velocity of the residual current field and ζ is the Lagrangian integral time scale of the random part of the Lagrangian motion; the dimensionless parameters υ and λ denote the ratio of the kinetic energies of the residual current field and of the tidal current and the ratio of the amplitude of tidal displacement and the Eulerian integral length scale of the residual current field, respectively. The former expression is similar to the one introduced by ARONS & STOMMEL (1951) with, however, a definite physical interpretation of the factor of proportionality. The latter expression is of a form generally used in turbulence theory. The theory is applied to the western Dutch Wadden Sea. By means of current velocity measurements in this area the relevant parameters of the characteristic residual current vortices have been estimated. It is shown that υ and λ are of the order of 2.10-2 and 2.6, respectively; this gives, for longitudinal diffusion, a value óf b(υ,λ) of the order of 0.16. The Lagrangian integral time scale appears to be about 3.5 times the tidal period, whereas the step length of tidal random walk has values of 1 to 2 km for half a tidal period. Longitudinal diffusion due to lateral oscillatory current shear is shown to be of minor importance. The longitudinal diffusion coefficients obtained from the theory are of the same order of magnitude as those derived empirically from the salinity distribution.