Can We Study the Linear Instability of Tidal Sand Waves Using z – Coordinate on a Rectangular Domain? Theoretically, No

Abstract:

In many analytical works describing the linear stability of sand ripples or sand waves, a z coordinate defined on a flat-bottomed “box” domain is utilised. In this framework, the boundary forcing enters the equation transferring the flat-bottom boundary condition onto the bedform’s surface using a Taylor expansion. This technique is similar to the “linear extrapolation” used in studying linear gravity waves and suffers the same problem of violating mass conservation (Couche and Conte, 1997). Furthermore, the z-coordinate does not correctly represent the hydrodynamic response to the local change in the due to the presence of the bedform ( are the water depth and the thickness of the tidal, oscillatory Stokes layer). The change in is driven by two factors, (i) local water depth itself and (ii) the eddy viscosity’s dependence on depth. Using a simplified analytical problem for the limiting case with ultra-long sand wave, it can be analytically shown that mass conservation and the factor (i) can only be properly modelled in a terrain-following coordinate defined on a wavy-bottomed domain. The full set of equations are derived for the – coordinate and employed in the study of the linear stability of sand waves. A detailed analysis of the equations reveals that the mechanisms responsible for the spurious excitation of ultra-long sand waves are directly related to mass conservation, while also influenced by the strength of the tide and the grain size. Despite being consistent with previous studies indicating that the eddy viscosity’s dependence on depth is crucial for inhibiting the ultra-long sand waves, we find that the depth-dependence might take the form of with so that the modification of by local depth change can be counteracted ( are velocity scale and depth scales, is a small-amplitude bed form). The physical significance of may be related to the fact that turbulence shear production is intensified to a greater extent on the leeside and in the troughs of the bed form than it is reduced at crests. Further discussion of will be provided.