Evolution and decay of a rotating flow over random topography

Abstract

The evolution and decay of a homogeneous flow over random topography in a rotating system is studied by means of numerical simulations and theoretical considerations. The analysis is based on a quasi-two-dimensional shallow-water approximation, in which the horizontal divergence is explicitly different from zero, and topographic variations are not restricted to be much smaller than the mean depth, as in quasi-geostrophic dynamics. The results are examined by comparing the evolution of a turbulent flow over different random bottom topographies characterized by a specific horizontal scale, or equivalently, a given mean slope. As in two-dimensional turbulence, the energy of the flow is transferred towards larger scales of motion; after some rotation periods, however, the process is halted as the flow pattern becomes aligned along the topographic contours with shallow water to the right. The quasi-steady state reached by the flow is characterized by a nearly linear relationship between potential vorticity and transport function in most parts of the domain, which is justified in terms of minimum-enstrophy arguments. It is found that global energy decays faster for topographies with shorter horizontal length scales due to more effective viscous dissipation. In addition, some comparisons between simulations based on the shallow-water and quasi-geostrophic formulations are carried out. The role of solid boundaries is also examined: it is shown that vorticity production at no-slip walls contributes for a slight disorganization of the flow.

Equilibrium Statistical Mechanics and Energy Partition for the Shallow Water Model

“…The equilibria described by Merryfield et al [20] were obtained in the framework of the energy-enstrophy theory, neglecting any other potential vorticity moments than the potential enstrophy. Similar states were described as minima of the potential enstrophy for the macroscopic potential vorticity field by Sanson [37].…”

Experiments and simulations on self-organization of confined quasi-two-dimensional turbulent flows with discontinuous topography

“…The shallow-water model is adopted here, however, since it is a more appropriate approximation for high steps. A comparison of the two models in the presence of abrupt topography is reported in Zavala Sansón et al 21 The simulations represent a rectangular domain with horizontal dimensions L ϫ ␦L with L = 0.5, and ␦ = 2 being the aspect ratio of the tank. A step-like topography divides the flow domain in two geometric squares with aspect ratio 1.…”