Two-dimensional turbulence above topography

Abstract

Using a one-layer quasi-geostrophic model, we study the effect of random monoscale topography on forced beta-plane turbulence. The forcing is a uniform steady wind stress that produces both a uniform large-scale zonal flow U (t) and smaller-scale macroturbulence characterized by both standing and transient eddies. The large-scale flow U is retarded by a combination of Ekman drag and the domain-averaged topographic form stress produced by the eddies. The topographic form stress typically balances most of the applied wind stress, while the Ekman drag provides all of the energy dissipation required to balance the wind work. A collection of statistically equilibrated numerical solutions delineates the main flow regimes and the dependence of the time-average of U on parameters such as the planetary vorticity gradient β and the statistical properties of the topography. We obtain asymptotic scaling laws for the strength of the large-scale flow U in the limiting cases of weak and strong forcing. If β is significantly smaller than the topographic PV gradient then the flow consists of stagnant pools attached to pockets of closed geostrophic contours. The stagnant dead zones are bordered by jets and the flow through the domain is concentrated into a narrow channel of open geostrophic contours. In most of the domain the flow is weak and thus the large-scale flow U is an unoccupied mean. If β is comparable to, or larger than, the topographic PV gradient then all geostrophic contours are open and the flow is uniformly distributed throughout the domain. In this open-contour case there is an "eddy saturation" regime in which U is insensitive to large changes in the wind stress. We show that eddy saturation requires strong transient eddies that act effectively as PV diffusion. This PV diffusion does not alter the kinetic energy of the standing eddies, but it does increase the topographic form stress by enhancing the correlation between topographic slope and the standing-eddy pressure field. Using bounds based on the energy and enstrophy power integrals we show that as the strength of the wind stress increases the flow transitions from a regime in which most of the form stress balances the wind stress to a regime in which the form stress is very small and large transport ensues.

Stochastic variability of oceanic flows above topography anomalies

“…[6] In the absence of direct external forcing, eddy fluxes tend to reduce large scale potential vorticity gradients, therefore driving anticyclonic flow above topographic bumps [Bretherton and Haidvogel, 1976]. One expects then that time-mean of the eddy term h eddy is positive (resp.…”

The Mid-Ocean Dynamics Experiment

“…Quite another approach was taken by BRETHERTON and HAIDVOGEL (1976) who observed that quasi two-dimensional flows dissipate enstrophy preferentially over energy. The tendency thus is towards a state of minimum enstrophy for given energy.…”

Crude closure for flow with topography through large-scale statistical theory

“…For a given energy there are in general multiple stationary solutions of (15) for di erent v alues of . Yet only the steady-states that lie on the branch > ; 1 are nonlinearly stable and thus are the most physically relevant moreover, they minimize the potential enstrophy for given energy ( 19]). Carnevale and Frederiksen 18] show that in the limit of in nite resolution the canonical equilibrium is statistically sharp and identical to the nonlinearly stable minimum enstrophy state.…”

“…The emergence of such states as averages over large time intervals in damped and driven numerical simulations has been studied extensively with many interesting results ( 1,19,20,21,22]). Here, in a complementary direction, we devise and test crude closure algorithms based on the energy-enstrophy statistical theory where the entire closure is based on a single predicted evolving parameter, the energy.…”