Beta-plane turbulence above monoscale 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.

A Barotropic Model of Eddy Saturation

“…We note, that the flow characteristics that were described in section 3 are not quirks of the simple sinusoidal topographies (3). The eddy saturation regime, the drag crisis, and multiple equilibria that are present here when geostrophic contours are open, have been all also recently found to exist in this model for a flow above a random monoscale topography with open geostrophic contours (Constantinou and Young 2017) and also above a multi-scale topography that has a k −2 power spectrum (not reported).…”

“…The model formulated in (2) is the simplest model that can be used to investigate beta-plane turbulence above topography driven by a large-scale zonal wind stress applied at the surface of the fluid. It has been used in the past for studying the interaction of zonal flows with topography (Hart 1979;Davey 1980;Holloway 1987;Carnevale and Frederiksen 1987) and recently by Constantinou and Young (2017) for studying the geostrophic flow regimes above random monoscale topography.…”

“…For example, using measurement from the cDrake experiment Donohue et al (2016) estimated that the bottom-velocity component of the ACC transport accounts for about 25% of the total transport. Constantinou and Young (2017) discussed the role of standing eddies using a simple barotropic model forced by an imposed steady wind stress and retarded by a combination of bottom drag and topographic form stress (Hart 1979;Davey 1980;Holloway 1987;Carnevale and Frederiksen 1987). Constantinou and Young (2017) used a random monoscale topography and argued that a critical requirement for eddy saturation is that the ratio of the planetary potential vorticity (PV) to the topographic PV is large enough so that the geostrophic contours (that is the contours of the planetary plus the topographic contribution to PV) are "open" in the zonal direction.…”

Eddy saturation of the Southern Ocean: a baroclinic versus barotropic perspective

“…After the flow equilibrates, we average (for at least 30 years) to obtain the time-mean fields. The results from the barotropic runs (single-layer configuration) are qualitatively similar to those of Constantinou and Young (2017) and Constantinou (2018). For very weak wind stress forcing (τ ≤ 0.02 N m −2 ) the transport increases linearly with wind stress.…”

Eddy Saturation of the Southern Ocean: A Baroclinic Versus Barotropic Perspective

“…One may obtain a lower bound for the volume‐averaged zonal velocity, and by imposing the extra restriction that the potential enstrophy power integral is balanced, one finds that the value of this lower bound increases with wind stress. Thus, for high enough wind stress values a transition to the upper branch must occur (Constantinou & Young, ). The same transition occurs here in baroclinic runs.…”