Ocean Barotropic Vorticity Balances: Theory and Application to Numerical Models

Abstract

The ocean is a highly turbulent fluid with a Reynolds number of planetary flows as high as a hundred billion. Indeed, the viscosity of saline water is so low that it only becomes a leading-order term of the momentum balance at the millimetric scale. Paradoxically, the Earth's rotation exerts such a large constraint on the flow that over scales larger than a few kilometers, it is rigidified and prevented from developing fully three-dimensional turbulence. Peter Rhines has commented that "Study of the effects of planetary rotation has convinced us that the oceans are less than totally fluid" (Rhines, 1986). Such a rigid behavior of large scale ocean circulation can be described by the so-called vertical vorticity balance (e.g., Vallis, 2006). Indeed, the Coriolis force resulting from the Earth's rotation drives an intense circulation about the local vertical axis which must be balanced in a steady state. This circulation is the result of both the variation of the Earth's spin about the local vertical axis with latitude (the so-called beta effect) and the vertical stretching of water masses subject to the Earth rotation (the so-called planetary vortex stretching). The involvement of both meridional and vertical ocean velocities in the torque of the Coriolis force means that the vorticity balance is a diagnostic tool to constrain both the horizontal and vertical flow.

A Scale‐Dependent Analysis of the Barotropic Vorticity Budget in a Global Ocean Simulation

“…The spatial structure and magnitudes of diagnosed bottom pressure torque (Figure B1f) agree well with results from Le Corre et al (2020) (see their Figure 7b), who used a terrain following vertical coordinate C-grid model (which is partially immune to the numerical issues identified by Styles et al (2022) and Waldman and Giordani (2023)). Furthermore, there is a fair consistency between the present results and bottom pressure torque diagnosed using B-grid model outputs (Hughes & De Cuevas, 2001;Yeager, 2015), which also do not suffer from numerical issues present in C-grid models (Styles et al, 2022).…”

“…We analyze the vorticity budget based on the depth‐integrated Boussinesq‐hydrostatic ocean primitive equations. Several studies have employed this vorticity budget approach to examine the role of surface wind stress, bottom pressure, and ocean eddies in governing the flow dynamics (e.g., Hughes & De Cuevas, 2001; Le Corre et al., 2020; Yeager, 2015), see Waldman and Giordani (2023) for a recent review. The complete vorticity budget of the depth‐integrated flow can be written as (see Appendix for derivation) $$$\beta \,V=\frac{J\left({p}_{\,\mathrm{b}},H\right)}{{\rho }_{o}}+\widehat{\boldsymbol{z}}\cdot \left(\frac{\nabla \wedge {\boldsymbol{\tau }}_{\,\mathrm{s}}}{{\rho }_{o}}-\frac{\nabla \wedge {\boldsymbol{\tau }}_{\,\mathrm{b}}}{{\rho }_{o}}+\nabla \wedge \mathcal{A}+\nabla \wedge \mathcal{B}\right)-f\frac{{Q}_{m}}{{\rho }_{o}}+f\,{\partial }_{t}\eta -\widehat{\boldsymbol{z}}\cdot \left(\nabla \wedge {\mathcal{U}}_{t}\right),$$$ where β = ∂ y f is the meridional derivative of the Coriolis parameter, V is the vertically integrated meridional velocity, z = η is the ocean free surface height, z = − H is ocean bottom, p b is bottom pressure, $$\nabla =\widehat{\boldsymbol{x}}\,{\partial }_{x}+\widehat{\boldsymbol{y}}\,{\partial }_{y}$$ is the horizontal gradient operator, and ρ o = 1,035 kg m −3 is the Boussinesq reference density.…”

“…Several studies have employed this vorticity budget approach to examine the role of surface wind stress, bottom pressure, and ocean eddies in governing the flow dynamics (e.g., Hughes & De Cuevas, 2001;Le Corre et al, 2020;Yeager, 2015), see Waldman and Giordani (2023) for a recent review. The complete vorticity budget of the depth-integrated flow can be written as (see Appendix A for derivation)…”

“…MOM6 is discretized using a C-grid and employs a vertical Lagrangian-remap method on a hybrid z*-isopycnal vertical coordinate to simulate the ocean state (Adcroft et al, 2019;Griffies et al, 2020). Hence, bottom pressure torque diagnosed in MOM6 is expected to suffer from these spurious forces (Styles et al, 2022;Waldman & Giordani, 2023). To diagnose physically relevant signals in bottom pressure torque, we need to account for these numerical errors.…”

Influence of topography and winds on the distribution of water masses on the Antarctic Continental Shelf

“…Sverdrup balance); in the return flow boundary layer, the advection of planetary vorticity is balanced by the curl of bottom friction (e.g. Waldman & Giordani 2023). As in the throughflow case, the top Ekman transport is prescribed, this sets off an identical top Ekman pumping and suction pattern where the resulting circulation is now constrained by walls on all sides.…”