An eddy closure for potential vorticity

Abstract

It is now over 40 years since a closure for the effects of mesoscale eddies in terms of Ertel potential vorticity was first proposed. The consequences of the closure that treats potential vorticity exactly the same as a passive tracer in isopycnal coordinates are explored in this paper. This leads to a momentum equation to predict the mean velocity. While the momentum equation is not unique due to the presence of an undefined potential function, the total energy equation is used to constrain its functional form. The inviscid form of the proposed eddy closure nearly conserves total energy; the error in conservation of total energy is proportional to the time derivative of the bolus velocity. The proposed eddy closure retains Kelvin’s circulation theorem with mean potential vorticity conserved along particle trajectories following the transport (mean + bolus) velocity field. The relative vorticity component of the potential vorticity being diffused along isopycnals leads to terms that look like viscous stress, but these terms do not satisfy two important conditions of standard viscous closures. A numerical model based on this closure is developed, and idealized simulations in a re-entrant zonal channel are conducted to evaluate the merit of the proposed closure. When comparing various eddy closures to an eddy-resolving reference solution, the closure that both transports and diffuses potential vorticity performs marginally better than its peers, particularly with respect to the core zonal jet structure and position. However, these favorable results are obtained only if a potential vorticity diffusion coefficient is used that is smaller than the coefficient used to compute the bolus velocity. Based on these results, we conjecture that extending eddy-closures to include potential vorticity dynamics is possible, but will require the use of a closure parameter that varies temporally and spatially.