Finite-amplitude baroclinic instability of a mesoscale gravity current in a channel

Abstract

A finite amplitude theory is developed for the evolution of marginally unstable modes for a mesoscale gravity current on a sloping bottom. The theory is based on a nonquasigeostrophic, baroclinic model of the convective destabilization of gravity currents which allows for large amplitude isopycnal deflections while filtering out harotropic instabilities. Two calculations are presented. First, a purely temporal amplitude equation is derived for marginally unstable modes not located at the minimum of the marginal stability curve. These modes eventually equilibrate with a new finite amplitude periodic solution formed.Second, the evolution of a packet of marginally unstable modes located at the minimum of the marginal stability curve is presented. These two models are dramatically different due to fundamental physical differences. For marginally unstable modes not located at the minimum of the marginal stability curve, it is possible to determine the evolution of a single normal mode amplitude. For the marginally unstable mode located at the minimum of the marginal stability curve the entire gravity current forms a nonlinear critical layer leading to an infinity of coupled amplitude equations. If this system is truncated, on an ad koc basis, to include only the fundamental harmonic and its accompanying mean flow, there exists a steadily-travelling solitary cold-core eddy solution.

Numerical simulations of the baroclinic dynamics of density‐driven coupled fronts and eddies on a sloping bottom

“…A partial answer to this question was provided by Mooney and Swaters [1996], who developed a wave-packet model for marginally unstable modes of unsheared gravity current solutions of the model. Unfortunately, since the steady current was unsheared in the transverse (i.e., cross-slope) direction, at the point of marginal instability the entire flow forms a nonlinear critical layer (similar to what happens in the Phillip's model of baroclinic instability; see, e.g., Warn and Gauthier [1989]), and thus determining the evolution of the marginally unstable mode required solving an infinite number of coupled amplitude equations.…”

“…Unfortunately, since the steady current was unsheared in the transverse (i.e., cross-slope) direction, at the point of marginal instability the entire flow forms a nonlinear critical layer (similar to what happens in the Phillip's model of baroclinic instability; see, e.g., Warn and Gauthier [1989]), and thus determining the evolution of the marginally unstable mode required solving an infinite number of coupled amplitude equations. Mooney and Swaters [1996] were unable to find an isolated eddy-like solution to these equations, except under the ad hoc approximation of simply neglecting all harmonics save the fundamental and the principal mean flow it generates. In this case the governing amplitude equations can be transformed into a sine-Gordon equation [see, e.g., Gibbon et al, 1979], which has a solitary wave solution which can be interpreted as a steadily traveling cold dome.…”

“…This asymmetry is an unique signature of the baroclinic destabilization of these currents. From a modal point of view the instability may be thought of as the coalescence of two topographic vorticity waves [Mooney and Swaters, 1996;Boss et al, 1996] which have been excited in the upper layer. The Swaters [1991] model was also able to reproduce a dipole-like instability similar to that observed by GKS.…”

Stability of meridionally-flowing grounded abyssal currents in the ocean

“…, which is the f -plane result of Mooney and Swaters [9]. Based on the nondimensionalizations introduced, it is possible to characterize the most unstable mode as having an along slope wavelength on the order of 94 km, an equatorward phase velocity on the order of 3 cm/s, a modal period on the order of 38 days and an e-folding amplification time on the order of 6 days (Swaters [3]).…”

Meridional flow of grounded abyssal currents on a sloping bottom in spherical geometry

“…These currents form an important component in the deep "leg" of the meridional overturning circulation in the oceans. The mesoscale dynamics of these currents has been described in a series of papers [1][2][3][4][5][6][7][8][9][10][11]. All of these studies have implicitly assumed either an f or β-plane approximation in which the dynamics is modelled in a Cartesian coordinate system with the implicit assumption that the horizontal length scales are not too much larger than the internal deformation radius (on the order of about 10-100 km in the ocean).…”