On the baroclinic instability of axisymmetric rotating gravity currents with bottom slope

Abstract

The baroclinic stability characteristics of axisymmetric gravity currents in a rotating system with a sloping bottom are determined. Laboratory studies have shown that a relatively dense fluid released under an ambient fluid in a rotating system will quickly respond to Coriolis effects and settle to a state of geostrophic balance. Here we employ a subinertial two-layer model derived from the shallow-water equations to study the stability characteristics of such a current after the stage at which geostrophy is attained. In the model, the dynamics of the lower layer are geostrophic to leading order, but not quasi-geostrophic, since the height deflections of that layer are not small with respect to its scale height. The upper-layer dynamics are quasi-geostrophic, with the Eulerian velocity field principally driven by baroclinic stretching and a background topographic vorticity gradient.Necessary conditions for instability, a semicircle-like theorem for unstable modes, bounds on the growth rate and phase velocity, and a sufficient condition for the existence of a high-wavenumber cutoff are presented. The linear stability equations are solved exactly for the case where the gravity current initially corresponds to an annulus flow with parabolic height profile with two incroppings, i.e. a coupled front. The dispersion relation for such a current is solved numerically, and the character istics of the unstable modes are described. A distinguishing feature of the spatial structure of the perturbations is that the perturbations to the downslope incropping are preferentially amplified compared to the upslope incropping. Predictions of the model are compared with recent laboratory data, and good agreement is seen in the parameter regime for which the model is valid. Direct numerical simulations of the full model are employed to investigate the nonlinear regime. In the initial stage, the numerical simulations agree closely with the linear stability characteristics. As the in stability develops into the finite-amplitude regime, the perturbations to the downslope incropping continue to preferentially amplify and eventually evolve into downslope propagating plumes. These finally reach the deepest part of the topography, at which point no more potential energy can be released.

Rotating dense currents on a slope. Part 1. Stability

“…The phase speed predicted by Choboter & Swaters (2000) also differs. They predict that the ratio of the phase speed to Nof speed should be approximately unity for a wide range of interaction parameters.…”

“…Thus, the abscissa C Nof /C H is equivalent to the slope parameters used by S91 and Choboter & Swaters (2000). In their asymptotic theory,s must be much smaller than unity.…”

“…Assuming a stationary upper layer, Choboter & Swaters (2000) predict that the flow is unstable for all µ and that N λ increases with µ. For example, they find N λ increases from 5 when µ = 1 to 15 when µ = 10.…”

Baroclinic instability of axially symmetric flow over sloping bathymetry

“…Other authors have attempted to incorporate this theory into eddy parameterizations over continental slopes (Stipa 2004b;Isachsen 2011). Choboter & Swaters (2000) used a non-QG asymptotic derivation to analyze the baroclinic instability of a double-frontal dense water layer over sloping topography in an annulus. Their explicit solutions are for a relatively narrow (1.5 Rossby radii apart at the bottom) coupled front on the bottom of an otherwise stationary fluid, whereas we investigate wider and more horizontally-uniform 2-layer flows here.…”

“…Since we model wider currents, we use the approximation that isopycnals do not intersect the bathymetry, consistent with the QG approximation. In addition, our focus here is deducing the influence of horizontal curvature on the instability, whereas Choboter & Swaters (2000) aimed to compare an existing theory for rectilinear bottom-trapped flow against laboratory experiments in a rotating tank.…”