Author:
CHOBOTER PAUL F.,SWATERS GORDON E.
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 characteristics
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
instability 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.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
13 articles.
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