Geoscience Reference
In-Depth Information
6
Instabilities of Shallow-Water Flows with Vertical Shear
in the Rotating Annulus
Jonathan Gula 1 and Vladimir Zeitlin 2
6.1. INTRODUCTION
shallow water may account for experimental observations.
In the outcropping configuration it was instructive to see
to what extent the simple two-layer shallow water-theory
reproduces the experiment.
The experiments mentioned above are not strictly
speaking shallow-water ones, although no pronounced
vertical structure was observed, as to our knowledge. The
results we present below may serve, nevertheless, to under-
stand the vertically averaged behavior of the full system.
Moreover, Williams et al. [2005] interpreted their exper-
iments in terms of shallow-water dynamics, referring to
Ford [1994]. As to the density currents, their instabilities
are traditionally studied with shallow-water models, as in
the classical paper by Griffiths et al. [1982]. Being stan-
dard in GFD, the two-layer shallow-water approximation
is a reasonable compromise between the realistic represen-
tation of the observed fluid flow and the computational
effort (and amount of resources) necessary for a full stabil-
ity analysis. It is, in addition, self-consistent and universal,
as, for example, the fine vertical structure of the flow may
vary from one experiment to another.
In Section 6.2 we present our results for the superrotat-
ing rigid-lid configurations (following Gula et al. [2009b],
where most of them were published). In Section 6.3 we
give new results for the free surface coniguration with out-
cropping, and in Section 6.4 we analyze the influence of
bathymetry on the instabilities.
There is a long tradition of experiments in differen-
tially rotating annuli in order to understand the baroclinic
instability and, more generally, the instabilities of fronts
in geophysical fluid dynamics (GFD) [ Hide , 1958; Fultz
et al. , 1959; Hide and Fowlis , 1965; Hart , 1972]. Recently
the interest in such experiments was revived in the context
of the so-called spontaneous emission of inertia-gravity
waves by balanced flows (see Ford [1994], O'Sullivan and
Dunkerton [1995], and the references in the special collec-
tion of Journal of Atmospheric Sciences on this subject,
Dunkerton et al. [2008]). Thus, short-wave patterns cou-
pled to the baroclinic Rossby waves were observed in
independent experiments [ Lovegrove et al. , 2000; Williams
et al. , 2005; Flór , 2007; Flór et al. , 2011] on instabilities of
the two-layer rotating flows in the annulus at high enough
Rossby numbers.
On the other hand, the classical experiments on unstable
density (coastal) currents by Griffiths and Linden [1982]
also used annular geometry and a two-layer setting, with
lighter fluid overflowing the denser one in the rotating
tank. Recently, similar experiments, but with sloping bot-
tom, were performed by Pennel et al. [2012].
Motivated by all these experiments, we undertook a
thorough stability analysis of a two-layer shallow-water
system in the rotating annulus both with the rigid lid
and with a free surface and outcropping interface. Our
main goal in the rigid-lid configuration was to check to
what extent the ageostrophic short-wave instabilities in
6.2. STABILITY OF FRONTS UNDER RIGID LID
1 Institute for Geophysics and Planetary Physics, University of
California, Los Angeles, California, United States of America.
2 Laboratoire de Météorologie Dynamique, École Normale
Supérieure; and Université Pierre et Marie Curie, Paris,
France.
A typical configuration used in laboratory experiments
by Williams et al. [2005] and Flór et al. [2011] is presented
in Figure 6.1. The annulus has an inner vertical sidewall
of radius r 1 , an outer vertical sidewall of radius r 2 , and a
 
 
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