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