Geophysical Research Letters
Can Large Oceanic Vortices Be Stable?
E. S. Benilov
Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
Observations show that radii of oceanic eddies often exceed the Rossby radius of deformation,
whereas theoretical studies suggest that such vortices should be unstable. The present paper resolves this
paradox by presenting a wide class of large geostrophic vortices with a sign-deﬁnite gradient of potential
vorticity (which makes them stable), in an ocean where the density gradient is mostly conﬁned to a thin
near-surface layer (which is indeed the case in the real ocean). The condition of a thin “active” layer is what
makes the present work diﬀerent from the previous theoretical studies and is of utmost importance. It turns
out that without it, the joint requirement that a vortex be large and have a sign-deﬁnite potential vorticity
gradient trivializes the problem by eliminating all vortices except nearly barotropic ones.
Mesoscale eddies play an important role in oceanic circulation, as they transport signiﬁcant amounts of water
a long way across the ocean. Satellite imagery shows that eddies exist in a highly variable ﬁeld, and yet they
retain their characteristics over distances of thousands kilometers.
The paradox associated with oceanic eddies was identiﬁed more than 30 years ago: observations show that
mesoscale eddies exist for years (Chelton et al., 2011; Lai & Richardson, 1977), whereas theoretical studies
suggest that they are unstable and should disintegrate within weeks. Stable vortices with typical oceanic
parameters have been found only in ﬂuids that are unbounded (Dritschel, 1988; Dritschel et al., 2005; Tsang &
Dritschel, 2014) or layered (Benilov, 2004; Benilov & Flanagan, 2008; Dewar & Killworth, 1995; Katsman et al.,
2003), whereas more realistic models—with rigid/free boundaries and continuous stratiﬁcation—have all
indicated instability (e.g., Mahdinia et al., 2017; Nguyen et al., 2012; Yim et al., 2016, and references therein).
The mechanism of the instability is as follows: since the radii of eddies may exceed the Rossby radius of defor-
mation by a factor of 2–4 (Chelton et al., 2011; Olson, 1991), the horizontal shear in the vortex is too weak
to suppress the baroclinic instability caused by the vertical shear (the role of horizontal shear in supressing
baroclinic instability was discussed by Benilov, 2003, see p. 320, paragraph 2).
One cannot help but wondering, however, why the paradox cannot be resolved by extending the quasi-
geostrophic (QG) criterion of Dritschel (1988) to bounded ﬂuids and ﬁnding (stable) vortices that satisfy it.
This appears to be a straightforward path, but no one has followed it to the end. There have been only four
attempts to exploit this idea. The theoretical results of Sutyrin (1989), however, have never been tested for
the “real” ocean, whereas the eddies examined by Sutyrin and Radko (2016) and Radko and Sisti (2017) were
relatively small (in both cases, the radius of the maximum swirl velocity was 21 km—i.e., comparable to, or
smaller than, typical values of
). Finally, Benilov (2017) examined eddies using an asymptotic model which
neglects (potentially unstable) short-wave disturbances.
To understand why the stability criterion of Dritschel (1988) has been underused, note that it requires that
the radial gradient of potential vorticity (PV) be sign deﬁnite. Then, as shown below, large vortices with
sign-deﬁnite PV gradient cannot generally have a sizable vertical shear, that is, they are nearly barotropic—
whereas mesoscale oceanic eddies are strongly baroclinic.
There seems to be only one exception to this rule: the large size, sign-deﬁnite PV gradient, and baroclinicity
can coexist in the same vortex —but only if the ocean’s density stratiﬁcation is conﬁned to a thin near-surface
layer (sometimes referred to as “active”).
To express the above claim in mathematical terms, introduce the radius
of the vortex, the Rossby radius
the Burger number
, and the active-to-passive depth ratio
. Then, if
(the large-vortex limit),
• A wide class of stable quasigeostropic
vortices in a continuously stratiﬁed
ocean is presented
• The stable vortices explain the
observed longevity of oceanic
E. S. Benilov,
Benilov, E. S. (2018). Can large
oceanic vortices be stable?
Letters, 45, 1948–1954.
Received 27 DEC 2017
Accepted 12 FEB 2018
Accepted article online 16 FEB 2018
Published online 28 FEB 2018
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