The Role of Curvature in Modifying Frontal Instabilities -- Short
Summary

In this study, we revisit the role of curvature in modifying frontal
stability. We first consider the statement “fq < 0 implies
potential for instability”, where f is the Coriolis parameter and q is
the Ertel potential vorticity (PV). This is true for any inviscid
baroclinic flow. It is also evident in the transition of a governing
equation for circulation within a front from elliptic to hyperbolic form
as the discriminant changes sign. However, for curved fronts, an
additional scale factor enters the discriminant owing to conservation of
absolute angular momentum, L, leading to Solberg’s (1936) generalization
of the Rayleigh criterion. In non-dimensional form, this expression also
generalizes the classical instability criterion of Hoskins (1974) by
accounting for centrifugal forces: modification of the front’s vertical
shear and stratification owing to curvature tilts the absolute vorticity
vector away from its thermal wind state and, in an effort to conserve
the product of non-dimensional PV (q’) and absolute angular momentum
(L’), this alters Rossby and Richardson numbers permitted for stable
flow. The criterion, Φ’=L’q’ < 0, is then investigated in
non-dimensional parameter space representative of low-Richardson-number
vortices. An interesting outcome is that, for Richardson numbers near
one, anticyclonic flows increase in q’, while cyclonic flows decrease in
q’. Though stabilization is muted for anticyclones (owing to
multiplication by L’), the de-stabilization of cyclones is robust, and
may help to explain an observed asymmetry in the distribution of
submesoscale coherent vortices in the global ocean.