The role of curvature in modifying frontal instabilities. Part I: Review of theory and presentation of a nondimensional instability criterion

In this study, we examine the role of curvature in modifying frontal stability. We first evaluate the classical criterion that the Coriolis parameter, f, multiplied by the Ertel potential vorticity (PV), q, is positive for stable flow and that instability is possible when this quantity is negative. The first portion of this statement can be deduced from Ertel’s PV theorem, assuming an initially positive fq. Moreover, the full statement is implicit in the governing equation for the mean flow, as the discriminant, fq, changes sign. However, for curved fronts in cyclo-geostrophic or gradient wind balance (GWB), an additional term enters the discriminant owing to conservation of absolute angular momentum, L. The resulting expression, Lq < 0, simultaneously generalizes Rayleigh’s (1917) criterion by accounting for baroclinicity and Hoskins’ (1974) criterion by accounting for centrifugal effects. In particular, changes in the front’s vertical shear and stratification owing to curvature tilt the absolute vorticity vector away from its thermal wind state; in an effort to conserve the product of absolute angular momentum and Ertel PV, this modifies gradient Rossby and Richardson numbers permitted for stable flow. This forms the basis of a non-dimensional expression valid for inviscid, curved fronts on the f -plane, which can be used to classify frontal instabilities. In conclusion, the classical criterion, fq < 0, should be replaced by the more general criterion, Lq < 0, for studies involving gravitational, centrifugal, and symmetric instabilities at curved density fronts. In Part 2 of the study, we examine interesting outcomes of the criterion applied to low-Richardson number fronts and vortices in GWB.