J. Fluid Mech. (1996), uol. 315, pp. 65-84 Copyright 0 1996 Cambridge University Press 65 Stability of a potential vorticity front: from quasi-geostrophy to shallow water By E. BOSS’, N. PALDOR2 AND L. THOMPSON’ School of Oceanography, University of Washington, Seattle, WA 98195-7940, USA ’Department of Atmospheric Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel (Received 12 October 1995 and in revised form 29 December 1995) The linear stability of a simple two-layer shear flow with an upper-layer potential vorticity front overlying a quiescent lower layer is investigated as a function of Rossby number and layer depths. This flow configuration is a generalization of previously studied flows whose results we reinterpret by considering the possible resonant interaction between waves. We find that instabilities previously referred to as ‘ageostrophic’ are a direct extension of quasi-geostrophic instabilities. Two types of instability are discussed : the classic long-wave quasi-geostrophic baroclinic instability arising from an interaction of two vortical waves, and an ageostrophic short-wave baroclinic instability arising from the interaction of a gravity wave and a vortical wave (vortical waves are defined as those that exist due to the presence of a gradient in potential vorticity, e.g. Rossby waves). Both instabilities are observed in oceanic fronts. The long-wave instability has length scale and growth rate similar to those found in the quasi-geostrophic limit, even when the Rossby number of the flow is O(1). We also demonstrate that in layered shallow-water models, as in continuously stratified quasi-geostrophic models, when a layer intersects the top or bottom bound- aries, that layer can sustain vortical waves even though there is no apparent potential vorticity gradient. The potential vorticity gradient needed is provided at the top (or bottom) intersection point, which we interpret as a point that connects a finite layer with a layer of infinitesimal thickness, analogous to a temperature gradient on the boundary in a continuously stratified quasi-geostrophic model. 1. Introduction Observations of frontal instabilities in the ocean (e.g. Watts & Johns 1982; Barth 1994) and laboratory experiments (Griffiths & Linden 1982; Griffiths, Killworth & Stern 1982) have motivated numerous studies of frontal instabilities within the framework of the shallow-water approximation (e.g. Killworth, Paldor & Stern 1984, hereafter referred to as KPS; Paldor & Ghil 1991). In geophysical flows, fronts are identified as regions of rapid changes in the surface density or isopycnal depth. In the ocean, such density changes occur over horizontal distances of the order of the deformation radius, and are associated with jets and even more rapid changes in potential vorticity (PV) (Hall & Fofonoff 1993). We will argue below that a density front can be viewed as a PV front. While the quasi-geostrophic approximation (QG) is not strictly applicable for density fronts because of the 0(1) change in the depth of isopycnal surfaces and the 0(1) Rossby number of the flow