Stability of a potential vorticity front: from quasi-geostrophy to shallow water

Abstract

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 boundaries, 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.

Saturn’s Northern Hemisphere Ribbon: Simulations and Comparison with the Meandering Gulf Stream

“…However, as both thin jet and contour dynamic models do not allow homogenization of PV, they do not allow for a complete study of the development of the instability's nonlinear saturation. There have also been many linear and quasi-linear stability analyses relevant to meandering jets (e.g., Meacham and Flierl 1991;Boss et al 1996;Paldor and Ghil 1997;Boss and Thompson 1999), which are useful for examining the physical mechanisms of meandering instabilities but nevertheless not applicable to nonlinear growth and saturation.…”

Shelf Impact on Buoyant Coastal Current Instabilities

“…When the coastal current tends to be surface advected (i.e., d goes to low values), the values of the growth rates of baroclinic instabilities decrease and the most unstable wavelength increases. The two-layer shallow-water model (Boss et al 1996;Sakai 1989;Gula and Zeitlin 2010b) was then used to take into account ageostrophic instabilities. Unlike the intermediate models (i.e., quasigeostrophic or frontalgeostrophic models ;Swaters 1993;Reszka and Swaters 1999), the shallow-water model takes into account finite Rossby numbers and fast wave motions.…”

“…However, these ageostrophic instabilities have large growth rates only for finite Rossby (or Froude) numbers (Sakai 1989;Gula and Zeitlin 2010b) and they are generally neglected for small Rossby number flows. The validity of quasigeostrophic models to describe unstable modes of outcropping fronts having large isopycnal deviation was studied by Boss et al (1996). Their linear stability analysis shows that the spatial structure of the frontal modes induced by the outcropping front differs from standard Rossby modes but does not change the characteristics of the low wavenumber instability.…”

Instability of a Front Formed by Two Layers with Uniform Potential Vorticity

“…Although a Kelvin mode behaves like a gravity wave for a large wavenumber, it has a feature of a Rossby wave when the wavenumber becomes small (Iga 1993(Iga , 1995. The remaining Rossby waves in such systems are also indicated from another standpoint by Boss et al (1996), who showed a outcrop of frontal surface (the same as closed boundary in Iga 1995) plays a part of Sfunction-like potential vorticity gradient and makes a Rossby wave even if potential vorticity is uniform in the interior. According to Sakai (1989), baroclinic instability is identified as an instability caused by resonance between two Rossby waves placed vertically.…”