Instability associated baroclinic critical layers in rotating stratified shear flow

Instability Associated Baroclinic Critical Layers in Rotating Stratified Shear Flow Chen Wang and Neil Balmforth Department of Mathematics, University of British Columbia chenwang@math.ubc. ca Abstract The notion of over-reflection has been used to rationalize the linear instability of shear flows that are directed and sheared in a horizontal plane but stratified and subject to rotation in the vertical. In the linear stability analysis of these types of flows, two types of critical levels may appear that translate to singular points in the corresponding differ- ential eigenvalue problem. The first type corresponds to the classical kind of critical level encountered for unstratified flows whereat the phase speed of a wave matches the local flow speed. The second type are more novel, arising where the D¨oppler-shifted wavespeed matches the natural gravity wavespeed. Such “baroclinic” critical levels are chacterized by unusual properties of reflection and transmission; we explore how this impacts linear over-reflectional instability using a combination of WKB theory and numerical solution of the linear eigenvalue problem. In particular, we examine how Kelvin waves riding on the wall of a channel become unstably coupled to gravity waves propagating on the other side of a baroclinic critical level. Introduction Stratified shear instability plays an important role in a number of contexts from geophysics and astrophysics. The particular variety we consider here is the instability associated with a base flow that directed and sheared in the horizontal plane, but stratified and subject to rotation in the vertical direction. This type of instability has been explored by, for example, Le Diz`es and Billant (2009) in cylindrical geometry and Vanneste and Yavneh (2007) for a local Cartesian tangent plane. The basic mechanism for instability is identified as due to the over-reflection of waves with different wave action (or energy, pseudo-momentum). Importantly, these works consider specific configurations for the basic flow that eliminates certain singular critical levels from the linear stability analysis; our purpose here is to reconsider how such levels can affect the stability problem. The classical type of critical levels in unstratified shear flow are associated with the positions where the phase speed of a wave c matches the local flow speed U (y), where, for a Cartesian system, the background flow is directed in x and sheared in y. In linear stability theory, these levels translate to singular points of the differential eigenvalue problem unless the vorticity gradient of the background flow vanishes there (U 00 (y) = 0). For the stratified shear flow problem considered here, the linear stability analysis develops two further potential singular points at the locations where the D¨oppler-shifted phase speed of the wave, c − U (y), matches the natural phase speed of gravity waves, ±N/k, with N the buoyancy frequency and k the streamwise wavenumber; i.e. (c−U ) 2 = N 2 /k 2 . In Vanneste and Yavneh (2007) and Le Diz`es and Billant (2009), these “baroclinic critical levels” are removed with a suitable selection of the basic state parameters. We avoid this selection here, but for simplicity remove the more traditional critical level by considering flow without a background vorticity gradient. VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016