# American Institute of Mathematical Sciences

March  2009, 2(1): 215-229. doi: 10.3934/krm.2009.2.215

## Three-dimensional instabilities in non-parallel shear stratified flows

 1 Department of Mathematics and Statistics, Center for Environmental Fluid Dynamics, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-1804, United States, United States 2 N/A, United States

Received  November 2008 Revised  November 2008 Published  January 2009

The instabilities of non-parallel flows ($\overline{U}(x_3)$, $\overline{V}(x_3), 0)$ ($\overline{V} \ne 0$) such as those induced by polarized inertia-gravity waves embedded in a stably stratified environment are analyzed in the context of the 3D Euler-Boussinesq equations. We derive a sufficient condition for shear stability and a necessary condition for instability in the case of non-parallel velocity fields. Three dimensional numerical simulations of the full nonlinear equations are conducted to characterize the respective modes of instability, their topology and dynamics, and subsequent breakdown into turbulence. We describe fully three-dimensional instability mechanisms, and study spectral properties of the most unstable modes. Our stability/instability criteria generalizes that in the case of parallel shear flows ($\bar{V}=0$), where stability properties are governed by the Taylor-Goldstein equations previously studied in the literature. Unlike the case of parallel flows, the polarized horizontal velocity vector rotating with respect to the vertical coordinate ($x_3$) excites unstable modes that have different spectral properties depending on the orientation of the velocity vector. At each vertical level, the horizontal wave vector of the fastest growing mode is parallel to the local vector ($d\overline{U}(x_3)/dx_3$, $d \overline{V}(x_3)/dx_3)$. We investigate three-dimensional characteristics of the unstable modes and present computational results on Lagrangian particle dynamics.
Citation: Alex Mahalov, Mohamed Moustaoui, Basil Nicolaenko. Three-dimensional instabilities in non-parallel shear stratified flows. Kinetic & Related Models, 2009, 2 (1) : 215-229. doi: 10.3934/krm.2009.2.215
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