Taylor-Caulfield instabilities in a layered stratified shear flow - eScholarship

Taylor-Caulfield Instabilities in a Layered Stratified Shear Flow Giordano Ponetti and Neil J. Balmforth Dipartimento di Matematica e Informatica, Universit`a di Palermo, Italy giordano.ponetti@dmi.unict. it Department of Mathematics, University of British Columbia, Canada njb@math.ubc. ca Abstract We study the stability of interfacial waves in a stratified shear flow where the density profile takes the form of a staircase of interfaces separating uniform layers, such as may arise from double-diffusive processes in lakes and oceans. Internal gravity waves riding on density interfaces can resonantly interact due to a background shear flow, resulting in the Taylor-Caulfield instability. The many steps of the density profile permit a multitude of interactions between different interfaces, and a rich variety of Taylor-Caulfield instabilities. We provide a linear instability analysis for a staircase with piecewise-constant density profile, allowing for an arbitrary number of interfaces. The staircase is embedded in a background linear shear flow. For long wavelength, weakly nonlinear structures in weakly stratified fluid, we study the onset of instability. Fully nonlinear states are studied numerically in the long-wave and weak stratification limit. Introduction We explore the Taylor-Caulfield instabilities (TCI) that may arise in a stratified fluid with a staircase density profile and background shear flow. The classical Kelvin-Helmholtz instability is the most well-known variety of stratified shear instability and is commonly interpreted at the resonant interaction of waves riding on vorticity interfaces. Holmboe waves (HWI) are the next most studied type of instability, arising when a wave on a vorticity interface resonantly interacts with a gravity wave lying at a density interface Holmboe (1962). KHI and HWI are considered robust instabilities in the sense that they have been observed in natural systems and experiments. Taylor (1931) also pointed out the possibility of a third instability in which waves riding on density interfaces could resonantly interact when their phase speeds become locked due to the Doppler-shifting effect of a background shear flow. This instability has been more elusive to observe, with the scientific community more concerned with why one might find basic states with layered density profiles in the first place. Taylor’s instability was quantified and studied in more detail by Caulfield (1994), in parallel with some first laboratory experiments Caulfield et al. (1995) and nonlinear simulations Lee and Caulfield (2001). Balmforth et al. (2012) provided a comprehensive study of TCI in the long-wave limit that two density interfaces were considered as a stratified “defect” in the shear flow. Here we explore the linear stability and nonlinear evolution of multiple TCI growing on layered density profiles. We consider a density staircase with equally spaced steps of equal size superposed on a background linear shear flow. We study normal mode solutions of the incompressible 2D Navier-Stokes and heat equations under the Boussinesq approximation. Our results show a predominance of near-neighbor interfaces TCI which are particularly strong in the limit of small bulk Richardson numbers and wavenumbers. This limit allows VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016