Instability, evolution, and mixing in stratified shear flow as a function of Richardson number

Instability, Evolution, and Mixing in Stratified Shear Flow as a Function of Richardson Number Joseph Werne (1) and B.A. Pettersson-Reif (2) NorthWest Research Associates, Boulder, CO, USA werne@cora.nwra. com Forsvarets Forskningsinstitutt (FFI), Kjeller, Norway bjorn.reif@ffi.no Abstract High-resolution direct-numerical simulations of the Kelvin-Helmholtz instability reveal flow morphologies and evolutions that depend strongly on the Richardson number Ri. The highest Ri = 0.2 case studied displays flat KH billows that become turbulent immediately upon billow formation, with turbulence appearing in the cores and then migrating laterally as time progresses. In contrast, the lowest Ri = 0.05 case exhibits coherent KH billows with round cross sections and rapid solid-body rotation that stabilizes them and delays the onset of turbulence in the billow cores. This results in a complex sequence of transitions at low Ri that include 1) layered billow formation, 2) secondary instability of billow edges, 3) vigorous turbulence in the braid region between billow cores, and 4) later development of turbulence in the cores. Despite these stark di↵erences, the final flow states exhibit nearly identical mid-layer stability profiles and shear/buoyancy timescale ratios N/S for all Ri, providing useful guidance for layer parameterization. Introduction Stratified shear turbulence influences the transport of momentum, heat, kinetic energy, and particulate matter for a host of important problems, such as the near-surface atmo- spheric and oceanic boundary layers, coupling across the tropopause and above in earth’s atmosphere, and the near-surface and sub-tachocline shear layers in the solar interior. To study the dynamics, evolution, and mixing in such layers, we employ high-resolution direct-numerical simulations (DNS) of the Kelvin-Helmholtz instability for a range of stratifications quantified by the Richardson number Ri. In order to examine equally vig- orous turbulent motions for all Ri studied here, we increase the Reynolds number Re with Ri so that similar length-scale ranges result for all simulations conducted. All simulations are carried out using the NWRA Triple code, which is a pseudo- spectral Boussinesq solver employing the third-order Runge-Kutta time-stepping algo- rithm of Spalart et al. (1991) to advance the coefficients for a spectral representation of laterally periodic Fourier modes and either sine or cosine series in the vertical, depending on the parity of the field variable being represented. An efficient 3D fast-Fourier transform algorithm is used to move between spectral and physical space (Werne et al., 2005). Problem Formulation Hyperbolic-tangent velocity U = U 0 tanh(z/h) and linear temperature T = z profiles are used to initiate shear flow in a 4 ⇥2 ⇥2 Cartesian geometry, where is the wavelength of the most unstable asymptotic linear mode. Here U 0 , h, and are the velocity amplitude, initial half-shear-layer depth, and mean background temperature gradient, respectively, VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016