Anisotropy of emergent large-scale dynamics in forced stratified shear flows

We report the results of a series of 3D direct numerical simulations of forced stratified shear flows in various computational domains using spectral element methods . We force the flow to relax back towards vertical hyperbolic tangent profiles of streamwise velocity and buoyancy, with characteristic half-depth d₀, half-velocity jump U₀, and half-buoyancy jump b₀, with a relaxation time t_r = 100τ where τ=d₀/U₀. We consider computational domains of vertical extent L_z=48 (scaled with d₀) with a range of (scaled) horizontal extents 16 < L_h< 512.

We simulate a fluid with Prandtl number Pr =ν/κ=1, where ν is the kinematic viscosity and κ is the buoyancy diffusivity, and set the initial bulk Reynolds number Re =U₀ d₀/ν=50 and initial bulk Richardson number Ri₀=b₀d₀/(U₀)^2=1/80 (corresponding to the initial minimum gradient Richardson number). At these parameters, the flow is initially (vigorously) unstable to a primary Kelvin-Helmholtz instability (KHI). We simulate the continuously forced flow over about 5000τ , and investigate the dynamically emergent length scales and turbulence properties of the statistically stationary flow, in particular the local turbulent flux coefficient Γ, as well as streamwise and spanwise Fourier spectra. We find that the shear layer half depth deepens markedly and converges to d~8 (i.e., Λ_z=16) with markedly increased minimum gradient Richardson number (~0.1) and Re ~ 400 with associated convergent vertical mixing properties only for L_h ~ 96 or larger. However, emergent dominant yet large-scale spanwise or streamwise flow structures appear to extend up to (scaled) L_y ~ 50 or even L_x ~ 115 and thus may be fully resolved for horizontal domains of extent L_h=256 or larger. Our observations demonstrate the marked anisotropy of characteristic emergent, significantly extended length scales, even for such apparently `weakly stratified' forced shear flows, and are consistent with the possibility that an `imprint' of the primary linear instability even continues to survive in such vigorously turbulent flows, as the streamwise flow structures have a scale consistent with the most unstable KHI for the converged deepened turbulent shear layer.