Nonlinear instability of elementary stratified flows at large Richardson number

Elementary stably stratified flows with linear instability at all large Ric hardson numbers have been introduced recently by the authors [J. Fluid Mech . 376, 319-350 (1998)]. These elementary stratified flows have spatially co nstant but time varying gradients for velocity and density. Here the nonlin ear stability of such flows in two space dimensions is studied through a co mbination of numerical simulations and theory. The elementary flows that ar e linearly unstable at large Richardson numbers are purely vortical flows; here it is established that from random initial data, linearized instabilit y spontaneously generates local shears on buoyancy time scales near a speci fic angle of inclination that nonlinearly saturates into localized regions of strong mixing with density overturning resembling Kelvin-Helmholtz insta bility. It is also established here that the phase of these unstable waves does not satisfy the dispersion relation of linear gravity waves. The vorti cal flows are one family of stably stratified flows with uniform shear laye rs at the other extreme and elementary stably stratified flows with a mixtu re of vorticity and strain exhibiting behavior between these two extremes. The concept of effective shear is introduced for these general elementary f lows; for each large Richardson number there is a critical effective shear with strong nonlinear instability, density overturning, and mixing for elem entary flows with effective shear below this critical value. The analysis i s facilitated by rewriting the equations for nonlinear perturbations in vor ticity-stream form in a mean Lagrangian reference frame. (C) 2000 American Institute of Physics. [S1054-1500(00)00801-6].