Stratified shear flow instabilities at large Richardson numbers

Abstract

Numerical simulations of stratified shear flow instabilities are performed in two dimensions in the Boussinesq limit. The density variation length scale is chosen to be four times smaller than the velocity variation length scale so that Holmboe or Kelvin-Helmholtz unstable modes are present depending on the choice of the global Richardson number Ri. Three different values of Ri were examined Ri = 0.2, 2, 20. The flows for the three examined values are all unstable due to different modes namely: the Kelvin-Helmholtz mode for Ri = 0.2, the first Holmboe mode for Ri = 2, and the second Holmboe mode for Ri = 20 that has been discovered recently and it is the first time that it is examined in the non-linear stage. It is found that the amplitude of the velocity perturbation of the second Holmboe mode at the non-linear stage is smaller but comparable to first Holmboe mode. The increase of the potential energy however due to the second Holmboe modes is greater than that of the first mode. The Kelvin-Helmholtz mode is larger by two orders of magnitude in kinetic energy than the Holmboe modes and about ten times larger in potential energy than the Holmboe modes. The results in this paper suggest that although mixing is suppressed at large Richardson numbers it is not negligible, and turbulent mixing processes in strongly stratified environments can not be excluded.

Stratified shear flow instabilities in the non-Boussinesq regime

“…We expect an analogous schematic to the one shown in Figure 8 should hold for the Holmboe problem. Similar effects should also be observed when smooth basic states [30][31][32][33][34][35] are considered in the non-Boussinesq regime. In terms of general applicability, since large-scale stratified flows tend to be dominated by buoyancy effects, non-Boussinesq effects are more likely to manifest for small-scale flows.…”

Optimal excitation of two dimensional Holmboe instabilities

“…We have checked by doubling the channel size that the eigenspectrum and the perturbation transient growth are converged. Similar insensitivity to the location of the channel walls has been previously reported by Alexakis [18]. The perturbation equations can be written compactly by introducing a streamfunction ψ so that the velocities are u = ∂ z ψ and w = −∂ x ψ, and the vertical displacement, defined through the relation:…”

“…Hazel and Smyth and Peltier had analyzed the character of the bifurcations that occur in the instability of the continuous profile as the Richardson number at the center of the shear layer increases. Alexakis [18,21,22] extended the analysis to even higher stratifications and was able to predict theoretically and show numerically that there are higher branches of Holmboe instabilities in these continuous profiles. Because of the geophysical and astrophysical interest we will concentrate our analysis of non-modal perturbation growth of shear layers at very high Richardson numbers which possess sparse islands of exponential instability of low growth rate.…”

“…Holmboe instabilities have been reproduced in laboratory experiments [5][6][7][8][9][10][11] and have been numerically simulated [12][13][14][15][16][17][18][19]. At finite amplitude these Holmboe modes * Electronic address: navidcon@phys.uoa.gr † Electronic address: pjioannou@phys.uoa.gr equilibrate into propagating waves and they can induce mixing in highly stratified environments [12,17,20].…”

“…Consequently, Holmboe instabilities present the intriguing possibility that they may be responsible for transition to turbulence and mixing in highly stratified flows. In that vein, Alexakis [18,21,22] has proposed that the surmised necessary mixing in order for a thermonuclear runway to occur and highly stratified white dwarfs form supernova explosions could be effected with Holmboe instabilities.…”