Highly stratified shear layers are rendered unstable even at high stratifications by Holmboe instabilities when the density stratification is concentrated in a small region of the shear layer. These instabilities may cause mixing in highly stratified environments. However these instabilities occur in limited bands in the parameter space. We perform Generalized Stability analysis of the two dimensional perturbation dynamics of an inviscid Boussinesq stratified shear layer and show that Holmboe instabilities at high Richardson numbers can be excited by their adjoints at amplitudes that are orders of magnitude larger than by introducing initially the unstable mode itself. We also determine the optimal growth that is obtained for parameters for which there is no instability. We find that there is potential for large transient growth regardless of whether the background flow is exponentially stable or not and that the characteristic structure of the Holmboe instability asymptotically emerges as a persistent quasi-mode for parameter values for which the flow is stable.
The mixing of shear layers and the development of turbulence is severely impeded when the layer is located in regions of large stable stratification. The stratification is usually quantified with the non-dimensional Richardson number defined as the ratio of the local Brunt-Väisälä frequency N 2 to the square of the local shear U ′ , Ri = N 2 /U ′2 . Large stratification corresponds to large Richardson numbers. The significance of the Richardson number for the stability of stratified flows has been underscored with a theorem due to Miles and Howard [1,2] which proves that if everywhere in the flow Ri > 1/4, the flow is necessarily stable to exponential instability. The essential instability that leads primarily to mixing in both stratified and unstratified shear layers is the Kelvin-Helmholtz (KH) instability [3]. The KH modes are eventually stabilized when the local Richardson number exceeds 1/4, but if the stratification is concentrated in narrow regions within the shear layer the Richardson number may locally become smaller than 1/4 and an instability can result although the overall stratification is very large. Under such conditions, a new branch of instability of stratified shear layers emerges as shown by Holmboe [4], consisting of a pair of propagating waves with respect to the center flow, one prograde and one retrograde. This new instability branch has been named the Holmboe (H) instability and it is physically very interesting because it persists at all stratifications and can lead to mixing in highly stratified shear layers.
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 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.
Holmboe [4] also introduced a new method for analyzing the perplexing instabilities that may arise with the introduction of stratification. He went beyond classical modal analysis and proposed that a fruitful program for predicting and obtaining a clear physical idea of the instability of stratified shear flows is to simplify the background flow to segments of piecewise constant vorticity, as in Rayleigh’s seminal work [23], and then consider the dynamics of the edge waves that are supported at each vorticity and density discontinuity. Holmboe showed that the flow becomes unstable when the edge waves propagate with the same phase speed. This method of analysis has been extended and used to clarify the detailed mechanism of instability in stratified flows [24][25][26][27][28] and has been shown recently by Carpenter et al. [29] to be capable of readily assessing the character of instability of nonidealized observed flows. Also this edge wave description has elucidated the dynamics of other instabilities that occur in geophysics [30][31][32][33] and astrophysics [34][35][36].
However, the efficacy of Holmboe modes to mix highly stratified layers can be questioned because the instability occurs only in limited regions of parameter space and the modal growth rate of the instability becomes exponentially small with stratification. Specific estimates of the asymptotic behavior of the growth rates and of the unstable band of wavenumbers has been recently obtained by Alexakis [22]. Also the introduction of viscosity can further reduce the growth rates of the Holmboe insta-bilities. It is the purpose of this work to go beyond the modal analysis and investigate the non-modal stability of stratified shear layers that are Holmboe unstable in order to assess the true potential of growth of stratified shear layers. We will achieve this using the standard methods of generalized stability theory [37,38].
Because shear layers are powerful transient amplifiers of perturbation energy, we expect that the Holmboe instabilities can be excited at enhanced amplitude by composite non-modal perturbations. Even in the case of an infinite constant shear flow which has a continuous spectrum with no inviscid analytic modes and all perturbations eventually decay algebraically with time, the nonmodal solutions constructed by Kelvin [39] demonstrate that the asymptotic limit is non-uniform and perturbation energy can transiently exceed any chosen bound in the inviscid limit; the same is true for bounded Couette flow as shown by Orr [40]. The Kelvin-Orr solutions can be extended to stratified flows [41,42] and these solutions can produce transient perturbation growth that can also exceed in the inviscid lim
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