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Numerical simulations of stratified shear flow instabilities are performed intwo dimensions in the Boussinesq limit. The density variation length scale ischosen to be four times smaller than the velocity variation length scale sothat Holmboe or Kelvin-Helmholtz unstable modes are present depending on thechoice of the global Richardson number Ri. Three different values of Ri wereexamined Ri =0.2, 2, 20. The flows for the three examined values are allunstable 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 hasbeen discovered recently and it is the first time that it is examined in thenon-linear stage. It is found that the amplitude of the velocity perturbationof the second Holmboe mode at the non-linear stage is smaller but comparable tofirst Holmboe mode. The increase of the potential energy however due to thesecond Holmboe modes is greater than that of the first mode. TheKelvin-Helmholtz mode is larger by two orders of magnitude in kinetic energythan the Holmboe modes and about ten times larger in potential energy than theHolmboe modes. The results in this paper suggest that although mixing issuppressed at large Richardson numbers it is not negligible, and turbulentmixing processes in strongly stratified environments can not be excluded.

Stratified shear flow instabilities at large Richardson numbers