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:

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... he 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.