Stability of two-dimensional Taylor–Green vortices in rotating stratified fluids

Abstract

The linear stability of the two-dimensional Taylor–Green vortices, which is a spatially periodic array of vortices, in rotating stratified fluids is investigated by local and modal stability analysis. Five types of instability appear in general: the pure hyperbolic instability, the strato-hyperbolic instability, the rotational-hyperbolic instability, the centrifugal instability and the elliptic instability. The condition for each instability and the estimate of the growth rate, which are useful in interpreting numerical results, are obtained in the framework of local stability analysis. Realizability of an instability is introduced to predict whether an unstable mode corresponding to an unstable region found in the local stability analysis exists at finite Reynolds numbers. In the absence of stratification, the pure hyperbolic instability is dominant for weak rotation; it is stabilized for strong rotation. For strong anti-cyclonic rotation, the elliptic instability or the centrifugal instability becomes dominant depending on the parameter values; further stronger rotation stabilizes both instabilities. For strong cyclonic rotation, the rotational-hyperbolic instability or the elliptic instability becomes dominant, although the growth rate is smaller than the anti-cyclonic cases. Strong stratification changes the stability properties. The strato-hyperbolic instability occurs for weak rotation. The rotational-hyperbolic instability and the elliptic instability are weakened under cyclonic rotation, while the latter survives and extends the unstable range under anti-cyclonic rotation. The pure hyperbolic instability and the centrifugal instability are less affected by stratification. The mode structures of each instability are in good agreement with the corresponding solution to local stability equations, confirming the physical mechanism of the instability.