Nonlinear stability of plane rotating shear flow under three—dimensional nondivergence disturbances

Abstract

Nonlinear stability criterion for plane rotating shear flow under three-dimensional nondivergence disturbances was obtained by using both variational principle and convexity estimate introduced by Arnold (1965) and Holm et al. (1985). The results obtained in this paper show that the effect of Coriolis force plays an important role in the nonlinear stability criterion, and the nonlinear stability property of the basic flow depends on both the distribution of basic states and the way the external disturbance acts on the states. The upper bound of the gradient of the mass density displacement from the equilibrium $$k^2 = \left| { \vee \left[ {\rho (\bar x,t)--\rho _e (\bar x)} \right]} \right|^2 /\left[ {\rho (\bar x,t)--\rho _e (\bar x)} \right]^2 $$ is determined by the basic states and one example was given to show the exact upper value ofk. The remarks on Blumen’s paper were also given at Section 4 of this paper.