On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing

Abstract

In this article, the spectral instability and the associated bifurcations of the shear flows of the 2D quasi-geostrophic equation with a generalized Kolmogorov forcing are investigated. To determine the linear instability of the basic shear flow, we write the corresponding eigenvalue problem as a system of finite difference equations whose nontrivial solutions are expressed in the form of continued fractions. By a rigorous analysis of these continued fractions, we prove the existence of a number Rc such that if the control parameter R, which is proportional to the Reynolds number and the intensity of the curl of forcing, is over Rc, then the basic shear flow loses its stability. To shed light on the bifurcation involved in the loss of stability of the basic shear flow, a natural method is used to reduce the quasi-geostrophic equation to a system of ordinary differential equations. Based on numerical experiments on the coefficients of this reduced system, we show that both supercritical and subcritical Hopf bifurcations occur depending on the frequency of the generalized Kolmogorov forcing. Moreover, we investigate the double Hopf bifurcations which occur at critical aspect ratios. Our results show that in the double Hopf bifurcation case, two periodic solutions, one stable and the other unstable, bifurcate on R>Rc.