Optimal Excitation of Asymmetric Perturbations on an Axisymmetric Barotropic Vortex: A Linear Singular-Value Analysis

Abstract

Abstract Singular vectors on a barotropic circular vortex consisting of three regions of piecewise-constant vorticity are investigated under the L2 norm to reveal the shape and growth rate of possible perturbations that may contribute to the formation of the hierarchal structure seen in natural and artificial vortices. Here, the analytical form of the singular values and the forward and backward singular vectors are derived for comparison with the corresponding eigenvalues and vectors. The results indicate no notable difference between the growing eigenvector and the singular vectors for Michalke and Timme’s profile, which is characterized by a ring of high vorticity. In contrast, for Syōno’s profile, which incorporates a negative vorticity region between a solidly rotating core and potentially rotating surroundings, the pattern of the forward singular vector giving the maximum growth in a specified time interval differs crucially from that of the growing eigenvector: whereas the radial peak of the streamfunction exists solely at the maximum-wind radius of the basic vortex in the growing eigenvector, it shifts outward in the growing forward singular vector. This implies the injection of considerable energy into the outer part of the basic vortex, rather than in the inner ring of the highest wind, facilitating the formation of hierarchal structure concentrated around the core region of the vortex.