A Formal Study of the Linear Initial-Value Problem of the Barotropic Shear Flow

Abstract

Analytic properties and asymptotic behaviour of the Fourier-Laplace transformed Green's function for the linear initial-value problem of the barotropic disturbance in the complex frequency-plane are investigated. Singularities of this Green's function are classified into three types; poles whose positions are determined by the usual eigenvalue problems; a pole which comes from the singularity of the governing equation; a cut which corresponds to the continuous mode and is drawn between two branch points whose positions are decided by the maximum and the minimum value of the basic flow.
Further it is shown that this Green's function asymptotically goes to zero in any direction of the complex frequency plane in reciprocal proportion to the absolute value of the complex frequency.