A study of absolute and convective instabilities with an application to the Eady model

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Abstract
The formalism for linear absolute and convective instabilities developed in plasma physics (Briggs, 1964; Bers, 1973) is extended. The conclusion is that any order saddle point as well as any singular point of a frequency as a function of a wavenumber Ω(k) may contribute to instability. Moreover, contributions may come from k-independent branches of the dispersion relation and from regular nonsaddle points of Ω(k). Accordingly, a variety of algebraic-exponential asymptotic behaviors, in particular, purely algebraic growths and sinusoidal oscillations, of a disturbance is possible. In shear flows and stratified flows instability may also be caused by the singularities associated with critical layers. It is shown that in such flows the asymptotic pattern of a growing disturbance may vary in the direction of the shear and/or stratification. Such variability of pattern may be present only in the presence of instabilities related to the interactions between critical layers and incident amplifying waves. Otherwise the pattern is the same to within a scalar factor. We devise a numerical method for investigating instabilities. The vertical structure boundary value problem, i.e. the nonhomogeneous form of Rayleigh's stability equation of linear stability theory, obtained by applying the Laplace-Fourier transform to the initial boundary value stability problem, is discretized. This leads to a two parameter, i.e. k and Ω, algebraic linear system, in which the coefficients are polynomials in k and Ω. Its solution is calculated symbolically. The determinant of this sytem is a polynomial D in k and Ω whose zeros approximate the union of the discrete and the continuous spectra of the problem. Let kn =kn (Ω) denote solutions of D(k,Ω+kV)=0. Images of the lines Im Ω=constant on the k-plane under the transformations k=kn (Ω) are analysed numerically on “pinching” points. These points determine the asymptotic response along the ray x/t∼V,t∞. A vertical variability of the response is determined from the solution of the algebraic system. The method described is applied to the Eady baroclinic instability model when the discretization is accurate to first and second order. The numerical evidence for the convergence of the method is presented. In all cases the discretization destabilizes relatively short waves, weakens the trapping effect near the horizontal boundaries and reduces the horizontal extent of a growing wave packet. All approximations to the Eady model studied are absolutely stable. Critical layers make no dominant contribution to the unstable disturbances, although a contribution from the continuous spectrum produces residual infinitesimal stationary Rossby waves behind the propagating growing pulse. The analysis of the vertical variability of the pulse indicates that at its leading edge the disturbance starts to develop at the top while in the wake of the pulse it is most persistent at the bottom.

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