Application of Vertical Normal Mode Expansion to Problems of Baroclinic Instability

Abstract

As an alternative to the finite difference method, we explore the use of the spectral method with normalmodes as the basis functions for discretizing dependent variables in the vertical direction in order to obtainnumerical solutions to time dependent atmospheric equations. The normal modes are free solutions to the timedependent perturbation equations linearized around the atmosphere at rest. To demonstrate the feasibility ofnormal mode representation in the spectral vertical discretization, the vertical normal mode expansion is appliedto the quasi-geostrophic potential vorticity equation to investigate the traditional baroclinic instability of Charneyand Green types on a zonal flow with a constant vertical shear. The convergence of the numerical solutions isexamined in detail in relation to the spectral resolution of expansion functions.We then extend the method of vertical normal mode expansion to solve the problem of baroclinic instabilityon the sphere. Two aspects are different from the earlier example. One is use of the primitive equations insteadof the quasi-geostrophic system and the other is application of normal mode expansions in the horizontal, aswell as vertical direction. First, we derive the evolution equations for the spectral coefficients of truncated seriesin three-dimensional normal mode functions by application of the Galerkin procedure to the global primitiveequations linearized around a basic zonal flow with vertical and meridional shear. Then, an eigenvalue-eigenfunction problem is solved to investigate the stability of perturbation motions superimposed on the 30' jetexamined earlier by Simmons, Hoskins and Frederiksen. From these two examples, it is concluded that thenormal mode spectral method is a viable numerical technique for discretizing model variables in the vertical.