Application of the Spectral Method to Solve the Meridional Circulation Equation in Spherical-Sigma Coordinates

Abstract

The equation governing the zonally averaged meridional circulation (W̄, v̄ in spherical-sigma coordinates is formulated including the semigeostrophic terms. This equation and the mass continuity equation are spectrally transformed in terms of Legendre polynomials. Finite differences are used to represent vertical derivatives. An iterative procedure is then used to solve the resulting spectral equations in which the quasi-geostrophic terms serve as the predictors and the semigeostrophic terms as the corrections. It is shown that this spectral iterative method is capable of solving the meridional circulation equation exactly when the forcing functions in this equation are prescribed in such a way that they are spectrally consistent with those in the zonally averaged momentum and thermodynamic energy equations. The computation time for the solution is comparable to that required for the conventional grid relaxation method. The discretization error incurred by the grid method is, however, noticeable when used with spectral model data. Given sufficiently coarse spectral truncation, such as to eliminate convectively unstable scales of atmospheric motion, it is shown that the meridional circulation equation can be solved when the moist static stability is used in place of the dry static stability and therefore the equation is hyperbolic over part of the domain.