A New Set of Orthonormal Modes for Linearized Meteorological Problems

Abstract

A new orthogonal decomposition based on the Schmidt decomposition approach has been applied to the barotropic equation linearized around the January 300-mb climatological flow. The Schmidt decomposition can be computed numerically performing a singular value decomposition of the numerical representation of the equation. The decomposition provides a set of positive real numbers whose minimum is linked to the singularity of the linear equation. A nonzero minimum singular value guarantees nonsingularity. Within the limits of the numerical precision and resolution used (R15 and R30) the nondivergent, global, barotropic equation linearized around the winter climatology is not singular, but it is very badly conditioned. The Schmidt decomposition gives two sets of orthonormal basis functions, and a possible interpretation is offered by expressing the covariance matrix of forced responses in terms of Schmidt modes. An interpretation of the basis is obtained by showing that one set corresponds to the EOF of the responses forced by random sources and the second basis to the forcings that excite that particular EOF.