EMPIRICAL NORMAL-MODES VERSUS EMPIRICAL ORTHOGONAL FUNCTIONS FOR STATISTICAL PREDICTION

The theory of empirical normal modes (ENMs) for a shallow water fluid is developed. ENMs are basis functions that both have the statistical properties of empirical orthogonal functions (EOFs) and the dynamical properties of normal modes. In fact, ENMs are obtained in a similar ma nner as EOFs but with the use of a quadratic form instead of the Eucli dean norm. This quadratic form is a global invariant, the wave activit y, of the linearized equations about a basic state. A general formulat ion is proposed for calculating normal modes from a generalized hermit ian problem, even when the basic stare is not zonal. The projection co efficients of the flow onto a few leading ENMs generally have a more m onochromatic behavior than that obtained for EOFs, which give them an intrinsically more predictable character. This property is illustrated by numerical experiments using the shallow water model on the sphere. It is shown, in particular, that the ENM coefficients, when used as p redictors in a statistical linear model, provide better predictions of the behavior of the shallow water atmosphere than EOF coefficients. I I is also shown that the choice of the basic state itself is crucial.