Empirical Normal-Mode Analysis of Atmospheric Data

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Abstract
The theory of empirical orthogonal functions (EOFs) is generalized in the context of normal modes on unidirectional sheared flows, at first using small disturbance to a monotonic potential vorticity basic state. A wave function is introduced (so called because of a partial analogy with quantum mechanical wave function), via the pseudomomentum, and is used to define the covariance matrix needed in an EOF analysis. The resulting new formalism comprises a fundamental, simpler and more physically insightful, version of EOF theory: it allows empirical reconstruction of the normal modes excited in an atmospheric time series, their respective variances, and phase speed relationship. This new approach permits quantitative and qualitative discussions of the underlying wave mechanisms present in a sheared flow. The theory is given for a hierarchy of models starting with the linearized quasigeostrophic equations. The extent to which these concepts carry over to nonlinear finite-amplitude disturbance is investigated. Included in these considerations are the nonlinear primitive equations in the hydrostatic approximation. The method is applied to 24 winters of the NMC dataset. The empirical results show the presence in the upper troposphere of normal modes, which oscillate in a statistical sense with their theoretically predicted natural frequencies. The normal modes are observed too with divergence e-folding times ranging between 2.5 and 4.5 days. The empirical normal-mode spectrum splits into a continuous and a discrete spectrum with oscillations of less than and greater than two weeks, respectively. The discrete spectrum is divided in forced meridional monopoles and meridional dipoles. In particular, the dipole with zonal wavenumber 2 shows a strong amplitude in phase space of the generalized EOF description for the North Atlantic blocking weather regimes.

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