Empirical stochastic models for the dominant climate statistics of a general circulation model

Two Markov models with different dynamics and forcing are used to model the transient eddy statistics of an idealized general circulation model (GCM). The first Markov model employs a physically based dynamical operator compo sed of the linearized primitive equations plus spatially uniform damping. T his model, when driven by spatially uncorrelated forcing, failed to produce reasonable fluxes and variances, despite tuning of the damping and forcing coefficients. This result contrasts with previous studies that have used t he linearized quasigeostrophic equations to model extratropical eddy statis tics. The second Markov model is constructed empirically from the time-lagged cov ariances of the GCM time series. This empirical Markov model could reproduc e the dominant covariances over a range of time lags, provided it contained a sufficiently large number of degrees of freedom. it could not, however r eproduce the time-lag evolution of the trailing EOFs contained in the model . The errors in the trailing EOFs displayed a systematic behavior that coul d be explained by assuming that the "effective noise" -the noise required t o reproduce the full covariances-is correlated over timescales comparable t o the smallest e-folding time of the eigenmodes. Under this assumption, the effective noise is not white, but, for sufficiently large model dimension, the dominant disturbances still can be modeled appropriately by a Markov m odel because their associated decorrelation;rates are small compared to the decorrelation rate of noise. This explanation is illustrated using a three -variable Markov model. These results suggest the following criteria for Ma rkov model estimation: the lag and number of EOFs should be chosen such tha t the least damped modes show little or no dependence on lag and that none of the imaginary eigenvalues are aliased (in a sense defined in the paper). The resulting Markov model for the dominant disturbances is not sensitive to EOF truncation or choice of time lag, except for the structure of the si ngular vectors and adjoints, and the ordering of the eigenmodes. The sensit ivity in singular vectors and adjoints is a plausible consequence of nonnor mality, as nonnormality of the underlying physical system leads to singular vectors differing considerably from the normal modes and EOFs. The stable eigenmodes resemble the leading EOFs used to construct the empir ical model, but differ considerably from the unstable eigenmodes of the lin earized GCM. This difference is attributed to nonlinear processes implicitl y represented in the Markov model. The dissipation and stochastic forcing w ere concentrated in the Tropics and subtropics, in contrast to the eddy var iance and fluxes, which were concentrated in subtropics and midlatitudes. T he associated singular vectors also were localized initially in the Tropics and subtropics, but eventually develop into robust extratropical disturban ces. Interestingly, the dominant singular vectors undergo a growth and deca y life cycle characteristic of the classic nonlinear life cycle, with time- averaged fluxes in close agreement with those diagnosed from the GCM climat ology The fact that the forcing, dissipation, and initial singular vectors are concentrated in the same vicinity suggests a dynamical feedback between Tropics and subtropics.