Can Quasigeostrophic Turbulence Be Modeled Stochastically?

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  • 1 Data Assimilation Office, NASA/Goddard Space Flight Center, Greenbelt, Maryland
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Abstract

Numerically generated data of quasigeostrophic turbulence in an equilibrated shear flow are analyzed to determine the extent to which they can be modeled by a Markov model. The time lagged covariances are collected into a matrix, Cτ, and are substituted into the fluctuation-dissipation relation for a first-order Markov model with white noise forcing
CτC0−1A
to determine whether the relation is satisfied for a single dynamic operator A. The dynamic operator obtained by inverting the relation was found to depend on time lag. In particular, for small time lags (τ < 1 day), the eigenvectors and imaginary eigenvalues were independent of time lag, while the damping rates increased linearly with time lag. It is shown analytically that precisely this discrepancy occurs when the relation is applied to data generated by a red noise Markov model using a time lag that is small compared to the decorrelation time of the noise. Although a fourth-order Markov model with white noise can more accurately reproduce the covariances, the result of inverting the fluctuation-dissipation relation for such a model implies that the spectrum of the noise involves a superposition of stochastic processes of different spectral characteristics, in which case the effective dissipation and stochastic excitation cannot be completely solved by inverting such generalized fluctuation-dissipation relations. Projecting the data onto the dominant EOFs can distort the dynamic operator and introduce discrepancies even when the underlying data rigorously satisfies the fluctuation-dissipation relation. Despite this confounding factor, the consistency of the results at each order suggests that the effective dissipation is composed of low-order cross-stream gradients of streamfunction and that the excitation is correlated in the cross-stream direction within only a few Rossby radii.

Abstract

Numerically generated data of quasigeostrophic turbulence in an equilibrated shear flow are analyzed to determine the extent to which they can be modeled by a Markov model. The time lagged covariances are collected into a matrix, Cτ, and are substituted into the fluctuation-dissipation relation for a first-order Markov model with white noise forcing
CτC0−1A
to determine whether the relation is satisfied for a single dynamic operator A. The dynamic operator obtained by inverting the relation was found to depend on time lag. In particular, for small time lags (τ < 1 day), the eigenvectors and imaginary eigenvalues were independent of time lag, while the damping rates increased linearly with time lag. It is shown analytically that precisely this discrepancy occurs when the relation is applied to data generated by a red noise Markov model using a time lag that is small compared to the decorrelation time of the noise. Although a fourth-order Markov model with white noise can more accurately reproduce the covariances, the result of inverting the fluctuation-dissipation relation for such a model implies that the spectrum of the noise involves a superposition of stochastic processes of different spectral characteristics, in which case the effective dissipation and stochastic excitation cannot be completely solved by inverting such generalized fluctuation-dissipation relations. Projecting the data onto the dominant EOFs can distort the dynamic operator and introduce discrepancies even when the underlying data rigorously satisfies the fluctuation-dissipation relation. Despite this confounding factor, the consistency of the results at each order suggests that the effective dissipation is composed of low-order cross-stream gradients of streamfunction and that the excitation is correlated in the cross-stream direction within only a few Rossby radii.
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