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- Author or Editor: Philip D. Thompson x
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Abstract
Starting with the vorticity equation for barotropic flow, we derive a system of stochastic differential equations that determines the time-evolution of the local variance of vorticity error originating in a large ensemble of initial states containing random and statistically isotropic initial errors. Those equations show that the local growth or decay of error variance depends primarily on the detailed structure of the true vorticity field; in general, the most rapid growth of error can be expected in concentrated regions of strong vorticity gradient.
Those stochastic differential equations provide the basis for a simple method of stochastic-dynamic prediction. It requires only a modest increase over the total volume of computation for deterministic prediction.
Abstract
Starting with the vorticity equation for barotropic flow, we derive a system of stochastic differential equations that determines the time-evolution of the local variance of vorticity error originating in a large ensemble of initial states containing random and statistically isotropic initial errors. Those equations show that the local growth or decay of error variance depends primarily on the detailed structure of the true vorticity field; in general, the most rapid growth of error can be expected in concentrated regions of strong vorticity gradient.
Those stochastic differential equations provide the basis for a simple method of stochastic-dynamic prediction. It requires only a modest increase over the total volume of computation for deterministic prediction.
Abstract
This note deals with a new class of solutions of the nondivergent barotropic vorticity equation. In general, these solutions require a complete representation in spherical harmonics and are therefore good comparison solutions for testing the accuracy of spectral methods of numerical integration.
Abstract
This note deals with a new class of solutions of the nondivergent barotropic vorticity equation. In general, these solutions require a complete representation in spherical harmonics and are therefore good comparison solutions for testing the accuracy of spectral methods of numerical integration.
Abstract
We propose here a method of “stochastic-dynamic” prediction that is computationally more efficient than integration of the full set of “second-moment” equations. This gain is achieved by omitting covariances between modes in different interacting triads, and by expressing intratriad covariances in terms of error variances, via the conditions for invariance of products of invariants. The resulting evolution equations for the error variances of all modal amplitudes constitute a closed system involving only those error variances.
To test the accuracy of this method, we have compared the predicted error variances with those calculated directly from an ensemble of 100 individual predictions, starting from an ensemble of 100 initial states containing random errors. These agree very well up to about the doubling time of total rms error, but later diverge as the effects of indirect interactions accumulate.
Abstract
We propose here a method of “stochastic-dynamic” prediction that is computationally more efficient than integration of the full set of “second-moment” equations. This gain is achieved by omitting covariances between modes in different interacting triads, and by expressing intratriad covariances in terms of error variances, via the conditions for invariance of products of invariants. The resulting evolution equations for the error variances of all modal amplitudes constitute a closed system involving only those error variances.
To test the accuracy of this method, we have compared the predicted error variances with those calculated directly from an ensemble of 100 individual predictions, starting from an ensemble of 100 initial states containing random errors. These agree very well up to about the doubling time of total rms error, but later diverge as the effects of indirect interactions accumulate.
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Abstract
This paper is concerned with calculating the nonlinear transfer and partition of energy in a triad of interacting modes of a two-dimensional viscous flow, driven by random sources and sinks of vorticity. Our approach to the problem lies in deriving a moment expansion from the Fokker-Planck equation for the equilibrium probability distribution of a large ensemble of such flows. For sufficiently small values of a set of dimensionless “ordering-parameters,” depending only on given external conditions, the moment expansion is convergent. A remarkably simple relation between the fourth and fifth moments leads to a tractable “sixth- moment discard” closure, which is in some respects similar to the “eddy-damped, quasi-normal” approximation, but has a clearer theoretical basis.
The accuracy of the “sixth-moment discard” closure is judged by comparing the theoretically-derived kinetic energy and energy-transfer spectra with those constructed from a large ensemble of numerical integrations of the original evolution equations. For nonhomogencous forcing corresponding to “ordering parameters” only slightly less than unity, the theoretical and numerical results agree to within about 5%.
Abstract
This paper is concerned with calculating the nonlinear transfer and partition of energy in a triad of interacting modes of a two-dimensional viscous flow, driven by random sources and sinks of vorticity. Our approach to the problem lies in deriving a moment expansion from the Fokker-Planck equation for the equilibrium probability distribution of a large ensemble of such flows. For sufficiently small values of a set of dimensionless “ordering-parameters,” depending only on given external conditions, the moment expansion is convergent. A remarkably simple relation between the fourth and fifth moments leads to a tractable “sixth- moment discard” closure, which is in some respects similar to the “eddy-damped, quasi-normal” approximation, but has a clearer theoretical basis.
The accuracy of the “sixth-moment discard” closure is judged by comparing the theoretically-derived kinetic energy and energy-transfer spectra with those constructed from a large ensemble of numerical integrations of the original evolution equations. For nonhomogencous forcing corresponding to “ordering parameters” only slightly less than unity, the theoretical and numerical results agree to within about 5%.
Abstract
From a statistical mechanical treatment of an ensemble of randomly forced two-dimensional flows of a viscous fluid, we derive two independent integral constraints on the form of the equilibrium energy spectrum. With a single hypothesis about the shape-preserving properties of the spectrum, those constraints determine the spectrum to within the value of a single universal dimensionless constant. In all other respects the argument is deductive and does not depend on closure approximations or hypotheses about the process of nonlinear energy transfer. The spectrum exhibits minus third-power dependence for, small scales, but minus first-power dependence for large scales. It is in good agreement with the results of detailed numerical integrations of the Navier-Stokes equations.
Abstract
From a statistical mechanical treatment of an ensemble of randomly forced two-dimensional flows of a viscous fluid, we derive two independent integral constraints on the form of the equilibrium energy spectrum. With a single hypothesis about the shape-preserving properties of the spectrum, those constraints determine the spectrum to within the value of a single universal dimensionless constant. In all other respects the argument is deductive and does not depend on closure approximations or hypotheses about the process of nonlinear energy transfer. The spectrum exhibits minus third-power dependence for, small scales, but minus first-power dependence for large scales. It is in good agreement with the results of detailed numerical integrations of the Navier-Stokes equations.
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Abstract
This paper concerns the problem of predicting the variance of a large ensemble of predictions evolving from an ensemble of slightly incorrect initial states, randomly distributed around a “most probable” initial state. Starting with the vorticity and thermodynamic energy equations for three-dimensional quasi-geostrophic flow, we derive evolution equations for the ensemble-averaged potential vorticity, the variance of deviations from that average, and the mean transport of those deviations. Under the assumption that the ensemble of initial error fields is statistically isotropic and homogeneous in horizontal planes, the resulting stochastic equations comprise a complete system with second-moment closure. Those equations provide the basis for a simple method of predicting the spatial distribution of the probable error of predictions.
Whether the local error-variance grows or decays is crucially dependent on the detailed structure and “local” scale of the “true” field of potential vorticity, relative to the characteristic scale of the error fields. If the local scale of the vorticity field is very large, the error-variance grows very slowly or may even decrease; if it is very small, the error variance grows rapidly.
The proposed method of stochastic-dynamic prediction involves only one inversion per time step, and thus requires only marginally more computation than deterministic prediction.
Abstract
This paper concerns the problem of predicting the variance of a large ensemble of predictions evolving from an ensemble of slightly incorrect initial states, randomly distributed around a “most probable” initial state. Starting with the vorticity and thermodynamic energy equations for three-dimensional quasi-geostrophic flow, we derive evolution equations for the ensemble-averaged potential vorticity, the variance of deviations from that average, and the mean transport of those deviations. Under the assumption that the ensemble of initial error fields is statistically isotropic and homogeneous in horizontal planes, the resulting stochastic equations comprise a complete system with second-moment closure. Those equations provide the basis for a simple method of predicting the spatial distribution of the probable error of predictions.
Whether the local error-variance grows or decays is crucially dependent on the detailed structure and “local” scale of the “true” field of potential vorticity, relative to the characteristic scale of the error fields. If the local scale of the vorticity field is very large, the error-variance grows very slowly or may even decrease; if it is very small, the error variance grows rapidly.
The proposed method of stochastic-dynamic prediction involves only one inversion per time step, and thus requires only marginally more computation than deterministic prediction.
Abstract
This is a study of the statistical behavior of a very low order “general circulation model,” consisting of a single finite-amplitude baroclinic wave interacting with a mean zonal shear flow, maintained against dissipation by differential heating. Starting with the equations for two-level quasi-geostrophic flow in a β-plane channel, we derive a closed system of evolution equations for five zonally-averaged quantities at 45° latitude—including net poleward heat transport, meridional kinetic energy and mean vertical shear (or mean horizontal temperature gradient).
This system of equations enables us to relate the equilibrium values of mean vertical shear between 250 and 750 mb and rms northward component of velocity at 500 mb to the rate of differential heating. The former is estimated to be 10.7 m s−1 at 45°S: Oort's observed statistics show that it is actually about 12 m s−1. The theoretical estimate of the rms northward component of velocity at 45°S is 11.0 m s−1. Oort's and Trenberth's statistics also give a value of about 11 m s−1.
For conditions at 45°S during the Southern Hemisphere summer season, numerical integrations of our model equations show that amplitude vacillations around the equilibrium state have a period of about 22.7 days, which compares favorably with the observed periods reported by Webster and Keller in addition to Randel and Stanford. This is added confirmation that a very simple model may provide a physically valid basis for understanding the dominant large-scale dynamical response to differential heating.
Abstract
This is a study of the statistical behavior of a very low order “general circulation model,” consisting of a single finite-amplitude baroclinic wave interacting with a mean zonal shear flow, maintained against dissipation by differential heating. Starting with the equations for two-level quasi-geostrophic flow in a β-plane channel, we derive a closed system of evolution equations for five zonally-averaged quantities at 45° latitude—including net poleward heat transport, meridional kinetic energy and mean vertical shear (or mean horizontal temperature gradient).
This system of equations enables us to relate the equilibrium values of mean vertical shear between 250 and 750 mb and rms northward component of velocity at 500 mb to the rate of differential heating. The former is estimated to be 10.7 m s−1 at 45°S: Oort's observed statistics show that it is actually about 12 m s−1. The theoretical estimate of the rms northward component of velocity at 45°S is 11.0 m s−1. Oort's and Trenberth's statistics also give a value of about 11 m s−1.
For conditions at 45°S during the Southern Hemisphere summer season, numerical integrations of our model equations show that amplitude vacillations around the equilibrium state have a period of about 22.7 days, which compares favorably with the observed periods reported by Webster and Keller in addition to Randel and Stanford. This is added confirmation that a very simple model may provide a physically valid basis for understanding the dominant large-scale dynamical response to differential heating.
