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C. E. Leith

Abstract

The first iteration of the recently developed nonlinear normal mode initialization procedure for primitive equation models leads to quasi-rotational dynamical and diagnostic equations agreeing with those of quasi-geostrephic theory in a simple Boussinesq f plane model. The proper initialization of a quasi-rotational model, however. requires a nonlinear modification of the geostrophic state traditionally used. Various generalizations are discussed briefly.

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C. E. Leith

Abstract

Climate change studies with general circulation models require the estimation of climatic means by calculation of finite-time averages. A simple stochastic model is used to estimate the standard error of such an estimation method as a function of averaging time. Consequences for long-range forecasting are also discussed.

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C. E. Leith

Abstract

The fluctuation-dissipation theorem of statistical mechanics, proved in this note for a system with two quadratic integrals of motion such as that of two-dimensional or geostrophic turbulence, relates, the mean response to impulsive external forcing of a dynamical system to its natural unforced variability. The utility of this theorem in providing an estimate of the response of climatic means to changing external influences is discussed.

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C. E. Leith

Abstract

Recent observations indicate that the planetary-scale motions of the atmosphere obey some of the laws of two-dimensional turbulence. The eddy-damped Markovian approximation to two-dimensional turbulence is applied to these motions to predict for an observed energy spectrum the nonlinear transfer rates, characteristic error spectra, and the rate of error growth. In this way estimates are derived of the predictability of the atmosphere and of the errors inherent in numerical models. The use of stochastic models for turbulence approximations is described in an Appendix.

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C. E. Leith

Abstract

The theoretical skill of Monte Carlo approximations to the stochastic dynamic forecasting technique proposed by Epstein is examined by means of an extension of earlier atmospheric predictability studies that used the test-field model of two-dimensional turbulence. The fundamental statistical hydrodynamical concept of an ensemble of phase paths evolving in a dynamical phase space is reviewed and used to define the statistical properties of a finite Monte Carlo sample. The application of a linear regression step to arrive at a final best estimate of the state of the atmosphere is also discussed. The resulting forecasts approach the climatological mean at forecast times so late that all skill has been lost.

For an ideal case with an observing resolution, hopefully achievable in the 1980s with satellite-based sensors, it is found that the. Monte Carlo procedure leads to the greatest improvement in mean-square vector wind forecast skill in the 6- to 10-day range. For another case corresponding roughly to present operational resolution the wind forecast skill is improved considerably in the 2- to 5-day range. Much of the improvement in mean-square skill is a consequence of the optimal filtering nature of the procedure which damps erroneous small scale structure in favor of the more predictable large scales.

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C. E. Leith and R. H. Kraichnan

Abstract

The test-field model of turbulence is used to compute the growth of prediction error for inertial-range turbulence in both three and two dimensions. It is found that initial uncertainty in high wavenumbers spreads through the entire inertial range according to a similarity behavior. For the energy inertial range, the time required for error to reach wavenumber k from very high wavenumbers is t=A ε −1/3k−2/3, where ε is the rate of energy transfer per unit mass and A≈10 in three dimensions or A≈2.5 in two dimensions. For the enstrophy inertial range in two dimensions the time for error to propagate from k′ down to kk′ (k and k′ both in the inertial range) is t≈4η −1/2{[ln(k′/ k 1)]2/3−[ln(k/ k 1)]2/3}, where η is the rate of enstrophy transfer and k 1 marks the bottom of the enstrophy inertial range. Error growth is also computed for a two-dimensional spectrum that fits the energy spectrum of planetary waves in the atmosphere. An initial state determined with a horizontal resolution feasible with a satellite-based observing system results in significant predictability of large-scale motions for more than a week. It is argued that the test-field model probably underestimates rather than overestimates predictability times.

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