ON STOCHASTIC DYNAMIC PREDICTION

I. The Energetics of Uncertainty and the Question of Closure

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  • 1 Air Weather Service, U.S. Air Force 1
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Abstract

In this, the first part of a two-part study, we examine the rationale of the stochastic dynamic approach to numerical weather prediction. Advantages of the stochastic dynamic method are discussed along with problems associated with the method. This method deals with the initial uncertainty by considering an infinite ensemble of initial states in phase space, relative frequencies within the ensemble being proportional to probability densities. The evolution of this ensemble in time, given by the stochastic dynamic equation set, is based upon the original deterministic hydrodynamic equation set. One may consider the latter set as a subset of the former. Insight into the nature of these equations is obtained by deriving the energy transformations associated with them. A simple baroclinic model is used to isolate the energy concepts and relations. The energetics yield qualitative and quantitative information on the nature of the growth of uncertainty. It is found that the baroclinic instability mechanism is responsible for most of the error growth as would be expected.

Previous predictability studies have considered that the simulation of the forces governing the atmosphere has been perfect. The effects of imperfect forcing can be viewed with the stochastic dynamic equations by adding another dimension to phase space for each parameter considered to be uncertain. The effect of the inclusion of this imperfect forcing is shown by the new energetic relations that result, and by numerical calculation of the changes in the growth of uncertainty.

The stochastic dynamic equations are faced with the same mathematical problem of “closure” found in analytical treatments of homogeneous isotropic turbulence; that is, an approximation concerning higher order moments must be made to close the system. A number of closure schemes are studied and it is found that the third moments, which are individually small, should nevertheless be retained. It is shown in the equations and verified by numerical calculations that the third moments do not affect energy conservation but affect energy conversion between uncertain components, with the eventual result of altering the forecast of the mean. An eddy-damped third moment scheme is found to give extremely accurate results when compared to Monte Carlo calculations.

Abstract

In this, the first part of a two-part study, we examine the rationale of the stochastic dynamic approach to numerical weather prediction. Advantages of the stochastic dynamic method are discussed along with problems associated with the method. This method deals with the initial uncertainty by considering an infinite ensemble of initial states in phase space, relative frequencies within the ensemble being proportional to probability densities. The evolution of this ensemble in time, given by the stochastic dynamic equation set, is based upon the original deterministic hydrodynamic equation set. One may consider the latter set as a subset of the former. Insight into the nature of these equations is obtained by deriving the energy transformations associated with them. A simple baroclinic model is used to isolate the energy concepts and relations. The energetics yield qualitative and quantitative information on the nature of the growth of uncertainty. It is found that the baroclinic instability mechanism is responsible for most of the error growth as would be expected.

Previous predictability studies have considered that the simulation of the forces governing the atmosphere has been perfect. The effects of imperfect forcing can be viewed with the stochastic dynamic equations by adding another dimension to phase space for each parameter considered to be uncertain. The effect of the inclusion of this imperfect forcing is shown by the new energetic relations that result, and by numerical calculation of the changes in the growth of uncertainty.

The stochastic dynamic equations are faced with the same mathematical problem of “closure” found in analytical treatments of homogeneous isotropic turbulence; that is, an approximation concerning higher order moments must be made to close the system. A number of closure schemes are studied and it is found that the third moments, which are individually small, should nevertheless be retained. It is shown in the equations and verified by numerical calculations that the third moments do not affect energy conservation but affect energy conversion between uncertain components, with the eventual result of altering the forecast of the mean. An eddy-damped third moment scheme is found to give extremely accurate results when compared to Monte Carlo calculations.

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