Variational method of data assimilation in the dynamic probabilistic modelling of climatic fields in the atmosphere

Abstract

Using a variational principle, we propose and numerically verify the method of dynamic probabilistic numerical modelling of weather conditions in the atmosphere. It is represented by the ensemble of possible independent realizations of space-time field complexes of hydrometeoelements. The realization ensemble satisfies the given set of statistical field characteristics in the atmosphere. Each realization of this ensemble satisfies the numerical model of hydrothermodynamics.We consider weather conditions in the atmosphere as the ensemble of possible realizations of the space-time fields of hydrometeoelements [9, 10]. The dynamic probabilistic method is used to model the realizations of this ensemble. Its essence is to unite the numerical models of atmosphere hydrothermodynamics and stochastic models in the framework of a single simulation process [3, 6, 7], using variational data assimilation.In the available climate numerical models based on complete hydrothermodynamic equations the result of climate modelling is obtained due to an ever increasing physical complication of the model itself, a detailed space-time resolution, different parametrizations (moisture, heat flows, boundary layers, etc.), and integration with respect to longtime period, which results in a quasiperiodic regime. In this paper we propose to directly model independent climatic realizations of space-time meteoelement fields with a set of statistical characteristics, which are optimally close to the corresponding characteristics of real fields as to a chosen quadratic quality functional. This approach closely resembles climate modelling by probabilistic models [9, 10] and allows one to combine the features of determinate numerical models of atmosphere dynamics and those of probabilistic models. The numerical hydrothermodynamic model is used as a space-time adjusting interpolant to fulfil and adjust the space-time realizations in an optimal sense. It allows one to obtain those climate characteristics which are not built into the original probabilistic model. They cannot even be reliably obtained by the probabilistic model itself, for example, in the problem of multidimensional correlation function interpolation. Unlike the global numerical climate models the model presented also allows one to obtain the corresponding ensembles of statistically independent climatic realizations for a separately taken region and for a short period of time. Clearly simultaneous use of hydrodynamic and probabilistic models, which are physically more complete, have a higher space-time resolution, and take into account the complex space-time features of real fields, must also lead to more exact climatic estimates."Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, RussiaThe work was supported by the Russian Foundation for Basic Research (98-05-65305, 99-07-90422) and the Project IG SB RAS-97, N30 'Study and modelling of global and regional climatic changes and pollutants transport ...

On one method of studying sensitivity to random fields perturbations in dynamic probabilistic modelling

“…This method is based on variational information assimilation described in detail in [3,4]. In the present work we essentially give the results of a continuation of this line of investigation.…”

“…In the present work we essentially give the results of a continuation of this line of investigation. As the climatic ensemble of the wind velocity field the authors used the ensemble which was obtained in [3,4] by using real information concerning the statistical structure and the corresponding assimilation methods with the aid of a hydrodynamical numerical model. The implementation ensemble satisfies the given set of statistical field characteristics in the atmosphere, each implementation of this ensemble satisfying a numerical hydrodynamical model.…”

“…The implementation ensemble satisfies the given set of statistical field characteristics in the atmosphere, each implementation of this ensemble satisfying a numerical hydrodynamical model. The qualitative and quantitative characteristics of this ensemble are also given in [3,4].…”

“…To solve equation (1.1)-(1.2) we use the explicit quasimonotonic numerical Van Leer scheme [4,7] of second-order accuracy on a uniform grid with steps ∆ x, ∆y, ∆t with respect to the variables x y, and t, respectively. We consider the grid functions Suppose in the domain G we have the set of space points ´x l y k µ, l 1 l ; k 1 k , at which the 'measured' values c u´xl y k µ of the function c´x y tµ are given at the instant t t. Then the problem is: in the entire set of solutions of (1.1)-(1.2), which is defined by the set of admissible initial functions g , we are to find a solution closest to the given values in a norm.…”

“…We consider the grid functions Suppose in the domain G we have the set of space points ´x l y k µ, l 1 l ; k 1 k , at which the 'measured' values c u´xl y k µ of the function c´x y tµ are given at the instant t t. Then the problem is: in the entire set of solutions of (1.1)-(1.2), which is defined by the set of admissible initial functions g , we are to find a solution closest to the given values in a norm. Therefore following [3,4] we consider the quality functional…”