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

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, Russia The 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 and transformation processes in the atmosphere of Siberia'. 436 V. A. Ogorodnikov and Λ V. Protasov Thus, as shown in [7], the general scheme for the numerical modelling of the climatic realization ensembles is as follows. With the available statistically significant real climatic information given as the corresponding statistical characteristics (distributions, norms, dispersions, correlations, etc.), the ensemble of the realizations of climatic random hydrometeoelement fields with given statistical structure is constructed by the statistical methods of modelling [3, 8, 10]. The variational assimilation problem for each realization of this ensemble is solved by the mathematical model of atmosphere hydrothermodynamics. Thus, the ensemble of new realizations is obtained. They differ from the initial ones within the accuracy of the solution to the assimilation problem and satisfy the properties inherent in the mathematical model. A detailed description of the joint dynamic probabilistic model and some of its characteristics are given in [7]. In the present work, in order to construct the ensemble of climatic realizations we use the model with substantial modifications. 1. NUMERICAL MODEL OF ATMOSPHERE DYNAMICS We used the modified model of variational data assimilation with time resolution of one hour, which is described in detail in [7]. The modification consisted in replacing a transport unit by a new computational transport unit at the corresponding stage of the employed method of splitting by physical processes, using the explicit quasimonotonic numerical second-order Van Leer scheme [12]. This allowed us to decrease the dissipativity of the numerical scheme by choosing the real coefficients of horizontal viscosity. Thus, it was possible to realistically reproduce the dispersions of the considered fields in the general dynamic probabilistic model. Due to its conditional stability the time step in the calculations was 15 minutes. Therefore in the general computational scheme in the regime of direct and adjoint modelling the corresponding unit was repeated four times. The construction of the corresponding adjoint numerical scheme for the Van Leer method presents a certain difficulty due to its substantial nonlinearity and the use of conditional expressions. Due to the consideration of special indicator functions it was possible to overcome this difficulty and successfully realize the corresponding adjoint computational unit in the variational data assimilation method. We discuss in greater detail the method of constructing adjoint equations, using the numerical difference Van Leer scheme for a transport equation. We consider the equation in the domain G = {[Ο, Χ] Θ [Ο,ί]} : 1 + ̂ = 0, *e[0,X], < 6 [ 0 , i ] (1.1) with the initial condition c(*,*)|t=o = <? (*). (1.2) In order to solve this equation we use the finite difference approximation proposed in [12] on a uniform grid with steps Δχ, At with respect to variables χ and t, respectively. We consider the grid functions c/ = c(xi,tj), u\ = u(xi,tj\ gi = g(xi) which are represented as the corresponding vector functions Vocational method of data assimilation 437 We write the finite difference analogue of equation (1.1) in the operator form c^ = LV'V), c° = g 3 ; = 1,. ,M. (1.3) Suppose that we have the set of space-time points in the domain G : The 'measured' values cu(xi,tm) of the function c(x,t) are given at these points. Then the problem is to find the solution, which is closest to the given values in a norm, among the total set of solutions of (1.1)-(1.2), which is specified by the set of admissible initial functions {g}. For this purpose we consider the quality functional J0 = £(Pc''-ci,/V-ci) (1.4) j€/e where (·, ·) denote the inner product of the vectors c{ of the measured data in the space RJ, P is the interpolation operator of the grid functions c j in the space f, and It is the set of points in time at which c{, j e Λ, are given. Thus, the problem reduces to seeking a minimum of the quadratic functional (1.4) under constraints in the form of equation (1.3). We seek a minimum with respect to the vector g of the initial data. By the Lagrange method of multipliers we consider the functional Ji(c'',c"',uO = 5>' LV,u''),c') + Λ (1-5) Jl where c* is an arbitrary vector of the Lagrange multipliers. When we use this method, the problem of seeking a conditional minimum of the fiinctional (1.4) reduces to seeking the stationary points of the functional (1.5). If we consider the arbitrary variations of this functional with respect to the vector g of the initial data, we obtain the expression Using the Gateaux differentiation operations, operations of transposition and change of the order of summation, we get where L*·, P* are adjoint operators in the Lagrange sense. Since the vector function c * is arbitrary, we choose it so that it satisfies the adjoint equation c·'= rV,c^u'')-2P*(Pc'-O, / e / t , c' = 0. Then the gradient of the functional (1.5) with respect to the vector g is expressed in terms of Thus, we can construct the iterative process of the minimization of the quality functional: