Numerical model of the circulation of the Black Sea and the Sea of Azov

The problem of mathematical modelling of the dynamics of the Black Sea and the Sea of Azov is considered with the use of the INMOM model developed at the Institute of Numerical Mathematics (INM) of the Russian Academy of Sciences. The model is based on the equations of general circulation written in a spherical σ -system of coordinates with a free surface boundary in the hydrostatics and Boussinesq approximations. The equations of sea dynamics are written in a symmetrized form. The numerical algorithm is based on the method of multicomponent splitting and has a flexible modular structure. Splitting with respect to physical processes and spatial coordinates is used. The problem is split into a series of energy-balanced subsystems called modules. Each particular module can be split further into modules of a simpler structure. The numerical experiment consists in the calculation of hydrophysical fields of the Black Sea and the Sea of Azov with the spatial resolution of ∼ 4 km, and 40 σ -levels non-uniformly distributed in depth are used in the vertical direction. Atmospheric forcing is calculated according to the EraInterim data, the calculation period is 3 years, from 2006 to 2008. The results of numerical simulation demonstrate good concordance with the observation data and also with the calculation of the Black Sea dynamics by the model of the Marine Hydrophysical Institute of the National Academy of Sciences of Ukraine. It is proposed to use the model presented here in the development of a monitoring system and for real-time forecast of water circulation in the Black Sea and the Sea of Azov. The problem of operative forecast [16] becomes nowadays one of the central problems of the mathematical modelling of seas and oceans. The base of real-time forecasts is a water circulation model for a given basin. At present, the system of modelling and real-time forecast for the dynamics of the Black Sea has been developed at the Marine Hydrophysical Institute of the National Academy of Sciences of Ukraine (MHI NASU) [16]. A finite difference model with explicit time integration schemes is used for this purpose (see [8–10]). The model adequately represents the structure of hydrophysical fields and is supplied with a block of observation data assimilation (see [10, 16]). The observation data assimilation is based on the application of the ∗Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 119333, Russia †Marine Hydrophysical Institute of the National Academy of Sciences of Ukraine, Sevastopol 99011, Ukraine The work was supported by the Program of the Presidium of the Russian Academy of Sciences No. 21.1 ‘The Black Sea as a simulation ocean model’, by the Federal Target Program ‘Scientific and Pedagogical Staff for Innovative Russia’, by grants of the Russian Foundation for Basic Research and by the Grant Council of the President of the Russian Federation. 96 V. B. Zalesny et al. Kalman filter. The system gives satisfactory forecast accuracy in an open sea with the largest errors near the frontal zones. The further improvement of the forecast accuracy requires the use of more efficient numerical calculation methods, modern schemes of observation data assimilation, and the ability to perform calculations on remote computers. The products of the Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS) in the field of sea circulation modelling based on the methods of splitting and adjoint equations [18] seem to be very promising for the prognostic system of the Black Sea. The application of such models allows one to improve the stability of calculations, increase the spatial resolution of the model, develop an efficient method of four-dimensional variational data assimilation. As the result, this should lead to a qualitative improvement of operative forecasts. The use of modern numerical methods, in particular, non-structured grids [7] may also provide us with the ability for an environment evolution prognosis in the coastal zone of the Black Sea within the framework of an integrated model. At present, simulation methods for the sea and ocean dynamics and algorithms of variational assimilation of observation data based on methods of splitting and adjoint equations are developed at the INM RAS (see [3, 18, 19, 27, 30]). The method of splitting has the two basic functions: first, it allows one to solve the problem in time efficiently; second, to construct a flexible, hierarchically developed information computing system (see [3, 20, 30]). The essence of the method is the representation of a complicated problem as a set of simpler modules. The model can be simplified, or physically enriched by rejecting or adding some modules. The method of adjoint equations is the base of analysis of complex systems (see [3, 19]). It essentially decreases the dimension of the space of solutions in the problem of the optimal choice of the initial parameters of the model aimed at bringing the model solution closer to the observation data. However, the following essential difficulty arises in the use of the method: it is necessary to construct and implement an adjoint model whose equations have a more complicated form comparing to the direct prognostic system. If the method of solution of the prognostic problem, or the form of equations, or their spatial approximation are changed, we have to change the adjoint analogue of the model too. This is rather laborious from the technological viewpoint. In our approach combining two methods, i.e., splitting and adjoint equations techniques, we are able to solve the variational assimilation problem more efficiently. We have to construct an adjoint analogue to a particular split module of the direct model. The direct model is composed from direct split problems. The adjoint one is composed from the corresponding modules of adjoint subproblems. Our approach simplifies the construction of the adjoint model and gives us the ability to calculate an algebraically precise gradient of the minimized cost function. In this paper we present the direct prognostic circulation model of the Black Sea and the Sea of Azov. Further we will construct the model of four-dimensional assimilation of observation data on its base. Numerical model of the circulation 97 1. Circulation model of the Black Sea and the Sea of Azov The model of hydrodynamics of the Black Sea and the Sea of Azov is the extension of the basic general circulation model of the World Ocean developed at the INM RAS (see [11, 30]). Let us present here a brief description of the problem of the dynamics of the Black Sea and the Sea of Azov and the methods of its solution. The dimensionless variable σ ∈ [0,1] is used in the model as the vertical coordinate: σ = z−ζ (x,y, t) H(x,y)−ζ (x,y, t) (1.1) where z is the vertical depth coordinate measured from the unperturbed sea surface towards the center of the Earth, H(x,y) is the depth of the sea, ζ (x,y, t) is the sea level deviation from its unperturbed state, (x,y) are the horizontal coordinates, in our case these are the longitude and latitude, respectively. It is worth noting that in the general case, INMOM can utilize any orthogonal coordinates, including those with a grid refinement in given subdomains. t is the time. In the notation of the system of primitive sea dynamics equations in the σ -system of coordinates, it is useful to introduce the function of geopotential surfaces expressed according to (1.1) as Z = (H−ζ )σ +ζ , Zσ ≡ ∂Z ∂σ . (1.2) Then the system of sea hydrothermodynamic equations takes the following form in this system of coordinates: Dtu−Zσ (l+ξ) v =− Zσ ρ0rx [ ∂ ∂x ( p− g