**1. Introduction**

At the present time, one of the rapidly developing fields is mathematical modelling of water areas of particular interest (seas, bays, open ocean areas) and open coastal regions. The relevance of this topic is justified by the need to assess anthropogenic impacts on marine areas and the consequences of such impacts. Moreover, a number of problems connected to the climate changes for a selected water area over several decades gain increasing interest. In order to simulate coastal ocean processes, involved with various types of physical phenomena occurring at a wide range of spatial and temporal scales, special approaches are required. To provide the correct representation of coastal ocean flows, regional models are created.

Regional ocean modelling has one unavoidable challenge relative to its global counterpart: open boundary conditions (OBC). By definition, a regional ocean model includes open boundaries over at least a part of its perimeter [1]. The "outer liquid" (open) boundary means the "water-to-water" boundary separating the considered area from an ocean. The result obtained both in long-term simulations and in operational forecasting directly depends on the method of setting the OBC. According to [2], in flow problems dominated by advection and/or wave motion, the OBC should allow phenomena generated in the domain of interest and coming from outside to pass through the boundary without undergoing significant distortion and without influencing the interior solution. One of the difficulties in setting such conditions is that there is no accurate information on external energy and

mass flows. If open boundaries are located in dynamically active areas, inaccurate accounting for this information leads to inconsistency of the results obtained with the observed fields of physical parameters. For the long-term climate modelling, the appropriate setting of boundary conditions at liquid boundaries is of particular importance.

There are different known methods [3–5] dealing with open boundaries in limited-area models. In a number of studies [6,7] the models are previously set up for a larger area and the results of the preliminary (diagnostic) simulations with these models are used to determine OBC. One-way and two-way grid-nesting techniques are devised to exchange information at the interface between fine-mesh regional models and coarse-mesh large-domain models. The use of averaged data for flows across the open boundary is also acceptable in some cases [8]. A large number of OBC have been proposed in the literature (see [9], for a review). The adaptive algorithm described in [3] is used in many recent studies. It is also possible to use sequential [1] or variational [5] data assimilation methods in order to reduce model-data misfit caused by unsatisfactory OBC. While there are many suggested methods, it is still a question of debate which methodology is suitable for a particular problem.

One of the approaches used for modelling multiphysics and multiscale coastal ocean processes is developing of hybrid models based on domain decomposition. The use of domain decomposition method (DDM) allows to reduce the solution process in the original domain to alternate solving the problem in subdomains, possibly having a simpler form, or apply different models in them (for example, the coupling of first and second order equations [10]). Particularly, it is possible to use DDM to combine models developed for individual phenomena at specific scales. For example, in [11,12] the idea of coupling a geophysical fluid dynamics model and a fully 3D fluid dynamics model in order to simulate multiphysics coastal ocean flows is presented and discussed. Implementation of DDM often requires "inner liquid" boundaries, separating the subdomains. In these subdomains one can obtain the results of simulations using meshes of different scales to achieve better approximation of boundary, bottom topography, etc. At the present time, the development of new algorithms and their effective implementation on multiprocessor computer systems can be attributed to the main goals of the domain decomposition. This makes DDMs be promising in mathematical modelling of processes in the oceans and seas [11–13]. Most studies on application of data assimilation (DA) together with DDM are related to developing parallel algorithms. Some studies [14–17] demonstrate the scalability of the domain decomposition approach and its mathematical consistency when applied in variational DA. Besides, DDM in variational DA problems is suitable not only for creating high-performance algorithms. Particularly, observational data could be available not in the whole modelling area but only in some subdomain in which variational DA procedure may be considered. In those cases the application of DDMs may be effective. This topic is a relatively new field in ocean modelling.

The approach described in this paper is based on [13,18–21]. In [18] the model based on primitive equations written in spherical *σ*-coordinates with a free surface in the hydrostatic and Boussinesq approximations is considered. For time approximation of the model the splitting method is used [22,23]. For this purpose the whole time interval is divided into subintervals; at each subinterval, the following subproblems named the steps of the splitting method are solved:

*Step 1.* The heat transfer problem.

*Step 2.* The salt transport problem.

*Step 3.* The problem of hydrological fields adaptation. It is solved in 3 substeps: (a) calculation of density and finding corrections to the velocity; (b) solving the problem of baroclinic adaptation; (c) solving the problem of barotropic adaptation (i.e., the system of linearized shallow water equations is considered).

The steps of the splitting method are formulated in [18]. The splitting method allows to consider the DA problem as a sequence of linear DA problems. In [19–21,24,25] an investigation of some of them is given. In [25] an iterative algorithm for solving the problem of variational assimilation of the temperature corresponding to the step 1 of the splitting method is considered. The numerical

experiments on the efficiency of the algorithm in the Baltic Sea area are carried out. In [19] an inverse problem of determining an unknown boundary function in the OBC for the simplest model of tides is studied. The results of the work are used in [21], where the algorithm for solving the problem of variational assimilation of the sea level anomaly at the liquid boundary corresponding to step 3 of the splitting method is considered and tested in the Baltic Sea circulation model. In [13] a new approach to formulation of DDM is discussed. This approach was applied to a convection-diffusion problem. The method described in [13] was implemented in the heat transfer block (step 1) of the Baltic Sea dynamics model.

The purpose of this work is to apply DDM to the variational DA problem. The method for solving the problem of restoring boundary functions at the "outer" and "inner" liquid boundaries based on the methods of variational DA and domain decomposition for the subproblem corresponding to a system of linearized shallow water equations (step 3-c) is studied. The problem of determining additional unknowns ("boundary functions") in the boundary conditions is considered as inverse and solved using well-known approaches [26,27]. The major problem in coastal ocean modelling is the reconciliation of model results with observational data. The approach described in the paper may be applied to regional ocean models in order to reduce a model-data misfit and the dependence of the results upon the unsatisfactory OBC. The choice of the approach to formulate the DDM in the present work is justified by a possibility of generalization, since the models in the subdomains may differ. Therefore, the results of this paper may be promising for multiscale and multiphysics coastal processes simulation.
