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Russ. J. Numer. Anal. Math. Modelling, Vol. 23, No. 1, pp. 39–61 (2008) DOI 10.1515/ RJNAMM.2008.003 c© de Gruyter 2008 Existence and uniqueness of a solution to primitive equations with stratification ‘in the large’ G. M. KOBELKOV∗ and V. B. ZALESNY∗ Abstract — Existence and uniqueness of a weak solution for the system of primitive equations with stratification describing large-scale ocean dynamics, is proved ‘in the large’. The system is obtained from the original 3D primitive equations with viscosities depending on ρz. The character of these dependences

Russ. J. Numer. Anal. Math. Modelling, Vol. 24, No. 6, pp. 515–542 (2009) DOI 10.1515/ RJNAMM.2009.033 c© de Gruyter 2009 Existence ‘in the large’ of a solution to primitive equations in a domain with uneven bottom A. V. DRUTSA∗ Abstract — The existence and uniqueness theorems ‘in the large’ are proved for a system of primitive equations in the Cartesian coordinates in a domain with an uneven bottom. The original equations are slightly modified: some terms containing mixed derivatives are omitted because they are small. Namely, it is proved that for arbitrary

Journal of Applied Analysis Vol. 8, No. 2 (2002), pp. 153–200 MATHEMATICAL ANALYSIS AND OPTIMAL CONTROL PROBLEMS FOR THE PERTURBATION OF THE PRIMITIVE EQUATIONS OF THE OCEAN WITH VERTICAL VISCOSITY A. BELMILOUDI Received February 28, 2001 Abstract. In this paper we consider an oceanic domain in R3, in which there exists, at initial time, a current U0, a pressure p0 and a density ρ0. The perturbation U , p and ρ of the velocity, the pressure and the density are induced by a perturbation of the mean windstress. The equations are of Navier-Stokes type for the

References [1] Arthur Bousquet, Michele Coti Zelati, and Roger Temam. Phase transition models in atmospheric dynamics. Milan Journal of Mathematics, pages 1–30, 2014. [2] Arthur Bousquet, Gung-Min Gie, Youngjoon Hong, and Jacques Laminie. A higher order finite volume resolution method for a system related to the inviscid primitive equations in a complex domain. Numerische Mathematik, 128(3):431–461, 2014. [3] Chongsheng Cao and Edriss S. Titi. Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere

DOI 10.1515/rnam-2014-0012 | Russ. J. Numer. Anal. Math. Modelling 2014; 29 (3):145–165 Alexey V. Drutsa A dierence scheme for equations of ocean dynamics on unstructured grids Abstract: A dierence scheme on unstructured grids is constructed for the system of equations of large scale ocean dynamics. The properties of the grid problem and grid operators are studied, in particular, a series of a priori estimates and the theorem on existence and uniqueness of the solution are proved. Keywords: Primitive equations, ocean dynamics equations, nonlinear partial

primitive equations as the time step tends to zero. 1. Introduction The splitting method is one of the basic numerical solvers for complicated evol- utionary problems (see, e.g., [3, 6–8]). In this case one can discriminate splitting over spatial variables and over physical processes [3]. Justifications of those meth- ods for equations of parabolic type were given in [3, 8], whereas for the Navier– Stokes equations those methods were not mathematically justified for a long time. An approach to justification of splitting methods for the Navier–Stokes equations was proposed

variability of coastal currents. 1 The model of sea currents and tidal waves The model of sea hydrodynamics is based on the set of primitive equations written in the bipolar orthogonal coordinate system with approximations of hydrostatics and Boussinesq. The dimensionless variable σ ϵ [0, 1] is used as the vertical coordinate, which is defined as follows: σ = z − ζ H − ζ $$\sigma = {{z - \zeta } \over {H - \zeta }}$$ where z is the ordinary vertical coordinate, 𝜁 is the sea surface height (SSH), H is the sea depth. The model equations are written in the symmetrized

to increase the spatial resolution of the Gulf of Finland. The free surface, sigma-coordinate primitive equation model under the Boussinesq, continuity, and hydrostatic assumptions is solved numerically. The problem of estimation of the pollution of some ‘protected’ marine sub-area by a passive tracer by means of the introducing an adjoint equation for the sensitivity function is formulated. The sensitivity function species the contribution of each basin point to the total pollution of the ‘protected area’. Keywords: Mathematical model, Baltic Sea circulation

theoretical results for some widely used atmospheric models (barotropic, two-layer quasigeostrophic, a system of ‘primitiveequations). In fact, most of the atmospheric and climate models are the results of some approximations to original systems (which are, in general, systems of partial differential equations). The problems of closeness for characteristics of original and approximating models are studied in the corresponding section of the paper. The central problem of the modern climate theory is the problem of the climate sensitivity to small perturbations of system

EstonianMarine Institute, University of Tartu. The model is based on primitive equations written in spherical σ coordinates with a free surface in the hydrostatic and Boussinesq approximations. The structure of numerical algorithm is described. The algorithm is based on the method of multicomponent splitting and includes splitting by physical processes and spatial coordinates. The equations of sea dynamics are written in a symmetrized form. The problem is split into several energetically balanced subsystems (splitting by physical processes). Each subsystem can be additionally