Mathematical Modelling of Ocean Circulation

The problems of ocean dynamics present more and more com­ plex tasks for investigators, based on the continuously sophisti­ cation of theoretical models, which are applied with the help of universal and efficient algorithms of numerical mathematics. The present level of our knowledge in the field of mathemat­ ical physics and numerical mathematics allows one to give rather complete theoretical analysis of basic statements of problems as well as numerical algorithms. Our task is to perform such analy­ sis and also to analyze the results of calculations in order to improve our knowledge of the mechanism of large-scale hy­ drological processes occurring in the World Ocean. The new level of numerical mathematics has essentially influenced , the formation of new solution methods of ocean dynamics prob­ lems, among which an important one is the splitting method, which has been already widely practised in various fields of science and engineering. A number of monographs by N. N. Yanenko, A. A. Samarsky, G.!. Marchuk (Rozhdestvensky and Yanenko 1968; Samarsky and Andreyev 1976; Marchuk 1970, 1980b) and others are devoted to the description of this methods. But the methods of the splitting theory require extensive creative work for their application to concrete problems, which are peculiar, as a rule, in problem formulation. The success of the application of these methods is related to the deep understanding of the essence of the described processes. In the last decades fundamental works of Arakawa, K.

1 Formulation of the Problem, Transformation of Equations and Elaboration of Ocean Circulation Models.- 1.1 Initial Equations, Their Simplification and Transformations.- 1.1.1 Initial Equations and Boundary Conditions.- 1.1.2 Simplification of Equations and Boundary Conditions for Large-Scale Flows.- 1.1.3 Velocity in the Quasi-Geostrophic Model: Calculation Formulas.- 1.1.4 Equations for Integral Functions in Off-Equatorial Current Models.- 1.1.5 Evaluation of the Order of Magnitude of the Quantities in the Equations for Integral Functions.- 1.1.6 Equations for the Calculation of Sea-Surface Topography at the Basin Boundary in an Off-Equatorial Currents Model.- 1.2 Diagnostic Sea Current Models.- 1.2.1 Quasi-Geostrophic Model for the Calculation of Sea-Surface Topography and Flow Velocity.- 1.2.2 Quasi-Geostrophic Model for the Calculation of the Total Mass Transport Stream Function and Flow Velocity.- 1.2.3 A Non-Linear Model for the Calculation of Sea-Surface Topography and Flow Velocity.- 1.2.4 The List of the Main Correlations in the Spherical System of Coordinates and for the Southern Hemisphere.- 1.3 Some Numerical Methods of Solving Simplified Equations of Hydrodynamics.- 2 The Simplest Methods of Difference Approximation and Constructed Equations Solution.- 2.1 The Construction of Difference Grids.- 2.2 The Methods of Approximation and Equation Solutions.- 2.2.1 The Methods of Calculating the Sea-Surface Topography (SST) at the Basin Boundary.- 2.2.2 Methods of Difference Approximation and Solutions of the Equations of Quasi-Geostrophic Models.- 2.2.3 The Methods of Difference Approximation and Solutions of the Equations of a Non-Linear Model.- 3 Numerical Methods of Solving Ocean Dynamics Problems.- 3.1 The Construction and Methods of Solving Simplified Problems of Ocean Dynamics.- 3.2 The Operator Representation of the Problem and the Principal Algorithm of the Splitting.- 3.3 The Evolutional Statement of the Problem.- 3.4 The Difference Schemes for the Equations of Motion.- 3.5 The Approximation of Adaptation Equations by Spadai Variables.- 3.6 The Approximation of the Adaptation Equations by Time.- 3.7 The Choice of the Parameters for Approximation in the Simplest Model.- 3.8 The Organization of the Numerical Algorithm.- 4 The Stationary Problems of Ocean Dynamics.- 4.1 The Statement of the Linearized Problem of the Ocean Climatic Condition.- 4.2 The Simplest Model of the Stationary Ocean Currents.- 4.3 The Ocean Dynamics Model, Taking into Account the Wind-Driven Currents.- 4.4 The Difference Operators of the Ocean Dynamics Problem and the Methods of Approximation.- 4.5 The Iterative Processes for Solving the Ocean Dynamics Difference Equations for the Barotropic Component.- 4.6 The Solution of the Difference Equations of the Ocean Dynamics Baroclinic Component.- 4.7 The Modified Iterative Process.- 4.8 The Simplest Model of Ocean Dynamics, Taking into Account the Non-Linear Turbulent Exchange.- 4.9 The Statement of Several Non-Linear Problems.- 4.10 The Problem of Non-Stationary Adjustment of Flow Fields to Atmospheric Disturbances.- 4.11 The Formation of the Thermocline in the Ocean.- 5 The Analysis of the Results of Calculations.- 5.1 On the Results of Diagnostic Calculations of the Currents in Different Oceanic Basins.- 5.2 The World Ocean Surface Topography and the Surface Gradient Currents.- 5.2.1 Density and Wind Fields for Diagnostic Calculations. Filtration of the Fields.- 5.2.2 Specified Data and Peculiarities of Calculating World Ocean Currents.- 5.2.3 The Peculiarities of the World Ocean Surface Topography and the Surface Gradient Currents for the Summer Season.- 5.3 The Large-Scale Circulation and Seasonal Variation of the World Ocean Waters.- 5.3.1 The Circulation in the World Ocean Surface and Intermediate Layers in the Summer Season.- 5.3.2 The World Ocean Deep and Bottom Layer Water Circulation.- 5.3.3 The Vertical Structure of the World Ocean Currents.- 5.3.4 Seasonal Variations of the World Ocean Surface Topography and the Surface Gradient Currents.- 5.3.5 Seasonal Variations of the Upper Layer Water Circulation of the World Ocean.- 5.3.6 The Structure and Seasonal Variations of the Vertical Circulation of the World Ocean Waters.- 5.4 The Hydrodynamic Adjustment of the Ocean Temperature, Salinity, Density and Flow Fields.- 5.5 The Diagnostic Calculations of Flows and the Adjustment of the Hydrological Elements of the North Atlantic.- 5.5.1 The Diagnostic Calculations of Flows by the Quasi-Geostrophic Model.- 5.5.2 The Diagnostic Calculations of Flows by the Non-Linear Model.- 5.5.3 The Adjustment of the Temperature, Salinity, Density and Flow Fields by the Quasi-Geostrophic Model.- 5.5.4 The Adjustment of the Temperature and Flow Fields by the Non-Linear Model.- 5.6 The Diagnostic Calculations of Flow in the Equatorial Belt of the Ocean.- 5.6.1 The Equatorial Atlantic Flow Calculations by the Quasi-Geostrophic Model: the Assessment of the Validity Limits of the Models.- 5.6.2 The Calculations of Flows of the Equatorial Belts in the Atlantic and Indian Oceans by the Non-Linear Model.- 5.7 The Calculation of Flows in the Black Sea Offshore Zone.- 5.7.1 The Diagnostic Calculations of Flows for the Summer and Fall Seasons.- 5.7.2 Numerical Experiments on the Calculation of the Vertical Velocity Component.- 5.7.3 The Adjustment of the Density and Flow Fields.- References.