Mathematical Modelling of Ocean Circulation

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.