The Mathematics of Marine Modelling: Water, Solute and Particle Dynamics in Estuaries and Shallow Seas

Preface
Contents
Contributors
1 Basic Equations of Marine Flows
1.1 Mathematical Description of Fluids
1.1.1 Fluids as Continuous Media
1.1.2 Integral and Differential Formulations
1.1.3 Averaging of Turbulent Flows
1.2 Governing Equations
1.2.1 Volume Conservation
1.2.2 Salt Conservation
1.2.3 Heat Balance
1.2.4 Momentum Balance
1.2.5 Common Formulations and Closures
1.3 Summary
References
2 Water Waves in Isotropic and Anisotropic Media: A comparison
2.1 Introduction
2.2 Gravity Waves
2.2.1 Surface Gravity Waves in Homogeneous Fluids
2.2.2 Gravity Waves in Heterogeneous Media
2.3 Inertial Waves
2.3.1 Waves in Shear Flows
2.3.2 Waves in Rotating Basins
2.3.3 Three-dimensional Effects
2.4 Discussion
2.4.1 The Linear Shear Flow as `Problematic' Equilibrium
2.4.2 Waves in Anisotropic Media
2.4.3 Mixing Due to Wave Focusing and Mean Flows
2.5 Conclusion
References
3 A Review of Nonlinear Boussinesq-Type Models for Coastal Ocean Modeling
3.1 Introduction
3.2 The Water Wave Problem
3.2.1 Dispersive Properties of the Linear Waves
3.2.2 Scaling of Variables and Operators
3.2.3 Nondimensionalization of Equations
3.2.4 Green–Naghdi Equation
3.3 A Finite Element Discretization of the Green-Naghdi Equation
3.3.1 Notation
3.3.2 Functional Setting
3.3.3 Variational Formulation and Solution Procedure
3.4 Numerical Results
3.5 Conclusions
References
4 Tides in Coastal Seas. Influence of Topography and Bottom Friction
4.1 Introduction
4.2 Model Formulation
4.3 Fundamental Wave Solutions
4.3.1 Derivation with Klein-Gordon Equation
4.3.2 Kelvin Wave
4.3.3 Poincaré Waves
4.3.4 Wave Solutions with a Transverse Topographic Step
4.4 Amphidromic Patterns in Semi-enclosed Basins
4.4.1 Superposition of Two Kelvin Waves
4.4.2 Solution to Extended Taylor Problem
4.4.3 Application to Basins Around the World
4.5 Discussion
4.6 Conclusions
References
5 Variational Water-Wave Modeling: From Deep Water to Beaches
5.1 Introduction
5.2 Derivation of Luke's Variational Principle
5.3 Transformed Luke's/Miles' Variational Principles with Wavemaker
5.3.1 FEM and Mesh Motion
5.3.2 Numerical Results: Comparison with Wave-Tank Experiments
5.4 Coupling Water Waves to Shallow-Water Beach Hydraulics
5.4.1 Numerical Results: Damping of Waves on the Beach
5.5 Summary and Conclusions
References
6 Quasi-2D Turbulence in Shallow Fluid Layers
6.1 Introduction
6.2 Two-Dimensional Turbulence
6.2.1 Inertial Ranges in 2D Turbulence
6.2.2 2D Turbulence: The Early Years
6.2.3 Coherent Structures and 2D Turbulence
6.3 2D Turbulence in Square, Rectangular and Circular Domains
6.3.1 Simulations of 2D Turbulence in Domains with No-Slip Walls
6.3.2 Quasi-Steady Final States: Laboratory Experiments
6.3.3 Forced 2D Turbulence on Confined Domains
6.4 Interaction of Vortices with Walls
6.4.1 No-Slip Walls as Vorticity Sources
6.4.2 Vorticity Production by Dipole-Wall Collisions
6.5 Review of 2D Turbulence Experiments in Shallow Fluids
6.5.1 Laboratory Experiments in Shallow Fluid Layers
6.5.2 2D Turbulence with Rayleigh Friction
6.5.3 Secondary Flows in Quasi-2D Turbulence in Thin Fluid Layers
6.5.4 Concluding Remarks
6.6 Summary
References
7 Turbulent Dispersion
7.1 Introduction
7.2 Model Requirements
7.3 Model Development
7.4 Reduction to One Dimension with Boundaries
7.5 Application to Dispersion in Turbulent Jets
7.5.1 Turbulent Round Jet
7.5.2 Turbulent Planar Jet
7.6 Turbulent Flow along a Wall—The Logarithmic Velocity Profile
7.7 Application to the Marine Ekman Layer
7.7.1 Surface Ekman Layer
7.7.2 Bottom Ekman Layer
7.8 Conclusions
References
8 Spreading and Mixing in Near-Field River Plumes
8.1 Introduction
8.2 Dynamical Regions
8.3 A Simple Near-Field Plume Model
8.4 Complications to The Simple Plume Model
8.4.1 Local Mixing Parameterization
8.4.2 Plume Frontal Mixing
8.4.3 Rotation and Return to Geostrophy
8.5 Conclusions
References
9 Lagrangian Modelling of Transport Phenomena Using Stochastic Differential Equations
9.1 Introduction
9.2 Stochastic Differential Equations
9.2.1 Introduction
9.2.2 Îto Stochastic Integrals
9.2.3 Îto Stochastic Differential Equations
9.2.4 Îto's Differentiation Rule
9.2.5 Stratonovich Stochastic Differential Equations
9.2.6 Fokker-Planck Equation
9.3 Particle Models for Marine Transport Problems
9.4 Numerical Approximation of Stochastic Differential Equations
9.5 Test Cases for Marine Transport Problems
9.5.1 Simple Vertical Diffusion
9.5.2 One Dimensional Water Column Including a Pycnocline
9.5.3 Multidimensional Diffusion in an Unbounded Domain
9.6 Conclusion
References
10 Morphodynamic Modelling in Marine Environments: Model Formulation and Solution Techniques
10.1 Introduction
10.2 Morphodynamic Modelling Approaches
10.3 Process–Based Models
10.3.1 Mathematical Formulation of Simulation Models
10.3.2 Mathematical Formulation of Exploratory Models
10.4 Solution Procedure
10.4.1 Initial Value Approach
10.4.2 Bifurcation Approach
10.5 Example: Morphodynamics of Tidal Inlet Systems
10.5.1 Introduction
10.5.2 Cross–Sectionally Averaged Morphodynamic Equilibria
10.5.3 Depth–Averaged Morphodynamic Equilibria
10.6 Summary and Conclusions
References
11 Wetting and Drying Procedures for Shallow Water Simulations
11.1 Introduction
11.2 Governing Equations
11.3 Space Discretization
11.3.1 Finite Volume Methods
11.3.2 Discontinuous Galerkin Schemes
11.4 Time Discretization
11.4.1 Explicit Time Integration
11.4.2 Implicit Time Integration
11.5 Concluding Remarks
References
Appendix Index
Index