Numerical Modeling of Ocean Dynamics

Zygmunt Kowalik, T S Murty

• Publisher: WSPC
• Year: 1993
• ISBN:
  9789814365666
• Topic: Physical Sciences
• Subtopic: Oceanography

While there are several excellent books dealing with numerical analysis and analytical theory, one has to practically sift through hundreds of references. This monograph is an attempt to partly rectify this situation. It aims to introduce the application of finite-difference methods to ocean dynamics as well as review other complex methods. Systematically presented, the monograph first gives a detailed account of the basics and then go on to discuss the various applications. Recognising the impossibility of covering the entire field of ocean dynamics, the writers have chosen to focus on transport equations (diffusion and advection), shallow water phenomena — tides, storm surges and tsunamis, three-dimensional time dependent oceanic motion, natural oscillations, and steady state phenomena. The many aspects covered by this book makes it an indispensable handbook and reference source to both professionals and students of this field.

Contents:

• Formulation of the General Equations:
  • Equations of Motion, Continuity and Diffusion
  • Two-Dimensional Equations
  • Application of the Stream Function
  • Viscosity in the Turbulent Flow
• Transport Equations:
  • Mathematical Rudiments
  • Boundary and Initial Conditions
  • Basic Numerical Properties
  • Explicit Versus Implicit Numerical Schemes
  • Centered Numerical Schemes
  • Computational Errors: Diffusion and Dispersion
  • Computational and Physical Modes of the Numerical Solution
  • Diffusive Processes
  • Application of the Higher Order Computational Schemes to the Advective Equation
• Two-Dimensional Numerical Models:
  • Basic Problems
  • Numerical Solution of the System of Equations
  • Step by Step Approach to the Construction and Analysis of Simple Numerical Schemes
  • Two-Dimensional Models
  • Numerical Filtering
  • Grid Refinement
  • Simulation of Long Wave Run-Up
  • Finite-Differencing of the Time Derivative
  • Finite-Differencing of the Space Derivative
  • Treatment of Open Boundaries
  • Treatment of the Nonlinear Advective Terms
  • Moving Boundary Models and Inclusion of Tidal Flats
  • Nested Grids and Multiple Grids
  • Stretched Coordinates and Transformed Grid Systems
• Three-Dimensional Time-Dependent Motion:
  • Introduction
  • Numerical Modeling of the Fjord Circulation
  • Three-Dimensional Motion in the Shallow Seas
  • Three Dimensional Modeling Utilizing the Mode Splitting and Sigma Coordinate
  • General Circulation Model — Rigid Lid Condition
  • A Three-Dimensional Semi-Implicit Model
  • A Two-Layer Model
  • Mode Splitting and Reduced Gravity Model
  • Quasi-Geostrophic Models
  • Streamfunction Models
  • The Bidston Models
  • Haidvogel et al.'s Model
• Normal Modes:
  • Introduction
  • Seiches
  • The Normal Mode Approach
  • Solutions for Lakes and Bays with Uniform and Variable Depth
  • Systems with Branches
  • Resonance Calculation for Irregular-Shaped Basins
  • Secondary Undulations
  • Helmholtz Mode
  • Open Boundary Conditions
  • Numerical Models for Resonance Calculations
  • Kelvin Waves, Sverdrup Waves and Poincaré Waves
  • Influence of Ice Cover on Normal Modes
• Steady State Processes:
  • Oceanographic Examples
  • Numerical Approximation
  • Boundary Conditions
  • Numerical Methods
  • Direct Methods
  • Higher Order Accuracy Schemes for Convection-Dominated Flow
• Appendix 1
• References
• Subject Index

Readership: Oceanographers, coastal engineers and environment scientists.

CHAPTER I

FORMULATION OF THE GENERAL EQUATIONS

1. Equations of motion, continuity and diffusion

1.1 General equations

Numerical models describing the fields of velocity and density in the ocean are based on the system of hydrodynamical-thermodynamical equations which incorporate the law of conservation of momentum, mass and energy. These equations will be derived in the Cartesian coordinate system although at times a spherical coordinate system will be used. A right-handed rectangular coordinate system with the origin located at the undisturbed level of the free surface is introduced.

The coordinate system is such that the x axis points towards east, the y axis points towards north, and the z axis points upwards, towards zenith. The components of velocity along these three axes will be respectively denoted by u, v, and w. For convenience and brevity we will use also a tensor summation notation, in which the axes are denoted by x₁, x₂, and x₃ and the velocity components by u₁, u₂, and u₃. In this notation, an index appearing twice in a term implies summation over all three index values.

The equation expressing the conservation of momentum is written in tensor notation as

where ρ is the density of water, t time, p pressure, Ω_j, is the component of the Earth's angular velocity, and σ_ij are the components of stress due to molecular viscosity. Also note that ε_ijk = +1 if i, j, k are in cyclic order, ε_ijk = −1 if i, j, k are in anticyclic order, and ε_ijk = 0 if any pair or all three indices have the same value; δ_3i = 1 for i = 3 and δ_3i = 0 for otherwise.

If μ is the molecular viscosity, then the stress tensor, σ_ij, can be expressed in terms of the rate of deformation of a fluid element by the motion

The conservation of mass can be expressed through the continuity equation

which for an incompressible fluid is

For any other property of the fluid (e.g., salinity, temperature, etc.), the general form of the conservation equation holds,

Here, F_i are the components of the flux of the property Φ due to internal forces or pressures and q is the total internal source of the property Φ. Note that eq.(1.1) and (1.3) are special forms of eq.(1.4).

1.2 Separation of the flow into an average and variation around an average state

The above equation can not be solved exactly, except in a few simple cases. Even if such a solution can be derived, to understand particular phenomenon we have to understand which terms in the general equations are responsible for its occurrence. One way to simplify the above set and to obtain the equations controlling the particular phenomenon is to separate these equations into two sets: one set expressing the average motion and the other describing the departures from the average. This approach usually is used to separate the motion into the average and turbulent parts (Hinze, 1975). The meaning of an average and variation from it will depend on the averaging period; by changing the averaging time one can separate particular phenomenon. Because of the nonlinear nature of the above equations, this separation can not be achieved as two independent equations, and one has to contend with two sets, in each of which terms expressing interactions between average and variable motion appear.

Multiplying eq.(1.3) by u_i and adding to eq.(l.l) and using eq.(1.2) after ignoring compressibility of the water, we obtain

To achieve the above mentioned separation, an average with respect to time is used and the averaging process is denoted by a bar:

The next step is to express the velocity, pressure and density fields as an average and a variation:

The averaging process of eq.(1.5) and eq.(1.3) gives

Subtracting eq.(1.8) from eq.(1.5) and eq.(1.9) from eq.(1.3) the equations of motion and continuity for the variations are derived

Noting that the variations in time have zero mean by definition, the above equations can be considered again. The nonlinear interactions between the average and variation are expressed by term

in eq.(1.8). This term becomes

In water bodies, generally speaking, variations in density are very small in proportion to the average density, whereas the fluctuations in velocity can be of the same order as the mean velocity. Hence, the terms containing correlations between fluctuations of density and velocity are small compared with the term...