Numerical Ocean Circulation Modeling

by Haidvogel, Dale B.; Beckmann, Aike

• ISBN13: 9781860941146
• ISBN10: 1860941141
• Format: Hardcover
• Copyright: 1999-12-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

This book offers a comprehensive overview of the models and methods employed in the rapidly advancing field of numerical ocean circulation modeling. For those new to the field, concise reviews of the equations of oceanic motion, sub-grid-scale parameterization, and numerical approximation techniques are presented and four specific numerical models, chosen to span the range of current practice, are described in detail. For more advanced users, a suite of model test problems is developed to illustrate the differences among models, and to serve as a first stage in the quantitative evaluation of future algorithms. The extensive list of references makes this book a valuable text for both graduate students and postdoctoral researchers in the marine sciences and in related fields such as meteorology, and climate and coupled biogeochemical modeling.

Table of Contents

Preface	vii
Acknowledgements	xiii
The Continuous Equations	1	(36)
Conservation of Mass and Momentum	1	(5)
Conservation of Energy and Heat	6	(3)
The Effects of Rotation	9	(4)
The Equations in Spherical Coordinates	13	(2)
Properties of the Unapproximated Equations	15	(4)
Conservation of angular momentum	15	(1)
Ertel's theorem	16	(2)
Conservation of mechanical energy	18	(1)
The Hydrostatic Primitive Equations	19	(6)
The Boussinesq-approximation	20	(1)
Incompressibility	21	(1)
The hydrostatic approximation	21	(4)
Initial and Kinematic Boundary Conditions	25	(1)
Approximate Systems	26	(11)
The beta-plane (Cartesian) equations	27	(3)
Quasigeostrophy	30	(3)
The shallow water equations	33	(4)
The 1D Heat and Wave Equations	37	(56)
Approximation of Functions	38	(9)
Taylor series	38	(1)
Piecewise linear interpolation	39	(1)
Fourier approximation	40	(5)
Polynomial approximations	45	(2)
Approximation of Equations	47	(4)
Galerkin approximation	47	(1)
Least-squares and collocation	48	(1)
Finite difference method	48	(3)
Example: The One-dimensional Heat Equation	51	(4)
Convergence, Consistency and Stability	55	(3)
Time Differencing	58	(9)
The wave equation	60	(6)
The friction equation	66	(1)
The Advection Equation	67	(4)
Higher-order Schemes for the Advection Equation	71	(2)
Sources of Approximation Error	73	(13)
Phase error/damping error	73	(6)
Dispersion error and production of false extrema	79	(1)
Time-splitting ``error''	79	(1)
Boundary condition errors	80	(1)
Aliasing error/nonlinear instability	80	(4)
Conservation properties	84	(2)
Choice of Difference Scheme	86	(1)
Multiple Wave Processes	87	(2)
Semi-implicit Time Differencing	89	(1)
Fractional Step Methods	90	(3)
Considerations in Two Dimensions	93	(28)
Wave Propagation on Horizontally Staggered Grids	93	(16)
Inertia-gravity waves	95	(5)
Planetary (Rossby) waves	100	(6)
External (barotropic) waves	106	(2)
Non-equidistant grids, non-uniform resolution	108	(1)
Advection and nonlinearities (aliasing)	108	(1)
Time-stepping in Multiple Dimensions	109	(2)
Semi-implicit Shallow Water Equations	111	(1)
Elliptic Equations	112	(3)
Conservation of Energy and Enstrophy	115	(3)
Advection Schemes	118	(3)
Three-Dimensional Ocean Models	121	(42)
GFDL Modular Ocean Model (MOM)	123	(10)
Design philosophy	123	(1)
System of equations	123	(2)
Depth-integrated flow	125	(1)
Spatial discretization, grids and topography	125	(3)
Semi-discrete equations	128	(1)
Time-stepping	129	(1)
Additional features	130	(1)
Concluding remarks	131	(2)
S-coordinate models (SPEM/SCRUM)	133	(12)
Design philosophy	133	(1)
System of equations	133	(3)
Depth-integrated flow	136	(1)
Spatial discretization, grids and topography	136	(4)
Semi-discrete equations	140	(2)
Temporal discretization	142	(1)
Additional features	142	(2)
Concluding remarks	144	(1)
Miami Isopycnic Model (MICOM)	145	(7)
Design philosophy	145	(1)
System of equations	145	(2)
Depth-integrated flow	147	(1)
Spatial discretization, grids and topography	148	(1)
Semi-discrete equations	149	(1)
Temporal discretization	150	(1)
Additional features	150	(1)
Concluding remarks	151	(1)
Spectral Element Ocean Model (SEOM)	152	(10)
Design philosophy	152	(1)
System of equations	152	(1)
Depth-integrated flow	153	(1)
Spatial discretization, grids and topography	154	(3)
Semi-discrete equations	157	(2)
Temporal discretization	159	(2)
Additional features	161	(1)
Concluding remarks	162	(1)
Model Applications	162	(1)
Subgridscale Parameterization	163	(40)
The Closure Problem	164	(3)
Overview of Subgridscale Closures	167	(3)
First Order Closures	170	(3)
Constant eddy coefficients	170	(3)
Higher Order Closures	173	(3)
Local closure schemes	175	(1)
Non-local closure schemes	175	(1)
Lateral Mixing Schemes	176	(12)
Highly scale selective schemes	177	(1)
Prescribed spatially varying eddy coefficients	178	(3)
Adaptive eddy coefficients	181	(1)
Rotated mixing tensors	182	(1)
Topographic stress parameterization	183	(3)
Thickness diffusion	186	(2)
Vertical Mixing Schemes	188	(12)
The vertical structure in the ocean	190	(1)
Surface Ekman layer	190	(3)
Stability dependent mixing	193	(1)
Richardson number dependent mixing	193	(1)
Bulk mixed layer models	194	(2)
Bottom boundary layer parameterization	196	(3)
Convection	199	(1)
Comments on Implicit Mixing	200	(3)
Process-Oriented Test Problems	203	(38)
Rossby Equatorial Soliton	204	(3)
Effects of Grid Orientation on Western Boundary Currents	207	(12)
The free-slip solution	212	(2)
The no-slip solution	214	(5)
Gravitational Adjustment of a Density Front	219	(6)
Gravitational Adjustment Over a Slope	225	(8)
Steady Along-slope Flow at a Shelf Break	233	(5)
Other Test Problems	238	(3)
Simulation of the North Atlantic	241	(40)
Model Configuration	241	(5)
Topography and coastline	242	(1)
Horizontal grid structure	242	(1)
Initialization	243	(1)
Forcing	244	(2)
Spin-up	246	(1)
Phenomenological Overview and Evaluation Measures	246	(11)
Western boundary currents	248	(1)
Quasi-zonal cross-basin flows	249	(1)
Eastern recirculation and ventilation	250	(1)
Surface mixed layer	251	(1)
Outflows and overflows	251	(1)
Meridional overturning and heat transport	252	(1)
Water masses	253	(1)
Mesoscale eddy variability	253	(2)
Sea surface height from a rigid lid model	255	(2)
North Atlantic Modeling Projects	257	(3)
CME	257	(1)
DYNAMO	258	(1)
DAMEE	259	(1)
Sensitivity to Surface Forcing	260	(1)
Sensitivity to Resolution	260	(6)
Effects of Vertical Coordinates	266	(7)
Effects of Artificial Boundaries	273	(2)
Dependence on Subgridscale Parameterizations	275	(1)
Dependence on Advection Schemes	276	(5)
The Final Frontier	281	(4)
Appendix A Equations of Motion in Spherical Coordinates	285	(2)
Appendix B Equation of State for Sea Water	287	(2)
Appendix C List of Symbols	289	(4)
Bibliography	293	(17)
Index	310