The Shallow Water Wave Equations: Formulation, Analysis and Application

1. 1 AREAS OF APPLICATION FOR THE SHALLOW WATER EQUATIONS The shallow water equations describe conservation of mass and mo mentum in a fluid. They may be expressed in the primitive equation form Continuity Equation _ a, + V. (Hv) = 0 L(l;,v;h) at (1. 1) Non-Conservative Momentum Equations a M("vjt,f,g,h,A) = at(v) + (v. V)v + tv - fkxv + gV, - AIH = 0 (1. 2) 2 where is elevation above a datum (L) ~ h is bathymetry (L) H = h + C is total fluid depth (L) v is vertically averaged fluid velocity in eastward direction (x) and northward direction (y) (LIT) t is the non-linear friction coefficient (liT) f is the Coriolis parameter (liT) is acceleration due to gravity (L/T2) g A is atmospheric (wind) forcing in eastward direction (x) and northward direction (y) (L2/T2) v is the gradient operator (IlL) k is a unit vector in the vertical direction (1) x is positive eastward (L) is positive northward (L) Y t is time (T) These Non-Conservative Momentum Equations may be compared to the Conservative Momentum Equations (2. 4). The latter originate directly from a vertical integration of a momentum balance over a fluid ele ment. The former are obtained indirectly, through subtraction of the continuity equation from the latter. Equations (1. 1) and (1. 2) are valid under the following assumptions: 1. The fluid is well-mixed vertically with a hydrostatic pressure gradient. 2. The density of the fluid is constant.

Contenu
I. Introduction.- Areas of Application for the Shallow Water Equations.- Finite Element Methods for Solution of the Shallow Water Equations.- Methods for Analyzing Spatial Oscillations in Numerical Schemes.- Methods for Analyzing Stability of Numerical Schemes.- II. Equation Formulation.- Primitive Equation Form.- Wave Equation Form.- Generalized Wave Equation Form.- Linearized Form of the Continuity and Momentum Equations.- III. Fourier Analysis Methods.- Fourier Analysis: An Accuracy Measure.- Amplitude of Propagation Factors Arising from Second Degree Polynomials.- IV. Stability.- General Concepts.- Routh-Hurwitz and Liénard-Chipart.- Routh-Hurwitz and Orlando.- Factorization of Higher Degree Polynomials into Lower Degree Polynomials.- Determination of Stability for a Product of Polynomials.- V. Explicit Methods Using Various Spatial Discretizations.- Equal Node Spacing and Constant Bathymetry in One Dimension.- Application to Unequal Node Spacing.- Applications with Even Node Spacing and Variable Bathymetry.- Application to a Rectangular Grid.- VI. Implicit Methods.- Reducing the Number of Time Dependent Terms in the Matrix for the Wave Equation.- Explicit Treatment of the Coriolis Term in an Implicit Wave Continuity Equation.- Repeated Back Substitutions Replacing Decompositions.- The Generalized Wave Continuity Equation.- VII. Spatial Oscillations.- N-Dimensional Uniform Rectangular Grid.- N-Dimensional Nonuniform Rectangular Grid with Multi-Information Nodes.- Leapfrog Scheme and Wave Equation Formulation on Linear Elements.- Leapfrog Scheme and Wave Equation Formulation on Quadratic Elements.- The Use of Dispersion Analysis in Evaluating Numerical Schemes.- The 2?x Test: Assessing the Ability to Suppress Node-to-Node Oscillations.- VIII. TemporalOscillations.- Numerical Artifacts.- A Different Three Time Level Approximation of the Momentum Equations.- A Two Time Level Approximation of the Momentum Equations.- IX. Applications.- Application to Quarter Circle Harbor.- Application to the Southern Part of the North Sea I.- Application to the Southern Part of the North Sea II.- X. Conclusions.- A. Equivalent Formulations of Conditions Which Guarantee Roots of Magnitude Less than Unity.