The Shallow Water Wave Equations: Formulation, Analysis and Application

Abstract

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. 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. Temporal Oscillations -- 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.