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Unformatted text preview: Chapter 7 Numerical Dispersion of Linearized SWE This chapter is concerned with the impact of FDA and variable staggering on the fidelity of wave propagation in numerical models. We will use the shallow water equations as the model equations on which to compare various approximations. These equations are the simplest for describing wave motions in the ocean and atmosphere, and they are simple enough to be tractable with pencil and paper. By comparing the dispersion relation of the continuous and discrete systems, we can decide which scales of motions are faithfully represented in the model, and which are distorted. Conversely the diagrams produced can be used to decide on the number of points required to resolve specific wavelengths. The two classes of wave motions encountered here are inertia-gravity waves, and Rossby waves. The main reference for the present work is Dukowicz (1995). The plan is to look at dynamical system of increasing complexity in order to highlight various aspects of the discrete systems. We start by looking at 1D versions of the linearized shallow water equations, and unstaggered and staggered versions of the discrete approximation; in particular we constrast these two approaches for several high order centered difference scheme and show the superiority of the staggered system. Second we look at the impact of including a second spatial dimensional and include rotation but restrict ourselves to second order schemes; the discussion is instead focussed on the various staggering on the dispersive properties. Lastly we look at the dispersive relation for the Rossby waves. 7.1 Linearized SWE in 1D Since we are interested in applying Fourier analysis to study wave propagations, we need to linearize the equations and hold the coefficients of the PDE to be constant. For the shallow water equations, they are: u t + gη x = 0 (7.1) η t + Hu x = 0 (7.2) 95 96 CHAPTER 7. NUMERICAL DISPERSION OF LINEARIZED SWE 7.1.1 Centered FDA on A-grid The straight forward approach to discretizing the shallow water equation in space is to replace the continuous partial derivatives by their discrete counter-parts. The main question is what impact do the choice of variable staggering have on the dispersion relationship. We start by looking at the case where u and η are co-located. We also restrict ourselves at centered approximation to the spatial derivatives which have the form: u x | j = ∑ M m =1 α m ( u j + m- u j- m ) 2Δ x + O (Δ x 2 M ) (7.3) where M is the width of the stencil; the α m are coefficients that can be obtained from the Taylor series expansions (see equations 3.24,3.27, and 3.29. The order of the neglected term on an equally spaced grid is O (Δ x ) 2 M . A similar representation holds for the η derivative. The semi-discrete form of the equation is then: u t | j + g ∑ M m =1 α m ( η j + m- η j- m ) 2Δ x = 0 (7.4) η t | j + H ∑ M m =1 α m ( u j + m- u j- m ) 2Δ x = 0 (7.5) To compute the numerical dispersion associated with the spatially discrete system,...
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This note was uploaded on 01/08/2012 for the course MPO 662 taught by Professor Iskandarani,m during the Spring '08 term at University of Miami.
- Spring '08