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Unformatted text preview: MPO 662 – Problem Set 6 Abstract The aim of this homework is to perform an in-depth analysis of the energy conserving discretiza- tion of the shallow water equation presented in class. The analysis centers on the continuous and discrete equations, and a working code should be produced at the end. This code must be validated against a known test problem. The reference to consult for this problem is: Sadourny (1975) and which can be downloaded from the course web site. 1 Introduction The inviscid shallow water equations in vector form are: ∂ u ∂t + u · ∇ u + f k × u + g ∇ η = 0 (1) ∂η ∂t + ∇ · [( H + η ) u ] = 0 . (2) where u is the velocity vector, η the surface displacement, H the resting depth of the fluid, f is the Coriolis parameter, k the unit vertical vector, H the resting depth of the fluid, and g the gravitational acceleration. The instantaneous layer thickness is h = H + η . The resting depth can be a function of space, but not time; the continuity equation can thus be written as ∂h ∂t + ∇ · h u = 0 2 The continuous equations Defining the vorticity: ζ = ∇× u = v x − u y , and potential vorticity q = f + ζ h , show that the momentum equation can be re-written in the equivalent form: ∂ u ∂t + q k × h u + ∇ parenleftbigg u · u 2 + gη parenrightbigg = 0 (3) Show that this system of equation conserves the following quantities Total Volume: integraldisplay A η d A (4) Total Energy: integraldisplay A bracketleftBigg gη 2 2 + h u · u 2 bracketrightBigg d A (5) Potential Enstrophy: integraldisplay A h q 2 2 d A, q = f + ζ h (6) Hints: Taking the inner product of the momentum equation with h u to derive the energy equation. To derive the enstrophy conservation equation, first derive the equation for potential vorticity, multiply it by hq and derive the conservation equation for potential enstrophy. Use the vector identities in the appendix, 6 in your derivations. 1 a115 a115 u u a113 η j,k a115 a115 v v a99 a99 ζ ζ a99 a99 ζ ζ Figure 1: Arakawa C-grid configuration. The depth and pressure are defined at the center of cell ( j,k ). The x-component of the velocity is defined on vertical cell edges ( j + 1 2 ,k ) and the y-components at the centers of horizontal edges ( j,k + 1 2 ). The vorticity, coriolis force, and enstrophy are defined on cell corners ( j + 1 2 ,k + 1 2 ). 3 The discrete equations The equations are discretized using finite differences, and the variables are staggered according to the Arakawa C-grid shown in figure 1. Show that the following finite difference scheme conserves energy discretely: ∂u ∂t − q V x y + δ x Φ = 0 (7) ∂v ∂t + q U y x + δ y Φ = 0 (8) ∂η ∂t + δ x U + δ y V = 0 (9) The new variables are defined on the Arakawa C-grid as follows: x-Mass Fluxes U = h x u u-point (10) y-Mass Fluxes V = h y v v-point (11) Total Head Φ = gη + u 2 x + v 2 y 2 η-point (12) Potential vorticity q = δ x v − δ y u + f h xy ζ-point (13) The operator δ x a and...
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- Spring '08
- Energy, Shallow water equations, Equatorial Rossby Soliton