Laplace Tidal Equations - Horizontal Structure | Massachusetts Institute of Technology - Edubirdie

University:
Massachusetts Institute of Technology
Course:
12.802 | Wave Motions in the Ocean and Atmosphere
Academic year:

2024
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61

Pages:

2
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umfikie3qn

Laplace Tidal Equations - horizontal structure Equations After separation of variables of the original linearized, hydrostatic, pressure coordinate equations (with the "standard approximation" of neglecting the horizontal part of the rotation) 1 are: ∂u/∂t + f x u = -grad Φ ∂Φ/∂t + gH_eq div u = 0 Here f is f(φ)ê_r. The vertical structure of the horizontal velocities satisfies the equation: ∂/∂p (1/S ∂/∂p F) = -1/gH_eq F with S = -Π ∂̄θ/∂p. The solution to this, with suitable boundary conditions, defines the equivalent depths H_eq. Solution of the horizontal eqns. If we take ∂/∂t of the momentum equations and subtract it from f x the same equations, we find: ∂²u/∂t² + f x u = -grad Φ + f x grad Φ For motions with frequency ω, the velocity is given by: u = 1/(f²-ω²) (grad Φ + f x grad Φ) Taking the divergence and substituting in the mass equation yields: ∂²Φ/∂t² - gH_eq div (1/(f²-ω²) grad Φ + 1/(f²-ω²) f x grad Φ) = 0 Using the vector identity curl Φ v = grad Φ x v + Φ curl v gives: ∂²Φ/∂t² - gH_eq div (1/(f²-ω²) grad Φ + gH_eq div (1/(f²-ω²) curl f Φ) = 0 For the spherical case, with f = 2Ω sin φ, the last term simplifies to: gH_eq (1/a cos φ) ∂/∂λ [Φ (1/a) ∂/∂φ (f/(f²-ω²))] If we put in Φ ~ exp(imλ - iωt), we end up with a single equation for the meridional structure, with the frequency acting somewhat like the eigenvalue. Small scale, high frequency If we consider small scale waves in the sense that grad Φ/Φ >> grad f/f, we can drop the last term and pull the f²-ω² out to find [1 - gH_eq/(f²-ω²)∇²] ∂Φ/∂t = 0 which has a zero frequency mode and a mode with ω² = f² + gH_eq k² where the wavenumber is defined by the solutions of ∇²Φ = -k²Φ as in the barotropic problem. These are the gravity waves. Low frequency The other mode has a low frequency (at least in the small scale limit) and satisfies (∂/∂t - gH_eq div (1/f²) grad) ∂Φ/∂t - gH_eq (1/a cos φ) (β/f²) ∂Φ/∂λ = 0 (at least away from the equator). In the case where the scales are still small, we can approximate this by [∂/∂t (∇² - f²/gH_eq) + β/(a cos φ) ∂/∂λ] Φ = 0 a form of the Rossby wave equation. The local phase speed c = a cos φ(ω/m) is given by c = -β/(k² + f²/gH_eq)

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