﻿ 2D Internal Gravity Waves
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

 2D Internal Gravity Waves

J.R. Holton in his An Introduction to Dynamic Meteorology (p.199) gives the linearized version of the governing equations for a two-dimensional internal gravity wave system as:

#### (∂/∂t + u(∂/∂x))u' + (1/ρ0)(∂p'/∂x) = 0 (∂/∂t + u(∂/∂x))w' + (1/ρ0)(∂p'/∂z) - (g/θ)θ' = 0 ∂u'/∂x + ∂w'/∂z = 0 (∂/∂t + u(∂/∂x))θ' + w'(dθ/dz) = 0

The solution for w' in this system of equations was found by Holton to be of the form w'=Re[We] where W is a complex constant and φ=kx+mz-νt. The system of equations is homogeneous so the solutions for u', p' and θ' must be of the same functional form as that of w'; i.e.,

#### u' = Re[Ueiφ] p' = Re[Peiφ] θ' = Re[Θeiφ]

where U, P and Θ are complex constants.

When the forms, Ue, Pe and Θe are substituted into the equation system the result is a set of linear complex equations in U, P and Θ; i.e.,

#### (uk-ν)U +(k/ρ0)P = 0 i(uk-ν)W +(im/ρ0)P - (g/θ)Θ = 0 mW + kU = 0 (dθ/dz)W + i(uk-ν)Θ = 0

This system of linear homogeneous equation has only the trivial solution unless the determinant of the coefficient matrix is zero. The coefficient matrix is

#### | 0   (uk-ν)   k/ρ0     0       | | i   0 (im/ρ0)     -g/θ   | | m   k 0     0       | | (dθ/dz)   0 0   i(uk-ν) |

The evaluation of a fourth order determinant seems formidable but there are enough zero elements to make the evaluation quite easy. The value of the determinant D is given by

#### -(uk-ν)[-m(-m(uk-ν)/&ρ0] + (k/ρ0)[k(-(uk-ν)2 + (g/θ)(dθ/dz)] = - [(uk-ν)2(k2+m2) - g(d ln(θ)/dz)]/ρ0

If the determinant is to be zero it must be that

#### (uk-ν)2 = gk2(d ln(θ)/dz)/K2

where K2=(k2+m2).

Let g(d ln(θ)/dz) be denoted as N2. Then the allowable frequencies ν are given by

#### ν = uk ±Nk/K

When the determinant condition is satisfied the solutions for U, P and Θ can be expressed in terms of W.

From the third equation it is found that

In summary

#### U = -(m/k)W P = -[ρ0(uk-ν)m/k2]W Θ = i[(dθ/dz)/(uk-ν)]W

Taking into account the dispersion relation these reduce to:

#### U = -(m/k)W P = ±[ρ0Nm/(k(k2+m2)1/2)]W Θ = ±i[(dθ/dz)(k2+m2)1/2)/(Nk)]W

 HOME PAGE OF applet-magic HOME PAGE OF Thayer Watkins