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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[Ue]
p' = Re[Pe]
θ' = Re[Θe]

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

  U = -(m/k)W.
From the fourth equation
Θ = -[(dθ/dz)/(i(uk-ν))]W.
From the first equation then
P=[ρ0(uk-ν)/k]U =-[ρ0(uk-ν)m/k2]

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

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