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 Incoming links: Atmospheric internal gravity wave

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 3 / f / Solution 2026-09-28

Let

I = \int w^{2} d z .

Because the Rayleigh quotient is stationary with respect to first-order changes of its eigenfunction, only the explicit dependence of

l^{2}

on

c = ω / k

contributes when the quotient is differentiated. Thus

2 k d k = \frac{1}{I} \int \frac{\partial l ^{2}}{\partial c} w^{2} d z d c,

where

\frac{\partial l ^{2}}{\partial c} = \frac{2 N ^{2}}{( u - c ) ^{3}} - \frac{u _{zz}}{( u - c ) ^{2}} .

It follows that

\frac{d c}{d k} = \frac{2 k I}{\int ( \partial l ^{2} / \partial c ) w ^{2} d z} .

Since

ω = k c

, the horizontal group velocity is

\frac{\partial ω}{\partial k} = c + \frac{2 k ^{2} I}{\int [ \frac{2 N ^{2}}{( u - c ) ^{3}} - \frac{u _{zz}}{( u - c ) ^{2}} ] w ^{2} d z} .

For a stationary ridge wave,

c = 0

. Assume

\overline{u} > 0

, so downstream is the positive

x

direction. The stated inequality is

l^{2} > - \frac{N ^{2}}{u ^{2}} ⟺ \frac{2 N ^{2}}{u ^{3}} - \frac{u _{zz}}{u ^{2}} > 0.

The denominator in the group-velocity formula is then positive, and

\partial ω / \partial k > 0 .

The atmospheric internal gravity waves generated by the ridge consequently carry their wave packet downstream.

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Vertical trapping of an atmospheric gravity wave Created 2026-09-28 Updated 2026-10-05

An atmospheric internal gravity wave is vertically trapped when its squared vertical wavenumber changes from positive to negative with altitude. A decrease of the Scorer parameter, caused for example by increasing wind speed or decreasing buoyancy frequency, creates a turning level and an evanescent upper region.

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