Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 4 / c

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- Solution c

Solution

Let

c = \frac{N}{∣ m ∣}, D = \frac{ω ^{2}}{c ^{2}} - k^{2} .

Solving zonal momentum and incompressibility for

\overset{u}{^}

and

\hat{ϕ}

in terms of

\overset{v}{^}

gives

\overset{u}{^} = \frac{i}{D} (\frac{β y ω}{c ^{2}} \overset{v}{^} - k \overset{v}{^}_{y}),

\hat{ϕ} = \frac{i}{D} (k β y \overset{v}{^} - ω \overset{v}{^}_{y}) .

Substitution into meridional geostrophic balance cancels the terms involving

D

and yields

\overset{v}{^}_{yy} - \frac{β ^{2} y ^{2} m ^{2}}{N ^{2}} \overset{v}{^} = \frac{k β}{ω} \overset{v}{^} .

ξ = \frac{β ∣ m ∣}{N} y .

The equation becomes

\overset{v}{^}_{ξξ} - ξ^{2} \overset{v}{^} = \frac{k N}{ω ∣ m ∣} \overset{v}{^} .

Using the Hermite polynomial eigenvalues gives

ω_{n} = - \frac{N k}{( 2 n + 1 ) ∣ m ∣}, n = 1, 2, \dots,

and

\overset{v}{^}_{n} (y) = V_{n} H_{n} (ξ) e^{- ξ^{2} /2} .

The associated pressure amplitude is

\hat{ϕ}_{n} = \frac{i ( k β y v ^ _{n} - ω _{n} v ^ _{n, y} )}{ω _{n}^{2} m ^{2} / N ^{2} - k ^{2}} .

For

n = 0

, the denominator used to solve for

\overset{u}{^}

and

\hat{ϕ}

vanishes because

ω_{0}^{2} m^{2} / N^{2} = k^{2}

. The inversion therefore assumed precisely the condition that excludes that degenerate case; it must be analyzed separately and is not a member of this Equatorial Rossby wave family.

Solved by gpt-5.6-sol high.

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