f_{0} ψ_{z t}^{'} + N^{2} w^{'} = 0.

Impermeability means

w^{'} = 0

z = - H

and

z = 0

. For a normal mode with nonzero frequency this is equivalent to the Neumann conditions

\hat{ψ}^{'} (- H) = 0, \hat{ψ}^{'} (0) = 0,

where the primes on

\hat{ψ}

z

. Equivalently,

ψ_{z}^{'} = 0

at both rigid boundaries.

Solved by gpt-5.6-sol high.

ii

Solution

q_{t}^{'} + β ψ_{x}^{'} = 0.

For

K^{2} = k^{2} + l^{2}

- ω [- K^{2} \hat{ψ} + \frac{f _{0}^{2}}{N ^{2}} \hat{ψ}^{''}] + β k \hat{ψ} = 0.

Thus the vertical structure equation is

\hat{ψ}^{''} - \frac{N ^{2}}{f _{0}^{2}} (K^{2} + \frac{β k}{ω}) \hat{ψ} = 0 .

The rigid-boundary eigenfunctions are

\hat{ψ}_{n} (z) = cos \frac{nπ z}{H}, n = 0, 1, 2, \dots .

ω_{n} (k, l) = - \frac{β k}{k ^{2} + l ^{2} + \frac{f _{0}^{2}}{N ^{2}} ( \frac{nπ}{H} ) ^{2}} .

The

n = 0

n \geq 1

are baroclinic vertical modes.

Solved by gpt-5.6-sol high.

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