w = 0

removes the quadratic term and gives

w_{t} = Δ^{3} w - R Δ_{1} w .

The three homogeneous boundary conditions select the vertical normal modes

W_{j} (z) = A_{j} sin (jπ z), j = 1, 2, \dots .

Take a horizontal Laplacian eigenfunction satisfying

Δ_{1} f + a^{2} f = 0.

W_{j} f

- (a^{2} + j^{2} π^{2})

. The growth rate is consequently

s (a, R, j) = R a^{2} - (a^{2} + j^{2} π^{2})^{3} .

Neutrality occurs at

R_{c, j} (a) = \frac{( a ^{2} + j ^{2} π ^{2} ) ^{3}}{a ^{2}} .

For fixed nonzero

a

, this increases strictly with

j

, so the first unstable vertical mode is

j = 1

and

R_{c} (a) = \frac{( a ^{2} + π ^{2} ) ^{3}}{a ^{2}} .

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