= Solution
For an <axisymmetric vertical mode of a shearing sheet> in this <Boussinesq approximation>, the perturbations have no horizontal gradients, so background-shear advection vanishes. The linear <Coriolis acceleration> and perturbation advection of the background shear give
$$
su'_x-2\Omega u'_y=-N^2\theta',\qquad
su'_y+\frac\Omega2u'_x=0,\qquad
su'_z=-\frac{ik}{\rho_0}P',
$$
while <thermal conduction> and <incompressibility> give
$$
(s+\beta)\theta'=u'_x,\qquad iku'_z=0,\qquad \beta=\xi k^2>0.
$$
Here $\beta$ is the <thermal diffusion rate of a Fourier mode>, not the cooling exponent in Question 2. Since $k\ne0$, <incompressibility> forces $u'_z=0$, and the vertical momentum equation then forces $P'=0$ for this nonzero-<wavenumber> amplitude.
The three remaining amplitudes satisfy
$$
\begin{pmatrix}s&-2\Omega&N^2\\\Omega/2&s&0\\-1&0&s+\beta\end{pmatrix}
\begin{pmatrix}u'_x\\u'_y\\\theta'\end{pmatrix}=0.
$$
A nontrivial solution requires the <determinant> to vanish, giving the <convective overstability> <dispersion relation>
$$
\boxed{(s+\beta)(s^2+\Omega^2)+N^2s=0,\quad\text{or}\quad
s^3+\beta s^2+(N^2+\Omega^2)s+\beta\Omega^2=0.}
$$
No division by $s$ or $s+\beta$ was needed, so the <thermal energy mode of a shearing sheet> is retained. The notation $N^2$ denotes a signed <buoyancy frequency> squared and can be negative for an adverse stratification; it must not be restricted to the square of a real positive frequency.
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