= Solution
{c}
For an <inviscid flow>, the force per unit volume is $\mathbf F=-\nabla p-\rho g\hat{\mathbf z}$. The gravitational <acceleration> has zero <curl>. Using the <curl> product rule,
$$
\nabla\times\left(-\frac1\rho\nabla p\right)
=-\nabla(\rho^{-1})\times\nabla p
=\frac{\nabla\rho\times\nabla p}{\rho^2}.
$$
The supplied two-dimensional <vorticity equation> therefore becomes
$$
\boxed{\partial_t\boldsymbol\omega+\mathbf v\cdot\nabla\boldsymbol\omega
=\frac{\nabla\rho\times\nabla p}{\rho^2}.}
$$
The out-of-plane <vorticity> has no stretching term in this two-dimensional setting. The source is <baroclinic vorticity generation>: differently directed <mass density> and <pressure gradients> generate rotation even from zero initial <vorticity>.
For an explicit sign check, set $\omega_y=\partial_z u-\partial_x w$, with $z$ upward. A thin smoothed interface has $\rho\simeq\bar\rho(z-\eta(x,t))$, with $\bar\rho'>0$. If $P(x,t)$ denotes the total <pressure> evaluated on the interface, then within the thin layer $p_x+\eta_xp_z\simeq P_x$ and
$$
\frac{(\nabla\rho\times\nabla p)_y}{\rho^2}
\simeq\frac{\bar\rho'}{\rho^2}P_x.
$$
Integrating through that layer and taking its thickness to zero gives the <vortex sheet> strength $\gamma=u_2-u_1$ and its linear evolution:
$$
\partial_t\gamma=\left(\frac1{\rho_1}-\frac1{\rho_2}\right)P_x.
$$
For a growing Fourier mode, the potentials from the preceding calculation with $\alpha$ replaced by $k$ give
$$
P-p_*=-\frac{2\rho_1\rho_2}{\rho_1+\rho_2}g\eta,\qquad
\boxed{\partial_t\gamma=-2\mathcal A g\eta_x.}
$$
Thus the right-hand descending slope of a light-fluid crest produces positive $\omega_y$, while the left-hand ascending slope produces negative $\omega_y$. Their induced <velocity> raises the light-fluid crest and lowers the heavy-fluid trough, increasing the slope and generating more <vorticity>. This is the positive feedback underlying <Rayleigh-Taylor instability>.
\Image[/past-exam-of-the-fluids-course-of-the-university-of-oxford/2022-rayleigh-taylor-feedback.png]
{title=Baroclinic feedback and finite-depth growth of an unstable density interface}
One can close this mechanism quantitatively. For $\eta=H(t)\cos(kx)$, the <kinematic boundary conditions> give $\gamma=2\dot H\coth(kh)\sin(kx)$. Combining with the sheet-generation equation gives
$$
\ddot H=\mathcal A gk\tanh(kh)H.
$$
For $kh\ll1$, the lids confine motion throughout both layers and horizontal redistribution is slow, yielding $\sigma\sim\sqrt{\mathcal A gh}\,k$. For $kh\gg1$, motion is localized within depth $k^{-1}$ on each side. Less fluid must accelerate at shorter <wavelength>, giving $\sigma\sim\sqrt{\mathcal A gk}$. Since $k_n=n\pi/b$, the ideal <growth rate> increases with $n$ with no short-wave cutoff.
The sharp-interface result requires $k\delta\ll1$ for the smoothed layer. When $k\delta$ becomes order one, it no longer applies. A smooth adverse <mass density> gradient has a local overturning time of order $[g\bar\rho'/\bar\rho]^{-1/2}$ rather than the unbounded sharp-interface rate. Finite molecular <diffusion> also relaxes small-scale <mass density> perturbations on a time of order $(Dk^2)^{-1}$, and <viscosity>, if included, damps small-scale <velocity>. Weak <diffusion> motivates the thin-layer picture; it does not justify extrapolating the discontinuous <dispersion relation> to arbitrary $n$.
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