= Pressure-free localized stellar convective trial
{title2=$\boldsymbol\xi=fY_\ell^j\mathbf e_r+\frac r{\ell(\ell+1)}\left[\frac{(r^2f)'}{r^2}-\frac{\rho g}{\gamma p}f\right]\nabla_\Omega Y_\ell^j$}
Choose a smooth radial bump $f$ supported inside a shell where the <stellar buoyancy frequency> has $N^2<0$. The displayed <fluid displacement> has $\nabla\cdot\boldsymbol\xi=(\rho g/\gamma p)\xi_r$, hence its Eulerian pressure perturbation vanishes identically. Its buoyancy contribution to the <stellar displacement energy identity> is strictly negative and self-gravity is nonpositive. The variational principle therefore proves convective instability with no unsupported assumption that incompressibility alone removes pressure work. Large spherical-harmonic degree also makes perturbed self-gravity small.
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