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A direct elimination of pressure gives the same result as taking a double curl of the momentum equation. Incompressibility gives ${\nabla }^{2}p=\mathrm{Ra}\mathrm{Pr}{\partial }_z\theta$. Apply ${\nabla }^2$ to the vertical momentum equation and subtract this pressure derivative to obtainHere ${\nabla }_h^2={\partial }_x^2+{\partial }_y^2$ is the horizontal Laplacian. At $z=0,1$, impermeability gives $w=0$ and fixed temperature gives $\theta =0$. The stress-free boundary condition gives ${\partial }_{z}u={\partial }_{z}v=0$, since horizontal derivatives of the boundary value $w=0$ vanish. Differentiating incompressibility once in $z$ therefore givesFor a horizontal Fourier mode $X=e^{i(k_{x}x+k_{y}y)}$, ${\nabla }_h^{2}X=-{\lambda }^{2}X$ with real $\lambda =\sqrt{k_x^2+k_y^2}$. More generally the nonnegative spectrum of $-{\nabla }_h^2$ follows by integration by parts for periodic or square-integrable horizontal disturbances. Substituting the separated normal mode gives, with $D=d/dz$,The boundary conditions admit a complete Fourier sine series in $z$. For a nonzero horizontal wavenumber, take $W=W_n\sin (n\pi z)$, $\Theta ={\Theta }_n\sin (n\pi z)$, and set $a_n^2={\lambda }^2+n^2{\pi }^2$. The two amplitude equations areThus the stress-free convection growth-rate polynomial isIts discriminant is $(\mathrm{Pr}-1{)}^2{a}_n^4+4\mathrm{Ra}\mathrm{Pr}{\lambda }^2/a_n^2$, so both growth rates are real for $\mathrm{Ra}>0$. For $\mathrm{Ra}<0$, its two coefficients after the leading term are positive: their sum gives ${\sigma }_1+{\sigma }_2=-(1+\mathrm{Pr})a_n^2<0$, and their product is $\mathrm{Pr}(a_n^4-\mathrm{Ra}{\lambda }^2/a_n^2)>0$. Real roots are both negative; nonreal roots have real part $-(1+\mathrm{Pr})a_n^2/2<0$. Hence stable heating from above damps every mode. Horizontally uniform thermal modes and uncoupled horizontal viscous modes are also purely damped; impermeability and incompressibility exclude a horizontally uniform nonzero $w$.

Because the unstable-side roots are real, the onset is stationary, an exchange of stabilities in stress-free convection. Setting $\sigma =0$ gives the free-slip convection neutral curveWith $x={\lambda }^2$, its derivative has the sign of $2x-n^2{\pi }^2$. Its minimum is at $x=n^2{\pi }^2/2$, with value $27n^4{\pi }^4/4$. Minimizing also over $n\ge 1$ gives the critical values