= Conduit equation
{title2=$A_T+(A^2[1-(A_T/A)_Z])_Z=0$}
The conduit equation evolves the positive cross-sectional area $A$ of a <viscous buoyant conduit>. Its scaling uses axial length $a_0/\sqrt{8\lambda}$ and <velocity> $\Delta\rho ga_0^2/(8\lambda\mu_o)$, where $\lambda=\mu_i/\mu_o$. Linearization at $A=1$ gives $\omega=2k/(1+k^2)$: <phase velocity> is upward but <group velocity> changes sign at $|k|=1$. Nonlinear elevation waves obey the <solitary-wave amplitude-speed relation for the conduit equation>.
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