 Conduit equation

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 $\Delta \rho g{a}_0^2/(8\lambda {\mu }_o)$, 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.