Dike

A geological dike is a sheet-like body or conduit formed when magma enters a fracture cutting across the surrounding rock. A vertical narrow dike can be modeled as a two-dimensional channel for coupled flow, heat transfer and wall melting.

With equal solid and liquid densities, magma entering a dike at constant flux $Q_0$ has $bw=Q_0$: melt added by the walls exactly supplies the increasing cross-sectional storage. If wall heat flux is ${\alpha }_T\rho c_{p}w(T-T_m)$, both walls give $b_t=2{\alpha }_T{c}_{p}w(T-T_m)/L$. Combining this with sensible-heat conservation admits $b=B(t)f(\eta )$, $\eta =2{\alpha }_{T}z/B$, $B^2=4{\alpha }_T{Q}_{0}t/S$. The shape equations are $f(f-\eta f^{\prime })=\vartheta$ and $(S-\eta f){\vartheta }^{\prime }=-(S+\vartheta )\vartheta /f$.

New wall melt enters at the wall melting temperature $T_m$. Besides supplying its latent heat, the flowing magma must warm this added melt. Relative to $T_m$, conservative heat storage is $b(T-T_m)$, so differentiating it retains $(T-T_m)b_t$ and gives the displayed dilution term. Omitting that term requires an additional approximation, such as a small sensible-to-latent heat ratio; it does not follow from equal densities alone.