Magma

Magma is molten or partially molten rock within a planet. Its dynamic viscosity, cooling and interaction with the surrounding elasticity influence transport and storage.

A magma intrusion is the emplacement of magma into surrounding rock. A shallow, approximately axisymmetric intrusion can lift an elastic plate roof, with spreading determined by viscous peeling of an elastic plate or fracture toughness.

With the axisymmetric clamped-plate deflection and fixed front curvature ${\kappa }_f>0$, a magma intrusion hasThese relations describe the advancing fracture branch, with roof bending rather than hydrostatic pressure controlling the interior load.

Let $V_T$ be total magma volume shared between a reservoir and an advancing toughness-controlled magma intrusion, $p_r=E(V_T-V)$, and conduit conductance $\beta >0$. With $A=\sqrt{8\pi /3}B{\kappa }_f^{3/2}$,Equilibria satisfy $\sqrt{V}(V_T-V)=A/E$. Two positive equilibria exist when $A/E<2V_T^{3/2}/(3\sqrt{3})$; the smaller is unstable and the larger is stable under this volume equation. An advancing solution between them approaches the larger. The singular pressure as $V\to 0$ prevents this late-time model from nucleating an intrusion by itself.

If a positive vapour-tip pressure magnitude $p_T$ sets $h_f\sim p_T{l}_p^4/B$, matching $h_f/l_p^2$ to the interior curvature ${\kappa }_i$ gives $l_p\sim (B{\kappa }_i/p_T{)}^{1/2}$. The viscous peeling of an elastic plate law then yields $\dot{R}\sim B^{3/2}{\kappa }_i^{7/2}/(\mu p_T^{1/2})$. For constant injected volume flux $Q$, $V=Qt$ and ${\kappa }_i\sim V/R^4$, henceOrder-one coefficients require solving the local peeling profile. The scaling assumes a front region much shorter than the intrusion radius.