Magma Chamber Cycle

What is modelled

Magma flows into a spherical chamber at a steady rate and raises its pressure. The ground above it swells (the pattern a volcano observatory measures with GPS or radar). When the overpressure reaches the tensile strength of the rock, a conduit opens and magma flows out through it; the flow drains the chamber until the conduit closes, and the cycle starts again. The plume height of the eruption comes from an empirical fit.

Equations

• Chamber volume change ΔV = π ΔP a³ / G (shear modulus G = 10 GPa, Poisson ratio 0.25).
• Mogi point source: uplift uz = (1 - ν) ΔV d / [π (r² + d²)3/2], horizontal ur = (1 - ν) ΔV r / [π (r² + d²)3/2], peak (1 - ν) ΔP a³ / (G d²).
• Recharge: dP/dt = (K / V) q with an effective bulk modulus K = 10 GPa; failure at the tensile strength.
• Conduit flow (Poiseuille) Q = π R&sup4; ΔP / (8 μ L), drain dP/dt = -(K / V) Q, so ΔP decays with τ = 8 μ L V / (π R&sup4; K).
• Plume height (Mastin et al. fit) H = 2.00 Q0.241 km for Q in m³/s of dense rock.

Checks run in the page

• The uplift volume integrated over the ground against 2 (1 - ν) ΔV, and the peak uplift formula.
• The parabolic conduit velocity profile integrated against the Poiseuille formula.
• Recharge time from stepping the pressure against V T / (K q).
• Erupted volume and drain time from integrating the falling flow against the closed forms.

Honest simplifications

The chamber is a point source in a uniform elastic half-space, the magma is a single incompressible liquid with no gas or crystals, the conduit is a straight pipe with constant viscosity and the plume height fit is used far outside the range where it was calibrated. The ground motion in the picture is magnified by the factor shown. The 3D monitor has no real units; its slider labels map only loosely to the quantities here.

• Choose Fast recharge and watch the eruptions come more often.
• Choose Viscous plug: the thick magma needs a long time to drain and the plume is small.
• Choose Deep small chamber: the ground bulge is wide and low, the pattern that makes deep sources hard to detect.