Volcano Cross-Section (2D): Magma Chamber Pressure, Conduit Flow, Plume Height and Lava Flow Lab

Presets

Live readouts

Graphs

Chamber overpressure and the open/close thresholds

Gas fraction up the conduit (fragmentation at 0.75)

Plume height versus ash flux (current eruption in red)

Volume flux out of the vent

Numerical checks

running...

How to use

Pick a preset (Hawaiian, Strombolian, dome, Vulcanian, Plinian, Pinatubo-class) and watch the chamber pressure rise in the graph until the vent opens. Try: (1) Plinian column, then drag Dissolved water down to 0.3: the gas fraction curve flattens, fragmentation disappears and the eruption turns into a lava flow; (2) raise Melt viscosity by 3 decades: the exit speed collapses and the chamber takes far longer to drain (the relaxation time in the readouts); (3) double the Conduit radius: flux rises 16 times, the plume only about 1.9 times; (4) turn Wind to 35 m/s and compare how far the grains fall; (5) use Trigger eruption or Recharge pulse. Space pauses, R resets.

The science

The chain from chamber to column. A magma chamber of volume V_c is recharged at a rate Q_in. Because magma and the surrounding rock are compressible, the overpressure ΔP above the lithostatic value grows as

d(ΔP)/dt = (K / V_c) (Q_in - Q_out)

with K an effective bulk modulus (here 1 GPa). When ΔP reaches the strength of the roof the vent opens; the eruption stops when ΔP falls to a chosen fraction of that value. This gives repeating cycles of charging and discharging.

Conduit flow. For a cylindrical conduit of radius a and length L filled with a Newtonian melt of viscosity η, the Hagen-Poiseuille law gives

u = ΔP a² / (8 η L), Q = π a² u = π a⁴ ΔP / (8 η L)

The fourth power of the radius is why a few metres of conduit width separate a trickle from a catastrophe. The speed is capped at 80 m/s (a stand-in for choked flow of a gas-rich mixture). Without recharge the overpressure then decays as exp(-t/τ) with τ = 8 η L V_c / (π a⁴ K), the relaxation time shown in the readouts.

Viscosity. η = 10^{log η₁₀₀₀ + (1000 - T)/120 Pa s is a rule-of-thumb temperature dependence (roughly one decade per 120 C), starting from a value chosen for the composition: basalt about 10², andesite 10⁴-10⁶, rhyolite and domes 10⁷-10¹⁰.}

Gas and fragmentation. Water dissolves in melt as w = k √p (Henry's law, sqrt form, k = 4.11e-6 Pa^-1/2). As magma rises, p falls, water exsolves and the gas volume fraction follows from the ideal gas law:

n = w_total - k √p, φ = (n / ρ_g) / (n / ρ_g + (1 - n) / ρ_m), ρ_g = p M / (R T)

When φ reaches about 0.75 the foam fragments into gas plus ash, the signature of an explosive eruption. If φ stays below 0.75 at the vent, the magma flows out as lava (effusive). The slider "water kept to the vent" mimics open-system degassing: degassed magma erupts quietly.

Column height. The rise of a buoyant plume depends only on the heat (mass) flux. Two empirical fits to historic eruptions are drawn side by side:

H = 2.00 V^0.241 (Mastin 2009), H = 1.67 V^0.259 (Sparks 1997)

H in km, V in m³/s of dense-rock equivalent ash-forming magma. A tenfold increase of flux raises the column by only about 1.7 times. Wind is not in these fits; in the picture it bends the column and carries fine ash away.

Lava flow. A Newtonian sheet of thickness h on a slope θ moves with mean speed u = ρ g sinθ h² / (3 η) (Jeffreys). Imposing the flux Q = u w h in a channel of width w gives h = (3 η Q / (w ρ g sinθ))^1/3. The flow stops when it has cooled too far: the Graetz number u h² / (κ L) falls to about 300 (Pinkerton and Wilson), so L = u h² / (300 κ) with thermal diffusivity κ = 10^-6 m²/s.

Ash fallout. A grain of diameter d reaching the column top at height H falls at its terminal speed (Stokes for fine grains, quadratic drag for coarse grains, whichever is smaller) while the wind carries it x = U H / v_t downwind. Fine ash therefore travels hundreds of km; lapilli lands within tens of km.

Checks in the page. (1) numerical integration of the Poiseuille profile against the closed-form flux; (2) the two plume-height fits against each other over five decades of flux; (3) depth-integrated lava profile against the imposed flux; (4) the pressure after one relaxation time against ΔP₀/e; (5) erupted volume against the chamber volume change (V_c ΔP / K); (6) the fragmentation-depth root against φ = 0.75; (7) stepped ash fall against U H / v_t.

Honest simplifications. Laminar Newtonian conduit flow (real magma is two-phase and shear-thinning, and the speed cap is a placeholder for choking); constant conduit radius and a vertical straight conduit; the pressure in the conduit is the weight of the melt column only (gas weight and friction ignored); one chamber, constant K; no crystal content, no vent erosion, no pyroclastic currents or collapse; the plume heights are empirical fits, not a fluid-dynamic plume model; uniform wind; lava is isoviscous and ignores crust formation and lateral spreading; time is sped up by a large and varying factor while charging. Not predictive: use it for intuition about how viscosity, gas, conduit width and chamber size change the style of an eruption.