Eruption column height to mass eruption rate calculator: power law and plume theory, erupted mass and DRE volume (free)

Turn a volcanic column height into a mass eruption rate, or the other way round, with an editable power law (Mastin et al. 2009 by default) and with Morton-Taylor-Turner buoyant plume theory. Get the erupted mass over a duration and its dense-rock-equivalent volume.

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Height
Scaling
Rate, mass
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I know Column height above the vent (km) Vent elevation (km above sea level) Magma density, dense rock (kg/m³)

Heights are the top of the column, measured from the vent. Type large numbers as 1e7 or 1.5×10^7. The dense-rock density turns mass into a dense-rock-equivalent (DRE) volume; 2500 kg/m³ is a round value, change it for your magma.

Result

Data

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Empirical power law

H = a · V^b, with H in km above the vent and V the DRE volume flux in m³/s. The defaults a = 2.00, b = 0.241 are the fit of Mastin et al. (2009, Journal of Volcanology and Geothermal Research 186) to well-observed eruptions; type other coefficients to use another fit in the same form.

a b

Buoyant plume theory (Morton, Taylor and Turner)

A plume from a point source of buoyancy flux F₀, entraining air at α times its rise speed, in air of steady buoyancy frequency N, rises to H = C · α^−1/2 F₀^1/4 N^−3/4 (C from integrating the plume equations, shown with the result). F₀ = g · Q · c_p · ΔT / (ρ_a · c_a · T_a): the heat the erupted mass brings, turned into buoyancy. The defaults are round textbook values.

Entrainment coefficient α (top-hat) Buoyancy frequency N (1/s) Magma heat capacity c_p (J/(kg·K)) Magma minus air temperature ΔT (K) Air density ρ_a (kg/m³) Air heat capacity c_a (J/(kg·K)) Air temperature T_a (K)

This is the simple Boussinesq plume: it leaves out wind, moisture, the dense gas jet near the vent and the change of air with height, which is why it and the power law can differ by tens of percent.

Erupted mass and volume

Duration (hours) Deposit bulk density (kg/m³, optional)

Assumes the rate stayed steady for the whole time. The bulk volume is the volume of the loose deposit, for comparing with a tephra volume.

What it does

The height of an eruption column is one of the few things you can see during an eruption, and it rises with the mass of magma erupted per second. This calculator converts column height ↔ mass eruption rate in two independent ways, then turns the rate into the erupted mass over a duration and its dense-rock-equivalent (DRE) volume:

an empirical power law H = a · V^b, H in km above the vent and V the DRE volume flux in m³/s. The default a = 2.00, b = 0.241 is the widely used fit of Mastin et al. (2009); both numbers are editable, so you can use another fit written in the same form;
buoyant plume theory (Morton, Taylor and Turner 1956): a turbulent plume that entrains air as it rises through stably layered air stops at H = C · α^−1/2 F₀^1/4 N^−3/4.

How to use it

Choose whether you know the column height (km above the vent) or the mass eruption rate (kg/s), and type it.
Type the vent elevation, to get heights above sea level, and the dense-rock density of the magma.
Change the power-law coefficients or the plume theory values if you want: the entrainment coefficient α, the air’s buoyancy frequency N (about 0.01 per second in the troposphere), and the heat values that give the buoyancy flux F₀ = g Q c_p ΔT / (ρ_a c_a T_a).
Type the duration and, if you like, a deposit bulk density to compare with a mapped tephra volume.
Read both answers, see them on the chart of height against rate, and download the report or the curves.

How the plume theory part works

The page integrates the three top-hat plume equations for volume, momentum and buoyancy flux from a point source, upwards until the plume’s momentum runs out. In scaled units the top is at 1.819 and the level of neutral buoyancy (where the plume spreads) at 1.382, about 0.76 of the top. With the buoyancy flux written as the physical flux F₀ in m⁴/s³, that gives C = 1.366. Because every quantity enters as a power, height grows only as the fourth root of the rate: doubling the column height takes 16 times the rate.

Worked example

A 15 km column above the vent with the defaults:

Power law: V = (15/2.00)^1/0.241 = 4,275 m³/s DRE, so 1.07 × 10⁷ kg/s at 2,500 kg/m³.
Plume theory: F₀ = 1.45 × 10⁸ m⁴/s³, so 3.68 × 10⁶ kg/s, with neutral buoyancy at 11.4 km.
Over 3 hours the power law rate erupts 1.15 × 10¹¹ kg, 4.62 × 10⁷ m³ DRE (0.0462 km³).

Going the other way, 1 × 10⁷ kg/s gives 14.8 km by the power law and 19.3 km by plume theory. The two differ because the simple plume leaves out wind, moisture and the jet near the vent; real eruptions scatter widely around any fit.

Frequently asked questions

Which answer should I use? The power law is fitted to real eruptions and is the usual first estimate. The plume theory shows how the result depends on the atmosphere and on the physics. Treat both as order-of-magnitude estimates.

Does wind matter? Yes: strong wind bends plumes over and lowers them, so a bent-over plume gives a too-low rate from either relation.

Is anything sent anywhere? No. Everything runs in your browser and nothing leaves your device.

Where the numbers come from

Power law form H = a·V^b with default coefficients a = 2.00, b = 0.241 from Mastin, L. G. et al. (2009), A multidisciplinary effort to assign realistic source parameters to models of volcanic ash-cloud transport and dispersion during eruptions, Journal of Volcanology and Geothermal Research 186, 10-21 (two coefficients quoted with credit, editable).
Plume theory: Morton, B. R., Taylor, G. I. and Turner, J. S. (1956), Turbulent gravitational convection from maintained and instantaneous sources, Proceedings of the Royal Society A 234, 1-23; equations integrated by fibub.
Heat capacities, temperatures, air density, buoyancy frequency, entrainment coefficient and densities: round textbook values chosen by fibub, editable.

Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.

Last updated 2026-11-09.