A model for steady flows of magma-volatile mixtures

A general one-dimensional model for the steady adiabatic motion of liquid-volatile mixtures in vertical ducts with varying cross-section is presented. The liquid contains a dissolved part of the volatile and is assumed to be incompressible and in thermomechanical equilibrium with a perfect gas phase, which is generated by the exsolution of the same volatile. An inverse problem approach is used – the pressure along the duct is set as an input datum, and the other physical quantities are obtained as output. This fluid-dynamic model is intended as an approximate description of magma-volatile mixture flows of interest to geophysics and planetary sciences. It is implemented as a symbolic code, where each line stands for an analytic expression, whether algebraic or differential, which is managed by the software kernel independently of the numerical value of each variable. The code is versatile and user-friendly and permits to check the consequences of different hypotheses even through its early steps. Only the last step of the code leads to an ODE problem, which is then solved by standard methods. In the present work, the model is applied to the study of two sample cases, representing the ascent of magma-gas mixtures on the Earth and on a Jupiter’s satellite, Io. Both cases lead to approximate but realistic descriptions of explosive eruptions, by taking pressure curves as inputs and outputting conduit shapes together with mixture density, temperature and velocity along the ducts.

💡 Research Summary

The paper presents a comprehensive one‑dimensional, steady‑state, adiabatic model for the ascent of magma‑volatile mixtures in vertical conduits whose cross‑section can vary with height. The liquid phase (magma) is treated as incompressible, containing a dissolved fraction of the volatile component, while the exsolved volatile forms a perfect‑gas phase. Both phases are assumed to be in thermomechanical equilibrium, which allows the authors to couple the gas equation of state with the solubility law of the volatile in the melt. The governing equations—mass continuity, momentum balance, and energy conservation—are written in a form that explicitly includes the pressure gradient, gravity, and frictional losses, as well as the geometric term arising from a variable conduit area.

A distinctive feature of the work is the inverse‑problem formulation: the pressure distribution along the conduit is prescribed as an input datum, reflecting the fact that pressure profiles can often be inferred from geophysical observations or imposed by boundary conditions. Given this pressure curve, the model analytically derives expressions for the mass flux, velocity, temperature, mixture density, and the conduit area as functions of height. The derivation proceeds symbolically; each line of the implementation corresponds to an algebraic or differential expression that the underlying symbolic engine manipulates independently of numerical values. This architecture makes it possible to test alternative hypotheses (e.g., different solubility models, non‑adiabatic heating, variable gas composition) at early stages without rewriting the code. Only in the final stage does the problem reduce to a single ordinary differential equation for the conduit area, which is then solved numerically with standard ODE solvers such as Runge‑Kutta.

The authors validate the framework with two case studies. The first represents a terrestrial basaltic eruption, where the volatile phase consists mainly of H₂O and CO₂ dissolved in a basaltic melt. The prescribed pressure drops from atmospheric values at the vent to several megapascals at depth, reproducing the rapid decompression that drives explosive eruptions. The model predicts a strong nozzle‑like contraction of the conduit, a rapid acceleration of the mixture to near‑sonic or supersonic speeds, and a concomitant cooling of the magma due to gas expansion. The second case simulates an eruption on Io, Jupiter’s volcanically active moon, using a sulfur‑rich gas mixture and a silicate melt. Despite the very different gravity and ambient pressure, the model again yields realistic conduit shapes and flow parameters, illustrating its adaptability to planetary environments.

Key insights emerging from the analysis include: (i) the central role of the prescribed pressure curve in controlling conduit geometry and flow acceleration; (ii) the importance of thermodynamic equilibrium in linking gas exsolution to temperature and density evolution; (iii) the ability of a simple 1‑D framework to capture the essential physics of explosive eruptions, such as rapid gas expansion, nozzle formation, and supersonic flow. The symbolic implementation also highlights the model’s flexibility: users can readily replace the perfect‑gas assumption, introduce non‑adiabatic heat exchange, or incorporate more complex rheologies without altering the overall solution strategy.

Limitations are acknowledged. The incompressibility of the liquid phase, the perfect‑gas treatment of the volatile, and the one‑dimensional approximation neglect several potentially important effects, such as melt compressibility, gas‑melt slip, turbulence, and three‑dimensional conduit branching. Future work is suggested to extend the framework to multi‑dimensional geometries, to incorporate non‑adiabatic processes (e.g., heat loss to surrounding rock), and to model viscoelastic or shear‑thinning rheologies that are characteristic of real magmas.

Overall, the paper delivers a versatile, analytically transparent, and computationally efficient tool for exploring the dynamics of magma‑volatile ascent in both terrestrial and extraterrestrial volcanic settings, providing a solid foundation for more sophisticated multi‑phase flow models in the future.