Mathematical Geosciences

, Volume 50, Issue 4, pp 417–429 | Cite as

Timescale Monitoring of Vesuvian Eruptions Using Numerical Modeling of the Diffusion Equation

  • Julien Amalberti
  • Xavier Antoine
  • Pete Burnard


It is necessary to understand the diffusive behavior of the volatile phase in order to interpret the control mechanisms of the bubble size in volcanic rocks and the processes influencing the eruption itself. A simple numerical model is proposed, based on the time-dependency of the diffusion equation for a hollow sphere, to simulate incorporation of atmospheric noble gases in pumice for Vesuvian (i.e., Plinian) eruptions. During a Vesuvian eruption, melt fragments are ejected into the air. The resulting pumice samples collected after the eruption phase exhibit a significant incorporation of elementally and isotopically fractionated atmospheric noble gases. The noble gas content of the trapped gases then potentially provides useful data with which to constrain the timescales of cooling for the samples. The system can be adequately described as a diffusion process into a hollow sphere through the one dimensional diffusion equation for a spherically symmetrical geometry. Diffusion coefficients are time-dependent to include the effect of an exponential decay of temperature with time. The complexity of this system requires numerical resolution of the diffusion equation due to the diffusion coefficient temperature dependency. The outer boundary condition is fixed with a given noble gas concentration via an inhomogeneous Dirichlet boundary condition, while the inner boundary condition is set as a flux-free boundary. The numerical model allows the noble gas content, and thus the noble gas elemental and isotopic ratios entering the bubble, to be modeled as a function of time, hollow sphere thickness, diffusion coefficients, initial and final temperatures, and quench rate.


Diffusion processes Vesuvian eruption Noble gas Numerical modeling 



This work benefited from financial support of the Agence Nationale de la Recherche (DEGAZMAG project, contract no. ANR 2011 Blanc SIMI 5-6 003) and Région Lorraine. This is a CRPG contribution.


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Copyright information

© International Association for Mathematical Geosciences 2018

Authors and Affiliations

  • Julien Amalberti
    • 1
    • 3
  • Xavier Antoine
    • 2
  • Pete Burnard
    • 1
  1. 1.Centre de Recherche Pétrographique et GéochimiqueVandoeuvre CedexFrance
  2. 2.Institut Elie Cartan de LorraineUniversité de Lorraine, Inria Nancy-Grand EstVandoeuvre-lès-Nancy CedexFrance
  3. 3.Department of Earth and Environmental SciencesUniversity of MichiganAnn ArborUSA

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