Mathematical Geosciences

, Volume 50, Issue 4, pp 417–429

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

• Julien Amalberti
• Xavier Antoine
• Pete Burnard
Article

## Abstract

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.

## Keywords

Diffusion processes Vesuvian eruption Noble gas Numerical modeling

## Notes

### Acknowledgements

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.

## References

1. Amalberti J, Burnard P, Laporte D, Tissandier L, Neuville DR (2016) Diffusion of noble gases (He, Ne, Ar) in silicate glasses close to the glass transition: distinct diffusion mechanisms. GCA 172:107–126Google Scholar
2. Anderson AT (1975) Some basaltic and andesitic gases. Rev Geophys Space Phys 13:37–55
3. Behrens H, Zhang Y (2001) Ar diffusion in hydrous silicic melts: implications for volatile diffusion mechanisms and fractionation. EPSL 192:363–376
4. Carroll MR, Stolper EM (1991) Argon solubility and diffusion in silica glass: implications for the solution behavior of molecular gases. Geochim Cosmochim Acta 55(1):211–225
5. Crank J (1975) The mathematics of diffusion, 2nd edn. Clarendon Press, Oxford, p p413Google Scholar
6. Dodson MH (1973) Closure temperature in cooling geochronological and petrological systems. Contrib Miner Petrol 40(3):259–274
7. Drapper DS, Carroll MR (1995) Argon diffusion and solubility in silicic glasses exposed to an Ar–He gas mixture. EPSL 132:15–24
8. Gonnermann HM, Mukhopadhyay S (2007) Non-equilibrium degassing and a primordial source for helium in ocean–island volcanism. Nature 449:1037–1040
9. Jaupart C (2000) Magma ascent at shallow levels. In: Sigurdsson H, Houghton B, McNutt S, Rymer H, Stix J (eds) Encyclopedia of volcanoes. Academic Press, pp 237–245Google Scholar
10. Kaneoka I (1980) Rare gas isotopes and mass fractionation: an indicator of gas transport into or from a magma. Earth Planet Sci Lett 48:284–292
11. Matsuda J, Matsubara K, Yajima H, Yamamoto K (1989) Anomalous Ne enrichment in obsidians and darwin glass: diffusion of noble gas in silica–rich glasses. GCA 53:3025–3033Google Scholar
12. Paonita A, Martelli M (2007) A new view of the He–Ar–CO2 degassing at mid-ocean ridges: homogeneous composition of magmas from the upper mantle. GCA 71:1747–1763Google Scholar
13. Pinti D, Wada N, Matsuda J (1999) Neon excess in pumice: volcanological implications. J Volcanol Geotherm Res 88:279–289
14. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, New YorkGoogle Scholar
15. Proussevitch A, Sahagian D (2005) Bubbledrive-1: a numerical model of volcanic eruption mechanisms driven by disequilibrium magma degassing. J Volcanol Geoth Res 143(1):89–111
16. Ruzié L, Moreira M (2010) Magma degassing process during Plinian eruptions. J Volcanol Geotherm Res 192:142–150
17. Sparks RS (2003) Dynamics of magma degassing. Volcan Degassing 213:5–22Google Scholar
18. Sparks RS, Wilson J, Hulme G (1978) Theoretical modeling of the generation, movement, and emplacement of pyroclastic flows by column collapse. J Geophys Res 83:1727–1739
19. Tait S, Jaupart C, Vergniolle S (1989) Pressure, gas content and eruption periodicity of a shallow, crystallising magma chamber. EPSL 92:107–123
20. Toramaru A (1995) Numerical study of nucleation and growth of bubbles in viscous magmas. J Geophys Res 100:1913–1931
21. Van Orman JA, Grove TL, Shimizu N (1998) Uranium and thorium diffusion in diopside. EPSL 160:505–519
22. Whitman AG, Sparks RSJ (1986) Pumice. Bull Volcanol 48:209–233
23. Zhang Y, Cherniak DJ (2010) Diffusion in minerals and melts. Rev Mineral Geochem 72:p103Google Scholar

© 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