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Modelling intrusions through quiescent and moving ambients

Published online by Cambridge University Press:  20 April 2015

Christopher G. Johnson ,
Andrew J. Hogg ,
Herbert E. Huppert ,
R. Stephen J. Sparks ,
Jeremy C. Phillips ,
Anja C. Slim  and
Mark J. Woodhouse
Christopher G. Johnson
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Andrew J. Hogg
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Herbert E. Huppert
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK School of Earth Sciences, University of Bristol, Queens Road, Bristol BS8 1RJ, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia
R. Stephen J. Sparks
Affiliation:
School of Earth Sciences, University of Bristol, Queens Road, Bristol BS8 1RJ, UK
Jeremy C. Phillips
Affiliation:
School of Earth Sciences, University of Bristol, Queens Road, Bristol BS8 1RJ, UK
Anja C. Slim
Affiliation:
School of Mathematical Sciences and School of Geosciences, Monash University, Melbourne, Victoria 3800, Australia
Mark J. Woodhouse
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK School of Earth Sciences, University of Bristol, Queens Road, Bristol BS8 1RJ, UK
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Abstract

Volcanic eruptions commonly produce buoyant ash-laden plumes that rise through the stratified atmosphere. On reaching their level of neutral buoyancy, these plumes cease rising and transition to horizontally spreading intrusions. Such intrusions occur widely in density-stratified fluid environments, and in this paper we develop a shallow-layer model that governs their motion. We couple this dynamical model to a model for particle transport and sedimentation, to predict both the time-dependent distribution of ash within volcanic intrusions and the flux of ash that falls towards the ground. In an otherwise quiescent atmosphere, the intrusions spread axisymmetrically. We find that the buoyancy-inertial scalings previously identified for continuously supplied axisymmetric intrusions are not realised by solutions of the governing equations. By calculating asymptotic solutions to our model we show that the flow is not self-similar, but is instead time-dependent only in a narrow region at the front of the intrusion. This non-self-similar behaviour results in the radius of the intrusion growing with time $t$ as $t^{3/4}$, rather than $t^{2/3}$ as suggested previously. We also identify a transition to drag-dominated flow, which is described by a similarity solution with radial growth now proportional to $t^{5/9}$. In the presence of an ambient wind, intrusions are not axisymmetric. Instead, they are predominantly advected downstream, while at the same time spreading laterally and thinning vertically due to persistent buoyancy forces. We show that close to the source, this lateral spreading is in a buoyancy-inertial regime, whereas far downwind, the horizontal buoyancy forces that drive the spreading are balanced by drag. Our results emphasise the important role of buoyancy-driven spreading, even at large distances from the source, in the formation of the flowing thin horizontally extensive layers of ash that form in the atmosphere as a result of volcanic eruptions.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Stratified flows
Type
Papers
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Journal of Fluid Mechanics , Volume 771 , 25 May 2015 , pp. 370 - 406
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© 2015 Cambridge University Press 

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