Skip to main content Accessibility help
Hostname: page-component-5cfd469876-tkzrn Total loading time: 0.257 Render date: 2021-06-23T13:17:31.905Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Modelling intrusions through quiescent and moving ambients

Published online by Cambridge University Press:  20 April 2015

Christopher G. Johnson
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Andrew J. Hogg
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Herbert E. Huppert
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
School of Earth Sciences, University of Bristol, Queens Road, Bristol BS8 1RJ, UK
Jeremy C. Phillips
School of Earth Sciences, University of Bristol, Queens Road, Bristol BS8 1RJ, UK
Anja C. Slim
School of Mathematical Sciences and School of Geosciences, Monash University, Melbourne, Victoria 3800, Australia
Mark J. Woodhouse
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


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.

© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below.


Abraham, G., Karelse, M. & Van Os, A. G. 1979 On the magnitude of interfacial shear of subcritical stratified flows in relation with interfacial stability. J. Hydraul. Res. 17 (4), 273287.CrossRefGoogle Scholar
Akar, P. J. & Jirka, G. H. 1994 Buoyant spreading processes in pollutant transport and mixing. Part I: lateral spreading with ambient current advection. J. Hydraul. Res. 32, 815831.CrossRefGoogle Scholar
Akar, P. J. & Jirka, G. H. 1995 Buoyant spreading processes in pollutant transport and mixing. Part II: upstream spreading in weak ambient current. J. Hydraul. Res. 33, 87100.CrossRefGoogle Scholar
Alavian, V., Jirka, G. H., Denton, R. A., Johnson, M. C. & Stefan, H. G. 1992 Density currents entering lakes and reservoirs. ASCE J. Hydraul. Engng 118 (11), 14641489.CrossRefGoogle Scholar
Amen, R. & Maxworthy, T. 1980 The gravitational collapse of a mixed region into a linearly stratified fluid. J. Fluid Mech. 96 (1), 6580.CrossRefGoogle Scholar
Ansong, J. K. & Sutherland, B. R. 2010 Internal gravity waves generated by convective plumes. J. Fluid Mech. 648, 405434.CrossRefGoogle Scholar
Baines, P. G. 2013 The dynamics of intrusions into a density-stratified crossflow. Phys. Fluids 25, 076601.CrossRefGoogle Scholar
Barenblatt, G. I. 1996 Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press.CrossRefGoogle Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209248.CrossRefGoogle Scholar
Bolster, D., Hang, A. & Linden, P. F. 2008 The front speed of intrusions into a continuously stratified medium. J. Fluid Mech. 594, 369377.CrossRefGoogle Scholar
Bonadonna, C., Genco, R., Gouhier, M., Pistolesi, M., Cioni, R., Alfano, F., Hoskuldsson, A. & Ripepe, M. 2011 Tephra sedimentation during the 2010 Eyjafjallajökull eruption (Iceland) from deposit, radar, and satellite observations. J. Geophys. Res. 116 (B12), B12202.CrossRefGoogle Scholar
Bonadonna, C. & Phillips, J. C. 2003 Sedimentation from strong volcanic plumes. J. Geophys. Res. 108 (B7), 2340.CrossRefGoogle Scholar
Bonnecaze, R. T., Hallworth, M. A., Huppert, H. E. & Lister, J. R. 1995 Axisymmetric particle-driven gravity currents. J. Fluid Mech. 294, 93122.CrossRefGoogle Scholar
Bursik, M. I., Carey, S. N. & Sparks, R. S. J. 1992a A gravity current model for the May 18, 1980 Mount St. Helens plume. Geophys. Res. Lett. 19 (16), 16631666.CrossRefGoogle Scholar
Bursik, M. I., Sparks, R. S. J., Gilbert, J. S. & Carey, S. N. 1992b Sedimentation of tephra by volcanic plumes. I. Theory and its comparison with a study of the Fogo A plinian deposit, Sao Miguel (Azores). Bull. Volcanol. 54 (4), 329344.CrossRefGoogle Scholar
Cadet, D. 1977 Energy dissipation within intermittent clear air turbulence patches. J. Atmos. Sci. 34, 137142.2.0.CO;2>CrossRefGoogle Scholar
Chapman, C. J. 2000 High Speed Flow. Cambridge University Press.Google Scholar
Chen, J.-C.1980 Studies on gravitational spreading currents. PhD thesis, California Institute of Technology.Google Scholar
Collini, E., Soledad Osores, M., Folch, A., Viramonte, J. G., Villarosa, G. & Salmuni, G. 2013 Volcanic ash forecast during the June 2011 Cordón Caulle eruption. Nat. Hazards 66, 389412.CrossRefGoogle Scholar
Costa, A., Folch, A. & Macedonio, G. 2013 Density-driven transport in the umbrella region of volcanic clouds: implications for tephra dispersion models. Geophys. Res. Lett. 40, 48234827.CrossRefGoogle Scholar
Dacre, H. F., Grant, A. L. M., Hogan, R. J., Belcher, S. E., Thomson, D. J., Devenish, B. J., Marenco, F., Hort, M. C., Haywood, J. M., Ansmann, A., Mattis, I. & Clarisse, L. 2011 Evaluating the structure and magnitude of the ash plume during the initial phase of the 2010 Eyjafjallajökull eruption using lidar observations and NAME simulations. J. Geophys. Res. 116, D00U03.CrossRefGoogle Scholar
Dellino, P., Gudmundsson, M. T., Larsen, G., Mele, D., Stevenson, J. A., Thordarson, T. & Zimanowski, B. 2012 Ash from the Eyjafjallajökull eruption (Iceland): fragmentation processes and aerodynamic behavior. J. Geophys. Res. 117, B00C04.CrossRefGoogle Scholar
Devenish, B. J., Francis, P. N., Johnson, B. T., Sparks, R. S. J. & Thomson, D. J. 2012 Sensitivity analysis of dispersion modeling of volcanic ash from Eyjafjallajökull in May 2010. J. Geophys. Res. 117, D00U21.CrossRefGoogle Scholar
Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 2742.CrossRefGoogle Scholar
Faust, K. M. & Plate, E. J. 1984 Experimental investigation of intrusive gravity currents entering stably stratified fluids. J. Hydraul. Res. 22 (5), 315325.CrossRefGoogle Scholar
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455493.CrossRefGoogle Scholar
Folch, A. 2012 A review of tephra transport and dispersal models: evolution, current status, and future perspectives. J. Volcanol. Geotherm. Res. 235, 96115.CrossRefGoogle Scholar
Francis, P. N., Cooke, M. C. & Saunders, R. W. 2012 Retrieval of physical properties of volcanic ash using Meteosat: a case study from the 2010 Eyjafjallajökull eruption. J. Geophys. Res. 117, D00U09.CrossRefGoogle Scholar
Garvine, R. W. 1984 Radial spreading of buoyant, surface plumes in coastal waters. J. Geophys. Res. 89 (C2), 19891996.CrossRefGoogle Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.Google Scholar
Gratton, J. & Vigo, C. 1994 Self-similar gravity currents with variable inflow revisited: plane currents. J. Fluid Mech. 258, 77104.CrossRefGoogle Scholar
Grundy, R. E. & Rottman, J. W. 1986 Self-similar solutions of the shallow-water equations representing gravity currents with variable inflow. J. Fluid Mech. 169, 337351.CrossRefGoogle Scholar
Harris, T. C., Hogg, A. J. & Huppert, H. E. 2002 Polydisperse particle-driven gravity currents. J. Fluid Mech. 472, 333371.CrossRefGoogle Scholar
Hatcher, L., Hogg, A. J. & Woods, A. W. 2000 The effects of drag on turbulent gravity currents. J. Fluid Mech. 416, 297314.CrossRefGoogle Scholar
Hazen, A. 1904 On sedimentation. Trans. Am. Soc. Civ. Engrs 53, 4588.Google Scholar
Herzog, M., Oberhuber, J. M. & Graf, H.-F. 2003 A prognostic turbulence scheme for the nonhydrostatic plume model ATHAM. J. Atmos. Sci. 60 (22), 27832796.2.0.CO;2>CrossRefGoogle Scholar
Hobbs, P. V., Radke, L. F., Lyons, J. H., Ferek, R. J. & Coffman, D. J. 1991 Airbourne measurements of particle and gas emissions from the 1990 volcanic eruptions of Mount Redoubt. J. Geophys. Res. 96, 1873518752.CrossRefGoogle Scholar
Hogg, A. J. & Woods, A. W. 2001 The transition from inertia to bottom-drag-dominated motion of turbulent gravity currents. J. Fluid Mech. 449, 201224.CrossRefGoogle Scholar
Holasek, R. E., Self, S. & Woods, A. W. 1996a Satellite observations and interpretation of the 1991 Mount Pinatubo eruption plumes. J. Geophys. Res. 101 (B12), 2763527655.CrossRefGoogle Scholar
Holasek, R. E., Woods, A. W. & Self, S. 1996b Experiments on gas–ash separation processes in volcanic umbrella plumes. J. Volcanol. Geotherm. Res. 70, 169181.CrossRefGoogle Scholar
Hoult, D. P. 1972 Oil spreading on the sea. Annu. Rev. Fluid Mech. 4, 341368.CrossRefGoogle Scholar
Jacobson, T., Milewski, P. & Tabak, E. G. 2008 Mixing closures for conservation laws in stratified flows. Stud. Appl. Maths 121 (1), 89116.CrossRefGoogle Scholar
Johnson, C. G. & Hogg, A. J. 2013 Entraining gravity currents. J. Fluid Mech. 731, 477508.CrossRefGoogle Scholar
Kotsovinos, N. E. 2000 Axisymmetric submerged intrusion in stratified fluid. ASCE J. Hydraul. Engng 126 (6), 446456.CrossRefGoogle Scholar
Koyaguchi, T., Ochiai, K. & Suzuki, Y. J. 2009 The effect of intensity of turbulence in umbrella cloud on tephra dispersion during explosive volcanic eruptions: experimental and numerical approaches. J. Volcanol. Geotherm. Res. 186 (1–2), 6878.CrossRefGoogle Scholar
Kristiansen, N. I., Stohl, A., Prata, A. J., Bukowiecki, N., Dacre, H., Eckhardt, S., Henne, S., Hort, M. C., Johnson, B. T., Marenco, F., Neininger, B., Reitebuch, O., Seibert, P., Thomson, D. J., Webster, H. N. & Weinzierl, B. 2012 Performance assessment of a volcanic ash transport model mini-ensemble used for inverse modeling of the 2010 Eyjafjallajökull eruption. J. Geophys. Res. 117 (D20), D00U11.CrossRefGoogle Scholar
Kurganov, A. & Tadmor, E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160, 241282.CrossRefGoogle Scholar
Lemckert, C. J. & Imberger, J. 1993 Axisymmetric intrusive gravity currents in linearly stratified fluids. ASCE J. Hydraul. Engng 119, 662679.CrossRefGoogle Scholar
Maurer, B. D. & Linden, P. F. 2014 Intrusion-generated waves in a linearly stratified fluid. J. Fluid Mech. 752, 282295.CrossRefGoogle Scholar
Maxey, M. R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441465.CrossRefGoogle Scholar
Miller, T. P. & Casadevall, T. J. 2000 Volcanic ash hazards to aviation. In Encyclopedia of Volcanoes (ed. Sigurdsson, H.), pp. 915931. Academic.Google Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234 (1196), 123.CrossRefGoogle Scholar
Oswalt, J. S., Nichols, W. & O’Hara, J. F. 1996 Meteorological observations of the 1991 Mount Pinatubo eruption. In Fire and Mud: Eruptions and Lahars of Mount Pinatubo, Philippines (ed. Newhall, C. G. & Punongbayan, R. S.), University of Washington Press.Google Scholar
Parker, G., Fukushima, Y. & Pantin, H. M. 1986 Self-accelerating turbidity currents. J. Fluid Mech. 171, 145181.CrossRefGoogle Scholar
Pouget, S., Bursik, M., Webley, P., Dehn, J. & Pavolonis, M. 2013 Estimation of eruption source parameters from umbrella cloud or downwind plume growth rate. J. Volcanol. Geotherm. Res. 258, 100112.CrossRefGoogle Scholar
Prata, A. J. & Prata, A. T. 2012 Eyjafjallajökull volcanic ash concentrations determined using spin enhanced visible and infrared imager measurements. J. Geophys. Res. 117, D00U23.CrossRefGoogle Scholar
Richards, T. S., Aubourg, Q. & Sutherland, B. R. 2014 Radial intrusions from turbulent plumes in uniform stratification. Phys. Fluids 26, 036602.CrossRefGoogle Scholar
Sarna-Wojcicki, A. M., Shipley, S., Waitt, R. B., Dzurisin, D., Hays, W. H., Davis, J. O., Wood, S. H. & Bateridge, T. 1980 Areal distribution, thickness, and volume of downwind ash from the May 18, 1980, eruption of Mount St. Helens. In The 1980 Eruptions of Mount St. Helens, Washington (ed. Lipman, P. W. & Mullineaux, D. R.), Open-File Report 80-1078, vol. 1250. US Geological Survey.Google Scholar
Schumann, U., Konopka, P., Baumann, R., Busen, R., Gerz, T., Schlager, H., Schulte, P. & Volkert, H. 1995 Estimate of diffusion parameters of aircraft exhaust plumes near the tropopause from nitric oxide and turbulence measurements. J. Geophys. Res. 100 (D7), 1414714162.CrossRefGoogle Scholar
Schumann, U., Weinzierl, B., Reitebuch, O., Schlager, H., Minikin, A., Forster, C., Baumann, R., Sailer, T., Graf, K., Mannstein, H., Voigt, C., Rahm, S., Simmet, R., Scheibe, M., Lichtenstern, M., Stock, P., Rüba, H., Schäuble, D., Tafferner, A., Rautenhaus, M., Gerz, T., Ziereis, H., Krautstrunk, M., Mallaun, C., Gayet, J.-F., Lieke, K., Kandler, K., Ebert, M., Weinbruch, S., Stohl, A., Gasteiger, J., Gross, S., Freudenthaler, V., Wiegner, M., Ansmann, A., Tesche, M., Olafsson, H. & Sturm, K. 2011 Airborne observations of the Eyjafjalla volcano ash cloud over Europe during air space closure in April and May 2010. Atmos. Chem. Phys. 11 (5), 22452279.CrossRefGoogle Scholar
Slim, A. C. & Huppert, H. E. 2011 Axisymmetric, constantly supplied gravity currents at high Reynolds number. J. Fluid Mech. 675 (1), 540551.CrossRefGoogle Scholar
Sparks, R. S. J. 1986 The dimensions and dynamics of volcanic plumes. Bull. Volcanol. 48, 315.CrossRefGoogle Scholar
Sparks, R. S. J., Bursik, M. I., Carey, S. N., Gilbert, J. S., Glaze, L., Sigurdsson, H. & Woods, A. W. 1997 Volcanic Plumes. John Wiley & Sons.Google Scholar
Sparks, R. S. J., Carey, S. N. & Sigurdsson, H. 1991 Sedimentation from gravity currents generated by turbulent plumes. Sedimentology 38 (5), 839856.CrossRefGoogle Scholar
Sparks, R. S. J., Moore, J. G. & Rice, C. J. 1986 The initial giant umbrella cloud of the May 18th, 1980, explosive eruption of Mount St. Helens. J. Volcanol. Geotherm. Res. 28, 257274.CrossRefGoogle Scholar
Spinetti, C., Barsotti, S., Neri, A., Buongiorno, M. F., Doumaz, F. & Nannipieri, L. 2013 Investigation of the complex dynamics and structure of the 2010 Eyjafjallajökull volcanic ash cloud using multispectral images and numerical simulations. J. Geophys. Res. 118 (10), 47294747.Google Scholar
Stevenson, J. A., Loughlin, S., Rae, C., Thordarson, T., Milodowski, A. E., Gilbert, J. S., Harangi, S., Lukács, R., Højgaard, B., Árting, U., Pyne-O’Donnell, S., MacLeod, A., Whitney, B. & Cassidy, M. 2012 Distal deposition of tephra from the Eyjafjallajökull 2010 summit eruption. J. Geophys. Res. 117 (B9), B00C10.CrossRefGoogle Scholar
Suzuki, Y. J. & Koyaguchi, T. 2009 A three-dimensional numerical simulation of spreading umbrella clouds. J. Geophys. Res. 114 (B3), B03209.CrossRefGoogle Scholar
Thorpe, S. A. 2010 Turbulent hydraulic jumps in a stratified shear flow. J. Fluid Mech. 654, 305350.CrossRefGoogle Scholar
Ungarish, M. 2005 Intrusive gravity currents in a stratified ambient: shallow-water theory and numerical results. J. Fluid Mech. 535, 287323.CrossRefGoogle Scholar
Ungarish, M. 2006 On gravity currents in a linearly stratified ambient: a generalization of Benjamin’s steady-state propagation results. J. Fluid Mech. 548, 4968.CrossRefGoogle Scholar
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. Chapman and Hall/CRC.CrossRefGoogle Scholar
Ungarish, M. & Huppert, H. E. 2002 On gravity currents propagating at the base of a stratified ambient. J. Fluid Mech. 458, 283301.CrossRefGoogle Scholar
Ungarish, M. & Zemach, T. 2007 On axisymmetric intrusive gravity currents in a stratified ambient – shallow-water theory and numerical results. Eur. J. Mech. (B/Fluids) 26 (2), 220235.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons..Google Scholar
Woodhouse, M. J., Hogg, A. J., Phillips, J. C. & Sparks, R. S. J. 2013 Interactions between volcanic plumes and wind during the 2010 Eyjafjallajökull eruption, Iceland. J. Geophys. Res. 118, 92109.CrossRefGoogle Scholar
Woodman, R. F. & Rastogi, P. K. 1984 Evaluation of effective eddy diffusive coefficients using radar observations of turbulence in the stratosphere. Geophys. Res. Lett. 11 (3), 243246.CrossRefGoogle Scholar
Woods, A. W. 1988 The fluid dynamics and thermodynamics of eruption columns. Bull. Volcanol. 50 (3), 169193.CrossRefGoogle Scholar
Woods, A. W. & Kienle, J. 1994 The dynamics and thermodynamics of volcanic clouds: theory and observations from the April 15 and April 21, 1990 eruptions of Redoubt Volcano, Alaska. J. Volcanol. Geotherm. Res. 62 (1), 273299.CrossRefGoogle Scholar
Wu, J. 1969 Mixed region collapse with internal wave generation in a density-stratified medium. J. Fluid Mech. 35 (3), 531544.CrossRefGoogle Scholar
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Modelling intrusions through quiescent and moving ambients
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Modelling intrusions through quiescent and moving ambients
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Modelling intrusions through quiescent and moving ambients
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *