Skip to main content Accessibility help

Modelling intrusions through quiescent and moving ambients

  • Christopher G. Johnson (a1), Andrew J. Hogg (a1), Herbert E. Huppert (a1) (a2) (a3) (a4), R. Stephen J. Sparks (a2), Jeremy C. Phillips (a2), Anja C. Slim (a5) and Mark J. Woodhouse (a1) (a2)...


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.



Hide All
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.
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.
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.
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.
Amen, R. & Maxworthy, T. 1980 The gravitational collapse of a mixed region into a linearly stratified fluid. J. Fluid Mech. 96 (1), 6580.
Ansong, J. K. & Sutherland, B. R. 2010 Internal gravity waves generated by convective plumes. J. Fluid Mech. 648, 405434.
Baines, P. G. 2013 The dynamics of intrusions into a density-stratified crossflow. Phys. Fluids 25, 076601.
Barenblatt, G. I. 1996 Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press.
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209248.
Bolster, D., Hang, A. & Linden, P. F. 2008 The front speed of intrusions into a continuously stratified medium. J. Fluid Mech. 594, 369377.
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.
Bonadonna, C. & Phillips, J. C. 2003 Sedimentation from strong volcanic plumes. J. Geophys. Res. 108 (B7), 2340.
Bonnecaze, R. T., Hallworth, M. A., Huppert, H. E. & Lister, J. R. 1995 Axisymmetric particle-driven gravity currents. J. Fluid Mech. 294, 93122.
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.
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.
Cadet, D. 1977 Energy dissipation within intermittent clear air turbulence patches. J. Atmos. Sci. 34, 137142.
Chapman, C. J. 2000 High Speed Flow. Cambridge University Press.
Chen, J.-C.1980 Studies on gravitational spreading currents. PhD thesis, California Institute of Technology.
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.
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.
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.
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.
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.
Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 2742.
Faust, K. M. & Plate, E. J. 1984 Experimental investigation of intrusive gravity currents entering stably stratified fluids. J. Hydraul. Res. 22 (5), 315325.
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455493.
Folch, A. 2012 A review of tephra transport and dispersal models: evolution, current status, and future perspectives. J. Volcanol. Geotherm. Res. 235, 96115.
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.
Garvine, R. W. 1984 Radial spreading of buoyant, surface plumes in coastal waters. J. Geophys. Res. 89 (C2), 19891996.
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.
Gratton, J. & Vigo, C. 1994 Self-similar gravity currents with variable inflow revisited: plane currents. J. Fluid Mech. 258, 77104.
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.
Harris, T. C., Hogg, A. J. & Huppert, H. E. 2002 Polydisperse particle-driven gravity currents. J. Fluid Mech. 472, 333371.
Hatcher, L., Hogg, A. J. & Woods, A. W. 2000 The effects of drag on turbulent gravity currents. J. Fluid Mech. 416, 297314.
Hazen, A. 1904 On sedimentation. Trans. Am. Soc. Civ. Engrs 53, 4588.
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.
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.
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.
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.
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.
Hoult, D. P. 1972 Oil spreading on the sea. Annu. Rev. Fluid Mech. 4, 341368.
Jacobson, T., Milewski, P. & Tabak, E. G. 2008 Mixing closures for conservation laws in stratified flows. Stud. Appl. Maths 121 (1), 89116.
Johnson, C. G. & Hogg, A. J. 2013 Entraining gravity currents. J. Fluid Mech. 731, 477508.
Kotsovinos, N. E. 2000 Axisymmetric submerged intrusion in stratified fluid. ASCE J. Hydraul. Engng 126 (6), 446456.
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.
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.
Kurganov, A. & Tadmor, E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160, 241282.
Lemckert, C. J. & Imberger, J. 1993 Axisymmetric intrusive gravity currents in linearly stratified fluids. ASCE J. Hydraul. Engng 119, 662679.
Maurer, B. D. & Linden, P. F. 2014 Intrusion-generated waves in a linearly stratified fluid. J. Fluid Mech. 752, 282295.
Maxey, M. R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441465.
Miller, T. P. & Casadevall, T. J. 2000 Volcanic ash hazards to aviation. In Encyclopedia of Volcanoes (ed. Sigurdsson, H.), pp. 915931. Academic.
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.
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.
Parker, G., Fukushima, Y. & Pantin, H. M. 1986 Self-accelerating turbidity currents. J. Fluid Mech. 171, 145181.
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.
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.
Richards, T. S., Aubourg, Q. & Sutherland, B. R. 2014 Radial intrusions from turbulent plumes in uniform stratification. Phys. Fluids 26, 036602.
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.
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.
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.
Slim, A. C. & Huppert, H. E. 2011 Axisymmetric, constantly supplied gravity currents at high Reynolds number. J. Fluid Mech. 675 (1), 540551.
Sparks, R. S. J. 1986 The dimensions and dynamics of volcanic plumes. Bull. Volcanol. 48, 315.
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.
Sparks, R. S. J., Carey, S. N. & Sigurdsson, H. 1991 Sedimentation from gravity currents generated by turbulent plumes. Sedimentology 38 (5), 839856.
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.
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.
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.
Suzuki, Y. J. & Koyaguchi, T. 2009 A three-dimensional numerical simulation of spreading umbrella clouds. J. Geophys. Res. 114 (B3), B03209.
Thorpe, S. A. 2010 Turbulent hydraulic jumps in a stratified shear flow. J. Fluid Mech. 654, 305350.
Ungarish, M. 2005 Intrusive gravity currents in a stratified ambient: shallow-water theory and numerical results. J. Fluid Mech. 535, 287323.
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.
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. Chapman and Hall/CRC.
Ungarish, M. & Huppert, H. E. 2002 On gravity currents propagating at the base of a stratified ambient. J. Fluid Mech. 458, 283301.
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.
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons..
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.
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.
Woods, A. W. 1988 The fluid dynamics and thermodynamics of eruption columns. Bull. Volcanol. 50 (3), 169193.
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.
Wu, J. 1969 Mixed region collapse with internal wave generation in a density-stratified medium. J. Fluid Mech. 35 (3), 531544.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Modelling intrusions through quiescent and moving ambients

  • Christopher G. Johnson (a1), Andrew J. Hogg (a1), Herbert E. Huppert (a1) (a2) (a3) (a4), R. Stephen J. Sparks (a2), Jeremy C. Phillips (a2), Anja C. Slim (a5) and Mark J. Woodhouse (a1) (a2)...


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed