# Thermal structure of a gas-permeable lava dome and timescale separation in its response to perturbation

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, B07201, doi:10.1029/2008JB006198, 2009 Click Here for Full Article Thermal structure of a gas-permeable lava dome and timescale separation in its response to perturbation Peter D. Hicks,1 Adrian J. Matthews,1,2 and Mark J. Cooker1 Received 24 October 2008; revised 27 March 2009; accepted 21 April 2009; published 1 July 2009. [1] The thermal boundary layer at the surface of a volcanic lava dome is investigated through a continuum model of the thermodynamic advection-diffusion processes resulting from magmatic gas flow through the dome matrix. The magmatic gas mass flux, porosity, and permeability of the rock are identified as key parameters. New, theoretical, nonlinear steady state thermal profiles are reported, which give a realistic surface temperature of 210°C for a region of lava dome surface through which a gas flux of 3.5 103 kg s1 m2 passes. This contrasts favorably with earlier purely diffusive thermal models, which cool too quickly. Results are presented for time-dependent perturbations of the steady states as a response to changes in surface pressure, a sudden rockfall from the lava dome surface, and a change in the magmatic gas mass flux at depth. Together with a generalized analysis using the method of multiple scales, this identifies two characteristic timescales associated with the thermal evolution of a dome carapace: a short timescale of several minutes, over which the magmatic gas mass flux, density, and pressure change to a new quasi-steady state and a longer timescale of several days, over which the thermal profile changes to a new equilibrium distribution. Over the longer timescale, the dynamic properties of the dome continue to evolve, but only in slavish response to the ongoing temperature evolution. In light of this timescale separation, the use of surface temperature measurements to infer changes in the magmatic gas flux for use in volcanic hazard prediction is discussed. Citation: Hicks, P. D., A. J. Matthews, and M. J. Cooker (2009), Thermal structure of a gas-permeable lava dome and timescale separation in its response to perturbation, J. Geophys. Res., 114, B07201, doi:10.1029/2008JB006198. 1. Introduction and lead to degassing-induced crystallization and an in- crease in viscosity [Sparks, 1997]. The bubbles can also [2] Lava domes are steep-sided, mounds of lava, formed form interconnected pathways through the lava, making it by extrusion of highly viscous lava from volcanic vents. permeable to gas flow. An internal gas pressure gradient Their rheology is due to the high silica content of the source drives the gas upward through the conduit and the dome magma; typically dacitic, rhyolitic, or andesitic volcanoes above, which then escapes into the atmosphere. Heat located in subduction settings. There are two main styles of advection by this gas provides an important mechanism of lava dome growth: endogenous and exogenous. An endog- heat transfer within a dome, in addition to diffusion. enous dome forms by internal emplacement of magma at the [4] Lava domes have several associated hazards. Under base of the dome from the magma conduit below. It has a certain circumstances, the high internal pressures can lead to layered internal structure. In exogenous dome growth, fresh failure, resulting in explosions, dome collapse and pyro- magma forces its way to the surface, leading to surface clastic flows [Fink and Kieffer, 1993; Calder et al., 2002]. features such as spines and lobes. Dome growth can switch The overpressures that lead to these failures can be triggered between these two styles on the same volcano, e.g., at by sudden increases in magmatic gas flux [Sparks, 1997]. Unzen [Nakada et al., 1995] and Soufrière Hills Volcano Collapse can also be triggered by gravitational instability if [Watts et al., 2002]. a dome outgrows the confines of the volcano summit and if [3] As magma ascends in the conduit, its depressurization it cannot be supported by the flanking talus slopes [Sato et leads to the exsolution of dissolved volatiles and the al., 1992]. External triggers can also lead to hazardous production of magmatic gas. These gases form bubbles, volcanic activity [Neuberg, 2000]. In particular, intense rainfall has been shown to initiate explosions, dome collap- ses and pyroclastic flows [Mastin, 1994; Yamasato et al., 1998; Matthews et al., 2002; Carn et al., 2004; Barclay et 1 School of Mathematics, University of East Anglia, Norwich, UK. 2 al., 2006]. Hence an understanding of the internal structure School of Environmental Sciences, University of East Anglia, of a dome and the conditions that can lead to failure are Norwich, UK. important from a natural hazards perspective. Copyright 2009 by the American Geophysical Union. [5] Due to the hostile conditions around active lava 0148-0227/09/2008JB006198$09.00 domes, basic physical parameters are difficult to measure. B07201 1 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 Blackbody surface temperatures of around 200 – 350°C model that includes both the diffusion of heat in the dome have been inferred from measurements of the infrared carapace, and the advection of heat by magmatic gas flow radiation emitted from a dome surface [Dzurisin et al., through the surface carapace layer of a permeable dome. 1990; Oppenheimer et al., 1993; Urai, 2000]. Surface Together with a parameterization of energy fluxes at the rockfalls expose hotter rock immediately below. Similarly dome surface, the model then generates steady state surface observations of incandescence, lead to estimated temper- temperatures, and temperature and pressure profiles within a atures near 650°C at a depth of approximately one metro dome, in good agreement with observations. For example, [Sparks et al., 2000; Watts et al., 2002]. The temperature in a region of the lava dome surface through which a gas increases further with depth within a surface carapace layer, mass flux of 3.5 103 kg s1 m2 passes, the predicted with a typical thickness of 10– 30 m [Iverson, 1990]. Below steady state surface temperature is 210°C. To investigate the this, the temperature is expected to be isothermal from theo- stability of a dome to external forcing, the steady state retical considerations [Melnik and Sparks, 1999] with a temperature and pressure solutions can then be subjected to a value near 830°C [Barclay et al., 1998]. range of time-dependent perturbations. These could include: [6] The total volume of a dome can be measured by [10] 1. a change in surface pressure due to the passage of photogrammetric-GPS techniques, and the growth rate in- a weather system; ferred [Herd et al., 2005]. The composition and mass flux of [11] 2. removal of surface material due to rockfall; magmatic gases emitted from the dome surface can be [12] 3. a change in the flux of magmatic gas at depth; detected by remote spectroscopic techniques [Francis et al., [13] 4. the impact of rainfall on the surface of a dome. 2000; Edmonds et al., 2001]. Additionally, chemical compo- [14] Perturbations 1 – 3 will be examined during the sition and bulk physical properties of field specimens of dome course of this paper, and their implications for lava dome rock can be measured in the laboratory. These include density, failure discussed. For each perturbation, the time-dependent specific heat capacity, thermal conductivity, porosity and evolution of the lava dome temperature, and the magmatic permeability [Sigurdsson, 2000; Spera, 2000; Couch et al., gas pressure, density and volume flux profiles will be 2001]. In summary, many key parameters of volcanic domes investigated. The model reveals a characteristic timescale are known incompletely, and with large error bounds. Several separation between the thermal response of the lava dome of these parameters, such as surface temperature and mag- and the dynamic response of the magmatic gas. This is matic gas flux, also exhibit considerable natural variability, investigated further using the technique of multiple-scale both temporally and spatially over a dome. analysis. Perturbations to volcanic domes by rainfall (num- [7] Models of dome growth and evolution should predict ber 4 in the list above) have been shown in previous parameter values that fall within the range of these obser- modeling studies to lead to instability and failure [Matthews vations. For example, in a one-dimensional model of the and Barclay, 2004; Elsworth et al., 2004; Simmons et al., flow of magmatic gas through a lava dome, Woods et al. 2004; Taron et al., 2007]. The response to rainfall of the [2002] calculated the steady state profiles of gas pressure as volcanic dome model developed here will form the basis of a function of depth within a permeable dome. Woods et al. a subsequent paper. [2002] predict realistic pressure profiles within a dome, below the carapace region. However, their model was 2. Development of a Model for a Gas-Permeable assumed to be isothermal and neglected the role of the Lava Dome thermal boundary layer upon the pressure profile. One- dimensional models of the thermal structure of an imper- [15] In this section, we develop a thermodynamic model meable lava dome have been considered by Matthews and of the temperature evolution of a lava dome carapace, Barclay [2004]. As there was no advection of heat by combined with the magmatic gas flow. magmatic gases, the only mechanism for vertical heat flow 2.1. Assumptions in the interior was diffusion. This led to unrealistically rapid cooling and low surface temperatures. [16] We consider an idealized lava dome consisting of a [8] Several models of lava dome growth have also been porous rock matrix (hereafter referred to as ‘‘matrix’’) proposed, employing either asymptotic studies exploiting whose physical properties correspond to those of a typical the typical small aspect ratio of dome height to horizontal andesite. The void spaces, which incorporate both the extent [Balmforth et al., 2000, 2004; Dragoni et al., 2005] effects of cracks and vesicles are assumed to be isotropically or numerical simulations using the finite element level set distributed. The porous medium is assumed to be rigid, method [Hale and Wadge, 2003; Bourgouin et al., 2007; while magmatic gas is forced upward through the connected Hale et al., 2007]. These models were successful in gener- void spaces. ating realistic dome geometries and free surface evolution [17] Two key properties in any model for porous media for endogenous growth. However, these models were either flow are the porosity (the proportion of the lava dome which isothermal or, in cases where temperature was allowed to is not occupied by andesite) and the permeability of the vary, the only heat transfer mechanism was thermal diffu- andesite which is a measure of the ease of gas flow through sion in the dome matrix. As a consequence, they did not the pore spaces and fracture network. [Melnik and Sparks, simulate realistic thermal profiles. Again, this is likely to be 2002] report laboratory measurements of porosity and due to the assumption of an impermeable dome and the permeability for a range of andesite samples from the omission of a magmatic gas flux and its associated heat Soufrière Hills Volcano. These range from a porosity of advection. 0.023 and a permeability of 6 1016 m2 for dense [9] In this paper, we build on previous modeling studies ‘‘glassy’’ fragments to a porosity of 0.72 porous and a of energy fluxes through a volcanic dome. We develop a permeability of 4 1012 m2 for pumice. In the absence of 2 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 Table 1. Typical Physical Properties of Andesite Used in the emission of magmatic gas is not uniformly distributed over Model the surface of a dome, with cracks and fissures in the Property Value structure of a dome matrix acting to channel above-average kr thermal conductivity 2.6 W m1 K1 magmatic gas fluxes to certain regions of a dome, while cr specific heat capacity 1000 J kg1 K1 large regions of the dome periphery may have little or no rr density 2600 kg m3 magmatic gas passing through them. Our contention is that f porosity 0.2 the flux of heat resulting from the advection of magmatic K permeability 1011 m2 gas is vital in describing the thermal structure of a dome, and therefore in order to generate lava dome surface temperatures corresponding to the maximum values recorded in the literature, it makes sense to consider the thermal structure of regions of small horizontal extent, but detailed information regarding the composition and struc- through which a magmatic gas flux, perhaps an order of ture of the dome it is difficult to estimate bulk properties magnitude above the background average may flow. from these measurements. The range of permissible perme- ability values used in our investigations is further limited by 2.2. Derivation of Model Equations the requirement that the magmastatic gas pressure cannot [21] We consider a system of one-dimensional conserva- dramatically exceed the magmastatic overburden pressure tion laws describing the flow of energy, mass, and momen- for extended periods of time. Costa [2006] fitted the tum in the direction along the outward normal to the lava classical Kozeny-Carman relationship it these samples and dome surface and at an angle a to gravity. This is the ~z axis in general the complicated lava dome evolution will pro- where ~z = 0 is at the dome surface, the atmosphere above duce large spatial inhomogeneities in both porosity and the dome is in the region ~z > 0 and the dome itself is in the permeability. However, for simplicity and following the region ~z < 0. The lower boundary of the model is at ~z = L earlier work of Woods et al. [2002], we shall assume for some positive constant L (as shown in Figure 1). This homogeneous distributions of porosity and permeability. system connects the behavior of the solid dome matrix These can be interpreted as either global averages over phase (whose properties are represented by a subscript r for the whole dome or localized representations of the features ‘‘rock’’) to the properties of the magmatic gas (whose and regions making up the dome. The values of the physical properties are represented by subscript g for ‘‘gas’’). properties of andesite are given in Table 1. [22] In the model equations variables with a tilde () are [18] The matrix and the magmatic gas passing through the dimensional quantities, variables with a caret (^) are non- void spaces are assumed to be in local thermal equilibrium, dimensional quantities and symbols with neither a tilde nor which means that at any point they share one common a caret are dimensional, physical constants. The porosity (f) temperature. This allows the conservation of energy within and density (rr) of the matrix are assumed to be constant, both the matrix and the magmatic gas to be written as a while the density (~ rg) and volume flux per unit cross- single equation. Energy is transferred through the solid sectional area of void space, or equivalently the average carapace of the lava dome by a combination of diffusion flow speed of magmatic gas (~vg) vary. Under these assump- in the matrix and magmatic gas, and advection of the tions, conservation of energy allows us to write an equation magmatic gas. The composition of a sample of magmatic governing the combined rock matrix and magmatic gas gas from the Soufrière Hills Volcano was shown to be in temperature T~ , itself a function of the level ~z and time ~t, excess of 90% water by volume [Hammouya et al., 1998]. in the form For simplicity, we assume that the thermal conductivity, specific heat capacity and viscosity of the magmatic gas take @ hn o i the same values as the equivalent properties of water vapor. ~g T~ ð1 fÞcr rr þ fcg r @~t The values of these physical properties are given in Table 2. @ @ @ T~ [19] Typical dome structures are approximately isothermal þ fcg r ~ ~g ~vg T ke ¼ 0; ð1Þ @~z @~z @~z below a surface carapace region of approximately 30 m thickness [Iverson, 1990]. In the model developed the where the specific heat capacities of the matrix and temperature at the base of the carapace region is maintained magmatic gases are cr and cg, respectively. The effective at 1100 K (the temperature of fresh magma extrusion from thermal conductivity ke, is assumed to satisfy the parallel the conduit [Barclay et al., 1998]), but the behavior of the model ke = (1 f)kr + fkg, with kr and kg being the thermal model is insensitive to the position of this interface. conductivities of the andesite and magmatic gas phases, [20] The total magmatic gas flux can be estimated from respectively. The effective thermal conductivity allows us to measurements of the sulfur dioxide (SO2) gas flux. This is combine the thermal diffusion in andesite and magmatic gas highly variable on many timescales. Typical values at the Soufrière Hills Volcano lie in the range 0.1– 50 kg s1 [Edmonds et al., 2003a, 2003b]. The mixing ratio of SO2 in Table 2. Typical Physical Properties of Magmatic Gas Used in the magmatic gas is approximately 15:1, hence the total mag- Model (Based on Physical Properties of Water Vapor) matic gas flux lies in the range 1.5 – 750 kg s1. The surface Property Value area of the Soufrière Hills Volcano dome is approximately kg thermal conductivity 0.016 W m1 K1 5 105 m2 [Carn et al., 2004], giving an average mass flux cg specific heat capacity 2026 J kg1 K1 per unit area of up to 1.5 103 kg s1 m2. However, the mg viscosity 2.82 104 Pa s 3 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 ability K, the porosity f, and the viscosity mg of the magmatic gas. The second term on the right-hand side of equation (3) is the correction due to gravity with g representing the acceleration due to gravity and a the angle between the vertical and the normal to the lava dome surface (Figure 1). This system of governing equations becomes closed with an equation of state for the magmatic gas. Magmatic gas is assumed to behave like an ideal gas and is governed by ~ rg Rg T~ ; pg ¼ ~ ð4Þ where Rg = 461.5 J kg1 K1 is the ideal gas constant for water vapor. 2.3. Boundary Conditions [24] To complete this system of equations, we need boundary conditions at the surface (~z = 0) and at some fixed level (~z = L) inside the lava dome. At the dome surface the difference between the surface temperature T~ s(t) = ~ z = 0, t) and the temperature of the atmosphere Ta generates T(~ a flux of heat away from the volcano which is a combination of radiative and convective (sensible) components [Neri, 1998; Carn et al., 2004] @ T~ ke ¼ s s T~s4 Ta4 þ Sh ra ca U T~s Ta ; ð5Þ @~z ~z¼0 Figure 1. Schematic diagram showing diffusive heat where in the radiative heat flux term s = 0.95 is the fluxes in the dome matrix and magmatic gas and advective emissivity of the andesite, s = 5.67 108 W m2 K4 is heat fluxes in the magmatic gas in a dome interior. Surface the Stefan-Boltzmann constant and in the convective heat heat fluxes are due to radiation, atmospheric convection, flux term Sh = 2.0 103 is the surface roughness or the and magmatic gas advection. All heat fluxes are assumed to aerodynamic transfer coefficient, ra = 1.0 kg m3 is the be perpendicular to the lava dome surface and at an angle a atmospheric density, ca = 1004 J kg1 K1 is the specific to the vertical direction. heat capacity of the air and U is the wind speed over the surface of the volcano. In this equation the diffusion of thermal energy up through the matrix and magmatic gas is into one term. For an isotropic porosity distribution and balanced by the radiative transfer of energy to and from the constant thermal conductivities, the effective thermal atmosphere, and a sensible heat flux (atmospheric convec- conductivity and the first derivative of the final term in tion due to wind across the lava dome surface). equation (1) can be interchanged. The second term in [25] Heat is advected upward the dome by the magmatic equation (1) corresponds to thermal advection in the gas flux. Hence there is also an advective gas flux away magmatic gas. from the lava dome surface as heat is carried by the [23 ] Mass conservation of magmatic gas rising up escaping magmatic gas [Carn et al., 2004]. This term could through the permeable matrix is governed by be included on the right-hand side of equation (5). However, in the lava dome interior there is also an advective heat flux @~rg @ to the surface in the magmatic gas. This term would be f þf ~g ~vg ¼ 0: r ð2Þ @~t @~z written on the left-hand side of equation (5). The mass of magmatic gas crossing the lava dome surface is conserved, Conservation of momentum for the magmatic gas occupy- and the temperature of the magmatic gas is continuous ing the void spaces of the porous medium is assumed to be across the interface. Hence the advective heat flux due to the given by Darcy’s law which relates the volume flux to the magmatic gas escaping from the surface is equal to the heat pressure gradient in the magmatic gas. If the total pressure flux from below, and these two advective heat fluxes cancel in the magmatic gas is ~pg, then (see Figure 1). This means that the advection of heat by magmatic gases does not contribute directly to the surface K @~pg energy balance. However, it has a direct impact on the ~vg ¼ þr ~g g cos a ; ð3Þ interior temperature profile, which then governs the sur- fmg @~z face temperature, and the loss of energy to the atmosphere where the constant of proportionality between the volume via the radiative and convective (sensible) heat fluxes in flux and pressure gradient includes the andesite perme- equation (5). 4 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 [26] Boundary conditions are also required at the bottom This means the nondimensional ideal gas equation has the of the carapace region. At a level ~z = L (the bottom of form the rigid carapace), the temperature is maintained at T~ (~z = L, t) = 1100 K, the andesite temperature at depth inferred pg ¼ Q^ ^ ^g T^ ; rg þ dT^ þ r ð12Þ by Barclay et al. [1998]. Additionally, further conditions on density are required at ~z = L. In the steady state where model the mass flux is constant and in the time-dependent model the density gradient is constant (with the exception Ta of the investigation into the forced mass flux variation at Q¼ ¼ 0:372; ð13Þ ½T depth). 2.4. Nondimensionalization of Model Equations rg [27] To assist analysis of this model it is first nondimen- d ¼ h ai ¼ 3:34: ð14Þ sionalized. The thermal boundary layer is assumed to have rg thickness L and this is used to nondimensionalize the length scales in the model, with the nondimensional (careted) [28] As well as choosing the temperature scale across the spatial coordinate ^z satisfying ^z = ~z/L. The advective thermal boundary layer it is convenient to specify the mass timescale for the magmatic gas flux is used to nondimen- flux of magmatic gas across the thermal boundary layer sionalize time. The nondimensional timescale ^t is h i K pg 2 ^t ¼ vg ~t =L; ð6Þ mg ¼ f rg vg ¼ ; ð15Þ mg LRg ½T where [vg] is a typical value for the magmatic gas mass flux. Properties of the magmatic gas and the lava dome where ~vg = [vg]^vg and [vg] = K/mgf is the typical temperature vary across the thermal boundary layer. This dimensional scale for the velocity. This is chosen to balance motivates nondimensional forms for magmatic gas pressure the volume flux with the pressure gradient in Darcy’s law rg) and temperature (T^ ) which are related to (^pg), density (^ (equation (3)). Combining (15) and (11) gives a typical their dimensional counterparts through pressure scale across the thermal boundary layer necessary to drive any prescribed mass flux through the pore spaces of the lava dome. ~pg ¼ pa þ pg ^pg ; ð7Þ [29] Having specified the dimensional scales in the prob- lem they are substituted into the field equations giving nondimensional conservation laws for energy, mass, and T~ ¼ Ta þ ½T T^ ; ð8Þ momentum. These are h i @ hn o i @ n o 1 @ 2 T^ ~rg ¼ rga þ rg r ^g ; ð9Þ ^g T^ þ e 1þe dþr ^g ^vg T^ dþr ¼ 0; @^t @^z Pe @^z2 ð16Þ where the dimensional scale for pressure [pg] is the difference between the atmospheric pressure pa = 9 105 Pa at the surface and the pressure at level ~z = L necessary to drive a @^rg @ h i þ ^g ^vg ¼ 0; dþr ð17Þ mass flux per unit cross-sectional area of mg through a @^t @^z dome. Similarly, the dimensional scale for temperature [T] = 802 K is the difference between the atmospheric tempera- and ture Ta = 298 K and the temperature at level ~z = L, which ~ z = L, t) = 1100 K. The dimensional scale for the pg @^ is T(~ ^vg ¼ b dþ^ rg ; ð18Þ magmatic gas density [rg], is the difference in magmatic gas @^zg density as a parcel of magmatic gas is moved from ~z = L respectively, where the nondimensional numbers present in to the surface, where it is subjected to atmospheric pressure this system of equations are and temperature. With this scaling for the magmatic gas density, the ideal gas equation defines rga through h i rg Lg cos a pa b¼ ; ð19Þ rga ¼ : ð10Þ pg Rg T a h i Similarly, the ideal gas equation can be used to relate the fcg rg dimensional scales themselves through e¼ ; ð20Þ ð1 fÞcr rr h i pg ¼ rg Rg ½T ; ð11Þ corresponding to the ratio of the pressure gradient to the effect of gravity in Darcy’s law and the ratio of the specific 5 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 Table 3. Values of Nondimensional Numbers in the Model for L = 20 m and mg = 3.5 103 kg s1 m2 a Symbol Value Nondimensional Number Equations Q 0.372 ratio of atmospheric temperature to temperature difference (13) across thermal boundary layer d 1.10 ratio of atmospheric density to density difference across (14) thermal boundary layer b 5.31 104 influence of gravity on the magmatic gas (19) e 1.16 104 ratio of magmatic gas specific heat capacity to andesite (20) specific heat capacity Pe 9.19 105 magmatic gas Péclet number (21) B 13.7 coefficient of radiative heat flux in thermal boundary condition C 96.4 coefficient of convective heat flux in thermal boundary condition for a wind speed of 5 m s1 a The values of the physical constants contributing to these numbers are given in Tables 1 and 2. heats of magmatic gas to andesite, respectively. A Péclet equation (16) reduces to the nondimensional classical heat number measures the ratio of thermal advection to thermal equation diffusion in the lava dome @ T^ 1 @ 2 T^ ¼ : ð25Þ @^t Pe @^z2 vg Lð1 fÞcr rr Pe ¼ : ð21Þ An equivalent equation coupled to the boundary condition ke (equation (23)) was studied by Matthews and Barclay [2004]. In such a model, cooling continues below observed [30] The nondimensional surface pressure and tempera- values of lava dome surface temperature on a timescale of ture are ^pg(0, ^t ) = 0 and T^ (0, ^t ) = T^ s(^t) respectively and the days, leading to unrealistic predictions of the full tempera- nondimensional surface density is ture profile. [32] Regions on the surface of a lava dome can maintain dT^s ð^tÞ temperatures between 200 and 350°C [Dzurisin et al., 1990; ^g ð0; ^t Þ ¼ r ; ð22Þ Q þ T^s ð^t Þ Oppenheimer et al., 1993; Urai, 2000] for several months. Therefore we shall look at the steady state behavior of while the temperature satisfies the nondimensional version equation (25), in which the temperature T^ = T^ (^z), is a of the heat flux boundary condition (5): function of position only and does not vary with time. In this case the lava dome temperature satisfies @ T^ ¼ H T^s ; ð23Þ @ 2 T^ @^z z¼0 0¼ : ð26Þ @^z2 where the nondimensional heat flux Temperature profiles resulting from equation (26) are linear functions of position, and yet must also satisfy the boundary ^ 6T^s2 4T^s3 T^s4 conditions (equation (23)) and T(1) = 1. This implies the ^ ^ H Ts ¼ B 4Ts þ þ 2 þ 3 þ C T^s : ð24Þ Q Q Q nondimensional temperature is related to the nondimen- sional level through Here B = ssLTa3/ke measures the importance of radiative T^ ¼ 1 H T^s ð^z þ 1Þ; ð27Þ cooling to a dome and C = ShracaUL/ke measures the importance of convective (sensible) cooling. Notice, C where the nondimensional surface heat flux H(T^ s) is given depends linearly on the wind speed over the lava dome by equation (24). Here the value of the surface temperature surface and Sh, the lava dome surface roughness. This T^ s is unknown, but can be found as the smallest positive completes the nondimensionalization of the model, with root of the quartic polynomial typical values of the dimensionless parameters given in Table 3 for L = 20 m and mg = 3.5 103 kg s1 m2. For T^s ¼ 1 H T^s : ð28Þ these parameter values, the advective timescale for the evolution of the model, [vg]/L = 434.6 seconds. [33] The level ~z = L here marks the transition between the surface carapace region and the lower isothermal part of 3. Thermal Structure of an Impermeable Lava the dome (at 1100 K). Therefore the steady state surface Dome temperature can be calculated as a function of the carapace thickness L (Figure 2). Typical measurements of the lava [31] We can consider the evolution of a solid imperme- dome carapace thickness suggest values of L between 10 able lava dome, which has no void spaces, and hence no and 30 m [Iverson, 1990]. However, this purely diffusive magmatic gas flow. In this case, the energy conservation model only predicts physically realistic lava dome surface 6 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 notational brevity w ePe and from equation (32) and the boundary conditions we find that the steady state temperature profile is T^ ð^zÞ ¼ 1 w1 H T^s ew^z ew ; ð33Þ for ^z 0. Notice, as in the purely diffusive model, equation (33) depends on the surface temperature T^ s. This is given by the smallest positive root of the quartic equation T^s ¼ 1 w1 H T^s ð1 ew Þ; ð34Þ by which the full profile is recovered. [ 35 ] The steady state temperature profile depends (through Pe) on mg, the mass flux per unit surface area of Figure 2. Steady state surface temperatures in an lava dome, with the relationship between [vg] and mg being impermeable dome, as a function of carapace thickness L, given by equation (15). For ‘‘standard values’’ of carapace for radiative heat fluxes only (U = 0) and radiative and thickness with L = 20 m, a mass flux mg = 3.5 103 kg convective heat fluxes with U = 2, 5, and 10 m s1. s1 m2, and surface wind speed U = 5 m s1, the surface temperature predicted in steady state is 210°C, which is in temperatures above 200°C for values of L less than one excellent agreement with the observed temperature range of meter. For values of L greater than 5 m, the predicted steady 200 – 350°C [Dzurisin et al., 1990; Oppenheimer et al., state surface temperature is less than 100°C regardless of 1993]. Variations in these parameters produce a range of the wind speed over the dome. This discrepancy illustrates steady state surface temperatures (and interior profiles). that equation (25), a purely diffusive model of lava dome Typical carapace thicknesses for lava domes have been cooling, neglects some physical process which acts to heat measured at 10 to 30 m by Iverson [1990]. The maximum the lava dome carapace. In the next section we show that wind speed experienced around the Soufrière Hills Volcano thermal advection by the magmatic gas is the missing heat dome is up to 35 m s1, which would occur during the transfer mechanism. passage of a strong hurricane. However, we are interested in the typical steady state profiles through the dome and therefore we consider the effect of a much slower wind 4. Steady State Thermal Structure of a Porous speed of 5 m s1, which is typical of wind speeds measured Lava Dome With Magmatic Gas Flow by a nearby automated weather station [Edmonds et al., [34] We now seek a model of the thermal structure of a 2003a]. As with the purely diffusive model shown in lava dome in which heat advection in the flux of magmatic Figure 2, increase in wind speed above this value will lower gases through the void spaces is included. Initially, we look the steady state surface temperature. The time-dependent ^ z), ^vg(^z), ^pg(^z) and ^ for steady states profiles of T(^ rg(^z). In the surface temperature variation resulting from wind speed steady state, derivatives with respect to time disappear, so increases associated with the passage of a hurricane will equations (16), (17), and (18) reduce to be discussed in section 6.1. Variability is also present in the magmatic gas mass flux per unit surface area of lava dome, @ n o 1 @ 2 T^ resulting from variation in the total gas mass flux, the size e ^g ^vg T^ ¼ dþr ; ð29Þ @^z Pe @^z2 and surface area of the lava dome, and the effective region through which gas is emitted. The total gas mass flux of @ h i 217 kg s1 and dome surface area of 494000 m2, reported ^g ^vg ¼ 0; dþr ð30Þ by Carn et al. [2004] for Soufrière Hills Volcano on @^z 20 March 2000 corresponds to a spatially uniform mass flux per unit surface area of 4.4 104 kg s1 m2 (and a @^pg corresponding steady state surface temperature of just ^vg ¼ b dþr ^g ; ð31Þ 65°C). However, neither the gas flux nor the surface @^z temperature are uniformly distributed across the surface of respectively. Immediately, as a result of the nondimensiona- a lava dome, with regions of lava dome cooler than this and lization, the steady state mass conservation (equation (30)) temperature hot spots consistent with values recorded by is integrated to give (d + r ^g)^vg = 1. Hence the steady state Dzurisin et al. [1990] and Oppenheimer et al. [1993]. energy conservation (equation (29)) is reduced to an equation [36] Figure 3 illustrates the change in the steady state involving only temperature: surface temperature as mg, L and U are varied. An increase in mg leads to an increase in the predicted steady state @ T^ 1 @ 2 T^ surface temperatures T^ s. This is due to the increased gas e ¼ 0: ð32Þ @^z Pe @^z2 mass flux, advecting additional thermal energy to the upper Temperature profiles generated by this equation are ex- reaches of the lava dome. In the limit mg ! 0, our new ponential functions of ^z and once again they must satisfy the model of the thermal structure of the lava dome carapace boundary conditions (equation (23)) and T(1)^ = 1. For simplifies to the purely diffusive model of an impermeable 7 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 carapace thicknesses of L = 1, 2, 5, 10, and 20 m and for surface wind speeds of U = 0 and 5 m s1. In the absence of a sensible heat flux from the surface, corresponding to no wind over the lava dome (C = 0 in 24), heat is transferred into the atmosphere by the thermal boundary condition solely as the result of radiation. The addition of heat transfer by atmospheric convection forced by a wind of 5 m s1 lowers the steady state surface temperature by approximately 50°C, when compared to the purely radiative case. However, for typical mg both models of surface heat flux correspond to physically realistic steady state surface temperatures. [37] For realistic values of mg the model is not strongly sensitive to the choice of L. Iverson [1990] reports solid carapace thicknesses for Mount St. Helens in the range of 10 to 30 m. Figure 3 shows that for a typical value of mg = 3.5 103 kg s1 m2 the steady state lava dome Figure 3. Steady state surface temperature in a permeable temperature is nearly the same for the values of L illustrated. dome, as a function of magmatic gas mass flux per unit At constant mg, the steady state surface temperature remains area. Carapace thickness of L = 1, 2, 5, 10, and 20 m, with unchanged even if L is increased significantly. The reason the temperature maintained at 1100 K below the carapace. for this can be seen by looking at the full temperature profile right across the carapace. matrix described in section 3. The steady state surface [38] The full steady state temperature profiles for L = 20 m temperatures predicted in the limit mg ! 0, for different and for a range of mass fluxes are shown in Figure 4a. choices of L are the same as those given in Figure 2. The These profiles exhibit thermal boundary layers (due to the resulting steady state surface temperatures are shown for Figure 4. Steady state profiles for (a) temperature, (b) gas density, and (c) total gas pressure as a function of level for magmatic gas mass fluxes per unit area between 0.5 103 (solid thick line) and 7.5 103 kg m2 s1 in increments of 1.0 103 kg m2 s1. The inset in Figure 4a shows the detail of the temperature profile just below the surface at z = 0. 8 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 exponential form of equation (33)), with the most rapid magmatic gas density profiles are in excellent agreement variations in temperature occurring just below the lava with each other. For all carapace thicknesses and magmatic dome surface. In addition to predicting physically realistic gas fluxes of interest, b 1 and consequently we lava dome surface temperatures, the thermal boundary layer subsequently neglect the influence of gravity on the in the temperature profiles indicates that if the top meter of magmatic gas mass flux. the matrix is removed (say as the result of a small rockfall), [40] Figure 4b shows the resulting steady state magmatic then the temperature of the freshly exposed magma can gas density profiles for L = 20 m corresponding to the same easily reach 600 –830°C. This is also consistent with visual range of mg as that used in Figure 4a. Density profiles have observations of incandescence after the removal of roughly a minimum below the lava dome surface, the value and a meter of material from the actual lava dome surface position of which is a function of the magmatic gas mass [Sparks et al., 2000; Watts et al., 2002]. The temperature flux through the dome. Having calculated the magmatic gas profile illustrates why the model is insensitive to the value density we can find the magmatic gas volume flux using of L: the thickness of the thermal boundary layer is (d + r^g)^vg = 1. Necessarily, each profile of magmatic gas significantly less than the value of L chosen for realistic volume flux has a maximum at the same level as the mass fluxes. Below the thermal boundary layer the temper- minimum in the density profile. Clearly, the magmatic gas ature profiles predicted by equation (33) are approximately mass flux is the same at all levels. Finally, using the isothermal. This indicates that including thermal advection magmatic gas density profile and the lava dome temperature in the magmatic gas is sufficient to maintain realistic steady we can calculate the magmatic gas pressure using equation (12). state surface temperatures even if below the level ~z = L, The corresponding pressure profiles are shown in Figure 4c. beneath the rigid carapace the matrix does not convect in Pressure profiles are monotonic, with everywhere a pressure order to maintain a constant temperature at the bottom of the gradient driving magmatic gas upward through the dome carapace. and out into the atmosphere. For fixed values of porosity [39] In steady state we can also calculate profiles for mag- and permeability, an increase in the steady state magmatic matic gas pressure, density, and volume flux. The total mass gas mass flux per unit surface area of lava dome produces flux per unit area through a dome is constant (d + r ^g)^vg = 1. an increase in the magmatic gas pressure gradient. Hence For small b (as defined by equation (19)), it can be shown for the standard parameter values of mg = 3.5 103 kg (see Appendix A) that an analytical form exists for the s1 m2, and L = 20 m, the magmatic gas pressure is 3.04 density profile, which is given by 105 Pa (three times atmospheric pressure) at a level 20 m below the dome surface. Note that this value is approxi- pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2w2 HðTs Þew^z þ a þ b^z mately 60% of the value of the magmastatic pressure of ^g ¼ d þ r ; ð35Þ rrg~z = 5.10 105 Pa (thick line in Figure 4c). Hence the Q þ T^ internal gas pressure in this steady state are comparable to where the weight of overlying rock. If the magmatic gas mass flux through the matrix is increased further, then the magmatic 2 gas pressure exceeds the magmastatic overburden pressure a ¼ d þ ^rg;s ðQ þ Ts Þ2 2w2 HðTs Þ; ð36Þ in a region just below the surface. If the magmatic gas pressure exceeds the magmastatic overburden pressure plus (the harder to estimate) dome-scale tensile strength of the b ¼ 2 Q þ 1 þ w1 HðTs Þew ; ð37Þ matrix, then it is possible that the dome matrix may sustain damage. This could widen the cracks through the dome and and the magmatic gas density at the surface r ^g,s, can be increase the porosity. Eventually, if sufficient damage to the found by evaluating equation (12) at the surface and it is matrix occurs, dome failure could be expected. given by 5. Multiple-Scale Analysis dT^s [41] The key parameters which determine the multiple ^g;s ¼ r : ð38Þ timescale behavior of a lava dome are e and 1/Pe. In Q þ T^s section 3 it was shown that taking the limit of e ! 0 with 1/Pe 6¼ 0 leads to a model of a lava dome which cools For a carapace thickness L = 20 m and a magmatic gas mass unrealistically quickly. Therefore models in which both e flux mg = 3.5 103 kg s1 m2, we find b = 5.31 104. and 1/Pe are small, and of a similar order of magnitude are The nondimensional parameter b (defined by equation (19)) considered (see Table 3). We take Pe1 = Ae, where A = measures the relative importance of gravity to magmatic gas O(1), which allows a multiscale response of this system in pressure gradient on the magmatic gas flux. The small value two disparate timescales. Following the standard multiscale of b indicates we should expect the magmatic gas flux to be analysis [e.g., Hinch, 1991] we can look for the multiscale largely driven by the magmatic gas pressure gradient with evolution of the model equations by letting physical prop- the influence of gravity being largely insignificant. This can erties such as temperature satisfy be confirmed by comparing the analytical b = 0, magmatic gas density profile (equation (35)), to the numerically calculated magmatic gas density profile with b = 5.31 T^ ð^z; ^t Þ ¼ T^ ^z; ^ts ; ^tf ; ð39Þ 104. It can also be shown [Hicks, 2008] that the two 9 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 for two separate timescales: coordinate system with L = 20 m this corresponds to one grid point every 8 cm across the dome carapace. ^tf ¼ ^t ; ð40Þ [45 ] Before perturbing the steady state profile, the analytically predicted profiles are run through the time- a fast O(1) timescale and dependent code for a period of 20 model days to generate numerically stabilized profiles. These numerically stabilized ^ts ¼ e^t ; ð41Þ profiles differ slightly from the analytical predictions of the steady states, given by equations (33) and (35), with the a slow O(e1) timescale. With L = 20 m and mg = 3.5 maximum variation occurring in the temperature profile just 103 the corresponding fast timescale ^t f = 434.6 seconds below the surface, where the large temperature gradients and the slow timescale ^t s = 43.2 days. make the finite difference approximation least accurate. The [42] Derivatives with respect to time can be expressed as numerically stabilized surface temperature is typically 2% higher than that predicted by the analytical steady state profile. This numerically stabilized profile is then used as @ @ @ ¼ þe : ð42Þ the initial profile for the subsequent perturbations. @^t @^tf ^ts @^ts ^tf 6.1. Influence of the Passage of a Low-Pressure We now substitute this relationship into equations (17) and Weather System (16), and look for regular perturbation expansions of the [46] Volcanic activity has been reported to be influenced form g^ = g^0 + e^ g 1 + e2g^2 + , where the symbol g^ is by atmospheric pressure variations at Pavlof Volcano in the ^ successively T, ^vg, ^pg and r ^g. Matching the coefficients of e Aleutian Islands on timescales of a year [McNutt and at leading order governs the behavior of the system on the Beavan, 1987; McNutt, 1999] and at Stromboli Volcano, fast timescale Italy over timescales of a few hours [Neuberg, 2000]. This motivates the question: what is the characteristic response of rg;0 @ @^ @^p a lava dome to atmospheric modulation? Our primary case g;0 dþr ^g;0 ¼ 0; ð43Þ @^tf @^z @^z study is the Soufrière Hills Volcano in the Caribbean, a region frequently affected by hurricanes. As an idealized example the effect of the pressure variation associated with @ T^0 the passage of a hurricane over the lava dome is investigat- ¼ 0: ð44Þ ed. Surface pressure variations alone will be considered and @^tf the effect of increased wind speed and surface sensible heat flux, and significant rainfall (which often accompanies Equation (43) states that while the density can evolve on the hurricanes) will not be included. Modeling the effect of fast timescale (as in the full system), the temperature in rainfall adds significant additional complexity and is not equation (44) is unchanged on this fast timescale. Therefore included here [Taron et al., 2007; Hicks, 2008], although its changes in temperature only occur over the much longer ^t s- effect on lava dome stability are significant [Mastin, 1994; timescale. To investigate temperature variations analytically, Yamasato et al., 1998; Matthews et al., 2002; Carn et al., higher order terms in the perturbation expansion must be 2004]. considered. Rather than undertaking this very complicated [47] The passage of the low-pressure weather system is analysis, the full system is investigated numerically, in modeled by assuming a new dimensional surface pressure section 6. Theoretically, variations in lava dome tempera- pg,s(~t ) which changes over time, decreasing from a constant ~ tures will only occur over much longer timescales than atmospheric pressure pa and returning to pa after time T . variations in magmatic gas volume flux, pressure, and density. 8 < p Dpg 1 cos 2p~t ; 0 ~t T ; a pg;s ð~t Þ ¼ ~ 2 T 6. Time-Dependent Perturbations of a Porous : pa ; otherwise; Lava Dome [43] To investigate the relative lengths of the fast mag- where Dpg is the amplitude of the variation from atmo- matic gas timescale and the slow temperature evolution spheric pressure and T is the length of time it takes for the timescale, we shall conduct three numerical investigations hurricane to pass over the lava dome and the atmospheric corresponding to physically realistic perturbations of a lava pressure to return to its original value. This section shows dome. In the next three subsections, we shall explore results from a typical model run for a very strong hurricane sequentially the effects of an atmospheric pressure variation, that takes T = 12 hours to pass over the volcano, with pa = a small rockfall from the dome surface, and a change in 90000 Pa at 1000 m elevation, and a minimum pressure magmatic gas mass flux at depth upon the steady state lava pa Dp = 80000 Pa. dome temperature profile. [48] Figure 5a shows the evolution of the pressure at the [44] To rewrite the system of equations in a form ame- surface. The resulting disturbance in the dome interior nable to numerical analysis, we shall work solely in the magmatic gas pressure (Figure 5b) closely follows the small b limit and assume the role of gravity has a negligible perturbation of atmospheric pressure. In Figure 5b, the impact on the magmatic gas flux. The equations are absence of a visible lag between the perturbation and investigated using a standard finite difference procedure the resulting disturbance indicates that the pressure varies on a regular grid with 250 points. In the dimensional rapidly in response to the surface pressure perturbation, with 10 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 This indicates that, like the pressure, variations in density and volume flux are transmitted over the short timescale of a few minutes. [49] The evolution of the temperature (Figure 6) is markedly different to the profiles of pressure, density, and gas volume flux shown previously. Surface temperature variations are quite small, up to only 4°C. Temperature changes at depth are smaller still. The surface temperature of the dome does not return to its original value immediately after the pressure perturbation has passed over the dome: the temperature distribution is still recovering many hours after the hurricane has completely departed. Hence temperature variations occur on a much longer timescale than gas pressure, density or volume flux changes, as predicted. [50] To study the influence of wind speed variation, a second test was run where the convective heat flux from the surface varied between 5 m s1 and 30 m s1. A sinusoidal evolution of the wind speed was used to match the pressure evolution (equation (45)) and to mimic the high winds associated with the passage of a hurricane. The increased wind speed and associated higher surface heat fluxes pro- duced significantly greater temperature changes, with a sur- face temperature minimum of 110°C recorded 40 minutes after peak wind speeds. This compares with a steady state surface temperature after an indefinite period of 30 m s1 winds of 100°C. The surface temperature does not approach prehurricane levels until many hours after the hurricane has passes and surface pressure and wind speed are back to normal. Some changes in the pressure profiles are also evident in this period, but in this phase further pressure evolution is determined by the much slower temperature profile recovery. 6.2. Influence of Small Rockfalls [51] The surface of a lava dome is unstable and affected by frequent small rockfalls. These are on a much smaller scale than dome collapse events, but locally small rockfalls allow material to be gradually removed from the surface thus exposing hotter magma from below. The effect of this on the steady state profiles described in section 4 is simulated by an idealized instantaneous removal of the top metre of rock and examination of the subsequent Figure 5. Evolution of lava dome, as a dry, low pressure weather system with a minimum pressure of 80,000 Pa moves over the summit of the volcano. (a) Surface pressure (in 105 Pa), (b) pressure with contours every 0.1 105 Pa, (c) magmatic gas density with contours every 0.05 kg m3, and (d) magmatic gas volume flux per unit surface area with contours every 0.01 m s1. See legends for shading. information transmitted over a short timescale of the order of a few minutes. The deviation from the steady state profile decreases with depth, indicating that the disturbance is localized close to the dome surface. The corresponding variations in the profiles of r and v are shown in Figure 5c and Figure 5d, respectively. Again, we see that the variations in density and magmatic gas volume flux last as long as the variation in the surface pressure, with the Figure 6. As in Figure 5, showing the evolution of lava greatest disturbances occurring just below the dome surface. dome surface temperature. 11 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 Figure 7. Evolution of lava dome, after the top 1 m is removed due to a rockfall from the surface. (a) Temperature in °C, (b) gas density in kg m3, and (c) pressure in Pa at 3 seconds, 30 seconds and 5 minutes, 50 minutes, and 500 minutes after the instantaneous removal of 1 meter from the lava dome surface. evolution of the lava dome. To justify sudden removal of to magmatic gas pressure and density are shown in Figure 8. rock in the model we note that rockfalls do occur on much Although a lava dome surface temperature in excess of shorter timescales than even the shortest timescale over 800°C and the surface heat flux given by equation (5) which the magmatic gas pressure, density, and volume flux results in rapid cooling, the changes in temperature again evolve. take place much more slowly than the simultaneous evolu- [52] Figure 7 shows the evolution of the lava dome tion of the magmatic gas pressure, density, and volume flux. temperature, and magmatic gas density and pressure profiles Hence 30 minutes after the rockfall, the profile of gas at 3 seconds, 30 seconds, and 5, 50 and 500 minutes after pressure and density have equilibrated with the current the instantaneous removal of the top one metre of andesite temperature (indicated by approximately constant isopleths from the dome. Two thicker lines are plotted in Figure 7, in Figure 8). However, the temperature profile is still corresponding to the steady state profile prior to the rockfall adjusting as show by Figure 7. While the lava dome is (dashed) and the final steady state profile after the rockfall cooling over several days, the magmatic gas pressure, (solid). The removal of 1 m of andesite exposes magma density, and volume flux quickly adapt to the changing with a temperature in excess of 800°C, as observed in actual temperature profiles, changing on the order of a few rockfall events. This is a 500°C increase locally in the minutes. surface temperature. [53] The instantaneous removal of surface rock brings 6.3. Influence of Magmatic Gas Mass Flux Variation about an instantaneous discontinuity in the pressure profile, [54] The steady state profiles calculated in section 4 as magma previously at depth is suddenly exposed on the assume that the magmatic gas mass flux through the lava surface. Over short timescales the pressure discontinuity is dome is constant. However, significant temporal variations smoothed out, with the new surface pressure being carried in the magmatic gas fluxes from the Soufrière Hills Volcano rapidly into the dome interior, producing physically reason- have been recorded [Edmonds et al., 2001]. Hence steady able magmatic gas volume fluxes. Isopleths corresponding state solutions may not be applicable and the response to a time-dependent mass flux is investigated. 12 of 16

B07201 HICKS ET AL.: HEAT FLOW IN A GAS-PERMEABLE LAVA DOME B07201 Figure 8. As in Figure 7. (a) Density with contours every Figure 9. Evolution of lava dome, when at ~t = 0 the 0.02 kg m3 and (b) pressure with contours every 0.05 magmatic gas mass flux per unit area changes from 3.5 105 Pa. See legends for shading. 103 to 5.5 103 kg m2 s1 at a depth of ~z = 20 m. (a) Density with contours every 0.05 kg m3 and (b) pressure [55] An idealized experiment is carried out, where the with contours every 0.25 105 Pa. See legends for shading. steady state profiles are perturbed by instantaneously in- creasing the magmatic gas flux per unit surface area of lava timescale of about 10 minutes, it takes a period of approx- dome, 20 m below the surface, from 3.5 103 to 5.5 imately 5 days (the slow ^t s-timescale) for the surface 103 kg m2 s1, at ~t = 0. This corresponds to an increase in temperature to approach its new value. This is predicted the steady state surface temperature from 210 to 260°C. by the analysis at the start of this section with equations (43) Such an increase in mass flux could be an idealization of the and (44) implying the temperature will change slowly while effect of the system being charged with a basaltic input at the magmatic gas pressure, density, and volume flux evolve depth [Edmonds et al., 2001]. much more rapidly. Over the longer timescale the magmatic [56] Immediately after the mass flux is increased, there is gas pressure, density, and volume flux continue to change in a discontinuous jump in pressure at the lower boundary. slavish response to the very slow temperature variations. This is shown by the rapidly changing isopleths at this point in a time-level section of density (Figure 9a) and pressure 7. Discussion (Figure 9b). The rapidly changing isopleths are an imme- diate consequence of Darcy’s law (equation (18)) as a [58] Models of the thermal structure of a lava dome that greater pressure gradient is required to drive the increased rely solely on diffusion in the solid matrix for the transport mass flux up through the dome. However, by ~t = 10 minutes the effect of the increased mass flux through the dome has been transmitted to the surface, and the isopleths once again appear approximately constant in time, albeit with a higher pressure gradient. By this stage the changes in the pres- sure corresponding to the fast ^t-timescale have been com- pleted and any further change in pressure occurs due to the temperature evolution alone, over the much longer timescale. [57] The analytical steady state profiles predict a lava dome surface temperature of 260°C for mg = 5.5 103 kg m2 s1, which is an increase of almost 50°C from the initial surface temperature. However, 10 minutes after the instantaneous increase in mg, the lava dome surface tem- perature has barely changed from its initial value. Figure 10 shows the surface temperature as a function of time as it varies between the two analytical steady state temperatures. Notice, in marked contrast to the magmatic gas pressure and Figure 10. As in Figure 9, showing the evolution of lava density which evolve to quasi steady states over the fast ^t f dome surface temperature. 13 of 16

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