Effect of Viscous Dissipation on Nonisothermal High-Viscosity Magma Flow in a Volcanic Conduit

Transcription

1 Fluid Dynamics, Vol. 39, No. 6, 2004, pp Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp Original Russian Text Copyright 2004 by Barmin, Vedeneeva, and Mel nik. Effect of Viscous Dissipation on Nonisothermal High-Viscosity Magma Flow in a Volcanic Conduit A. A. Barmin, E. A. Vedeneeva, and O. E. Mel nik Received June 3, 2004 Abstract A two-dimensional nonisothermal model of magma flow in a volcanic conduit is proposed. The model makes it possible to investigate the effect of the processes of viscous dissipation and heat conduction on the magma flow. It is established that the effect of these processes is significant, particularly in the case of high flow rates. It is shown that in this case the conduit resistance calculated from the Poiseuille formula widely used in one-dimensional models is highly overestimated. This is related to the formation of a strongly heated fluid layer with reduced viscosity in the near-wall conduit zone. Within the framework of the proposed model it is possible to describe eruptions with flow rates which are several times higher than the flow rates obtained within the framework of one-dimensional models. Keywords: viscous dissipation, temperature-dependent viscosity, conduit resistance, explosive volcanic eruptions, magma flow rate. In volcanic eruptions magma located in the Earth s crust ascends to the surface. The magma contains molten rocks (melt) and dissolved gas. It is assumed that the magma is concentrated in a magma chamber located at a depth of several kilometers and then erupts through a volcanic conduit. A distinction is made between extrusive (lava) and explosive eruptions. In extrusive eruptions, liquid magma (lava) flows out from the volcanic vent. The explosive eruptions characteristic of high-viscosity magmas are much more destructive. In this case a gas flow carrying particles of fragmented melt breaks through to the surface. When magma ascends from a magma chamber, the magma is converted into a bubbly liquid since gas bubbles are formed and grow in the magma due to the release of dissolved gas. In explosive eruptions the bubbly liquid is fragmented with the formation of a gas-particle dispersion when certain critical conditions are reached, whereas in extrusive eruptions no fragmentation occurs. In explosive eruptions, assuming that the transition zone from bubbly fluid flow to gas-particle dispersion flow is narrow [1], the magma conduit flow is considered to contain two zones, namely, a bubbly fluid flow zone and a gas-particle dispersion flow zone, separated by a fragmentation front. In both zones the flow is assumed to be one-dimensional and isothermal, and in the bubbly fluid and gas-particle dispersion zones the flows are assumed to be laminar and turbulent, respectively [1 3]. The resistance of the volcanic conduit walls in the bubbly fluid flow zone is calculated using the Poiseuille formula. This approximation is valid for a laminar incompressible fluid flow with constant viscosity. In the present study we propose a two-dimensional nonisothermal (in the bubbly fluid flow zone) model of a steady-state magma flow in a circular cylindrical conduit which makes it possible to take the processes of viscous dissipation and heat conduction into account. The magma viscosity is assumed to depend on the temperature and the concentration of dissolved gas. Problems of restructuring viscous fluid channel flows with temperature-dependent viscosity were considered in [4 7]. In [4, pp ] a steady-state fluid flow in a circular cylindrical pipe was considered under the assumption that the velocity has a single component directed along the pipe axis and the temperature is independent of the radial variable. Analytical solutions were found for three specific temperature-dependent viscosities. In [5 7] an exponential temperature dependence of the viscosity was used /04/ Springer Science+Business Media, Inc.

2 864 BARMIN et al. In [5] a time-dependent viscous fluid flow in a circular pipe with a given constant wall temperature was considered. The convective terms were neglected and it was assumed that the pressure gradient was equal to a given constant value and the velocity had a single longitudinal component. All the quantities depended only on the radius and time. The effects of restructuring of the velocity profile from a parabolic to a rectangular profile and heating of the fluid near the pipe walls were obtained for a fixed pipe cross-sectional area. In [6 7] steady-state fluid flows in a flat slot with walls maintained at constant temperature and given temperature and velocity profiles at the inlet were considered. As in [4, 5], in [6] it was assumed that the velocity had only a single longitudinal component. The problem was solved with reference to magma flow in lava tubes. In [6] the solution was constructed by means of asymptotic methods and in [7] numerically. In [6, 7] it was shown that the fluid slot flow evolves downstream from a flow with a parabolic velocity profile given at the inlet to a flow with a flat velocity profile characteristic of ideal fluid flow. In this case hot boundary layers with a high velocity gradient are formed on the walls. Thus, the flow is restructured as in [5]. From the safety standpoint the question of predicting the intensity of a volcanic eruption is the most important. Therefore, as distinct from [4 7], in the present study we solve the boundary-value problem of determining the magma flow rate for a given pressure drop at the inlet to and outlet from a conduit of given length. The viscosity is assumed to depend not only on the temperature but also on the concentration of the dissolved gas. In this case we will take a real temperature and concentration dependence of the viscosity obtained on the basis of modern experimental data [8]. As distinct from [6, 7], the conduit is assumed to be circular and cylindrical rather than a flat slot. We will study the effect of the processes of viscous dissipation and heat conduction on the eruption dynamics. In the case of high flow rates the conduit resistance differs qualitatively from that used in onedimensional isothermal models: when the magma ascends in the conduit the resistance decreases, whereas in the one-dimensional models it increases. As compared with the one-dimensional isothermal model, a significant increase in eruption intensity is obtained. 1. MATHEMATICAL MODEL We will consider a steady-state magma flow in a circular cylindrical conduit which in the case of explosive eruptions is assumed to consist of two zones: a bubbly fluid flow zone and a gas-particle dispersion flow zone separated by a fragmentation front perpendicular to the axis of the conduit. In the case of extrusive eruptions the entire conduit is occupied by bubbly fluid. In the bubbly fluid flow zone the magma is incompressible viscous heat-conducting multicomponent fluid consisting of a melt, a gas dissolved in the melt, and gas bubbles whose effect on the magma compressibility is neglected. The magma viscosity depends on the temperature and the concentration of the dissolved gas. The temperature varies due to the processes of viscous dissipation and heat conduction. In the bubbly fluid flow zone the magma flow is assumed to be laminar, two-dimensional, and axisymmetric. This flow takes place until a fragmentation threshold is reached in the magma, namely, a critical condition in satisfying which the magma is converted from a bubbly fluid into a gas-particle dispersion. In the gas-particle dispersion flow zone the magma is a perfect gas carrying melt particles containing dissolved gas. The velocities of the carrier gas and the particles are assumed to be identical. Since in the gas-particle dispersion flow zone the flow is turbulent [1 3], we will consider only cross-section-average flow characteristics and the flow will be assumed to be one-dimensional. In the gas-particle dispersion flow zone we will neglect the variation of the mean temperature of the magma and the resistance of the volcanic wall conduit [2]. The system of dimensionless equations describing the magma flow in the bubbly fluid flow zone can be obtained from the Navier-Stokes equations for the case of narrow conduits with a length L much greater than their radius R when the flow is fairly slow, i.e., for Reynolds numbers Re L/R. In cylindrical coordinates the system has the form:

3 EFFECT OF VISCOUS DISSIPATION (rw) + v r r z = 0 (1.1) p r = 0, p z = α 1 ( rµ v ) k (1.2) r r r w T ( ) v 2 r + v T z = βµ + γ 1 ( r T ) (1.3) r r r r µ(c g, T )= 1 µ 0 10 A(c 0,T ) (1.4) A(c g, T )= ln(c g ) ln(c g ) (T + 1)T 0 ( ln(c g )) (1.5) c g = c g0 p (cg0 p < cgm ), c g = c gm (c g0 p cgm ), c g0 = C f p0 α = µ 0 ρ m v 2 0 = 2 ρ m v 0 R p 0 Re Eu, k = ρ mv 2 0 gr p 0 v 2 = 2 0 Eu Fr 2 (1.6) v 2 0 v 2 0 β = µ 0 = 1, γ = κ ρ m v 0 R ct 0 Re ct 0 ρ m cv 0 R = 1 Pe Here, (1.1) is the continuity equation, (1.2) is the momentum equation, (1.3) is the heat transfer equation, (1.4) is an empirical formula for the magma viscosity [8], (1.5) is the solubility law, and (1.6) are dimensionless parameters of the problem. The z and r axes are directed along the conduit and the radius, respectively, z [0; L/R], r [0; 1]; v and w are the longitudinal and radial velocity components in the bubbly fluid flow zone; p, µ, T and c g are the pressure, the viscosity, the magma temperature increment with respect to the given temperature at the conduit inlet, and the mass concentration of the gas dissolved in the magma, respectively; v 0, p 0, T 0,andµ 0 are characteristic dimensional values of the velocity, pressure, temperature and viscosity used in obtaining the dimensionless equations (1.1) (1.3): p 0 and T 0 are the lithostatic pressure and temperature at the conduit inlet, µ 0 is a viscosity determined from formula (1.4) for given p 0 and T 0 (when p = 1andT = 0wehaveµ = 1); c gm is the maximum concentration of the gas dissolved in the magma at which bubbles are nucleated; ρ m is the dimensional phase density of the melt; c, κ, andc f are the dimensional specific heat and magma thermal conductivity and gas solubility coefficients, which are assumed to be constant; and g is the acceleration of gravity. In going over to dimensionless variables, we will take the conduit radius R as the characteristic length. Since the gas density in the bubbles is much less than the melt density and the bubble concentration in the bubbly fluid flow zone is relatively low, we assume that the magma density is equal to the melt density ρ m. At the conduit inlet we specify a parabolic velocity profile and constant pressure and temperature z = 0: v(r, 0)=2v a (1 r 2 ), w(r, 0)=0, p(0)=p in, T (r, 0)=0 where v a is the cross-section-average dimensionless velocity of the magma flow, which by virtue of the incompressibility is constant in the bubbly fluid flow zone. The velocity determines the dimensionless magma mass flow rate Q = πv a and is an unknown quantity to be determined. On the axis of the conduit we impose the conditions of flow symmetry r = 0: v w (0, z)=0, r T (0, z)=0, r (0, z)=0 r on the wall the no-slip condition for the velocity and the adiabaticity or constant temperature condition for the temperature r = 1: v(1, z)=0, w(1, z)=0, T (1, z)=0 (T (1, z)=const = 0) r

4 866 BARMIN et al. In the case of isothermal boundary conditions the conduit wall temperature is assumed to be equal to the magma temperature at the conduit inlet. The transition from bubbly fluid flow to gas-particle dispersion flow takes place when the bubble concentration in the magma reaches a critical value taken as the fragmentation condition α g = α gf [9]. We note that in the above model of the magma flow in the bubbly fluid zone we neglected a number of effects which can affect the eruption dynamics. We neglected the magma compressibility due to the presence of bubbles. Prior to fragmentation, the typical bubble concentration in the magma reaches 50 70%; therefore the real velocity before fragmentation can be 2 3 times greater than the velocity of a homogeneous fluid. Thus, the assumption that the magma is incompressible is a very strong one and will be surmounted in further investigations. The concentration of the gas dissolved in the magma is calculated from the equilibrium solubility law which cannot be used at high magma flow velocities when the diffusion lag in bubble growth has a significant effect [10]. Idealized boundary conditions were used for the conduit wall temperature. For example, in certain eruptions the magma intrudes into a cold conduit and the thermal losses to the surrounding rocks can be significant. The system of dimensionless equations for cross-section-average quantities describing the magma flow in the gas-particle dispersion flow zone has the form: (ρ g α g +(1 α g )c g )v = c gm v a (1.7) (1 α g )(1 c g )v =(1 c gm )v a (1.8) δρv dv dz = dp dz kρ, δ = ρ mv 2 0 p 0 = 2 Eu, ρ = ρ gα g +(1 α g ) (1.9) ρ g = ρ g0 p, ρ g0 = p 0 ρ m R g T 0 (1.10) Here, c g can be determined from formula (1.5), equations (1.7) and (1.8) are the continuity equations for the gas and the melt, respectively, (1.9) is the momentum equation for the entire mixture, and (1.10) is the equation of state of the gas. ρ g is the phase gas density, α g is the volume concentration of the free gas, v is the cross-section-averaged flow velocity in the gas-particle dispersion flow zone, ρ is the mixture density; and R g is the gas constant. In going over to dimensionless variables we take the true melt density ρ m as the characteristic density. In the gas-particle dispersion zone the variation of the average magma temperature can be neglected and we can assume that the average magma temperature is equal to the average post-fragmentation temperature [2]. Since for realistic parameters numerical simulations of the flow in the bubbly liquid zone show that the magma is heated only in a very narrow near-wall region and the temperature increases by not more than a factor of three, we can assume that the average gas-particle dispersion temperature is equal to the given temperature T 0 at the conduit inlet. If the dependence p(z) is one-to-one, the system of equations (1.7) (1.10) can be reduced to the form: dp dz = 1 ( 1 f (p), f (p)= v(p) + δ dv ) (1.11) k v a dp v a v(p)= ρ g (p)α g (p) +(1 α g (p)), ρ g(p)=ρ g0 p (1.12) α g (p)= C(p) ρ g (p) +C(p), C(p)=c gm c g (p) 1 c gm (1.13) where the dependence c g (p) can be determined from formula (1.5). In order for the dependence c g (p) to be one-to-one it is sufficient that f (p) 0 on the entire pressure interval considered. From the mass conservation law for the entire mixture ρv = v a we have

5 EFFECT OF VISCOUS DISSIPATION 867 dv dp = d dp ( ) va = 1 M 2, M = v ρ δv a a, a = 1 dp δ dρ (1.14) where M is the Mach number and a is the dimensionless speed of sound. In accordance with (1.11) and (1.14), f (p)=0whenm 2 = 1. Thus, the dependence p(z) is one-to-one until the flow velocity reaches the local speed of sound. From Eq. (1.11) we can find the location of the fragmentation front z f : z f = L pout R p f f (p)dp, p out = p(l/r) (1.15) where p is the pressure at which the critical bubble concentration is reached: α g = α gf. The fragmentation pressure p f can be found from the condition α g = α gf and Eq. (1.13) with allowance for (1.5) p f = c g0 (1 α gf )+ c 2 g0 (1 α gf )2 + 4α gf ρ g0 (1 c gm )c gm (1 α gf ) 2α gf ρ g0 (1 c gm ) 2 The pressure p f is independent of the magma flow velocity. At the conduit outlet the pressure p out is specified by the formula p out = p a (p p a ), p out = p (p > p a ) where p a is the dimensionless atmospheric pressure; p is a root of the equation f (p)=0 whose value depends on the magma flow rate Q = πρv = const = πv a as on a parameter. In accordance with our numerical experiments, for parameters corresponding to the magma flow in a volcanic conduit the function f (p) has a single root on the possible pressure interval. In Fig. 1 we have plotted a graph of the location of the fragmentation front z f as a function of the magma flow rate Q for the values of the parameters given below, which are characteristic of the magma flow in a volcanic conduit. Thus, the problem of determining the magma flow rate Q as a function of the pressure p in given at the conduit inlet is formulated. The dependence Q(p in ) can be nonunique; therefore, we will find the unique inverse dependence p in (Q) [2]. The problem was solved numerically. The pressure p in at the conduit inlet was found for a given flow rate Q using the ranging method. This method consists in selecting the unknown pressure p in at the conduit inlet from the condition of equality of the sum of the lengths of the bubbly fluid and gas-particle dispersion zones to a given conduit length. In each ranging stage the length of the bubbly fluid zone was determined from the solution of the system of partial differential equations (1.1) (1.3) and the length of the gas-particle dispersion zone from formula (1.15). The system of equations (1.1) (1.3) in the pressure, two velocity components, and temperature was solved using the finite-difference method separately in each calculated conduit cross-section progressively downstream until the fragmentation pressure p f was reached. In each computational cross-section the unknown pressure was found the ranging method from the no-slip condition on the wall for the radial velocity component. At a fixed pressure in the calculation cross-section the velocity and the temperature were found using the matrix sweep method [11].

6 868 BARMIN et al. Z f Q p in Fig. 1 Fig. 2 Fig. 1. Relative location of the fragmentation front Z f = z f /(L/R) as a function of the magma flow rate Q Fig. 2. Magma flow rate Q as a function of the pressure p in at the conduit inlet for adiabatic and isothermal conduit wall conditions and an extrusive eruption (curves 1 and 2, respectively). Curve 3 corresponds to the same dependence obtained within the framework of the one-dimensional isothermal model; Q 0 = 1.9, Q 1 = 2.6, and Q 2 = CALCULATION RESULTS In what follows, we will give the numerical calculations for the following values of the parameters characteristic of the magma flow in a volcanic conduit: R = 25 m, L = 5000 m, v 0 = 1.0 m/s, p 0 = ρ m gl + p atm = MPa, g = 9.8 m/s 2, ρ m = 2500 kg/m 3, µ 0 = Pa s, T 0 = 1123 K, C f = Pa 1/2, c gm = 0.05 c = 1200 J/(kg K), κ = 0.8 J/(m s K), α gf = 0.7 This corresponds to the following dimensionless quantities: Re = 1.7, Eu = , Fr = , Pe = , v 2 0/cT 0 = Initially, we will consider the problem for a fixed length of the bubbly fluid zone equal to L/R. Thisflow corresponds to magma conduit flow without fragmentation, i.e., to an extrusive type of eruption. In Fig. 2 we have plotted graphs of the magma flow rate Q as a function of the pressure p in at the conduit inlet for adiabatic and isothermal wall conditions (curves 1 and 2). The broken curve 3 represents the same dependence for isothermal flow when the velocity profile is parabolic and the viscosity depends only on the concentration of the gas dissolved in the magma. All three curves almost coincide for low excess pressures at the conduit inlet (with respect to the lithostatic pressure). In the case of the nonisothermal model, flows with higher flow rates, as compared with the isothermal model, are realized with increase in the pressure p in, the flow rates being higher for the adiabatic wall condition than for the isothermal condition. In the nonisothermal model the dependence Q(p in ) is nonunique: there are pressures p in which correspond to two flow rates Q 1 and Q 2, i.e., for a given conduit length and pressure drop two flow regimes with significantly different flow rates can take place. There is a critical pressure p in such that at higher pressures at the conduit inlet a solution does not exist. In order to explain the results obtained we will consider the dynamics of the magma conduit flow. In Fig. 3 we have plotted graphs of the radial distributions of the longitudinal velocity component v, the temperature T, and the viscosity µ (r is the radius) in various conduit cross-sections in the near-wall zone for a flow rate Q 2 and the isothermal wall condition.

7 EFFECT OF VISCOUS DISSIPATION 869 µ (1 r) 10 4 Fig. 3 Fig. 4 Fig. 3. Velocity, temperature, and viscosity distributions in the near-wall zone for the conduit cross-sections z = 0, 2.5, 50, and 200 (curves 1 4) andq 2 = The cross-section z = 200 corresponds to the conduit outlet Fig. 4. Velocity distribution over the conduit cross-sections z = 0, 5, 10, and 200 (curves 1 4) for Q 2 = The cross-section z = 200 corresponds to the conduit outlet The magma viscosity varies due to two competing processes. On the one hand, the pressure decreases downstream and, accordingly, the concentration of the gas dissolved in the magma also decreases (see (1.5)) and the magma viscosity increases. On the other hand, heat is released due to viscous friction. The heat release is most intensive in the neighborhood of the walls, where the velocity gradients are high as a result of the no-slip conditions. The heat flows from the walls into the interior of the conduit due to heat conduction; however, the heat transfer process is weak for the parameters considered. Thus, a narrow heated fluid layer, in which the temperature increases downstream and the fluid viscosity decreases, is formed in the neighborhood of the conduit walls. As a result, the parabolic velocity profile (given at the conduit inlet and corresponding to isothermal flow) begins to change and approaches the rectangular profile characteristic of ideal fluid flow (Fig. 4). In this case the pressure drop required for the magma to flow at a given flow rate decreases (vanishing for the ideal fluid). In Fig. 5 we have shown the variation of the conduit resistance (stress tensor components τ rz (1, z) = (µv r) r=1 ) along the z axis of the conduit for flow rates Q 1 and Q 2. For comparison purposes, for the flow rate Q 2 we have shown the variation of the conduit wall resistance obtained using the isothermal model from the Poiseuille formula, which in dimensionless variables has the form: τ rz (1, z)= 4αv a µ(c g, 0). For low flow rates the processes of dissipative heat release are not significant, the temperature varies only slightly, the velocity profile remains similar to a parabolic profile, and the changes in the conduit wall resistance due to viscous dissipation are almost negligible (Fig. 5). Therefore, in Fig. 2 curves 1 and 2 almost coincide with curve 3 obtained within the framework of the one-dimensional isothermal model. However, for higher flow rates, even in the initial conduit cross-sections, the temperature almost doubles in the vicinity of the walls due to viscous dissipation and the velocity profile becomes almost rectangular (Fig. 4), the conduit resistance decreasing (Fig. 5). Therefore, for a fixed pressure drop the magma flow rate is much higher (Fig. 2).

8 870 BARMIN et al. τ rz Fig. 5. Conduit resistance τ rz (1, z)=(µv r) r=1 for a pressure p in = at the conduit inlet and flow rates Q 1 = 2.6and Q 2 = (curves 1 and 2, respectively). Curve 3 corresponds to the same dependence obtained for the one-dimensional isothermal model and Q 0 = 1.9 z Thus, in Fig. 2 the lower-flow-rate regime corresponds to a weakly heated magma. The flow is similar to an isothermal flow with a parabolic velocity profile. The conduit resistance is due to the high viscosity. The higher-flow-rate regime corresponds to a magma strongly heated in the near-wall zone. The velocity profile is flat over almost the entire flow region. The conduit resistance is due to the higher flow rate. For the adiabatic wall condition no heat flows out from the conduit; therefore, the viscous dissipation effect is more significant. The presence of several steady-state eruption regimes at a fixed pressure in the magma chamber is also observed within the framework of other volcanic conduit magma flow models, for example, in the one-dimensional isothermal model of eruption of a compressible magma [1 3]. Frequently, during a single eruption several types of pumice are formed simultaneously [12, 13], some of which have traces of strong heating and deformation (fragmented, partially melted crystals). The formation of a strongly heated layer in the vicinity of the wall and the weak variation of the temperature at the centre of the conduit may explain this phenomenon. The higher-flow-rate (Q 2 ) regimes correspond to the explosive type of eruption [14]; therefore, in this case it is necessary to solve the complete system of equations with allowance for magma fragmentation and the presence of a gas-particle dispersion flow zone. In Fig. 6 we have plotted graphs of the magma flow rate Q as a function of the pressure p in at the conduit inlet analogous to those given in Fig. 2 but corresponding to the explosive type of eruption. In Fig. 7 we have plotted graphs of the conduit resistance (in the bubbly fluid flow zone) for low Q 5 and high Q 2 flow rates in the case of isothermal conduit wall conditions. At low pressures p in at the conduit inlet (low flow rates) all three curves in Fig. 6 lie close together. As the pressure p in increases, flows with flow rates several times higher than the flow rates obtained from the isothermal model are realized within the framework of the nonisothermal model. However, in the case of explosive eruptions the nonuniqueness of the dependence Q(p in ) disappears. Within the framework of the nonisothermal model no solution corresponding to cold magma in the near-wall zone exists even for low magma flow rates (Figs. 5 and 7). At low flow rates the bubbly fluid flow zone occupies a small part of the conduit (Fig. 1). Therefore, in the case of low flow rates a fixed pressure drop between the fragmentation pressure p f and the pressure p in at the conduit inlet takes place over a much smaller distance. This makes it possible to have eruptions at small values of p in. With increase in the flow rate the length of the gas-particle dispersion flow zone decreases almost to zero and the bubbly fluid occupies almost the entire conduit. The pressure p in at the conduit inlet is bounded by the strength of the rocks forming the volcanic conduit walls and cannot exceed the lithostatic pressure by more than MPa ( in dimensionless variables). Therefore, using Fig. 6, we can estimate the maximum possible flow rate. Clearly, within

9 EFFECT OF VISCOUS DISSIPATION 871 Q τ rz p in z Fig. 6 Fig. 7 Fig. 6. Magma flow rate Q as a function of the pressure p in at the conduit inlet for the adiabatic and isothermal conduit wall conditions and an explosive eruption (curves 1 and 2, respectively). Curve 3 corresponds to the same dependence for the one-dimensional isothermal model; Q 2 = 125.6, Q 3 = 62.8, Q 5 = 22, p 3 = 0.983, and p 4 = 1.48 Fig. 7. Conduit resistance τ rz (1, z) =(µv r) r=1 in the bubbly fluid flow zone for flow rates Q 5 = 22 and Q 2 = (curves 1 and 2, respectively). The vertical broken lines correspond to magma fragmentation τ rz p z z Fig. 8. Conduit resistance τ rz (1, z) =(µv r) r=1 in the bubbly fluid flow zone (a) and pressure p(z) (b) for a pressure at the conduit inlet p in = p 3 = and flow rate Q 3 = 62.8 (curves 1). The same dependences are plotted for the onedimensional isothermal model for p in = p 3 and Q 4 = 33 < Q 3 (curves 2) andforp in = p 4 = 1.48 > p 3 and Q 3 (curves 3). The vertical broken lines (a) and the break points on the curves (b) correspond to magma fragmentation the framework of the model considered it can be several times higher than within the framework of the isothermal model. In Fig. 8 we have plotted profiles of the resistance (in the bubbly fluid flow zone) (a) and the pressure (b) in the case of isothermal boundary conditions on the conduit wall for a flow rate Q 3 corresponding to the pressure p 3 at the conduit inlet. For comparison, we have also plotted the resistance and pressure profiles obtained within the framework of the isothermal model for a flow rate Q 4 < Q 3 at the same pressure p 3 at the conduit inlet and a flow rate equal to Q 3 corresponding to a pressure p 4 > p 3 at the conduit inlet. For curves 1 and 2 the pressures at the conduit inlet are equal; therefore, the initial viscosities are also equal but the flow rate for curve 2 is significantly lower. For curves 1 and 3 the flow rates are equal but for curve 3 the pressure at the conduit inlet is higher and, therefore, the initial viscosity is lower. Thus, for both curves 2 and 3, obtained within the framework of the isothermal model, at the beginning of the conduit the resistance is lower than for curve 1 (Fig. 8a). In the isothermal case the conduit wall resistance increases

10 872 BARMIN et al. downstream due to an increase in the viscosity, whereas in the nonisothermal case the conduit resistance decreases sharply due to viscous dissipation leading to restructuring of the velocity profile and a decrease in viscosity. Accordingly, in both isothermal cases the pressure decreases downstream more rapidly than in the nonisothermal model (Fig. 8b). Since the fragmentation pressure p f depends on neither the flow rate nor the pressure at the conduit inlet, for curves 1 and 3 the location of the fragmentation front is the same, while for curve 2 fragmentation takes place at a greater depth. From Fig. 8b it is clear that, within the framework of the nonisothermal flow model, in the bubbly fluid there is no strong increase in the pressure gradient like that in the isothermal model. Certain fragmentation criteria [15 16] are based on the assumption of a strong increase in the pressure gradient before fragmentation. The results obtained show that these criteria need correcting. Summary. The two-dimensional nonisothermal model makes it possible to explore the effect of the processes of viscous dissipation and heat conduction on the magma flow in a volcanic conduit. This effect turns out to be significant. It is shown that eruptions with flow rates several times higher than the flow rates obtained within the framework of the isothermal model can take place. In the case of high flow rates the behavior of the conduit resistance differs qualitatively from the resistance calculated from the Poiseuille formula widely used in one-dimensional models. As the magma ascends in the conduit the resistance decreases rather than increases. The dependence obtained for the variation of the resistance along the conduit can be used to correct the one-dimensional models. The boundary value problem of determining the magma flow rate as a function of the given pressure at the conduit inlet is solved. In the case of extrusive eruptions this dependence is nonunique: there is an interval of conduit inlet pressures such that eruptions with low and high flow rates can take place at the same pressure. REFERENCES 1. Yu. B. Slezin, Dynamics of the dispersion regime of volcanic eruptions. 2. Flow rate instability condition and nature of catastrophic explosive eruptions, Vulkanologiya i Seismologiya, No. 1, 23, (1984). 2. A. A. Barmin and O. E. Mel nik, Eruption dynamics of high-viscosity gas-saturated magmas, Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza, No. 2, 49 (1993). 3. A. W. Woods and T. Koyaguchi, Transition between explosive and effusive eruptions of silicic magmas, Nature, 370, 641 (1994). 4. S. M. Targ, Fundamental Problems of Laminar Flow Theory [in Russian], Gostekhizdat, Moscow, Leningrad (1951). 5. A. L. Gonor and A. A. Chulkov, Theory of hydrodynamic thermal explosion, Dokl. Akad. Nauk SSSR, 316, 856 (1991). 6. H. Ockendon, Channel flow with temperature-dependent viscosity and internal viscous dissipation, J. Fluid Mech., 93, 737 (1979). 7. A. Costa and G. Macedonio, Viscous heating in fluids with temperature-dependent viscosity: implications for magma flows, Nonlinear Processes in Geophysics, 10, 545 (2003). 8. K. U. Hess and D. B. Dingwell, Viscosities of hydrous leucogranite melts: a non-arrhenian model, Amer. Mineralogist, 81, 1297 (1996). 9. R. S. J. Sparks, The dynamics of bubble formation and growth in magmas: a review and analysis, J. Volcanol. and Geotherm. Res., 3, 1 (1978). 10. M. Mangan, L. Mastin, and T. Sisson, Gas evolution in eruptive conduits: combining insights from high temperature and pressure decompression experiments with steady-state flow modeling, J. Volcanol. and Geotherm. Res., 129, 23 (2004). 11. A. A. Samarskii and A. V. Gulin, Numerical Methods [in Russian], Nauka, Moscow (1989). 12. M. Rosi, P. Landi, M. Polacci, A. Di Muro, and D. Zandomeneghi, Role of conduit shear on ascent of the crystalrich magma feeding the 800-year-b.p. Plinian eruption of Quilotoa Volcano (Ecuador), Bull. Volcanology, 66, No. 4, 307 (2004).

11 EFFECT OF VISCOUS DISSIPATION M. Polacci, P. Papale, and M. Rosi, Textural heterogeneities in pumices from the climactic eruption of Mount Pinatubo, 15 June 1991, and implication for magma ascent dynamics, Bull. Volcanology, 63, No. 2 3, 83 (2001). 14. S. Carey and H. Sigurdsson, The intensity of Plinian eruptions, Bull. Volcanology, 51, No. 1, 28 (1989). 15. P. Papale, Strain-induced magma fragmentation in explosive eruptions, Nature, 397, No. 6718, 425 (1999). 16. O. Melnik, Dynamics of two-phase conduit flow of high-viscosity gas- saturated magma: large variations of sustained explosive eruption intensity, Bull. Volcanology, 62 No. 3, 153 (2000).