Journal of Volcanology and Geothermal Research 97 (2000) 215–231 www.elsevier.nl/locate/volgeores

Energetics of gas-driven limnic and volcanic eruptions

Y. Zhang*

Department of Geological Sciences, The University of Michigan, Ann Arbor, MI 48109-1063, USA

Abstract This paper lays the foundation for the rigorous treatment of the energetics of gas exsolution from a gas-containing liquid, which powers gas-driven volcanic and limnic eruptions. Various exsolution processes (reversible or irreversible, slow or rapid) are discussed, and the maximum amount of kinetic energy derivable from a reversible gas exsolution process is obtained. The concept of dynamic irreversibility is proposed for discussing the kinetic energy available from irreversible gas exsolution processes. The changes of thermodynamic properties during gas exsolution processes are derived. Density– relations for gas–liquid mixtures are presented, including empirical relations for irreversible gas exsolution. The energetics of gas-driven eruptions through both fluid and rigid media, including the role of and the role of chamber expansion work, are investigated. For reversible processes, the energetics can be used to discuss the dynamics of gas-driven eruptions, leading to maximum erupting velocities and maximum eruptible fractions. For irreversible processes, empirical relations and parameters must be employed. The exit velocities of the Nyos eruption and the 18 May 1980 eruption of Mount St. Helens are modeled by incorporating possible irreversibility. ᭧ 2000 Elsevier Science B.V. All rights reserved.

Keywords: energetics; gas-driven limnic eruptions; volcanic eruptions

1. Introduction . It is hence of great interest to investigate the general aspects of thermodynamics A gas-driven eruption is powered by the exsolution and energetics of gas exsolution from a liquid. Some of dissolved gas or gases in a liquid, such as volcanic volcanic eruptions are not gas-driven. For example, eruptions driven by H2O exsolution from magma (Self gas exsolution is often not critical in basaltic eruptions et al., 1979; Wilson et al., 1980) and limnic eruptions (MORB and Hawaiian type eruptions) because driven by CO2 exsolution from water (Kling et al., basaltic melts typically contain low H2O and CO2. 1987; Sigurdsson et al., 1987; Sigvaldason, 1989; These eruptions are not the subject of this paper. Sabroux et al., 1990; Zhang, 1996). The violence of Because gas-driven limnic and volcanic eruptions such an eruption is caused by the large amount of are rapid and large in scale compared to heat diffusion kinetic energy that is ultimately harvested from the distance, heat loss from the bubbly liquid during an thermal energy stored in the initially gas-saturated eruption is expected to be negligible and the process is liquid. As a gas-saturated liquid is decompressed, it roughly adiabatic. Therefore, this paper concentrates vesiculates and the bubbly liquid system expands. on the energetics of the adiabatic gas exsolution from Under the right conditions, the volume expansion a liquid, including: (1) the amount of kinetic energy causes rapid upward acceleration, leading to an derivable from the process; and (2) thermal effects, to lay the foundation for the rigorous treatment of the * Fax: ϩ1-313-763-4690. energetics of gas-driven eruptions. My treatment from E-mail address: [email protected] (Y. Zhang). the energetics point of view also leads to some simple

0377-0273/00/$ - see front matter ᭧ 2000 Elsevier Science B.V. All rights reserved. PII: S0377-0273(99)00164-X 216 Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231

Nomenclature a Activity of a component (subscript) in a phase (superscript) C Concentration in kg mϪ3. C ˆ r if there is only one component in the phase C Molar isobaric heat capacity D Diffusivity f The ratio of mass of the entrained liquid to that of the original erupting gas–liquid g Acceleration due to Earth’s gravity H & H Enthalpy and partial molar enthalpy Id Dynamic irreversibility defined in Eq. (2b) KE Kinetic energy KE Kinetic energy per unit mass k1 & k2 Constants defined in Eq. (21). k1 is dimensionless k3 & k4 Constants defined in Eq. (23). k3 is dimensionless k5 & k6 Dimensionless constants defined in Eq. (27) n Number of moles P Pressure PE Potential energy PE Potential energy per unit mass Ϫ Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 R Gas constant in J kg 1K . R is 461.5 J kg K for H2O, and 188.9 J kg K for CO2 Rd Dynamic reversibility defined in Eq. (2a) S & S Entropy and partial molar entropy T U Internal energy u Erupting velocity V Volume W Work done by the system X Mole fraction of a component a Isobaric thermal expansivity d The mass fraction of the gas phase e A coefficient defined in Eq. (16) k Heat (or thermal) diffusivity l Ostwald coefficient of a gas in a liquid r Density, or density of the gas–liquid mixture if without super or subscript Z ϵ PgV/(RT)); it characterizes the nonideality of the gas Z ˆ 1 for an ideal gas) Subscript A Component A (volatile). A1 and A2 are used to refer to two volatile components B Component B (nonvolatile) max Maximum 0 Initial state s At saturation t Total amount in the liquid and gas phases Superscript g Gas phase l Liquid phase 0 Pure standard state Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 217 equations describing the dynamics of eruptions. These point of view, slowness or rapidness of a process dynamic equations are similar to the equations in and is irrelevant, only the reversibility or irreversi- are alternatives to earlier work in modeling eruption bility is important. However, from a dynamic point dynamics (e.g. Sparks, 1978; Wilson et al., 1980; of view, whether the process is slow or rapid is very Jaupart and Tait, 1990). important because the amount of kinetic energy In this paper, the kinds of gas exsolution processes obtained from a uniform gas exsolution process is will be discussed first and the concept of dynamic crucial to understanding explosive eruptions. Slow reversibility is introduced. Then the changes of processes, whether reversible or not, do not generate thermodynamic properties during an eruption are kinetic energy that is important to gas-driven derived. Then the equations of state for gas–liquid eruptions. mixtures are obtained for several cases, including In an explosive eruption due to gas exsolution from the case for two volatile components. Using these a liquid, the pressure increases with depth as dP ˆ results, the energetics of both types of gas-driven r g ϩ a† dh where P is the pressure, r the density of eruptions (CO2-driven limnic eruptions and H2O- the gas–liquid mixture, g the acceleration due to driven volcanic eruptions) are investigated. Finally, Earth’s gravity, a the upward acceleration of the disequilibrium between the gas and liquid phases are erupting mass, and h the depth. That is, the total modeled and applied to eruption and the system pressure consists of two parts: (1) the static 1980 Mount St. Helens eruption. pressure owing to the outside pressure (such as atmos- pheric) and the overlying weight of the column; and (2) the acceleration pressure (similar to the dynamic 2. Thermodynamics of gas exsolution from a liquid pressure, e.g. White, 1991) owing to the acceleration of the gas–liquid mixture. The acceleration pressure 2.1. Kinds of gas exsolution processes increases the kinetic energy of the system. In a slow process, either the acceleration pressure is negligible Gas exsolution from a liquid can be accomplished or the viscosity is very high. In a rapid process, the in many ways. One way is through formation of acceleration pressure is significant. That is, only when bubbles that ascend relatively slowly to the surface the total system pressure is greater than the static and escape from the liquid without carrying liquid surrounding pressure, is kinetic energy produced. droplets. This will be referred to as degassing, as in The greater the difference, the greater the kinetic fumaroles and in beer and soft drinks. This slow energy. The following examples illustrate several degassing process is not considered here. Following cases of rapid or slow, and reversible or irreversible Zhang et al. (1997), an eruption is defined as the ejec- processes: tion of a significant amount of liquid from the (1) A classical reversible process is a slow system. Even though there is always some relative exsolution process in which the static surrounding motion between the liquid and gas phases, the pressure is always roughly the same as the system focus in this paper is on the uniform exsolution of pressure and the system itself is always at equilibrium. a gas from a liquid without significant flow of For gas exsolution from a liquid, if the static surround- one phase relative to the other. That is, on a scale ing pressure decreases very slowly due to slow ascent larger than individual gas bubbles or liquid drops of the bubbly magma, gas bubbles would grow slowly but smaller than the diameter of the erupting con- and the gas and liquid phases in the system would be duit, the gas–liquid mixture is regarded as uniform, in equilibrium (no concentration gradient of the vola- either a bubbly liquid (in which the liquid phase is tile component in the liquid away from bubble walls). continuous and bubbles are not), or a drop-laden or The result is a classical reversible process, producing dusty gas (in which the gas phase is continuous). This no kinetic energy. A reversible adiabatic process is characteristic will be referred to as uniform gas isentropic. exsolution. (2) The general conditions for a reversible adiabatic A uniform gas exsolution process can be rapid or gas exsolution process dS ˆ 0† are: (a) the system slow, reversible or irreversible. From a thermodynamic pressure equals the static surrounding pressure plus 218 Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 the acceleration pressure; (b) the gas and liquid phases H is the enthalpy. Step 2 is for the system to adiabatically are in bulk equilibrium (not just surface equilibrium); and slowly (hence complete dynamic irreversibility) and (c) viscous dissipation is negligible. Hence, one expand through slow bubble formation and growth may conceive a rapid gas exsolution process that is to its final state (P, T, V, bubbly liquid). For step 2, dH roughly reversible. For a rapid decrease of the static ˆ d U ϩ PV†ˆdQ ϩ dW ϩ P dV ϩ V dP ˆ P Ϫ † ϩ ˆ d ˆ d ˆ surrounding pressure in a gas–liquid system, if the Psur dV V dP 0 because Q 0, W Ϫ ; ˆ ˆ : number density of bubbles is high or the diffusivity Psur dV P Psur and dP 0 Therefore, the of the gas component in the liquid is large (such as second step of this adiabatic process is enthalpic, CO2 in water), diffusion may be able to maintain bulk which can be used to determine the final tempera- equilibrium between the gas and liquid phases (i.e. no ture of the system. The total enthalpy change in concentration gradients from bubble wall to the inter- the process is the same as the enthalpy change in ior of the liquid). If furthermore the liquid viscosity the first step: does not impede bubble growth and flow ascent, the D ˆ l Ϫ †; † gas exsolution process would be reversible. Such a H V0 P P0 1 l reversible rapid exsolution process due to rapid where V0 is the initial liquid volume (see Nomen- decrease in the surrounding pressure will be referred clature for notations). to as a dynamic reversible process, to distinguish it (4) All gas-driven eruptions are expected to be irre- from the slow reversible process above. A dynamic versible to some degree owing to the presence of reversible process generates a significant amount of concentration gradients and viscous dissipation but kinetic energy. not dynamically completely irreversible. As the static (3) A very slow exsolution process is not neces- surrounding pressure decreases, bubbles grow rapidly sarily reversible. For example, exsolution of H2O but typically not rapid enough to maintain bulk equi- from a hydrous rhyolitic glass at relatively low librium between the liquid and gas phases. The rapid (such as 550ЊC) is slow (that is, bubble growth of bubbles leads to volume expansion and growth is slow) and irreversible. If a piece of hydrous density decrease, resulting in rapid upward motion rhyolitic glass containing 2.0% H2O is heated to (an eruption) of the gas–liquid mixture and an 550ЊC at 1 bar (equivalent to decompressed to 1 bar increase in both kinetic and potential energies. at 550ЊC), bubbles nucleate and grow slowly and the The kinetic energy obtained from such a process glassy liquid deforms slowly to accommodate bubble is between that of cases (2) and (3) above. That growth. The slowness is due to the high viscosity of is, other conditions being equal, the maximum the liquid (ϳ1010 Pa s) and small diffusivity of the gas amount of kinetic energy is derived from a dyna- component in the liquid. This process is irreversible mically reversible process. because bubble growth occurs in a strong concentra- From the above discussion of different types of tion gradient and because there is viscous flow. This processes, I propose to define dynamic reversibility process will be referred to as “dynamically comple- (Rd) and irreversibility (Id) as follows: tely irreversible” since it is irreversible and since no R ˆ KE=KE ; 2a† kinetic energy is generated. If the process is adiabatic, d max entropy of the system increases. Even though the I ϵ 1 Ϫ R ˆ 1 Ϫ KE=KE ; 2b† determination of entropy increase is not straightfor- d d max Յ Յ ; ward for this process, the enthalpy change for an adia- where 0 Rd 1 KE is the kinetic energy derived batic and dynamically completely irreversible process from a given process, and KEmax is the maximum can be determined as follows. Starting from the initial kinetic energy derivable from the reversible process equilibrium state (P0, T0, V0, liquid, no gas phase), from the same initial state to the same final pressure. accomplish the process in two steps: step 1 is For case (1) above (a slow reversible process), the to release the system pressure adiabatically from P0 dynamic reversibility is undefined because both KE to P without any exsolution (due to slow diffusion) so and KEmax are zero. For case (2) above (a dynamic ˆ ˆ : that the system changes to (P, T1, V1, liquid, no gas reversible process), Rd 1 and Id 0 For case (3) ˆ 2 =2 † ˆ ; phase). For step 1, dH H P S dP V dP where above (a dynamically completely irreversible process), Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 219

ˆ ˆ : g g l l g g Rd 0 and Id 1 For case (4) above (a general ˆ ‰  ; † Ϫ  ; ; †Š Ϫ ‰  ; † nA SA T P SA T P XA nA;0 SA T0 P0 dynamic irreversible process), Id is between 0 Ϫ l ; ; l †Š ϩ ‰ l ; ; l † and 1. The above definition of dynamic irreversi- SA T0 P0 XA;0 nA;t SA T P XA bility quantifies the relation between dynamic Ϫ l ; ; l †Š ϩ l ‰ l ; ; l † irreversibility and kinetic energy generated in a SA T0 P0 XA;0 nB;0 SB T P XB gas-driven eruption. Ϫ l ; ; l †Š: SB T0 P0 XB;0 (3)

2.2. General thermodynamic considerations g ‰ g ; † Ϫ l ; ; l †Š The terms nA SA T P SA T P XA and g g l l n ‰S T ; P † Ϫ S T ; P ; X ; †Š represent the Consider a binary system in which A is the volatile A;0 A 0 0 A 0 0 A 0 entropy change owing to the exsolution of component component and B is assumed to be perfectly non- A from the liquid under isothermal, isobaric, and equi- volatile. That is, the gas phase is purely A. There librium conditions. Therefore, they can be written in are two phases in the system: gas and liquid. The terms of exsolution (vaporization) enthalpy. For temperature is assumed to be uniform in the system example, but it may vary as a function of time. That is, the gas and liquid phases are assumed to be in thermal equi- H g T; P† Ϫ H l T; P; Xl † Sg T; P† Ϫ Sl T; P; Xl †ˆ A A A : librium since bubbles in liquid (or drops in gas) are A A A T small and since the Lewis number Le ˆ D=k p 1 4† where D is the chemical diffusivity of the gas com- k Since the gas phase is assumed to be pure A, P is the ponent in the liquid and is the diffusivity of l heat. The variation of the liquid density with vola- same as the partial pressure of A and XA is a function tile concentration, temperature and pressure is of T and P. Hence the above difference in the partial ignored when considering the density of the molar entropy is a function of T and P only. If the gas bubbly liquid. is ideal (hence enthalpy of the gas is independent of Consider a gas-saturated liquid plus pressure) and the liquid phase is an ideal mixture (hence no mixing enthalpy), Eq. (4) can be simplified an equilibrium gas phase at an initial state (T0, l ; l ; g † l as: P0, nA;0 nB;0 nA;0 where nA;0 is the initial number of moles of A in the liquid (subscript 0  g;0 † Ϫ  l;0 ; † g ; † Ϫ l ; ; l † Ϸ HA T HA T P means initial; subscript A or B means component S T P SA T P XA A T A or B; superscript l or g means the liquid or gas phase; and superscript 0 will be used to DH 0 T; P† ˆ ex ; 5† indicate the pure standard state). The initial con- T l centration of A in the liquid is expressed as CA;0 0 Ϫ where DH is the standard-state molar enthalpy of in kg m 3. The concentration can also be ex l : exsolution (or evaporation) of component A. Note expressed as molar fraction XA;0 The final state l l g l g l g that mixing entropy does not enter into Eq. (5), is (T, P, n ; n ; ; n † with n ; ϩ n ˆ n ϩ n ˆ A B 0 A A 0 A;0 A A nor does the pressure dependence of gas entropy. n ; : Using entropy as an example, the change A t Hence, the dependence of the exsolution entropy in a thermodynamic property can be found as and enthalpy on pressure is due to: (1) the pres- follows: sure dependence of the molar enthalpy of the gas phase, which is weak (e.g. at 800ЊC, enthalpy of D ˆ ; ; g ; l ; l † Ϫ ; ; g ; l ; l † Ϫ1 Ϫ1 S S T P nA nA nB;0 S T0 P0 nA;0 nA;0 nB;0 H2O fluid decreases by 8.4 J mol bar at P Ͻ 500 bar; Burnham et al., 1969); and (2) the pres- ˆ g g ; † ϩ l l ; ; l † ϩ l l ; ; l † nASA T P nASA T P XA nB;0SB T P XB sure and compositional dependence of the partial molar enthalpy of the dissolved gas component in Ϫ g g ; † Ϫ l l ; ; l † nA;0SA T0 P0 nA;0SA T0 P0 XA;0 the liquid, which is also expected to be weak. l l l Therefore, the exsolution enthalpy is not expected Ϫ n ; S T ; P ; X ; † B 0 B 0 0 B 0 to depend strongly on pressure, especially at low 220 Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 . The specific case for the exsolution equation is:  enthalpy of water (a multi-species component) from g   l l  l n DH n ; C ϩ n ; C silicate melts, and its pressure dependence, will be dS ˆ d A ex ϩ A t A B 0 B dT discussed elsewhere. T T l ; ; l † Ϫ l ; ; l † "# ! ! The terms SA T P XA SA T0 P0 XA;0 and 2 l 2 l l l l l SA l SB l S T; P; X † Ϫ S T ; P ; X ; † in Eq. (3) represent ϩ n ; Ϫn ; dX B B B 0 0 B 0 A t 2 l B 0 2 l A the entropy variation in the liquid phase as a function XA P;T XB P;T of temperature, pressure and composition. They can Ϫ a l  l ϩ l a l  l † : † be simplified as follows: nA;t AVA nB;0 BVB dP 8 For a reversible adiabatic process, dS ˆ 0; i.e. the l ; ; l † Ϫ l ; ; l † SB T P XB SB T0 P0 XB;0 right-hand side of Eqs. (7) and (8) is zero, which can be used to calculate the final T from T0, P0, ˆ‰l ; ; l † Ϫ l ; ; l †Š ϩ ‰ l ; ; l † SB T P XB SB T P0 XB SB T P0 XB and P. Following the same procedure, DH can be found: Ϫ l ; ; l †Š ϩ ‰ l ; ; l † SB T P0 XB;0 SB T P0 XB;0 D Ϸ g D  ; † Ϫ g D  ; † ϩ l  l H n Hex T P n ; Hex T0 P0 nA;tCA Ϫ l ; ; l †Š A A 0 SB T0 P0 XB;0 ϩ g  l † Ϫ † ϩ l Ϫ al † Ϫ †: n ; CB T T0 V0 1 0T0 P P0 l B 0 ˆ Ϫ †al  l ϩ  l T ϩ aB;0 ; 9† P0 P BVB CB;0 ln Rln l (6) T0 aB The differential form is: where a is the isobaric thermal expansivity, C is the Ϸ ‰ g D  ; †Š ϩ ‰ l  l ϩ g  l † dH d nA Hex T P nA;tCA nB;0CB dT molar isobaric heat capacity, a the activity (the depen- ϩ l Ϫ al † : † dence of the activity on temperature has been V0 1 0T0 dP 10 ignored), and R the gas constant in J molϪ1 KϪ1. Replacing Eqs. (5) and (6) into Eq. (3), the expression The change in the internal energy of the system can for S is: be found to be: "#DU ˆ DH Ϫ PV Ϫ P V †: 11† D  0 ; † D  0 ; † 0 0 D Ϸ g Hex T P Ϫ g Hex T0 P0 S nA nA;0 T T0 The work done by the system is: Z T ˆ ; † ϩ  l ϩ g  l † W Psur dV 12 nA;tCA nB;0CB ln T0 ! l l where Psur is the surrounding pressure (not necessarily aA;t l aB;0 ϩ n ; Rln ϩ n ; Rln ϩ P Ϫ P† the same as the system pressure). For an adiabatic A t al B 0 al 0 A B process, W ˆ ϪDU: The work done by the system l l  l l l l  l is related to the kinetic energy.  n ; a V ϩ n ; a ; a V †: (7) A t A A B 0 B 0 B B Expressions (7), (9), (11) and (12) for DS, DH, DU, and W are general expressions and independent of the The last term in Eq. (7) is often negligible (i.e. the type of processes (whether the processes are reversi- effect of thermal expansion and compression of the ble or not). They depend on both T and P. For a liquid is often negligible). The mixing terms involving specific process, there is a relation between T and P. the logarithm of activity ratios can be rewritten as Hence given the final P, the values of T, DS, DH, DU, DGmix/T (that is nonzero even for ideal mixtures) and W can be calculated if the molar exsolution where G is the Gibbs free energy, not DHmix/T (that enthalpy, heat capacity, mixing entropy and the solu- is zero for ideal mixtures) as in Sahagian and Prous- bility relation are known. sevitch (1996). The differential form of the above Table 1 summarizes equations for calculating Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 221

Table 1 a final temperature. Then calculate solubility coeffi- Thermodynamics of adiabatic gas exsolution from a solution. Initial cient, and the molar exsolution enthalpy of A. The saturation state: (T0, P0, C0); final equilibrium state: (T, P, C); no ˆ Ϫ concentration of A in the liquid is then calculated initial gas phase. Given P, find V (or V V V0) and T (or ˆ Ϫ from the solubility coefficient. Next the total T T T0), S, H, U, and W amount of A in the gas phase can be calculated 1. Regardless of the process (reversible or irreversible): from a mass balance. Then DS and DH can be calcu- lated. For a reversible adiabatic process, vary T so that Ml ng ϩ ng †RT V ˆ ϩ A B DS ˆ 0 and the resultant state is the final state. For a rl P Á completely dynamically irreversible process, vary T D ˆ l P0 T Ϫ l l V V0 0 by ignoring component B in the vapor so that DH ˆ V P Ϫ P † (Eq. (1)), which is equiva- P T0 0 0 P Á lent to: DV Ϸ lV 0 Ϫ 1 if l is constant and T/T Ӎ 1 0 P 0 P Á P D 0 ; † P l g  g  l  l g  l l l T g Hexi T P l ai;0 D ; † Ϫ D ; † ϩ ; ϩ † DS ˆ n ; C ; ln ϩ n ϩ R n ; ln nA Hex T P nA;0 Hex T0 P0 nA tCA nB;0CB i i 0 P i T i i T i i 0 al P 0 i ϩ Ϫ † l al  l;  Ϫ † Ϸ ; P0 P i ni;0 P;iVi T T0 0 (13) P D ˆ l Ϫ † ϩ gD 0 ; † ϩ l Ϫ al † Ϫ †: H CP0 T T0 i ni Hexi T P V0 1 0T0 P P0 according to Eq. (9). Knowing the final T, all other D ˆ D Ϫ Ϫ †ˆϪ ˆ Ϫ : U H PV P0V0 W PsurdV thermodynamic properties of the final state can be calculated. Some worked numerical examples for 2. Reversible process (either slow or rapid): calculation of temperature and thermodynamic func- T is solved from: DS ˆ 0: tions are given below. Worked example 1, limnic eruption of Lake Nyos. P W Ϸ lP V ln 0 assuming l is constant in the temperature and Assume that the bottom water (208 m depth) was rev 0 0 P pressure range ˆ initially saturated with CO2. Therefore P0 : ; ˆ : ; ˆ : P0 1=2 1=2 2 13 MPa Pout 0 09 MPa T0 296 K Assume W Ϸ l P V ln Ϫ k P † V ‰1 Ϫ P=P † Š Ϫ Ϫ rev 0 0 0 P 2 0 0 0 D ˆ : 1; l ˆ : 1 HexCO2 19 3 kJ mol CP0 4 184 kJ kg = Ϫ Ϫ for l ˆ k ϩ k =P†1 2: 1; al ˆ : 1; g ˆ : 1 2 K P0 0 00023 K and nA;0 0 Evaporation of water is ignored. If the process is reversible, the ˆ † : 3. Dynamically completely irreversible process Rd 0 final T is 291.7 K with DT ˆ Ϫ4:3K: The total work Ϫ1 D ˆ Ϫ g D ; †= l;0: done by the system (ϪDU ) is 5.7 kJ kg . This work T is solved from: T nA Hex T P CP includes the expansion work and the work done to ˆ D Ӎ l = Ϫ l †: W P V V0 0P0T T0 P increase the kinetic energy of the system (which could reach 4.1 kJ kgϪ1; Zhang, 1996). To examine D Ͼ S 0. the relative importance of each term in Eq. (7), the Subscripts indicate component A, B, or i; superscripts indicate first term (brackets) on the right-hand side of Eq. (7) is the phase (l, liquid; g, gas). Work done by the system is W. Subscript Ϫ61.1, the second term 47.4, the third mixing term 0 indicates the initial state. Superscript 0 indicates the standard 13.2, and the last term 0.47 J kgϪ1 KϪ1. Hence the last l : D state. The heat capacity of the whole liquid solution is CP Hex Ϫ1 term is not important but the mixing terms are signi- is the molar exsolution enthalpy (19.3 kJ mol for CO2 exsolution from water). Activity of a component is a. Thermal expansion coef- ficant. If the process is completely dynamically irreversible, using Eq. (13), the final T would be 292.7 K with DT ˆ Ϫ3:3K: The total work done by the thermodynamic properties of the final state and the the system would be 1.6 kJ kgϪ1 and there would be total work done by the system. Given initial condi- no kinetic energy. The entropy of the system increases tions, a process (providing a relation between final T by 14.0 J kgϪ1 KϪ1. and P), and final P, the general procedure Worked example 2, CO2 exsolution from water in ˆ ; for calculating the final T and changes in thermo- Experiment #89 of Zhang et al. (1997). P0 496 kPa ˆ ; ˆ : : dynamic properties is as follows. Give a unit mass Pout 11 kPa and T0 295 35 K If the process is of a gas–liquid mixture (such as one kg). Guess reversible, the final T is found to be 294.24 using 222 Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231

Ϫ1 Ϫ1 Ϫ1 Ϫ1 Eq. (7) with DT ˆ Ϫ1:1K: The work done by the (188.9 J kg K for CO2 and 461.5 J kg K for Ϫ1 g system is 1.7 kJ kg and kinetic energy is generated. H2O), T is in K, and Z ( ϵ P V/(RT )) characterizes the If the process is completely dynamically irreversible, nonideality of the gas Z ˆ 1 for an ideal gas). For the final T is 294.54 K using Eq. (13). The work H2O vapor at Ն 800ЊC and Յ 3 kbar, ͉Z Ϫ 1͉ Ͻ 0:15 Ϫ1 done by the system is 0.4 kJ kg and no kinetic (Burnham et al., 1969). For CO2 gas at 20ЊC and energy is generated. The entropy increases by Յ 25 bar, ͉Z Ϫ 1͉ Յ 0:15 (Angus et al., 1976). 4.3 J kgϪ1 KϪ1. Hence the ideal gas assumption will be used in these cases. For CO2 at 800–1500ЊC and Յ3 kbar, Z can be 2.3. Equation of state for a gas–liquid mixture during significantly greater than 1 and, to a first-order reversible exsolution approximation, can be expressed as: Ϸ ϩ e ; † The equation of state of a gas–liquid mixture, i.e. Z 1 P 16 r Ϫ Ϫ the dependence of (the density of a gas–liquid where e Ϸ 0.3 kbar 1 at 1100 K and 0.25 kbar 1 at mixture) on P (the system pressure) and T (that 1600 K (Sterner and Pitzer, 1994). depend on P), is essential for the calculation of The general expression for d is: volume work done by a gas–liquid system through l l r C ; C gas exsolution. The temperature effect on is d ˆ d ϩ A 0 Ϫ 1 Ϫ d ϩ d † A ; 17† Յ 0 rl 0 rl small (the effect is 20% on the gas density and 0 r smaller on ) and complicated. Hence the effect l where CA is the concentration of the gas component A is ignored. The general relation is derived below Ϫ in the liquid (in kg m 3). Combining Eqs. (14) and for a gas–liquid mixture for both reversible and (17) to eliminate d leads to: irreversible exsolution. Assume that initially, the ! mass fraction of the gas phase is d and the rl rl Ϫ l =rl 0 0 ˆ 0 1 CA;0 0 Ϫ d 0 system pressure is P0 at which the gas in the liquid is at r rl Ϫ l =rl 1 CA saturation. A gas phase exsolves as P is reduced. At P, ! d l l l l l the mass fraction of the gas phase is and that of the r C ; =r Ϫ C =r Ϫ d: ϩ 0 d ϩ A 0 0 A : 18† liquid is 1 From Wilson et al. (1980): rg 0 Ϫ l =rl 1 CA 1 d 1 Ϫ d ˆ ϩ ; 14† Using the approximations that the liquid density is r r g rl Ϫ l =rl Ϸ ; constant, and 1 CA;0 0 1 then: g l where r, r and r are the densities of the gas– l l l l r d r ϩ C ; C liquid mixture, the gas and the liquid. The density ˆ 1 Ϫ d ϩ 0 A 0 Ϫ A r 0 r g r g of the liquid is assumed to be constant (ignoring ! T or P effect). Wilson et al. (1980) used constant rl ZP Cl mass fraction of the gas d (d at fragmentation) ˆ 1 ϩ d Ϫ 1 ϩ l 0 Ϫ A : 19† 0 r g 0 Z Pg r g to model the dynamics of explosive eruptions. In 0 the following discussion, d is allowed to increase The above equation is the most general equation of from the continuous exsolution of the gas from the state for a gas–liquid mixture and applies whether or liquid. not equilibrium between the gas and liquid phases is reached. Therefore, the above equation also applies to 2.3.1. Equation of state for a gas–liquid mixture when irreversible processes. Because Eq. (19) does not the gas phase contains only one component, whether explicitly relate r to P, other constraints (such as or not equilibrium is reached between the gas and equilibrium between the gas and liquid, see below; liquid phases the case for irreversible processes will be discussed The density of the one-component gas is: in a later section) must be used to relate r to P. rg ˆ P g= ZRT†; 15† 2.3.2. Equilibrium between the gas and liquid phases where R is the gas constant in J kgϪ1 KϪ1 If chemical equilibrium between the liquid and gas Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 223 phases is always maintained, then: process if equilibrium is reached and if there is only one component in the gas phase. For generality, the l = g ˆ l =r g ˆ l; † CA CA CA 20 above and below expressions are given in terms of gas-phase pressure. The gas-phase pressure is the l where is the Ostwald solubility coefficient that may system pressure for dusty or drop-laden gas flows depend on both temperature and pressure. (after fragmentation). For most bubbly flows (before For H2O gas (assumed to be ideal) in silicate liquid, fragmentation), the gas-phase pressure can also be l depends on pressure as: taken as the liquid and system pressure (but see Prous- = sevitch et al., 1993, for possible differences between l ˆ k ϩ k =Pg†1 2; 21† 1 2 gas phase pressure and liquid pressure owing to dynamic pressure, surface tension pressure, etc.). where k1 and k2 are the two constants. For H2O gas in rhyolitic melt at 850ЊC, using data of Blank et al. Section 4 discusses cases in which the gas pressure (1993), Silver et al. (1990), Kadik et al. (1972) and may differ from the liquid pressure. ˆ : ˆ Khitarov and Kadik (1973), k1 0 085 and k2 : 107 2 Pa: 2.3.4. Three special cases for the equation of state for For CO2 gas in liquid water at Յ 25 bar, using data a gas–liquid mixture when the gas phase contains of Weiss (1974), Wiebe and Gaddy (1940), Weast only one component and when equilibrium is reached (1983) and Dean (1985), l ˆ 1at18ЊC, and the between the gas and liquid phases temperature dependence is roughly: One special case of Eq. (24) is for ideal gas and for d ˆ : 0 0 3:12Ϫ4042=Tϩ911269=T2 l ˆ e 22† rl 0 Ϸ Ϫ l ϩ l P0 : † 1 0 g 25 at 0–50ЊC and Յ 20 bar. r P For the nonideal CO in silicate melt, l depends on 2 The above equation is applicable to, for example, CO2 CO gas pressure as: 2 or CH4 in water at relatively low pressures. A second special case is for H O in silicate melts with Z ˆ 1 l ˆ k ϩ k Pg; 23† 2 3 4 and l given by Eq. (21):  where k3 and k4 are the two constants. For CO2 in rl rlRT 1 Pg k 1=2 ϳ Њ ˆ : ˆ ˆ Ϫ d Ϫ ϩ d ϩ l 0 Ϫ 2 : basaltic melt at 1300 C, k3 0 00244 and k4 1 0 k1 0 0 Ϫ Ϫ r Pg Pg Z Pg 1:933 × 10 11 Pa 1 from experimental data of Pan 0 † et al. (1991), Pawley et al. (1992) and Dixon et al. 26

(1995). A third special case is for CO2 in silicate melts at high pressures with Z given by Eq. (16) and l given by Eq. 2.3.3. Equation of state for a gas–liquid mixture when (23): the gas phase contains only one component and when rl Pg equilibrium is reached between the gas and liquid ˆ k ϩ k 0 Ϫ k Pg; 27† r 5 6 g 4 phases P If chemical equilibrium between the liquid and gas ˆ Ϫ d Ϫ ϩ e g; ˆ l = ϩ where k5 1 0 k3 k6 P0 and k6 0 Z0 phases is always maintained, the equation of state for d rl = g 0 RT P0 are two dimensionless constants. the gas–liquid mixture can be found by combining Eqs. (19) and (20): 2.3.5. Equation of state for a gas–liquid mixture when rl rl g the gas phase contains two components and when ˆ Ϫ d ϩ d ZRT ϩ l Z P0 Ϫ l; † 1 0 0 0 24 equilibrium is reached between the gas and liquid r Pg Pg Z 0 phases where the effect of T on the gas-phase density is Natural silicate melts often contain two major ignored (i.e. T/T0 Ϸ 1). Eq. (24) is a general equation volatiles, CO2 and H2O. Following the procedures of state for a gas–liquid mixture for a reversible above, the EOS for the reversible exsolution of two 224 Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 gases from a liquid is: and (4) the role of buoyancy for eruption through a fluid medium (positive term). The acceleration pres- rl 0 ˆ Ϫ ˆ ˆ 1 sure can be expressed as: Pacc P Pstatic r Ϫ ϩ †; P Pout Pwt where P, Pstatic, Pout, and Pwt are, l l l l respectively, the system pressure, the static pressure, A1;0PA1;0 A2;0PA2;0 A1PA1 A2PA2 ϩ Ϫ Ϫ the outside pressure (such as the atmospheric Z ; R Z ; R Z R Z R ϩ A1 0 A1 A2 0 A2 A1 A1 A2 A2 ; pressure), and the pressure due to the overlying weight = † ϩ = † R PA1 ZA1RA1 PA2 ZA2RA2 of the erupting column. Because Pd(1/r) is the total 28†  expansionR work done by the unit mass W†; and r where the first subscript of l, Z and P indicates the gas Pwtd(1/ ) is work done per unit mass to increase component A1 or A2, and the optional second its own potential energy (PE ˆ gh where h is the height from depth 1 to depth 2), Eq. (30a) can also subscript 0 indicates the initial state. PA1 and PA2 in Eq. (28) are related by: be written as: Z =r ˆ l l 1 at P P2 A1;0ZA1PA1;0 A2;0ZA2PA2;0 ˆ Ϫ Ϫ † =r† ϩ  ; Ϫ l ˆ Ϫ l : 29† KE P Pout Pwt d 1 Wextra A1 A2 =r ˆ ZA1;0PA1 ZA2;0PA2 1 at P P1 30b† If initially there are bubbles in the magma, the satur- ation pressure (Ps) at which bubbles would completely or dissolve into the magma can be found from the initial Z 1=r 2 ˆ  Ϫ =r† Ϫ ϩ  ; † volume or mass fraction of the gas phase. The KE W Pout d 1 PE Wextra 30c =r equation of state are then similar to Eq. (28) but the 1 1 l initial properties P0, 0 and Z0 must be replaced by the or l saturation properties Ps, s and Zs. Z =r 1 2 These equations of states will be used in the next ˆ Ϫ † =r† Ϫ ϩ  : † KE P Pout d 1 PE Wextra 30d =r section to calculate the energetics of gas-driven 1 1 eruptions. Because the maximum amount of total expansion work is done in a reversible process, the maximum 3. Energetics of gas-driven eruptions amount of kinetic energy is obtained when: (1) the process is reversible; and (2) the system pressure is 3.1. General considerations much greater thanR the static pressure. Ϫ † =r† Integration of P Pout d 1 in Eq. (30d) by The expansion work against the acceleration parts leads to: Z =r pressure increases the kinetic energy of the erupting 1 2 Ϫ Ϫ Ϫ † =r†ˆ P2 Pout Ϫ P1 Pout mass. The increase of kinetic energy per unit erupted P Pout d 1 =r r r mass KE ˆ u2=2† due to the acceleration pressure as 1 1 2 1 Z the pressure decreases from P to P is: P2 1 2 Ϫ dP : Z =r ˆ (31) 1 at P P2 r ˆ =r† ϩ  ; † P1 KE Pacc d 1 Wextra 30a 1=r at PˆP 1 Replacing Eq. (31) into Eq. (30d) leads to: r where Pacc is the acceleration pressure, 1/ is the Z P2 Ϫ volume per unit mass, and W is the work done ϩ ϩ dP Ϫ  Ϫ P2 Pout extra KE PE r Wextra r to the unit mass by the rest of the system or by the P1 2  32† surroundings. Wextra may include: (1) the effect of Ϫ ϩ P1 Pout ˆ : friction (negative term); (2) work done by magma r 0 expansion in the magma chamber to the erupting 1 mass (positive term); (3) the effect of entrainment of The velocity (or kinetic energy) of a gas-driven ˆ shallow water, magma, or wall rock (negative term); eruptions at the exit (i.e. P2 Pout) can be obtained Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 225 from: Z Pout Ϫ ϩ ϩ dP Ϫ  ϩ P1 Pout ˆ : KEexit PEexit r Wextra r 0 P1 1 33†

3.2. Differences between eruptions through a fluid medium and those through a rigid medium

A gas-driven eruption may be discharged either through a fluid medium, such as a limnic eruption through water and an eruption through a lava lake, Fig. 1. Magma chamber shape and possible buoyancy effect: (a) no or through a rigid medium, such as most volcanic buoyancy effect; and (b) there is a buoyancy effect. The melt in the chamber above the entrance level of the eruption conduit pushes the eruptions through rocks. There are several differences melt up into the conduit to the height z1. for eruptions through a fluid medium and those through a rigid medium: (1) The relation between the local system pressure to the energy of the erupting column. However, buoy- and the local hydrostatic or lithostatic pressure. The ancy does not play a direct role unless the magma local system pressure in the “conduit” for an eruption chamber is shaped in such a way for buoyancy to through a fluid medium is expected to be roughly the play a role (Fig. 1) because rocks may withstand same as the hydrostatic pressure (Zhang, 1996). significant stress and do not deform continuously at Therefore, for an eruption through a fluid conduit, the time scale of a violent eruption. The elastic stress although there is no definite chamber, the local system distribution in the rocks may conspire to produce pressure is known a priori to be roughly equal to the either positive or negative buoyancy, the sign of hydrostatic pressure and the source pressure stays the which depends on the detailed geology of the system same (P0). However, for an eruption through a rigid (L. Wilson, personal communication). medium, the local system pressure in the magma (3) The role of entrainment. For eruptions through a chamber and in the erupting column does not neces- fluid medium, the effect of entraining shallow water or sarily equal the lithostatic pressure (Jaupart and Tait, shallow magma may reduce the erupting velocity 1990) because rocks may withstand significant stress significantly. For eruptions through a rigid medium, and do not deform instantaneously at the time scale of there may be entrainment of wall rocks but the effect a violent eruption. To a first-order approximation, the may be small. To rigorously treat shallow water chamber volume does not vary with time. As eruption entrainment in a limnic eruption, the concentration goes on, the mass, density and pressure in the magma of CO2 in the entrained water, and much other detailed chamber decreases. The pressure decrease in the information, must be known. Without considering magma chamber affects the eruption dynamics and such details, a simple treatment of the effect of shal- may eventually lead to the collapse of the overlying low water entrainment is to assume that the entrained rocks into the chamber and the formation of a caldera. shallow water only adds mass to the erupting column (2) The role of buoyancy and chamber processes. but does not affect the equilibrium between the gas For eruptions through a fluid medium, buoyancy plays and liquid phases due to the imperfect mixing between a role so that lifting a parcel of liquid (mass m) from a entrained water and original high-CO2 water. If so, the lake bottom to the lake surface (without exsolution) kinetic energy per unit original mass is the same but requires little work, roughly mgh(Dr)/r where h is the the velocity decreases as: height increase, r the average density of water, r the = u ˆ‰2KE= 1 ϩ f †Š1 2; 34† density difference between the bottom water and the surface water (Dr/r is typically of order 0.001, a where f is the ratio of the entrained mass to the origi- small number). For eruptions through a rigid medium, nal mass. the gas exsolution in the magma chamber contributes (4) Material properties. The diffusivity of CO2 in 226 Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 water is Ն 200 times greater than that of H2Oin example, the eruption of CO2-saturated water may rhyolitic magma (Zhang et al., 1991). Therefore, reach an exit velocity of 52 m sϪ1 if starting from limnic eruptions are expected to be more dynamically 100 m saturation depth, 86 m sϪ1 if starting from reversible. The viscosity of hydrous rhyolitic magma 200 m depth. If the mass of the entrained shallow (Hess and Dingwell, 1996) is ϳ8 orders of magnitude water is about the same as the mass of the erupting greater than that of water. Therefore, buoyant rise of water from bottom, and if shallow water entrainment bubbles in liquid, which tends to decrease the erupt- is treated in a simple way as in Eq. (34), the exit ibility, plays a larger role in limnic eruptions. velocity would be 37 m sϪ1 if the eruption starts from 100 m saturation depth, 61 m sϪ1 if it starts 3.3. Dynamics of reversible gas-driven eruptions from 200 m depth. through a fluid medium A limnic eruption can be terminated when insuf- ficiently saturated water is drawn into the eruption Because buoyancy plays an important role in gas- conduit. Once such conditions arise, the initial driven eruptions through a fluid medium, the momentum of the batch of water is not enough for it surrounding does work to raise the liquid. When the to rise to a shallow enough depth for bubbles to grow initial liquid is bubble-free, the initial liquid in (equivalent to too small a trigger). Without buoyancy neutrally buoyant and buoyancy work matches the induced by bubble growth, the batch of water, being  Ϸ potential energy. That is, Wextra PEexit in Eq. (33) slightly denser than the surrounding water, will stop for eruptions through a fluid medium (limnic erup- rising and sink. The eruption thus will not continue. tions and eruptions through a lava lake). Therefore, That is, a limnic eruption is not expected to com- for a gas-driven eruption through a fluid medium that pletely degas the whole lake. can deform rapidly to maintain steady state and hydro- For H2O-driven basaltic eruptions through a lava static pressure in the conduit, recognizing that the lake, ignoring shallow magma entrainment and ϩ r initial saturation pressure is Pout 0gh and that the using Eq. (26), integration of Eq. (35) leads to: pressure at the initial depth does not change with time, s Ϫ †=r ˆ Ϫ †=r ˆ r =r ˆ ˆ P1 Pout 1 P0 Pout 0 0gh 0 gh 1 P P P k ; u2 Ϸ 0 l ln 0 Ϫ k ϩ k Ϫ 2 2 PEexit Eq. (33) reduces to the Bernoulli equation: r 0 1 1 2 0 P P0 P0 Z Pout dP r ! KE ϩ PE ϩ ˆ 0: 35† exit exit r k2 P P0 ϩ 2 : 37† P P0 For CO2-driven limnic eruptions, assuming: (1) nucleation sites are abundant in lake water and CO2 If a batch of basaltic melt containing 0.5% of total bubbles start to nucleate with only small supersatur- H2O is injected into a lava lake 100 m deep ation, leading to reversible exsolution; and (2) CO2 (26.5 bar), the injected melt would be at saturation. gas is ideal, Eq. (25) is applicable with ll0. Carrying Upon rising, bubbles form and grow in the batch, out the integration in Eq. (35) using Eq. (25) leads to causing it to rise more rapidly. The maximum exit (Zhang, 1996) velocity is ϳ100 m sϪ1. Entrainment of shallow  magma and irreversibility would reduce this exit 1 2 Ϸ l P0 P0 Ϫ ϩ P ; † u r ln 1 36 velocity. 2 0 P P0 where u is the ascent velocity and becomes the exit 3.4. Dynamics of reversible gas-driven eruptions ˆ : velocity at P Pout Because the process is assumed through a rigid medium to be reversible and shallow fluid entrainment is ignored, u calculated from Eq. (36) is the maximum For gas-driven eruptions through a rigid medium ascent velocity. The above equation has been used to (most violent volcanic eruptions because the wall investigate the dynamics of CO2-driven water erup- rocks can withstand stress and cannot relax in the tion from Lake Nyos and and to show time scale of a violent eruption), buoyancy does not that such eruptions can be violent (Zhang, 1996). For help the eruption. However, the expansion work done Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 227 by the bubbly magma inside the chamber, minus the Hence, expansion work against the outside pressure, is =r† Ϫ  d 1 P1 Pout supplied to the erupting column. To evaluate this W ˆ Ϫr P Ϫ P † ͉rˆr ˆ : extra 1 1 out dr 1 r extra work on the erupting column, consider a thin 1 † disk-shaped magma chamber so that the magma 39 chamber can be characterized by a single pressure Therefore, for any given chamber pressure of P1,Eq.(33) P1. Initially, the magma chamber pressure (the again reduces to the Bernoulli equation: saturation pressure) is P0. The pressure in the Z Pout magma chamber (and conduit) can be different from ϩ ϩ dP ˆ : † KEexit PEexit r 0 40 the lithospheric pressure. The fall of rock pieces into P1 the chamber does not change significantly the volume If friction is important, its effect can be added (e.g. of the chamber since the new magma chamber Eq. (1) in Wilson et al., 1980). Note that even though includes the space left by the falling rock. Elastic eruptions through a fluid medium and those from a deformation of the lithosphere does not change the rigid medium are both described by the Bernoulli volume of the chamber significantly. Hence, before equation, the lower limit of the integration is different. the total collapse of the overlying rocks to form a In Eq. (35) for an eruption through a fluid medium, the caldera, which probably restores the magma chamber lower limit of integration is a constant P ; whereas in pressure to lithostatic pressure and seals the conduit, 0 Eq. (40) for an eruption through a rigid medium, the the chamber volume may be regarded as constant. lower limit is the instantaneous magma chamber pres- Therefore, M =r ˆ M =r where M and r are the 1 1 0 0 1 1 sure P that decreases with time. A gas-driven explo- mass and density of the bubbly magma in the cham- 1 sive eruption can stop if either (1) P is so small that ber. As the eruption proceeds and magma is removed, 1 KE solved from Eq. (40) is equal to or less than bubbly magma in the chamber expands to fill space, exit zero or (2) the difference between the lithostatic pres- and pressure and density in the magma chamber sure and the magma chamber pressure is large enough decreases. to lead to the collapse of rocks into the chamber, Assume that the eruption process is quasi-steady so resulting in a caldera. Hence the above equation can that the variation of the mass and flow velocity in the be used to calculate the maximum fraction of magma conduit is negligible (or when the conduit volume is that can be erupted for a given gas–magma system negligible compared to either the erupted volume or using reversible equation of state. the magma chamber volume). The extra work done by the expansion of the magma in the chamber minus the work against the outside pressure is supplied to the 4. Some models for treating irreversibility erupted magma. (Kinetic energy variation inside the magma chamber is ignored since the process is In estimating the exit velocity using the Bernoulli assumed to be quasi-steady.) Hence for the eruption equation ignoring fluid/rock entrainment and friction of mass dM, magma chamber mass changes from M1 with the conduit, the density of the bubbly liquid as a to M1 Ϫ dM and its pressure changes from P1 to function of the system pressure must be known so that P1 Ϫ dP. The work done by the magma chamber to the integration can be carried out. For a reversible a unit erupted mass is: process, the gas and liquid phases are always in Z =r Ϫ equilibrium and the gas pressure is the same as the 1 1 at P1 dP  ˆ M1 Ϫ † 1 r Wextra P Pout d uniform liquid pressure. Hence can be calculated. dM =r r 1 1 at P1 However, irreversibility can arise because of: (1) Z Ϫ P1 dP =r† chemical disequilibrium between the gas and liquid M1 d 1 ˆ P Ϫ P † dP: 38† phases due to finite diffusivity of the gas component dM out dP P1 in the liquid (e.g. water concentration in erupted =r ˆ =r ; = ˆ r = r: Since M1 1 M0 0 M1 dM 1 d Since dP1 is pumice does not correspond to equilibrium at 1 bar Ϫ †Â infinitesimally small, the integration equals P1 Pout pressure; J. Blank, personal communication); and ‰ =r†= Š ˆ Ϫ †ˆϪ Ϫ † =r† ˆ : d 1 dP P P1 dP P1 Pout d 1 P P1 (2) the pressure difference between the gas and the 228 Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 liquid due to dynamic pressure that depends on visc- concentration in the liquid is linear to the equilibrium osity (Proussevitch et al., 1993). Hence the relation concentration. between r and P is complicated and future numerical Model 1a is a one-parameter linear model. A calculations or experiments may elucidate the best nonlinear model containing two parameters for a approach. Below, I suggest several empirical models one-component gas can be written in the form of for estimating the relation between density and l ˆ g l ϩ Ϫ g † l ϩ g l Ϫ l † pressure. C 1Ceq 1 1 C0 2 Ceq Ceq;out  Ϫ l = l †; † 4.1. Model 1 1 Ceq C0 44

l In the first class of models, chemical disequilibrium where Ceq;out is the equilibrium concentration of the between the gas and liquid phases is considered but gas component in the liquid at the final exit the pressure difference between the gas phase and the pressure (Pout), and g 2 is another constant that is Ϫ g ; Ϫg †= Ϫ l = l † g = Ϫ system is ignored. Assume that the initial state is between min 1 1 1 1 Ceq;out C0 and 1 1 l = l †: g l always at equilibrium saturation and that the gas Ceq;out C0 The condition for 2 is to insure that: (1) C l pressure is always the system pressure (ignoring decreases monotonically as Ceq decreases with eruption surface and dynamic effects). Hence gas phase density (that is, the gas concentration in the liquid must decrease l l is the same as treated before. However, because of with time monotonically); and (2) C is always Ceq (i.e. disequilibrium between the gas and liquid phases, the gas concentration in the liquid is always saturated or the liquid may contain more gas component than the supersaturated). Model 1b reduces to model 1a when g ˆ : l l equilibrium concentration. That is, the gas phase 2 0 In model 1b, C is a parabolic function of Ceq contains less gas mass than at equilibrium (d Յ d eq). and the parabolic curve passes through two points: the l ˆ l ; l ˆ l † Hence the density of the bubbly liquid lies between initial point Ceq C0 C C0 and the final point l ˆ l ; l ˆ l †: that of the equilibrium bubbly liquid and that of pure Ceq Ceq;out C Cout Using Eq. (44), the equation liquid. Therefore, one way to write the equation of of state becomes: state for irreversible gas exsolution is:  rl 0 ˆ Ϫ gl ϩ gl P0 Z Ϫ flϪ l ZPout rl g Ϫ g 1 0 out 0 1 1 1 r P Z Z P ˆ rl ϩ rl ; where 0 Յ g Յ 1; 0 out r r 0 l 0  eq r Z0P 41†  1 Ϫ : (45) ZP0 where g 1 is a constant and r eq is the density of the mixture at equilibrium (Eqs. (24)–(28)). 4.2. Model 2 If there is only one volatile component, using r eq from Eq. (24) with d ˆ 0; Eq. (41) becomes: 0 In the second class of models, the gas pressure may g ± rl P Z deviate from the system pressure (P P) owing to 0 ˆ 1 Ϫ g l ϩ g l 0 : 42† surface and dynamic pressure, but the gas concen- r 1 1 0 P Z 0 tration in the liquid is always in equilibrium with g ˆ ; When 1 1 the above equation reduces to Eq. (24) the gas pressure. Similar to the above development, d ˆ : with 0 0 Eq. (42) can be rewritten in the form of model 2a can be defined as: actual average gas concentration in the liquid (Cl) f Ϫ f † l ˆ 1 1 1 1 which is not the equilibrium concentration Ceq ˆ ϩ ; where 0 Յ f Յ 1: 46† l = †: l ± l ; g 1 P ZRT When CA CA;eq comparison of Eq. P P P0 d ˆ † (19) (with 0 0 and Eq. (42) leads to: The equation of state becomes: l ˆ g l ϩ Ϫ g † l : † C 1Ceq 1 1 C0 43 rl ZgP 0 Ϸ 1 Ϫ l ϩ l 0 : 47† r 0 g That is, model 1a means that the actual average gas Z0P Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 229 One may also define a complicated two-parameter would be 56 m sϪ1, enough to rise to ϳ160 m above model. lake surface. Since there is no direct observation on the exit velocity, except for an estimate of minimum 4.3. A combination model column height of 120 m based on the distribution of dead animals (Sigvaldason, 1989), all the above More general models can be constructed by incor- scenarios are possible. porating both the pressure difference and chemical disequilibrium between the gas and liquid phases. For example, incorporating both models 1a and 2a, 4.5. Applications to Mount St. Helens the equation of state can be written from Eq. (19) For explosive volcanic eruptions, there may be both (with d ˆ 0† as: 0 pressure difference and chemical disequilibrium between the gas and liquid phases. The 18 May rl ZgP ˆ 1 Ϫ g l ϩ g l 0 ; 48† 1980 eruption of Mount St. Helens may be modeled r 1 1 0 g Z0P as follows. Use or assume the following pre-eruptive g conditions: T ˆ 1200 K; P ˆ 2:2 kbar; P ; ˆ where P is given by Eq. (46). 0 0 H2O 0 : ; ; ˆ : ; 1 32 kbar PCO2 0 0 88 kbar initial H2Ot content of ˆ : ; 4.4. Applications to Lake Nyos eruption 4.6% (Rutherford et al., 1985), Pout 0 7 bar the initial depth of 8.3 km (from the initial pressure and Zhang (1998) examined bubble growth rates using a density of 2700 kg mϪ3), and no initial bubbles in ˆ : l experimental data and whether the gas and liquid the magma and hence Ps P0 for H2Ot in the phases can be in equilibrium during limnic eruptions. dacitic melt is taken to be 0.085 ϩ (107.4 Pa/P)so He suggested that equilibrium is possible if number ˆ : ; that at PH2O;0 1 32 kbar the equilibrium H2Ot in 10 Ϫ3 density of bubbles is much greater than 10 m . the melt is 4.6%. l for CO2 in the dacitic melt is Since heterogeneous bubble nucleation density in taken to be the same as that in basaltic melt. The lake water is expected to be high, it is possible that nonideality of CO2 is accounted for by using Z ˆ 1 ϩ equilibrium between the gas and liquid phases is 0:29P at 1200 K where P is in kbar (Sterner and maintained. However, if equilibrium is not main- Pitzer, 1994). The small and complicated nonideality tained, the effect of disequilibrium on the eruption of H2O is ignored. If friction is ignored and the process dynamics can be modeled as follows. For limnic erup- is assumed to be reversible, the calculated maximum tions, since liquid viscosity is low, gas pressure is not initial exit velocity would be 542 m sϪ1. The bubbly expected to deviate significantly from the liquid magma velocity in the conduit is 210 m sϪ1 at frag- pressure and irreversibility is most likely owing to mentation if fragmentation is assumed to occur at 74% chemical disequilibrium between the gas and liquid vesicularity. Hence calculating exit velocity by phases. Hence model (1a) or (1b) can be used. For assuming that the bubbly magma velocity is zero at g ˆ : example, using model (1a) with 1 0 8 during fragmentation may significantly underestimate the Lake Nyos eruption (meaning that as the water exits exit velocity. If friction is ignored but the exsolution lake surface, it still contains an average CO2 concen- process is irreversible, using a combination model g ˆ : tration corresponding to 5 bar), the exit velocity can with 1 0 75 (meaning that the erupted pumice be calculated from: still contains 0.9% H O ) and f ˆ 0:5 (meaning  2 t 1 1 P P P that the gas pressure at the exit is 1.4 bars), the calcu- u2 Ϸ g l 0 ln 0 Ϫ 1 ϩ ; 49† lated initial exit velocity is 325 m sϪ1 and the dynamic 2 1 r P P 0 0 reversibility is 0.4. Direct observations of the exit to be 79 m sϪ1 (instead of 89 m sϪ1 for reversible velocities are variable. The observed minimum ˆ g ˆ Ϫ1 eruption). The dynamic reversibility is Rd 1 choked exit velocity is 100–110 m s (Kieffer, 0:8: If furthermore entrainment of shallow water is 1981) and the velocity increased to 325 m sϪ1 at or treated simply using Eq. (34) with f ˆ 1 (i.e. mass after the exit (Kieffer, 1981). Another report of the of entrained shallow water is about the same as the observed velocity for a cloud is mass of the erupting bottom water), the exit velocity 150 m sϪ1 from photographic and seismic-station 230 Y. Zhang / Journal of Volcanology and Geothermal Research 97 (2000) 215–231 evidence (Friedman et al., 1981; Voight, 1981). both limnic and volcanic eruptions. Such empirical Hence, the eruption process was irreversible. The approaches can also model the observed exit velo- velocity 325 m sϪ1 is close to the estimated velocity city of the 18 May 1980 eruption of Mount St. g ˆ : incorporating irreversibility with 1 0 75 and Helens. f ˆ : : g f 1 0 5 Other combinations of 1 and 1 can also produce an exit velocity of ϳ325msϪ1.If Acknowledgements friction is significant, the g 1 and f 1 values can be greater. The smaller observed velocity (150 m sϪ1) would require either significant fric- This research is partially supported by NSF grants EAR-93-04161 and EAR-94-58368. 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