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Thermodynamics and MassTransport in Multicomponent,Multiphase H2O Systemsof Planetary InterestXinli Lu and Susan W. KiefferDepartment of Geology, University of Illinois, Urbana, Illinois 61801;email: xinlilu@uiuc.edu, skieffer@uiuc.edu

Annu. Rev. Earth Planet. Sci. 2009. 37:449–77

First published online as a Review in Advance onFebruary 10, 2009

The Annual Review of Earth and Planetary Sciences isonline at earth.annualreviews.org

This article’s doi:10.1146/annurev.earth.031208.100109

Copyright c© 2009 by Annual Reviews.All rights reserved

0084-6597/09/0530-0449$20.00

Key Words

geothermal systems, cryogenic systems, thermodynamics, fluid dynamics,clathrates, Mars, Enceladus, sound speed

AbstractHeat and mass transport in low-temperature, low-pressure H2O systemsare important processes on Earth, and on a number of planets and moonsin the Solar System. In most occurrences, these systems will contain othercomponents, the so-called noncondensible gases, such as CO2, CO, SO2,CH4, and N2. The presence of the noncondensible components can greatlyalter the thermodynamic properties of the phases and their flow propertiesas they move in and on the planets. We review various forms of phase dia-grams that give information about pressure-temperature-volume-entropy-enthalpy-composition conditions in these complex systems. Fluid dynamicmodels must be coupled to the thermodynamics to accurately describe flowin gas-driven liquid and solid systems. The concepts are illustrated in de-tail by considering flow and flow instabilities such as geysering in moderngeothermal systems on Earth, paleofluid systems on Mars, and cryogenicice-gas systems on Mars and Enceladus. We emphasize that considerationof single-component end-member systems often leads to conclusions thatexclude many qualitatively and quantitatively important phenomena.

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INTRODUCTION: PLANETARY CONTEXT

A fundamental observation on terrestrial H2O geothermal systems, and even on cryogenic H2Osystems, is that H2O is rarely the only component in the system. CO2, CO, N2, NH3, CH4,H2S, SO2, C2H2 (acetylene), C3H8 (propane), C2H6 (ethane), and other gases appear in varyingconcentrations both in liquid and solid H2O systems. The systems are thus more typically multi-component than single component. Freezing, melting, vaporization, condensation, sublimation,and deposition occur in these systems, so they are additionally multiphase. Any dynamic flowprocesses in such systems are inherently more complex than in single-component systems (Wallis1969a,b,c,d). We illustrate these complexities here by focusing on flow from reservoirs in single-and multicomponent aqueous systems with both liquid and solid reservoirs.

Conditions conducive to multicomponent, multiphase flow occur on, and in, a wide variety ofSolar System bodies, including Earth (Figure 1a,b), Mars, Jupiter, Neptune, Saturn, and comets,as well as numerous icy satellites, such as Enceladus (Figure 1c). We present thermodynamicdata and methods for analyzing such systems at low temperatures and illustrate with a few exam-ples how combining the thermodynamics with fluid dynamics models leads to complex fluid flowphenomena that differ from the single-component counterparts. The eruption of a pure H2Ogeyser (or one with only minute traces of other gases) is driven by processes related to the boil-ing conditions of H2O, e.g., Old Faithful geyser (Kieffer 1984, Ingebritsen & Rojstaszer 1993),although it is possible that even Old Faithful has significant noncondensible gases (Hutchinsonet al. 1997). Eruptions of CO2-rich geysers are driven by conditions of gas exsolution ratherthan boiling of H2O (Lu 2004; Lu et al. 2005, 2006). Serious errors can be made by assumingthat a partial pressure due to an H2O component is equal to the total pressure of a mixture.In engineering, it is well known that drilled wells into gas-rich liquid are much more difficultto manage and contain than those that contain only H2O liquid or solid ice. Ignoring gases in

a b c

Figure 1(a) Old Faithful geyser, Yellowstone National Park. Photo by Susan W. Kieffer. (b) A discharging geothermalwell from a CO2-rich source area, Karawau, New Zealand. Note supersonic plume at top of wellhead. Photoby Susan W. Kieffer. (c) Multiple jets in the plume emerging from Enceladus, the frigid satellite of Saturn.Photo from the NASA Cassini Mission.

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analysis of fracture flow can lead to erroneous conclusions about the nature of the flow or reservoirproperties.

Our initial perspective from Earth is from studies of low-temperature geothermal systems(White 1967; Rinehart 1980a,b; Henley et al. 1984; Economides & Ungemach 1987). We includethe even lower-temperature clathrate systems (Kvenvolden 1993). Clathrates are ices that containforeign gas molecules in cages of H2O molecules. Both geothermal fluids and clathrates havesignificant amounts of non-H2O components—dissolved gases ranging from approximately 0.01%to several tens of percent (Giggenbach 1980, Henley & Ellis 1983, Sloan 1998). We especiallyfocus on those systems that are below the boiling point of pure water in which dissolved gasesplay a major role in the fluid dynamics (Lu et al. 2005, 2006). Similar dynamics were involvedin the explosive eruptions of Lake Nyos, Cameroon (Zhang & Kling 2006). In the frozen form,the formation of methane clathrates in glaciers (Miller 1961) and on the ocean floor (Sloan 1998,Max et al. 2006), and their degassing upon either heating or release from high pressure, involvesmulticomponent, multiphase processes.

Pressure (P), temperature (T), and composition (x) relations of these systems can be found in thefluid inclusion literature (Bodnar 2003). However, quantitative analysis of dynamics in geothermaland planetary systems requires knowledge of other valuables, such as specific entropy (s), enthalpy(h), and volume (v), as well as different choices of independent and dependent variables, and theseare the focus of this review.

For planetary applications, we have focused on two bodies that are the current topics of intenseexploration: Mars and Enceladus, a satellite of Saturn. On Mars, CO2 and H2O in solid, gas,and possibly liquid and clathrate forms are the dominant volatile components in the crust andat the icy poles (Miller & Smythe 1970, Milton 1974, Longhi 2006). Methane (CH4) has beenobserved in the atmosphere (Formisano et al. 2004) and is of interest because it could indicateeither life or active volcanic processes. The behavior of subsurface ices versus latitude and seasonis of great interest for Mars, and the recent arguments for “seeps” of a fluid (Malin & Edgett 2000,Hartmann 2001) versus granular flow (Shinbrot et al. 2004) have made the studies of volatiles ofcurrent interest. It is also possible that there is a liquid water–CO2 fluid at shallow depths, or aliquid water–CH4 fluid (Lyons et al. 2005).

The discovery of a large plume emanating from the frigid south pole of Enceladus (Figure 1c)during the Cassini spacecraft mission to Saturn has raised the fundamental question, “What kindof reservoir produces these plumes?” Porco et al. (2006) used a single-component H2O model(“Cold Faithful”) to suggest that the reservoir is liquid water. The proposal that liquid water couldbe as close as 7 m to the surface and related models for a deeper “ocean” (Nimmo et al. 2007)have generated much speculation that Enceladus could host near-surface life and thus might bean astrobiological target in near-term Solar System exploration.

Kieffer et al. (2006) pointed out that the Cold Faithful model cannot account for the pres-ence of at least two of the three noncondensible gases in the observed abundances (N2 andCH4)1; the third major gas, CO2, could be in solution in liquid water, but only at depths greaterthan 20 km. They proposed instead a multicomponent model (dubbed “Frigid Faithful”) inwhich cold decomposing clathrates host all of these gases (as well as the other minor organic

1The substance observed by the INMS instrument on the first Cassini encounter with Enceladus in 2006 has a molecularweight of 28. Initially it was attributed to N2 (Waite et al. 2006). During later encounters in 2008, J.H. Waite (NASA/JPL/SwRI news conference of 3/26/08, available at http://saturn.jpl.nasa.gov; unpublished as of the time of preparation of thisreview) proposed that it was CO, which also has a molecular weight of 28. Either molecule forms clathrates with roughlysimilar properties and thus the conclusions here are unaffected by this uncertainty.

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constituents seen in the plume). This system forms a self-sealing advection machine that operatesat a much lower temperature than required for liquid water (Gioia et al. 2007). Initially, the dif-ference between the Cold Faithful and Frigid Faithful scenarios appears to be temperature, butmore fundamentally, the difference is the behavior of single- versus multicomponent systems.

Although ammonia (NH3) has not been detected on Enceladus, there has been much specu-lation that it has existed in the interior in the past, and possibly at present (Matson et al. 2007).Clathrate and ammonia hydrates have long been suspected to play key roles in the evolution andpresent state of Titan (Lunine & Stevenson 1985, 1987). We do not discuss this complex systemin this review.

Analysis of multicomponent, multiphase systems is complicated, and systems are often sim-plified in composition to a single component. In this review, we discuss the thermodynamicphase diagrams that apply to a number of multicomponent, multiphase systems using variouscoordinate systems that are appropriate to the particular problems of fluid dynamics consid-ered, with applications to Earth, Mars, and Enceladus. We focus on depressurizing processes asa fluid moves from a high-pressure reservoir to a lower-pressure environment, and we show howdiffering conclusions are reached from analysis of single- versus multicomponent systems. Ourapproach brings in some analyses that are common in engineering practice and explains thesein the context of geologic and planetary problems. In many ways, this review complements, anddoes not overlap, that of Zhang & Kling (2006) by considering the complex thermodynamics ofmultiphase multicomponent systems, combining the thermodynamics with fluid dynamics mod-els of flow in cracks, and extending the work to cryogenic conditions and to other planets andmoons.

THERMODYNAMIC PROCESSES AND APPROPRIATE PHASEDIAGRAMS FOR ANALYSIS

When analyzing a complex, multicomponent, multiphase process, the first and most basic step isto understand the first and second laws of thermodynamics and then to map the behavior requiredby the first and second laws onto relevant phase diagrams to get an estimate of the behavior ofthe system. However, these two steps must be combined with calculation of another fundamentalparameter, the sound speed, to constrain the behavior of a system. Sound speed is the velocity atwhich small disturbances in pressure or density are propagated through a material. The relationof flow velocities to the sound velocity determines the flow regime.

If flow velocities are significantly lower than the sound speed, flow is subsonic and, therefore,nearly incompressible. This assumption underlies much of traditional geophysical fluid dynamicsin understanding oceans, atmospheres, and Earth’s interior. If flow velocities are greater than thesound speed, flow is supersonic, and compressibility and nonlinear effects, such as shock waves,must be considered in analyses. Physical phenomena and analytic approaches are very different inthe two different flow regimes. We demonstrate that these effects are important in decompressionof geothermal and volcanic systems on Earth and other planets. As we demonstrate here, thesound speed in multiphase systems can be very low and this allows supersonic flow regimes notencountered in single-component regimes to be possible.

When the fluid velocity equals the sound speed, the phenomenon of “choking” plays a major,and controlling, role in high-velocity fluid dynamics because choked conditions control the massflux from depressurizing systems. Low sound speed means that the flow would choke at rela-tively low velocities and thus low mass fluxes. Choking phenomena are found in many terrestrialgeothermal exploration and exploitation situations, and they control mass flux and, therefore,power production and economic potential.

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Thermodynamic Processes

Standard thermodynamic variables used in many analyses are state variables such as pressure,temperature, density, entropy, and enthalpy. Fluid dynamic analyses require additionally somefirst partial derivatives of these variables, such as the sound speed. These derivatives have somefundamentally different properties in multiphase and multicomponent systems.

First law, second law, and sound speed. For an open system with fluids flowing in and out, thegeneral form of the first law is as follows (Huang 1988a):

Qsyst = Esyst + Ws + mout

(h + V 2

2+ gz

)out

− min

(h + V 2

2+ gz

)in

, (1)

where Qsyst is the rate of the heat gained by the system, Esyst is the rate of change of stored energywithin the system, Ws is rate of work (power) done by the system (except flow work), m is massflow rate of the fluid, h is specific enthalpy, V is velocity of the fluid, and z is the elevation. Thesubscripts in and out represent the thermodynamic state of the fluid flowing in and out of thesystem, respectively. Below in this review, we use subscripts such as this, or 1, 2, 3, etc., to indicatethe values of variables along flow paths without further definition where obvious. Enthalpy h is

h = u + Pv = u + Pρ

, (2)

where u is specific internal energy, and ρ is density.For a steady-state steady flow with no power done by the system, mout = min = m, Esyst = 0,

and Ws = 0. Considering a thermodynamic process from state 1 to 2 (Adkins 1983), Equation 1becomes

q = (h2 − h1) + 12

(V2

2 − V21

) + g(z2 − z1), (3)

where q is the heat gained by the system per unit mass. In the cases we discuss here, the depres-surizing process is very fast, and the heat transfer rate to or from the surrounding of the system ismuch smaller compared with other terms in Equation 3, so q = 0 can be assumed for the energybalance. In many decompression-driven flows on Earth and other planetary bodies, the potentialenergy term g(z2 − z1) is also negligible in Equation 3. When heat transfer q and initial velocityV1 are also negligible, the exit velocity V2 in Equation 3 can be expressed as

V2 =√

2 (h1 − h2). (4)

The general form of the second law of thermodynamics for an open system (Huang 1988b) canbe expressed mathematically as(

dSdt

)irr

= dSdt

+∑out

ms −∑

in

ms −∑ Qk

Tk, (5)

where ( dSdt )irr is the rate of entropy production (the so-called path function) owing to internal

irreversibility; dSdt is the rate of the entropy change within the open system;

∑out ms and

∑in ms

are total entropy contents of the outgoing and incoming streams, respectively; and Qk is the heattransfer rate at temperature Tk (k is the index indicating the different heat sources and theirtemperatures).

The second law of thermodynamics requires ( dSdt )irr > 0 for any irreversible process (i.e., for

all real processes). ( dSdt )irr = 0 only for reversible ideal processes. Irreversibility can be caused by

heat transfer driven by a temperature difference, free-expansion owing to a pressure difference

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(not quasi-static), and dissipation (Thompson 1972) in a viscous flow (viscous work going intodeformation of a fluid particle). In steady-state conditions, dS

dt = 0.If heat transfer is negligible or the process is adiabatic, Equation 5 becomes(

dSdt

)irr

=∑out

ms −∑

in

ms ≥ 0. (6a)

Note that, in Equation 6a, “>” corresponds to an irreversible process and “ = ” is for a reversibleprocess. For one stream of fluid from state 1 to 2, Equation 6 can be further expressed as

s2 − s1 ≥ 0. (6b)

Although not usually thought of as a fundamental thermodynamic parameter in the context ofP-v-T-s phase relations, the sound speed is an essential parameter for analysis of moving materialsin planetary systems where pressure gradients can be great, and it is directly related to the staticvariables discussed above. Sound speed is a partial derivative of two fundamental static variables,pressure and density, and is related to the bulk modulus, K (which is a measure of incompressibility,and K = 1/β, where β is the compressibility):

c2 = Kρ

=(

∂P∂ρ

)s. (7)

This derivative is taken for constant entropy conditions because sound wave propagation involvesinfinitesimally small and rapid amplitude changes assumed to be adiabatic and reversible, i.e., con-stant entropy. Although defined in terms of the static variables K and ρ, the sound speed is a dynamicvariable and plays an important role in the simple wave equation. Some finite amplitude processes,such as rapid decompression of a pressurized reservoir, are also well approximated by the adiabaticand reversible isentropic condition (Kieffer & Delany 1979, pp. 1612–15). We show in sectionsbelow how the sound speed varies in multicomponent, multiphase systems, and emphasize that itis a fundamental parameter that must be considered in modeling flow in these complex systems.

Single-Component H2O System: Phase Diagrams

Temperature-entropy diagrams. Although pressure-temperature (P-T) diagrams are com-monly used in Earth and planetary sciences, this representation of the phase relations does notshow details of the two-phase region. Therefore, in many common flows where a flow under-goes a phase change, the P-T diagram is limited. The temperature-entropy (T-s) diagram ismuch more useful in analyzing the thermodynamic processes if the system is undergoing phasechanges. These diagrams are commonly used in engineering and geothermal applications (ther-modynamic analysis of power cycles, steam turbine thermal process design, flow in geothermalwells, and geothermal steam separation processes), and the phase diagrams (or their equivalentMollier charts) are widely available for some substances. However, T-s diagrams typically do notinclude the low-temperature range and have had only limited application in planetary sciences.Although Kieffer (1982) generated T-s diagrams for SO2, S, and CO2 for very-low-temperatureconditions, we could not find one for H2O at low temperatures. We present one for temperaturesdown to 145 K in Figure 2.

Why are T-s diagrams useful? Because, as discussed above, many flow processes—to firstorder—are adiabatic and reversible, i.e., isentropic. An isentropic process is a vertical line on theT-s diagram, and thus many properties can be inferred directly from such a representation. Theproperties of each phase are represented and the properties of the mixture can be calculated fromthe proportions of the phases.

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–3 –1 1 3 5 7 9 11 13 15 17 19 21

–100

–50

0

50

100

150

200

Tem

pera

ture

(°C)

Temperature (K)

Pres

sure

(bar

)

250

300

350

400

450

150

200

250

300

350

400

450

Tem

pera

ture

(K)

500

550

600

650

700

0

S + L

L

S

V1

1'

100 300 500 700

102

A

B

CD

Entropy (kJ/kg–K)

Specific volumePressureEnthalpy

Vapor mass fraction

108 107 2 x 106

0.005

0.05

0.5

10

206 m3/kg

26146

5.687 x 107

105 611 10

1 Pa

1.468 x 10-3

8.328 x 10-5

200016001200800400–300 kJ/kg

x = 0.2

L + V

0.40.6 0.8

S + V

0.2 0.4 0.6 0.8

Vapor

Solid

Liquid

100

10–2

10–4

10–6

10–8

10–10

a b

Figure 2(a, b) Diagrams of water system. Generated with Engineering Equation Solver, EES (Klein 2007). Thermodynamic properties fromHarr et al. (1984). Correlations are valid up to a pressure of 10,000 bar. Properties of ice T < 273.15 K and pressures above thesaturation vapor pressure of ice are based on Hyland & Wexler (1983) for temperatures from 173.15 K to 473.15 K. Data below−100◦C (with vapor mass fraction shown by dashed curves) are extrapolated. In (b), A is the triple point, C is the critical point. AC is thevapor-pressure curve and AB is the sublimation-pressure curve. The melting-point temperature for ice Ih decreases with increasingpressure, AD. For pressure increases from triple point to 300 bar, the melting point slightly decreases approximately 2 K; however,when the pressure continues to increase to 2000 bar, the melting point decreases by approximately 18 K (Li et al. 2005, Luscher et al.2004, Saitta & Datchi 2003, Longhi 2001, Chou et al. 1998). This is because water expands on freezing, causing the density of ice to beless than liquid water, which is different from most substances for which the solid density is greater than that of the liquid (Moore1962). In (a), the data are extrapolated down to 145 K. Isobars, constant specific volume lines, isenthalpic lines, and x lines (massfraction of saturated water vapor) in a two-phase region are all presented. The liquid-vapor (L + V) two-phase region corresponds tothe vapor-pressure curve AC in (b). The solid-vapor (S + V) two-phase region corresponds to the sublimation-pressure curve AB in(b). The curve AD in (b) corresponds to many nearly horizontal lines in the solid-liquid region (S + L) in (a), with temperature veryclose to 273.16 K for pressures less than 300 bar. These lines are almost identical and look like a horizontal line in (a). Process 1-1′ ismentioned in the text.

Two typical isentropic decompression processes in the two-phase region are shown inFigure 3a (the high-temperature part of Figure 2). Process 1–1′ is decompression from a saturatedliquid; process 2–2′ is decompression from a saturated vapor. If the reservoir condition (either 1or 2) is known, then the mass fraction of steam (or liquid) can be determined by the lever rule, theenthalpy can be calculated, and an estimate of flow velocity can be calculated from Equation 4, as-suming that the flow has not choked. Conversely, if the exit velocity and the vapor mass fraction (x1′ )are known, state 1′ can be determined and hence the corresponding reservoir condition (state 1)can be inferred. Similar arguments apply to any initial state within the two-phase region insteadof on the phase boundary.

For an adiabatic process, if the changes of kinetic energy and potential energy are negligiblecompared with the change of the enthalpy, the process is isenthalpic. In Figure 3a, the process3–3′ is isenthalpic, a flow path typical of that in a geothermal well. One of the critical questions

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c

a b

–1.0 0.1 1.2 2.3 3.4 4.5 5.6 6.7 7.8 8.9 10.00

100

200

300

400

100

20

10

6

1 bar

1

1'

3

3'

3"

2

2' 2"

4 5

Critical point

L V

L + V

1

23

0

4

P1

P1

P1, T1

P2, T2

P4, T4

P3, T3

P2

P2

P3

P3

P4

P4

L V

Critical point

–2.5 –2 –1.5 –1 –0.5 0 0.5 1–120

–80

–40

0

40

80 353

313

273

233

193

153

10

L

S

2

3

230 K

258 K

1

2'

2"

243 K1'

1"

3'

3"

Tem

pera

ture

(°C)

Tem

pera

ture

(°C)

Tem

pera

ture

(K)

Tem

pera

ture

Entropy (kJ/kg–K)

EntropyEntropy (kJ/kg–K)

x = 0.2 0.4 0.6 0.8

L + V

L + VS + L

x = 0.01

0.2

S + V5.687 x 107

m3/kg

26146

206

10

–500kJ/kg

–450–400

–350 –300 –250 –200 –150 –100 –50 081

612

1 Pax = 0.05

x = 0.1

x = 0.15

Specific volumePressureEnthalpy

Vapor mass fraction

Figure 3Different depressurizing processes in the two-phase regions. (a) Process 1–1′ is an isentropic depressurizing process from reservoir state1 (saturated liquid, P = 20 bar and x = 0) to and exit state 1′ (liquid-vapor two-phase region, P = 1 bar). Process 2–2′ shows anisentropic depressurizing process from saturated vapor (state 2, P = 20 bar and x = 1) to the two-phase region (state 2′, P = 1 bar).From the lever rule, or Equation 8 in the text, states 1′ and 2′ can be estimated to have 19% and 83% vapor content, respectively.(b) Schematic diagram showing different depressurizing processes in liquid region. (c) Depressurizing processes in a low-pressure andlow-temperature region. Data below −100 ◦C (with vapor mass fraction shown by dashed curves) are extrapolated. L, liquid; V, vapor; S,solid.

in reservoir engineering is, “Can the enthalpy of the reservoir be inferred from measurements ofthe properties at the well head?”

Given the enthalpies of the saturated liquid and vapor of the outflow, hf and hg, and the massfraction of vapor in the outflow, x3′ , and the assumption that the process is isenthalpic, then thereservoir enthalpy, h3 = h3′ , can be inferred from the lever rule:

x3′ = h3 − hf

hg − hf. (8)

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In the liquid-alone region, these processes can only be shown at an exaggerated scale(Figure 3b). 0–1 is an isobaric process, 0–2 is isothermal, 0–3 is isenthalpic, and 0–4 is isen-tropic. On or near the surface of planets and moons where the surface pressure and temperatureare very low, the depressurizing process will involve the low-temperature ice-vapor part of thephase diagram (Figure 3c, the enlarged low temperature region of Figure 2).

We can illustrate the usefulness of this representation for examining the plumes erupting fromEnceladus. The plumes were reported to have a mass ratio of solid/vapor of 0.42, correspondingto a fraction vapor of x = 0.7 (70%)2 (Porco et al. 2006). What kind of reservoir and processproduces these plumes? We illustrate the processes suggested for the single-component reservoirhere, and then contrast these results with a multicomponent analysis below.

In a single-component H2O system, two processes are available to produce the mixture in theplume: sublimation of ice with subsequent recondensation of some of the vapor, and boiling ofliquid. Consider first the recondensation scenario path 1–1′ in Figure 2. The initial state chosenfor our example is the sublimated vapor on the phase boundary (state 1 in Figure 2) correspondingto 225 K. We chose the final state to correspond to the relatively low temperature of 145 K. Fromthe lever rule, a plume in equilibrium at this final state has a composition of that reported, x ≈ 0.7.Porco et al. (2006, p. 1398) were considering the masses of ice and vapor to be comparable, whichwould correspond to x ≈ 0.5. It can be seen from Figure 2 that this is a more difficult compositionto obtain from sublimation under plausible Enceladus conditions. If the kinetic limitations are suchthat the final temperature of reaction would be approximately 180–190 K, it is difficult to obtainx ≈ 0.5 − 0.7, because the final state is more vapor rich x ≈ 0.8 − 0.9. Note that this correspondsto the recalculated conditions mentioned in Footnote 2.

Instead, the behavior of a reservoir of boiling water at 273 K was considered as the mostplausible candidate for producing the plume. Three potential reservoirs that correspond to theCold Faithful model are illustrated in Figure 3c: (a) a liquid water reservoir just at triple pointconditions (process 1–1′-1′ ′) corresponding to initial conditions proposed by Porco et al. (2006),(b) a liquid water reservoir 20◦ above the triple point (process 2–2′-2′ ′), and (c) a 50%–50% mixedice-liquid slurry at conditions very close to triple point (3-3′-3′ ′). The final state is assumed to be145 K. From this T-s diagram, it is easily seen that these conditions only produce approximately10% vapor; a higher final temperature of 180–190 would produce even less vapor. As noted byPorco et al., a mixture forming from a liquid reservoir is mostly ice. If all ice was entrained, theplume should contain more ice than observed, and more than 90% of the ice must be left behind.

The two different conclusions (boiling water versus recondensation of sublimated vapor) areat the root of the current controversy over the subsurface conditions at Enceladus’ south pole. Wediscuss below how the presence of noncondensible gases also affects this conclusion.

T-v phase diagrams to illustrate nonequilibrium effects in condensing vapor flow. Nonequi-librium thermodynamic processes occur in condensing flow because supersaturation of the vaporis required for the nucleation of condensed phases. These effects become pronounced even attemperatures characteristic of terrestrial geothermal regions (e.g., ∼200◦C) and may be especiallyprominent at the low temperatures of interest for planetary surface conditions (e.g., see Kieffer1982 for a discussion of this in the context of SO2 volcanism on Io; Ingersoll et al. 1985, 1989 forsublimation dynamics on Io; and Byrne & Ingersoll 2003 for a discussion of sublimation of Marspolar caps).

2We picked the example to address the mass fraction reported by Porco et al. (2006). Kieffer et al. (2009) subsequentlyrecalculated the mass fraction and show that it is significantly higher than this value (x > 0.8).

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673

623

573

523

473

423

373

323

273

223

173

123

73–4 –2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

–200

–150

–100

–50

0

50

100

150

200

250

300

350

400Critical point

273 K

A (473 K, 15.5 bar)

D E

G

C J B

L L + V V

S S + V V

F (375 K, 1.013 bar)

I (190 K, 3.5 x 10-7 bar)

Two-phase equilibrium expansion (T0 = 473 K)Saturation line

Two-phase equilibrium expansion (T0 = 190 K)Metastable vapor expansion (T0 = 473 K)Metastable vapor expansion (T0 = 190 K)Schematic real expansion processafter supersaturation

Tem

pera

ture

(°C)

Tem

pera

ture

(K)

log specific volume (m3/kg)

Figure 4Temperature-volume semilog plot for H2O. L, liquid; V, vapor; S, solid. See text for discussion of lettered points.

Although the topic of nonequilibrium flows with condensation could be the subject of a separatereview article because so many new computational and experimental techniques have shed lighton this problem, the basic concepts can be illustrated with yet a different phase diagram: thetemperature-volume (T-v) diagram introduced by Zel’dovich & Raizer (1966) and applied to SO2

condensation in the context of Io by Kieffer (1982). This diagram is useful because expansions arewell described by the specific volume of the system, and kinetic effects can be specified in termsof temperature or temperature differences.

We illustrate the phenomena for the specific case of decompression of H2O from a typicalgeothermal reservoir at 200◦C, 15.5 bars (Figure 4). Decompression from icy conditions followsas a subset of this case (discussed below starting at 190 K), and we then apply the analysis tononequilibrium condensing flow in fractures on Enceladus.

An equilibrium expansion process is one end-member of possibilities in which the mix-ture composition is always the equilibrium composition at given conditions (curve A-F-B ofFigure 4). Condensation of droplets must occur rapidly enough for equilibrium to be maintained.The expansion path lies so close to the saturation curve boundary that it is nearly indistinguishable,even in semilog coordinates.

If condensation does not occur (e.g., if there is insufficient supersaturation of the vapor, lackof nucleation sites, or insufficient time to form nucleation centers), then the fluid remains in thevapor phase and expands isentropically. The fluid can be treated as an ideal gas. For a given initialcondition (T0, P0, v0 at point A) and a given final temperature T, the specific volume v and pressureP along the isentropic metastable process can be determined using the perfect gas law equationsfor v(T) and P(T), giving the metastable vapor isentrope, path A-D-C.

There are significant differences between equilibrium and metastable expansion. For example,the pressure at 375 K is six times higher for a metastable expansion than for an equilibrium one, and

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the corresponding volumes are different by a factor of 6. A real expansion process is usually a pathbetween these two limiting processes, shown schematically by the path A-D-E-G in Figure 4,where we have arbitrarily illustrated a metastable vapor isentrope to 100◦C supersaturation.

The enthalpy drop for the metastable expansion from A to D is 234 kJ kg−1, less than the297 kJ kg−1 enthalpy drop of the equilibrium expansion from A to F. Conduit velocities, whichare directly proportional to the enthalpy change, are correspondingly less (684 m s−1 versus771 m s−1).

These effects become more extreme as the temperature difference between initial and final stateincreases, as shown by the spread between curves A-D-C and A-F-B in Figure 4. The specificvolume difference between the two processes at 348 K is approximately 1 order of magnitude,whereas it becomes 5 orders of magnitude at 223 K. This volume difference is mainly due to thedifference in system pressures at a given temperature.

At the triple point, ice is the stable phase in equilibrium with vapor and water. Vapor expansionat a temperature below its freezing point can be different from the behavior in the liquid-vaportwo-phase region. For the temperature range analogous to that on the south pole of Enceladus,curves I-J (blue curve) and I-B (blue dots) in Figure 4 show the metastable vapor expansionand the two-phase equilibrium expansion, respectively, from the initial state of saturated vapor at190 K. At a condition with a very low temperature and pressure, the specific volume becomes verylarge and the spacing between molecules is so rarified that the vapor may not find solid nuclei togrow on.

On Enceladus, nonequilibrium conditions may be dominant because of the very low temper-ature. To illustrate the effects of nonequilibrium conditions on condensation in low-temperatureflows, consider the formation of a micron-sized ice particle. Assuming the mass flux onto thesurface of an ice particle is ρvvrms(2π )−1/2 (Ingersoll 1989), the radius growth rate of a sphericalparticle in a condensing flow is

drdt

=(

ρv

ρice

)vrms(2π )−1/2, (9)

where the root mean square molecular velocity vrms = (3RT/M )1/2, R is the universal gasconstant = 8.314 J mol-K−1, T is temperature in Kelvin, M is molar mass of vapor in kg mol−1,and ρv and ρice are the vapor and ice densities in kg m−3, respectively.

In a low-temperature vapor that is flowing fast and has a relatively short flow path, such aswater vapor escaping from fractures on Enceladus, it is more likely that flow would follow anonequilibrium process (I-J in Figure 4) than an equilibrium process. If an expanding vaporfollows a nonequilibrium path, such as I-J, the vapor density at any given temperature is greaterthan that at equilibrium condition (about 1 order of magnitude at 175 K). Growth rates willtherefore be higher, possibly by an order of magnitude, and conclusions based on growth ratesin single-component systems may tend to underestimate possible growth rates. However, thepresence of noncondensible gases in conduits, as advocated by Kieffer et al. (2006), will alsoinfluence growth rates, so many factors must be considered.

Sound speeds in multiphase systems: s-ρ phase diagrams. Finally, we consider a fourth phasediagram that illustrates yet another thermodynamic property: the sound speed. The sound speedof end-member phases can be quite high: 1435 m s−1 for liquid water and 470 m s−1 for steam.However, the sound speed of a mixed liquid-gas fluid can be as low as a meter or a few meters persecond (Figure 5). The speed of sound in a multiphase system depends on whether heat and masstransfer are maintained in equilibrium as the small pressure pulses associated with the sound wavepass through the fluid. If equilibrium is not maintained, the sound speed drops by approximately

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ba

Water-steam phasesnot in thermodynamic

equilibrium

Water-steam phasesin thermodynamic

equilibrium

200 bars (366° C)

100 bars (311° C)

50 bars (264° C)10 bars (180° C)5 bars (152° C)1 bar (100° C)

200 bars (366° C)

100 bars (311° C)

50 bars (264° C)

10 bars (180° C)5 bars (152° C)

1 bar (100° C)

10,000

1000

100

Soun

d sp

eed

(m s

–1)

Soun

d sp

eed

(m s

–1)

η (mass fraction, steam)

η

10

10–5 10–4 10–3 10–2 10–1 100

10–5 10–4 10–3 10–2 10–1 100

1

1000

100

10

1

10–4

10–3

10–2

10–1

Mas

s fr

actio

n

Pressure

1 bar

300250

30025020017515012510075

2550

75100

125

50

25100

100

10–1

10–2

10–3

10–4

1 5

Entropy (Joule g–1 K–1)

Den

sity

(g c

m–3

)

10

(V)

(L+V)

(L)

Critical point

1Φ

2Φ

H20

Figure 5(a) Sound speed of water-steam mixtures not in thermodynamic equilibrium (top) and in thermodynamicequilibrium (bottom). Reproduced from Kieffer 1977. (b) Entropy-density phase diagram for H2O, withcontours of pressure and mass fraction. Reproduced from Kieffer and Delany 1979.

one order of magnitude (Figure 5a, top), whereas if it is maintained, the sound speed drops bynearly two orders of magnitude (Figure 5a, bottom). It can be as low as 1 m s−1 (magnitude varieswith pressure).

The sound speed depends on the mass fraction of vapor in the liquid-vapor system. The fluidcan be either a liquid with gas bubbles (boiling liquid) or a vapor with liquid droplets (aerosol ormist). What causes this phenomenon? The definition of sound speed has two terms (Equation 7): adirect proportionality to the compressibility, K, and an inverse proportionality to the density, ρ. Ina boiling two-phase mixture, the density is nearly that of the liquid phase, but the compressibilityis that of the gas phase—much lower than that of the liquid phase. These two effects dramaticallydiminish the sound speed (left side of Figures 5a). However, as the vapor mass fraction increasestoward the aerosol state, the mixture density decreases and the effect is reversed (right side ofFigure 5a).

Referring again to Equation 7, note that the sound speed (squared) is (δP/δρ)s, which suggeststhat changes of sound speed between liquid, gas, and (gas + liquid) phases (both boiling liquid andaerosol phases) can be visually seen on a plot of entropy versus density with contours of constant

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pressure (Figure 5b) (Kieffer & Delany 1979). This plot is analogous to a normal topographicmap where contours of elevation (z) are plotted against two directions, x and y: (δz/δy)x. Thus,when a s-ρ graph is read as a topo map and P is considered equivalent to z, the topography of as-ρ graph visually illustrates the large variations in sound speed that characterize the single andmultiphase regions (Figure 5b). When traversed in a vertical direction at constant entropy, thespacing between isobars on such a plot reflects the magnitude of the sound speed—large spacingindicates small gradients reflecting low sound speeds, and tight spacing indicates large gradientsand high sound speeds.

Multicomponent, Multiphase Liquid-Gas Systems

Unfortunately, it is difficult to make similar representations of these phase diagrams in multicom-ponent systems because solubility of the gas into the liquid or solid adds additional complexity.We use the commonly occurring systems, including H2O-CO2 and H2O-CH4, to illustrate howHenry’s law and solubility curves are used in the study of multiphase and multicomponent flows.If gas dissolves in a liquid in equilibrium, the molar concentration of a solute gas in a solution,Xaq, is proportional to the partial pressure of that gas above the solution, Pg:

Pg = KHXaq. (10)

KH is the Henry’s law constant on the molar concentration scale, which can be determined byexperiments, Xaq:

Xaq = ng,aq

ng,aq + nl, (11)

where ng,aq is the number of moles of the gas in aqueous phase and nl is the number of moles ofliquid water.

Henry’s law is accurate if concentrations or partial pressures are relatively low. Deviations fromHenry’s law become noticeable when concentrations and partial pressures are high, but first-ordereffects can be considered even in this case by using Henry’s law. Detailed descriptions of theHenry’s coefficient are given in Lu (2004).

H2O-CO2 and H2O-CH4 systems. The H2O-CO2 system is a typical aqueous system foundin many geothermal systems on Earth, and it could be present on Mars (Lyons et al. 2005, Okubo& McEwen 2007, Than 2007) or other planetary bodies, such as Enceladus as discussed below.

Several experimental studies on the solubility of CO2 in water were carried out by previousresearchers (Ellis & Golding 1963, Malinin 1974; a review of work is in Lu 2004). Using Malinin’scorrelation, Lu (2004) analyzed the thermodynamics of the H2O-CO2 system and generatedsolubility curves versus temperatures and total system pressures (Figure 6a,b). As is well known,and can be seen from these figures, at temperatures less than 160◦C, the CO2 solubility increaseswith decreasing temperature, but at temperatures above 160◦C, the CO2 solubility increases withincreasing temperature.

Aqueous methane systems H2O-CH4 may exist on Mars (Lyons et al. 2005). The solubility ofCH4 is much less than that of CO2 (Figure 6c). For example, at 50◦C and 300 bar, the solubility ofCO2 is approximately 20 times that of CH4. The lowest solubility region of CH4 is at approximately100◦C, with a wide, flat curve range, whereas CO2 has its lowest solubility at approximately 150◦C,with a narrower, flat curve range.

Based on recent spectroscopic detections of CH4 in the Martian atmosphere, Lyons et al. (2005)have reexamined the role of C-O-H fluids in the Martian crust. They concluded that there may bereservoirs of H2O-CH4 fluids at depths less than 9.5 km in the crust, with H2O-CO2-rich fluids

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0 50 100 150 200 250 300 3500

20,000

40,000

60,000

80,000

100,000

120,0000.14

0.12

0.1

0.08

0.06

0.04

0.02

0

300 bar200 bar100 bar

1000 bar

600 bar

900 bar800 bar700 bar

500 bar400 bar

A

B

C

0 50 100 150 200 250 300 3500

0.10

0.20

0.30

0.40

200 bar

1000 bar800 bar600 bar400 bar

A

B

C

50 bar

20 40 60 80 100 120

2000

4000

6000

8000

10,000

10 bar (total pressure)

8

6

4

2

1

Q

R

a b

c

Solu

bilit

y of

CO

2 (m

ole

frac

tion

)So

lubi

lity

of C

O2 (

mol

e fr

acti

on)

Solu

bilit

y of

CO

2 (pp

m)

Temperature (°C)

Temperature (°C)

Temperature (°C)

Solu

bilit

y of

CH

4 (m

ole

frac

tion

)

Solu

bilit

y of

CH

4 (pp

m)

Figure 6(a) Solubility of CO2 at different pressure and temperature conditions. (b) Solubility of CO2 in alow-temperature and low-pressure region. (c) Solubility of CH4 at different pressure and temperatureconditions. See text for discussion of lettered points.

at greater depths. This suggests two possible scenarios for fluid migration in the Martian crust:(a) upward flow of H2O-CO2 mixtures from a deeper reservoir, and (b) upward flow of H2O-CH4

mixtures from shallow reservoirs. The existence of these reservoirs has strong implications forplanetary degassing because the volatiles exsolve as the fluids ascend (Lyons et al. 2005). WhenH2O-CH4-rich fluids decompress and cool, they may encounter a two-phase region.

Slow ascent of the shallow fluids would allow degassing of a vapor-rich phase, providingmethane to the atmosphere (Figure 6c). The illustrative process line A-B-C in this figure corre-sponds to the initial and final temperatures and lithostatic pressures of Lyons et al. (2005), withan initial concentration of CH4 of 0.1 mole fraction. Upon ascent from the reservoir, the flowremains liquid with the gas in solution from A to B, ∼8.5 km, and 320◦C, but degassing occurs

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at B as proposed by Lyons et al. (2005). From B to C, the flow is a liquid-gas two-phase mixture,unless flow separation occurs, for example, the liquid stagnates and the gas rises as a separatephase.

Upward flow from a H2O-CO2-rich reservoir at a depth greater than 9.5 km could also producetwo-phase flow as shown in Figure 6a, where the conditions for the process line A-B-C were againbased on Lyons et al. (2005). Degassing occurs at point B at shallower and cooler conditions,∼6 km and 210◦C, than for the H2O-CH4-rich fluids.

Sound speed in multicomponent liquid-gas systems. Sound speeds in liquid-gas systems suchas H2O-CO2 or H2O-CH4 are low because they have the increased compressibility of the gassyliquid, but the sound speeds do not drop as low as for those in liquid-vapor systems becausethere is no mass transfer between the two phases at the time scales of acoustic wave propaga-tion. Therefore, to first order the H2O-air system provides an illustration of the magnitude ofthe multiphase effect. These systems mimic the nonequilibrium H2O-steam system shown inFigure 5a (top) (details in Kieffer 1977) and therefore we do not discuss this further here.

Multicomponent, Multiphase Solid-Gas Systems

At lower temperatures in multicomponent systems, the solid ice phase may contain dissolved gases.In general, these systems are called clathrates, from Latin and Greek roots meaning lattice. Whenthe ice matrix is H2O they are called gas hydrates; we use the terms interchangeably. With specificgases they are called, for example, methane hydrate, carbon dioxide hydrate, etc., and if there aremultiple gases, simply mixed hydrate or mixed clathrate.

Forty seven natural-gas hydrate locations are known on Earth, in permafrost and in oceansediments (Sloan 1998), and they are potential natural gas resources. Gas hydrates can exist onother planetary bodies where pressure, temperature, and other conditions are favorable for theirformation (Miller 1961, Kargel & Lunine 1998, Gautier & Hersant 2005, Hand et al. 2006). Thepossible existence of CO2 clathrates on Mars has been proposed by many researchers (Miller &Smythe 1970, Longhi 2006, Kargel et al. 2000), and it has been suggested that methane clathratesmay supply CH4 to the Martian atmosphere (Prieto-Ballesteros et al. 2006).

The pressure-temperature decomposition (Pd-Td) curves for clathrate stability and decompo-sition depend on the nature of the gas and gas mixture. The phases involved are hydrate (H), ice(I), and gas (G) (i.e., the decomposition reaction is H = G + I). The decomposition curves (Pd-Td

curves) of the N2, CH4, and CO2 gas hydrates, together with the data sources used to generatethe curves, are shown in Figure 7a.

Energy balance for the decomposition of a clathrate reservoir. In the section on T-sdiagrams, above, we considered the energy and mass balances for a single-component ice sourcefor the plume of Enceladus. Consider a parallel analysis for decomposition of a clathrate reservoirto produce the plume. If a clathrate reservoir with temperature lower than 273 K is pierced by afracture (or well), the clathrates will decompose (hydrate → H2O ice + gases) and material willleave the system.

To analyze this, consider the energy balances (Figure 7b) with flow through a control volume,CV. Denoting (ha,d −hr) as �Hdis, the enthalpy of dissociation of the clathrate hydrate (Yamamuro& Suga 1989, Kang et al. 2001), the energy balance of the system is

Qin + Qsource − Qout = mout,g�Hdis + mout,vLsub + moutPr

ρr. (12)

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Longhi (2005)Miller & Smythe (1970)

Sloan (1990)

Sloan (1990)

Sloan (1990)

Kuhs et al. (1997)

Makogon & Sloan (1994); also in Sloan (1998)

ba

80 100 120 140 160 180 200 220 240 260 280

Mixed hydrate

Vout

Clathrate reservoir

H2O/CO2ice cap

System boundary(control volume)

out

+ gz2V2

h +

(hr, Tr, ur, r)ρ

ha,d

Plume

z

3 x 102

102

101

100

10–1

10–2

10–3

10–4

10–5

N2 hydrate

CH4 hydrateCO2 hydrate

(I + G)

(H)

Qout.

Qin.

Qsource

.

P d (b

ar)

Td (K)

mout.

Figure 7(a) Phase equilibrium (H = I + G) for N2, CH4, CO2, and mixed gas clathrates at temperature below 273 K. The composition of themixed gas clathrate is that of the plume emanating from Enceladus’ south pole in the first Cassini encounter (Waite 2006).(b) Schematic diagram of the derivation of energy balance of a clathrate reservoir with outgoing fluids due to decomposition.

The left side of Equation 12 is the net heat gained by the system, whereas the right side is theheat that is required to dissociate the clathrate hydrate and to accelerate the outgoing fluids, butincludes a term for possible sublimation of ice as well. Assuming steady-state conditions, the leftside of Equation 12 is equal to the right side.

The rate of heat leaving the system (Qout) from the south pole of Enceladus is estimated tobe 3 to 7 GW (Spencer et al. 2006). Based on the mole fractions of N2, CH4, and CO2 (Waiteet al. 2006) and the values of enthalpies of dissociation of pure N2, CH4, and CO2 hydrates (Kanget al. 2001, Yamamuro & Suga 1989), the enthalpy of dissociation of the mixed hydrate �Hdis is�Hdis = 890 kJ kg−1 of the gas mixture.

To estimate the magnitude of the terms on the right side, we use an approximate densityρr = 1000 kg m−3. For an order-of-magnitude reservoir pressure we use Pr = 5 bar (Kieffer et al.2006). The molar ratio of the mixture of gases (N2, CH4, and CO2) to H2O vapor is approximately1:10 (Waite et al. 2006), which is equivalent to the mass ratio of 1:7 for the given molar fractionsdetected. The total mass flow rate of H2O in the plume is mout � 100 kg s−1 (Hansen et al. 2006),thus the mass flow rate of the gas mixture (N2, CH4, and CO2) is mout,g ∼ 15 kg s−1. The firsttwo terms on the right side of Equation 12 are ∼1 × 10−2 GW and 2 × 10−4 GW, very smallcompared with the observed energy flux.

Taking the mass rate of the sublimated H2O vapor, mout,v, as 350 kg s−1, and the latent heat forthe sublimation of ice Lsub = 2800 kJ kg−1, we can estimate the third term as 0.84 GW. The threeterms are ∼1 GW, small compared with the radiated heat. This led Kieffer et al. (2006) to proposethat the plume represents a small leak on a massive advection system in which large amounts ofvapor are condensing on the walls releasing latent heat, which contributes an additional radiatedenergy component.

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Sound speeds in multicomponent, multiphase solid-gas systems. We conclude our reviewof sound speeds in multicomponent, multiphase systems by summarizing the effects of particlesentrained in gases on the sound speed. We do not attempt to cover densely packed solid-gasmixtures, such as granular materials (see Jaeger et al. 1996 for a review of the behavior of thesecomplex substances), or the more subtle effects of embedded particles or bubbles in solid H2O.

Two effects are important in determining the sound speed of gas-solid mixtures: mass loadingof the gas by the particles and heat transfer from the particles to the gas. To first order, it is assumedthat the particles move with the gas, the so-called pseudogas approximation (Wallis 1969b). For aperfect gas, Equation 7 gives the sound speed as

c =(

γ RTM

) 12

, (13)

where γ is the isentropic exponent for the gas, and M is the molecular weight. For a perfect gas,the sound speed is a direct function of temperature and γ , and inversely proportional to molecularweight.

If the mass ratio of solids/gas is denoted by m, then the effect of the mass loading alone inthe pseudogas approximation is to change R/M from that of the gas to R/M(1+m) (Wallis 1969b,equation 2.17), that is, mass loading mimics increasing molecular weight and decreases the soundspeed. If heat transfer is important, then γ , the ratio of heat capacities, is changed to

γ = cp,gas + mcs

cv,gas + mcs, (14)

where cp,gas and cv,gas are the heat capacities at constant pressure and constant volume for the gas,and cs is the averaged heat capacity for the solid phase. For significant mass loading, γ tends towardunity. These effects are shown in Figure 8a, which compares the behavior of an H2O-solid dusty

0.1 1 10 100

104

103

102

101

400 K600 K1200 K

400 K600 K750 K

ba

SF6 (vapor)-solid mixture

H2O (vapor)-solid mixture

1200

1000

800

600

400

200

1300

1100

900

700

500

30000 2 4 6 8 10

Tem

pera

ture

(K)

Tem

pera

ture

(°C)

Entropy (Joule g–1 K–1)

1Φ

2Φ1 bar

1 ba

r

2510

0500

25

100

(L + V)

(V)(L)

m = 25m = 4m

= 0.2m

= 0

Critical pointCritical pointCritical point

Kimberlite

c (m

/s)

m (mass ratio solids : vapor)

Figure 8(a) Dependence of sound speed on mass loading, temperature, and composition. (b) Temperature-entropy phase diagram for H2Ocomparing the thermodynamic history of the H2O during isentropic ascent of the fluid alone (m = 0 path) with isentropic ascent of amixture containing weight ratio, m, of solids to vapor (reproduced from Kieffer 1984). L, liquid; V, vapor; S, solid.

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gas with that of a heavy-molecular-weight gas SF6. SF6 could provide a good analog for dusty gasbehavior for m > 5.

Particles entrained in a gas also affect the thermodynamics of an ascending gas phase, as isillustrated in Figure 8b (compare with Figure 3). If no heat is transferred from the particles tothe gas phase, the gas expands adiabatically and isentropically as discussed in ThermodynamicProcesses and Appropriate Phase Diagrams for Analysis, above (m = 0 path in Figure 8b). Inmany cases, the initial conditions are such that adiabatic expansion would cause the fluid to expandfrom the vapor or supercritical initial state into the two-phase field (e.g., m = 0, or m = 0.2 inFigure 8b). However, if sufficient particles transfer heat to the vapor, the expansion is no longernearly adiabatic and the thermodynamic paths veer toward the vapor field. The amount of heatavailable for transfer to the gas depends on the mass loading (m). For high values of mass loading(m > 10), the expansion tends to be more isothermal than isentropic, and two-phase conditionsare avoided.

Choked versus Unchoked Flow

When a fluid flows through a pressure gradient in pipes or fractures, the flow velocity might exceedthe sound speed. Normally, for pure liquids such situations do not arise in geologic settings becausethe sound speeds are a kilometer to a few kilometers per second.

However, the situation is very different for liquid-gas or gas-solid systems. Fluids easily attainthe low velocities (meters to a few hundred meters per second) in a number of situations, particu-larly in geothermal and volcanic settings. In pipe, fracture, or nozzle flow, when the fluid velocityreaches sonic velocity, the flow chokes and mass flux is limited. In such settings, full decompressionto the ambient pressure may occur downstream of the choke point, which is often underground(e.g., in plumes). It has been suggested that Old Faithful geyser is choked (Kieffer 1989); Plinianvolcanic eruptions and steam blasts are choked (Buresti & Casarosa 1989, Mastin 1995); and thatchoked flow played a role in triggering long-period seismic events at Redoubt Volcano, Alaska(Morrissey & Chouet 1997), and in eruptions on Mars (Wilson & Head 2007). In such cases, flowin the conduit only partially decompresses erupting material, and the remaining decompressionto ambient pressure occurs downstream of the choke point in an underexpanded (overpressured)plume (Kieffer 1982).

Determination of choked conditions is crucial for understanding the relation of plumes to thereservoirs and conduits that feed them. Transition from subsonic conduit flow to supersonic plumeflow is complicated and includes many nonlinear processes.

DYNAMICS AND NUMERICAL SIMULATIONS:EXAMPLES OF GAS-LIQUID SYSTEMS

Thermodynamic analysis as reviewed above is very useful in defining relations between ini-tial and final states and for estimating flow properties given certain thermodynamic paths. Butdynamical models are required when dealing with materials in motion for determining vari-ous properties (velocity, pressure, temperature, and density, etc.) of the material as functionsof space and time. The coupling of the thermodynamic equations of state with conservationlaws for motion results in complex equations that almost always require numerical solution.Compared with the number of single-component models, there are relatively few multiphase,multicomponent models, and they always contain many embedded assumptions. Here and inthe next section, we illustrate several specific examples for both gas-liquid flows and gas-solidsystems.

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Fluid flow in fractures may be steady or not. Steady fracture flow is indicated by the dischargesof liquid hot springs, or vapor/wet vapor fumaroles. However, two-phase flows may be unsteadyfor at least two reasons. They may break down internally into various flow regimes (homogeneous,bubbly, slug, annular, mist, etc.), which we do not consider here. They may also break down becauseheat and mass discharges are not balanced, for example, the cyclic phenomenon of geysering, whichis observed both naturally and industrially (Rinehart 1980a,b; Jiang et al. 1995). Understandingsources of cyclicity in fractures that feed plumes, and separating these effects from atmosphericprocesses that may contribute to plume unsteadiness, is important for interpreting plumes andtheir relation to the subsurface.

We use the words vapor and gas very specifically in this section: gas means a nonconden-sible constituent of the liquid, and vapor means the vapor phase, which may contain gas (e.g.,CO2) in addition to the condensable phase (e.g., the H2O). Gas-rich liquids can bubble3 in re-sponse to pressure or temperature changes at fluid temperatures well below the boiling pointof pure water. For example, in an H2O-CO2 system (Figure 6b), at 1 bar pressure pure wa-ter boils at 100◦C, but aqueous CO2 solutions will bubble (degas, fizz, flash) at only 70◦C(point R) if the CO2 concentration is greater than 500 ppm. At 8 bar, pure water boils at170◦C, but if the CO2 concentration is more than 4000 ppm, the solution bubbles at only90◦C at the same pressure (point Q). Unlike pure H2O systems, the composition of the bub-ble is determined by gas solubility and the latent heat associated with partial pressure of H2O isnegligible.

The exsolution of gases from gas-water aqueous systems can cause hazardous engineeringsituations and the gas plays a crucial role in the low-temperature geysers. To analyze this complexsystem and to describe the flow behavior, it is necessary to develop a multiphase, multicomponentflow model, with mass transfer between the phases being considered.

Partial formulations of the flow equations have been provided for pure water systems withphase change or water-air mixtures without mass transfer at the two-phase interface (e.g., Wallis1969c). However, for systems with dissolved gases, a source term must be introduced into thegoverning equations to account for the mass exchange between the phases and components. Thiswas provided recently for geysering flow (Lu 2004, 2006).

Conservation Equations

For one-dimensional, time-dependent flow in a vertical pipe, the mass conservation equations forthe liquid and gas phases are

∂

∂t[ρL(1 − α)] + ∂

∂z[ρL(1 − α) vL] = SLG (15)

and∂

∂t(ρG α) + ∂

∂z(ρG α vG) = −SLG, (16)

where ρ, α, and v are density, void fraction, and velocity, respectively; t is time; and SLG

is the total mass exchange rate (including CO2 and water vapor) between liquid and gaseousphases. Expressions for the mass exchange SLG can be found in Lu (2004) and Lu et al. (2006). The

3We avoid using boiling to describe this phenomenon because boiling usually refers to a rapid vaporization of a pure liquid,which typically occurs when a liquid is heated to its boiling point. We introduce the equivalent words bubbling, fizzing, orflashing.

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momentum conservation equation is

∂P∂z

=(

∂P∂z

)G

+(

∂P∂z

)A

+(

∂P∂z

)F, (17)

where(

∂P∂z

)G,

(∂P∂z

)A, and

(∂P∂z

)F are the three components of pressure gradients of gravitation,

acceleration, and friction, respectively (Wallis 1969b).These equations differ from the equations for boiling flow in a pipe discussed by Wallis with

regard to the mass exchange between the phases and components. Additional equations requiredto close the model include (a) a drift flux model for coupling the velocities of the gas and liquidphases for bubbly and slug flow regimes, such as that proposed by Zuber & Findlay (1965) orWallis (1969a,d); (b) a temperature profile along the flow path or energy equation for determiningthe temperature profile; (c) a pressure distribution along the flow path at initial state; and (d )equations for determining the thermophysical and fluid properties.

Model Implementation and Verification: Te Aroha Geyser, New Zealand

For verification of this formulation, Lu et al. (2004, 2006) carried out experimental and numericalstudies for flow in a vertical geothermal well at Te Aroha, on the north island of New Zealand.The well is 70 m deep and 10 cm in diameter. The temperature at the bottom of the well isapproximately 90◦C and approximately 70◦C on the top. The fluid in the well is water with highlevels of dissolved CO2 (∼3000 ppm). The well can be shut down, but when it is fully opened tothe atmosphere, the flow is not steady, but cyclic, with a small geyser several meters high eruptingapproximately every 12 min.

For the verification, the inlet boundary at the bottom was assumed to have a constant flux. Theoutlet boundary condition at the wellhead is defined by atmospheric pressure, 1 bar at sea level onEarth, i.e., the flow was not choked. The initial pressure distribution along the well is determinedfrom the momentum balance, assuming that pure water is flowing up the well.

The thermodynamic processes that occur as the fluid flows from the bottom of the well (pointQ) to the top (point R) are shown by the path Q-R in Figure 6b. As the fluid rises up the well,the decompression causes the CO2 solution to become supersaturated and so the CO2 comes outof solution. This degassing process forms a cyclic two-phase flow.

Close agreement was obtained between the analytical and measured results with regardto the major variables, such as void fraction and pressure variations at different depths(Figure 9a,b) of the well, cycle period, water level variations at the top part of the well, andflash point (the start of degassing) variations at the lower part of the well (Lu et al. 2005, 2006).Sensitivity studies showed the range of conditions under which the flow would and would not besteady.

The study showed that the supersaturated CO2 solution, instead of superheated water, causesbubble generation and intermittent flashing, which induces geysering in the well. The sensitivitystudies showed that a larger inflow rate does not make a positive contribution in generatinggeysering flow. On the contrary, too large an inflow rate may cause the CO2-driven geysering todisappear and be replaced by a steady gas-liquid two-phase flow.

Geysering is a unique phenomenon of transient two-phase flow with intermittent dischargecharacteristics that happen only when some key parameters have particular values that match oneanother. In pure H2O systems, boiling conditions determine the conditions; in multicomponentsystems, gas exsolution determines them.

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An Example of How Gravity Conditions Influence the Behavior of CO2-H2OAqueous Flow: Earth versus Mars

There is strong evidence for fracture-controlled paleofluid flow on Mars (Okubo & McEwen2007). How would Te Aroha behave if it were erupting on Mars, which has approximately one-third the gravitational field of Earth? Would it be geysering flow?

If the Martian atmospheric pressure was lower than ∼1 bar when the flow occurred, the properanalog would be discharging terrestrial geothermal wells. The wellhead pressures of open flowinggeothermal wells are often tens of bars. In this case, the flow is choked at the exit plane and aflared, noisy supersonic plume forms downstream of the exit plane (Armstead 1983, p. 123, plate 9).The noise results from both edge noise and internal shock waves in the tulip-shaped plume. Byanalogy, we can assume that flow from a fracture on Mars would be choked if the atmosphere wereat low pressure unless the flow itself erodes the conduit (Kieffer 1982).

To isolate the effect of gravity alone, we assume that in the past the atmosphere may have beenat pressures of approximately 1 bar. The key parameters determining flow rate are then gravity andinflow rate. If the conduit is the same length as that of Te Aroha (70 m), the reservoir pressure is ap-proximately one-third of that on Earth. After a transient initial phase of approximately 2000 s, thegeysering behavior stops and the flow becomes a continuous gassy-liquid discharge (Figure 9a,b,

ba

dc

1.0

2.0

3.0

4.0

5.0

6.0

0 500 1000 1500 2000 2500 3000Time (s)

0 500 1000 1500 2000 2500 3000Time (s)

Pres

sure

var

iati

ons

(bar

)

50 m (Earth)

50 m (Earth)

50 m (Mars)

50 m (Mars)

20 m (Mars)

20 m (Mars)

30 m (Mars)

30 m (Mars)

40 m (Earth)

40 m (Earth)

30 m (Earth)

30 m (Earth)

20 m (Earth)20 m (Earth)

0

0.10

0.20

0.30

0.40

0.50V

oid

frac

tion

0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 70001.5

2.5

3.5

4.5

5.5

6.5

150 m

120 m

90 m

60 m

0

0.1

0.2

0.3

120 m

90 m

60 m

40 m (Mars)

40 m (Mars)

Pres

sure

var

iati

ons

(bar

)

Time (s)

Voi

d fr

acti

on

Time (s)

150 m

Figure 9Numerical simulation and comparison of terrestrial and Martian gravity conditions on the behavior of CO2-H2O aqueous flow:(a) pressure variations and (b) void fractions. Numerical simulation of pressure variations and void fractions on Mars (scenario 2, seetext): (c) pressure variations and (d ) void fractions.

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curves marked with Mars). The void fraction is, however, greater for the Martian analog because ofthe lower gravitation. On Earth, gravitational pressure suppresses bubble formation below a certainlevel (the flash point, the start of degassing) that is changing during a cycle (Lu et al. 2005, 2006).The reduced hydrostatic pressure on Mars results in (a) more CO2 released from the solution atdifferent depths in a upward flow, and (b) the same mass of gases in fluid occupying more volumethan on Earth and hence a bigger value of void fraction as shown in Figure 9b. As a result, the inter-mittent discharge (geysering) on Earth has turned into a continuous two-phase discharge on Mars.

However, if the geyser were extended to 210 m depth so that the reservoir has the samehydrostatic pressure as Te Aroha, a different phenomenon is observed (Figure 9b,c). The geyseringphenomenon remains under Martian conditions, but the period changes to 37 min instead of12 min. The hydrostatic pressure matches the given CO2 concentration and the inflow rate andhence causes geysering to happen again. The hydrostatic pressures at 60 m, 90 m, 120 m, and150 m on Mars are almost equal to those on Earth at 20 m, 30 m, 40 m, 50 m, respectively, i.e.,the rate of change of hydrostatic pressure is approximately one-third of that on Earth. The timefor the whole changing process in each cycle (period) mainly depends on the rate of change ofhydrostatic pressure. The lower rate of change in pressure results in a longer cycle period. Weconclude that geysering flow is as likely to have occurred on Mars as on Earth.

DYNAMICS AND NUMERICAL SIMULATIONS: EXAMPLESOF GAS-ICE SYSTEMS (CLATHRATES)

Because sublimation of ice in the single-component H2O system under terrestrial and planetaryconditions has been the subject of much literature (see, for example, Brown & Cruikshank 1997,Brown et al. 1988, and references therein), we concentrate here on reviewing the dynamics ofmulticomponent ice-gas systems, namely, clathrates. No comprehensive mathematical or numer-ical model has yet been formulated for simulating the clathrate reservoir degassing process underthe low-pressure and low-temperature conditions of other planets. Models have been developedonly in the past few years for terrestrial degassing of clathrates. They are typically 1D, or ax-isymmetric, models for simulating methane hydrate reservoir dissociation and predicting naturalgas production either by heating or depressurizing reservoir hydrates ( Ji et al. 2001, Goel et al.2001, Moridis et al. 2004, Sun et al. 2005). All models are for temperatures greater than 273 K,where the dissociation of the methane hydrate causes the hydrate to decompose into methane gasand water. Effects that are considered in the models include reservoir temperature and pressure,zone permeability, heat transfer into the decomposing zone, kinetics of decomposition, thermalparameters of the materials, and flow regime.

The only reservoir on Earth for which there are data to validate such models is the Messo Yakhifield in Siberia (Makogon 1997), and refinements of the available data in planetary settings arenot sufficiently abundant to allow verification for extrapolations. More importantly, the terrestrialclathrates decompose from hydrate into gas plus liquid water, whereas the planetary clathrates areso cold that decomposition would be into gas plus ice, for which there are no formulations. Toexamine clathrate decomposition under cold, low-pressure conditions, we modified the model ofJi et al. (2001) to apply to decomposition of the clathrate into ice and gas. We chose this modelbecause of its simplicity and because it is easy to use for a parametric study when extended tolow-temperature conditions, but we emphasize that many effects not considered in the modelneed to be included in future studies.

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Mas

s flux

of g

as m

ixtu

re (k

g/m

2 –s)

Years

Days

Mas

s flux

of g

as m

ixtu

re (k

g/m

2 –s)

Years

Within 365 days (porosity = 0.2)

After 1 year (porosity = 0.2)

Zone 1(Gas + Water)

Zone 2(Hydrate + Gas)

Well

Decomposition front Flow

Impermeable rock

TdPdγ

x(t)l

0 10 20 30 40 50 60 70 80 90 100

10–10 60 120 180 240 300 360

10–2

10–3

10–4

10–5

10–6

10–7

10–3

10–4

10–5

10–6

10–7

10–8

After 1 year (porosity = 0.3)

After 1 year (porosity = 0.1)

0 10 20 30 40 50 60 70 80 90 100

a

cb

(Pr = 5 bar, Tr = 190 K, Pg = 2.5 bar)

Pr = 41 bar, Tr = 250 K, Pg = 20.5 bar

Pr = 1 bar, Tr = 160 K, Pg = 0.5 bar

Pr = 0.1 bar, Tr = 135 K, Pg = 0.05 bar

Figure 10(a) Schematics of one-dimensional hydrate reservoir degassing model on Earth (from Ji et al. 2001). (b) Calculated degassing fluxesowing to decomposition of a hypothetical clathrate hydrate reservoir on Enceladus. Fluxes of the nominal reservoir (top axis in daysapplies only to the top curve) and sensitivity analysis of porosity (dashed curves). (c) Sensitivity analysis for different reservoir pressureand temperature conditions.

Conservation Equations

Consider natural gas production from methane hydrate due to depressurization of a porouspermeable clathrate zone by injection of a production well into the zone (Figure 10a) (see detailsin Ji et al. 2001). The governing equations of mass, momentum, and energy, including heat transferand the effects of throttling (isenthalpic process), were linearized by Ji et al. (2001) to give a setof self-similar solutions for temperature and pressure distributions in the reservoir. This resultsin a system of coupled algebraic equations for the location of the decomposition front expressed by

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the distance of the decomposition front from the well, l(t), and the temperature (Td) and pressure(Pd) at the front. The distance from the well l(t) is

l(t) = √γ t, (18)

where γ is a constant in m2/s.For a given reservoir pressure and temperature and the pressure at the well, Td, Pd, and γ can

be determined through an iterative scheme by simultaneously satisfying three equations: (a) equi-librium pressure and temperature relation on the decomposition front (i.e., Td-Pd decompositioncurve), (b) linearized energy equation at the decomposition front, and (c) the combined equation ofmomentum and mass balances at the decomposition front ( Ji et al. 2001). From Td, Pd, and γ , thepressure and temperature distributions in the reservoir are calculated from the self-similar solu-tions of the linearized momentum and energy equations by satisfying the corresponding boundaryconditions.

For Enceladus, the question is, “Can this process produce the observed mass fluxes?” Thevolumetric production rate Qv (degassing rate) and its time dependence are ( Ji et al. 2001)

Qv(t) = k1

μg

P2d − P2

g

Pg

1erfα1

12√

πχ1t, (19)

which depends on the phase permeability of gas in zone 1 (dissociated hydrate zone) k1, thepressure of the gas in the fracture Pg is the pressure of the gas in the well, the viscosity of the gas,μg. Two variables that appear here are mostly introduced for ease of calculation:

χ1 = k1Pg

�1μg(20a)

and

α1 =√

γ

4χ1, (20b)

one of which includes the porosity on the reservoir, �1. Sensitivity of the natural gas productionfrom hydrate by depressurization to variations of reservoir parameters can be found in Ji et al.(2001). The mass flux Qm(t) (degassing rate) of the gas mixture is then Qm(t) = Qv(t)/ρg.

Model Implementation: Plumes on Enceladus

The general principles of clathrate degassing by depressurization for Enceladus were illustratedby Kieffer et al. (2006) using an extension of the Ji et al. (2001) model. The model was extendedby Fortes (2007) and Halevy & Stewart (2008). The gas composition was modified to reflect themixed gases of the plume, using a simplified mixing law. Decomposition into liquid + gas assumedby Ji et al. was changed to reflect decomposition into ice + gas. The degassing fluxes Qm werecalculated for two plausible reservoirs: one shallow and cool (5 km, 5 bar, 190 K) and one deeperand warmer (40 km, 41 bar, 250 K) (Figure 10b,c). The pressure in the fracture, Pg, was assumedto be half of the reservoir pressure based on the assumption that when gas flows in complicatednetworks, pressure conditions are commonly determined by narrow constrictions where the flowchokes to sonic conditions of pressure, temperature, and flow velocity (Mach number = 1) (Kieffer1989). Sensitivity studies, and the assumed role of entrained ice particles, were considered.

The time-variable flux for these, shown in Figure 10b,c, is of the order of that measured in theplume, 10−7 to 10−6 kg s−1 m−2. The rapid time variability implied by the process (opening andclosing of small fractures) is similar to the time scale of variabilities observed by the instrumentson Cassini (<1 month). Much refinement of models needs to be done as data provide more

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constraints on interior processes on Enceladus (Kieffer & Jakosky 2008), but results from thesecalculations suggest that degassing of clathrates on Enceladus can produce gases at roughly the ratesobserved.

SUMMARY POINTS

1. Mixed phases in a single-component system can have complicated properties, such assound speed in boiling liquids or aerosols. These properties complicate fluid motions.The flow of H2O as a pure liquid phase or a pure vapor phase alone in a conduit is muchsimpler than the flow of boiling or gassy fluid, which can choke at sonic velocity, and candevelop flow instabilities such as geysering, under certain conditions of heat and masstransport. Serious errors can be made by assuming that a partial pressure due to an H2Ocomponent is equal to the total pressure of a mixture.

2. Multiphase, multicomponent systems are inherently more complicated than multiphase,single-component systems. Counterintuitive phenomena may occur, such as gases beingmore soluble in H2O ice (clathrate) than in H2O liquid. Even phenomena that super-ficially appear to be similar can have significant differences when examined in detail.Geysering, for example, in a cool CO2-driven system is induced by a periodic bubblegeneration that is caused not by the superheated water, as it is in single-componentsystems, but by the supersaturated CO2 solution.

FUTURE ISSUES

1. The differences between multiphase, multicomponent systems and multiphase, single-component systems need to be kept in mind in quantitative interpretations of multicom-ponent systems; for example, in efforts to infer reservoir characteristics on the planetsor moons from surface or plume observations. Resolution of the controversy about thenature of the reservoir feeding Enceladus’ plume depends on accounting for all of thegas components (CO2, N2, CH4, plus trace simple organics) as well as the phases.

2. Creation of a thermodynamic database that would allow the exploration of the variousthermodynamic processes discussed in the first part of this paper and the various dy-namic processes discussed in the latter part is essential for understanding the processesin the colder planetary settings of Mars and of some moons. This includes the standardthermodynamic properties and their derivatives, the properties of metastable H2O attemperatures below 0◦C, the standard thermodynamic properties of clathrate phases, aswell as data for clathrates in cold saline environments.

3. Extension and development of degassing models to colder planetary conditions is muchneeded, as are kinetic data.

4. Thinking and teaching about multicomponent systems is needed to educate future Earthand planetary scientists about the similarities and dissimilarities of multicomponent andsingle-component systems. The use of single-component approximation can lead to con-clusions that are not only quantitatively wrong, but also qualitatively wrong. Degassing,depressurizing, and flow instabilities occur for fundamentally different reasons in the twosystems.

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DISCLOSURE STATEMENT

The authors are not aware of any biases that might be perceived as affecting the objectivity of thisreview.

ACKNOWLEDGMENTS

This research was supported by NASA NNX06AJ10G and NASA NNX08AP78G as well asCharles R. Walgreen, Jr. Endowed Chair funds to SWK.

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Annual Reviewof Earth andPlanetary Sciences

Volume 37, 2009Contents

Where Are You From? Why Are You Here? An African Perspectiveon Global WarmingS. George Philander � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Stagnant Slab: A ReviewYoshio Fukao, Masayuki Obayashi, Tomoeki Nakakuki,and the Deep Slab Project Group � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �19

Radiocarbon and Soil Carbon DynamicsSusan Trumbore � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �47

Evolution of the Genus HomoIan Tattersall and Jeffrey H. Schwartz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �67

Feedbacks, Timescales, and Seeing RedGerard Roe � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �93

Atmospheric Lifetime of Fossil Fuel Carbon DioxideDavid Archer, Michael Eby, Victor Brovkin, Andy Ridgwell, Long Cao,Uwe Mikolajewicz, Ken Caldeira, Katsumi Matsumoto, Guy Munhoven,Alvaro Montenegro, and Kathy Tokos � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 117

Evolution of Life Cycles in Early AmphibiansRainer R. Schoch � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 135

The Fin to Limb Transition: New Data, Interpretations, andHypotheses from Paleontology and Developmental BiologyJennifer A. Clack � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 163

Mammalian Response to Cenozoic Climatic ChangeJessica L. Blois and Elizabeth A. Hadly � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 181

Forensic Seismology and the Comprehensive Nuclear-Test-Ban TreatyDavid Bowers and Neil D. Selby � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 209

How the Continents Deform: The Evidence from Tectonic GeodesyWayne Thatcher � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 237

The Tropics in PaleoclimateJohn C.H. Chiang � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 263

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AR374-FM ARI 27 March 2009 18:4

Rivers, Lakes, Dunes, and Rain: Crustal Processes in Titan’sMethane CycleJonathan I. Lunine and Ralph D. Lorenz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 299

Planetary Migration: What Does it Mean for Planet Formation?John E. Chambers � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 321

The Tectonic Framework of the Sumatran Subduction ZoneRobert McCaffrey � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 345

Microbial Transformations of Minerals and Metals: Recent Advancesin Geomicrobiology Derived from Synchrotron-Based X-RaySpectroscopy and X-Ray MicroscopyAlexis Templeton and Emily Knowles � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 367

The Channeled Scabland: A RetrospectiveVictor R. Baker � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 393

Growth and Evolution of AsteroidsErik Asphaug � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 413

Thermodynamics and Mass Transport in Multicomponent, MultiphaseH2O Systems of Planetary InterestXinli Lu and Susan W. Kieffer � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 449

The Hadean Crust: Evidence from >4 Ga ZirconsT. Mark Harrison � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 479

Tracking Euxinia in the Ancient Ocean: A Multiproxy Perspectiveand Proterozoic Case StudyTimothy W. Lyons, Ariel D. Anbar, Silke Severmann, Clint Scott,and Benjamin C. Gill � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 507

The Polar Deposits of MarsShane Byrne � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 535

Shearing Melt Out of the Earth: An Experimentalist’s Perspective onthe Influence of Deformation on Melt ExtractionDavid L. Kohlstedt and Benjamin K. Holtzman � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 561

Indexes

Cumulative Index of Contributing Authors, Volumes 27–37 � � � � � � � � � � � � � � � � � � � � � � � � � � � 595

Cumulative Index of Chapter Titles, Volumes 27–37 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 599

Errata

An online log of corrections to Annual Review of Earth and Planetary Sciences articlesmay be found at http://earth.annualreviews.org

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