Post-solidification cooling and the age of Io's lava flows

Icarus 176 (2005) 123–137 www.elsevier.com/locate/icarus

Post-solidification cooling and the age of Io’s lava flows Ashley Gerard Davies ∗ , Dennis L. Matson, Glenn J. Veeder, Torrence V. Johnson, Diana L. Blaney Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109-8099, USA Received 16 August 2004; revised 7 January 2005 Available online 25 March 2005

Abstract The modeling of thermal emission from active lava flows must account for the cooling of the lava after solidification. Models of lava cooling applied to data collected by the Galileo spacecraft have, until now, not taken this into consideration. This is a flaw as lava flows on Io are thought to be relatively thin with a range in thickness from ∼1 to 13 m. Once a flow is completely solidified (a rapid process on a geological time scale), the surface cools faster than the surface of a partially molten flow. Cooling via the base of the lava flow is also important and accelerates the solidification of the flow compared to the rate for the ‘semi-infinite’ approximation (which is only valid for very deep lava bodies). We introduce a new model which incorporates the solidification and basal cooling features. This model gives a superior reproduction of the cooling of the 1997 Pillan lava flows on Io observed by the Galileo spacecraft. We also use the new model to determine what observations are necessary to constrain lava emplacement style at Loki Patera. Flows exhibit different cooling profiles from that expected from a lava lake. We model cooling with a finite-element code and make quantitative predictions for the behavior of lava flows and other lava bodies that can be tested against observations both on Io and Earth. For example, a 10-m-thick ultramafic flow, like those emplaced at Pillan Patera in 1997, solidifies in ∼450 days (at which point the surface temperature has cooled to ∼280 K) and takes another 390 days to cool to 249 K. Observations over a sufficient period of time reveal divergent cooling trends for different lava bodies [examples: lava flows and lava lakes have different cooling trends after the flow has solidified (flows cool faster)]. Thin flows solidify and cool faster than flows of greater thickness. The model can therefore be used as a diagnostic tool for constraining possible emplacement mechanisms and compositions of bodies of lava in remote-sensing data.  2005 Elsevier Inc. All rights reserved. Keywords: Jupiter; Io; Volcanism; Lava flow; Lava lake; Thermal; Modeling

1. Introduction Io is the most volcanically active body in the Solar System. It is important to understand the details of cooling of lava on Io in order to better characterize the role of volcanism in the transfer of heat from the interior to the surface. Several previous studies have derived ages for volcanic surfaces by applying cooling models to temperatures determined from remote observations of infrared flux. These models generally balance heat loss from the surface of a lava * Corresponding author. Fax: +1-818-393-4445.

E-mail address: [email protected] (A.G. Davies). 0019-1035/$ – see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2005.01.015

body with the supply of heat released from solidification and subsequent cooling which is conducted through a thickening solid crust. A model temperature therefore implies not only an age for the surface, but also the thickness of the crust that has formed. A feature of several of these earlier models (Davies, 1996; Howell, 1997; and Keszthelyi and McEwen, 1997: see Table 1) is a constant supply of latent heat from the solidifying lava. These “latent-heat dependent,” or LHD, models are versatile and can be used on both finite-thickness bodies (such as lava flows) in their active phase, and semi-infinite bodies, such as deep lava lakes. However, after a lava flow solidifies, these LHD models are no longer applicable. Use of these models for times after the lava has solidified would

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Table 1 Comparison of lava flow models and applications

Flow surface temperature

Head and Wilson, 1986 (1)

Carr, 1986

Davies, 1996a

Howell, 1997

(2)

(3)

Y

Y

Y

Y

Y

Y

Whole active flow area

(4)

Keszthelyi and McEwen, 1997 (5)

Davies, 1996,b Davies et al., 2001 (6)

Davies CASc This paper (7)

Y

Y

Y

Y

Y

Multiple units

Y

Y

Post-eruption cooling of flow with molten interior

Y

Y

Post solidification cooling Basalt

Y

Y Y

Y

Y

Y

Komatiite

Y

Y

Y

Y

Y

Y

a Davies (1996) used a shrinking isothermal area and an additional flow model component to fit IRTF data of the 1990 Io thermal outburst (see Veeder et al., 1994; Blaney et al., 1995). b Davies et al. (2001) used a combination of two Davies (1996) flow components to fit Galileo NIMS and SSI data of the Pillan 1997 eruption and Pele’s thermal emission. c Cooling after solidification, modeled by Carr (1986) and in this analysis. New work is highlited in grey boxes.

result in ages that are too old, crusts that are too thick, and inaccuracies in estimates of the rates of lava supply and in assessing the role of effusive activity in Io’s overall thermal budget. We add a cautionary note that from a mechanical point of view a lava flow may be taken to be “solid” when a mass fraction (>50% of crystals) is reached and the flow can no longer move. In this paper we are dealing with heat content and it is convenient to refer to a flow as being “solid” when 100% crystallization is attained, that is, when all of the latent heat has been exchanged. Thus, we refer to the chemical or mineralogical solidus rather than to the point at which the flow is mechanically “solid.” This distinction is not necessary for the topic treated in this paper. However, it is a critical distinction when calculating accurate run-out lengths for flows. An important factor affecting the utility of the previous models is the thickness of lava flows on Io. Recent analyses indicate that Io’s lava flows are relatively thin (see Table 2). Thin flows limit the time over which LHD models can be applied, indicating the need for a more complete consideration of cooling history after lava solidification. In this paper we present a model that can be applied after a lava flow has completely solidified, picking up at the point in the cooling process where the LHD models are no longer applicable. We call this new model ‘CAS,’ for cooling after solidification. CAS is described in Section 5, and in detail in Appendix A. The characteristics of CAS and previous models are compared in Table 1. We have used the CAS model to derive the cooling history of lava flows of different compositions during and after the solidification process. We then compare the results with the LHD (semi-infinite) case and discuss the consequences of the interpretation of ages and emplacement history of several eruptive events on Io.

2. Lava composition and volcanism on Io The surface temperature verses time profiles produced from all of the pre- and post-solidification cooling models are dependent on the input values for the thermal and physical characteristics of the lava and the eruption environment. Most importantly, lava composition must be specified. As shown in Table 3, high-temperature silicate melts (with surface temperatures in excess of 900–1000 K) have been detected many times on Io from Earth-based observations (e.g., Johnson et al., 1988; Veeder et al., 1994; Blaney et al., 1995; Davies, 1996; Stansberry et al., 1997; Howell et al., 2001), and from spacecraft. Voyager’s Infrared Imaging Spectrometer (IRIS) data indicated color temperatures as high as 854 K at Pele (Pearl and Sinton, 1982). Carr (1986) modeled IRIS data as the combination of radiation from a lava channel, an active flow, and an inactive flow and fit Voyager IRIS spectra of hot spot thermal emission with this model. When the results were combined with a study of observed topography that concluded that Io’s crust was composed primarily of silicic material of strength capable of supporting observed caldera slopes (Clow and Carr, 1980), Carr concluded that silicate volcanism was prevalent on Io. Carr also predicted a highly-differentiated crust with basalt volcanism originating from close to the surface and a more peridotite-like composition of magmas originating from greater depths. The Voyager IRIS data for Pele, Loki and Sengen Patera were subsequently modeled as silicate eruptions by Howell (1997). Still, prior to the arrival of the Galileo spacecraft, it was not known whether the dominant style of volcanism on Io was sulfur or silicate in nature. Additional conclusive silicate-melt-temperature results were obtained by the Galileo spacecraft (Davies et al., 1997, 1999, 2000, 2001; Davies, 2003a; Lopes et al., 2001; McEwen et al., 1997, 1998a, 1998b). Observations from terrestrial observatories

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Table 2 Flow and lava lake crust thickness estimates Location

Latitude, N

Longitude, W

Thickness (m)

Instrument, observation

Reference

Lava flows Pillan Patera Prometheus Amirani Monan Tupan Zamama Gish Bar Zal Malik Mani

−11 −1 27 19 −17 21 16 43 −35 19

242 155 112 106 141 173 89 78 127 122

8–10 0.7 0.9 0.4 0.4 1.8 1.3 0.6 1.3 0.7

SSI, direct measurement from shadows NIMS, inferred from total thermal emission “ “ “ “ “ “ “ “

Williams et al., 2001b Davies, 2003a “ “ “ “ “ “ “ “

Lava flow or lava lake crust Loki Patera 12 Loki Patera 12

310 310

5–10 m after 300 days 0.8–2.6 m after 10–80 days

Rathbun et al., 2002 Davies, 2003c

Loki Patera

310

7 m after 540 days

Modeling basalt crust formation NIMS, inferred from Oct. 2001 temperature and age data Modeling of basalt crust on magma sea

12

Matson et al., 2004

Table 3 Lava liquidus temperature data for Io Temperature (K)

Location

Instrument

Reference

600 654 900 1200 1225 1200–1600 1100+ 1400+ 1500+ 1450–1600+ 1450–1600+ 1800–1825

Disc-integrated Pele ∼35 W Loki region ∼35 W Loki region Zamama Various locations Various locations Pele Pele Pillan

Ground-based VGR IRIS Ground-based Ground-based Ground-based Ground-based GLL SSI + NIMS Ground-based Ground-based GLL SSI + NIMS GLL SSI; CAS ISS GLL SSI + NIMS

Sinton, 1980 Pearl and Sinton, 1982 Johnson et al., 1988 Veeder et al., 1994 Blaney et al., 1995 Davies, 1996 Davies et al., 1997 Stansberry et al., 1997 Spencer et al., 1997 Davies et al., 2001 Radebaugh et al., 2004 McEwen et al., 1998a

1870+

Pillan

GLL SSI + NIMS

Davies et al., 2001

Notes

Aug. 1986 outburst Jan. 1990 outburst Aug. 1986 outburst Jan. 1990 outburst Joint NIMS-SSI analysis

Joint NIMS-SSI analysis SSI Clear/1 micron ratio; NIMS 3T fit to combined Pele + Pillan spectrum Joint NIMS-SSI analysis

Key: VGR = Voyager; GLL = Galileo; CAS = Cassini; NIMS = Galileo Near Infrared Mapping Spectrometer; SSI = Galileo solid state imaging experiment; ISS = Cassini imaging system; IRIS = Voyager Infra-Red Imaging Spectrometer.

using adaptive optics (AO) systems continue to detect silicate temperature eruptions (e.g., Marchis et al., 2001, 2002; de Pater et al., 2004). The dominance of silicate volcanism as the primary thermal source on Io was further bolstered by the discovery of ultra-high-temperature volcanism (implied melt temperatures of 1800 to >1870 K) and thus a magnesium-rich composition for at least some magma on Io (McEwen et al., 1998a; Davies et al., 2001). Ultra-hightemperature volcanic activity was an important discovery of the Galileo mission. Following from Carr (1986), a possible composition-based dichotomy was noted between Pele and Pillan Patera (Davies et al., 2001). What of sulfur volcanism? Ionian sulfur volcanism appears to be of a secondary nature, resulting from remobilization of sulfur-rich deposits (originally formed by the devolatilization of magma) by primary, silicate magmas. Such remobilization has been observed on Earth at Mauna Loa, Hawai’i (Greeley et al., 1984), at Siretoko-Iosan in Japan (Watanabe, 1940) and other locations. Possible sulfur vol-

canism has been seen at Emakong Patera on Io (Williams et al., 2001a). This paper therefore uses both basaltic and ultramafic thermo-physical values in the cooling models. Typical basaltic and ultramafic composition-dependent model input variables are shown in Table 4.

3. Volcanic cooling before solidification Models of cooling and thermal emission from emplaced lava bodies have been applied to remotely-sensed thermal emission data for Earth (Head and Wilson, 1986; Keszthelyi and Denlinger, 1996) and for Io (Carr, 1986; Davies, 1996; Howell, 1997; Keszthelyi and McEwen, 1997). These latter models and their variants are compared in Table 1. Subsequent references to the different models use the numbers that appear in the first row of this table. Heat loss mechanisms from lava flows on Io are shown schematically in

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Table 4 Input parameters used in Davies (1996) flow model [model (3) in Table 1] Liquidus temperature (K) Environment temperature (K) Substrate temperature (K) Surface heat loss mechanism Latent heat of fusion (J kg−1 ) Specific heat (J kg−1 K−1 ) Emissivity (W m−2 K−4 ) Thermal conductivity (W m−1 s−1 ) Liquid density (kg m−3 ) Solid density (kg m−3 )

Basalt

Ultramafic “komatiite”

1200–1600 3 105–130 Radiation 4 × 105 1500 1.0 0.9–1 2600 2800

1900 3 105–130 Radiation 8 × 105 1200 1.0 1.0 2800 2845

Fig. 1. The Davies (3) and (6), Howell (4), and Keszthelyi and McEwen (5) models have similarities in their approach: the distribution of temperatures on the surface of a lava body is a function of the age of the surface, i.e., the oldest material is at the coolest temperatures. By integrating over the entire area of the lava body, the total thermal emission can be determined. These models also predict the temporal evolution of thermal emission from the lava. Once the composition-dependent thermo-chemical parameters of the lava are chosen, the Davies models (3) and (6) produce the age and area of the thermal source, the Howell (4) model produces a rate of areal increase and an expression of the age of the eruption, and the Keszthelyi and McEwen model (5) produces a similar ‘survivability’ time (tau), an expression of the age of a lava surface. In all cases, a surface-

integrated thermal emission spectrum is produced. As more and more data from both Io and Earth are analyzed, different eruption signatures (Davies, 1996, 2001; Howell, 1997) and emission temporal evolutions (Davies, 2001, 2003a, 2003b; Davies et al., 1997, 2000, 2001; Keszthelyi et al., 2001; Davies and Keszthelyi, 2001) are recognized, indicative of different eruption styles and mass eruption rates. Despite their limitations, these relatively simple models have been successfully applied to Galileo Near Infrared Mapping Spectrometer (NIMS) and solid state imaging experiment (SSI) data. For instance, the Davies (3) model has been used to determine areal coverage rates and eruption durations (Davies et al., 2000, 2001). A full treatment of post-solidification cooling in these cases is not critical since NIMS and SSI are most sensitive to new, hot lava surfaces (with ages typically seconds-to-weeks) that are young enough that the flows still have molten interiors. In a more specialized case, the Davies (3) and Howell (4) models are also particularly well suited for modeling thermal emission from the volcanic feature Loki Patera, which for many years exhibited the temporal thermal emission characteristics of a huge lava lake (Rathbun et al., 2002), a situation very close to the ideal semi-infinite case for LHD cooling models. These models have been used to determine the age distribution and crust thickness of Loki Patera volcanics from NIMS and Photo-Polarimeter Radiometer (PPR) data (Spencer et al., 2000; Lopes-Gautier et al., 2000; Davies, 2003c). In the case of Pele, identified as a long-lived

Fig. 1. Heat loss mechanisms for a lava flow on Io. Heat is lost from the upper surface by radiation into a vacuum. Heat is conducted upward from the molten interior of the flow through an upper crust that thickens with time. Heat is lost by radiation through cracks in the crust, which expose the hot interior. Heat is lost by conduction downward through the base of the crust. As the flow progresses, the surface crust thickens. Temperature T1 is greater than temperature T2, which is in turn greater than temperature T3.

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127

active lava lake (Davies et al., 2001), a very young surface age is always derived from NIMS and SSI data (Davies et al., 2001; Radebaugh et al., 2004), which is indicative of the surface crust constantly being disrupted by vigorous overturning. The lava lake at Pele, and possible the lava lake or magma sea at Loki Patera (Rathbun et al., 2002; Davies, 2003c; Matson et al., 2004) are cases where the geometry of the semi-infinite space model is close enough to reality as to allow its use at all times.

4. Flow thickness estimates Measurements and estimates of ionian flow thicknesses are shown in Table 2. The most closely-studied and bestdocumented eruption on Io to date, where flows were seen to be emplaced, took place in the vicinity of Pillan Patera during the summer of 1997 (Williams et al., 2001b; Davies et al., 2001; Keszthelyi et al., 2001). Flows from this massive eruption covered over 5000 km2 in a few weeks, and were observed periodically by Galileo NIMS as they cooled over the next two and a half years (Davies et al., 2001). Measurements of shadows in Galileo SSI images yielded direct measurements of flow thicknesses on Io. This was the first measurement of active lava flows on any planetary body other than Earth. The Pillan flows were found to be ∼8–10 m thick (Williams et al., 2001b) (Fig. 2). At other locations on Io, including Prometheus (Fig. 3), Amirani, Gish Bar, Culann, Tupan, Zamama, and Maui, estimates of volumetric fluxes and areal coverage rates from Galileo NIMS data (Davies, 2001, 2003a; Davies et al., 2000), in conjunction with areal coverage rates derived from high-resolution SSI images (McEwen et al., 2000; Keszthelyi et al., 2001), have estimated flow thickness in the range of ∼0.1–5 m (Davies et al., 2000; Davies, 2003a). Pa-

Fig. 2. Lava flows at Pillan Patera, Io. This high-spatial-resolution (19 m/pixel) mosaic shows a complex collection of lava flows, pits, domes and possibly rafted plates of lava, emplaced during the eruption of June 1997 that was observed by Galileo (McEwen et al., 1998a; Davies et al., 2001; Williams et al., 2001b). Shadow measurements of the edge of an individual lava flow (arrowed) indicate that the flow thickness ranges from 8 to 11 m (Williams et al., 2001b). NASA base image is number PIA02536.

Fig. 3. Part of the Prometheus flow field as seen by the Galileo SSI in February 2000 (Orbit I27). At least three populations of flows within the main flow field take up the lower left-hand portion of the image. These flows are (A) relatively hot, young, low-albedo flows that are thought to be the most recently emplaced; (B) cooler, older, lighter flows on which volatiles have begun to condense; and (C) higher-albedo, cooler ‘background’ flows. Surface albedo increases with time as sulfur allotropes and SO2 condense on cool surfaces, sulfur at ∼400 K, and SO2 at ∼130 K. NASA base image is number PIA02557.

hoehoe and open-channel flows on Io appear to be relatively thin. What of other forms of lava emplacement? CAS can be used to better understand Loki Patera, Io’s most powerful volcano. To explain the observed thermal emission Loki Patera is either being supplied with lava at a huge rate, or is a huge lava lake, perhaps even the surface expression of a magma ‘sea’ (Matson et al., 2004). A high-spatial-resolution observation obtained by Galileo NIMS in October 2001 of most of the southern half of Loki Patera revealed a surface temperature distribution consistent with a “resurfacing wave” propagating across the floor of the patera (Davies, 2003c); see Fig. 4. This temperature distribution was consistent with the ‘foundering lava lake crust’ suggested by Rathbun et al. (2002), although it did not rule out resurfacing of the patera floor by lava flows. The Davies (2003c)

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Fig. 4. Davies (2003c) age map of part of the floor of Loki Patera is shown here superimposed on the best Voyager image of Loki. Ages are derived from crust temperatures resulting from a two-temperature fit to high-spatial-resolution NIMS data obtained in 2001 (31INTHLOKI01). Davies (1996) cooling model is used to calculate the age for each pixel from its temperature. Crust thicknesses are also calculated, which in this case range from 0.8 to 2.6 m, the lava crust being thinnest where the surface is hottest and youngest. The age distribution of the surface implies resurfacing at a rate of about 1 km per day from the SW corner towards the NE. This age distribution is consistent with the “foundering lava lake crust” model suggested by Rathbun et al. (2002), although resurfacing by lava flows cannot be ruled out (Davies, 1996, 2003c).

analysis used the Davies (3) cooling model to determine the age distribution of the crust and the crustal thickness, assuming a semi-infinite model. The derived temperatures and estimated crust thicknesses constrain minimum flow thickness for the ‘resurfacing by flows’ scenario. Whatever the mechanism of resurfacing, a crust ranging from 0.8 to 2.6 m is suggested across a large portion of the floor of Loki Patera at that time (October 2001) for the lava lake scenario, with an age distribution from 10 days (crustal thickness of 0.8 m) to 80 days (crust thickness of 2.6 m). Assuming a basalt composition, the maximum thickness of the lava lake crust reaches ∼7 m after 540 days (Matson et al., 2004), the resurfacing period proposed by Rathbun et al. (2002). The estimate for minimum lava flow thickness is therefore about 12 m (the combined top and base thicknesses: see Table 5) for flows that are still thermally buffered by latent heat release yielding the same surface temperature as the lava lake model. If the Loki Patera flows are in fact thinner than 12 m, then the cooling of the flow is more rapid. The implications of thinner flows are discussed below.

Unfortunately, there were no successful follow-up Galileo observations of Loki at similar spatial resolution, so the subsequent cooling trend for this resurfacing event was not determined. The cooling trends of flow and lava lake diverge with time, based on the supply of latent heat, and assuming the process is not interrupted by emplacement of new flows or replacement of the lake surface. This change in cooling behavior is the feature to search for in observational data to constrain eruption mechanism. However, resurfacing happens frequently at Loki, so it has not been possible to chart cooling trends for long enough to observe this divergence. As noted above, LHD models cannot be applied after the solidification of the lava body. For instance, the modified Stefan model of Head and Wilson (1), which is incorporated into the Davies (3) and Howell (4) models, balances the loss of heat by radiation from the upper surface of the flow with the conduction of heat from the interior, through a surface crust that thickens with time. The lava body is treated as a semi-infinite half-space, with heat lost from the upper sur-

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129

Fig. 5. Cooling profiles through the upper crust, molten interior, and base crust and substrate as a function of surface temperature for a 10-m-thick ultramafic flow on Io. The crust temperature profiles are generated using Eqs. (A.4) (top crust) and (A.5) (base crust and substrate).

face only. Heat loss is buffered by the release of latent heat as the lava solidifies from the upper surface downwards. However, lava flows in all physical cases have finite thickness, and as discussed above, in many cases on Io are relatively thin. Thus a more complete treatment of cooling must include the effects of cooling from the point of complete solidification, where cooling is faster than in the still-liquid case. This aspect of cooling has not been revisited since the seminal work of Carr (2). Additionally, an emplaced lava flow cools both from the top and the base. A stagnant lava flow will solidify from the base upwards, as well as from the top downwards. Base-crust formation further reduces the time over which the aforementioned semi-infinite-space (LHD) models can be applied.

5. Cooling after solidification (CAS) model To address the problem of thin cooling flows we have developed a model for finite-thickness lava bodies after they have completely solidified. We do this with a finite-element model of lava flow cooling (CAS). We test it against observed temperatures of flows of known thickness. CAS is described in detail in Appendix A. In short, by selecting thermal and physical parameters for both lava and underlying material, cooling profiles can be generated for lava bodies before and after solidification using a model that takes into account heat loss by radiation from the lava body upper surface, and by conduction to the ground beneath (Fig. 1). CAS uses the Schmidt graphical method (e.g., Lydersen, 1979) to model the evolving temperature profile in the solid, but still hot, lava body.

Figure 5 shows temperature profiles through the upper crust, the still-molten isothermal interior, and the base crust and subsurface, as a function of upper surface temperature, for a 10-m-thick flow of ultramafic composition on Io. The flow is completely solidified when upper and lower crusts meet. In this example, the flow has solidified when the surface temperature is 280 K, which takes ∼450 days of cooling. These temperature profiles are generated using Eqs. (A.4) and (A.5). Figure 6 shows cooling profiles after flow total-solidification point, as generated using the CAS model. The importance of the solidification of thin flows and subsequent cooling can be highlighted as follows: after 25 years of cooling, the surface temperature of this 10-m-thick ultramafic flow is 117 K (a detectable temperature on Io’s surface with instruments such as the Galileo PPR). A semiinfinite-space body takes over 1500 years to cool to this temperature. The presence of thin flows reduces cooling times to these low temperatures by orders of magnitude. 6. Results of model application Flow thickness and cooling curves as a function of time for ultramafic and basaltic flows are shown in Fig. 7a (basaltic) and Fig. 7b (ultramafic). Using the values in Table 4 for basaltic and ultramafic compositions, Table 5 shows the time taken for the flow to totally solidify. Heat loss is buffered by the release of latent heat until complete solidification. For ultramafic lava, models have been run for flow thicknesses from 0.1 to 100 m. Also shown are the thicknesses of the top and base crusts, and the surface temperature at the time of total solidification. These are the input variables needed to run CAS. Table 6 shows cooling times

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(a)

(a)

(b)

(b) Fig. 6. Temperature profiles (a) as a function of surface temperature through a 10-m-thick flow of ultramafic composition in an ionian environment. Profiles are derived using the cooling after solidification (CAS) model. The flow reaches total solidification when the crusts forming from the top and base surfaces meet at a point near the middle of the flow. In this particular case, this point is reached when the surface temperature of the flow reaches 287 K. Subsequent cooling curves are shown for each successive 10 K drop in surface temperature, down to a surface temperature of 97 K. (b) Shows an enlargement of the surface detail [inset box in (a)] with temperature and cooling times in years (numbers in parentheses).

as a function of surface temperature for 1- and 10-m thick basaltic and a wide range of ultramafic flows down to surface temperatures of 90 K, with values for the semi-infinite case for purposes of comparison. 6.1. Surface temperature dependence on flow thickness and composition; crustal thicknesses; flux through base and top crusts From Table 5, it takes approximately 4 days for a 1-m-thick basalt flow to completely solidify on the sur-

Fig. 7. Surface temperatures as a function of time for flows of different thickness and composition. (a) Shows cooling profiles for basalt flows of 1and 10-m thickness. Also is shown the Carr (1986) surface cooling curve for a 10-m-thick basalt flow with a 5% porosity. (b) Shows temperature profiles for ultramafic flows for thicknesses of 0.1, 1, 10, and 100 m. The thinner the flow, the faster cooling diverges from the semi-infinite case, which may be suitable for lava lake thermal modeling with time. The small vertical bar is the PPR surface temperature derived for the Pillan flows 2.3 years after emplacement (Spencer et al., 2000).

face of Io. If the flow is 2 m thick, the solidification time is 15 days. We note that we are not modeling a brecciated flow top. A 10-m-thick flow takes one year to completely solidify. For ultramafic (komatiite) compositions the cooling times are similar: 5 days for a 1-m-thick flow, 19 days for a 2-m-thick flow, and 448 days for a 10-m-thick flow. Total solidification time for ultramafics is approximately 20% longer than for basalts. Formation of crustal thickness as a function of time is shown in Fig. 8. Initially, the base crust thickens faster than the upper crust as the flow quenches against the cold substrate. Soon, radiation cooling becomes dominant and the upper crust now thickens faster than the base. If thermal erosion of the substrate is taking place, then the base crust may not form until the flow stops and stagnates. Figure 9 shows the flux through base and top crusts for a 10-m-thick ultramafic flow. As time passes after complete solidification, the heat that has flowed into the substrate eventually returns, coming up through the

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Table 5 Solidification times for different lava flowsa Lava type

Flow thickness (m)

Time to total solidification (ts )

Base thickness (m)

Top thickness (m)

Surface temp. (Ts ) at time ts (K)

Basalt Basalt Basalt Basalt Basalt Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic Ultramafic

1 2 5 8 10 0.1 0.2 0.5 1 2 5 8 10 20 30 40 50 75 100

3.8 days 15 days 92 days 228 days 364 days 1.25 h 4.8 h 29 h 5 days 19 days 112 days 282 days 448 days 5 years 11 years 19 years 30 years 67 years 121 years

0.46 0.90 2.25 3.57 4.45 0.046 0.09 0.22 0.44 0.86 2.13 3.40 4.20 8.4 12.6 17 21 31 42

0.54 1.10 2.75 4.43 5.55 0.054 0.11 0.28 0.56 1.14 2.87 4.90 5.80 11.6 17.4 23 29 44 58

447 382 309 277 237b 805 693 565 483 410 331 296 280 237 215 201 190 172 160

a Thermal conductivity k = 1 W m−1 K−1 ; temperature of host T = 105 K; emissivity e = 1. H b For emissivity = 0.9, the surface temperature at total solidification point is 262 K.

Table 6 Temperatures and cooling times for solid and partially molten flows Flow surface Time (s) temperature Basalt (K) 1m

10 m

Semi-infiniteb 0.1 m

0.5 m

1m

5m

10 m

50 m

100 m

Semi-infiniteb

1800 1600 1400 1200 1000 900 800 700 600 500 400 300 200 130 105 100 90

n/a n/a 1 2 150 470 1560 5680 2.38 × 104 1.23 × 105 8.66 × 105 1.01 × 107 1.24 × 108 5.15 × 108 1.05 × 109 1.36 × 109 1.84 × 109

n/a n/a 1 2 150 470 1560 5680 2.38 × 104 1.23 × 105 8.66 × 105 1.01 × 107 2.98 × 108 1.03 × 1010 5.85 × 1010 8.70 × 1010 2.05 × 1011

0.35 3.2 18 96 592 1610 4761 1.58 × 104 6.15 × 104 1.96 × 105 4.15 × 105 9.95 × 105 3.41 × 106 1.62 × 107 4.65 × 107 6.24 × 107 1.30 × 108

0.35 3.2 18 96 592 1610 4761 1.58 × 104 6.15 × 104 2.97 × 105 9.56 × 105 2.40 × 106 8.17 × 106 3.52 × 107 8.88 × 107 1.13 × 108 2.08 × 108

0.35 3.2 18 96 592 1610 4761 1.58 × 104 6.15 × 104 2.97 × 105 1.97 × 106 1.66 × 107 6.38 × 107 2.43 × 108 5.03 × 108 6.05 × 108 9.19 × 108

0.35 3.2 18 96 592 1610 4761 1.58 × 104 6.15 × 104 2.97 × 105 1.97 × 106 2.18 × 106 1.50 × 108 6.18 × 108 1.17 × 109 1.38 × 109 2.01 × 109

0.35 3.2 18 96 592 1610 4761 1.58 × 104 6.15 × 104 2.97 × 105 1.97 × 106 2.18 × 106 6.17 × 108 4.12 × 109 7.92 × 109 9.29 × 109 1.39 × 1010

0.35 3.2 18 96 592 1610 4761 1.58 × 104 6.15 × 104 2.97 × 105 1.97 × 106 2.18 × 106 6.17 × 108 9.76 × 109 2.39 × 1010 2.87 × 1010 3.99 × 1010

0.35 3.2 18 96 592 1610 4761 1.58 × 104 6.15 × 104 2.97 × 105 1.97 × 106 2.18 × 106 6.17 × 108 2.07 × 1010 1.17 × 1011 1.73 × 1011 4.06 × 1011

n/a n/a 1a 2 150 470 1560 5680 2.38 × 104 1.23 × 105 6.94 × 105 2.09 × 106 7.37 × 106 3.29 × 107 8.50 × 107 1.65 × 108 2.04 × 108

Ultramafic 0.35 3.2 18 96 592 1610 4761 9.03 × 103 1.55 × 104 2.65 × 104 5.13 × 104 1.19 × 105 4.34 × 105 2.83 × 106 1.29 × 107 2.02 × 107 6.26 × 107

a Grey cells indicate a partially-molten flow. White cells indicate a completely solidified flow. b Semi-infinite cases are included to highlight the magnitude of the differences that develop between finite-thickness and semi-infinite bodies with time.

base. It is then conducted to the surface, and radiated into space. 6.2. Comparison with Galileo data results We now compare the predictive model output with observations of cooling of flows on Io. During the close Io fly-by in October 1999 (Galileo orbit I24) high-spatial-resolution SSI observations (at ∼19 m/pixel) allowed flow thicknesses

to be determined from shadow measurements: the 1997 Pillan flows appeared to be ∼8–10 m thick (McEwen et al., 2000; Williams et al., 2001b). The implied average mass eruption rates were in excess of 104 m3 /s (Davies et al., 2001; Williams et al., 2001b), with the peak volumetric eruption rate almost certainly higher. Table 7 shows the range of temperatures that have been derived from modeling Pillan cooling. From high-resolution Galileo PPR data obtained during the October 1999 fly-by, Spencer et al. (2000) derived a 17-micron brightness temperature of 200 K, ±20 K, from

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Table 7 Surface temperature for 2.3 year-old flows

Fig. 8. Formation of surface (top) crust (dashed line) and base crust (dotted line) as a function of time for a basaltic composition flow in an ionian environment. The solid/liquid interfaces move towards the center of the lava body. Semi-infinite-space (LHD) cooling models can only be applied to the point when the top and base crusts meet. Each thickness of lava flow has a different total solidification time (solid line). Observed thicknesses of Pillan 1997 flows and Prometheus flows are shown.

Fig. 9. Flux through the top and base of a 10-m-thick ultramafic flow on Io. Heat loss by radiation from the top of the flow dominates cooling. Energy flows through the base of the flow down into the crust. The flow cools sufficiently after ∼1500 days for this energy to flow back up through the base into the body of the flow, where it is conducted to the surface.

a scan across the Pillan flows 2.3 years after they had been emplaced. The thickness, age and temperature observations allow the CAS model to be tested against observational data, and compared with results from the Davies (1996) ‘semi-infinitespace’ model. The CAS prediction for 10-m-thick flows of ultramafic composition, using the values in Table 4, is that the surface temperature of the flows would be 249 K after 2.3 years (7.25 × 107 s) for an already solid 10-m-thick flow, and 260 K for the semi-infinite case (necessitating a flow in excess of 13-m thick to still maintain a molten interior at this time). These values are slightly greater than the 200 ± 20 K

Model

Flow surface temperature (K)

Reference

Howell semi-infinite half-space (h/s) Davies semi-infinite h/s ultramafic Davies semi-infinite h/s basalt CAS, ultramafic, 10 m thick CAS, ultramafic, 8 m thick CAS, ultramafic, 1 m thick CAS, basalt, 10 m thick CAS, basalt, 8 m thick CAS, basalt, 1 m thick

245a 260b 237 249 230 110 228 216 115

Spencer et al., 2000 Davies et al., 2001 Davies et al., 2001 This paper This paper This paper This paper This paper This paper

a Requires ∼13 m depth of solidification, implying that flows are thinner than ∼13 m to cool to observed temperatures, and not considering formation of a base crust. b Upper crust thickness is 8.5 m.

measured by PPR. An 8-m-thick ultramafic flow would solidify after 282 days (2.44 × 107 s) and in the remaining 1.53 years would cool to 230 K, tantalizingly close to the observed PPR temperature range. Similarly, a basalt flow 8 m thick would completely solidify in 228 days (1.96 × 107 s), and in the subsequent 1.52 years the surface would cool to 216 K. A 10-m-thick basalt flow solidifies and cools to 228 K in the same time. The CAS model output is closer to observational data than the older ‘semi-infinite’ models. The implication of the above comparison is that the basalt and ultramafic flow models do not quite cool fast enough to match observations, although the difference is only on the order of 20 K at most. Several possible reasons for the discrepancy are: (1) the model thermo-physical input parameters are not correct and need adjusting; (2) the PPR-derived temperature fit was made from long-wavelength infrared data and is not sensitive to hotter components of the flow surface; (3) the PPR field of view (∼2 km) includes cold, non-lavaflow surface material, which lowers the derived temperature; and (4) the actual surface temperature distribution is more complicated than the model predicts. It is also possible that (5) the bulk of the flows at Pillan were toward the thinner end of the 8–10 m range derived from SSI data for a single flow. A small decrease in thickness (less than 10%) brings the modeled temperatures into parity with the observed temperatures using the current input values. Although it is not possible to determine whether cases (2) to (5) are the causes of the discrepancy, it is useful to explore required modifications of thermal input parameters to match the PPR observations [case (1), above]. It is possible to derive a set of parameters that reproduce the PPR data of October 1999 (Galileo Orbit I24). This provides a yardstick for fitting other NIMS and PPR data (see Rathbun et al., 2004). Surface ages can therefore be derived for a number of different lava emplacement regimes. Figure 10 shows variation in cooling rate caused by varying thermal conductivity using values of 1, 2 and 3 W m−1 K−1 . After cooling for ∼280 days (2.4 × 107 s) surface temperatures cover a spread of ∼50 K. After 2.3 years (7.25 × 107 s), the age

Post-solidification cooling and the age of Io’s lava flows

133

7. Conclusions

Fig. 10. Variation in cooling rate as a function of varying input variable value, in this case, thermal conductivity. Cooling curves are shown for the semi-infinite space (Davies, 1996) and for a 10-m-thick ultramafic flow, with profiles derived using the CAS model. Reducing thermal conductivity (k) speeds up cooling of the surface. Temperatures after 2.3 years of cooling range from ∼255 K (for k = 1 W m−1 K−1 ) to ∼300 (for k = 3 W m−1 K−1 ). By varying this and other input values we can constrain the Davies (1996) and CAS models through fitting both remote data of ionian eruptions and terrestrial eruption data. Note that the CAS model can have different sensitivities to thermal parameters than the semi-infinite-space model.

of the Pillan flows when observed by PPR, the temperature spread is ∼45 K. It may also be possible to constrain possible compositions from the necessary input parameters needed to explain observed cooling behavior. Additionally, surface temperatures from NIMS data (see, for example, Davies et al., 2001; Davies, 2003c) lead to independent surface age estimates, and these results can be compared with the two cooling model outputs. 6.3. Other factors affecting cooling Cooling of a flow surface with time is affected by other factors, especially at temperatures lower than the temperatures being modeled here. A flow surface is heated by solar insolation (50 W/m2 ) when in sunlight, although this heat is subsequently lost at night. The black body equilibrium temperature with solar insolation at Io is ∼130 K, but we are primarily concerned in this paper with modeling volcanic temperatures close to 220 K. The methodology used is adequate for modeling the Pillan eruption under consideration. Additionally, when the surface temperature drops below ∼400 K, sulfur can condense on the surface, and when the surface temperature drops to ∼130 K, SO2 will condense on the surface (Keiffer, 1982). If these layers are thick, the flow underneath will be insulated and heat loss is slowed. A detailed examination of this effect is reserved for the future because it has little effect on the considerably higher-temperature, shorter-term heat-transfer processes in this study.

With the exception of Carr (1986), previous models of cooling of lava flows and their use in interpreting remote sensing data are applicable to the case where heat loss is buffered by the release of latent heat from ongoing solidification of a liquid phase. Such models are no longer accurate once the lava body has completely solidified. While this is not necessarily a problem with lava lakes, which can be treated as a semi-infinite liquid space, even thick flows eventually solidify. Since individual flows on Io are apparently not very thick (typically less than 10 m in depth), the effects of cooling after solidification are required to determine age as a function of temperature. The application of the finite-element model presented in this paper allows cooling after solidification to be modeled, using the depthtemperature profile generated by the “semi-infinite-space” models (Davies, 1996; Howell, 1997) as the starting point. Our first attempts to use this model produce close fits to observed data: refinement of input parameters can proceed from this point to more closely match observations. This new approach allows modeling of lava bodies of different morphologies. These analyses can be used to quantify cooling behavior for lavas emplaced by different mechanisms. Thermal structure and temporal evolution, observable remotely, should be diagnostic of emplacement mechanism. This approach has been successfully applied to effusive emplacement of insulated flows (see Davies et al., 1997), open-channel flows with fire-fountaining, seen at Pillan in 1997 (Davies et al., 2001), the lava lake at Pele (Davies et al., 2001), and the flow fields at Prometheus and Amirani (Davies, 2003a). Now, CAS can be applied to derive flow thicknesses and flow ages from observed surface temperature. This is particularly important when considering the large flow fields that have been emplaced on Io, such as at Prometheus, Amirani, and Isum. Here, old, relatively cool flows occupy large areas for a long time, and these contribute most of the observed power emitted at thermal infrared (>8 microns) wavelengths, detectable using radiometry (e.g., PPR). The young flows, with active emplacement of lava, are most detectable by instruments such as NIMS and SSI, but these thermal sources are short-lived and fleetingly detectable by short-wavelength (<5 microns) instruments on a geological time scale. Relatively thin flows, such as those inferred at Prometheus, solidify and subsequently cool much faster than thick flows (e.g., the Pillan 1997 flows). Model runs show that flows of basaltic composition on the surface of Io cool faster than flows with ultramafic composition. Active lava lakes (e.g., Pele, and possibly Loki Patera) exhibit periodic or episodic overturning, resetting the surface temperature and cooling clock, and then cooling until the next overturn or disruption event. Complete solidification and subsequent cooling of a stagnant lava lake or ponded flow should also be identifiable from a time series of observations of thermal emission. Analogous CAS techniques can be applied to

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terrestrial datasets, obtained from Earth-orbiting platforms (such as ASTER on the Terra spacecraft, and the Hyperion spectrometer on the Earth Observing-1 (EO-1) spacecraft), from aircraft (e.g., TIMS), as well as from field (ground) observations, with the cooling model taking into account differences in cooling processes due to Earth’s atmosphere and higher surface temperature (see Davies, 1996). Additional modeling of thermal emission trends from Loki Patera will quantify the expected differences in cooling trends expected for different emplacement mechanisms by considering that (a) with the flow model, new flows are being emplaced on older, but still warm, flows, and (b) with a lava lake the supply of heat through the crust is a function of crust thickness only. It may be that relatively high-spatial resolution (160 km/pixel) and high temporal resolution observations attainable with Adaptive Optics of Loki Patera’s volcanic activity (see Marchis et al., 2002, 2005; de Pater et al., 2004) over the course of a resurfacing cycle of ∼540 days (Rathbun et al., 2002) may provide sufficient constraint to choose between a lava lake or flow model for Loki Patera. From within the jovian system, high-temporalresolution observations of Loki Patera should be a priority for the next mission to Jupiter (e.g., the Jupiter Icy Satellite Orbiter, or JIMO) in order to resolve this question concerning Io’s most important volcano. JIMO will have the opportunity to observe Io throughout the nominal four-year mission at Jupiter, a period covering a number of Loki Patera resurfacing cycles. Finally, the CAS approach allows a more stochastic approach to modeling the continual emplacement of lava flows on Io. CAS enables calculation, for any set of input parameters (flow composition, flow thickness, subsurface composition and sub-flow thermal gradient) of the subsequent thermal behavior of a flow. Integrating the model predictions across the surface area of the lava body allows the total thermal emission from a flow or lake to be determined. Flows can be erupted one on top of another and the resulting regional, and even global, heat flow averages determined. 8. Summary With the discovery that lava flows on Io are relatively thin, most previous models of volcanic thermal emission variability are seen to be limited to relatively short times after emplacement when the lava is still partially molten. Our new model has been developed for determining the surface temperature of lava flows, lakes, and seas on Io to include times after solidification. The CAS model is directly testable because it accurately predicts an observable quantity (thermal emission) as a function of (a) flow thickness, (b) lava composition, and (c) effusion mechanism. Acknowledgments The cooling after solidification finite element model was first developed for lava bodies as part of AGD’s doctoral

work at Lancaster University, UK. A.G.D. thanks Lionel Wilson and Harry Pinkerton for their invaluable guidance during that time. Subsequently, part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract to NASA. The authors gratefully acknowledge the support of the NASA Planetary Geology and Geophysics Program. We thank Laszlo Keszthelyi and an anonymous reviewer for their helpful comments and suggestions.

Appendix A. Numerical modeling of the cooling and solidification process To determine the thickness of crust on a lava flow as a function of both time and surface temperature, the model of Head and Wilson (1986) uses a variation of the solution to the Stefan model of formation of ice on a freezing lake. From Head and Wilson (1986) the thickness C (m) of the crust at time t is given by √ C = 2λ κt, (A.1) where κ is thermal diffusivity of the lava (m2 s−1 ), and λ is a dimensionless quantity such that √ exp(−λ2 ) L π = , (A.2) c(TS − T0 (t)) λ erf(λ) where Ts is the magma liquidus temperature (K), T0 (t) is the surface temperature (K) of the flow at time t, c is specific heat capacity (J kg−1 K−1 ), L is latent heat capacity (J kg−1 ) and erf is the error function. The heat flow through the surface of the lava flow is equal to the temperature gradient at the surface multiplied by the thermal conductivity of the lava, and must be equal to the total heat loss F (W m−2 ) by radiation from the upper surface such that k(TS − T0 (t)) F= √ , πκt erf(λ)

(A.3)

where k is thermal conductivity (W m−1 K−1 ). A caveat must be inserted at this point. As noted by Keszthelyi and Denlinger (1996) the Head and Wilson model is not a true analytical model to the process of cooling and solidification of a lava flow as it relies on the repeated application, at different surface temperatures, of the Stefan problem for a fixed surface temperature. Initially the solution of Head and Wilson is close to that produced by numerical models, but, with the passage of time, the Head and Wilson model produces longer cooling times than those derived from the numerical solution. Davies (1996) overcame this problem by selecting thermo-physical parameters that reproduced observed cooling rates. Use of the Head and Wilson model only becomes problematical when very low surface temperatures are modeled. As described above, as lava flows on Io are thought to be relatively thin, and the surface of lava lakes are relatively young (and therefore hot), the utility of semi-infinite-space models is limited. The cooling of very

Post-solidification cooling and the age of Io’s lava flows

thick flows using results from Head and Wilson should be regarded with caution. The processes of cooling from the base of the flow, the growth of a base crust, and cooling after solidification, are not affected. The use of, and results from, the model described in the next section are numerically valid. The method of solution is as follows. A surface temperature T0 (t) is selected and the left-hand side of Eq. (A.2) solved. The right-hand side of Eq. (A.2) is solved iteratively to find λ, which is used with the value of T0 (t) in Eq. (A.3) to find time t, the time from commencement of cooling. The derived values of t and λ are then used in Eq. (A.1) to find the crustal thickness at time t for surface temperature T0 (t). Figure 8 shows the thickness of top and base crusts for the semi-infinite case, on Io, for a silicate of basaltic composition (using the thermo-physical parameters given in Table 4). The model assumes a substrate of the same composition as the lava. Although initially the crust forms more quickly at the base, quenched as it is against the cold (105 K) substrate, after less than 20 min the upper crust is thicker than the base crust, and remains so until complete solidification. Figure 8 allows the selection of a flow thickness and determination of the top and base thicknesses as a function of time. As soon as the flow is solid, this approach cannot be applied further, as heat loss is no longer buffered by release of latent heat. We now determine the temperature profile in the lava body at the point of solidification. This is the starting point for subsequent modeling of the cooling process. Crustal thickness and cooling time are found from Fig. 8, and these are used in Eq. (A.1) to determine λ for both top and base crusts. Temperature Tx in the upper crust as a function of depth x is given by   
 x TS − T0 (t) + T0 (t). Tx = (A.4) erf √ erf(λ) 2 κt The temperature in the base crust and substrate at a distance y from the total solidification plane within the lava flow is given by   
 y TS − Th + Th , Ty = (A.5) 1 + erf √ 1 + erf(λ) 2 κt where Th is the original temperature of the crust underlying the flow, taken to be 105 K. Using the temperature profile constructed using Eqs. (A.4) and (A.5), we use the Schmidt graphical method (e.g., Lydersen, 1979). Here, the differential forms of the Fourier equation for the conduction of heat ∂T d 2T =k 2 ∂t dx are replaced with the finite differences

(A.6)

∂T Tx,(t+t) − Tx,t (A.7) → , ∂t t ∂ 2T ((T(x+x),t − Tx,t )/x) − ((Tx,t − T(x−x),t )/x) → 2 x ∂x (A.8)

135

Fig. 11. Derivation of successive cooling profiles in a cooling body using the Schmidt graphical method. In this example the surface temperature on either side of the slab is fixed at T0 . The starting profile, T0 –a2–a3–a4–a5–T0 at time t is used to generate a profile at time t + t . This profile is T0 –b2–b3–b4–b5–T0 . Point b2 is the average of T0 and a3. Point b3 is the average of a4 and a2. Point b4 is the average of a5 and a3. Profile c, at time t + 2t , is similarly constructed from profile b. The relationship between the width of each element x and time increment t is dependent on thermal conductivity, and shown in Eq. (A.11).

T(x+x),t − 2Tx,t + T(x−x),t . (A.9) x 2 When inserted into Eq. (A.6), these differences yield =

Tx,(t+t) − Tx,t t =k (T(x+x),t − 2Tx,t + T(x−x),t ), (A.10) x 2 where t (time increment) and x (distance increment) are such that t can be chosen to give t = 0.5. x 2 Equation (A.10) therefore reduces to

k

(A.11)

1 Tx,(t+t) = (T(x+x),t + T(x−x),t ). (A.12) 2 The right-hand side of Eq. (A.12) contains only temperatures at time t, and the left-hand side gives the temperature at time t + t. The use of Eq. (A.12) for a slab, showing how a new temperature profile is constructed from the previous temperature profile, is shown in Fig. 11. In this example, the surfaces of the slab are kept at a constant temperature. With a constant ambient or environmental temperature TE and a constant heat transfer coefficient, h, the heat flow through the surface of the slab per unit area, Q, is given by Q=

Tx=0 − TE , k/ h

(A.13)

where k is the thermal conductivity of the lava. k values for basaltic and ultramafic compositions are given in Table 4. Equation (A.13) implies that an extension of a temperature curve from the surface of the slab will intercept the Te line

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so

   k x − + TE . AA(1) = tan θ h 2

(A.17)

The surface temperature T0 (t) is known at the initiation of the process, and for subsequent time steps is found from AA(2) + AA(1) (A.18) . 2 The flux through the surface of the flow is found by multiplying the surface temperature gradient by the thermal conductivity of the silicate material. As the flow surface cools, the surface flux, and therefore the heat transfer coefficient, also changes. These quantities are recalculated at each time step.

T0 (t) =

References

Fig. 12. Cooling of slab with defined coefficient of heat transfer h at surface. The new surface temperature T0 is the result of finding the balance between the transfer of heat to the surface and heat loss by radiation, itself a function of surface and environment temperature (TE ). Equations (A.14)–(A.16) in Appendix A are used to calculate a new value of h at every time interval (t ).

at a distance k/ h from the surface, as shown in Fig. 12. In order to include the surface temperature in the calculations, the first interval is located at a distance x/2 outside the surface. A FORTRAN program (COOLASOL) uses successive applications of Eq. (A.12) to determine the cooling of a lava body after starting from the temperature profile at solidification point. As an example, for a time interval t of 2000 s, and using a value of k = 7 × 10−7 m2 s−1 , x is found from Eq. (A.11) to be 0.052915 m. For the upper surface of the flow, the radiative heat transfer coefficient, h, is defined by h=

Q σ T0

(t)4 /(T

0 (t) − TE )

,

(A.14)

where σ is the Stefan–Boltzmann constant, T0 (t) is the flow upper surface temperature, and TE is the temperature of the environment into which the flow is radiating. The finite-element analysis method is highly versatile as it can use any temperature profile as a starting point. As shown in Fig. 12 (from Davies, 1988), the initial values are set up as follows: tan θ =

AA(2) − T0 (t) x/2

(A.15)

therefore, k/ h =

T0 (t) − TE tan θ

(A.16)

Blaney, D.L., Johnson, T.V., Matson, D.L., Veeder, G.J., 1995. Volcanic eruptions on Io: heat flow, resurfacing and lava composition. Icarus 113, 220–225. Carr, M.H., 1986. Silicate volcanism on Io. J. Geophys. Res. 91, 3521– 3532. Clow, G.D., Carr, M.H., 1980. Stability of sulfur slopes on Io. Icarus 44, 729–733. Davies, A.G., 1988. Sulfur–silicate interactions on the jovian satellite, Io. Ph.D. thesis. Lancaster University, Lancaster, Lancashire, UK. Davies, A.G., 1996. Io’s volcanism: thermo-physical models of silicate lava compared with observations of thermal emission. Icarus 124, 45–61. Davies, A.G., 2001. Volcanism on Io: the view from Galileo. Astron. Geophys. 42 (2), 10–15. Davies, A.G., 2003a. Volcanism on Io: estimation of eruption parameters from Galileo NIMS data. J. Geophys. Res. 108, 5106–5120. Davies, A.G., 2003b. The pulse of the volcano: discovery of episodic volcanism at Prometheus on Io. Lunar Planet. Sci. XXXIV. Abstract 1455 [CD-ROM]. Davies, A.G., 2003c. Temperature, age and crust thickness distributions of Loki Patera on Io from Galileo NIMS data: implications for resurfacing mechanism. Geophys. Res. Lett. 30, 2133–2136. Davies, A.G., Keszthelyi, L.P., 2001. Eruption styles on Io as seen by Galileo: a comparison with Earth volcanism. Jupiter Conference, Boulder, CO, June 2001. Davies, A.G., McEwen, A.S., Lopes-Gautier, R.M.C., Keszthelyi, L., Carlson, R.W., Smythe, W.D., 1997. Temperature and area constraints of the South Volund Volcano on Io from the NIMS and SSI instruments during the Galileo G1 orbit. Geophys. Res. Lett. 24, 2447–2450. Davies, A.G., Keszthelyi, L.P., the Galileo NIMS and SSI Teams, 1999. Galileo observations of Io’s volcanism: constraints on lava temperature and eruption style. Abstract. In: Fall AGU Meeting, EOS Trans. 80 (46), F638. Davies, A.G., Lopes-Gautier, R.M.C., Smythe, W.D., Carlson, R.W., 2000. Silicate cooling model fits to Galileo NIMS data of volcanism on Io. Icarus 143, 211–225. Davies, A.G., Keszthelyi, L.P., Williams, D., Phillips, C.B., McEwen, A.S., Lopes, R.M.C., Smythe, W.D., Kamp, L.W., Soderblom, L.A., Carlson, R.W., 2001. Thermal signature, eruption style and eruption evolution at Pele and Pillan on Io. J. Geophys. Res. 106 (E12), 33,079–33,104. de Pater, I., Marchis, F., Macintosh, B.A., Roe, H.G., Le Mignant, D., Graham, J.R., Davies, A.G., 2004. Keck AO observations of Io in and out of eclipse. Icarus 169, 250–263. Greeley, R., Theilig, E., Christensen, P., 1984. The Mauna Loa sulfur flow as an analog to secondary sulfur flows (?) on Io. Icarus 60, 189–199.

Post-solidification cooling and the age of Io’s lava flows

Head, J.W., Wilson, L., 1986. Volcanic processes and landforms on Venus: theory, predictions and observations. J. Geophys. Res. 91, 9407–9466. Howell, R.R., 1997. Thermal emission from lava flows on Io. Icarus 127, 394–407. Howell, R.R., Spencer, J.R., Goguen, J.D., Marchis, F., Prange, R., Fusco, T., Blaney, D.L., Veeder, G.J., Rathbun, J.A., Orton, G.S., Grocholski, A.J., Stansberry, J.A., Kanner, G.S., Hege, E.K., 2001. Ground-based observations of volcanism on Io in 1999 and 2000. J. Geophys. Res. 106, 33,129–33,140. Johnson, T.V., Veeder, G.J., Matson, D.L., Brown, R.H., Nelson, R.M., Morrison, D., 1988. Io: evidence for silicate volcanism in 1986. Science 242, 1280–1283. Keiffer, S.W., 1982. Dynamics and thermodynamics of volcanic eruptions: implications for the plumes on Io. In: Morrison, D. (Ed.), Satellites of Jupiter. Univ. Arizona Press, pp. 647–723. Keszthelyi, L., Denlinger, R., 1996. The initial cooling of pahoehoe flow lobes. Bull. Volcanol. 58, 5–18. Keszthelyi, L., McEwen, A.S., 1997. Thermal models for basaltic volcanism on Io. Geophys. Res. Lett. 24, 2463–2466. Keszthelyi, L.P., McEwen, A.S., Phillips, C.B., Milazzo, M., Geissler, P., Williams, D., Turtle, E., Radebaugh, J., Simonelli, D., the Galileo SSI Team, 2001. Imaging of volcanic activity on Jupiter’s moon Io by Galileo during GEM and GMM. J. Geophys. Res. 106, 33,025–33,052. Lopes, R.M.C., Kamp, L.W., Doute, S., Smythe, W.D., Carlson, R.W., McEwen, A.S., Geissler, P.E., Keiffer, S.W., Leader, F.E., Davies, A.G., Barbinis, E., Mehlman, R., Segura, M., Shirley, J., Soderblom, L.A., 2001. Io in the near infrared: Near-Infrared Mapping Spectrometer (NIMS) results from the Galileo flybys in 1999 and 2000. J. Geophys. Res. 106, 33,053–33,078. Lopes-Gautier, R., Doute, S., Smythe, W.D., Kamp, L.W., Carlson, R.W., Davies, A.G., Leader, F.E., McEwen, A.S., Geissler, P.E., Keiffer, S.W., Keszthelyi, L., Barbinis, E., Mehlman, R., Segura, M., Shirley, J., Soderblom, L.A., 2000. A close-up look at Io in the infrared: results from Galileo’s Near-Infrared Mapping Spectrometer. Science 288, 1201–1204. Lydersen, A.L., 1979. Fluid Flow and Heat Transfer. Wiley, New York. Marchis, F., Prange, R., Fusco, T., 2001. A survey of Io’s volcanism by adaptive optics observations in the 3.8 micron thermal band (1996– 1999). J. Geophys. Res. 106, 33,141–33,160. Marchis, F., dePater, I., Davies, A.G., Roe, H.G., Fusco, T., Le Mignant, D., Deschamps, P., Mackintosh, B.A., Prange, R., 2002. High-resolution Keck adaptive optics imaging of violent activity on Io. Icarus 160, 124– 131. Marchis, F., Le Mignant, D., Chaffee, F.H., Davies, A.G., Kwok, S.H., Prange, R., de Pater, I., Amico, P., Campbell, R., Fusco, T., Goodrich, R.W., Conrad, A., 2005. Keck AO survey of Io global volcanic activity between 2 and 5 microns. Icarus. In press. Matson, D.L., Davies, A.G., Veeder, G.J., Rathbun, J.A., Johnson, T.V., 2004. Loki Patera as the surface of a magma sea. Lunar Planet. Sci. XXXV. Abstract 1882 [CD-ROM]. McEwen, A.S., Simonelli, D.P., Senske, D.R., Klassen, K.P., Keszthelyi,

137

L., Johnson, T.V., Geissler, P.E., Carr, M.H., Belton, M.S.J., 1997. High temperature hot spots on Io as seen by the Galileo solid-state imaging experiment. Geophys. Res. Lett. 24, 2443–2446. McEwen, A.S., Keszthelyi, L., Spencer, J.R., Matson, D.L., LopesGautier, R., Klassen, K.P., Johnson, T.V., Head, J.W., Geissler, P., Fagents, S., Davies, A.G., Carr, M.H., Breneman, H.H., Belton, M.J.S., 1998a. Very high-temperature volcanism on Jupiter’s moon Io. Science 281, 87–90. McEwen, A.S., Keszthelyi, L.P., Geissler, P., Simonelli, D.P., Carr, M.H., Johnson, T.V., Klassen, K.P., Breneman, H.H., Jones, T.J., Kaufman, J.M., McGee, K.P., Senske, D.A., Belton, M.J.S., Schubert, G., 1998b. Active volcanism on Io as seen by the Galileo SSI. Icarus 135, 181–219. McEwen, A.S., Belton, M.J.S., Breneman, H.H., Fagents, S.A., Geissler, P., Greeley, R., Head, J.W., Hoppa, G., Yaeger, W.L., Johnson, T.V., Keszthelyi, L., Klassen, K.P., Lopes-Gautier, R., McGee, K.P., Milazzo, M.P., Moore, J.M., Pappalardo, R.T., Phillips, C.B., Radebaugh, J., Schubert, G., Schuster, P., Simonelli, D.P., Sullivan, R., Thomas, P.C., Turtle, E.P., Williams, D.A., 2000. Galileo at Io: results from highresolution imaging. Science 288, 1193–1198. Pearl, J.C., Sinton, W.M., 1982. Hot spots on Io. In: Morrison, D. (Ed.), Satellites of Jupiter. Univ. of Arizona Press, Tucson, AZ, pp. 724–755. Radebaugh, J., McEwen, A.S., Milazzo, M., Keszthelyi, L., Davies, A.G., Turtle, E., Dawson, D., 2004. Observations and temperatures of Io’s Pele Patera from Cassini and Galileo spacecraft images. Icarus 169, 65–79. Rathbun, J., Spencer, J.R., Davies, A.G., Howell, R.R., Wilson, L., 2002. Loki: a predictable volcano? Geophys. Res. Lett. 29 (10), 84–88. Rathbun, J., Spencer, J.R., Tamppari, L.K., Martin, T.Z., Barnard, L., Travis, L.D., 2004. Mapping of Io’s thermal radiation by the Galileo Photopolarimeter–Radiometer (PPR) instrument. Icarus 169, 127–139. Sinton, W.M., 1980. Io’s 5-micron variability. Astrophys. J. 235, L49–L51. Spencer, J.R., Stansberry, J.A., Dumas, C., Vakil, D., Pregler, R., Hicks, M., Hege, K., 1997. A history of high-temperature Io volcanism: February 1995 to May 1997. Geophys. Res. Lett. 24, 2451–2454. Spencer, J.R., Rathbun, J.A., Travis, L.D., Tamppari, L.K., Barnard, L., Martin, T.Z., McEwen, A.S., 2000. Io’s thermal emission from the Galileo Photopolarimeter–Radiometer. Science 288, 1198–1201. Stansberry, J.A., Spencer, J.R., Howell, R.R., Dumas, C., Vakil, D., 1997. Violent silicate volcanism on Io in 1996. Geophys. Res. Lett. 24, 2455– 2458. Veeder, G.J., Matson, D.L., Johnson, T.V., Blaney, D.L., Goguen, J.D., 1994. Io’s heat flow from infrared photometry: 1983–1993. J. Geophys. Res. 99, 17,095–17,162. Watanabe, T., 1940. Eruptions of molten sulfur from the Siretoko–Iosan volcano, Hokkaido, Japan. Jpn. J. Geol. Geogr. 17, 289–310. Williams, D.A., Greeley, R., Lopes, R., Davies, A.G., 2001a. Evaluation of sulfur flow emplacement on Io from Galileo data and numerical modelling. J. Geophys. Res. 106 (E12), 33,161–33,174. Williams, D.A., Davies, A.G., Keszthelyi, L.P., Greeley, R., 2001b. The Summer 1997 eruption at Pillan Patera on Io: implications for ultrabasic lava flow emplacement. J. Geophys. Res. 106 (E12), 33,105–33,120.