The Effects of Melt on Impact Craters on Icy Satellites and on the Dynamics of Io's Interior

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Authors Elder, Catherine Margaret

Publisher The University of Arizona.

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Link to Item http://hdl.handle.net/10150/556825 THE EFFECTS OF MELT ON IMPACT CRATERS ON ICY SATELLITES AND ON THE DYNAMICS OF IO’S INTERIOR

Catherine Margaret Elder

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF PLANETARY SCIENCES

In Partial Fulﬁllment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

2015 2

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Catherine Margaret Elder, titled The eﬀects of melt on impact craters on icy satellites and on the dynamics of Io’s interior and recommend that it be accepted as fulﬁlling the dissertation requirement for the Degree of Doctor of Philosophy.

Date: 7 April 2015 Adam Showman

Date: 7 April 2015 Shane Byrne

Date: 7 April 2015 Isamu Matsuyama

Date: 7 April 2015 Alfred McEwen

Date: 7 April 2015 Randy Richardson

Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the ﬁnal copies of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulﬁlling the dissertation requirement.

Date: 7 April 2015 Dissertation Director: Adam Showman 3

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulﬁllment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: Catherine Margaret Elder 4

ACKNOWLEDGEMENTS

First, I thank my advisor, Adam Showman, for all his help and career advice over the years. Adam is always excited about science with an enthusiasm that few professors have, so meeting with him to discuss my research was always fun. In addition to Adam, I would like to thank Shane Byrne and Christopher Hamilton for mentoring and helping me navigate the search for a postdoc position. Thanks to Randy Richardson for always being the person on my committee to step back and ask what I need to do next to continue making progress. And thanks to Alfred McEwen and Isamu Matsuyama for providing many useful suggestions and helping me see the implications of my research in a broader context. I would also like to thank my other co-authors. Most recently, Paul Tackley, who taught me how to use his mantle convection code and helped me interpret the results. Thanks to Jay Melosh and Veronica Bray who were my first advisors when I started graduate school. Veronica in particular worked closely with me on the project and encouraged me to develop career skills such as writing proposals and reviewing papers. My path to LPL started at Cornell where I was welcomed into the planetary science community by the faculty and the graduate students. The graduate students treated me as a peer and gave me a glimpse into what grad school would be like. They are still some of the people I most look forward to seeing at conferences. Among the faculty, I would especially like to thank Jim Bell for helping me select classes, summer internships, and ultimately a grad school. He also led the first geology trip I ever attended where I saw an impact crater and volcanoes for the first time. Also thanks to all my physics buddies at Cornell who made problem sets a lot more fun and do-able. While in grad school, I have been lucky enough to have the support of many friends near and far. I thank them all for their support and making life more fun. In particular, I would like to thank the LPL grad students who were here at the same time as me for their advice, camaraderie, and friendship. Finally, I would like to thank my family: my sister, Jean, for her friendship and support, and my parents. I am very thankful that my parents believed me when I was nine years old and told them that I wanted to be an astronomer when I grew up. They bought me a telescope even though light pollution in Boston is terrible, gave me astronomy books, and took me to ‘Astronomy day’ at the Boston Museum of Science every year. And Casey, who I met early in grad school and who has been my main source of moral support and encouragement since then. I am so happy that whether I have a good day or a bad day, I always get to go home to Casey and the cats. 5

DEDICATION

To my parents, who always encouraged my interest in astronomy. 6

TABLE OF CONTENTS

LISTOFFIGURES ...... 9

LISTOFTABLES ...... 11

ABSTRACT ...... 12

CHAPTER1 Introduction ...... 14 1.1 Planetaryheatloss ...... 14 1.1.1 Mantleconvection...... 15 1.1.2 MarsandtheMoon...... 16 1.1.3 Earth ...... 17 1.1.4 Venus ...... 17 1.1.5 Io...... 18 1.2 Surface processes as a window to the interior ...... 19 1.2.1 Surface expression of mantle convection ...... 20 1.2.2 Crater morphology as a probe of the subsurface ...... 21 1.3 Mapofthedissertation...... 23

CHAPTER 2 The theoretical plausibility of central pit crater formation via meltdrainage ...... 24 2.1 Introduction...... 24 2.2 Meltvolume...... 25 2.3 Fracturevolumeestimate...... 28 2.4 Meltdrainage ...... 31 2.4.1 Fracturedistribution ...... 32 2.4.2 Drainage through plane parallel fractures ...... 33 2.5 Results...... 41 2.6 Discussion ...... 46 2.6.1 Diﬀerences between rock and ice ...... 46 2.6.2 Limitationsandassumptions...... 47 2.6.3 The role of viscous dissipation ...... 50 2.6.4 Mars...... 51 2.6.5 Othericysatellites ...... 51 2.7 Conclusions ...... 54 7

TABLE OF CONTENTS – Continued

CHAPTER 3 Melt migration in Io’s convecting mantle ...... 56 3.1 Introduction...... 56 3.2 Themodel...... 61 3.2.1 Meltsegregation ...... 61 3.2.2 Convective scaling laws ...... 65 3.2.3 Melt segregation and convection ...... 71 3.3 Results...... 72 3.4 Discussion ...... 80 3.4.1 ComparisontoEarth ...... 80 3.4.2 Comparisontoobservations ...... 83 3.5 Conclusions ...... 84

CHAPTER 4 Convection in Io’s partially molten mantle ...... 86 4.1 Introduction...... 86 4.2 Model ...... 90 4.2.1 Rheology ...... 92 4.2.2 Composition...... 93 4.2.3 Meltinganderuption ...... 93 4.2.4 Tidalheating ...... 96 4.3 Results...... 97 4.3.1 Statisticalsteadystatebehavior ...... 97 4.3.2 Tidal heating distribution ...... 105 4.3.3 Eruptionparameterization ...... 106 4.3.4 Inﬂuence of reference viscosity ...... 108 4.3.5 Inﬂuence of permeability ...... 109 4.3.6 Heatingrate...... 110 4.4 Discussion ...... 112 4.5 Conclusion...... 118

CHAPTER 5 Comparison of the one-dimensional model and the two- dimensional numerical simulations ...... 120 5.1 Directcomparison...... 120 5.2 Convective scaling laws ...... 122 5.3 Meltmigrationmodel...... 126 5.4 SummaryofComparison ...... 128

CHAPTER 6 Conclusions and Future Work ...... 130 6.1 Centralpitcraterformation ...... 130 6.2 DynamicsofIo’smantle ...... 132 8

TABLE OF CONTENTS – Continued

6.2.1 Melt generation and migration in an upwelling column of man- tleonIo...... 133 6.2.2 Mantle convection in Io’s partially molten mantle ...... 134 6.2.3 Comparison of results to observations ...... 135

REFERENCES...... 137 9

LIST OF FIGURES

2.1 Isiscentralpitcrater ...... 24 2.2 Asketchofcentralpitcraterformation ...... 26 2.3 Widthsofimpactgeneratedfractures ...... 34 2.4 Asketch ofmelt drainagethroughfractures ...... 36 2.5 Dependence of drainage velocity on fracture width ...... 38 2.6 Solidification dependence on initial temperature ...... 44 2.7 Fractured, melted, drained and observed pit volumes ...... 45 2.8 Volume of melt able to drain on different bodies ...... 46 2.9 Volumeofslushabletodrain ...... 47 2.10 Volume of melt able to drain on different icy satellites ...... 52 2.11 Volume of melt generated on different icy satellites ...... 52

3.1 AsketchofIo’smantle ...... 60 3.2 Radialprofilesforthenominalcase ...... 74 3.3 Radial profiles for 25% eruption efficiency ...... 80 3.4 The effect of internal heating on radial profiles ...... 81

4.1 Io’ssurface ...... 88 4.2 Non-uniformtidalheatingratefields ...... 98 4.3 Radial profiles for a non-uniform tidal heating rate ...... 99 4.4 Radial profiles for a uniform tidal heating rate ...... 100 4.5 Uniformtidalheatingratefields ...... 103 4.6 Heatfluxcontributions ...... 105 4.7 Heat flux contributions over 100,000 yrs ...... 105 4.8 Melt fraction radial profiles for different eruption parameterizations . 108 4.9 Melt fraction radial profiles for different permeability constants . . . . 110 4.10 Radial profiles for different heating rates ...... 113 4.11 Temperature fields at different heating rates ...... 114 4.12 Melt fraction fields at different heating rates ...... 115 4.13 Heat flux contributions at different heating rates ...... 116

5.1 Two dimensional melt fraction and vertical velocity fields...... 121 5.2 Horizontally averaged radial melt profiles of two-dimensional simulations...... 122 5.3 One dimensional model melting profile ...... 123 10

LIST OF FIGURES – Continued

5.4 Melt fraction, melt velocity, and volcanic heat ﬂux as functions of temperatureandmantlevelocity...... 126 11

LIST OF TABLES

2.1 Constants - Impact melt on Ganymede ...... 28 2.2 Constants-Impactmeltdrainagerock ...... 41 2.3 Constants - Impact melt drainage water ...... 42

3.1 Rheologicalconstants...... 73 3.2 Sensitivityofresults ...... 76 3.3 Sensitivity of results to velocity scaling law ...... 77 3.4 Sensitivity of results to melt fraction in the boundary layer ...... 78

4.1 Modelparametervalues ...... 91 4.2 Results for different heating rate distributions ...... 106 4.3 Results for different eruption parameterizations ...... 107 4.4 Results for different Tlith‘s ...... 109 4.5 Results for different reference viscosities ...... 109 4.6 Results for different permeability constants ...... 111 4.7 Resultsfordifferentheatingrates ...... 113

5.1 One-dimensional and two-dimensional comparison ...... 125

6.1 Results and observations comparison ...... 136 12

ABSTRACT

Over the last fifty years, our knowledge of the Solar System has increased expo- nentially. Many planetary surfaces were seen for the first time through spacecraft observations. Yet the interiors of most planetary bodies remain poorly studied. This dissertation focuses on two main topics: the formation of central pit craters and what this reveals about the subsurface volatile content of the target material, and the mantle dynamics of Io and how they relate to the extensive volcanism on its surface. Central pit craters are seen on icy satellites, Mars, the Moon, and Mercury. They have terraced rims, flat floors, and a pit at or near their center. Several formation mechanisms have been suggested. This dissertation assesses the feasibility of central pit crater formation via drainage of impact melt through impact-generated fractures. For impacts on Ganymede, the expected volume of melt and volume of fracture space generated during the impact and the volume of melt able to drain before fractures freeze shut all exceed the observed central pit volumes on Ganymede. This suggests that drainage of impact melt could contribute to central pit crater formation on Ganymede. Molten rock draining through solid rock fractures will freeze shut more rapidly, so this work suggests that impact melt drainage is unlikely to be a significant factor in the formation of central pit craters on rocky bodies unless a significant amount of volatiles are present in the target. Io is the most volcanically active body in the Solar System. While volcanoes are most often associated with plate tectonics on Earth, Io shows no signs of plate tectonics. Previous work has suggested that Io could lose a significant fraction of its internal heat through volcanic eruptions. In this dissertation, I investigate the relationship between mantle convection and magma generation, migration by porous flow, and eruptions on Io. I couple convective scaling laws to a model solving the 13 two-phase flow equations applied to a rising column of mantle. I show that Io has a partially molten upper mantle and loses the majority of its internal heat through volcanic eruption. Next, I present two-dimensional numerical simulations that self-consistently solve the two-phase flow equations including mantle convection and magma generation, migration by porous flow, and eruption. These simulations produce a high heat flux due to volcanic eruption, a thick lithosphere, a partially molten upper mantle, and a high eruption rate—all consistent with observations of Io. This model also reveals the eruption rate oscillates around the statistical steady state average eruption rate suggesting that the eruption rate and total heat flux measurements from the past 35 years may not be representative of Io’s long term behavior. 14

CHAPTER 1

Introduction

1.1 Planetary heat loss

According to the accretionary model for the formation of the Solar System, planets grow by accreting planetesimals. Most of the planetesimals’ kinetic energy is converted to heat during the impacts (de Pater and Lissauer, 2010). In larger bodies, this heat warms the body enough to enable heavier metals to sink through the lighter silicates, resulting in a differentiated body. As the metal sinks towards the center of the body, it releases potential energy, which is converted to heat (de Pater and Lissauer, 2010). Thus most planetary bodies start with a hot interior. Rocky planets experience additional heating through the decay of radioactive isotopes (e.g. uranium, thorium, and potassium; Schubert et al., 2001). In differentiated rocky planetary bodies, as the core cools, it loses its heat to the base of the planetary mantle. Tidal dissipation can also heat planetary mantles (Peale et al., 1979). On Earth, more than half of the energy currently in the mantle comes from the decay of radioactive isotopes (Schubert et al., 2001). All of these sources heat the mantle enough that rock can behave viscously. The surface of planetary bodies is significantly cooler than their interiors even for bodies with dense atmospheres. With a warm viscous interior and a cold top surface, convection is more efficient than conduction at removing heat from a planetary mantle . Although mantle convection likely occurred on most planetary bodies at some point in their history, how it affects the surfaces we observe today varies greatly. Venus and Earth are similar in size and presumably similar in composition, yet Earth has plate tectonics and Venus does not. Though Io is very small, it is heated by tidal heating, but it does not have plate tectonics either. This dissertation will discuss how most of Io’s internal heat is removed from its mantle by volcanic eruptions. 15

The Moon and Mars, also relatively small, do not show signs of signiﬁcant recent tectonism and volcanism which implies that their interiors are no longer convecting. Now, we are starting to discover rocky exoplanets. Petigura et al. (2013) found that 22% of sun-like stars are expected to have an Earth sized planet in their habitable zones. However, we are currently unable to predict how these exoplanets might lose their internal heat. We must now try to identify the main factors that cause a planet to evolve into something like Earth, Venus, or Io.

1.1.1 Mantle convection

Whether heat will be transported by conduction or convection can be estimated using the non-dimensional Rayleigh number: g∆ρd3 Ra = , (1.1) ηκ where ρ is the density of the fluid, g is the acceleration due to gravity, ∆ρ is the density contrast across the fluid, d is the depth of the fluid layer, η is the viscosity of the fluid, and κ is the thermal diffusivity. In most planetary mantles, the density contrast is primarily due to the temperature contrast between the cold surface and the warm mantle interior. Warm rock is less dense than cold rock and the density contrast can be written as α∆T where α is the thermal expansivity and ∆T is the temperature contrast. The Rayleigh number compares the timescale for heat transport by advection to the timescale for heat transport by diffusion. The timescale for heat transport by advection depends on both the buoyancy force driving flow and the viscous drag of the fluid. If heat can diffuse across the domain faster than it can be transported by advection across the domain, then the heat will be lost through conduction rather than convection. In a vigorously convecting fluid layer, the fluid interior is well mixed and close to isothermal. A cold top surface causes the formation of a boundary layer between the surface and the interior. If the fluid layer is heated from below, a hot boundary layer develops at the base of the layer as well. Most of the temperature drop between the bottom and the top occurs across these boundary layers when the Rayleigh 16 number is high. Cold rock is more dense than hot rock, so the boundary layers are gravitationally unstable. Thus portions of the top boundary layer break off and sink through the interior and portions of the hot bottom boundary layer break off and rise through the interior. In the case of an internally heated viscous fluid layer, only the cold top boundary layer is present and the sinking of cold plumes forces motion in the rest of the layer (Schubert et al., 2001). Planetary mantles are heated though some combination of internal heating by the decay of radioactive isotopes and tidal dissipation and heating from below due to a cooling core. In the above description, we have assumed the fluid has a constant viscosity, but the viscosity of rock is highly temperature dependent. On many planets, this causes the formation of a cold stagnant lid which is too rigid to flow even on geologic timescales. Much of the temperature contrast between the hot interior and the cold surface occurs across this stagnant lid which is too rigid to participate in convection. A thermal boundary layer, which can participate in convection, develops between the stagnant lid and the isothermal mantle. It is the temperature contrast across this thermal boundary layer that drives convection. Beneath the stagnant lid, convection is very similar to the simpler case of isoviscous convection.

1.1.2 Mars and the Moon

Mars and the Moon do not show any evidence of currently active tectonics or volcanism, so if the deep interior is still convecting, it is no longer able to alter the surface. On small bodies, the ratio of surface area to volume is higher than on large bodies, so heat loss is more efficient, and they cool faster. Crustal differentiation may have been more efficient on the Moon and Mars than on the Earth which would have fractionated radiogenic isotopes into the crust (Schubert et al., 2001). These argu- ments and the observations of the surface imply that convection has likely ceased in the interiors of Mars and the Moon (Schubert et al., 2001). The evidence of volcanism and tectonics in the early history of Mars suggests that convection used to occur. In particular, the large volcanoes in the Tharsis mound imply pressure release melting in a large mantle plume (Schubert et al., 2001). 17

On the Moon, Apollo samples allow us to more easily establish the timeline of the evolution of the surface. The oldest rocks on the lunar surface are in the lunar highlands. Apollo samples indicate that the lunar highlands crystalized by 4.0 Ga (Schubert et al., 2001). The mare basalts, darker material ﬁlling near side large impact basins, were emplaced 3.8-3 Ga. They resemble typical terrestrial basalts in their major element composition. However, isotope and concentration ratios for lunar basalts vary more than terrestrial mid-ocean ridge basalts, which implies distinct source regions and less mixing in the lunar mantle. Mare volcanism could have been produced either by in situ radioactive heating or by pressure release melting in ascending mantle material (Schubert et al., 2001). Although most volcanism on the Moon ceased 3 Ga, recent high resolution images from the LRO mission found 70 small (100-5,000 m across) anomalous volcanic regions which crater distributions suggest are younger than 100 million years (Braden et al., 2014). This relatively recent volcanism, suggests a long decline in lunar volcanism.

1.1.3 Earth

On Earth, which is still actively convecting, the convection cells extend to the planet’s surface. The tectonic plates are the cold upper thermal boundary layer and subduction is the downwelling of a portion of that boundary layer (Schubert et al., 2001). Convection in Earth’s mantle is driven by the decay of radioactive isotopes, long-term secular cooling, and heat from the core. More than half of the heat budget comes from the decay of radioactive isotopes (Schubert et al., 2001). Heat- ing the cold subducted plates removes approximately 70% of the mantle heat ﬂux. The ascent of hot mantle plumes removes much of the remaining heat, but partial delamination of the cold lithosphere could also contribute (Schubert et al., 2001).

1.1.4 Venus

Venus and Earth are similar in size and expected to be similar in composition. This implies that they have a similar internal heat budget. However, Venus does 18 not currently have plate tectonics (Schubert et al., 2001). If Venus were losing its internal heat through stagnant lid convection rather than plate tectonics, to lose the heat currently being produced by the decay of radioactive isotopes, it would require a lithosphere thinner than 25 km (Schubert et al., 2001). However, such a thin lithosphere does not seem likely based on the high topography and pristine (not viscously relaxed) craters observed on Venus’s surface (Schubert et al., 2001). The spatial distribution of craters on Venus is not statistically diﬀerent from a random distribution (Schaber et al., 1992). This implies that Venus’s whole surface was resurfaced at the same time. Crater counts suggest that this global resurfacing event occurred 300–700 Mya (Herrick, 1994; Schaber et al., 1992; Strom et al., 1994). This resurfacing event has been interpreted to correspond to either the cessation of plate tectonics or near-global catastrophic subduction (Schubert et al., 2001). Based on the lack of plate tectonics and Venus’s estimated lithospheric thickness, it seems likely that heat has been building up in Venus’s interior ever since this resurfacing event (Nimmo and McKenzie, 1998). Thus, Venus could lose heat through stagnant lid convection punctuated by occasional episodes of catastrophic overturn (Turcotte, 1993). Coronae have been suggested as a possible example of present day subduction on Venus, which could be the initiation of a new episode of resurfacing (Sandwell and Schubert, 1992; Turcotte, 1993). Recent numerical simulations of mantle convection and magmatism show that episodic lid overturn interspersed by periods of quiescence oﬀer the simplest match to observations of Venus compared to stagnant lid convection with heat loss through volcanic eruptions (Armann and Tackley, 2012).

1.1.5 Io

Io is similar in size to the Moon, so it is expected to have lost its heat from accretion and differentiation. However, the Laplace resonance with Europa and Ganymede pumps up Io’s orbital eccentricity, which causes the tides induced by Jupiter to stretch and compress Io through its orbit (Peale et al., 1979; Yoder and Peale, 1981). This tidal flexing heats Io’s interior and currently produces a surface heat flux 19 of 3 ± 1 W m2 (Veeder et al., 2004). Solid state stagnant lid convection cannot produce this high surface heat flux if heat loss is in equilibrium with heat production through tidal dissipation (Moore, 2003). Stagnant lid convection would also require a thin lithosphere which is inconsistent with the tall mountains on Io’s surface (O’Reilly and Davies, 1981). However, Io does not show any signs of plate tectonics either. This dissertation will show that partial melting of Io’s mantle and heat loss through volcanic eruptions in addition to convection beneath a thick stagnant lithosphere can explain Io’s internal heat loss and match observational constraints.

1.2 Surface processes as a window to the interior

On Earth, seismic tomography allows us to image Earth’s mantle. We can now detect the cold roots of continental shields and even cold subducting slabs (Fowler, 2006). However, minimal if any seismic data exists for all other planetary bodies. Several other methods can also probe planetary interiors. Radar can be used to probe the near subsurface (e.g. ice on Mars (Bramson et al., 2014)). Density variations in the subsurface can be detected by taking careful measurements of a body’s gravity field and removing the contribution from topography. This has been done with great success for the Moon by GRAIL (Zuber et al., 2013), but the gravity fields of many planetary bodies are not known to a precision that can probe the details of the interior. However, simply measuring the moment of inertia, the Love number, the radius and density of a planetary body can be used to estimate the thickness of layers of significantly different densities for example the size of a metallic core. However, even these values are uncertain for many bodies in our Solar System (e.g. Anderson et al., 2001). Magnetometers can be used to detect either a dynamo produced by a body or an induced magnetic field. The presence of a dynamo on a rocky planetary body reveals that it has a convecting liquid layer in its metallic core. The presence of an induced magnetic field reveals a layer of high conductivity which based on other knowledge about the specific planetary body could indicate a salty liquid water layer (e.g. Europa (Kivelson et al., 2000)) or a layer of partially molten 20 rock (e.g. Io (Khurana et al., 2011)). Although gravity and magnetic measurements offer many important insights into the interiors of planetary bodies, many questions remain. Tectonism and volcanism are byproducts of mantle convection, so observing the surfaces of planetary bodies can provide additional constraints on the structure and evolution of their mantles.

1.2.1 Surface expression of mantle convection

Even on Earth, seismic tomography capable of resolving subducting plates is a relatively recent tool, because it requires a global network of digital seismic stations (Fowler, 2006). Much of evidence supporting plate tectonics comes from observing the surface of the Earth. As soon as the first precise global maps were published, people noticed that the continents seemed to fit together like puzzle pieces. More detailed geologic studies showed that mountain belts, rock types, and fossil types on the coast of Africa continued on the coast of South America (Schubert et al., 2001). However, plate tectonics was not widely accepted until magnetic field anomalies parallel to mid-ocean ridges were discovered. New ocean crust is formed at a mid- ocean ridge, and as it crystalizes and cools through the Curie temperature, it is magnetized by Earth’s magnetic field. Drilling projects confirmed that the age of the oceanic crust increases with distance from the mid-ocean ridge (Maxwell et al., 1970). Now it is also recognized that on Earth, volcanism and earthquakes occur primarily at plate boundaries (Isacks et al., 1968). The Hawaiian Islands are another surface expression of plate tectonics. The age of the islands increases with distance from the currently active island, because the islands sit on top of a plate that moves over a mantle plume (Wilson, 1963). These types of features observed on another planet would now be considered possible evidence for plate tectonics. Io does not show any signs of plate tectonics, but its surface features do provide insight into its interior. The most dramatic characteristic of Io’s surface is extensive volcanism. This volcanism contributes to a high surface heat flux (Veeder et al., 2004). Io’s extensive volcanism, high surface heat flux, and lack of plate tectonics implies that these volcanoes may be very important to the evolution of Io’s mantle. 21

Io’s surface also hosts many tall mountains which do not appear to be volcanic (Schenk and Bulmer, 1998). The high resurfacing rate on Io has lead to the theory that forced subsidence of surface material causes compression in the lithosphere and forces the uplift of tall mountains (Schenk and Bulmer, 1998). These tall mountains suggest that Io must have a thick lithosphere capable of supporting topography as high as 18 km (O’Reilly and Davies, 1981; Schenk et al., 2001). Such a lithosphere would limit conductive heat flow, so that conduction alone could not produce Io’s high observed surface heat flux (O’Reilly and Davies, 1981). The distribution of these volcanic and tectonic mountains could also hold clues about the distribution of internal heating in Io’s mantle (Segatz et al., 1988). Tidal heating through the whole depth of the mantle would produce a high surface heat flux at Io’s poles and a lower surface heat flux near the sub-Jovian and anti-Jovian points. Tidal heating primarily in a partially molten asthenosphere would produce a maximum surface heat flux near the equator and a very low surface heat flux at the poles (Segatz et al., 1988). Convection can decrease the spatial variation of surface heat flux, but variations of a few percent are still expected (Tackley et al., 2001; Tackley, 2001). Recent analysis of the distribution of volcanoes on the surface supports the asthenospheric heating model, but finds a 30–60◦ eastward offset of the volcanoes from the predicted location of maximum surface heat flux (Hamilton et al., 2013), which raises more questions about Io’s mantle and its relationship to the surface.

1.2.2 Crater morphology as a probe of the subsurface

Crater morphology is another tool that can be used to study the subsurface of planets. Small craters probe just the near subsurface while large craters will ex- cavate deeper into the planet, sometimes excavating a large fraction of the crust (e.g. Wieczorek et al., 2013). Terracing in craters indicates a change in the strength of the target with depth. Bramson et al. (2014) measures the depth of terraces in craters in Arcadia Planitia Mars where radar sounding data has detected a subsurface radar-transparent layer. By comparing the depth of the terraces to the 22 delay time between the surface and subsurface radar reflections, Bramson et al. (2014) determined the wave velocity and thus, dielectric constant of the subsurface radar-transparent layer and found that it is consistent with pure water ice. Larger craters can be used to probe deeper into the subsurface. Central peaks in complex craters form when the target rock is lifted during the impact. Thus on Europa, central peak craters can be used to estimate a minimum thickness of the outer ice shell (Turtle and Pierazzo, 2001). However, two of the largest impact features on Europa, Tyre and Callanish, do not have central peaks. Tyre and Callanish are flat circular features surrounded by concentric rings of fractures that do not have central peaks or raised rims, but are surrounded by radially distributed secondary craters (Moore et al., 2001). Moore et al. (2001) concluded that Tyre and Callanish formed in a target that was mechanically weak at depth. This mechanically weak layer could either be a liquid water ocean or brine-rich convecting ice (Moore et al., 2001). These represent only a few examples of how crater morphology has been used to study the subsurface of planetary bodies, but they demonstrate the utility in understanding the formation mechanisms specific to different crater morphologies. Central pit craters are a type of complex crater most common on icy satellites and Mars that exhibit terraced rims, flat floors, and a pit at or near the center (Passey and Shoemaker, 1982). The formation mechanism for this type of crater is unknown, but their presence primarily on icy satellites and Mars implies that some ice is required for their formation. However, recently this type of crater has been identified on the Moon and Mercury which are not thought to harbor any subsurface volatiles (Xiao and Komatsu, 2013; Xiao et al., 2014). Several mechanisms have been proposed to explain the formation of central pit craters including collapse of a central peak in weak ice (Passey and Shoemaker, 1982), impact into a layered target (Greeley et al., 1982), explosive release of volatiles (Carr et al., 1977), and drainage of impact melt or impact related debris (Croft, 1981). In this dissertation, we test the hypothesis that central pit craters can form via drainage of impact melt into impact generated fractures. Understanding the conditions that lead to the formation of this type of crater could potentially lead to a method able to determine locations 23 on planetary bodies that contained subsurface volatiles at the time of impact or it could indicate a subsurface layer of a different strength that prohibits the formation of a central peak.

1.3 Map of the dissertation

This dissertation will attempt to add to our understanding of crater morphology across the Solar System and mantle convection in Io. In chapter 2, I evaluate the feasibility of central pits forming in craters due to the drainage of impact melt into impact generated fractures. I do this by estimating the volume of melt produced during an impact, the volume of fracture space generated during an impact, and the volume of melt able to drain before freezing fractures shut and comparing these to the observed volume of central pits in craters on Ganymede. Chapter 3 presents a 1-dimensional model of melting in Io’s mantle. Here, I couple a 1-dimensional model of melt generation and migration in an ascending column of mantle material to convective scaling laws. The melt generation and migration model depends on the velocity and temperature of the upwelling column. The convective scaling laws depend on the melt fraction in the mantle and the amount of heat lost due to volcanic eruption. Thus by coupling the convective scaling laws and the melting model, I can calculate the amount of magma in Io’s mantle and how much heat it loses through volcanic eruption rather than conduction through a stagnant lid. In chapter 4, I present two-dimensional numerical simulations of mantle convection that include the generation, migration, and eruption of magma in Io’s mantle. Here I investigate the distribution of magma in Io’s mantle, the development of a lithosphere, and the relative importance of heat loss due to volcanic eruption and conduction through the stagnant lithosphere. Chapter 5 compares the results of the models presented in chapters 3 and 4. Chapter 6 summarizes the main conclusions of this dissertation and discusses possible avenues of future work. 24

CHAPTER 2

The theoretical plausibility of central pit crater formation via melt drainage

The material in this chapter has been published in the journal Icarus: Elder, C. M., V. J. Bray, and H. J. Melosh (2012). The theoretical plausibility of central pit crater formation via melt drainage. Icarus, 221(2), pp. 831-843.

2.1 Introduction

Central pit craters occur most commonly on Mars and Jupiter’s icy moons, Ganymede and Callisto. They exhibit terraced rims, flat floors and a pit at or near the center (Passey and Shoemaker, 1982). An example of a central pit crater is shown in figure 2.1. On Ganymede, they typically occur in craters with diameters ranging from 50-80 km (Bray et al., 2012). Central pit craters are not found on all icy satellites, so they are not considered to be a consequence purely of cratering in ice (Passey and Shoemaker, 1982). However, the lack of central pit craters on most rocky bodies besides Mars suggests that some ice is required. An understanding of why central pits form in some impact craters but not all may thus provide insight into the quantities and location of ice on terrestrial planets and satellites.

Figure 2.1: Galileo SSI image of Isis crater and its topographic proﬁle. The crater is a 73 km diameter central pit crater on Ganymede. North is up in the image, and the proﬁle was taken from the southwest to the northeast across the center of the crater. 25

Several mechanisms have been proposed to explain the formation of central pits, including impact into a layered target (Greeley et al., 1982), collapse of a central peak in weak ice (Passey and Shoemaker, 1982), explosive release of volatiles (Carr et al., 1977) and drainage of impact melt or impact related debris (Croft, 1981). In this paper, we investigate the feasibility of drainage. Croft (1981) originally suggested that drainage of brecciated rock, ice and melt-water could produce central pits in crater floors (figure 2.2). However, drainage of large quantities of solid brecciated material could lead to pits in all large lunar craters which is not observed. Therefore, more recent work restricts the draining material to actual molten material (Bray, 2008; Elder et al., 2010; Senft and Stewart, 2011). Here, we suggest that central pits could form by drainage of impact melt if the volume of fracture space and volume of melt generated both exceed the volume of the pit and if a significant amount of melt can drain before the fractures freeze shut. Furthermore, if drainage of impact melt is a viable central pit formation mechanism, it must form pits in craters in ice or ice-rock mixes, but not those in volatile-poor targets. We investigate this hypothesis by estimating the volume of impact melt able to drain into sub-crater fractures in craters with diameters ranging from 5 to 200 km on Ganymede. First, in section 2.2, we calculate the melt volume generated during an impact. In section 2.3, we estimate the amount of fracturing that occurs in an impact event. Finally, in section 2.4, we estimate the volume of melt able to drain into the fractures before they freeze shut. We compare the observed volumes of central pits on Ganymede from Bray et al. (2012) to the volume of melt generated, the volume of fractures generated, and the volume of melt able to drain before freezing. Finally, we compare the volume of impact melt expected to drain before melt solidification for craters on Earth, the Moon, and Ganymede.

2.2 Melt volume

The volume of material that melts during an impact depends on the temperature of the target, impact angle, target porosity, target material, impactor material, im- 26

Figure 2.2: A) A sketch of a post-impact pre-drainage crater on Ganymede. B) A sketch of a crater on Ganymede after the melt has drained into the fractures depict- ing empty fractures (grey), filled fractures (black), and undrained melt remaining in the pit (black). Fractures may not be filled completely and some melt may remain in the floor of the pit. Melt could drain anywhere in the crater, but most drainage will occur in the center of the crater where the melt is concentrated and the fracture density is highest. pactor size, and impactor velocity (Bjorkman and Holsapple, 1987; Pierazzo et al., 1997; Pierazzo and Melosh, 2000; Artemieva and Lunine, 2005; Kraus et al., 2011; Barr and Citron, 2011). Melt is found on the terraced rims and on the floors of complex craters (Melosh, 1989). On Earth impact melt deposits often form coher- ent melt sheets in discrete lenses or annular sheets (Grieve and Cintala, 1992). In the case of impacts into ice, hydrocode modeling suggests that the melt forms a “hot plug” of material with a size and location similar to central pits (Bray, 2008; Senft and Stewart, 2011). In order to compare the volume of melt available for drainage to the observed volumes of central pits on Ganymede, we require a scaling law for the volume of melt as a function of final crater diameter. Direct observations of impact melt deposits are available only on Earth. Estimating impact melt volumes on Ganymede therefore requires other methods. Computer hydrocodes can simulate impact crater formation in a range of target materials which allows for the 27 investigation of melt production on different planetary bodies including icy satellites (Ahrens and Okeefe, 1977; Pierazzo et al., 1997; Artemieva and Lunine, 2003, 2005; Kraus et al., 2011; Barr and Citron, 2011). To find the relationship between the volume of melt generated and the final crater diameter, we combine scaling laws relating melt volume to projectile size, transient crater diameter to projectile size, and transient crater diameter to final crater diameter. Kraus et al. (2011) used hydrocode simulations to develop scaling laws for the mass of melt produced in impacts into icy targets as a function of the mass and velocity of the impactor. These scaling laws depend on the impact angle, the porosity, the target temperature, and the melting energy. We use this scaling law and values given in table 2.1 to find the ratio of the mass of impact melt generated to the mass of the impactor. Assuming the target and projectile have the same density, this ratio is equivalent to the ratio of the volume of impact melt generated to the volume of the projectile. We assume a spherical projectile and use pi scaling laws to relate the projectile diameter to the apparent crater diameter for a 45 degree impact angle (Melosh, 1989; Kraus et al., 2011). The pi scaling laws depend on the gravity,

impact velocity, and parameters CD and β which must be determined by laboratory or numerical experiments (Melosh, 1989; Kraus et al., 2011). Combining these two scaling laws and using values given in table 2.1 yields the volume of melt generated as a function of the apparent crater diameter. The apparent crater diameter is related to the transient crater diameter by a factor of 1.3 (Melosh, 1989; Holsapple, 1993; Kraus et al., 2011). We relate the transient crater diameter to the final crater diameter using the theoretical scaling law presented in Bray et al. (2012). This relationship has also been determined using hydrocode modeling (Kraus et al., 2011) and previous observations of craters (McKinnon and Schenk, 1995). These three different scaling laws yield similar results for impacts into ice. We use Bray et al. (2012)’s results, because they include a larger survey of craters on Ganymede than previous studies, and they represent the minimum estimate of transient crater diameter which yields the minimum estimate of melt generated. If Bray et al. (2012)’s scaling law implies a sufficient volume of melt, the other scaling laws will as well. 28

Table 2.1: Constants used to calculate the amount of impact melt generated on Ganymede. Value Units Reference Target Temperature 110 K average based on Squyres (1980) Melt energy of ice 8.2×105 J/kg at150K Kraus et al. (2011) Gravity 1.428 m/s2 Murray and Dermott (2000) CD foriceat150K 2.5 ± 0.4 Kraus et al. (2011) β foriceat150K 0.16± 0.02 Kraus et al. (2011)

We find the fraction of total impact melt that remains in the crater using equation 28 in Kraus et al. (2011). We find that, for Ganymede, the volume of melt remaining in a crater of final diameter D is

V = (4.8 × 10−4)D3.1(1+0.2log(D)) (2.1)

where D is in kilometers and V is in cubic kilometers. Here we have assumed the most likely impact angle, 45 degrees, the typical impact velocity at Ganymede, 20 km s−1 (Zahnle et al., 2003), the density of ice for both the target and the impactor, 1000 kg m−3 and no porosity. We also calculated the volume of impact melt expected for vertical impacts, impacts at 30 degrees, impact velocities of 12 km s−1 and 60 km s−1, impactor densities of 3000 kg m−3 and 8000 kg m−3, and porosities of 20%. These values do not change the volume calculated using equation 2.1 by more than a factor of 2. Note that equation 2.1 is speciﬁc to Ganymede. Melt volume depends only on projectile size, impact velocity, impact angle and material properties (e.g. Kraus et al. (2011)). However, because the ﬁnal crater diameter, D, depends on the gravitational acceleration, g, the relation between V and D does depend on g. Combining scaling laws thus introduces a gravity dependence and makes equation 2.1 unique to Ganymede.

2.3 Fracture volume estimate

Impact induced fractures and voids have been inferred beneath many terrestrial impact craters based on gravity observations (Innes, 1961; Pilkington and Grieve, 29

1992; Henkel et al., 2010) and seismic studies (Ahrens et al., 2002; Koeberl et al., 2007). Geologic mapping studies of eroded craters where the central uplift is exposed show that the intensity of faulting increases towards the center of the crater (e.g. Kenkmann, 2002; Kenkmann et al., 2005; Riller et al., 2010). Geologic mapping studies which measure block size find that the block size increases with increasing distance from the crater center (e.g. Kenkmann et al., 2006, 2010). This suggests that the fracture density likely decreases with increasing distance from the crater center. All the material beneath an impact crater will be fractured, but the largest volume of void space will be located beneath the center of the crater. This void space could hold the material that appears to be missing from the center of central pit craters if the volume of the void space beneath the impact crater is large enough to accommodate a volume of melt at least as large as the volume of the central pit (see figure 2.2). In this section, we estimate the volume of void space beneath a crater of a given size. Fracturing occurs due to the compressive shock wave, the release of the rar- efaction wave, and by shear deformation during expansion of the transient cav- ity (Collins et al., 2004). Numerical modeling studies are only beginning to attempt to estimate the volume of fracture void space generated during an impact (Collins et al., 2011). Fractures have been detected beneath impact craters by measuring the gravity anomalies over craters (Pilkington and Grieve, 1992; Dvorak and Phillips, 1977) but only over terrestrial and lunar craters. Due to the lack of geophysical data for the Galilean satellites, fracture volumes beneath craters on Ganymede must be deduced from our knowledge of crater-related fracturing on the Earth and the Moon. The negative gravity anomalies observed over terrestrial craters could be caused by lower density post-impact sediments or lower density impact melt sheets, but the main cause of the low density region is due to fracturing and brecciation of the target rocks (Pilkington and Grieve, 1992). Assuming this, Gauss’s law implies that the gravity anomaly should approximately equal the acceleration due to gravity of the mass of rock that could fill the empty space (Innes, 1961). As a result, the 30 volume of impact generated fracture space can be estimated for terrestrial craters from the observed gravity anomalies by assuming a density for the target material. Pilkington and Grieve (1992) compiled gravity anomalies for 58 Terrestrial impact craters. We assumed a target material density of 2670 kg m−3 based on typical surface density assumptions (Hayford and Bowie, 1912). The target density will vary by location, but our results do not change for rock densities above 2600 kg m−3. A fit to these observations suggests that a terrestrial crater with a diameter D (km) 3 would have a sub-crater fracture volume, Vf (km ) of approximately:

2.5 Vf =0.01D (2.2)

The amount of fracturing that occurs during an impact could plausibly vary with the gravity or strength of the target body. The gravity anomalies associated with four lunar craters were measured and reported as mass deﬁciencies by Dvorak and Phillips (1977). We calculated the fracture volume beneath these lunar craters by assuming a crustal density of 2800 kg m−3 for the Moon (Neumann et al., 1996). The observations suggest that a lunar crater with diameter D (km) would have a sub-crater fracture volume (km3) of approximately:

2.9 Vf =0.03D (2.3)

The lunar and terrestrial relationships between fracture volumes and crater diameters follow similar trends, but the fracture volume beneath lunar craters is larger by 3D0.4. Gravity anomalies have only been measured over 4 lunar craters, so the derived scaling law could differ significantly from reality. Therefore, we use the scaling law based on terrestrial observations and apply the results to Ganymede. The amount of fracturing caused by an impact into ice could be different than the amount of fracturing caused by an impact into rock. Unfortunately gravity anomalies have not been measured over impact craters on any body other than the Earth or the Moon, so changes in fracture volume due to surface composition cannot be directly measured. However, fracturing in ice can be compared to fracturing in 31 rock using Weibull parameters. The Weibull distribution uses two material constants to relate the number of flaws in a material to the tensile strain it experiences (Melosh et al., 1992). It can also be used to predict the distribution of fragment sizes (Melosh et al., 1992). The Weibull parameters for ice (Stewart and Ahrens, 1999) are nearly equivalent to the Weibull parameters for Dresser Basalt (Lindholm et al., 1974), so we expect the fracture distribution and number of fractures in ice and rock to be similar.

2.4 Melt drainage

The melt layer in a complex crater overlies a layer of breccia and fractured bedrock (Melosh, 1989) which could allow melt to partially drain. Pseudotachylites are a possible example of this on Earth. They are dike like structures which consist of angular and rounded wall-rock fragments enveloped by a microcrystalline and spo- radically glassy matrix that crystallized from melt (Riller et al., 2010). They were long thought to have formed in situ from the frictional melting of the local rock. However, recent work asserts that the composition of the fracture wall rock diﬀers from the composition of the matrix in many locations at both Sudbury and Vre- defort impact structures (Speers, 1957; Dressler, 1984; Rousell, 2003; Lieger et al., 2009). Considering this geochemical evidence, Lieger et al. (2009) suggested that these veins formed when impact melt was injected into fractures formed during the impact. The viscosity of water is much lower than the viscosity of molten rock, so this process may occur more readily on icy satellites where the impact melt is water. If this is the case, the draining impact melt could leave behind a central pit. Fracture density is the highest in the center of the crater (Kenkmann, 2002), so drainage would be most likely to occur there. Some of the fracture volume will be located beneath the periphery of the crater as well. However, the fractures beneath a crater are likely interconnected, so even if melt drains primarily in the center of the crater as we assume in this study, it may be able to spread out laterally through the sub-crater fracture network and utilize the whole fracture volume as it drains. 32

We investigate the physics of melt drainage through vertical plane parallel fractures to determine if the volume able to drain is large enough to form a central pit and to compare the volume of melt expected to drain in an icy target to the volume expected to drain in a rocky target.

2.4.1 Fracture distribution

It is not possible to measure the distribution of fracture widths beneath impact craters on Ganymede, Mars, or the Moon. Even on Earth very few impact craters have fractures exposed. The Vredefort Dome in South Africa is an impact structure which has been eroded down 8-11 km exposing rocks containing pseudotachylites uplifted during the impact event (Riller et al., 2010). Estimates of its original diameter range from 200 km to 300 km (Grieve and Therriault, 2000). Riller et al. (2010) measured the width of concentric and radial pseudotachylite dikes at Vre- defort and plotted them against the distance from the crater center at which they occur. They do not report an average pseudotachylite width, but visual inspection of figure 4 in Riller et al. (2010) indicates an average width of approximately 5 cm and a range of fracture widths from 0 to 3 m. Kenkmann and Ivanov (1999) find pseudotachylite veins and dikes of a similar size range at the Rochechouart impact crater in France: < 1mm to several decimeters in width. These distributions might not be representative of all crater diameters and all target materials. We speculate that if an impact forms a smaller crater, it may form narrower fractures as well which would make our assumed average fracture width an upper limit for central pit craters which are typically smaller than Vredefort. Based on the similar Weibull parameters of basalt (Lindholm et al., 1974) and ice (Stewart and Ahrens, 1999), we assume the fracture distributions on rocky planets and icy satellites are similar. Due to the lack of detailed measurements of this sort on other planets and other terrestrial craters we adopt a Vredefort-type size range and assume a Poisson distribution of the frequency of fracture widths (figure 2.3). The total number of fractures beneath an impact crater of diameter D can be estimated using the total fracture volume, the depth of the fracture region, the 33 average fracture width, and the fracture length. Based on a logarithmic fit to data in table 1 of Ahrens et al. (2002) with an R2 value of 0.94, the depth of the fractured

region in kilometers, dfrac, can be estimated by:

dfrac =2.6ln(D) − 2.1 (2.4)

where D is the crater diameter in kilometers. We do not expect lithostatic pressure to affect fracture depth on Ganymede. The maximum depth of a crevasse in a glacier can be estimated based on strain rate, creep parameters, density and gravity (Cuffey and Paterson, 2010). Using the same strain rate as Stewart and Ahrens (1999), creep parameters for ice at 228 K (Cuffey and Paterson, 2010), the density of water ice, and the acceleration due to gravity on Ganymede, we find that fractures on Ganymede could remain open to depth of approximately 2 × 104 km which far exceeds the depth of the fractured region expected for the largest crater we consider, 200 km diameter, based on equation 2.4. We solve drainage in a two-dimensional fracture so the fracture length affects only the number of fractures not how much can drain. For simplicity, we assume the length of the fracture is the crater diameter, D. We find the total number of fractures by dividing the fracture volume (equation 2.2) by the fracture depth (equation 2.4), the average fracture width (5 cm) and the assumed fracture length (D): 77D1.5 N = (2.5) ln(D) − 0.82 where D is the crater diameter in kilometers. The total number of fractures multiplied by the frequency of a given fracture width calculated from the Poisson distribution gives the number of fractures expected for a given fracture width (figure 2.3).

2.4.2 Drainage through plane parallel fractures

We have assumed the same fracture distribution for each body, but diﬀerences in material properties may lead to a larger volume of impact melt draining into fractures on certain bodies. We consider several end-member cases: liquid water 34

3000 D = 5 km D = 50 km D = 100 km 2500 D = 150 km D = 200 km 2000

1500

1000 Number of Fractures 500

0 −3 −2 −1 0 1 10 10 10 10 10 Fracture width (m)

Figure 2.3: The distribution of fracture widths for craters with diameters 5 km, 50 km, 100 km, 150 km, and 200 km. We consider a range of fracture widths from 0.001 to 3 m as observed at Vredefort and an average fracture width of 5 cm also observed at Vredefort.

flowing through water ice on Ganymede, and molten rock flowing through solid rock on Earth and the Moon. As melt flows through a fracture, several processes com- pete to determine whether the hot melt will cool and solidify against the cooler wall rock or heat and melt the wall rock. This process has been investigated previously in the similar problem of magma buoyantly ascending through an open dike (Bruce and Huppert, 1989, 1990; Schubert et al., 2001). Bruce and Huppert (1989)’s description of the thermodynamics of magma ascending through a dike also applies to impact melt draining through a fracture (figure 2.4). The flowing liquid advects heat along the channel while at the same time heat is conducted out of the channel into the colder surroundings (Bruce and Huppert, 1989). This loss of heat can allow the liquid to release latent heat and solidify against the walls which could completely block the channel (Bruce and Huppert, 1989). How- ever, the continual supply of heat from the flowing liquid may heat the wall rock (Bruce and Huppert, 1989). This can melt the channel walls and widen the dike. Whether fluid flowing through a channel solidifies against the channel walls or melts the channel walls depends on the relative temperatures, material properties, and fracture width. Bruce and Huppert (1989, 1990) model the thermal effects of hot magma flowing through a dike. Petford (1994) gives a simple equation for the 35 rate of solidification and the rate of melt back based on suggestions made in Bruce (1989). Although these simple equations are not based on the full set of governing equations they reproduce the results of Bruce and Huppert (1989, 1990) to within 15% (Petford, 1994). Our goal is not to precisely predict the volumes of central pits, rather it is to show that melt drainage is a plausible formation mechanism for central pit craters by comparing the melt drainage on Ganymede to melt drainage on Earth and the Moon. Therefore the more rigorous calculations of Bruce and Huppert (1989, 1990) are not necessary for this study and we use the simple equations in Petford (1994). The liquid water draining on Ganymede has a much lower viscosity than the molten rock draining on Earth and the Moon, so the melt on Ganymede may be able to travel through fractures much more quickly than the melt on Earth and the Moon. Furthermore, although we consider a pure liquid melt in our calculations, post-impact melt is more realistically represented by a mix of water and solid ice fragments. The edges and lower levels of the melt pool experienced lower pressures and temperatures during the impact, so the base of the melt will contain a pro- portionally higher solid ice fraction. However, the lower density ice component will readily rise, producing a stratified impact melt pool with a lower layer of ice-free water insulated by an icy top layer. Therefore our assumption of ice-free impact melt water entering fractures is reasonable. For rock, the solid crystals will have a higher density than the molten rock causing them to settle to the bottom of the melt pool, drain into the fractures with the melt, and possibly clog the fractures which further reduces the amount of melt able to drain after impacts into rocky targets. For our model, we assume melt flows through a vertical plane parallel channel as illustrated in figure 2.4. The mean velocity of laminar flow in a one-dimensional channel is determined by the balance between the pressure gradient and the gradient of the shear force between the fluid and the channel walls (Turcotte and Schubert, 2002). The pressure gradient driving the flow of melt draining through a vertical fracture with rigid walls is due the force of gravity. For laminar flow, the mean velocity,u ¯, 36

Figure 2.4: A sketch of the evolution of melt flowing through a two-dimensional vertical fracture where b is the initial fracture width, h is the fracture width at some time t, and d is the depth the melt reaches in the time it takes for the fracture to freeze shut (analogous to the sketch of the buoyant ascent of magma in Bruce and Huppert (1989)). A) As the melt flows through, it begins to freeze to the fracture walls and eventually freezes shut. B) At first the melt freezes to the fracture walls, but as hot melt continues flowing into the fracture it starts to melt back the fracture walls. 37 in the channel is h2ρ g u¯ = l (2.6) 12µ

where h is the width of the fracture at time t, ρl is the density of the liquid, g is the acceleration due to gravity, and µ is the viscosity of the ﬂuid. For Reynolds numbers over 103 where the Reynolds number is

ρ uh¯ Re = l (2.7) µ ! the flow will be turbulent. The average turbulent velocity is typically found by using experimental data to find a relationship between the friction factor and the Reynolds number (Dean, 1978; Abe et al., 2001). Although different experiments yield different fits, we found that the various suggested relationships between friction factor and Reynolds number are usually the same to within 20% (Abe et al., 2001; Beronov et al., 2006). This difference is significant when studying the intricacies of turbulent flow, but any of the relationships between friction factor and Reynolds number in Dean (1978); Abe et al. (2001); Beronov et al. (2006) would provide a reasonable estimate for the average turbulent velocity in an impact generated fracture to be used in our calculations. We use the classic relationship between friction factor and Reynolds number for duct flow presented by Dean (1978) to calculate the velocity for any flow with a Reynolds number above 103. We find that the average velocity of the bulk flow parallel to the channel walls is:

1 7 g 4/7 ρ / u¯ = l h5/7 (2.8) 0.073 µ ! Using the constants given in tables 2.2 and 2.3, we can determine when the velocity will be laminar and when the velocity will be turbulent for each planet. Figure 2.5 shows the laminar velocity and turbulent velocity as a function of fracture width and the Reynolds number at each fracture width for each planet considered. The transition between laminar and turbulent ﬂow occurs at a Reynolds number of 103 which is denoted by a vertical line. These plots show that the ﬂow of silicate impact melt through fractures on the Earth and the Moon will be laminar in most fractures. 38

The flow of water impact melt through fractures on Ganymede will be turbulent for most fractures. The Poisson distribution (figure 2.3) implies a negligible number of fractures smaller than 0.01 m or larger than 0.1 m. Therefore for the purposes of this study, we suppose that melted rock flowing through fractures on the Earth and the Moon is laminar at all widths and melted ice flowing through fractures on Ganymede is turbulent at all widths.

Ganymede (ice) Earth (rock) Reynolds Number Reynolds Number −2 0 2 3 5 7 4 10 10 10 8 10 10 10 10 10 Laminar Laminar Turbulent Turbulent 2 106 10

0 104 10

−2 102 10 Velocity (m/s) Velocity (m/s) −4 100 10

−2 −6 10 10 10−3 10−2 10−1 100 10−3 10−2 10−1 100 Fracture width (m) Fracture width (m) Moon (rock) Reynolds Number −2 0 2 4 10 10 10 10 Laminar Turbulent 102

100

10−2 Velocity (m/s) 10−4

−6 10 10−3 10−2 10−1 100 Fracture width (m)

Figure 2.5: The laminar and turbulent velocity as a function of fracture width (bottom axis) for Ganymede, Earth, and the Moon. The top axis shows the Reynolds number at diﬀerent fracture widths. The transition between laminar and turbulent ﬂow which occurs at a Reynolds number of 103 is marked with a vertical dashed line.

For conditions where advection is not important, the depth of solidification can be calculated using Petford (1994)’s equation 5a and the velocity of the fluid. We used equation 5a in Petford (1994) to find the time it takes for the fracture to 39

freeze shut in the absence of advection, τf ,

2 2 b S∞ τ = (2.9) f 4κ

where b is the initial fracture width, κ is the thermal diffusivity, and S∞ is the Stefan number of the solid, L S∞ = (2.10) c(Tf − T∞) where L is the Latent heat, c is the specific heat capacity, Tf is the freezing temperature, and T∞ is the initial temperature of the solid fracture walls. We then found the depth of solidification by integrating the velocity over this length of time for both laminar flow, 4 2 ρ gb S∞ d = l (2.11) 288µκ and turbulent flow

1 7 4/7 / 2 g ρ 49S∞ d = l b19/7 (2.12) 0.073 µ ! 456κ ! To determine when advection will prevent solidification Petford (1994) compares the rate of solidification to the rate of advection at the top of the fracture. For melt drainage this would be equivalent to the bottom of the fracture. However, we found that the depth the melt reaches before solidifying in the absence of advection (equations 2.11 and 2.12) is much shallower than the total fracture depth (equation 2.4). Therefore we compare the rate of solidification to the rate of advection at the depth the melt would reach in the absence of advection. That depth represents the distance melt at the freezing temperature would travel before losing all of its latent heat to the fracture walls. Although latent heat will not actually be lost before the melt cools, the melt has a certain amount of energy it must lose before solidifying. By comparing the rates of solidification and melt back at the depth of solidification, we are asking are the fracture walls cold enough to take in all of the latent heat and advected heat? If the answer is yes, conduction dominates and the fracture will freeze shut. If the answer is no, the rate of melt back will be higher than the rate of solidification, the fracture walls will start to melt back and the fracture will never 40 freeze shut. Melt back will not start until the walls are hot enough that conduction out of the melt is negligible, but once heated, the walls cannot cool down, so if melt back starts, melt will continue to drain until the fracture is completely full. We compare the rate of solidification at time, t = τf , to the rate of meltback at depth, H = d, using equations 2.9, 2.11, 2.12, and Petford (1994) equations 5a and 6b. We find that the equations simplify such that the rate of solidification will be greater than the rate of meltback for both laminar and turbulent flow if

4/3 S∞ 1 >C (2.13) Sm where C is a constant, S∞ is given by equation 2.10 and Sm, the stefan number of the ﬂuid, is L Sm = (2.14) c(Tm − Tf )

where L is the latent heat, c is the specific heat capacity, Tm is the temperature of the melt, and Tf is the freezing temperature. Thus whether a fracture eventually solidifies or flow continues until the fracture fills completely depends on the initial temperature of the melt and the solid and the values of the constants latent heat, specific heat capacity, and the freezing temperature. This is only true where the depth of the fracture is greater than the depth at which the melt solidifies (the fracture is effectively infinite). If that is not the case, drainage will stop when the fracture is full. Using this approximation, narrow fractures freeze before wider fractures as one might expect, but if the melt is hot enough and/or the solid walls are hot enough, no fracture will freeze shut. After determining whether a fracture freezes shut or flow continues until the fracture is full, we calculated the volume that could drain into each fracture. We have assumed the fracture length is equal to the crater diameter for this two- dimensional problem (section 2.4.1), so if the fracture freezes shut, the volume that drains into one fracture is the width of that fracture times the depth it drains before solidification times the diameter of the crater. If melt is able to continue flowing, the fracture will fill completely and the volume is simply the width of the fracture times the depth of the fracture (equation 2.4) times the crater diameter. The total 41

Table 2.2: Constants for Rock Symbol Constant (units) Value Reference −3 ρs Density of solid rock (kg m ) 2800 Neumann et al. (1996) −3 ρl Density of molten rock (kg m ) 2600 Blatt et al. (2006) µ Viscosity of molten rock (Pa s) 100 (Schubert et al., 2001) k Thermal Conductivity of solid Rock 2.5 average between 300K and (W/mK) 1400K calculated from Whittington et al. (2009) c Heat Capacity of Rock (J/kgK) 1200 average between 300K and 1400K calculated from Whittington et al. (2009) cl Heat Capacity of Molten Rock 1000 Turcotte and Schubert (J/kgK) (2002) 2 κ Thermal Diﬀusivity (m /s) k/(ρsc) Turcotte and Schubert (2002) L Latent Heat (J/kg) 4 × 105 Turcotte and Schubert (2002) Tf Melting temperature (K) 1673 Pierazzo et al. (1997) C Constant in equation 2.13 1.5−4/3 C−4/3 Petford (1994)

volume that can drain beneath a crater is equal to the number of fractures of a certain width times the volume that can drain into a fracture of that width summed over all fracture widths.

2.5 Results

The inequality presented in equation 2.13 depends on the specific heat capacity, the latent heat, a constant, the initial temperature of the draining liquid, the initial temperature of the solid fracture walls, and the freezing temperature. We use this equation and constants given in tables 2.2 and 2.3 to make figure 2.6. If the initial temperature of the liquid and the initial temperature of the solid are both below the line, the fractures will solidify. If the temperatures are both above the line, the fractures will never freeze shut and the melt will drain until it has filled the fractures. For impacts into rock, we consider a temperature range from the freezing temperature to the vaporization temperature for molten rock and a temperature 42

Table 2.3: Constants for Water Symbol Constant (units) Value Reference −3 ρs Density of solid ice (kg m ) 927.4 Lide (2004) −3 ρl Density of liquid water (kg m ) 999.8426 Lide (2004) µ Viscosity of liquid water (Pa s) 7 × 10−4 average between 273K and 373K calculated from Korson et al. (1969) k Thermal Conductivity of ice 3 average between 110K (W/mK) and 273K calculated from Klinger (1980); Desch et al. (2009) c Heat capacity of ice (J/kg K) 1470 average between 110K and 273K calculated from Desch et al. (2009) cl Heat capacity of water (J/kg K) 4200 Lide (2004) 2 κ Thermal Diffusivity (m /s) k/(ρsc) Turcotte and Schubert (2002) L Latent Heat (J/kg) 3.3 × 105 Schroeder (2000) Tf Melting temperature (K) 273 C Constant in equation 2.13 1.5−4/3 C−4/3 Petford (1994) 43 range from the Earth’s surface temperature to the freezing temperature for the solid rock fracture walls. We expect a solid temperature close to 1000 K (Ivanov, 2004) and a melt temperature of 2000 K (Bray et al., 2010, and references therein). For impacts into ice, the highest possible temperature of draining water depends on the amount of material above the top of the fracture, because the vaporization temperature of water varies with pressure. If the currently observed central pits were originally full of impact melt, the maximum amount of material above the top of the fracture is controlled by the maximum depth of a central pit. Using the central pit depth scaling law from Bray et al. (2012) we find that a 200 km crater would have a 3 km deep central pit. Central pits that are not shallowed by the presence of a central dome are usually found in craters smaller than 90 - 100 km (Schenk, 1993), so this is likely an overestimate of maximum pit depth for typical central pit craters. The material at the bottom of a 3 km pit would be under a pressure of 4.3 × 106 Pa which is below the critical point for water (2.2 × 107 Pa). The vaporization temperature at this pressure is 535 K (cf. Schroeder, 2000)), so we consider a temperature range from the freezing point of water which does not vary much with pressure to 535 K for post-impact melt and a temperature range from 100 K, just below the surface temperature of Ganymede, to the freezing temperature of water for the solid ice fracture walls on Ganymede. As with rock, we expect the temperature of the post-impact melt to be below the vaporization temperature and the post-impact solid to be above the pre-impact surface temperature. We suggest a melt temperature of 300 K and a solid temperature of 200 K are likely post-impact conditions for Ganymede. In sections 2.2, 2.3, and 2.4, we estimated the volume of melt generated, the volume of fracture space available, and the volume of melt able to drain before freezing the fractures shut. In figure 2.7, we plot all these volumes as a function of crater diameter and compare them to the volumes of central pits observed on Ganymede from Bray et al. (2012) using constants in table 2.3, an initial melt temperature of 300 K, and an initial solid temperature of 200 K. Central pits observed on Ganymede are most commonly found in craters with diameters ranging from 50-80 km but are 44

Figure 2.6: The boundary between initial temperature conditions that lead to fractures solidifying and fractures never solidifying for rock (left) and water (right). Low initial melt and solid temperatures will lead to solidification whereas high initial melt and solid temperatures will allow melt to continue flowing without solidification. The ‘x’ marks the initial temperatures we expect for each material (Tm = 2000 K, T∞ = 1000 K for rock; Tm = 300 K, T∞ = 200 K for water). noted over a broader range of crater sizes (Bray et al., 2012). Here we consider craters with diameters from 5-200 km. The volume of impact melt remaining in the crater (equation 2.1) and the volume of fracture space available (equation 2.2) both exceed the observed volume of central pits. The volume of melt able to drain before freezing fractures shut also exceeds the observed central pit volumes (figure 2.7). This implies that a central pit could form even if some melt remains in the crater and some fractures remain empty. Note for craters smaller than approximately 60 km the volume of melt able to drain exceeds the volume of melt in the crater which suggests that at small crater diameters, central pit volume is limited by the amount of melt available and at larger crater diameters, it is limited by the volume able to drain. These calculations suggest that impact melt draining through impact generated fractures is a viable hypothesis for the formation of central pits on Ganymede. This hypothesis could be tested by measuring gravity anomalies over central pit craters on Ganymede. If the pits form by drainage into fractures, the gravity anomaly over the crater should be either lower than gravity anomalies over similar sized craters on Earth or non-existent if the fractures fill completely. 45

4 10 Observed Pit Volumes Estimate of Volume Able to Drain Total Fracture Volume 3 10 Volume of Melt in Crater )

3 2 10

1 10 Volume (km

0 10

−1 10 1 2 10 10 Crater Diameter (km)

Figure 2.7: The estimated volume of melt able to drain (-), volume of fracture space (- · -), and volume of melt remaining in the crater (- -) as a function of crater diameter on Ganymede. These volumes are compared to observations of the volume of central pits on Ganymede (x’s) measured in Bray et al. (2012).

Impact melt drainage could form central pits in craters on Ganymede, but we do not observe central pits in craters on most rocky bodies, so if the melt drainage hypothesis is correct, it must not be able to form central pits on the Earth or the Moon. In ﬁgure 2.8, we compare the volume of water (temperature 300 K) able to drain into solid ice (temperature 200 K) on Ganymede to the volumes of melted rock (temperature 2000 K) able to drain through solid rock (temperature 1000 K) on the Earth and the Moon. As expected, more impact melt is able to drain if the melt is water instead of molten rock. The volume of melt able to drain on Ganymede exceeds the volume of melt able to drain on Earth and the Moon by several orders of magnitude and is only slightly below the total fracture volume. We expect central pits to form in craters of the size range we consider at any post-impact temperatures on Ganymede. On Earth and the Moon, very little molten rock is able to drain even for the largest craters considered. If the initial temperature of the melt and/or the solid were a few hundred degrees warmer, the fractures would not freeze shut, so the volume able to drain would equal the fracture volume. This would most likely lead to a central pit, so we expect that the post-impact temperatures of the solid and the impact melt in impacts into pure rock are cool enough that the fractures freeze shut. 46

4 10 Fractures Ganymede Earth Moon 2 10 ) 3

0 10 Volume (km −2 10

−4 10 1 2 10 10 Crater Diameter (km)

Figure 2.8: The volume of melt expected to drain on Ganymede (- -), Earth (- · -), and the Moon (··) and the total fracture volume (–) as a function of crater diameter.

2.6 Discussion

2.6.1 Diﬀerences between rock and ice

Many of the constants used in the equations in this paper have different values for different materials but most vary only by a factor of 2 or 3 between water and silicate melt. However, viscosity is 105 times larger for molten rock than it is for water. This allows water to flow into fractures much faster than molten rock. Furthermore, any solid particles present in a melt pool will float in the case of ice in water whereas for rock the solid crystals will sink to the bottom of the melt pool so that in reality, the material draining is only partially molten and has a higher effective viscosity than pure molten rock. Jaedicke et al. (2008) found that for a volume fraction of snow particles of 0.6, the effective viscosity of water slush is 67 Pa s. Marsh (1981) found that magma typically stops flowing when the crystal volume fraction exceeds 0.55 which corresponds to a critical viscosity of 104.6 Pa s. As expected, these higher viscosities lead to a lower volume of material draining into fractures (see figure 2.9). The difference between molten rock and partially molten rock is greater than the difference between liquid water and water slush for two reasons. First, the depth at which the fracture freezes shut is inversely proportional to viscosity for the laminar flow of molten rock and inversely proportional to viscosity to the 47 one seventh power for the turbulent flow of liquid water (equations 2.12 and 2.11). Second, fewer fractures fill completely in the case of water slush than liquid water whereas no fractures fill completely for either molten rock or partially molten rock. Another notable difference between icy targets and rocky targets is the melting temperatures. The heat capacity of ice and rock are fairly similar, but rock can experience a much larger temperature contrast between solid and liquid because of its higher melting temperature. These factors combine such that several orders of magnitude more melt can drain in an icy target than in a rocky target. This is well demonstrated by the contrast in melt volume between Ganymede and the Moon which have similar accelerations due to gravity and differ only in composition in our model of the problem. The difference in volume able to drain on the Earth and the Moon shows that gravity also affects how much impact melt can drain, but this is a relatively minor difference compared to the differences caused by the target material.

4 10 Fractures Ganymede 2 Ganymede slush 10 Moon Moon slush )

3 0 10

−2 10 Volume (km −4 10

−6 10

1 2 10 10 Crater Diameter (km)

Figure 2.9: The volumes of melt expected to drain on Ganymede (- -) and the Moon (· ·) compared to the volumes of slush expected to drain on Ganymede (- · -) and the Moon (- -) and the total fracture volume (–).

2.6.2 Limitations and assumptions

The vertical plane parallel fractures we considered are a very idealized scenario. In reality, fractures beneath an impact crater will not be vertical or planar. Most 48 likely the melt will drain through a brecciated region where channels twist and turn and widen and narrow. However, based on the scale of fractures seen at Vredefort and Sudbury (Riller et al., 2010) we expect the brecciated region to more closely resemble fractures than the microcrystalline pore space found in a solid rock. To estimate the effects of non-vertical fractures on the volume of melt able to drain we consider the concept of tortuosity. Tortuosity is defined as the ratio of the length of a channel and the length of the medium containing the channel (Epstein, 1989). This affects our assumptions for the fracture distribution. The total fracture volume and the depth of the fractured region are based on observations, so assuming portions of the fracture have horizontal components increases the volume of individual fractures and decreases the total number of fractures. With fewer fractures available, less melt will drain. Tortuosity also affects the drainage of impact melt because the pressure gradient in non-vertical portions of a fracture is lower which slows the velocity and reduces the amount of melt able to drain before freezing. The change in pressure experienced by the fluid between the top and bottom of the fractured

region is ∆P = ρgdfrac for both vertical fractures and tortuous fractures. However, dP the pressure gradient for vertical fractures is dz = ρg and the pressure gradient for tortuous fractures could be approximated by dP = ρg where the tortuosity, τ = L , dz τ dfrac and L is the length along the fracture. With this modiﬁed pressure gradient, the velocity in a tortuous fracture is

h2ρ g u¯ = l (2.15) 12µτ

for laminar flow and 1 7 g 4/7 ρ / u¯ = l h5/7 (2.16) 0.073τ µ ! for turbulent flow. We estimate the volume of impact melt that will drain into fractures with a tortuosity of two by doing our calculations from section 2.4 using the modified velocities (equations 2.15 and 2.16). We find that 1/4 as much melt drains when the flow is laminar (molten rock impact melt) and approximately 1/3 as much melt drains when the flow is turbulent (liquid water impact melt). 49

A different assumption for the average fracture width could also increase or decrease the total volume able to drain. Doubling the average fracture width would decrease the total number of fractures by a factor of two because the total fracture volume is not affected. For laminar flow, doubling the average fracture width would increase the volume that drains into one fracture by a factor of 32, so the total volume drained would increase by a factor of approximately 16. For turbulent flow, the volume that drains into one fracture would increase by a factor of approximately 13, so the total drainage volume would increase by approximately 6.5. However, in the case of Ganymede some of the wider fractures are able to fill completely before solidifying, so doubling or halving the average fracture width changes the total drainage volume by a factor of less than 6. In the case of molten rock draining into solid rock, a smaller average fracture width would lead to even less melt draining. However, if the average fracture width was double our assumed width (10 cm instead of 5 cm) and all our other assumptions were the same, more than one order of magnitude more melt would drain. The volume of melt able to drain on the Earth for an average fracture width of 10 cm is still less than the observed volumes of all but two central pits on Ganymede. However, we do not think it very likely that the average fracture width is larger than 5 cm. In section 2.4, we argued that if the average fracture width scales with crater diameter, it would decrease with decreasing crater diameter. Vredefort is larger than any of the central pit craters observed on Ganymede or Mars, so we find it unlikely that the average fracture width is larger than 5 cm beneath craters of a size that could have central pits. Several other assumptions could also increase or decrease our estimate of how much melt is able to drain. We treated the fracture walls as a semi-infinite half space, but in reality, the fractures may be close together reducing the amount of cold solid able to absorb heat from the draining melt. This would increase the amount of melt able to drain. It is also likely that the fractures do not have a constant width. This could cause the flow to slow if the fractures get narrower with depth leading to a smaller total volume draining. However, the fractured region is likely 50 a complicated network of interconnected channels. The melt will flow through the path of least resistance choosing wide fractures when they are available. We also made the assumption that the draining material is pure liquid, but it could be a slurry of liquid and solid instead which would increase the viscosity of the fluid and possibly decrease the temperature. This would decrease the volume of melt able to drain. Although our calculations are not precise enough to predict the volumes of central pits, the assumptions should overestimate (or underestimate) the drainage volume by similar amounts on all bodies. More melt can drain on Ganymede than Earth or the Moon, so it is likely that central pits can form via drainage of water into water ice on Ganymede, but they cannot form via drainage of molten rock into solid rock fractures on the Earth or the Moon.

2.6.3 The role of viscous dissipation

Under some circumstances, viscous dissipation can prevent solidiﬁcation of magma buoyantly rising through a dike (Turcotte, 1990; Carrigan et al., 1992). One can estimate the maximum change in temperature of the ﬂuid due to dissipation by assuming all of the potential energy is converted to temperature change:

gdfrac = c∆T (2.17)

To get the upper limit, we calculate dfrac using equation 2.4 for the largest crater we consider in this paper, a 200 km diameter crater. Using the specific heats of water and molten rock given in tables 2.2 and 2.3 we found that the maximum temperature change due to dissipation, ∆T , of the draining fluid would be 4 K for Ganymede, 118 K for Earth, and 19 K for the Moon. The temperature change due to dissipation for Ganymede and the Moon is less than 15% the initial temperature difference between the liquid and the freezing temperature. The temperature change due to dissipation estimated for melt draining through fractures on Earth is 36% the initial temperature difference between the liquid and the freezing temperature. However, in this calculation we have assumed that all of the potential energy goes into dissipation that raises the temperature of the melt. In reality, only a small 51 fraction of the potential energy will go into dissipative heating, so this effect is negligible on all three bodies.

2.6.4 Mars

Mars is the only rocky body that features central pits. Some craters on Mars also exhibit central pits on top of central peaks which are not seen on Ganymede (Barlow, 2006). If central pits on Mars were formed via molten rock draining through solid rock, we would expect the pit volumes to fall somewhere in between the estimated drainage volumes on Earth and the Moon, because gravity is the only diﬀerence. However, Mars has ice beneath its surface (Byrne et al., 2009) and may have had more in the past (Baker et al., 1991). This ground ice would be melted and/or vaporized during an impact (Ivanov and Pierazzo, 2011). However, in large craters (D > 100 km) the central uplift will be made of mostly dry rocks initially located below the aquifer (Ivanov and Pierazzo, 2011). This may explain why central pits do not occur in craters with diameters larger than 114 km on Mars (Barlow, 2006). In Martian craters smaller than 100 km in diameter, central pits could form through the removal of volatile materials near the crater center by vaporization during the impact or if liquid water pools in the center of the crater, it could drain without solidifying since the high initial fracture wall temperature will not allow the ﬂuid to cool. We expect that the volume of central pits in Martian craters is controlled only by the availability of ground ice in the target rock. The amount of ground ice available on Mars is expected to vary both with latitude and with time (Baker et al., 1991; Byrne et al., 2009). A more detailed investigation into the amount of ground ice available on Mars would be needed to predict the volume of the central pits on Mars.

2.6.5 Other icy satellites

The presence of central pits in craters on Mars, Ganymede, and Callisto and the lack of central pit craters on most rocky bodies suggests that ice is required 52

3 10 observations Ganymede Titan Triton Pluto 2 Rhea ) 10 Mimas 3

Volume (km 10

0 10 2 10 Crater Diameter (km)

Figure 2.10: The volume of melt expected to drain on Ganymede (–), Titan (- · -), Triton (- · -), Pluto (··), Rhea (- -), and Mimas(- -) as a function of crater diameter compared to the observed central pit volumes (x’s) on Ganymede.

3 10 observations Ganymede Titan Triton Pluto 2 Rhea ) 10 Mimas 3

Volume (km 10

0 10 2 10 Crater Diameter (km)

Figure 2.11: The volume of melt remaining in a crater after the impact on Ganymede (–), Titan (- · -), Triton (- · -), Pluto (··), Rhea (- -), and Mimas(- -) as a function of crater diameter compared to the observed central pit volumes (x’s) on Ganymede. to form central pit craters. However, central pit craters are not seen on all icy satellites. A few craters on Titania, Rhea, and Tethys resemble central pit craters but all occur in craters with diameters over 300 km (Schenk, 1993). Europa’s larger craters are likely inﬂuenced by a subsurface ocean (Moore et al., 1998); Triton has a relatively young surface (Schenk and Zahnle, 2007); and Titan has few craters compared to Ganymede or Mars, and the current crater population is based on Cassini RADAR data covering only 33% of the surface (Neish and Lorenz, 2012). 53

Therefore, we do not expect to see central pits on any of those large icy satellites, but one might expect to see central pit craters more frequently on other icy satellites. Using equation 2.12 with the gravity changed for each satellite, we calculate how much melt could drain on Rhea, Mimas, Triton, Pluto, and Titan which range from just barely smaller than Ganymede to much smaller than Ganymede. We compare these volumes to the observed central pit volumes on Ganymede and the amount of melt that could drain on Ganymede (figure 2.10). They differ in surface temperature as well which might affect post-impact temperature. Figure 2.10 uses a solid post-impact temperature of 200 K and a melt temperature of 300 K. Reducing the solid temperature to 100 K would decrease the volume of melt able to drain by less than an order of magnitude. Assuming the actual volume of melt able to drain on Ganymede is closer to the observed pit volumes than it is to our calculation and that the other drainage volumes overestimate reality by a similar amount, we suggest that significant amounts of melt could drain on satellites larger than Rhea. Although large volumes of melt could drain on mid-sized icy satellites, the volume of melt produced in an impact will vary between satellites due to differences in gravity, temperature, and typical impact velocity. The amount of melt generated in an impact scales with projectile size (Kraus et al., 2011), but the final crater diameter depends on gravity in addition to projectile size (Melosh, 1989), so on the smaller moons, the crater will be bigger whereas the melt volume will be the same for a given projectile size. This suggests that central pits will occur in larger craters on satellites with lower gravity (Senft and Stewart, 2011). We repeated our calculation in section 2.2 for Titan, Triton, Pluto, Rhea, and Mimas (figure 2.11). To do this we adjusted the gravity, the typical impact velocity (Zahnle et al., 2003), and the surface temperature for each body. Figure 2.11 shows that the volume of melt remaining in a crater on these satellites decreases with gravity with the exception of Pluto where impacts produce less impact melt than impacts on smaller body Rhea because the typical impact velocities on Pluto are much lower than those on satellites. The volume of melt generated also depends on impactor composition. We have assumed ice projectiles in our calculations for all icy bodies, but if the impactor 54 was rocky or metallic, it would generate a larger volume of melt (Barr and Citron, 2011). Our estimates on the volume of melt produced and the volume able to drain seem to suggest that central pits should form on most icy satellites. However, these calculations over-estimate the volume of melt able to drain on Ganymede. This suggests that the actual volume of melt able to drain into fractures beneath craters on other icy satellites is also lower than we estimate. Thus central pits would be unlikely on satellites the size of Rhea or smaller. Triton has a relatively young surface (Schenk and Zahnle, 2007), so that may explain the lack of central pit craters there. Perhaps a continued search for impact craters on Titan could reveal central pits on that body as well. A significant volume of melt could drain into fractures beneath a crater on Pluto, but the volume of melt remaining in the crater is probably not enough to form a central pit. Some satellites could have central pits in larger craters than the largest central pit craters on Ganymede. On Ganymede, craters larger than approximately 60 km in diameter have bright domes on the floor of a central pit (Schenk, 1993). These craters are known as central dome craters and are thought to form when the impact affects a deeper more viscous layer (Schenk, 1993). If this deep viscous layer does not exist on other icy satellites, central pit craters at larger crater diameters could be possible.

2.7 Conclusions

Several mechanisms have been proposed to explain the formation of central pit craters. In this study, we investigate the hypothesis that impact melt drains through fractures beneath the crater leaving behind a central pit at the surface. First, we estimate the volume of melt and volume of fracture space generated during an impact. To form a central pit via melt drainage, the volume of the melt generated and the volume of the fracture space both must exceed the volume of the central pits. We find that both the expected volume of void space and the expected melt volume exceed the observed volume of central pits on Ganymede. We also estimate 55 the volume of melt able to drain into fractures before freezing the fractures shut. We find that our upper limit on the volume of melt able to drain into fractures on Ganymede also exceeds the volumes of the pits observed on Ganymede. We conclude that melt drainage is a plausible formation mechanism for central pit craters on Ganymede. Central pits are seen in craters on some icy satellites and on Mars. They are not seen on Earth, the Moon, or other rocky planets. To investigate this, we calculated the volume of water draining through fractured ice on Ganymede, and molten rock draining through fractured solid rock on Earth and the Moon. We find that water drains readily through fractures on Ganymede, but very little molten rock is able to drain on Earth and the Moon. This suggests that if the impact target contains a sufficient amount of ice and the ice melts during the impact, a significant amount of the water can drain into fractures beneath the crater and leave behind a central pit. A central pit will not form if the target is dry rock because only a small volume of melt will drain into fractures beneath the crater. However, some melt will drain and solidify in the fractures. Pseudotachylites could be evidence of that process on Earth. 56

CHAPTER 3

Melt migration in Io’s convecting mantle

3.1 Introduction

Io is the most volcanic body in the solar system because it has a significant orbital eccentricity of 0.0041, which causes the strength of Jupiter’s gravitational pull to vary as the distance between the two bodies varies. This periodically stretches and compresses Io generating friction in its interior. Io’s orbit is maintained at this eccentricity by the Laplace resonance with Europa and Ganymede which pumps up Io’s eccentricity as tidal dissipation damps it down (Peale et al., 1979). This leads to melt erupting onto the surface, a surface heat flux of 3 ± 1 W m−2 (Veeder et al., 2004), and erasure of impact craters at rate of at least 1 mm year−1 (Johnson et al., 1979). If the melt erupting onto the surface is derived from partial melting of a mostly solid mantle, Io should be chemically differentiated with a low density siliceous crust and eruption temperatures less than 1375 K (Keszthelyi and McEwen, 1997). However, observations by the Near Infrared Mapping Spectrometer (NIMS) and the Solid State Imager (SSI) on Galileo found many hot spots with temperatures significantly above 1300-1450 K which implies an ultramafic composition rather than the basaltic composition typical for present-day terrestrial eruptions (McEwen et al., 1998). By applying a lava fountain model to the highest reported eruption temperature, the SSI observation of the 1997 eruption at Pillan, (Keszthelyi et al., 2007) found that the lower limit on the eruption temperature for that eruption is ∼1613 K. A recent ground based campaign detected a powerful eruption on Io in August of 2013. Modeling of the outburst spectrum places a lower limit of 1200-1300 K on the eruption temperature, but the best-fit model spectrum suggests a temperature of 1900 K or higher suggesting ultramafic lava at that eruption as well (de Kleer et al., 57

2014). Spectral data of dark material which is correlated with detectable hot spots suggests that they contain magnesium-rich orthopyroxene which is consistent with the composition inferred from high eruption temperatures (Geissler et al., 1999). The spectral data indicates that most dark materials have the same composition, and most measured eruption temperatures on Io are lower limits, which suggests that the high temperature measured at the June, 1997 eruption at Pillan could be the typical eruption temperature for all silicate volcanism on Io (Keszthelyi et al., 2007). The high eruption temperatures and the spectral data disprove the theory that Io’s mantle is extremely chemically differentiated as would happen through partial melting of a mostly solid mantle. Instead Io’s mantle must be well mixed. (Keszthelyi et al., 1999) suggest that this could be achieved in a partly crystallized global magma ocean which heats the base of the crust and remixes that material into the mantle. Galileo and Hubble Space Telescope (HST) observations place constraints on the amount of melting in Io’s mantle. Data from the magnetometer on the Galileo spacecraft could not be explained by interactions between plasma and Io’s atmosphere alone. Khurana et al. (2011) showed that a partially molten asthenosphere at least 50 km thick and at least 20% molten beneath a 50 km thick solid lithosphere would have a high enough conductivity to produce an induced magnetic field that matches the magnetometer observations. However, such a conducting layer should also affect the oscillations of Io’s oxygen and sulfur UV aurora, and Hubble Space Telescope observations did not detect such an effect (Roth, 2012). Io’s aurora is created when plasma rotating with Jupiter’s magnetosphere interacts with the atmosphere surrounding Io which orbits Jupiter more slowly than Jupiter’s magnetosphere. Thus Io’s aurora could be affected by variability in Io’s atmosphere (including volcanic plumes) in addition to a possible induced magnetic field and variations in the plasma environment. Many previous studies have attempted to determine if Io is in thermal equilibrium such that the observed surface heat flux balances the tidal heating generated in the mantle. Ojakangas and Stevenson (1986) developed a simple model for Io’s 58 coupled thermal and orbital evolution in which they parameterized convective heat flow and tidal heat production as power laws of temperature. They determine the time evolution of the model through two nonlinear equations that determine the eccentricity and the temperature. They find that the solution is unstable unless the increase in temperature is stopped when it reaches the solidus of the mantle, because they expect volcanism will transport excess heat when the mantle temperature is above the solidus. This leads to periodic thermal states where Io oscillates between a cold interior in which tidal dissipation exceeds the rate of heat loss and a hot interior in which tidal dissipation is less than heat loss (Ojakangas and Stevenson, 1986). Fischer and Spohn (1990) argue that magma will not be removed instanta- neously as assumed by Ojakangas and Stevenson (1986), so they develop a coupled thermal and orbital evolution model similar to Ojakangas and Stevenson (1986) in which the effects of partial melt on mean mantle viscosity and the shear modulus are parameterized and used in the tidal dissipation calculations. They find three possible stages in Io’s evolution: 1) a stable phase in which Io is in a hot, high-dissipation state and tidal heat production equals surface heat flow which is stable with respect to small perturbations in temperature and/or orbital eccentricity, but eventually becomes unstable; 2) next, Io passes through transitory oscillations similar to those predicted by Ojakangas and Stevenson (1986); and 3) a stable state of low interior temperature and a low internal heating rate that is in approximate equilibrium with heat loss (Fischer and Spohn, 1990). The stable high dissipation state proposed by (Fischer and Spohn, 1990) was investigated in more detail by Moore (2003). He used a tidal dissipation model to calculate the amount of heating generated in Io’s mantle and convective scaling laws to calculate the heat flux conducted through Io’s lithosphere both for a given mantle temperature. He then identified the mantle temperature at which the heat generated in the interior by tidal dissipation was perfectly balanced by the heat transported to the surface by convection. He found that at this mantle temperature, Io’s surface heat flux would be an order of magnitude lower than Io’s observed surface heat flux. Thus either the amount of heat generated in Io’s mantle by tidal dissipation is not 59 in equilibrium with the heat transported to the surface by convection, some other heat loss mechanism such as melt segregation dominates (Moore, 2003), or tidal dissipation models are not correctly calculating the amount of tidal heating. Models balancing convection and tidal dissipation have all suggested that Io may not currently be in thermal equilibrium (Ojakangas and Stevenson, 1986; Fischer and Spohn, 1990; Moore, 2003). However, the orbital evolution of the Galilean satellites due to tidal dissipation can be measured through the astrometrically observed positions of the Galilean satellites. Lainey et al. (2009) analyzed over one hundred years of this data and found that the rate of tidal dissipation in Io is in good agreement with the observed surface heat flux. This suggests that Io is currently in thermal equilibrium, and some other heat transport mechanism must be operating on Io to keep it in the high dissipation equilibrium proposed by Fischer and Spohn (1990). Several models have suggested that the melting of mantle material and the upward migration of this molten rock could contribute significantly to Io’s global heat loss (e.g. O’Reilly and Davies, 1981; Moore, 2001). Noting that tall mountains on Io imply a thick lithosphere O’Reilly and Davies (1981) suggested that the thick lithosphere could exist if a significant fraction of Io’s heat flow out of the mantle is transported by advection through volcanoes instead of the more common conduction through the lithosphere. Moore (2001) attempted to determine whether melt migrating through the mantle could carry all of Io’s tidally generated heat towards the surface. He assumed that the energy from tidal dissipation goes directly into melting mantle rock which migrates upwards as the solid sinks down according to Darcy’s law and mass conservation laws. He solved these equations over a 500 km thick partially molten zone beneath a 50 km thick lithosphere and found that at melt fractions below 20% all of Io’s tidally generated heat could be transported by melt segregation alone (Moore, 2001). These previous models of heat loss from Io’s mantle have included heat transport through either melt migration (O’Reilly and Davies, 1981; Moore, 2001) or mantle convection (e.g. Ojakangas and Stevenson, 1986; Fischer and Spohn, 1990; Moore, 60

Figure 3.1: A sketch of Io’s mantle in which cold narrow plumes sink through Io’s partially molten mantle while the rest of the mantle rises and begins to melt when it reaches the solidus temperature.

2003), but not both. Here we present a one-dimensional model which includes both mantle convection and melt segregation. We envision that plumes from Io’s cold dense lithosphere sink through its partially molten mantle. These plumes warm as they sink and mix with the mantle material. In upwelling portions of the mantle, tidal heating heats the material as it rises towards the cold surface, and it begins to partially melt through decompression melting and tidal heating (ﬁgure 3.1). This melt is less dense than the solid, so it buoyantly ascends between the solid grains. Here we investigate this problem by adapting a model for partial melting and melt migration beneath a mid-ocean ridge to Io and couple this to convective scaling laws for a partially molten material to calculate the upwelling velocity and mantle temperature. This allows us to self consistently solve for the amount and distribution of magma in Io’s mantle and the relative amounts of heat transport via conduction through a stagnant lid and volcanic eruption. This model is described in section 3.2. In section 3.3, we present the results of model and their sensitivity to uncertainty in the model and the rheology of Io’s mantle. In section 3.4, we compare our results to observational constraints. 61

3.2 The model

3.2.1 Melt segregation

The behavior of partially molten rock is a well studied problem in the context of the Earth (e.g. Stevenson and Scott, 1991, and references therein). If a rock is heated or the pressure is reduced such that the temperature rises above the solidus temperature, melt will begin to form at the contacts between mineral grains of different compositions and form an interconnected network. The melt is less dense than the residual solid matrix. This drives relative motion between the liquid and the solid matrix leading to a process involving two-phase flow (McKenzie, 1984). McKenzie (1984) derived governing equations for the buoyancy driven percolation of a liquid through a porous solid matrix from the conservation of mass, momentum, and energy. These equations have been widely applied but often focus on the melt migration and matrix motion while neglecting the generation of melt. How much melt is generated is one of the key questions that must be answered to understand Io’s mantle, so we turn to a model developed by Hewitt and Fowler (2008) that was used to model melt generation and two-phase flow in a column of mantle rising beneath a mid-ocean ridge. Hewitt and Fowler (2008) include both melt production and melt segregation according to the equations for Darcy flow through a deforming solid (Fowler, 1985; Hewitt and Fowler, 2008). In the partially molten region conservation of mass is described by ∂φ m + ∇· (φu)= (3.1) ∂t ρl ∂φ m − + ∇· ((1 − φ)V )= − (3.2) ∂t ρs

where φ is the melt fraction, u and V are the melt and solid velocities respectively, ρl

and ρs are the melt and solid densities respectively, and m is the rate of converting solid to melt (Hewitt and Fowler, 2008). Darcy’s law describes the motion of the two phases relative to each other

kφ φ(u − V )= (−∇pl + ρlg) (3.3) ηl 62

where ηl is the melt viscosity, pl is the pressure of the melt, g is the acceleration due

to gravity, and kφ, the permeability, is given by h2φn k = (3.4) φ b where h is the grain size, and n and b depend on the texture (Stevenson and Scott, 1991). The permeability of partially molten zones in planetary mantles is uncertain both because the distribution of pathways through the solid matrix is unknown and because, the permeability might change as the system evolves(Stevenson and Scott, 1991). Permeability can be calculated explicitly for idealized materials ranging from parallel tubes with circular cross sections on a square lattice of spacing h to packed spheres. For these two examples, n = 2 and n = 3 respectively (Stevenson and Scott, 1991). Textures observed in partial melts with a low melt fraction most closely resembled an isotropic network of tubes with triangular cross sections. Therefore, we use the values suggested by Stevenson and Scott (1991) for that texture: b = 1000 and n = 2. The conservation of the total momentum is given by

∇· ((1 − φ)σs)+ ∇· (φσl)+(φρl + (1 − φ)ρs)g =0 (3.5)

where the melt stress tensor, σl, is described by

σl = −plδ (3.6)

where δ is the Kronecker delta and we have assumed the ﬂuid supports negligible

deviatoric stress (Hewitt and Fowler, 2008). The solid stress tensor, σs is given by

σs = −psδ + τ (3.7)

where ps is the solid pressure and τ is:

∂Vi ∂Vj 2 τij = ηs + − ∇· V δij (3.8) ∂xj ∂xi 3 !

where ηs is the viscosity of the solid rock (Hewitt and Fowler, 2008). The viscosity of the solid rock will depend on the temperature of the rock. We calculate this using 63 the Arrhenius viscosity law described later in section 3.2.2 with a melt fraction of zero. The relationship between the pressures of the two phases, or compaction, is given by (Hewitt and Fowler, 2008):

η p − p = − s ∇· V. (3.9) s l φ

Melt production is determined through conservation of energy described by

∂ ∂ ∂ mL + ρlcφ ∂t + u · ∇ T + ρsc(1 − φ) ∂t + V · ∇ T − αTφ ∂t + u · ∇ pl ∂ −αT (1 − φ) ∂t + V · ∇ ps = ∇· (k∇T )+Φ+ E (3.10)

where L is the latent heat, cp is the specific heat capacity, T is the temperature, α is the thermal expansion coefficient, k is the thermal conductivity, and Φ is the viscous dissipation (Hewitt and Fowler, 2008). Beneath sea floor spreading zones on Earth, melting occurs primarily because of decompression melting as convection moves material upwards. Decompression melting resulting from convection will also be important on Io, but on Io, the rising material is also being tidally flexed, so it is important to include internal heating in the energy conservation equation, which we do with the parameter E. As in Hewitt and Fowler (2008), we assume that the thermal conductivity and the specific heat capacity are the same in each phase. By requiring thermodynamic equilibrium at the boundary between the partially molten region and the subsolidus material, Hewitt and Fowler (2008) derive the boundary condition:

φ(ps − pl)=0 (3.11)

which requires that either the melt fraction φ or the pressure diﬀerence ps − pl is zero at the top and bottom boundaries of the partially molten zone. As in Hewitt and Fowler (2008), we assume the former is true at the bottom boundary and the latter applies to the upper boundary because melting begins at the bottom of the partially molten zone and migrates upwards. Although magma will form dikes and magma chambers and erupt to the surface, how these dikes and magma chambers form and how much of the magma will erupt is not well understood, so as in 64

Hewitt and Fowler (2008), we take the simplest thermodynamically viable solution and assume that the melt solidiﬁes at the top of the partially molten zone. The buoyancy of magma relative to solid rock means that the most interesting physics will happen in the vertical direction. We take the one dimensional version of equations (3.1)-(3.10) where y represents the depth such that y = 0 is the surface. As in Hewitt and Fowler (2008), we choose the depth at which the litho-

static solidus ﬁrst intersects the adiabatic geotherm, ym, to be the reference depth

at which the lithostatic pressure is pm and the lithostatic solidus is Tm. Internally heated convecting mantles are nearly adiabatic beneath the top boundary layer, so if the temperature of the upwelling rock is known, ym can be determined using the temperature of the convecting mantle and the equation for the solidus in the absence of partial melting. In the absence of partial melting, the pressure proﬁle would be lithostatic and the solidus would be:

Tm = To +Γps = To +Γρsgym (3.12)

where Γ is the Clapeyron slope and To is the solidus at the surface. We determine

To using the peridotite solidus determined experimentally by Hirschmann (2000).

Then if Tm is the temperature of the isothermal mantle, ym is the depth at which

melting begins. Within the partially molten zone, the solidus, Ts, is:

TS = To +Γpl. (3.13)

To solve the problem, Hewitt and Fowler (2008) deﬁne the vertical coordinate z = y − yb such that the partial molten zone is conﬁned by 0

where the subscripts t and z refer to partial derivatives with time and vertical position respectively, and the two unknowns are the dimensionless melt fraction, φ,

and the dimensionless eﬀective pressure, N = ps −pl. ǫ, Wo, δ, P and m0 are scaling parameters deﬁned in Hewitt and Fowler (2008). The boundary conditions in this dimensionless one-dimensional form are

W = Wo, φ =0at z =0 (3.16)

N =0at z =1 (3.17)

We solve equations (3.14) and (3.15) in the partially molten zone, 0

3.2.2 Convective scaling laws

The temperature and ascent velocity of the column of rock were treated as free parameters in Section 3.2.1, but these parameters can be estimated using convective

1Hudson (1998) found this to be the most accurate finite differencing scheme for numerically solving the conservation law with a source term. 66 scaling laws. Convective scaling laws that relate the surface heat flux due to conduction through the stagnant lid to the average mantle temperature Ti can be derived from boundary layer stability analysis (e.g., Turcotte and Schubert, 2002, chapter 6). Therefore if the flux of heat conducted out of a subsolidus mantle through the stagnant lid is known, we can calculate the average mantle temperature. Previous work with convective scaling laws has not included the effects of partial melting. The presence of partial melt in a planetary mantle will affect the density contrast driving convection, the viscosity of the mantle material, and the balance of the internal heating rate with the heat loss. For the purposes of deriving convective scaling laws, we assume that the cold upper boundary layer is solid and that the melt is evenly distributed throughout the mantle beneath the cold top layer. In reality, melt segregation causes melt to accumulate in larger fractions near the top of the partially molten region and melt will be present in the solid lithosphere in the form of magma chambers and dikes, but convective scaling laws describe only the average behavior of the mantle.

We assume that the total heat ﬂux observed at Io, Fobsv, is equal to the heat

conducted from the mantle through the stagnant lid to the surface, Fc, plus the heat

ﬂux from volcanic eruptions, Fm:

Fobsv = Fc + Fm (3.18)

The heat flux from volcanic eruptions includes both the latent heat released as the melt solidifies and the heat released as the now solidified rock cools from the melting temperature to the surface temperature. We estimate this heat flux as the product of the mantle-to-surface magma mass flux and the energy released by solidification and cooling per unit mass. The mechanics of melt moving from a partially molten mantle to the surface is complicated and not well understood, so we make the simple approximation that the mass flux of magma erupting onto the surface is equal to

ρlwlφtop where ρl is the density of the magma, wl is the velocity of the magma at

the top of the partially molten zone, and φtop is the fraction of magma at the top of 67 the partially molten zone. This leads to a heat ﬂux of

Fm = feρlwlφtop(L + c∆T ) (3.19)

where L is the latent heat of solidification, cp is the specific heat, and ∆T is the temperature difference between the solidus at the surface and the surface temperature, and fe is the fraction of melt able to leave the mantle. Varying fe allows us to test the sensitivity of our results to eruption efficiency. The presence of molten rock beneath the solid lithosphere enhances the density contrast between the cold lithosphere and the warm interior. We account for this increased density contrast by including the effects of melt fraction in the density contrast used to calculate the Rayleigh number: g∆ρd3 Ra = (3.20) i κη where g is the acceleration due to gravity, ∆ρ is the density contrast between the mantle and the surface, d is the thickness of the mantle, κ is the thermal diffusivity, and η is the viscosity of the mantle. The density contrast including a contribution from partial melting can be written as

∆ρ = ρs[α∆T + β∆φ] (3.21) where ρs is the density of the solid, α is the thermal expansivity, ∆T is the temperature contrast between the cold lid and the warm interior, β = − 1 ∂ρ = ρs ∂φ T,P − − ρl ρs is an expansion coefficient for the bulk mixture associated with variations ρs in melt fraction, and ∆φ = φ − 0 is the melt contrast between the solid lid and the partially molten interior. Partial melting in the mantle will also affect its viscosity. The viscosity of a planetary mantle is most often described by the temperature dependent Arrhenius viscosity law (e.g. Reese et al., 2005). The effects of pressure in Io’s mantle should be small, so we neglect the effects of pressure. Here we also include an additional term that incorporates the partial melt dependence (Moore, 2003): µ h m E η = exp exp (−Bmeltφ) (3.22) 2A B ! RTi 68 where µ is the shear modulus, A is a constant, h is the grain size, B is the Burgers vector, m is a constant, E is the activation energy, R is the gas constant, Ti is the mantle temperature, Bmelt is a constant, and φ is the melt fraction. Previous studies using convective scaling laws to describe the average behavior of the mantle rely on the fact that a vigorously convecting mantle is close to isothermal. By including the effects of partial melting we are introducing some uncertainty, because the magma is not evenly distributed within the mantle. In fact, the lower mantle is expected to be solid, and the partially molten upper mantle is expected to have an increasing melt fraction with decreasing depth. Thus, a solid plume sinking from the top boundary layer will first sink through a layer of high melt fraction (low density and low viscosity). As it continues to sink, the melt fraction in the surrounding material will decrease (density and viscosity will increase). Finally, if the plume is still dense compared to the surrounding mantle, it will sink through the solid lower mantle. To quantify this uncertainty, we test two cases: 1. ∆φ = φavg −0

and η(φ = φavg) where φavg is the average melt fraction of the whole mantle; 2.

∆φ = φavg mz − 0 and η(φ = φavg mz) where φavg mz is the average melt fraction of the partially molten portion of the mantle. On many planetary bodies, temperature-dependent viscosity and a cold surface lead to the formation of a viscous immobile top layer that does not participate in the convection below. A thin boundary layer, known as the rheological sublayer, that does participate in convection, develops between the stagnant lithosphere and the interior of the convecting mantle. It is the density contrast across this rheological boundary layer that drives convection. The melt fraction within this rheological sublayer is unknown a priori. Magma in the mantle may be capable of migrating via porous ﬂow until it reaches the more rigid stagnant lithosphere at which point it begins moving primarily through fractures. This would lead to a partially molten rheological sublayer, or perhaps the advection of magma toward the stagnant lithosphere would warm the base of the lithosphere allowing it to become mobile and forming a solid rheological boundary layer. We test the latter scenario by considering a case in which ∆φ in equation (3.21) is assumed to be zero. The temperature 69 contrast across the rheological sublayer has been well studied in previous work (e.g., Solomatov and Moresi, 2000; Reese et al., 2005). It is ∆T ∆T = (3.23) rh θ where θ, the Frank-Kamenetskii parameter, describes the viscosity contrast across the lithosphere: d ln η ∆T E θ = ∆T = 2 (3.24) dT RT i

(e.g. Reese et al., 1999).

In subsolidus stagnant lid convection, the heat ﬂux conducted through the stagnant lid is given by: ∆T F = k Nu (3.25) c d where k is the thermal conductivity and Nu is the Nusselt number:

−4/3 1/3 Nu = aθ Rai (3.26)

(e.g. Turcotte and Schubert, 2002; Solomatov and Moresi, 2000; Reese et al., 2005). Using equation (3.23), equations (3.25) and (3.26) can be re-written for a solid mantle as: 3 1/3 ∆Trh gρsα∆Trhd Fc = k (3.27) d κη ! Thus the Rayleigh number in equation (3.26) cannot simply be replaced by the Rayleigh number for a partially molten mantle (equation 3.20). Instead, we note −1/3 1/3 that for a partially molten mantle θ Rai should be replaced by 1 3 ρ g(α ∆T + β∆φ)d3 / s θ (3.28) κη ! and the Nusselt number becomes: 1 3 ρ g(α ∆T + β∆φ)d3 / Nu = aθ−1 s θ (3.29) κη ! Reese et al. (2005) ﬁnds that a in equation (3.26) is equal to 0.67. Applying the numerically derived scaling parameters to a problem including partial melt introduces some uncertainty because Reese et al. (2005)’s numerical simulations were for 70 convection in a solid mantle. We therefore test the sensitivity of our results to the constant a. For a known melt fraction and heat ﬂux via volcanic eruptions, we can then calculate the temperature of the mantle by solving equations (3.18), (3.22), (3.24), (3.25), and (3.29). Scaling laws for the velocity in the mantle are derived by equating the work done by viscous forces to the work done by the buoyancy forces that are driving convection (e.g., Solomatov, 1995; Schubert et al., 2001; Turcotte and Schubert, 2002). For an internally heated mantle in the absence of partial melt, this results in a relationship between the average mantle velocity, Vi, the Rayleigh number, and the Frank-Kamenetski parameter:

βu κ Rai Vi = arh (3.30) d θ where arh and βu are constants (e.g., Solomatov and Moresi, 2000). arh and βu diﬀer depending on whether the viscous dissipation occurs primarily within the 1 2 rheologic boundary layer (βu = 2 ) or throughout the whole mantle (βu = 3 ) (Solomatov and Moresi, 2000). Solomatov and Moresi (2000) ﬁnd that more than

half of the dissipation occurs in the rheological boundary layer making βu = 1/2 a better ﬁt to their numerical data for time-dependent internal heating convection. We test both velocity scaling laws presented by Solomatov and Moresi (2000):

arh =0.053 and βu =2/3 (scaling law I); and the better ﬁt, arh =0.38 and βu =1/2 (scaling law II). The presence of melt aﬀects the buoyancy force on solid plumes descending through the partially molten mantle. However, by including the melt contrast in the buoyancy force and using the derivation method outlined in Schubert et al. (2001, chapter 8), one can show that the β∆φ component of the density contrast will be multiplied by a factor much less than 1 and the melt contrast contribution becomes negligible in the velocity scaling law. We therefore use equation (3.30) with ∆φ in

Rai equal to zero. Equation (3.30) represents the average mantle velocity in an internally heated mantle while the melt migration model in Section 3.2.1 requires the velocity of 71 an ascending mantle column. An internally heated mantle will have no hot lower boundary layer and thus no hot ascending plumes (Turcotte and Schubert, 2002, chapter 6). Rather cold plumes sinking from the top boundary layer will drive convection. In steady state boundary-layer theory, this is assumed to drive coun- terrotating cells (Turcotte and Schubert, 2002, chapter 6). If this steady state assumption is correct, equating the ascension velocity with the average mantle velocity is a reasonable assumption. However, mantle convection is more likely to be strongly time-dependent with random cold plumes delivering cold material to all parts of the mantle (Solomatov and Moresi, 2000). Furthermore, the convective motions in Io’s mantle may be more complicated than those of most planetary bodies as tidal dissipation is not expected to be uniformly distributed throughout the mantle (Segatz et al., 1988). Non-uniform tidal heating may drive flow patterns where upwelling occurs primarily in the areas of higher tidal dissipation and downwelling occurs primarily in the areas of lower tidal dissipation (section 4.3). However, understanding the details of the effects of non-uniform tidal heating requires 3-dimensional mantle convection simulations coupled with a tidal dissipation model that can calculate the spatially varying internal heating rate based on the spatially varying rheological properties of the mantle and tidal stresses. Since this has not yet been done, we assume that Io’s average mantle velocity can be calculated from the convective scaling laws calculated by Solomatov and Moresi (2000). To test the sensitivity of our model to these assumptions, we test cases where the ascending velocity is five times higher or five times lower than the velocity predicted by equation 3.30

3.2.3 Melt segregation and convection

The two-phase ﬂow model (Section 3.2.1) depends on the temperature of the con-

vecting mantle, Ti, and the velocity of the ascending column of mantle which we

approximate as Vi. The convective scaling laws (Section 3.2.2) can be used to calculate these parameters if the melt fraction in the mantle, φ, and the heat lost due to

volcanic eruption, Fm are known. Both the two-phase ﬂow model and the convective 72 scaling laws depend on the internal heating rate which drives the internal activity. Thus for a given internal heating rate, we can iterate between the two-phase ﬂow model and the convective scaling laws to solve for the structure of the mantle.

3.3 Results

Io’s observed surface heat flux is 3 ± 1 W m−2 which corresponds to a total internal heating rate of 1.3±0.4×1014 W(Veeder et al., 2004). Using the values in Table 3.1 and an internal heating rate of 1.3×1014 W, we find that Io’s mantle has an average temperature of approximately 1520 K and an average mantle velocity of less than 1 cm year−1. This leads to the formation of a 200 km layer of partial melt which is on average 7% molten. This corresponds to melting less than 2% of the whole mantle. Figure 3.2 shows the radial profiles of the effective pressure (the difference between solid pressure and liquid pressure), N, the percent molten, φ, the velocity of the solid, W , and the velocity of the magma, w. The top of the partially molten zone is 49% magma that travels with a velocity of approximately 3 cm year−1 while the solid matrix sinks with a velocity of 2 cm year−1 to accommodate a higher melt velocity. The heat flux due to conduction through the stagnant lid is 0.0074 W m−2, and the remainder of the 3 W m−2 total surface heat flux is transported by volcanic eruptions. This confirms that Io loses the majority of its internal heat through the heat pipe mechanism of planetary heat loss. The exact values for the parameters in Table 3.1 are uncertain even for Earth’s mantle which has been extensively studied compared to Io, where there are very few constraints on rheology. We test the sensitivity of our model to this uncertainty by varying the values of the uncertain parameters in table 3.1. Table 3.2 shows the range of values tested and the resulting range in values of the average melt fraction in the partially molten zone, φ, the mantle temperature, T , the depth of onset of

melting, dφ, and the ratio of heat lost due to magmatic eruption to heat lost due to

conduction through the stagnant lithosphere, Fm/Fc. We ﬁnd that our results are not sensitive to the melt density (for a ﬁxed density contrast equal to 500 kg m−3), 73

Table 3.1: Rheology constants Parameter Symbol Value

Io’s radius (km) Rs 1821 a Radius of Io’s core (km) Rcore 900 Acceleration due to gravity (m s−2) g 1.796 Surface temperature (K) Ts 110 Universal gas constant (J K−1 mole−1) R 8.314 −1 b 5 Activation energy (J mole ) EA 3 × 10 Pre exponential factor (s−1)b A 8.7 × 1015 Grain size (m) h 10−3 Grain size exponentb m 2.5 Shear modulus (GPa)b µ 80 Burger’s vector (nm)b B 0.5 Melt fraction coeﬃcient Bmelt 25 Heat capacity (J/kgK) cp 1200 Thermal conductivity (W/mK) k 4 Thermal expansivity (K−1) α 3 × 10−5 2 Thermal diﬀusivity (m /s) κ k/(ρsc) 3 Density of solid rock (kg/m ) ρs 3300 3 Density of molten rock (kg/m ) ρl 2800 Latent heat (J/kg) L 5 × 105 Clapeyron slope (K/Pa)c Γ 10−7 Tortuosity factord b 1000 Viscosity of molten rock (Pa s) ηl 5 aAnderson et al. (2001), bKarato and Wu (1993), cHewitt and Fowler (2008), dStevenson and Scott (1991) 74

A B C D 0 0 0 0 −20 −20 −20 −20 −40 −40 −40 −40 −60 −60 −60 −60 −80 −80 −80 −80 −100 −100 −100 −100

y (km) −120 −120 −120 −120 −140 −140 −140 −140 −160 −160 −160 −160 −180 −180 −180 −180 −200 −200 −200 −200

0 50 0 50 −3 −2 −1 0 1 0 10 20 30 N (MPa) φ (%) W (cm yr−1) w (cm yr−1)

Figure 3.2: A) Effective pressure (solid pressure minus liquid pressure); B) percent of material that is molten; C) solid matrix velocity; and D) melt velocity versus depth in Io’s mantle. the density contrast (for a fixed solid density equal to 3000 kg m−3), the channel geometry’s effects on permeability (b in equation (4.8)), the magma viscosity, the mantle viscosity’s dependence on melt fraction (Bmelt in equation (3.22)), or the core radius. Varying those parameters over a range of reasonable values changes the average mantle melt fraction by less than a factor of two; changes in the temperature and thus depth of onset of melting are minimal; and the ratio of heat flux due to volcanic eruptions to heat flux conducted through the stagnant lithosphere is close to 400 for all values tested. Our results do change depending on our assumptions about the grain size which affects both the permeability and the mantle viscosity (Table 3.2). Although grain size in the mantle can vary with space and time due to dynamic recrystallization and grain grown (Karato and Wu, 1993), we assume a constant grain size. We find that 5 mm grains instead of 1 mm grains increase the viscosity of the mantle (Equation 3.22) which leads to a higher mantle temperature and thus a deeper onset of melting. The average melt fraction within the partially molten zone is approximately 7% for grain sizes 1-5 mm, but the deeper onset of 75 melting for larger grains leads to more melting total in the mantle. The higher mantle temperature with larger grains also leads to a higher conductive heat flux out of the mantle, but heat loss due to volcanic eruptions is still 300 times higher than heat flux due to conduction through the stagnant lithosphere. For most of our models, we assume that tidal heating is evenly distributed throughout the whole mantle. The distribution of tidal heating does not affect the results of the convective scaling laws (Section 3.2.2). However, the melt migration model could yield different results for different vertical distributions of tidal heating. In our nominal model (Figure 3.2), we assume that the tidal heating rate (E in equation 3.10) is equal to Io’s total heat output divided by the whole mantle volume. We also try a model in which the tidal heating rate, E, is equal to Io’s total heat output divided by the volume of the partially molten zone (assuming the partially molten zone is a global layer of uniform thickness). Assuming the heating occurs only in the partially molten zone leads to a higher melt fraction (Table 3.2, E) but a lower melt velocity relative to the average mantle velocity. This lower melt velocity leads to a slightly lower eruption rate, so the mantle remains slightly warmer leading to a higher conductive heat flux through the stagnant lithosphere and an overall decrease in the ratio between heat flux due to volcanic eruptions and heat flux due to conduction through the stagnant lithosphere (Table 3.2). In addition to uncertainty in the material properties of and tidal heating distribution in Io’s mantle, convective scaling laws are a simplification of mantle convection, so we test the sensitivity of our results to several key values in the convective scaling laws. The relationship between the Nusselt number and the Rayleigh number has been verified numerically (e.g. Solomatov and Moresi, 2000; Reese et al., 2005), and these numerical studies determined the scaling factor, a, in equation (3.26). Since these numerical studies were not specific to Io and did not include partial melt, we vary a. This has a negligible effect on the melt fraction, the mantle temperature, and the depth of the onset of melting. However, a directly relates the Rayleigh number to the heat flux due to conduction through the stagnant lithosphere (Equations 3.25 and 3.29), so a higher value of a increases the heat flux due to conduction through 76

Table 3.2: Sensitivity to assumptions where φ is the melt fraction in the partially molten zone, T is the average mantle temperature, dφ is the depth at which melting begins and Fm/Fc is the ratio of the heat ﬂux transported by magmatic eruptions to the heat ﬂux transported by conduction through the stagnant lid.

Values φ (%) T (K) dφ (km) Fm/Fc −3 ρl 2300 - 2800 kg m 7.2-8.4 1519-1522 216-248 nochange ∆ρ 300 - 700 kg m−3 7-9.7 1514-1530 222-252 379-422 b 100-1000 4.9-7.2 1522-1523 216-217 404-454 ηl 1-10Pas 5.3-8.9 1522-1523 216-217 379-447 Bmelt 3-40 7-7.3 1519-1527 211-224 384-447 Rcore 600-900km 6.7-7.2 1522-1523 216-218 404-447 hvisc 1-5mm 6.6-7.2 1522-1730 216-567 308-404 E 1.7 × 10−9−4.3 × 7.2-12.9 1522-1534 216-237 282-404 10−9 W kg−1 a 0.53-0.67 no change no change no change 404-516 fe 0.25-1 7.2-7.8 1522-1709 216-531 84-404

the lithosphere even if convective vigor does not change (Table 3.2). We also test the sensitivity of our results to the assumed velocity scaling law both because of possible uncertainty in the scaling law itself and because of the uncertainty in the velocity structure of Io’s mantle discussed in section 3.2.2. We try both velocity scaling laws presented by Solomatov and Moresi (2000) and find that this has a minimal effect on our results (Table 3.3). We also try a case in which the velocity is a factor of five higher and a case in which the velocity is a factor of five lower. The average mantle velocity has a strong dependence on temperature (Equation 3.30), so dividing the velocity in equation (3.30) by five requires higher mantle temperatures to get a similar convective velocity. This higher temperature leads to a deeper onset of partial melting and a higher conductive heat flux (Table 3.3). In general, the velocities reported in table 3.30 are quite low compared to expected convective velocities on Earth. The applicability of the velocity scaling law will be discussed further in section 5. In section 3.2.2, we explained that the melt fraction in the mantle decreases with depth and that it is unknown if the rheological sublayer is solid or partially molten. 77

Table 3.3: Sensitivity to velocity Scaling law where φ is the melt fraction in the partially molten zone, T is the average mantle temperature, Vi is the average mantle velocity, dφ is the depth at which melting begins and Fm/Fc is the ratio of the heat ﬂux transported by magmatic eruptions to the heat ﬂux transported by conduction through the stagnant lid. −1 φ (%) T (K) Vi (m s ) dφ (km) Fm/Fc −11 Vi lawII 7.2 1522 6.7×10 216 404 −11 Vi lawI 7 1528 5.5×10 227 389 −10 (Vi law II)×5 9.2 1490 2.5×10 161 516 −11 (Vi law II)÷5 6.4 1568 2.1×10 293 282

This leads to uncertainty when calculating the Rayleigh number due to both the viscosity and the density contrast’s dependence on melt fraction. The rheological boundary layer could be solid or partially molten and dense plumes could sink through the whole mantle or just the partially molten portion of the mantle (Section 3.2.2), so we test three cases: 1) a solid rheological boundary layer over a mantle

with an average melt fraction of φavg (∆φ = φavg − 0 and η(φavg)); 2) a partially

molten rheological boundary layer over a mantle with an average melt fraction of φavg

(∆φ = 0 and η(φavg)); and 3) a solid rheological boundary layer over partially molten mantle using the average melt fraction in the partially molten zone rather than the

average melt fraction in the whole mantle (∆φ = φavg mz − 0 and η(φavg mz)) Our results do not change signiﬁcantly between these three cases. In particular, table 3.4 shows that the density contrast between the solid rheological boundary layer and the average melt fraction of the whole mantle has a negligible eﬀect on the melt fraction in the partially molten zone, the average mantle temperature, the average mantle velocity and the depth of the onset of melting. In these cases, the average melt

fraction in the whole mantle (φavg) is 1.7%, so β∆φ is approximately 0.0024. α∆Trh is approximately 0.0019, so although the temperature contrast and the melt fraction contrast have a similar contribution to the density contrast between the rheological sublayer and the adiabatic mantle, the Rayleigh number would only decrease by a factor of two if instead β∆φ was zero. The Nusselt number is proportional to the 78

Table 3.4: Sensitivity to the melt fraction assumed when calculating the density contrast between the rheological sublayer and the convecting mantle (equation 3.21) and the viscosity (equation 3.22). φavg indicates the melt fraction averaged over the whole mantle and φavg mz indicates the melt fraction averaged over the partially molten portion of the mantle. The values reported are the melt fraction in the partially molten zone, φ, the average mantle temperature, T , the average mantle velocity, Vi, the depth at which melting begins, dφ and the ratio of the heat ﬂux transported by magmatic eruptions to the heat ﬂux transported by conduction through the stagnant lid, Fm/Fc. −1 ∆φ η(φ) φ (%) T Vi (m s ) dφ Fm/Fc (K) (km) −11 φavg − 0 η(φavg) 7.2 1522 6.7×10 216 404 −11 0 η(φavg) 7.2 1522 6.7×10 216 535 −10 φavg mz − 0 η(φavg mz)8.0 1505 1.3×10 187 183

Rayleigh number to the 1/3 power (Equation 3.29), so a factor of two decrease in the density contrast leads to only a factor of 21/3 decrease in heat transport via conduction through the stagnant lithosphere. This small change in heat transport is accommodated by a negligibly small increase in the average mantle temperature and average mantle velocity. We also try a case in which we calculate the viscosity and the melt fraction’s contribution to the density contrast using the average melt fraction in the partially molten zone instead of the whole mantle. This has a larger eﬀect on the system, because it decreases the mantle viscosity and increases the Rayleigh number by a factor of about 7 (β∆φ =0.012), but it does not change the results signiﬁcantly (Table 3.4).

Finally, equation (3.19) with fe = 1 assumes that all of the magma that reaches the top of the partially molten zone erupts. In reality, some of this melt will never reach Io’s surface because it may freeze in the cooler lithosphere before reaching the surface or it may reach a point of neutral buoyancy where additional forces are needed to cause eruption. To test our sensitivity to this uncertainty, we consider several cases where only a fraction (fe) of the melt flux at the top of the partially molten zone actually leaves the mantle. When the efficiency of eruption is decreased, the mantle is not cooled as easily, so decreasing the fraction of melt allowed to erupt 79 leads to higher mantle temperatures, a deeper onset of melting and thus a higher overall mantle melt fraction (Table 3.2). Despite the decreased eruption efficiency and the higher mantle temperature, the heat flux due to volcanic eruption still exceeds the heat flux due to conduction through the stagnant lithosphere. Io loses most of its internal heat through volcanic eruption rather than conduction through the stagnant lithosphere for the full range of parameter and scaling law assumptions that we tested. We expect that the cases in which some melt is not permitted to erupt to the surface (fe < 1) are more realistic since it is unlikely that the eruption efficiency is 100%. Furthermore, the melt flux out of the upwelling column we consider should be higher than the average melt flux out of Io’s mantle since a melt flux of zero is expected above downwelling zones. Figure 3.3 shows the radial profiles of the effective pressure, N, the percent melt, φ, the velocity of the solid, W , and the velocity of the magma, w, when only 25% of the magma reaching the top of the partially molten zone is allowed to erupt. This lower eruption efficiency leads to a higher mantle temperature which increases both the depth of onset of melting to 530 km and the velocity of the mantle. The average melt fraction is 8% in the partially molten zone and 5% in the whole mantle. The melt fraction reaches a peak value of 15% at the top of the partially molten zone where it travels with a velocity of 43 cm year−1. Since we only allow 25% of this magma to erupt to the surface, this leads to an eruption rate of 1.6 cm year−1. The main effects of limiting the fraction of magma able to erupt are a higher mantle temperature and a higher mantle velocity. The higher mantle velocity leads to a higher melt velocity at the top of the partially molten zone but a lower melt fraction at the top of the partially molten zone. This is because we assume a constant tidal heating rate, so if a parcel of mantle material moves from the bottom of the mantle to the top in a shorter period of time, the total heating that parcel experiences is lower. 80

A B C D 0 0 0 0

−50 −50 −50 −50

−100 −100 −100 −100

−150 −150 −150 −150

−200 −200 −200 −200

−250 −250 −250 −250

y (km) −300 −300 −300 −300

−350 −350 −350 −350

−400 −400 −400 −400

−450 −450 −450 −450

−500 −500 −500 −500

0 5 10 15 0 5 10 15 −6 −4 −2 0 2 0 50 N (MPa) φ (%) W (cm yr−1) w (cm yr−1)

Figure 3.3: A) Eﬀective pressure (solid pressure minus liquid pressure); B) percent of material that is molten; C) solid matrix velocity; and E) melt velocity versus depth in Io’s mantle for an eruption eﬃciency of 25%.

3.4 Discussion

3.4.1 Comparison to Earth

Our results predict significantly more melting on Io than Hewitt and Fowler (2008) predicted beneath mid-ocean ridges on Earth. We find both a thicker partially molten zone and a higher average melt fraction within that partially molten zone. The partially molten zone is thicker, because the solidus increases with pressure which is lower in Io’s mantle due to Io’s low gravity. Thus for similar mantle temperatures, the mantle temperature will equal the solidus deeper in the mantle of small body like Io (equation 3.13). In addition, tidal dissipation continues to heat mantle material as it rises through the partially molten zone leading to a higher overall melting rate in Io’s mantle. Io’s low gravity also decreases the buoyancy force driving melt solid segregation which leads to a lower maximum melt velocity. We have also assumed a smaller grain size than Hewitt and Fowler (2008) which decreases the permeability of the 81 solid matrix and thus also contributes to the lower melt velocity. Despite the slower melt solid segregation, high melt fractions lead to a mass flux of melt high enough that the solid matrix must sink to accommodate the upward migration of melt. Our simulations also predict a higher effective pressure (solid pressure minus liquid pressure) on Io than on Earth. We calculate the solid viscosity according to the Arrhenius law (equation 3.22) using the mantle temperature calculated by convective scaling laws whereas Hewitt and Fowler (2008) assume a solid viscosity of 1019 Pa s. With the Arrhenius law and the mantle temperature from convective scaling laws we find a solid viscosity of 5×1020 Pa s. This higher solid viscosity makes the solid matrix more difficult to deform decreasing compaction and increasing the effective pressure (equation 3.9).

A B C D

0 0 0 0 E = 0 W kg−1 E = 1.7x10−9 W kg−1 −50 −50 −50 −50

−100 −100 −100 −100

−150 −150 −150 −150 y (km)

−200 −200 −200 −200

−250 −250 −250 −250

−300 −300 −300 −300 0 5 10 0 5 10 −5 0 5 0 50 100 N (MPa) φ (%) W (cm yr−1) w (cm yr−1)

Figure 3.4: A) Eﬀective pressure (solid pressure minus liquid pressure); B) percent of material that is molten; C) solid matrix velocity; and D) melt velocity versus depth in Io’s mantle assuming the same mantle temperature (1500 K) and upwelling velocity (3 cm year−1) and model constants as Hewitt and Fowler (2008) for an internal heating rate of 0 W kg−1 and 1.7 × 10−9 W kg−1.

The differences between the partially molten zones on Io and Earth resulting from Io’s small size and internal heating can be seen more clearly if we consider melt migration using the same rheological values, mantle velocity and mantle temperature 82 as Hewitt and Fowler (2008). Figure 3.4 shows the effective pressure, melt fraction, solid matrix velocity, and melt velocity for two cases: one with an internal heating rate of 0 W kg−1, and one with an internal heating rate of 1.7 × 10−9 W kg−1. In both cases, instead of using convective scaling laws, we have assumed a mantle temperature of 1500 K and a mantle velocity of 10−9 m s−1 as Hewitt and Fowler (2008) did for Earth. This is not an equilibrium solution for Io, but rather a direct comparison of melt migration on Earth and Io. We have also used the values given in Table 1 of Hewitt and Fowler (2008) which differ slightly from the values in Table 3.1. However, we used the acceleration due to gravity and surface temperature appropriate for Io as given in our Table 3.1. These results show that if Io and Earth had the same mantle temperature and the same upwelling velocity, Io would have a thicker partially molten zone, because the solidus temperature depends on the pressure, which at a given depth is lower on Io than on Earth due to Io’s lower gravity. Io’s lower gravity also reduced the buoyancy force driving melt solid segregation. This reduces the effective pressure (here we use the same solid viscosity as Hewitt and Fowler (2008)).The reduced the effective pressure results in a reduced magnitude of the solid velocity gradient. The magnitude of the gradient of the melt velocity is also reduced and the maximum melt velocity is lower. The melt fraction gradient does not change with gravity, so the thicker partially molten zone results in more total melting. The main effects of Io’s small size on melt generation and migration are a deeper onset of melting, a lower effective pressure, and a lower melt velocity. Figure 3.4 also shows that adding internal heating to the ascending column increases the amount of melting. The increased melting leads to more rapid melt solid segregation (equation 3.3) which decreases the solid matrix velocity and increases the melt velocity. The increase in melt-solid segregation is large enough that the solid velocity becomes negative at the top of the partially molten zone to accommodate the high mass flux of melt. The steeper solid velocity gradient leads to a higher effective pressure than the case with no internal heating. However, the higher melt fraction prevents a large increase in effective pressure (equation 3.9), and the 83 effective pressure is still lower than that in the partially molten zone on Earth. The comparison of the two cases shown in figure 3.4 show that the main consequences of internal heating on mantle material rising through a partially molten zone are increased melting and increased melt-solid segregation.

3.4.2 Comparison to observations

Khurana et al. (2011) estimated that the electrical conductivity of a partially molten layer 50 km thick and 20% molten would best explain the Galileo magnetometer detection of an induced magnetic field whereas Roth (2012) did not see any influence from a conducting layer in the rocking of Io’s auroral spots. In most of our models, we find the partially molten zone is about 7% molten. However, the melt fraction is highest near the top of the partially molten zone where it is also easier to detect (Khurana et al., 2011). In the simulation shown in Figure 3.2, the top 50 km of the partially molten zone is an average of 15% molten. If instead of assuming Io’s total heat output is generated evenly throughout the mantle, we assume it is generated only in the partially molten zone (E in equation 3.10), then the average melt fraction in the partially molten zone is 13% and rises to 29% in the top 50 km. However, these examples assume 100% eruption efficiency which is unrealistic. If instead we assume only 25% of the magma that reaches the top of the partially molten zone can erupt (perhaps still an overestimate), the melt fraction in the partially molten zone is 8% and only reaches 12% molten in the top 50 km of the partially molten zone. We conclude that with certain assumptions, our model can produce the melt fractions inferred by Khurana et al. (2011), but for other assumptions, it cannot. A partially molten layer with low melt fractions might not have a high enough conductivity to be detectable, so the simulations with lower melt fractions would be consistent with the observations of Roth (2012). Given the uncertainty in our model we cannot say with certainty whether it better agrees with the melt fractions inferred by Khurana et al. (2011) or the low conductivity inferred by Roth (2012). Io’s high resurfacing rate can also be used to constrain activity in its interior. Johnson et al. (1979) found that a resurfacing rate of at least 1 mm year−1 is re- 84 quired to explain the lack of observed impact craters on Io’s surface. Our model predicts eruption rates of 15-20 mm year−1 which is consistent with the inferred minimum resurfacing rate. Decreasing the eruption efficiency does not significantly decrease the eruption rate (16 mm year−1 instead of 17 mm year−1), because melt velocities in the mantle increase significantly leading to similar eruption rates despite the lower efficiency. However, we expect our results to overestimate the actual eruption rate for several reasons. First, these calculations were done for an upwelling column of mantle which will have higher eruption rates than downwelling zones. Additionally, we neglect intrusion of magma into the lithosphere. Intrusion does remove magma from the mantle, but it will not cool to Io’s surface temperature so equation (3.19) would overestimate the surface heat flux due to intruding magma. Therefore, although 16-20 mm of magma may leave the top of upwelling zones in Io’s mantle every year, much of this magma may stall in the lithosphere or crust before reaching Io’s surface and erasing craters.

3.5 Conclusions

We adapted a model for melt generation and two-phase flow in an upwelling column of mantle beneath mid-ocean ridges to an upwelling column in Io’s tidally heated mantle. This model requires an assumed mantle velocity and temperature. We calculated the temperature and velocity of Io’s mantle using convective scaling laws adapted to include the effects of partial melting: increased density contrast, decreased viscosity, heat loss through volcanism which can all be calculated with the melt generation and two phase flow model. We couple the melt generation and two phase flow model and the convective scaling laws to self-consistently calculate the distribution of magma in Io’s mantle and the relative importance of heat transport via conduction through the stagnant lid and heat transport via volcanic eruptions. We confirm that Io is losing the majority of its internal heat through volcanic eruptions rather than conduction through a stagnant lithosphere. In all cases tested, the heat flux due to volcanic eruptions exceeds the heat flux due to conduction through 85 a stagnant lithosphere by approximately two orders of magnitude. Io’s mantle has a solid lower mantle and a partially molten upper mantle that is 5-13% molten. Melt- ing starts at a depth 200-500 km below the surface. Peak melt fractions can reach close to 100% magma near the top of the partially molten zone, but the percent molten quickly drops to the single digits deeper in the partially molten zone. Our results suggest that Io’s eruption rate is 1.5-2 cm year−1 which is sufficiently high to explain the lack of impact craters on Io’s surface. 86

CHAPTER 4

Convection in Io’s partially molten mantle

4.1 Introduction

Jupiter’s moon Io has a surface heat flux of 3±1 W m−2 (Veeder et al., 2004)—30 times that of Earth (Davies and Davies, 2010). This heat is produced through tidal dissipation in Io’s mantle. As this tidal dissipation works to circularize Io’s orbit, the Laplace resonance with Europa and Ganymede pumps up Io’s eccentricity allowing the tidal heating to continue (Peale et al., 1979). This high heating rate leads to rapid resurfacing. The lack of impact craters on Io’s surface suggests it is resurfaced at an average rate of at least 1 mm year−1 (Johnson et al., 1979). However, this resurfacing is not due to plate tectonics; instead, Io’s surface is covered in volcanoes (Figure 4.1) whose eruptions continuously bury older surface rock and force it to subside back towards the mantle. With no plate tectonics and a high surface heat flux, Io should have a very thin lithosphere through which heat is conducted to the surface, but tectonic mountains, as tall as 17.5 km (Schenk et al., 2001), suggest that Io has a thick lithosphere capable of supporting significant topography (O’Reilly and Davies, 1981). This led O’Reilly and Davies (1981) to suggest that Io loses its internal heat through volcanic eruptions by advecting magma to the surface where it then freezes and cools to the surface temperature burying the old surface. The forced subsidence of the surface continually advects cool rock downward and suppresses the conductive warming of the lithosphere (O’Reilly and Davies, 1981). Heat transport by volcanic eruption also greatly reduces the total heat flow due to conduction which is inversely related to the thickness of the lithosphere. These two factors could allow Io to have a high surface heat flux (primarily due to volcanic eruption) and a thick lithosphere. This type of planetary heat loss has been named ‘the heat pipe mechanism’. 87

If Io’s extensive volcanism has been active through most of its history and if the erupting magma is derived from partial melting of a mostly solid mantle, Io should be highly differentiated (Keszthelyi and McEwen, 1997). Such a differentiated mantle and crust should lead to eruption temperatures less than 1375 K (Keszthelyi and McEwen, 1997). However Galileo data from the Solid State Imager (SSI) camera and the Near Infrared Mapping Spectrometer (NIMS) found that many volcanic centers had eruption temperatures significantly above the 1300-1450 K temperature typical of terrestrial basalts (McEwen et al., 1998). Keszthelyi et al. (2007) applied a lava fountain model (instead of the previously used lava flow models) to the highest reported eruption temperature, the SSI observation of the 1997 eruption at Pillan and found that the lower limit on the temperature of that eruption is ∼1615 K. This is lower than previous estimates, but still implies an ultramafic rather than basaltic composition. A recent ground based campaign detected a powerful eruption on Io in August of 2013. Modeling of the outburst spectrum places a lower limit of 1200-1300 K on the eruption temperature, but the best-fit model spectra suggest an eruption temperature of 1900 K or higher (de Kleer et al., 2014). Spectral data from the Galileo mission is consistent with the ultramafic composition inferred from high temperature eruptions. Low albedo spots that changed between Voyager and Galileo observations tend to correspond to detectable hot spots, and many dark spots are confined to caldera interiors. Spectral data from these dark materials suggest that they contain magnesium-rich orthopyroxene which is consistent with high temperature eruptions (Geissler et al., 1999). The similar compositions of dark materials on Io suggest that actual eruption compositions and temperatures may be uniform. Most high temperatures measured at eruptions on Io are lower limits, so the high temperature estimate from the 1997 eruption at Pillan could be the typical eruption temperature for all silicate volcanism on Io (Keszthelyi et al., 2007). The implied ultramafic composition of lava erupting on Io’s surface disproves the theory that magmatic activity has extensively differentiated Io (Keszthelyi et al., 1999; Keszthelyi et al., 2007). Rather Io’s mantle must be well mixed. This could be achieved by a partially molten magma ocean which heats the base of the crust 88 and remixes that material into the mantle (Keszthelyi et al., 1999).

Figure 4.1: Global mosaic of images from the Galileo and Voyager missions. Credit NASA/USGS.

Previous models of heat transport in Io’s mantle have considered either solid state convection or melt migration, but both processes act together to transport heat through Io’s mantle. The first numerical simulations of mantle convection in Io were performed by Tackley et al. (2001) in three-dimensional spherical geometry and showed that higher Rayleigh number convection results in lower spatial variation of surface heat flux resulting from non-uniform tidal dissipation. These calculations considered internal heating Rayleigh numbers many orders of magnitude lower than what is expected to be appropriate for Io (Tackley, 2001). Tackley (2001) used two-dimensional calculations of convection in Io’s asthenosphere to reach higher Rayleigh number regimes and develop scalings that could be used to extrapolate to Io. These scalings predict spatial variations in surface heat flux of several percent (Tackley, 2001). Neither Tackley et al. (2001) or Tackley (2001) treat the melt solid segregation or explicitly include the effects of partial melting on viscosity. Moore (2001) took the opposite approach and considered heat transport by magma migration while ignoring advection of heat due to convection. Moore (2001) considered 89 a system in which tidal heating is balanced by melt production and the melt rises through the solid matrix according to Darcy’s law and mass conservation between the liquid and the solid with no net flow of material. He found that even with melt fractions below 20%, all of Io’s tidally generated heat could be transported by melt segregation alone. Further evidence indicating that Io’s mantle may contain a significant amount of magma in its mantle was discovered when the Galileo spacecraft detected an induced magnetic field at Io. Khurana et al. (2011) showed that a completely solid mantle does not have a high enough electrical conductivity to produce the observed induced magnetic field. The best fit to the data consists of a layer at least 20% molten and 50 km thick beneath a 50 km thick non-conducting crust (Khurana et al., 2011). The dipolar nature of the field and the lack of higher order spherical harmonics suggests that the signal cannot be produced by localized regions of melt; rather, the data is better explained by a global layer of interconnected melts (Khurana et al., 2011). However, a conducting layer should also affect the time-variable morphology of Io’s aurora (Roth, 2012). Hubble space telescope observations of Io’s aurora can be explained without the high-conductivity layer inferred by Khurana et al. (2011) (Roth, 2012) but are a less direct probe of the magnetic field than the magnetometer’s measurements. Here we present two-dimensional (2D) numerical simulations of mantle convection in Io that self consistently solve the two-phase flow equations (e.g. McKenzie, 1984) and calculate the heat lost through volcanic eruption in addition to conduction through the stagnant lid. First, in section 4.2, we describe the equations solved, the numerical model, and the parameters to which the model may be sensitive. In section 4.3, we present the results of our simulations and the sensitivity of those results to the uncertainty due to the assumptions described in section 4.2. Finally in section 4.4, we compare our results to observational constraints. By self-consistently solving the two-phase flow equations, we can reproduce Io’s 1) thick solid lithosphere; 2) partially molten mantle; and 3) high eruption rate. 90

4.2 Model

The physical model and solution method are similar to those used in previous models of Earth (e.g. Nakagawa and Tackley, 2005, 2010; Nakagawa et al., 2009, 2010), Mars (Keller and Tackley, 2009), and Venus (Armann and Tackley, 2012), with the added effects of magma migration through the mantle rather than instantaneous eruption. The infinite Prandtl number approximation is made and compressible anelastic approximations are applied. The set of equations (in Einstein notation), nondimensionalized to the mantle depth, D, thermal diffusion time-scale D2/κ where κ is the thermal diffusivity, and the temperature drop across the system ∆T , are

1. Conservation of mass: ∇·~ (ρ~v)=0 (4.1)

2. Conservation of momentum:

2 Razρˆ (C,z,T ) ∇·~ η vi,j + vj,i − vk,kδij − ∇~ p = (4.2) 3 ∆ρthermal

3. Conservation of energy

DT ρC = −Di αρT v + ∇~ (k∇T )+Φ+ ρH (4.3) p Dt s z where ρ is the density, ~v is the bulk velocity, η is the viscosity, p is the pressure, z is the vertical coordinate, C is the composition, T is the temperature, ∆ρthermal

is the fractional density variation with temperature (α∆T ), Cp is the speciﬁc heat, α is the thermal expansivity, k is thermal conductivity, Φ is the viscous dissipation and H is the internal heating rate. Dis, the surface dissipation number, and Ra, the Rayleigh number, are non-dimensional numbers deﬁned by:

αgD Dis = (4.4) Cp and ρgα∆TD3 Ra = . (4.5) ηκ 91

Table 4.1: Model parameter values Parameter Symbol Value

Planetary radius Rs 1821 km Core radius Rcore 941 km Gravity g 1.8 m s−2 Surface temperature Ts 100 K −1 −1 Speciﬁc heat cp 1200 J kg K Latent heat of melting L 600 kJ kg−1 Thermal conductivity k 3 W m−1 K−1 Thermal expansivity α 5 × 10−5 K−1 Internal heating rate H 1.4 × 10−9 W kg−1 18 Reference viscosity η0 10 Pa s −1 Activation energy EA 300 kJ mole Activation volume V 5 cm3 mole−1 Viscosity melt exponent B 3 −10 2 Permeability constant k0 3 × 10 m Permeability exponent n 2 Lithosphere base temperature Tlith 1400 K

The density, thermal expansivity, and thermal conductivity vary with depth. Solid- state phase transitions in the mantle are not included in this study, because they are unlikely to be important on Io given its small size. The assumed values of physical properties are listed in Table 4.1. If the temperature rises above the solidus, melting is calculated to keep the temperature on the solidus. When melt forms it migrates according to Darcy’s law:

kφ F = f(vliq − vsol)= (1 − f)g∆ρ (4.6) ηl

where F is the Darcy flux, f is the melt fraction, vliq is the velocity of the liquid, vsol is the velocity of the solid matrix, kφ is the permeability of the mantle, ηl is the melt viscosity, g is the acceleration due to gravity, and ∆ρ is the density contrast between the liquid and the solid. The surface boundary is free slip and isothermal. A surface temperature of 100 K is assumed. The core-mantle boundary is free-slip and insulating with zero flux, because heat flux from the core is expected to have negligible effects on Io’s 92 mantle compared to the heating from tidal dissipation. A reflecting boundary is used for both sides of the domain. The initial temperature field is adiabatic with a potential temperature of 1600 K between two thin boundary layers at the top and bottom of the domain. It also includes random temperature perturbations with an amplitude of 20 K. We solve equations (4.1)-(4.6) using the code StagYY (Tackley, 2008) in Carte- sian geometry. We consider a domain with the depth of Io’s mantle and a width approximately equal to the distance from Io’s equator to pole. StagYY uses a com- bined Eulerian-Lagrangian method with finite-volume discretization, a direct solver to calculate the velocities, and variable mass tracer particles to track the advection of composition and melt (Tackley and King, 2003). We use a grid resolution of 256 × 64 with 4 million tracers. Calculations are run to a statistical steady state in which heat flux, mean temperature, root mean squared velocity, and melt fraction fluctuate around a mean value that does not evolve with time.

4.2.1 Rheology

We assume the viscosity of the mantle material depends on temperature and pressure according to the Arrhenius model, with an additional term to account for the decrease in viscosity due to partial melting:

E + pV E η(T,p)= η0 exp − exp(−Bf) (4.7) RT RT0 where T is the temperature, p is the pressure, EA is the activation energy, V is the

activation volume, R is the gas constant, η0 is the reference viscosity at temperature

T0 = 1600 K and zero pressure, and B is a constant that describes the viscosity’s dependence on melt fraction, f. We use values for the activation energy, EA, and the activation volume, V , as determined in the laboratory by Karato and Wu (1993) for diﬀusion creep in dry olivine. We try a similar range of values for B as Moore (2003). The viscosity depends on grain size which is unknown, so we vary the reference viscosity from 1017 Pa-s to 1019 Pa-s. 93

4.2.2 Composition

Composition is assumed to vary between two end-members, basalt and harzburgite. The composition of Io’s mantle is not well constrained (e.g. Keszthelyi et al., 2007), so we assume a bulk composition similar to the bulk composition of Earth’s mantle, 20% basalt and 80% harzburgite (Xu et al., 2008). Initially, the composition is homogenous, but compositional variations arise due to melt-induced diﬀerentiation. The treatment of melt generation in StagYY is described in Armann and Tackley (2012, and references therein). However, Io’s high internal heating rates lead to melt fractions in its mantle that are higher than other rocky solar system bodies, so in this study, we allow both basalt and harzburgite to melt. If molten harzburgite encounters solid basalt, the basalt should melt and the harzburgite should solidify. We do not include this eﬀect in our initial study, but future work will include this.

4.2.3 Melting and eruption

The fraction of a grid cell that is molten is determined by the amount of melting required by the local composition, temperature, and pressure conditions and by the amount of melt which has migrated into the cell from adjacent cells. The density contrast between the liquid and the solid is also calculated according the composition of each phase and the local pressure and temperature conditions. The migration of the magma is determined by Darcy’s law (Equation 4.6). There is some degree of uncertainty in the appropriate values for both the permeability of the material and the melt viscosity. We test a range of values for melt viscosity. The permeability of the material has several sources of uncertainty: the grain size of the solid, the channel geometry, and the permeability’s dependence on the fraction of liquid. Permeability can be defined by a2f n k = (4.8) φ b where a is the grain size, b relates to the interconnectedness of the space between the grains, f is the melt fraction, and n depends on the melt fraction and the channel geometry. The exponent n has been estimated from both experimental studies and 94 idealized models and is typically assigned a value of 2 or 3 (Stevenson and Scott, 1991). We use n = 2 for most cases we consider but also try one case with n = 3. We combine the uncertainty in a and b into one parameter, the permeability constant, a2 −9 2 −11 2 0 k = b . We vary this permeability constant from 3 × 10 m to 3 × 10 m which for typical values of b corresponds to grain sizes ranging from approximately 0.1 mm to 1 mm. Darcy flow is a good description of the movement of magma through most of the mantle, but eventually, as the magma rises towards the surface, it forms macroscopic channels that can fracture the more rigid near-surface rock, forming dikes and sills. This movement of magma through the lithosphere to the surface occurs over different time and spatial scales than mantle convection and the mechanics of eruption is still an active area of research even on Earth, so we do not model the full process of eruption. Rather we try two methods to parameterize this process. The main method we use to parameterize eruption involves defining the base of the lithosphere

according to a certain temperature, Tlith, which is a free parameter. For appropriate values of Tlith, rock at a temperature less than Tlith will be too viscous to participate in convection and will be a part of the stagnant lithosphere. We assume that when melt reaches the base of the lithosphere as defined by Tlith, it erupts to the surface if it is less dense than the solid at every depth as it moves through the lithosphere. We also try cases in which eruption is parameterized such that any melt that would otherwise freeze instead erupts to the surface. This results in only magma at the top of the partially molten zone erupting. In statistical steady state, this magma would be re-supplied by migration of magma from below. We try this parameterization of eruption in two ways: all melt which would otherwise freeze instead erupts, and all basaltic melt which would otherwise freeze instead erupts. The latter prevents harzburgite (which is denser than basalt) from erupting. In all parameterizations, we assume that the movement of the magma and the resulting subsidence occurs only in the vertical direction. Therefore, erupting magma is moved to the surface grid cell directly above it, cooled to the surface temperature, and the solid lithosphere is displaced downwards to conserve mass. The new temperature profile resulting from 95 this displacement is calculated. The total mass of erupta is calculated by summing the erupta from each surface grid cell. The heat removed from Io by these magmatic eruptions consists of both the latent heat released upon solidification and the heat released when the now solidified magma cools to the surface temperature. The magmatic heat flux is calculated by dividing the heat removed by the total mass of erupta by the surface area and the time step. We do not include the effects of heat lost to cooler rock as the magma moves through the lithosphere. This is an important process to consider in future work, because in reality, only some of the melt moving through the lithosphere will erupt. Some of the melt will lose enough heat to the surrounding solid that it will freeze before erupting (e.g., O’Neill et al., 2007). On Earth, the volume of magma intruding exceeds the volume of magma erupting onto the surface at most volcanoes with a median value of five times more intrusion than eruption (White et al., 2006). Though the percentage erupting on Io may be higher if a constant flux of magma through the lithosphere keeps the lithosphere warmer than that of Earth’s, it is likely that a significant fraction of magma leaving Io’s mantle never reaches the surface. We expect that neglecting this effect leads to an overestimate of eruption rate and possibly an underestimate of the mantle melt fraction. It may also overestimate the lithospheric thickness because adding heat to the lithosphere may weaken it and allow the lower lithosphere to break off and participate in the convection below unless volcanic conduits are spaced widely enough that this does not affect the average behavior of the lithosphere. Improving the parameterization of eruption such that it accounts for intrusion and the heat transport between melt and solid as the magma moves through the lithosphere will be considered in future work.

4.2.4 Tidal heating

Io’s global average heat ﬂow measured through ten years of ground based observations is ∼2.5 W m−2 (Veeder et al., 1994). Galileo observations of higher than expected brightness temperatures at the poles led to a revised global average heat ﬂow estimate of 3±1 W m−2 (Veeder et al., 2004). If Io’s currently observed surface heat 96

flux is in equilibrium with the heating rate in Io’s mantle, the revised global average heat flow estimate corresponds to a heating rate of 1.8 ± 0.6 × 10−9 W kg−1. In our nominal models, we assume an average heating rate of 1.4 × 10−9 W kg−1, which is consistent with ground based estimates of Io’s global heat flow (Veeder et al., 1994) and still within the error bars of the post-Galileo estimates (Veeder et al., 2004). The magnitude of tidal heating depends on the tidal potential and the mantle rheology, so it can vary spatially. Dissipation in a homogeneous mantle produces maximum dissipation rates at the poles whereas dissipation in a 50-100 km thick asthenosphere produces maximum dissipation rates at the equator (Segatz et al., 1988). We do not attempt to determine the true distribution of tidal dissipation in Io’s mantle in this paper, but we note that it is non-uniform in both end-member case considered by Segatz et al. (1988) and in more recent tidal dissipation models (Tobie et al., 2005; Beuthe, 2013). To approximate this non-uniform heating, we adopt a heating rate varying with the horizontal position, x: πx H(x)= Havg 1 + cos (4.9) A where Havg is the average internal heating rate and A is the width of the domain. Io’s actual heating rate could vary with depth as well, but in this initial study we only consider horizontal variations in heating rate. We do also consider one case

in which the heating rate (Havg) is distributed uniformly throughout the domain to test the sensitivity of our results to internal heating distribution. Io’s high rate of volcanism is currently unique in the solar system. To understand how internal heating rate could aﬀect the mechanism through which an Io-sized body loses its internal heat, we consider internal heating rates ranging from 4.6 × 10−13 W kg−1 to our nominal value of 1.4 × 10−9 W kg−1.

4.3 Results

4.3.1 Statistical steady state behavior

Our simulations start with a solid mantle of homogeneous composition, which is heated as described in Section 4.2.4. The heating leads to convection, melting, melt 97 migration, and eruption, and eventually the system evolves to statistical steady state. Figure 4.2A shows an example of the temperature field in a simulation that has reached statistical steady state. It shows a cold lithosphere overlying a warm mantle. Figure 4.2B shows the melt fraction field for the same simulation with white contours indicating locations with 1% melt. The cold lithosphere is solid, the upper mantle is partially molten, and the lower mantle is solid. The horizontally averaged radial melt profile shows that the partially molten upper mantle reaches a maximum of 6% magma (Figure 4.3A). The partially molten layer extends from a depth of approximately 100 km below the surface to approximately 550 km below the surface. The vertical component of the mantle velocity field, also featuring lines indicating the locations of 1% melt, shows that very little melting occurs in the downwelling portions of the mantle (Figure 4.2C). The composition field (Figure 4.2C) shows that the solid portion of the mantle is pure harzburgite (some basalt is present in the magma) and the lithosphere is compositionally distinct. The lithosphere is a mix of harzburgite and basalt, but in reality, if molten harzburgite encounters solid basalt as it moves through the lithosphere, it should solidify and melt the basalt instead; a process which is not included in these simulations. Therefore, we expect the lithosphere to have a higher basalt fraction than predicted in these simulations. Through most of the mantle, the temperature profile in our simulations is typical for a convecting mantle. It is close to adiabatic (Figure 4.3B). However, contrary to typical mantle convection (e.g. Reese et al., 2005), a large part of the temperature drop occurs beneath the stagnant lid. We define the base of the stagnant lid as the depth at which the maximum gradient of the horizontally averaged radial profile of the root mean squared mantle velocity intersects the depth axis as in Reese et al. (2005, Figure 1). This is due to the non-uniform internal heating rate that we assumed for this simulation. Radial profiles of temperature and velocity resulting from a uniformly distributed internal heating rate show behavior more typical of stagnant lid convection (Figure 4.4). The velocity goes to zero at the same depth where the temperature begins to decrease. Figure 4.4 also shows that melt fraction increases as depth decreases in the mantle and decreases abruptly as depth decreases 98

Figure 4.2: Instantaneous fields after evolving to statistical steady state for the case using the values in table 4.1 and an internal heating rate of 1.4−9 W kg−1 distributed such that the heating rate is higher on the left side of the domain (equation 4.9). The fields shown are A) temperature ranging from 100 K (blue) to 2137 K (red); B) melt fraction ranging from solid material (blue) to 88% magma (red); C) the vertical (z) component of the mantle velocity (m s−1); and D) the composition of the solid component, C, where C = 1 indicates pure harzburgite. The domain is 880 km deep and 2772 km wide. The white lines in B, C, and D correspond to locations that are 1% molten. The melt fraction field shows that regions within the white lines have melt fractions greater than 1%. 99

A B C D E 0 0 0 horizontal 0 0 vertical −100 −100 −100 rms −100 −100

−200 −200 −200 −200 −200

−300 −300 −300 −300 −300

−400 −400 −400 −400 −400

−500 −500 −500 −500 −500 Depth (km) −600 −600 −600 −600 −600

−700 −700 −700 −700 −700

−800 −800 −800 −800 −800

−900 −900 −900 −900 −900 0 5 0 1000 2000 0 500 1000 3000 3200 3400 0 0.5 Percent Molten Temperature (K) Velocity (cm/yr) Density (kg/m3) Composition

Figure 4.3: The statistical steady state horizontal average radial proﬁles for the non-uniform tidal heating case of the A) percent of partial melting, B) temperature, C) root mean squared (rms) velocity, root mean squared horizontal velocity, and the mean of the absolute value of the vertical velocity, D) density, and E) basalt fraction of the solid component versus depth. The dashed horizontal line indicates the core mantle boundary and the solid horizontal line indicates the base of the stagnant lid which is deﬁned as the intersection of the maximum gradient of the (rms) velocity with the depth axis.

in the stagnant lid. Thus the peak melt fraction corresponds to the base of the stagnant lithosphere. The relationship between temperature, velocity, melt fraction and the base of the lithosphere in the non-uniform internal heating case can be understood through figure 4.2. The left side of the domain, which experiences a higher heating rate has a thinner lithosphere and high melt fractions at the base of the lithosphere. Even a small pocket of 100% magma can lead to a melt fraction of a few percent at that depth in the horizontally averaged radial profile. Similarly, areas of high melt fraction have a low viscosity allowing for high local velocities at shallow depths on the left side of the domain whereas material at that same depth on the right side of the domain is cool and stagnant. Between the heights of approximately 600 and 800 km, the horizontally averaged radial profiles have averaged over some lithosphere and some partially molten mantle (Figure 4.2). The values at these depths are not representative of every horizontal location at that depth. 100

A B C D E 0 0 0 0 0 horizontal −100 −100 −100 vertical −100 −100 rms −200 −200 −200 −200 −200

−300 −300 −300 −300 −300

−400 −400 −400 −400 −400

−500 −500 −500 −500 −500 Depth (km) −600 −600 −600 −600 −600

−700 −700 −700 −700 −700

−800 −800 −800 −800 −800

−900 −900 −900 −900 −900 0 5 0 1000 2000 0 200 400 3000 3200 3400 0 0.5 Percent Molten Temperature (K) Velocity (cm/yr) Density (kg/m3) Composition

Figure 4.4: The statistical steady state horizontal average for the uniformly distributed tidal heating case of the A) percent molten (%), B) temperature, C) root mean squared (rms) velocity, root mean squared horizontal velocity, and the mean of the absolute value of the vertical velocity, D) density, and E) basalt fraction for the solid component versus depth. The dashed horizontal line indicates the core mantle boundary and the solid horizontal line indicates the base of the stagnant lid which is deﬁned as the intersection of the maximum (rms) velocity gradient with the depth axis.

Convection is driven by density contrasts. In rocky mantles, these density contrasts are primarily due to temperature contrasts, but in the case of wide-spread partial melting, the presence of magma also contributes to density contrasts. We find that the depth of the minimum horizontal-mean density corresponds to the depth of the peak melt fraction (Figures 4.3 and 4.4). This depth of highest melt fraction also corresponds with the bottom of the stagnant lid1 which can be seen in the horizontal average radial profiles for the uniform heating distribution (figure 4.4). This is likely to be true in the non-uniform heating case as well, but the stagnant lithosphere and depth of maximum melt fraction both vary horizontally, so the horizontally averaged radial profiles do not show this. The simulations show the stagnant lithosphere is thick as predicted by previous

1Our vertical resolution is 64 cells and the base of the stagnant lid is calculated according the resulting velocity profiles and need not line up exactly with the center of the grid cell. This results in a slight offset between the highest melt fraction and the base of the stagnant lid in figure 4.4. 101

authors (e.g. O’Reilly and Davies, 1981). O’Reilly and Davies (1981) analyzed the importance of subsidence of the lithosphere resulting from volcanic eruption by considering the statistical steady-state heat balance in the lithosphere. Equation (2) in O’Reilly and Davies (1981) shows that the subsidence of the cold surface is balanced by conduction of heat into the lithosphere from the hot mantle below and internal heating in the lithosphere. We compare the relative importance of these three terms in our results. After eruption, StagYY conserves mass through compaction of the mantle and subsidence of the lithosphere in accordance with magma removal. The velocity of the resulting subsidence is not calculated explicitly, but the average resurfacing rate is calculated. As in O’Reilly and Davies (1981), we dT d2T equate the resurfacing rate and the subsidence rate. We calculate dz and dz2 in the lithosphere from the temperature profile shown in figure 4.3A. Then we integrate each term in equation (2) of O’Reilly and Davies (1981) over the thickness of the lithosphere. We find that the internal heating term is four orders of magnitude lower than advection and conduction terms. Thus downward advection of the lithosphere is in fact a key component of the heat balance in the lithosphere and allows for the development of a thick lithosphere as shown in O’Reilly and Davies (1981). Our simulations do not account for the movement of magma through the lithosphere or the stalling and solidification of magma in the lithosphere. These processes would result in magma losing heat to cooler surrounding rock and could work to warm the lithosphere more rapidly than conduction from below. Further- more, intrusion of magma mid-lithosphere would force subsidence of material significantly warmer than the surface material. These effects may decrease the relative importance of downward advection in the overall heat balance in the lithosphere. O’Reilly and Davies (1981) considered only a portion of the lithosphere far from volcanic vents where these processes would not contribute. Perhaps the presence of magma in Io’s lithosphere is sufficiently localized such that it can be ignored when considering the average behavior of the lithosphere, but the high rate of volcanism on Io merits a more detailed study testing this assumption in the future. Within the convecting mantle beneath the stagnant lid, mantle velocities are 102

high compared to the velocities expected for other bodies in the solar system (e.g. Solomatov and Moresi, 2000; Armann and Tackley, 2012). We find that convective velocities in Io’s mantle are hundreds of centimeters per year. The convection is more vigorous in the upper portion of Io’s mantle where magma is present (Figures 4.3 and 4.4) because partial melting decreases the mantle viscosity (Equation 4.7). This emphasizes that it is necessary to include both partial melting and convection when modeling Io’s mantle. We find that mantle velocities are a factor of a few lower when the internal heating is distributed uniformly rather than according to equation (4.9) (Figures 4.3 and 4.4). This is likely because the horizontal variation in heating rate drives horizontal flow. Figure 4.2C shows that non-uniform heating results in one main upwelling zone with high velocities on the left side of the domain and primarily downwelling on the right side of the domain. In the uniform heating case, plumes sinking through the mantle originate at random locations in the top thermal boundary layer and the upwelling and downwelling zones are uniformly distributed (Figure 4.5A). This suggests that non-uniform heating rates can have a strong effect on the geometry of convection cells and the vigor of convection in Io’s mantle. This conclusion should be tested with three-dimensional spherical geometry in future work. The velocity of convection in the mantle is related to the formation of magma. The locations of high vertical velocity are correlated with the locations of high melt fraction (Figures 4.2, 4.3, 4.4 and 4.5). Upwelling regions in the lower mantle are always found below regions of the upper mantle that are at least 1% molten. This is because as the solid rock in the lower mantle ascends, it melts due to decompression melting. Once some magma forms, that parcel of mantle is less dense and less viscous which may lead to further increases in velocity. As previous authors have pointed out, if Io has sustained its current eruption rate through most of solar system history, it should be compositionally differentiated (Keszthelyi and McEwen, 1997). We use simplified treatments of composition and eruption which cannot make detailed predictions for the composition of magma erupting on Io’s surface or the exact composition of Io’s crust. However, our results 103

Figure 4.5: Instantaneous fields after evolving to statistical steady state for the case using the values in table 4.1 and an internal heating rate of 1.4−9 W kg−1 distributed uniformly. The fields shown are A) temperature with temperatures ranging from 100 K (blue) to 2140 K (red); B) melt fraction ranging from solid material (blue) to 55% magma (red); C) the vertical (z) component of the mantle velocity (m s−1); and D) the composition of the solid component, C, where C = 1 indicates pure basalt and C = 2 indicates pure harzburgite. The domain is 880 km deep and 2772 km wide. The white lines in B, C, D, E correspond to locations that are 1% molten. The melt fraction field shows that regions within the white lines have melt fractions greater than 1%. 104 do show that, as expected, eruption leads to a compositionally distinct crust (Figures 4.2D, 4.3E, 4.4E and 4.5D). We find that all of the basalt is either in the crust or in the magma in the mantle. We expect that our simulations over-estimate the amount of harzburgite in the crust, because they do not account for the fact that if molten harzburgite is next to solid basalt, the basalt should melt and the harzburgite solidify. Therefore even when harzburgite is buoyant enough to erupt, once it rises to a depth where it encounters basalt, it is likely to solidify and melt the basalt which might then erupt. Therefore, our simulations may underestimate the amount of chemical differentiation. Our simulations also do not account for the possibility of magma stalling somewhere in the lithosphere before reaching the surface. Rather, magma that reaches the base of the lithosphere is moved to the surface. A more realistic scenario in which magma could cool and solidify in the lithosphere before erupting may not affect all compositions equally. Further work which includes the equilibration of magma composition and the exchange of heat between magma and the lithosphere as the magma moves through the lithosphere would be necessary to put firm constraints on the composition of Io’s crust and the relationship between the crust and the lithosphere.

Heat Flow Contributions 4 10

2 10

0 10 Magmatic Heat flux (TeraWatts) Conductive −2 10 0 0.02 0.04 0.06 0.08 0.1 0.12 Time (Gyrs)

Figure 4.6: Heat flux contributions from magmatic eruption (red) and conduction through the stagnant lithosphere (blue) in a simulation using the values in table 4.1 and a heating rate of 1.4−9 W kg−1 distributed as in equation (4.9). After 0.01 Gyr, both heat flux contributions have evolved to a state where the average value is no longer changing, but the magmatic heat flux oscillates around this average value. 105

Heat Flow Contributions 4 10

2 10

0 10 Magmatic Heat flux (TeraWatts) Conductive

−2 10 0 10 20 30 40 50 60 70 80 90 100 Time (1000 Years)

Figure 4.7: An example of how the heat flux contributions vary over 100,000 yrs where magmatic eruption is depicted in red and conduction through the stagnant lithosphere is depicted in blue. This shows the heat flux contributions from 0.0314 - 0.0315 Gyr in the same simulation as figure 4.6.

While in statistical steady state, Io loses two orders of magnitude more internal heat through volcanic eruptions than through conduction through the stagnant lithosphere (Figure 4.6). Although it does reach a statistical steady state, the heat flux due to volcanic eruptions oscillates around an average value whereas the heat flux due to conduction through the solid lithosphere is relatively constant with time. Figure 4.7 shows an example of the oscillation of the magmatic heat flux over a 100,000-year period and reveals that it oscillates over time scales less than a few thousand years. This implies that the heat flux observed from volcanoes on Io over the past 35 years may not be representative of the average over geologic time scales.

4.3.2 Tidal heating distribution

Figures 4.3 and 4.4 showed that the distribution of internal heating aﬀects the horizontally averaged radial proﬁles, because the non-uniform case leads to more partial melting and a thinner lithosphere on the side of the domain experiencing a higher internal heating rate. As expected, in the uniform heating rate case, the lithospheric thickness does not vary horizontally (Figure 4.5). The range of melt fractions is similar for both heating distributions. However, the peak horizontally averaged melt fraction is higher in the uniform heating case where the depth of the 106

uniform sinusoidal fpeak (%) 7.73 5.5 Df (km) 350 559 T (K) 1482 1533 Dlith (km) 261 150 13 14 Fmag (W) 9.22 × 10 1.09 × 10 12 12 Fcond(W ) 1.15 × 10 1.51 × 10 Eruptionrate(km/Gyr) 8,946 10,644

Table 4.2: The statistical steady state average peak melt fraction in the horizontally averaged radial melt profile (fpeak), thickness of the partially molten zone (Df ), mantle temperature (T ), thickness of the lithosphere (Dlith), heat flux due to magmatic eruption (Fmag), heat flux due to conduction through a stagnant lithosphere (Fcond), and eruption rate for uniformly distributed internal heating and an internal heating rate that decreases according to equation (4.9).

peak melt fraction does not vary horizontally. The onset of melting occurs at a similar depth for both tidal heating distributions. Thus because the non-uniform heating case has partial melting at shallower depths on the left side of the domain, the total vertical extent of the partially molten zone is larger in the non-uniform heating case (Table 4.2 and ﬁgures 4.2 and 4.5).

4.3.3 Eruption parameterization

Movement of magma from the mantle to the surface of a planet is an active area of research and is not fully modeled in the simulations presented here. Instead we try several parameterizations of eruption: ‘lithosphere’, any melt that reaches the base of the lithosphere erupts if it is less dense than the solid above it; ‘freezing’, any melt which would otherwise freeze is instead moved to the surface; ‘freezing basalt’, any molten basalt which would otherwise freeze is instead moved to the surface. We find that if all melt that would freeze instead erupts, heat is very efficiently transported out of the mantle. The resulting mantle temperature and peak melt fraction are significantly lower than those of the other two parameterizations and the ratio of the magmatic heat flux to the conductive heat flux exceeds 500 (Table 4.3). This is likely an underestimate of the temperature and melt fraction in the mantle because 107

Freezing Freezing Lithosphere basalt fpeak 1.29 13.4 5.5 Df (km) 469 520 559 T (K) 1231 1482 1533 Dlith (km) 488 202 150 14 14 14 Fmag (W) 3.50 × 10 1.87 × 10 1.09 × 10 11 11 12 Fcond(W ) 5.96 × 10 7.53 × 10 1.51 × 10 Eruptionrate(km/Gyr) 30,365 18,380 10,644

Table 4.3: The statistical steady state average peak melt fraction in the horizontally averaged radial melt profile (fpeak), thickness of the partially molten zone (Df ), mantle temperature (T ), thickness of the lithosphere (Dlith), heat flux due to magmatic eruption (Fmag), heat flux due to conduction through a stagnant lithosphere (Fcond), and eruption rate for three different parameterizations of eruption.

melt which would freeze at the base of the lithosphere would release its latent heat while at the top of the mantle whereas if instead this melt is forced to the surface the latent heat is released while no longer in contact with the mantle. This also results in a very thick lithosphere and partial melt only in the deep mantle (Figure 4.8). Eruption parameterizations ‘lithosphere’ and ‘freezing basalt’ yield similar results which suggests that when using the ‘lithosphere’ parameterization, harzburgite is either freezing before erupting or is too dense to erupt. In the ‘lithosphere’ eruption parameterization, we specify the base of the lithosphere according to the depth at which a certain temperature, Tlith, is reached. Here we deﬁne the base of the lithosphere according to temperature for numeri-

cal convenience. When we choose a suitable Tlith, this definition of the lithosphere agrees with that defined by the maximum gradient of the radial velocity profile. We aim to choose the highest temperature at which the silicate mantle is too viscous to participate in convection, but we test our sensitivity to this choice by trying

Tlith = 1200K − 1600K. Choosing a low value for Tlith results in a thin lithosphere

that facilitates conduction from the mantle to the surface. A high Tlith causes melt to erupt from deeper in the mantle and leads to such a thick lithosphere that half of the mantle is the lithosphere, and eruption rates are high, because any melt that 108

−200

−400 lithosphere freezing −600 freezing basalt Depth (km)

−800

−1000 0 5 10 15 Percent Molten

Figure 4.8: The horizontally averaged radial melt proﬁle for three diﬀerent parameterizations of eruption: ‘lithosphere,’ any melt that reaches the base of the lithosphere erupts if it is less dense than the solid above it (black line); ‘freezing,’ any melt which would otherwise freeze erupts (blue line); ‘freezing basalt,’ any basaltic melt which would otherwise freeze erupts (red line).

forms quickly reaches the base of the lithosphere and erupts (Table 4.4).

4.3.4 Inﬂuence of reference viscosity

The grain size and composition of Io’s mantle is unknown, so there is some uncertainty in the reference viscosity that should be used. To understand the eﬀect this uncertainty has on our results, we test three diﬀerent reference viscosities where the

reference viscosity is η0 in equation (4.7). A higher reference viscosity makes convection less efficient because it slows the convective velocities. This results in higher mantle temperatures and more magma generation (Table 4.5). However, lower bulk velocities with the same permeability also slow the transport of melt to the base of the lithosphere, so despite the increased melting, eruption rates are lower for high reference viscosities (Table 4.5). The lower eruption rates lead to a lower heat flux due to volcanic eruption, so the ratio of magmatic heat flux to conductive heat flux is lower for higher reference viscosities. 109

Tlith 1200K 1400K 1600K fpeak 8.50 5.5 6.35 Df (km) 563 559 469 T (K) 1692 1533 1284 Dlith (km) 24 150 461 13 14 14 Fmag (W) 1.86 × 10 1.09 × 10 2.86 × 10 12 12 11 Fcond(W ) 6.21 × 10 1.51 × 10 7.34 × 10 Eruptionrate(km/Gyr) 1,979 10,644 26,043

Table 4.4: The statistical steady state average peak melt fraction in the horizontally averaged radial melt profile (fpeak), thickness of the partially molten zone (Df ), mantle temperature (T ), thickness of the lithosphere (Dlith), heat flux due to magmatic eruption (Fmag), heat flux due to conduction through a stagnant lithosphere (Fcond), and eruption rate for different values of Tlith.

17 18 19 η0 10 Pas 10 Pas 10 Pa s fpeak 4.95 5.5 8.43 Df (km) 291 559 784 T (K) 1463 1533 1591 Dlith (km) 215 150 156 14 14 13 Fmag (W) 1.63 × 10 1.09 × 10 9.10 × 10 12 12 12 Fcond(W ) 1.25 × 10 1.51 × 10 1.46 × 10 Eruptionrate(km/Gyr) 15,974 10,644 9,134

Table 4.5: The statistical steady state average peak melt fraction in the horizontally averaged radial melt profile (fpeak), thickness of the partially molten zone (Df ), mantle temperature (T ), thickness of the lithosphere (Dlith), heat flux due to magmatic eruption (Fmag), heat flux due to conduction through a stagnant lithosphere (Fcond), and eruption rate for three different reference viscosities. 110

4.3.5 Inﬂuence of permeability

The permeability of Io’s mantle is unknown due to uncertainties in grain size, channel geometry, and dependence on melt fraction. To understand the effects of the uncertainty in grain size and channel geometry on our results we vary the permeability constant (see section 4.2.3) from 3× 10−9 m2 to 3× 10−11 m2. To understand the effects of the uncertainty in melt fraction dependence, we vary the exponent in equation (4.8). We try the commonly assumed values of n = 2 and n = 3 (Stevenson and Scott, 1991). A higher permeability should allow melt to move more efficiently through the mantle and reach the base of the lithosphere faster. We see the consequence of this through higher eruption rates for higher permeability constants (Table 4.6). These higher eruption rates lower the mantle temperature. The lower mantle temperature results in a thicker lithosphere (Table 4.6). The higher eruption rates and the lower mantle temperatures lead to a higher magmatic heat flux and a lower conductive heat flux for a higher permeability constant (Table 4.6). Increasing the melt fraction exponent n from 2 to 3 has a negligible effect on the results compared to the permeability constant (Table 4.6).

−200

−400 k0=3×10−11 −10 −600 k0=3×10 Depth (km) k0=3×10−9 −800

−1000 0 10 20 30 Percent Molten

Figure 4.9: The horizontally averaged radial melt proﬁle for three diﬀerent permeability constants: 3 × 10−11 m2 (black), 3 × 10−10 m2 (dashed blue), 3 × 10−9 m2 (dash dot red). The horizontal dashed black line represents the depth of the core mantle boundary 111

−9 −10 −10 −11 k0 3 × 10 3 × 10 3 × 10 3 × 10 n 2 2 3 2 fpeak 30.97 5.5 6.91 6.98 Df (km) 751 559 581 516 T (K) 1386 1533 1540 1574 Dlith (km) 238 150 154 155 14 14 14 14 Fmag (W) 2.73 × 10 1.09 × 10 1.16 × 10 1.03 × 10 12 12 12 12 Fcond(W ) 1.22 × 10 1.51 × 10 1.52 × 10 1.68 × 10 Eruptionrate(km/Gyr) 27,521 10,644 11,341 10,152

Table 4.6: The statistical steady state average peak melt fraction in the horizontally averaged radial melt profile (fpeak), thickness of the partially molten zone (Df ), mantle temperature (T ), thickness of the lithosphere (Dlith), heat flux due to magmatic eruption (Fmag), heat flux due to conduction through a stagnant lithosphere (Fcond), and eruption rate for different permeability constants and exponents.

4.3.6 Heating rate

Io is currently the only body in our solar system losing the majority of its internal heat through magmatic eruption. This eﬃcient mechanism of planetary heat loss likely dominates on any planet with a high internal heating rate (Moore and Webb,

2013). To test this we vary the internal heating rate in our simulations (Havg in equation 4.9) and compare the amount of melting and eruption for different internal heating rates. As expected, a higher internal heating rate leads to more melting (Figures 4.10A and 4.12 and Table 4.7). It also leads to more vigorous convection manifesting through higher convective velocities (Figure 4.10C). The different convective velocities corresponding to the different internal heating rates result in similar mantle temperatures for all internal heating rates that result in any melting (Figure 4.10B). The thickness of the lithosphere increases with decreasing internal heating rate. If the thickness of the lithosphere were controlled primarily by the forced subsidence of the lithosphere by eruption, the thickness of the lithosphere should decrease with internal heating rate because eruption rate increases with internal heating rate. This implies that the amount of heat that must be removed by conduction through the lithosphere has a strong influence on the thickness of the lithosphere as well. The heat lost due to magmatic eruption and the heat lost due 112

to conduction through the stagnant lid both decrease with decreasing internal heating rate (Table 4.7). Heat loss due to volcanic eruptions dominates over heat loss by conduction through a stagnant lid in any case where some magma is generated (Figure 4.13). Although magma production and eruption rate increase with internal heating rate, some magma is generated even at internal heating rates as low as that inferred for the Moon (Table 4.7, figures 4.10 and 4.12). Apollo 15 measured a Lunar surface heat flux of 0.021 W m−2 and Apollo 17 measured a heat flux of 0.016 W m−2 (Langseth et al., 1976), which corresponds to an internal heating rate of approximately 10−11 W kg−1. However, there is uncertainty in both the measured values and whether they are representative of the global average Lunar surface heat flux (Ziethe et al., 2009). If we take this internal heating rate to be accurate, our results imply that the Moon is losing most of its internal heat through volcanism. Although some recent volcanism has been observed on the Moon (Braden et al., 2014), observations do not suggest that it is currently losing most of its internal heat through volcanism. This could be due to uncertainty in free parameters such as reference viscosity, uncertainty in internal heating distribution (i.e. efficient crustal differentiation could have fractionated heat producing radiogenic isotopes into the crust (Schubert et al., 2001)), or it could imply that our parameterization of eruption overestimates eruption rates. Our assumption that magma reaching the base of the lithosphere will automatically erupt is likely a poor assumption for a 300 km thick lithosphere. We expect eruption efficiency is inversely related to lithospheric thickness. Thus our simulations likely overestimate eruption rates more strongly when the internal heating rate is low. However, if the thick lithosphere does suppress eruption, more heat will have to be removed by conduction through the lithosphere which will result in either a thinner lithosphere or a higher temperature contrast through the lithosphere. Future work with an improved eruption parameterization is needed to accurately estimate how the thickness of the lithosphere and the eruption rate vary with internal heating rate. 113

H(W/kg) 1.4 × 10−12 1.4 × 10−11 1.4 × 10−10 1.4 × 10−9 fpeak 0 3.64 5.49 5.5 Df (km) 0 583 459 559 T (K) 872 1342 1377 1533 Dlith (km) 345 311 278 150 12 13 14 Fmag (W) 0 4.66 × 10 2.07 × 10 1.09 × 10 11 11 11 12 Fcond(W ) 3.46 × 10 3.71 × 10 4.63 × 10 1.51 × 10 Eruptionrate(km/Gyr) 0 468 2,084 10,644

Table 4.7: The statistical steady state average peak melt fraction in the horizontally averaged radial melt profile (fpeak), thickness of the partially molten zone (Df ), mantle temperature (T ), thickness of the lithosphere (Dlith), heat flux due to magmatic eruption (Fmag), heat flux due to conduction through a stagnant lithosphere (Fcond), and eruption rate for different internal heating rates. A B C

0 0 0

−100 −100 −100

−200 −200 −200

−300 −300 −300

−400 −400 −400

−500 −500 −500 −12 Depth (km) H=10 −600 H=10−11 −600 −600 −10 −700 H=10 −700 −700 H=10−9 −800 −800 −800

−900 −900 −900 0 5 0 1000 2000 −5 0 5 Percent Molten Temperature (K) log(Velocity (cm/yr))

Figure 4.10: Horizontally averaged radial proﬁles of A) percent molten, B) temperature, and C) log of the root mean squared total mantle velocity for four diﬀerent internal heating rates, H: 10−12 W kg−1 (black), 10−11 W kg−1 (cyan), 10−10 W kg−1 (magenta), and 10−9 W kg−1 (blue). The horizontal black dashed line indicates the core mantle boundary and the solid colored lines indicate the base of the stagnant lid for each internal heating rate. 114

Figure 4.11: Instantaneous temperature ﬁelds after evolving to statistical steady state for internal heating rates H = 10−10 W kg−1 (top), H = 10−11 W kg−1 (middle), and H = 10−12 W kg−1 (bottom) distributed according to equation 4.9. The domain is 880 km deep and 2772 km wide.

4.4 Discussion

Our simulations typically result in a cool solid lithosphere 150 - 260 km thick. A lithosphere of this thickness could easily support the observed tall mountains on Io’s surface (Carr et al., 1998). However, the thickness of the lithosphere is 115

Figure 4.12: Instantaneous melt fraction ﬁelds after evolving to statistical steady state for internal heating rates H = 10−10 W kg−1 (top), H = 10−11 W kg−1 (middle), and H = 10−12 W kg−1 (bottom) distributed according to equation 4.9. At the lowest heating rate considered, there is negligible melting. The domain is 880 km deep and 2772 km wide. very sensitive to the eruption parameterization. When we parameterize eruption by allowing any melt that migrates up to a depth where the temperature is equal to Tlith to erupt, the thickness of the lithosphere becomes very sensitive to the

parameter Tlith. Picking a low value for Tlith keeps the magma in the mantle to 116

Figure 4.13: The ratio of heat ﬂux due to magmatic eruptions to heat ﬂux due to conduction versus internal heating rate. These simulations were run with a reference viscosity of 1018 Pa-s and a permeability constant of 3 × 10−10 m2 shallower depths resulting in a thin lithosphere whereas picking a high value for

Tlith can result in a lithosphere that extends through half of the mantle. When we parameterize eruption by allowing any melt which would otherwise freeze to instead erupt, a very thick lithosphere develops. This is likely because moving the ‘freezing’ magma to the surface instead of allowing it to freeze in situ removes more energy from the mantle to the surface and cools the mantle more efficiently. For both of the eruption parameterizations we have used thus far, we ignore the movement of magma through the lithosphere. As magma rises through the lithosphere, it will heat the surrounding solid material. This additional heating of the lithosphere could weaken the rock such that it breaks away and sinks through the mantle thus thinning the lithosphere. In addition to eruption parameterization, our lithospheric thickness predictions may not be realistic because we have only considered Cartesian geometry. In spherical geometry, subsidence due to rapid resurfacing may lead to significant compression in the lithosphere which could alter its structure. Although our simulations are consistent with the constraint that Io’s lithosphere must be thick enough to support significant observed topography, we do not make any definitive statement about the thickness of Io’s lithosphere due to uncertainty in the effects of eruption on lithospheric thickness. 117

In most of our simulations the highest value in the horizontally averaged radial melt profile is between 5% and 9% magma. Khurana et al. (2011) infer a 20% molten layer based on the induced magnetic field detected by the Galileo magnetometer. They also show a the model field produced by a global layer with 5% melt. This does not fit the observations as well as the model field produced by a global layer with 20% melt, but it is a much closer fit than a solid mantle. Thus we expect that our peak values of 5-9% magma may cause an induced field similar to the one observed by the Galileo magnetometer. However, our simulations show a partially molten layer with melt fraction decreasing with depth beneath a thick stagnant lithosphere. The magnetic signature of this distribution of magma has not been calculated, so further work is needed to accurately compare our results to magnetic field observations. Our parameterization of eruption allows any magma that reaches the base of the lithosphere to erupt to the surface. This is likely an over-estimate of eruption efficiency, so in reality, some of this magma erupting to the surface might instead be forced to remain in the mantle causing higher melt fractions than we currently predict. However, a decreased eruption efficiency does not guarantee significantly higher mantle melt fractions. It could instead lead to higher internal temperatures and convective velocities and only a slight change in melt fraction. Furthermore, the composition of Io’s mantle is uncertain, which causes uncertainty in both the solidus temperature and the electrical conductivities which affect melt fraction predictions by mantle convection modeling and magnetic field observations respectively. Improved eruption parameterization in mantle convection modeling and continued observational efforts should allow the two approaches to converge on Io’s true mantle melt fraction. Typical eruption rates in our simulations range from 9 mm yr−1 to 16 mm yr−1. This well exceeds the minimum resurfacing rate of 1 mm yr−1 required to bury impact craters (Johnson et al., 1979). Eruption rate estimates are also sensitive to the parameterization of eruption. Assuming melt erupts when it rises to a depth where the temperature is 1200 K instead of 1400 K leads to a low eruption rate of only 2 mm yr−1, because this assumption causes a thinner lithosphere which 118

allows conduction through the lithosphere to become more efficient. Alternatively, if we assume that all melt which would otherwise freeze instead erupts, eruption rates increase to as high as 30 mm yr−1. Although all simulations predict average eruption rates somewhere between 2 mm yr−1 and 30 mm yr−1, the eruption rate fluctuates around these average values in all simulations (Section 4.3.1), so the eruption rate observed over the past 35 years need not exactly match the long term average eruption rate. In general our simulations are consistent with all observational constraints on Io’s mantle. Although we do not provide a firm estimate on Io’s lithospheric thickness due to uncertainty in our parameterization of eruption, all of our simulations produce a lithosphere capable of supporting Io’s tall mountains. In most of our simulations, the lithosphere greatly exceeds the required thickness. Therefore, we expect that future simulations with an improved parameterization of eruption will also self-consistently generate a lithosphere thick enough to support Io’s mountains. All of our simulations also produce a partially molten layer. We find that the magma is distributed in a way not considered by previous electrical conductivity modeling aimed at reproducing the observed induced magnetic field. With certain assumptions, we can get results similar to Khurana et al. (2011)’s inference of a 20% molten layer of magma beneath a 50 km thick lithosphere. With other assumptions, our simulations result in a very thick lithosphere suppressing melting at detectable depths which could be consistent with the non-detection of Roth (2012). None of our simulations produce a partially molten layer where magma is evenly distributed. The buoyancy of magma relative to the solid mantle leads to melt fraction decreasing with depth below the base of the lithosphere. A melt fraction gradient was not considered in previous electrical conductivity modeling, so it is unknown if this would affect the expected induced magnetic field. All of our simulations lead to an eruption rate high enough to erase any impact craters on Io’s surface. We also find that the eruption rate oscillates with time which has implications for the interpretation of the currently observed eruption rate on Io. 119

4.5 Conclusion

The heat flux due to volcanic eruptions is at least 60 times greater than the heat flux due to conduction through the stagnant lithosphere on Io for an internal heating rate of 1.4 × 10−9 W kg−1 any reasonable choice of parameter values. For the most likely parameter values, the heat flux due to volcanic eruptions is 75 times greater than the heat flux due to conduction through the stagnant lithosphere. These numerical simulations of mantle convection that include melt generation, segregation and eruption confirm that Io is losing most of its internal heat through the heat pipe mechanism of planetary heat loss. The structure of Io’s mantle includes a cold solid lithosphere above a partially molten upper mantle and a solid and convecting lower mantle. Peak horizontally averaged melt fractions in the partially molten upper mantle fall between 5 % and 9 % for simulations using the most realistic values for parameters in table 4.1. Eruption rates cause resurfacing of 9-15 mm yr−1 for reasonable choices of parameter values. However, the eruption rate oscillates around the long term average eruption rate, so observations of Io’s eruption rate and thus total heat loss over the past 35 years may not be representative of Io’s long term behavior. 120

CHAPTER 5

Comparison of the one-dimensional model and the two-dimensional numerical simulations

5.1 Direct comparison

We now compare the results of the one-dimensional model (section 3) to the two- dimensional numerical simulations of Io’s mantle (section 4). We use the values given 20 in table 3.1 with a reference viscosity of η0 = 10 Pa s and a permeability constant −9 2 of k0 = 10 m which corresponds to a grain size of 0.85 mm and b = 722. With this choice of parameter values, the one-dimensional model (section 3) finds that Io has a mantle temperature of approximately 1500 K, an average mantle velocity of approximately 0.3 cm year−1, and an average melt fraction of 1.4% averaged over the whole mantle. In contrast, for these parameters, the two-dimensional numerical simulations (section 4) find an average temeprature of 1785 K, an average mantle velocity of 55 cm year−1, and an average melt fraction of 2% beneath the stagnant lithosphere. The average mantle temperature and velocity in the numerical simulations are significantly higher than the average mantle temperature and velocity as predicted by the one-dimensional model. However, the one-dimensional model assumes that the heat flux due to volcanic eruptions above an upwelling column is representative of the heat flux due to volcanic eruptions for the whole surface which is unrealistic. If instead we assume that the heat flux due to volcanic eruptions over an upwelling column of mantle is only representative of a quarter of the

surface (fe =0.25), then the one dimensional model predicts a mantle temperature of 1670 K and an average mantle velocity of 1.6 cm year−1. The melting and vertical velocity ﬁelds of the two-dimensional numerical simulations suggest that the distribution of partially molten upwelling columns of mantle material may comprise even less than one quarter of the mantle (Figure 5.1). Furthermore, the horizontally 121

averaged radial proﬁle of the root-mean squared mantle velocity suggests that the lowest velocities are in the lower mantle (Figure 5.2C), so underestimating the average mantle velocity may lead to a velocity more representative of the lower mantle material before it rises and begins partially melting.

Figure 5.1: Two-dimensional numerical simulations of mantle convection in Io using a uniformly distributed internal heating rate of 1.7 × 10−9 W kg−1, rheological 20 parameters as speciﬁed in table 3.1, a reference viscosity of η0 = 10 Pa s, and a −9 2 permeability constant of k0 = 10 m . A) The spatial distribution of melting where red indicates 100% molten and blue indicates 100% solid. B) The vertical component of the velocity in the mantle. Red indicates an upward velocity of 10−8 m s−1 and blue indicates a downward velocity of 10−8 m s−1. The white lines indicate contours of 1% melt.

Despite the differences in average mantle temperature and velocity, the radial melting profiles from the one-dimensional model and the two-dimensional simulations are similar (Figures 5.2A and 5.3). In both radial melt profiles, starting at approximately 500 km below the surface, the melt fraction rises with decreasing depth and reaches its highest value at the top of the partially molten zone. Finally, 122

0 A 0 B 0 C

-100 -100 -100

-200 -200 -200

-300 -300 -300

-400 -400 -400

-500 -500 -500 Depth (km) -600 -600 -600

-700 -700 -700

-800 -800 -800

0 10 20 0 1000 2000 0 100 200 300 Percent Molten Temperature (K) Velocity (cm/yr)

Figure 5.2: Horizontally averaged radial proﬁles of A) melting, B) temperature, and C) root mean squared velocity in a two-dimensional numerical simulation of mantle convection in Io’s mantle using a uniformly distributed internal heating rate of 1.7 × 10−9 W kg−1, rheological parameters as speciﬁed in table 3.1, a reference 20 −9 2 viscosity of η0 = 10 Pa s, and a permeability constant of k0 = 10 m .

both models predict that heat loss through volcanic eruptions will dominate over heat loss via conduction through a stagnant lithosphere. For this choice of rheological parameters, the two dimensional numerical simulations predict that 97% of the total surface heat flux will be from volcanic eruptions. In the one-dimensional model, when we assume that the volcanic heat flux above a one-dimensional partially molten rising mantle column is representative of one quarter of the surface, 99% of the total surface heat flux is due to volcanic eruptions. Thus both models confirm that heat loss due to volcanic eruptions is the dominant mechanism for heat loss on Io.

5.2 Convective scaling laws

If the heat loss due to conduction through the stagnant lithosphere and the average mantle melt fraction are known, convective scaling laws can determine the average mantle temperature and velocity (section 3.2.2). To test the accuracy of these convective scaling laws, we use the heat loss due to conduction through the stagnant 123

−100

−200

−300 Depth (km) −400

0 10 20 f (%)

Figure 5.3: Radial profile of melt in a rising column in Io’s mantle calculated using the method described in section 3, an internal heating rate of 1.7 × 10−9 W kg−1, rheological parameters as specified in table 3.1, fe =0.25, a grain size of 0.85 mm, and constant b = 722. lithosphere and average melt fraction (in the whole mantle) from a series of the simulations in section 4 to calculate the average mantle temperature and velocity with convective scaling laws. We then compare these calculated mantle temperatures and velocities to the average mantle temperature and velocity in the two-dimensional numerical simulations (Table 5.1). The average mantle temperatures calculated from the results of the two- dimensional numerical simulations do not show any simple correlation with the heat loss due to conduction through the stagnant lithosphere or the average mantle melt fraction. These simulations were specifically designed to determine the state of Io’s mantle and the sensitivity of those results to various rheological parameters and model assumptions, so the lack of a correlation between the average mantle temperatures with the average melt fractions or conductive heat loss in this data set does not necessarily mean that future work will not be able to derive convective scaling laws applicable to a partially molten mantle. Table 5.1 does show that the average mantle temperatures calculated from the results of the two-dimensional simulations and the temperatures calculated from convective scaling laws both increase with increasing reference viscosity. This is because a more viscous mantle will make convection less efficient and slow mantle cooling. In general, table 5.1 shows that convective scaling laws underestimate the average mantle temperature. We suggest 124

that, for a partially molten mantle, a = 0.2 (See equation 3.29) more accurately reproduces the average mantle temperatures found in two-dimensional simulations than the a = 0.67 of Reese et al. (2005). Results assuming a = 0.2 are also shown in table 5.1. We also find that the average mantle velocities calculated from the results of the two-dimensional numerical simulations do not show any simple correlation with the heat loss due to conduction through the stagnant lithosphere, the average mantle melt fraction, or the average mantle temperature. However, the average mantle velocity is also correlated with the assumed reference viscosity. Average mantle velocity decreases with increasing reference viscosity. This is also because a more viscous mantle makes convection less efficient and slows the movement of buoyantly unstable parcels of mantle material. As discussed in section 5.1, the average mantle velocity predicted by the convective scaling laws is significantly lower than the average mantle velocity calculated by the two-dimensional numerical simulations. In section 3, we favored velocity scaling law II from Solomatov and Moresi (2000), but table 5.1 shows that velocity scaling law I estimates higher mantle velocities that are closer to the velocities calculated by the two-dimensional simulations. This is in contrast to Solomatov and Moresi (2000) who found that velocity scaling law II more accurately reproduced their results. This difference may suggest that the presence of a significant amount of magma in a planet’s mantle may change the distribution of viscous dissipation in a convecting mantle. Although velocity scaling law I from Solomatov and Moresi (2000) predicts mantle velocities closer to those calculated from the results of the two-dimensional numerical simulations, it still un- derestimates the mantle velocity by approximately a factor of six (Table 5.1). Future work should focus on deriving convective scaling laws that describe convection in a partially molten mantle. 125 7 8 9 9 7 − − − − − 08 10 10 10 10 10 18 0402 =3 × × × × × 1 2 1 n 10 2 1 II) calculated i = ) and the whole 7 8 9 9 7 V c − − − − − F 11 10 10 10 10 10 − 18 0 × × × × × I and 1 2 10 1 k 10 3 2 i V 67. Finally, we show the . × 8 9 10 9 8 − − − − − =0 10 10 10 10 10 10 ) and average mantle velocity =8 − 18 i a 0 × × × × × T 5 8 10 7 k 10 1 9 = 8 8 10 9 8 − − − − − 9 10 10 10 10 10 − 18 0 ) in two-dimensional simulations presented × × × × × i 6 1 7 10 k 10 1 3 on. The convective scaling law results show V ), average melt fraction of the whole mantle c = 2 instead of F a = 7 8 9 9 7 − − − − − 10 10 10 10 10 17 17 × × × × × 0 1 2 η 1 10 10 2 1 nant lid ( = 8 8 10 9 8 − − − − − using the heat transported by conduction ( ng laws with 10 10 10 10 10 19 19 × × × × × 0 2 times a factor of 6. 6 1 η 10 7 10 1 8 . 3.2.2 =0 = 8 9 10 10 8 − − − − − a 10 10 10 10 10 20 20 × × × × × 0 3 5 η 10 5 10 7 6 ), and the average mantle velocity ( 8 8 9 9 7 i − − − − − T ) and the average mantle velocity (from both scaling laws: ). Also shown is the average mantle temperature ( i 10 10 10 10 10 18 T × × × × × avg 9 2 1 10 2 1 f 8 8 10 9 8 − − − − − 10 10 10 10 10 18 × × × × × uniform sinusoidal 0.0105 0.0073 0.0638 0.0247 0.0029 0.006 0.0072 0.0162 0.01 ) 6 1 indicates the reference viscosity assumed in each simulati − ) 7 0 ) 0.0287 0.0373 0.0368 0.0374 0.0314 0.0273 0.0248 0.0457 0. ) 1 ) 1 1 η 2 1 1 − ) 4 − . − − 1 4 − 6(m s × (W m Pas 10 (K) 1714 1768 2021 1895 1613 1713 1694 1791 1776 (K) 1517 1562 1757 1660 1438 1518 1502 1578 1567 (K) 1799 1798 1917 1858 1752 1786 1811 1780 1793 (m s I I (m s II (m s I (m s ), average mantle temperature ( c i i i i i i i i 0 avg V T V η V F T V f T V avg =0.2 f in section a ( Table 5.1: Heat transported via conduction through the stag Two dimensional results: the average mantle temperature ( mantle average melt fraction ( Convective scaling law results: (using scaling law I) calculated using the convective scali with the convective scaling laws described in section average mantle velocity from scaling law I and 126

5.3 Melt migration model

−10 −9.5 −9 −8.5 −8 −7.5 −10 −9.5 −9 −8.5 −8 −7.5 0.22 0.18 0.2

0.35 0.08 0.160.16 0.2 0.25 0.3 0.12 0.3 0.35 0.14 0.2 0.25 0.08 0.1 0.12 0.14 0.1 450 0.2 450 0.15 1650 1650 0.2 0.15 400 400 0.150.15

1600 350 1600 0.25 0.2 350 0.18 0.08 0.150.15 0.16 0.12 0.14 0.3 0.1 0.18 T (K) T (K)

0.2 d (km) d (km)

0.08 0.16

0.12 0.14 300 0.3 300 0.1

0.150.15 0.25 0.2

1550 1550 0.25 250 0.2 250 0.25 0.2 0.350.35 0.3 0.3 0.25 0.40.4 200 0.2 200 1500 0.08 0.08 1500 −10 −9.5 −9 −8.5 −8 −7.5 −10 −9.5 −9 −8.5 −8 −7.5 Log(V) (m s−1) Log(V) (m s−1) −10 −9.5 −9 −8.5 −8 −7.5 −10 −9.5 −9 −8.5 −8 −7.5 180 120 180 120 −8−8 −7.2 160 100 160 100

−7.4 40 60 80 −7.6−7.6 −7.4 20 140 −7.8 450 140 450 1650 1650

−8.2 400 400

−7.8

60 80 −8 350 20 40 350 1600 −8.4 1600 −8 −7.2 120

100 −7.6 −7.4 T (K) T (K) −7.2 120 −7.8 d (km) d (km) 100 −7.6 −7.4 300 20 300 40 60 80 −8.2 1550 −8.6 −8.2 1550 250 250 −8 −8.8 −7.8 −8.8 −8.4−8.4 −8 200 20 40 60 80 200 1500 1500 −10 −9.5 −9 −8.5 −8 −7.5 −10 −9.5 −9 −8.5 −8 −7.5 Log(V) (m s−1) Log(V) (m s−1)

Figure 5.4: One-dimensional model results for A) average melt fraction in the partially molten zone, B) melt fraction at the top of the partially molten zone, C) logarithm of melt velocity (m s−1) at the top of the partially molten zone, and D) heat ﬂux (W m−2) transported to the surface by volcanic eruptions all for diﬀerent mantle temperatures (y-axis) and upwelling velocities (logarithm of the upwelling velocity on the x-axis) using the values in table 3.1 and the melt generation and migration model described in section 3.2.1. The depth of the onset of melting (right vertical axis) is calculated with equation 3.13.

Comparing the results of section 3 to the results of section 4 leads to uncertainty in the ability of convective scaling laws to predict the mantle temperature and velocity. Here we calculate the steady state solutions of melt generation and two- phase ﬂow for a wide range of mantle temperatures and velocities. In this section, we have not self-consistently calculated the mantle temperature and velocity, so the solutions do not represent equilibrium in Io’s mantle. Rather they explore the relationships between mantle temperature, mantle velocity, melt generation, and melt migration. Figure 5.4 shows the average melt fraction in the partially molten 127

zone, the melt fraction at the top of the partially molten zone, the logarithm of the melt velocity at the top of the partially molten zone, and the heat transported out of the mantle via volcanic eruption using the parameter values given in table 3.1 for an internal heating rate of 1.7 × 10−9 W kg−1. Based on convective scaling laws and the results of two-dimensional numerical simulations we expect Io’s mantle velocity is between 10−10.5 and 10−7.5 m s−1 and its mantle temperature is between 1500 and 1750 K which corresponds to melting beginning at a depth between 200 and 700 km below the surface. A higher mantle velocity leads to a higher average melt fraction in the partially molten zone, because it increases the advection of warm material into the partially molten zone. Buoyancy drives the melt upwards and leads to an increasing melt fraction with decreasing depth. The melt fraction at the top of the partially molten zone is highest. At velocities higher than approximately 10−8.5 m s−1, a higher velocity leads to more melting and a higher melt fraction at the top of the partially molten zone. At velocities lower than approximately 10−9.5 m s−1, the melt fraction at the top of the partially molten zone increases with decreasing mantle velocity. This is because a low velocity allows the rising mantle material to experience more tidal heating as it moves through the partially molten zone compared to a parcel of mantle rising quickly. The heat flux transported out of the mantle by volcanic eruptions depends on the flux of melt out of the top of the mantle which depends on both the melt fraction and the melt velocity at the top of the partially molten zone (Equation 3.19). The melt velocity at the top of the partially molten zone increases with mantle temperature and mantle velocity. At mantle velocities above approximately 10−8.5 m s−1, both the melt velocity and the melt fraction at the top of the partially molten zone increase with increasing mantle velocity, so the heat flux transported via volcanic eruptions increases rapidly with increasing velocity. At mantle velocities lower than approximately 10−9.5 m s−1, melt velocity decreases with decreasing mantle velocity but the melt fraction at the top of the partially molten zone increases with decreasing mantle velocities, so changes in the amount of heat transported via 128

volcanic eruptions is minor. It is important to note that for an assumed rheology, internal heating rate, and eruption efficiency, there is a unique average melt fraction, mantle temperature and velocity that will lead to a surface heat flux that exactly balances the assumed internal heating rate. For the parameter values used in making figure 5.4 (Table 3.1

and fe = 1), that unique solution is T =1520 K and log V = −10.2. As discussed in section 3.2.2, it is unlikely that all melt that reaches the top of the partially molten zone will erupt to the surface, so although most of the values in figure 5.4D exceed Io’s observed surface heat flux, it shows that a decreased eruption efficiency would require a higher mantle temperature or a higher mantle velocity. Furthermore, if Io is not in thermal equilibrium, any of the mantle temperatures and velocities in figure 5.4 are possible. Within this range of mantle temperatures and velocities, for an internal heating rate of 1.7 × 10−9 W kg−1 assuming the values in table 3.1, the partially molten zone above upwelling regions in Io’s mantle is expected to be between 6 and 22% molten. The peak melt fraction, at the top of the partially molten zone is expected to be between 10 and 50%. Melting is not expected in downwelling portions of Io’s mantle, so this range of values is an upper limit.

5.4 Summary of Comparison

In this chapter, we have shown that although convective scaling laws underestimate mantle temperature and velocity, the average melt fraction in the partially molten zone only varies by a factor of 4 over a wide range of mantle temperatures and velocities. All of our results, one-dimensional and two-dimensional, show that Io has a solid lower mantle and a partially molten upper mantle. In the partially molten zone, melt fraction increases with decreasing depth and reaches its highest values at the top of the partially molten zone. We also ﬁnd that in both the one- dimensional model and the two-dimensional numerical simulations, heat loss due to volcanic eruptions is approximately two orders of magnitude higher than heat loss due to conduction through the stagnant lithosphere. Both models conﬁrm that Io 129 has a partially molten layer in its mantle and is losing the majority of its internal heat through volcanic eruptions. 130

CHAPTER 6

Conclusions and Future Work

Although we have now observed the surfaces of many planetary bodies in our Solar System, many questions remain about their subsurfaces. There are some methods (e.g. seismic tomography, precise gravity ﬁeld measurements) that can probe the subsurface in detail, but these methods have not been applied to many planetary bodies. This dissertation addresses two topics that may lead to insight on the subsurfaces of planetary bodies: 1. the conditions that lead to the formation of central pit craters and what that might indicate about the material properties of the impacted surface; and 2. the dynamics of Io’s mantle and how this relates to the extensive volcanism observed on Io’s surface.

6.1 Central pit crater formation

Central pit craters are complex craters that have a terraced rim, a flat floor, and a pit at or near their center (Passey and Shoemaker, 1982). They are seen most commonly on icy satellites and Mars (Passey and Shoemaker, 1982), but recently, they were also discovered on the Moon and Mercury (Xiao and Komatsu, 2013; Xiao et al., 2014). Several formation mechanisms have been suggested for central pit craters including: collapse of a central peak in weak ice (Passey and Shoemaker, 1982), impact into a layered target (Greeley et al., 1982), explosive release of volatiles (Carr et al., 1977), drainage of impact melt or impact related debris (Croft, 1981). If verified, the association of this type of impact crater with volatiles would suggest that subsurface ice was present on Mars at the locations of these craters when they formed. However, the recent detection of central pit craters on Mercury and the Moon suggests that they may be able to form without volatiles (Xiao and Komatsu, 2013; Xiao et al., 2014). 131

In this dissertation, I test the feasibility of central pit crater formation via impact melt drainage through fractures generated during the impact. I estimate the volume of fracture space generated during an impact, the volume of impact melt generated during an impact, and the volume of melt able to drain through fractures before freezing them shut. These volumes all exceed the observed central pit volumes on Ganymede, which suggests that impact melt drainage can form central pits in craters on Ganymede. I also considered this problem for impact craters on the Earth and the Moon. I found that liquid water flows through water ice fractures at a much higher velocity than molten rock draining through solid rock fractures. This results in a larger volume of impact melt draining before freezing fractures shut in craters in icy targets. I also showed that central pit craters are most likely on large icy satellites. The amount of melt generated in an impact scales with projectile size (Kraus et al., 2011), but the final crater diameter depends on gravity in addition to projectile size (Melosh, 1989), so on the smaller moons, a projectile of a given size will form the same volume of melt but a larger crater. Central pit craters are unlikely to form on icy bodies that orbit the sun instead of a giant planet. This is because the volume of melt generated during an impact increases with increasing impact velocity (Kraus et al., 2011), so more impact melt is expected in a crater on a satellite like Ganymede, deep in Jupiter’s gravitational well, than a body like Pluto where impact velocities are expected to be low. The work in this dissertation shows that central pits could form in craters on Ganymede due to impact melt draining through impact generated fractures, but it cannot explain the formation of central pits in rocky targets with no volatiles. Thus the discovery of central pit craters on the Moon and Mercury (Xiao and Komatsu, 2013; Xiao et al., 2014) is surprising. The work in this dissertation does not completely rule out the formation of central pits in pure rock targets. If the impact melt had a very low viscosity, then drainage velocities would be higher and the volume of melt able to drain before the fractures freeze shut would increase. Alternatively if the post impact temperature of the rock and/or impact melt were significantly higher than expected, draining impact melt could melt the surrounding rock instead 132

of freezing which would allow drainage to continue and increase drainage volume. Both a very low melt viscosity and very high post impact temperatures seem to require a unique scenario. Future hydrocode impact crater simulations could work to better constrain the post impact properties of impact melt and fractured target rock. However, the discovery of these craters in dry rocky targets could instead suggest that they do not form through impact melt drainage. Based on the results in this dissertation, drainage of impact melt in icy targets seems difficult to prevent, but multiple processes could contribute to the development of central pits in craters on Ganymede. Alternatively, central pit craters could form by different mechanisms on rocky and icy bodies. The differences in morphology of central pit craters on rocky and icy bodies supports this last hypothesis. Mars (Barlow, 2006), Mercury (Xiao and Komatsu, 2013), and the Moon (Xiao et al., 2014) all have ‘summit pit craters’ which are craters that have a pit at the top of their central peak. The presence of ‘summit pit craters’ on Ganymede is still debated (Bray, 2008; Alzate and Barlow, 2011). If ‘summit pit craters’ are not present on Ganymede, this may imply one mechanism that forms central pits and summit pits in craters on rocky bodies and another mechanism (impact melt drainage) that forms central pits in craters on icy bodies.

6.2 Dynamics of Io’s mantle

According to ten years of ground based observations, Io has an average surface heat flux of approximately 2.5 W m−2 (Veeder et al., 1994)—almost 30 times higher than that of Earth (Davies and Davies, 2010). Io shows no signs of plate tectonics (Lopes and Spencer, 2007), and previous studies have shown that its high surface heat flux cannot be explained by solid-state convection alone if tidal heat production is in equilibrium with convective heat loss (Moore, 2003). However, the analysis of one hundred years of astrometrically observed positions of the Galilean satellites suggests that the rate of energy generated through tidal dissipation in Io is in good agreement with the observed surface heat flux suggesting that Io is close to ther- 133 mal equilibrium (Lainey et al., 2009). Thus some other mechanism is required to transport heat from Io’s mantle to its surface. Volcanic eruption could be respon- sible for the transport of a significant fraction this heat (e.g. O’Reilly and Davies, 1981; Moore, 2001). However, previous studies of Io’s mantle have included only solid-state convection or melt migration through the mantle with no net velocity. In reality, mantle convection and melt migration are both important processes in the evolution of Io’s mantle, and they are strongly coupled to each other.

6.2.1 Melt generation and migration in an upwelling column of mantle on Io

In this dissertation, I adapted a one-dimensional two-phase flow model for decompression melting and melt migration in an upwelling column of mantle rock to Io. This model requires an assumed upwelling velocity and mantle temperature, so I couple it to convective scaling laws. Previous use of convective scaling laws has focused on solid-state convection (e.g. Reese et al., 2005), so I modified the scaling laws to include the effects of partial melting, including: an increased density contrast between the cold solid upper boundary layer and the warm partially molten interior, a decreased mantle viscosity, and an additional mode of heat transport out of the mantle —volcanic eruption. I found that for an assumed total mantle heating rate of 1.3 × 1014 W, Io has a partially molten upper mantle and loses the majority of its internal heat through volcanic eruptions independent of eruption efficiency. Future work should incorporate the effects of non-uniform tidal dissipation by coupling a tidal dissipation model to the two-phase flow model. The magnitude of tidal dissipation depends on the viscosity and shear modulus of the material both of which vary with temperature and melt fraction (e.g., Segatz et al., 1988; Fischer and Spohn, 1990; Moore, 2003). Thus based on the results presented in this dissertation, it is likely that the tidal dissipation rate could vary strongly with radius. Although the work presented in this dissertation finds the steady-state behavior of the mantle for a given internal heating rate, future work should focus on modifying the model so that the mantle dynamics can evolve in time allowing it to be coupled to an orbital evolution model. This thermal-orbital coupled model could 134 then be used to test the conclusions of Fischer and Spohn (1990) and investigate whether including heat transport via melt migration and volcanic eruption would change the nature of thermal-orbital evolution and prevent Io from evolving into an oscillatory state. It could also be coupled to thermal-orbital models of Europa (e.g. Hussmann and Spohn, 2004) and Ganymede (e.g. Bland et al., 2009) to study the evolution of the Laplace resonance which depends on the thermal evolution of each satellite.

6.2.2 Mantle convection in Io’s partially molten mantle

I used two-dimensional numerical simulations of mantle convection that self- consistently solve the two-phase flow equations including mantle convection, and magma generation, migration by porous flow, and eruption to investigate the effects partially melting on the evolution of Io’s mantle. As in the one-dimensional modeling, this naturally leads to a partially molten upper mantle. It also leads to the development of a thick stagnant lithosphere capable of supporting significant topography. Despite melt generation and migration, convection remains vigorous, which demonstrates that both processes operate together and both must be included to accurately model the dynamics of a partially molten mantle. This model also shows that although heat transport via volcanic eruption dominates over heat transport via conduction through the stagnant lithosphere, the rate of volcanic eruption oscillates with time around the statistical steady state average eruption rate. This suggests that the high surface heat flux observed on Io over the past 35 years may not be representative of recent geologic history. The work presented in this dissertation shows that a tidal heating rate that decreases horizontally leads to the development of large scale circulation and a lithosphere whose thickness varies horizontally. Future work should verify this conclusion with three-dimensional simulations and consider a larger range of tidal dissipation distributions including some that vary with depth as the lower viscosity in the partially molten zone will increase the rate of tidal heating. The surface heat flux resulting from different distributions of tidal heating could then be compared to ob- 135

servations of the distribution of volcanic centers (e.g. Hamilton et al., 2013) and the global heat ﬂow (e.g. Veeder et al., 2015). Future work could also focus on verifying convective scaling laws for a partially molten mantle. These could then be used for thermal-orbital evolution models as described above, and they could be applied to exoplanets where the uncertainties in heating rate, rheology, and interior structure may be too large to thoroughly explore with more time intensive multi-dimensional simulations.

6.2.3 Comparison of results to observations

Table 6.1 compares the results of both the one-dimensional melt generation and migration model and the two-dimensional numerical simulations to available observational constraints on Io’s mantle. Both find that the average melt fraction in the partially molten zone is lower than the global 20% molten layer which best explains Galileo magnetometer data. However, the global partially molten layer considered to explain the Galileo magnetometer data is 50 km thick beneath a 50 km thick crust and the magnetometer cannot probe more deeply into the mantle. Both my models find a partially molten layer much thicker than could be detected by the Galileo magnetometer, and the highest concentration of melt is at the top of this thick partially molten layer, so the Galileo magnetometer may be detecting just the top of a thicker partially molten layer. An eruption rate of at least 1 mm year−1 is required to erase any impact craters based on estimates for impactor flux (Johnson et al., 1979), and both my models find an eruption rate higher than 1 mm year−1 (Table 6.1) consistent with the lack of impact craters on Io’s surface. The one-dimensional melt generation and migration model does not make any predictions about the thickness of Io’s lithosphere, but the two-dimensional simulations do show the development of a thick cold solid lithosphere which is thick enough to support the observed topography on Io’s surface (Table 6.1). Finally, previous studies have suggested that most of Io’s internal heat may be transported to the surface by volcanic eruptions, but to test this one needs to include the effects of both mantle convection and melt generation, migration, and eruption. The one-dimensional model and two-dimensional 136

Table 6.1: Observational constraints, one-dimensional results (section 3), and two- dimensional results (section 4) for the amount of melting in the partially molten zone, the eruption rate, the lithospheric thickness, and the ratio of heat transported via conduction through the stagnant lithosphere to heat transported via volcanic eruption. Observations 1D model 2D simulations Melting ∼20% 5-13% 2-4% Eruption rate >1 mm yr−1 16.4 - 16.5 mm yr−1 9 - 16 mm yr−1 Lithospheric Thickness >30km 150-260km Fc/Fmelt low 0.0019-0.012 0.0045-0.016

simulations presented in this dissertation are the ﬁrst to self-consistently include both of these eﬀects, and both show that as predicted, Io is losing most of its internal heat through volcanic eruptions rather than conduction through a stagnant lithosphere (Table 6.1). 137

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