Corresponding author: M. E. Rumpf, Hawai‵i Institute of Geophysics and Planetology, University of Hawai‵i, 1680 East-West Rd., Honolulu, HI 96822, USA. (firstname.lastname@example.org)
 We have performed a series of numerical simulations of heat transfer between lunar lava flows and the underlying regolith with the goal of determining the depths in the substrate beneath which implanted extralunar volatiles would survive outgassing by the downward-propagating heat pulse. Exogenous materials of interest include solar wind and solar flare particles, and the cosmogenic products of galactic cosmic ray (GCR) particles emplaced in the lunar regolith early in Solar System history. Extraction and analysis during future lunar missions would yield information about the evolution of the Sun and inner Solar System environment. Particles implanted in regolith deposits may be protected from gardening and saturation if buried by a lava flow, but must be sufficiently deep in the regolith to survive the consequent heating. Our simulations include detailed treatments of lava and regolith thermophysical properties (thermal conductivity, specific heat capacity), which vary widely over the temperature range relevant to lava flows in the lunar environment (~200 to >1500 K). Simulations adopting temperature-dependent properties, together with a treatment of latent heat of lava crystallization, indicate that implanted volatiles would be fully preserved at depths greater than 20 cm beneath 1 m thick lava flows, ~60% deeper than predicted by simulations employing constant properties. These results highlight the necessity of appropriate prescription of material properties. Consideration of the range of lunar lava flow thicknesses allows us to determine the range of depths from which pristine samples may be recovered.
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 Exposed directly to space, the lunar surface is continuously bombarded with radiation and micrometeorites and has been disrupted on local to global scales by impacts of asteroid and comet debris. These impact events excavate and redistribute material. Over time this process fragments, comminutes, and mixes the surface material forming a poorly sorted regolith [McKay et al., 1991; Korotev, 1997; Lucey et al., 2006]. This surface regolith becomes implanted with solar wind and solar flare particles, and the cosmogenic products of galactic cosmic rays (GCR) as it forms [Wieler et al., 1996; National Research Council (NRC), 2007]. It may also contain early materials from the Earth and other planetary bodies that were deposited on the Moon as meteorites [Spudis, 1996; Armstrong et al., 2002; Ozima et al., 2005; Crawford et al., 2008; Ozima et al., 2008; Armstrong, 2010; Joy et al., 2011, 2012]. In contrast to the ever changing surface of Earth, the lunar surface has been exposed to such processes for billions of years. Having no atmosphere or magnetosphere and little internal geologic activity, the Moon has been passively recording events in the neighborhood of the Earth since soon after formation of the Earth-Moon system. Location and extraction of caches of exogenous materials during future lunar missions could provide valuable information on the variability of composition and flux of incoming materials over the lifetime of the Solar System.
 One of the goals of the Apollo missions was to decipher solar wind, cosmic ray, and bombardment history from volatiles returned in regolith core samples and soil scoops collected from the upper 3 m of the lunar regolith. However, because of many millions of years of exposure of the upper portion of the regolith to space, the records of exogenous processes in these samples are often highly complex and difficult to constrain [McKay et al., 1991; Joy et al., 2011]. In contrast, buried regolith layers (paleoregoliths) not exposed at the surface will have been protected from sustained bombardment by an overlying layer (Figure 1), thereby providing a snapshot of the exogenous particle population. Such a protective overlying layer might be provided by a lava flow, a flow of impact melt, an impact ejecta blanket, or a fall deposit from a pyroclastic eruption. A lava flow would protect the underlying paleoregolith and provide a rock sample to date isotopically, determining the enclosure age of the underlying regolith [McKay et al., 1989; Spudis, 1996; Crawford et al., 2007, 2010; McKay, 2009; Spudis and Taylor, 2009]. Although lava will potentially protect a paleoregolith and implanted materials from further bombardment, disruption, and gardening, it will also heat the regolith, potentially volatilizing the embedded particles and damaging the record of solar or galactic particle implantation and any fragments of volatile-bearing cometary or asteroid material. Heating experiments performed on a range of doped lunar and terrestrial samples, meteorites, and lunar simulants determined the temperature ranges over which specific volatiles are released [Gibson and Johnson, 1971; Simoneit et al., 1973; Haskin and Warren, 1991; Fegley and Swindle, 1993]. A summary of those results (Table 1) shows that, although some chemical species lightly adsorbed on mineral surfaces might be released at temperatures as low as ~570 K, many entrapped species and reaction-produced molecules will remain in the regolith until heated to a temperature range of 970–1600 K.
Table 1. Temperature Ranges at Which Volatiles Are Released From Regolith Materialsa
 To quantify the depths to which a lava flow will heat a substrate, we have developed a numerical model based on that of Fagents et al. . The current model improves upon the previous by including explicit treatment of temperature-dependent thermophysical properties of both lava and substrate. A common strategy used in many thermal models of lava flows is to omit the temperature and time dependence of material properties, instead using constant parameters to derive analytical solutions, to save computational time, or to help ensure the stability of numerical solutions. Such approximations are sufficient for first-order accuracy; however, for detailed descriptions of lava heat transfer, inclusion of property dependences is a key element. Thermal models of lunar and terrestrial lava flows benefit from the inclusion of temperature-dependent properties and a clear definition of substrate properties. The values of thermophysical properties will vary greatly over the temperature range that a lava flow experiences from eruption (at ~1450–1715 K) [Murase and McBirney, 1970; Taylor et al., 1991; Williams et al., 2000] to ambient temperature on Earth (~295 K; 22°C) or on the Moon (~200 K; −73°C) [Vaniman et al., 1991]. The thermophysical properties of both the the lava and substrate influence the temperature distribution throughout the system. However, the particulate regolith possesses contrasting properties to those of the lava; of particular significance is the low thermal conductivity. Addressing the unique properties of the substrate beneath a lava flow is, thus, important for understanding the heat budget of the system.
 In this paper, we investigate the effects of temperature- and depth-dependent definitions of material properties in a lava-substrate thermal model. Beginning with the constant thermophysical properties of the lava and substrate used by Fagents et al. , we introduce, in turn, treatments of latent heat of crystallization, temperature-dependent lava and regolith thermal conductivities and specific heat capacities, and depth-dependent regolith density. The effects of including each temperature- or depth-dependent property are first compared to the constant-value case of Fagents et al. , and then all variables are included in a detailed simulation to compare with the results of constant-value simulations. We also include a simulation employing a solid basalt substrate to investigate the effects of different substrates on the resulting temperature distributions, as well simulations that explore the sensitivity of regolith heating depths to diurnal variations in ambient temperature at the lunar surface. The computed maximum depths in the substrate reached by pertinent isotherms represent the minimum depths of accumulated regolith required to preserve a record of implanted solar particles during the period of regolith exposure. Assessment of temperature distributions produced by simulations including variable thermophysical properties therefore enables more reliable predictions of preservation depths.
2.1 Lava Flow Heat Transfer
 Heat transfer from a lava flow on a body with no atmosphere is governed by radiation to space, conduction to the substrate, internal conduction, advection and viscous dissipation (if in motion), and latent heat produced during crystallization. To derive temperature distributions as a function of time, these mechanisms are balanced using the conservation of energy. In one dimension, the energy equation is given for a stationary lava flow by
where ρ is density, h is enthalpy, t is time, z is depth, k is thermal conductivity, c is specific heat capacity, and Sh represents additional source terms (e.g., the rate of heat generation due to crystallization). This form allows for variable properties (ρ(z), k(T), and c(T), where T is temperature), but in the case of constant properties, the relation h = cT holds and equation (1) reduces to
where κ is thermal diffusivity, given by κ = k/ρc, and additional source terms are neglected.
 When equation (1) is applied to a computational domain representing a lava overlying a substrate, different properties may be defined for the lava and substrate materials, but the energy flux due to conduction across the lava-substrate interface must be conserved according to Fourier's law:
where subscripts lava and sub represent lava and substrate, respectively.
 The lava upper surface will cool by radiation alone in the vacuum environment of the Moon. Thus, a radiative boundary condition is applied to the flow surface, following the Stefan-Boltzmann law:
where qrad is the radiative heat flux, σ is Stefan-Boltzmann constant, ε is the emissivity of the material, Tsurf is the temperature of the flow surface, and T∞ is the far-field temperature of the environment into which the hot surface is radiating. (We adopt T∞ = 4 K for the Moon, but discuss in section 3.5 the effect of using a higher T∞). The fourth power dependency and large temperature difference at the surface boundary makes radiation an important heat loss mechanism. The surface of a lava flow will readily radiate heat until a sufficiently thick crust forms and insulates the interior of the flow, after which the cooling rate is controlled by conduction through the crust.
 Once initial and boundary conditions are specified, and appropriate material properties defined, equation (1) may be solved to derive the temperature distribution in the lava and substrate using the temperature-enthalpy relation c∂T/∂z = ∂h/∂z. For our purposes, the source term Sh represents the volumetric rate of energy generation due to latent heat of crystallization (L), which is released at variable rates between the liquidus and solidus temperatures as the different mineral phases crystallize.
2.2 Properties of Lunar Basalt
 The ability of our model to adequately represent lava-substrate heating depends upon use of appropriate material property definitions. Properties of many terrestrial materials are well documented under a wide range of conditions. However, definitions of lunar material properties, especially at high temperatures, are less common. A variety of approaches must be employed to appropriately treat lunar basalt and regolith in a numerical simulation. The most important properties controlling the cooling of a lava flow are thermal conductivity, specific heat capacity, and latent heat of crystallization [Fagents et al., 2010]. For our simulations, the lava is assumed to be degassed (i.e., relevant to the distal reaches of a flow), with a constant density of 2980 kg m−3, appropriate to a lunar basalt in a molten state [Murase and McBirney, 1973].
Murase and McBirney  measured the thermal conductivity of several igneous compositions in solid and liquid phases. We adopted the expression
as a simple fit to data measured for a synthetic lunar basalt for temperatures up to 1500 K. Thermal conductivity decreases from 2.7 to 0.67 W m−1 K−1 as the temperature increases from 200 to 1500 K; above this temperature, the trend reverses and conductivity increases with temperature [Touloukian et al., 1981; Čermàk et al., 1982; Büttner et al., 1998]. By comparison, Fagents et al.  used a constant conductivity of 1.5 W m−1 K−1, representing an average value for a synthetic lunar basalt over this temperature range.
 The specific heat capacity represents the energy needed to increase the temperature of a given mass of material by a certain amount or, conversely, the energy released when a given mass cools by a certain amount. Measurements of specific heat capacity of lunar basalts at high temperatures are sparse, so data taken from measurements of terrestrial basalts were used to describe the simulated lunar basalt. Keszthelyi  gives an expression for specific heat capacity based on laboratory data of Touloukian et al. (1989), which shows specific heat increasing from ~600 to 1100 J kg−1 K−1 according to
 At a transition temperature of 1010 K, the specific heat plateaus at 1100 J kg−1 K−1. This transition temperature, known as the Debye or Einstein temperature, represents the high temperature limit of specific heat capacity based on the crystal lattice structure of the material. The relationship in equation (6) agrees with trends of increasing specific heat capacity with temperature given by Touloukian et al.  and Čermàk et al. . For comparison, Fagents et al.  used a constant specific heat capacity of 1500 J kg−1 K−1 based on the data given by Hemingway et al. , Touloukian et al. ; Lange and Navrotsky , Büttner et al. , and Williams et al.  and which accounts for higher values reported in the melting interval.
Fagents et al.  used a simple method to account for latent heat of crystallization L [Carslaw and Jaeger, 1986], in which the specific heat capacity of the lava is adjusted over the solidification interval (ΔT = Tsol − Tliq) to an effective specific heat capacity, c′lava = clava + L/ΔT. For one end-member series of simulations, this expression was applied over the entire duration of flow cooling, therefore significantly overestimating both the latent heat produced during crystallization and the substrate heating depths. For comparison, another end-member series of simulations was run with no accounting for latent heat. These two end-member simulation series therefore provided upper and lower bounds on substrate heating depths, but did not accurately represent the release of latent heat as the lava crystallizes between liquidus and solidus.
 For the present study, we employed MELTS [Ghiorso and Sack, 1995; Asimow and Ghiorso, 1998] and rhyolite-MELTS [Gualda et al., 2012] to determine the range and the distribution of production of latent heat for a selection of mare basalts representing a range of compositions [Willis et al., 1972; Rhodes and Hubbard, 1973; Papike and Shearer, 1998; Smales et al., 1971; electronic appendix of Wieczorek et al., 2006]. Each composition released a total latent heat of 3–4 × 105 J kg−1 between the liquidus (~1450–1700 K) and solidus (~1200–1450 K) temperatures. In all cases, latent heat release was non-uniform within the crystallization interval. For our current model, we chose lunar sample 12038, a feldspathic basalt [Keil et al., 1971], as a representative composition. It released a total latent heat close to the average of all compositions (3.45 × 105 J kg−1) and had a liquidus (1470 K) similar to that of our chosen eruption temperature (1500 K), which thus ensures that our model captured the entire crystallization interval. To determine whether the latent heat release could be approximated as linear in our model, or whether it was better treated as a variable release across the solidification range, we compared the effects of two different crystallization rates. The first,
where fsol, the fraction of material that has solidified, represents a constant rate of crystallization between liquidus and solidus (Figure 2). The second crystallization rate varies across the solidification range as
These expressions closely match the crystallization rate derived from MELTS for this composition (Figure 2).
 Studies have suggested that emissivity of basaltic lava drops as low as 0.55 at very high temperatures [Abtahi et al., 2002; Burgi et al., 2002]. However, it is believed this transition occurs at temperatures well above our model eruption temperature of 1500 K [Ramsey and Lee, 2011; R. Wright, personal communication]. We therefore chose a constant emissivity (ε) of 0.99 [Crisp et al., 1990; Pieri et al., 1990; Flynn et al., 1993; Pinkerton et al., 2002].
2.3 Properties of Lunar Regolith
 The bulk density of the regolith describes the degree of compaction of the particles, such that increased density implies increased grain-to-grain surface contacts and therefore more efficient thermal conduction. Carrier et al.  give
where ρreg is density in kg m−3 and z is depth in meters, to describe the increase of regolith density beneath the lunar surface. Density is approximately 1300 kg m−3 at the surface and rapidly approaches a maximum of 1920 kg m−3 within about 3 m depth. Equation (9) is valid for present-day regolith thicknesses, but density profiles may have differed for ancient regoliths. However, given that the uppermost few tens of centimeters of regolith are the most critical for the heat transfer process, we retain the use of equation (9), since basic gardening processes would be the same for both young and old regoliths.
 Published data exist describing the behavior of various regolith thermal properties at temperatures less than 350 K. However, properties have rarely been measured at temperatures greater than 350 K. Thus, the temperature dependences of regolith properties must be extrapolated from measured values. Fountain and West  found that the thermal conductivity of lunar regolith depends on the density, or compaction, and the temperature of the sample. They used the relationship
to describe regolith thermal conductivity in a vacuum up to a temperature of 370 K. The first term, A, in equation (10) describes solid conduction between points of contact on the surface of regolith grains; the second term, B, represents radiation across pore spaces in the particulate substrate. Assuming that equation (10) holds at high temperatures, and using equation (9) to describe regolith density, a minimum thermal conductivity of 1.43 × 10−3 W m−1 K−1 occurs when density and temperature are at minimum values (1330 kg m−3 and 200 K, respectively). A maximum thermal conductivity of 0.253 W m−1 K−1 occurs if density and temperature are at maximum values (1920 kg m−3 and 1500 K, respectively), although in practice such high conductivities would never be achieved in our model scenario, because such high temperatures would not propagate to the depths corresponding to a density of 1920 kg m−3. In comparison, Fagents et al.  used a constant conductivity of 0.011 W m−1 K−1, based on in situ heat flow data reported by Langseth et al.  and Vaniman et al. .
Hemingway et al.  studied several lunar soils and found that
describes the regolith specific heat to within 10% in the temperature range 90–350 K. Values of creg increase from ~570 to 850 J kg−1 K−1 in this range. However, extrapolation of this equation to higher temperatures leads to an unphysical increase in the specific heat. Since specific heat is an intrinsic property, its value is determined by the composition of the material, it is assumed the temperature dependence of specific heat of the regolith is similar to that of the surrounding mare basalt, the primary source of the mare regolith. Thus, we adopt the form of equation (6), adjusted to match the value of equation (13) at 360 K, giving
to define regolith specific heat from 360 to 1010 K. We use the basalt transition temperature of 1010 K, where the regolith specific heat will plateau at a value of 1074 J kg−1 K−1. For comparison, a constant specific heat of 760 J kg−1 K−1 was used by Fagents et al.  based on the data of Horai et al. , Robie et al. , and Hemingway and Robie .
 The temperature of the lunar surface at the time of emplacement of a lava flow will depend on the timing within the lunar day-night cycle (a lunation) and with the latitude of emplacement. In situ measurements from a thermocouple buried in the regolith at the Apollo 17 landing site show surface temperature variation from 100 K to nearly 400 K throughout a lunation (29.5 Earth days). Diurnal temperature variations dampen to within a few degrees of ambient in the upper 2 cm of the regolith and are imperceptible at 80 cm depth [Langseth and Keihm, 1977]. With the exception of semi- and permanently shadowed polar regions, latitudinal differences in surface temperature fall within the day/night extremes [Dalton and Hoffmann, 1972]. To investigate the effects on regolith heating depths of varying surface temperatures, we ran simulations with initial regolith temperature profiles representative of the extremes of lunar day and night temperatures: a maximum daytime temperature of 400 K and a minimum nighttime temperature of 100 K. For both day and night simulations, initial regolith temperatures were set to converge to 200 K at ~80 cm depth.
2.4 Model Description
 Numerical simulations were performed using PHOENICS (Parabolic Hyperbolic Or Elliptic Numerical Integration Code Series), a computational fluid dynamics software program designed to simulate fluid flow, heat transfer, and chemical reaction processes (http://www.cham.co.uk). PHOENICS allows significant user freedom in the customization of simulations. Through a simple graphical interface, users can create models with unique geometries and material properties. For more complex simulations, for example, those involving temperature-dependent properties, the user can supply algebraic expressions to define characteristics specific to a model. We have previously used PHOENICS to model a variety of thermal and fluid dynamic problems [e.g., Fagents and Greeley, 2001; Fagents et al., 2000, 2010], and testing of the models against standard analytical solutions has demonstrated that our PHOENICS simulations have the requisite accuracy.
 In this study, we used a one-dimensional formulation to simulate a stationary lava flow at an initial temperature of 1500 K emplaced instantaneously on substrate material at ambient temperature (Figure 3). For our sensitivity tests, the flow thickness was set to 1 m to reduce computational time, although this thickness is consistent with meter-scale lobes identified at some locations on the Moon (e.g., Hadley Rille) [Vaniman et al., 1991]. For most runs, excluding those exploring diurnal temperature variations, the initial regolith temperature was set to 200 K, an average of global surface temperatures [Vaniman et al., 1991]. The computational grid contained 60–240 cells in the lava and 60–360 cells in the substrate, depending on modeled thicknesses (1–15 m). The temperature of the lower boundary of the computational domain (i.e., the bottom of the substrate) was free to vary. However, the depth of the substrate was chosen so that the heat pulse never reached the lower boundary in our simulations. Cell size was refined toward the lava surface and lava-substrate boundary to ensure model accuracy in zones of large temperature gradients. Simulations were run with a 30 min time step. A range of cell dimensions and time step sizes were explored to ensure optimal stability and convergence of the solution. The results were monitored to find the maximum depths of the 300, 500, and 700°C (573, 773, and 973 K) isotherms representing the temperatures at which specific extralunar particles will begin to volatilize (Table 1).
3.1 Temperature-Dependent Thermal Conductivity
 When lava conductivity, klava, varies inversely with temperature as in equation (5), the lower conductivity at high temperatures allows the core of the lava to remain near eruption temperature for a longer duration relative to the simulations employing a constant lava conductivity (Figures 4a and 4b). In contrast, as the lava surface cools radiatively to space, the conductivity of the lava increases, resulting in greater heat transfer with decreasing temperature; therefore, temperatures within a few tens of centimeters of the flow surface decrease at a higher rate than for the constant lava conductivity case (Figure 4b). As the flow continues to cool throughout, lava thermal conductivity increases, and temperatures drop more rapidly (Figures 4c and 4d). Overall, lava temperatures drop to near ambient more quickly than in the constant case. However, the initial retention of heat in the lava core and the increased thermal conductivity at the lava regolith interface cause more heat to be conducted into the regolith (Figure 4d), resulting in modest increases in the maximum penetration depths of the 773 and 973 K isotherms by 8% and 3%, respectively, over the constant-value case. The 573 K isotherm has a 2% shallower maximum depth than in the constant case due to the greater overall cooling rate of the lava.
 While the effects of temperature-dependent lava thermal conductivity are quite subtle, accounting for the temperature and density dependences of regolith thermal conductivity (kreg; equation (10)) have a marked effect on the amount of heat conducted from the lava into the regolith. At high temperatures, such as near the lava-regolith interface immediately after emplacement, the effect of the high temperature dominates the effect of the low density. Thus, the increased conductivity of the regolith allows for greater heat transfer than in the constant-value simulation. At shallow depths, the regolith is heated to higher temperatures than in the constant case (Figures 4a and 4b). In contrast, at low temperatures (i.e., deeper in the substrate), the conductivity drops to values lower than in the constant case, even given the greater regolith density at depth, leading to insulation against the heat pulse as it propagates downward. As the lava cools, the regolith also becomes less conductive near the less dense lava-regolith interface, allowing the available heat to remain within the substrate and to continue propagating downward in the regolith layer (Figures 4c and 4d). A bulge in the temperature profile appears within the regolith (Figure 4d), as the temperature in the regolith is warmer than in the lava. The increased heat within the regolith leads to greater penetration depths for all isotherms than in the constant-value case, increasing the maximum depths of the 573, 773, and 973 K isotherms by factors of 1.9, 2.3, and 2.8, respectively (Table 2).
Table 2. Maximum Penetration Depth and Time Taken to Reach Maximum Depth for Key Volatile Release Isotherms Beneath a 1 m Thick Lava Flow
573 K (300 °C)
773 K (500 °C)
973 K (700 °C)
All constant properties
Variable klava only
Variable kreg only
Variable klava and kreg
Variable clava only
Variable creg only
Variable clava and creg
Latent heat included, constant properties
All variable properties, no latent heat
All variable properties, latent heat included
Basalt substrate, variable properties, latent heat included
 When both lava and regolith conductivities vary with temperature, the lava conductivity controls the lava-regolith interface temperature, but isotherm depths within the substrate are dominated by the effects of the regolith conductivity, reaching approximately the same depths as the case in which the regolith conductivity was varied alone (Figures 4c and 4d). The effects of the inverse relationship between lava temperature and conductivity are seen in the prominence of the bulge in Figure 4d, as the regolith remains warmer than the rapidly cooling lava flow. With respect to the constant-value case, the 573, 773, and 973 K isotherm depths increase by factors of 1.8, 2.4, and 2.9, respectively (Table 2).
3.2 Temperature-Dependent Specific Heat Capacity
 The inclusion of a temperature-dependent lava specific heat capacity (clava; equation (6)) results in a lower specific heat at all temperatures than the constant-value case, but particularly at low temperatures. Lower specific heat implies less energy available within the lava, such that the entire flow cools more quickly than in the constant-value case (Figure 5). The increased cooling rate and the decrease in the available energy result in less heat being transferred into the regolith, leading to shallower maximum depths of all isotherms than in the case in which a constant lava specific heat was adopted; the 573, 773, and 973 K isotherms reach shallower maximum depths by 14%, 16%, and 17%, respectively (factors of ~0.83–0.86).
 At temperatures within 100 K above ambient, the temperature-dependent specific heat capacity of the regolith (creg; equation (14)) is less than the constant value adopted for comparison (760 J kg−1 K−1). Above this threshold, the specific heat increases to a maximum of 1074 J kg−1 K−1 at 1100 K. With a larger specific heat capacity, more energy is required to increase the temperature of the regolith, leading to shallower depths of heating and a small decrease in the temperature at the base of the lava flow (Figure 5). Each of the pertinent isotherms has a 23–24% shallower maximum depth (i.e., a factor of ~0.77) than for the constant-value model (Table 2). When both the lava and the regolith specific heat vary with temperature, the decreased total heat within the lava compounds the effects of increased regolith heat capacity, further reducing the maximum depths reached by each of the isotherms to 34–37% shallower (i.e., factors of 0.66–0.63) than in the constant case.
3.3 Latent Heat of Crystallization
 Including latent heat release in our simulation has a substantial effect on the total heat available in the system. However, the use of expressions for variable (equation (8)) versus linear (equation (7)) crystallization rates produced negligible differences in isotherm penetration depths. This implies that it is the total latent heat released over the cooling interval, and not the specific release pattern, that influences the maximum heating depths. Regardless of the equation used, as the lava cools and crystallizes through the solidification interval, the release of latent heat slows the cooling rate of the lava (Figure 6). This is apparent soon after emplacement, where radiative cooling from the upper surface leads to crystallization and release of latent heat in the upper portion of the flow (Figure 6a), maintaining warm temperatures for a longer duration than in the case where latent heat is not supplied. As the effects of surface cooling propagate downward in the lava and the core crystallizes, the latent heat maintains higher temperatures throughout the lava for a longer duration than if latent heat is not included in the model (Figures 6b–6d). The delayed lava cooling prolongs the duration of heat transfer to the regolith, thereby increasing the amount of heat introduced into the substrate and producing deeper penetration of the heat pulse (Figure 6d). The maximum depths of the 573, 773, and 973 isotherms increase by 19%, 28%, and 38% (factors of 1.2, 1.3, and 1.4), respectively, over the constant-value model (Table 2).
 Upon emplacement, a portion of the upper and lower surfaces (e.g., crusts) of the flow may cool sufficiently rapidly to quench to glass, thereby reducing the latent heat of crystallization released by the system. To account for this reduction, we ran a comparison model in which total latent heat released was reduced to 75% of the maximum amount of latent heat available within the lava, representing a lava with 25% quenching, which is likely a significant overestimation [Keszthelyi and Denlinger, 1996], and hence provides an extreme estimate of the effects of quenching. However, we find that, with 75% crystallization of the lava flow the maximum depths of the 573, 773, and 973 K isotherms decrease by only 4%, 5%, and 6%, respectively, from the maximum depths reached when the entire flow crystallized. This effect will be further reduced for a thicker lava flow, when the proportion of quenched crust will be less. Given that significantly reducing the latent heat release has only a minor effect on substrate heating, we have chosen to include 100% release of latent heat in order to fully represent the potential influence of solidification on the thermal budget of the lava-substrate system.
3.4 Regolith Density
 The increase in regolith bulk density from ~1300 kg m−3 at the surface to ~1920 kg m−3 at depths exceeding 3 m (equation (9)) [Carrier et al., 1991] has the potential to affect substrate heating in two ways: (i) by changing the bulk thermal diffusivity of the regolith (κreg = kreg/ρ creg) and (ii) by increasing the grain-to-grain contact areas, thus modifying regolith thermal conductivity (equation (10)). Regarding mechanism (i), we find that inclusion of the depth dependence in the simulations has little effect on the heating of the regolith, increasing the maximum depths of the 573, 773, and 973 K isotherms by less than 1 cm (3%, 5%, and 8%, respectively) (Table 2). Similarly, compaction of the surficial regolith by the weight of the overlying lava flow is unlikely to change the isotherm penetrations depths significantly.
 Effect (ii), on the other hand, is harder to constrain. However, Wechsler et al. (1972) show that, while both the solid conduction component (A in equation (10)) and the radiative coefficient (B in equation (10)) increase with increasing bulk density, radiative effects become a much greater influence on thermal conductivity as temperatures increase above ambient because of the T3 dependence. Therefore, the relatively modest effect of the increase in regolith density is swamped by the dependence of thermal conductivity on temperature. We include the density effect in our calculations solely based on its importance in equation (10).
3.5 Diurnal Surface Temperature Variations
 We find that varying the initial regolith temperature profile due to diurnal fluctuations produces negligible effects on regolith heating depths. Temperature differences of 100 K or more are only present in the top 1–2 cm of the regolith. This small region of temperature gradient is trivial compared to the energy introduced to the system by the overlying lava flow, emplaced at 1500 K. Adopting a nighttime profile with a regolith surface temperature of 100 K at the time of flow emplacement produces maximum depths of the 573, 773, and 973 K isotherms that are decreased by only ~0.1 cm (~1%) each. A daytime temperature profile with an initial regolith surface temperature of 400 K yields an increase in maximum depths of less than 0.5 cm (3%, 4%, and 5%, respectively).
 An additional effect of the diurnal temperature cycle is that, during the lunar day, solar irradiance on the surface of the lava will increase the effective ambient temperature to which it is radiating. We ran a comparison simulation using an effective ambient temperature of 250 K for T∞ in equation (4), following Keszthelyi , to investigate the resulting effect on the lava-substrate system. We find that, because of the form of the radiative heat flux, the initially high lava surface temperatures overwhelmingly dominate the heat flux, regardless of the value of T∞. Therefore, the specification of T∞ proves to be an insignificant factor in maximum penetration depths of the heat pulse. The change in surface temperature using T∞ = 4 versus 250 K does not affect the system until the surface of the lava cools to ~500 K, about 10 days after emplacement. At this time, the surface temperatures of the two cases (T∞ = 4 and 250 K) will deviate; however, this deviation does not reach the substrate until well after the heat pulse has reached maximum depths in the regolith and therefore does not affect our results (Figure S1).1
3.6 Comparison of Full Model to Constant-Property Model
 When all variable thermophysical parameters are included in the simulation there is a significant difference in the cooling behavior of the system compared to the simulation with constant parameters (Figure 7). Initially, the lower specific heat of the lava allows surface radiation to more rapidly cool the upper portion of the lava flow (Figures 7a and 7b). This cooling does not propagate to the center of the flow, as the release of latent heat and low thermal conductivity in the core act to maintain high temperatures. The increased thermal conductivity of the regolith at high temperatures transmits heat downward from the base of the flow leading to higher temperatures in the shallow regolith (Figures 7b and 7c). As the lava cools below the solidus (~1200 K) and latent heat is no longer released, the lava conductivity increases, and lava specific heat decreases. These effects compound to cool the lava more quickly than in the constant-property case (Figure 7d). Temperatures remain elevated in the regolith; as the top of the regolith cools, thermal conductivity decreases, creating a buffer against the warmer regolith at depth and producing a bulge in the temperature profile (Figure 7d). The maximum depths of the 573, 773, and 973 K isotherms are factors of 1.6, 2.2, and 2.9, respectively, deeper than for the constant-property case (Table 2).
3.7 Results for Solid Versus Particulate Substrates
 The variable-property model was modified to incorporate a solid basalt substrate of the same composition as the overlying lava, such that the substrate has a greater thermal conductivity, specific heat capacity, and density than if it were composed of regolith. Figure 8 shows a comparison of temperature profiles in regolith and solid basalt substrates. It can be seen that the differing substrate properties have a marked effect. The greater conductivity of the basalt substrate allows heat to be transported rapidly away from the lava-substrate interface to greater depths in the substrate. In addition, the greater specific heat capacity implies that the thermal energy delivered from the lava to the interface is unable to raise the substrate temperature to the same degree as would be the case for a regolith substrate. The sum effect of these two parameters is that solid substrate temperatures are in general elevated above ambient to lesser degrees, but to greater depths, than when the substrate consists of regolith (Figure 8). The regolith substrate is heated to higher temperatures but the overall extent of heating is confined to shallow depths. However, only the 573 K isotherm penetrates to greater depths in solid basalt than in regolith (by 10%); the 773 K isotherm penetrates just a few millimeters, and no portion of the substrate exceeds 973 K (Table 2).
3.8 Influence of Lava Flow Thickness
 To assess the influence on regolith heating of lava flow thickness, we ran simulations of 10 m lava flows with variable thermophysical properties for comparison with the 1 m case. Fagents et al.  showed that, when constant properties are used, the depth of heating scales with lava flow thickness, i.e., isotherms beneath a 10 m thick lava flow would penetrate 10 times deeper than those beneath a 1 m thick lava flow. When latent heat release and temperature-dependent properties are included in the 10 m simulations, the maximum depths of the 573, 773, and 973 K isotherms are only 9.5, 9.1, and 8.4 times the equivalent 1 m simulations. Thus, isotherm penetration depths scale close to but not quite linearly with flow thickness. This can be attributed to the significant nonlinearities introduced into the model by incorporating variable thermophysical properties.
4.1 Importance of Variable Material Properties
 The inclusion of temperature-dependent material properties is critical for adequately treating lava-substrate heat transfer and for the accurate assessment of the depths to which exogenous particles will be lost from the lunar regolith. Penetration of the key volatilization isotherms (Table 1) depends in a complex way on the variation with temperature of thermal conductivity, specific heat capacity, and the release of latent heat due to crystallization (Figure 9). Simulations including variable properties, as described by equations (5)–(14), predict that implanted solar wind ions will be disturbed and/or lost to depths 1.6 times greater than for the case in which all model parameters are kept constant (Table 2).
 In general, temperature-dependent thermal conductivity and specific heat capacity have competing influences on the temperature distribution in the regolith (Figure 9). However, the regolith thermal conductivity is the greatest single influence on heating depths, especially when one considers the effect of the T3 dependence (expressing radiative heat transfer across intergranular voids; equation (10)) over the temperature range of interest (200–1500 K). At the boundary with the lava, the high initial conductivity induced in the regolith allows for efficient transfer of heat into the substrate; then as the lava and the upper portion of the regolith cool, and regolith conductivity drops, it acts as an insulator, isolating the heat pulse in the regolith and allowing it to penetrate to greater depths (Figures 4d and 7d). Our results concur with those of Fagents and Greeley , who found that the influence of regolith properties (particularly thermal conductivity) was greater than the influence of lava properties in determining regolith temperatures at depth.
 To adequately model the thermal budget of a solidifying lava flow, it is necessary to include treatment of latent heat due to crystallization. As described in section 2.2, the predecessor [Fagents et al., 2010] to our current model did not specifically treat latent heat release, but adopted two end-member sets of constant lava properties in an attempt to place upper and lower limits on the expected range of heating depths. Simulations using the first set of properties, the constant-value results cited in this paper (Table 2), predicted maximum depths of 12.8, 7.0, and 3.7 cm for the 573, 773, and 973 K isotherms, respectively. The other end-member simulation approximated latent heat release by modifying the lava specific heat capacity, but applying it over the entire cooling duration. The resulting isotherm depths of 28, 14, and 7.8 cm are therefore significantly overestimated. In the current model, with latent heat treated explicitly and released exclusively between the liquidus and solidus, maximum isotherm depths of 15.3, 8.9, and 5.0 cm are predicted when all other properties are held constant (Table 2). These results fall predictably between those of the end-member simulations of Fagents et al. ; however, they are closer to those that did not account for latent heat in any way. Incorrect treatment of latent heat can therefore have a considerable effect on cooling simulations. Table 2 shows that maximum isotherm penetration depth increases by 28% (from 17.9 to 23 cm) when latent heat is included in the variable-property simulations.
 We found no significant difference in the heating depths predicted by simulations in which latent heat was distributed uniformly (equation (7)), compared to those in which latent heat was released non-uniformly (equation (8)), based on the crystallization pattern predicted by MELTS for a representative lunar basalt composition. In contrast, other studies have emphasized the importance of including non-uniform latent heat distributions [Lange et al., 1994; Patrick et al., 2004]. Based on our results, we argue that the need to incorporate non-uniform versus uniform latent heat release in a given lava flow thermal model will depend on the purpose of the model and the specific scenario being treated. Because we are interested in finding maximum isotherm depths after a significant period of heat transfer, the exact details of the temperature distribution on the shorter time scale of solidification are not relevant. Depending on the nature of the problem under investigation, we recommend that a range of latent heat distributions be explored to optimize the trade-off between accuracy and computational efficiency.
4.2 Specification of Substrate Material
 Our simulations produce significant differences in temperature profiles within a lava-substrate system when that substrate consists of solid basalt rather than particulate regolith. While the solid basalt scenario is not directly relevant to the search for paleoregoliths containing solar wind particles, the model results for different substrate types imply a need to specify the substrate material in models of terrestrial lava emplacement and cooling. The low thermal conductivity of particulate materials means that they insulate the base of the flow, allowing the lava core to retain heat for significantly longer than for solid substrates, thereby enhancing flow mobility. These effects will not be as extreme under terrestrial conditions because gas conduction by air in substrate pore spaces contributes a significant component to the effective conductivity of a particulate material [e.g., Wechsler and Glaser, 1965]. Nevertheless, the effective conductivity will generally be less than that of a solid substrate. During effusive eruptions on Earth, it is common for lava to flow over material other than solid basalt. For example, at Kilauea and Mauna Loa Volcanoes, Hawaii, there are many locations where lava has been emplaced on top of basaltic or reef-derived sand, soil, or explosive basaltic deposits (e.g., pyroclastic fragments). This range of compositions, water and organic contents, grain size, etc., may have a significant influence on lava-substrate heat transfer, flow mobility, and hence the length and areal coverage of the flow. It is prudent, therefore, to consider specification of a substrate material to treat conductive cooling from the flow base in models of lava flow emplacement and hazard prediction.
4.3 Limitations of Model
 We have conducted a study of the sensitivity of our model of lava-substrate heat transfer to the incorporation of variable thermophysical properties and have shown that they have important influences on lunar regolith heating. However, by opting to treat an instantaneously emplaced, stationary lava flow, we have not addressed the period of emplacement of the flow, which could significantly enhance substrate heating because of the additional heat advected by the moving lava, thus increasing the regolith depths required to retain implanted volatiles. Our choice not to treat the lava flow emplacement phase is in part based on an incomplete understanding of lunar lava flow emplacement mechanisms. Lunar flow units are areally extensive and produce individual lobes on the order of meters to tens of meters thick [e.g., Gifford and El-Baz, 1981; Wilhelms, 1987; Vaniman et al., 1991; Hiesinger et al., 2002; Hiesinger and Head, 2006; Robinson et al., 2012]. However, the long interval that has elapsed since volcanism was prevalent on the Moon means that flow morphologies and textures have become difficult to discern. It is therefore hard to determine the style(s) of emplacement of lunar mare lavas. While it is accepted that they are typically of large volume and low viscosity [e.g., Murase and McBirney, 1970; Taylor et al., 1991], it remains unclear whether they were emplaced as high effusion rate, turbulent sheet flows, or rather more modest, inflationary (pahoehoe-like) compound flow fields, or even perhaps some combination of the two mechanisms. Regardless of emplacement style, our model results are most relevant to the distal margins of flow lobes that would have had limited period of flow before coming to rest and solidifying; these areas would also be the most favorable for retrieving regolith samples.
 The results of our simulations are only as good as the underlying assumptions and information provided to the model in the form of the definitions of thermophysical properties. The model exhibits a particular sensitivity to the characterization of substrate thermal conductivity. Thermal conductivity of particulate materials varies in a complex way with temperature, particle size (and size distribution), bulk density (porosity), packing, composition, and the presence and pressure of any interstitial gas phase [e.g., Presley and Christensen, 1997a]. Past studies have investigated limited ranges of conductivity dependence for particulate materials [Cremers et al., 1970; Fountain and West, 1970; Wechsler et al., 1972; Langseth et al., 1976; Presley and Christensen, 1997a, 1997b; Huetter et al., 2008; Yuan and Kleinhenz, 2011], and data sets produced by different methods have different degrees of accuracy and reliability [Presley and Christensen, 1997a]. For example, experimental data are commonly derived for packed glass beads of uniform size and over very limited temperature ranges. This makes it particularly difficult to define a generally applicable conductivity function for the lunar regolith, given that regolith is a complex mixture whose physical properties (e.g., particle components and size distribution, and porosity) will naturally vary from site to site [Carrier et al., 1991]. Furthermore, while there are theoretical approaches to understanding thermal conductivity of particulate materials [e.g., Blumberg and Schlünder, 1995; Schotte, 1960; Watson, 1964; Wechsler et al., 1972], a thorough validation with physical data for natural particulate mixtures is lacking, particularly at the elevated temperatures relevant to our model scenario. The model's sensitivity to substrate thermal conductivity means that, while we have attempted to define a realistic temperature-dependent conductivity based on theory, it is unclear whether this approach adequately treats natural particulate mixtures over all ranges of conditions. The validity of this treatment therefore requires further exploration and would benefit from additional theoretical and experimental investigations, as well as physical validation. Nevertheless, we have established a robust methodology for modeling variable properties, and this is amenable to modification as better conductivity treatments are developed.
4.4 Implications for Lunar Exploration
 The majority of charged particles that impact the lunar surface are ions emitted by the Sun, primarily hydrogen, but also helium and small amounts of carbon, nitrogen, oxygen, and heavier elements [Haskin and Warren, 1991; McKay et al., 1991; Vaniman et al., 1991]. Because of their relatively low energies, these are usually implanted to depths of microns to millimeters within mineral grains [Haskin and Warren, 1991]. High-energy particles from more distant sources (e.g., GCRs) penetrate to meter-scale depths in solid material, producing a cascade of nuclear reactions and leaving linear tracks of crystal damage in the target rock [Vaniman et al., 1991; McKay et al., 1991; Goswami, 2001; Eugster, 2003; Lucey et al., 2006]; molecules of CO, CO2, H2O, and N2 are commonly created when the kinetic energy of the incoming particles is greater than the energy of reaction. Because of their greater penetration depths, the absence or presence of a regolith is not critical to the preservation of the GCR products. Thus, penetration into, and heating of solid basalt (either a pristine surface or underlying a regolith veneer) is perhaps more relevant to preservation of GCR records. The relatively restricted depth of the high-temperature domain within a solid basalt substrate (Figure 8) implies that GCR records would have remained undisturbed by the emplacement of the overlying flow without the need for an insulating regolith to assist preservation. Isolation of the particle tracks and daughter nuclei to individual flows within a vertical succession would allow dating of the GCR record in successive flows, thereby significantly improving the time resolution of GCR records.
 In the case of implanted solar wind materials and surviving fragments of hydrous-mineral bearing projectiles [e.g., Zolensky, 1997], the simulations presented in this contribution determine the depths to which 1–10 m thick lava flows will heat the underlying regolith to temperatures at which volatiles may be released. These depths represent the minimum thickness of regolith deposit needed for the particles to survive heating by a lava flow. When constant material properties are used in the model, a minimum of 12.8 cm of regolith (maximum heating depth of the 573 K isotherm) is needed to preserve a complete record of all implanted volatiles beneath a flow lobe 1 m thick. When temperature-dependent properties are included, at least a 20.4 cm thickness is needed for complete preservation. Simulations of 10 m thick lava flows show that maximum isotherm depths do not scale linearly with the thickness of the overlying flow, but rather the relative depth of heat pulse penetration decreases; a thickness of 194 cm is required, rather than 204 cm based on scaling up from the 1 m simulations.
 During the peak period of lunar mare volcanism, regolith is thought to have developed on fresh surfaces at a rate of 5 mm/Ma [Hörz et al., 1991] and may have reached 20 mm/Ma prior to 4 Ga [Crozaz et al., 1970; Duraud et al., 1975], although these production rates depend critically on our understanding of small impact crater formation [McEwen et al., 2005]. At such rates, it would take tens to hundreds of millions of years to accumulate enough regolith to preserve a complete record of extralunar volatiles beneath 1 m thick lavas. The thicknesses of lunar lava flows vary considerably, leading to a wide range in minimum regolith depths required to preserve exogenous materials, and hence time intervals required for sufficient regolith to develop. Our model can provide estimates of the range of depths within a regolith that must be sampled to extract a pristine suite of implanted volatiles, based on the thickness of the overlying flow. These depths can provide estimates of the interval that must elapse between successive flow field emplacements.
Hiesinger et al. [2000, 2002, 2003], and used Clementine multispectral imagery of several lunar maria to map boundaries between flow fields and to estimate emplacement ages. Intervals of over a billion years between emplacements of successive flow fields were found in most of the maria studied. In Oceanus Procellarum, a total age range of ~2.7 Ga was found, and adjacent and overlapping flows commonly exhibit several hundred million years between emplacements. The age gaps between successive flow fields are sufficiently large that individual flow units would have developed significant regolith cover [Hiesinger et al., 2000, 2002, 2003]. Clementine multispectral data were also used to define the thicknesses of spectrally distinct flow units excavated by small impact craters [Kramer, 2010; Weider et al., 2010]. More recently, high-resolution image data acquired by the Lunar Reconnaissance Orbiter Camera (LROC) [Robinson et al., 2010] have revealed details of lava flow characteristics that were previously undetectable with lower resolution data. Features such as impact craters, pit craters, and layered boulders exhibit tens of stratified units, on the order of 1–10 m in thickness, that are interpreted to be lava flow successions [e.g., Zanetti et al., 2011; Ashley et al., 2012; Robinson et al., 2012]. Although buried regoliths have not been directly observed in the image data, the LROC observations of thin, layered flow units and Clementine-based assessment of mare flow field ages imply that subsurface paleoregoliths should be present in the lunar maria. This may have been confirmed by the Lunar Radar Sounder instrument on Kaguya, which appears to have detected buried regolith layers [Ono et al., 2009]. As the new remote sensing data sets allow refinement of lava flow thicknesses, flow unit ages, and potentially, regolith layer thicknesses, our model will allow determination of the preservation potential of implanted volatiles at given sites within the lunar mare. These assessments can therefore be used to make recommendations for sites of future manned exploration of the lunar surface. As outlined by Crawford et al. , locating and sampling buried paleoregolith deposits would be greatly facilitated in the context of future human “sortie-class” missions to the lunar surface. These missions would preferably have the capacity for surface mobility of tens to hundreds of kilometers to reach proposed sample sites, and drilling to depths of at least tens of meters to obtain a useful succession of paleoregolith deposits and lava units. Sample analysis in situ or upon return to Earth could potentially yield valuable new information on, for example, the temporal variability in composition and strength of the solar wind and the changing galactic environment of the Solar System.
 The inclusion of temperature-dependent material properties is a crucial element in simulations describing the physics of cooling lava flows, as demonstrated by our comparisons of simulations of lava-substrate heat transfer using both constant and variable thermophysical properties. In our simulations, the use of temperature-dependent properties implies that solar wind volatiles would be disturbed at depths in the regolith that are 60% greater than the case in which constant properties are used. This increase in depth is driven by the inclusion of latent heat of crystallization and by the effects of a substrate thermal conductivity that is elevated at temperatures near eruption temperatures and then decreases as the regolith cools to ambient conditions.
 Substrate properties, particularly thermal conductivity, most strongly control the depth of penetration of key volatile-release isotherms into the substrate. This is especially marked for regolith materials, because of the T3 dependence of thermal conductivity due to radiation across the pore spaces within packed particulates. However, in addition to temperature, thermal conductivity is sensitive to many variables, including particle size, compaction, pore size, and the presence of any interstitial gas; any future modeling would therefore benefit from theoretical and experimental work that would lead to a more thorough characterization of thermal conductivity of regolith materials, particularly at elevated temperatures.
 The lunar regolith potentially holds important information about the history of our Solar System [NRC, 2007]. Solar wind particles, cosmic ray products, and other exogenous materials (e.g., debris from planets, asteroids, and comets) have great potential for advancing our understanding of the history of the Solar System, including that of the early Earth. Retrieving regolith deposits that have been protected from contemporary bombardment by overlying lava flows could be a key activity during future missions to the Moon. The work presented here provides the basis for identifying sites where such records might be preserved, which could, therefore, be targeted by future exploration activities.
Empirical constants in equation (10) (W m−1 K−1, W m−1 K−4)
c, clava, creg
Specific heat capacity, of lava, of regolith (J kg−1 K−1)
Effective specific heat capacity of lava (J kg−1 K−1)
Fraction of crystallized lava (-)
Enthalpy (J kg−1)
k, klava, kreg, ksub
Thermal conductivity, of lava, of regolith, of substrate (W m−1 K−1)
Latent heat of crystallization (J kg−1)
Radiative heat flux (W m−2)
Source term in energy equation (W m−3)
Radiative far-field temperature (~4 K)
Ambient temperature of lunar surface (200 K)
Lava liquidus temperature (K)
Lava solidus temperature (K)
Temperature at the lava flow surface (K)
Emissivity of lava (0.99)
Thermal diffusivity (m2 s−1)
Density, regolith density (kg m−3)
Stefan-Boltzmann constant (5.7 × 10−8 W m−2 K−4)
 This paper greatly benefited from comments by C. Dundas, M. Wieczorek, and an anonymous reviewer. This work was funded in part by the NASA Lunar Advanced Science and Exploration Research Program Grant NNX08AY75G. Financial support was provided for M.E.R through the NSF Graduate Research Fellowship Program. This is HIGP Publication 2002, SOEST Publication 8874, and LPI contribution 1722.
Auxiliary materials are available in the HTML. doi:10.1029/2012JE004131.