Dynamics of Subsurface Migration of Water on the Moon
Abstract
We investigate the dynamics of a water pumping mechanism driven by temperature variations in the lunar subsurface. The thermal environment at three polar sites and three sites in the Clavius region was simulated, taking into account local terrain and scattered radiation. A separate heat and mass transfer model was used to simulate depth-dependent, temperature-dependent, and pressure-dependent properties of the lunar subsurface. The results suggest that diurnally varying heat fluxes create suitable conditions for water migration at many sites across the lunar surface. Enabled by a constant supply to the lunar surface, water molecules typically migrate a few centimeters deep via the formation of distinct concentration peaks, and a downward flux driven by the repeated desorption and resorption. With a constant supply rate of 10−15 kg/(m2 s), the quantity of adsorbed water stored at Ga timescales reaches values on the order of 10−10 to 10−7 mol/m2 at the investigated sites. Based on our results, we present a new relation for the water migration depth that takes into account the ratios of surface temperatures. The results of a sensitivity analysis show that the desorption activation energy is a dominant factor for the quantity and depth of water migration. In addition to long-term accumulation of subsurface adsorbate, the model shows that temporarily captured water at shallower depths can be released during the lunar day at quantities of up to several µg m−2. This, as well as exposure of shallow water disturbed by impacts, may be a relevant source for surface water in illuminated areas.
Key Points
-
Our model combines 3D lunar surface temperatures and a detailed parametrization of heat and mass transfer properties of the subsurface
-
Water typically accumulates at depths <10 cm with a distinct peak and migration depth can be estimated based on surface temperature ratios
-
Water can be temporarily trapped at shallow depth even at mid-latitudes, where it is partially released to the exosphere during day
Plain Language Summary
Water bound to the lunar soil is released by the daily temperature cycle on the Moon. A portion of it is subsequently lost to space, but some of it is transported to deeper layers in the lunar soil. How much water will be transported within the soil depends on the temperature distribution, timescales, and how strongly the water bonds with the soil. Using a simulation model for the heat exchange at the lunar surface and the heat and mass transfer within the lunar soil, we show how pronounced this phenomenon is at six selected sites in the lunar south polar region and the Clavius region. We find that the quantities of accumulated subsurface water are generally low, on the order of several micrograms per unit area, and tend to be higher at the polar sites. Subsurface water can be found at depths of only a few centimeters and the depth of the peak concentration can be determined using a newly established relation for surface temperature ratios. Such shallow accumulated water is also partly released during some lunar days and could in principle be observed during day at the surface, even at warmer mid-latitudes, such as the Clavius region.
1 Introduction
The existence of lunar water has been confirmed through a variety of remote sensing data, but to date, ground truth remains to be provided. Extended in-situ measurements will allow determining the form and abundance of lunar water and thereby gaining knowledge about its origin, formation, stability, and mobility. Forms of water may vary from loosely adsorbed and highly volatile hydroxyl groups, to water molecules, to thick subsurface deposits of crystalline ice (e.g., Li et al., 2018, and references therein). Possible sources of lunar water are asteroid or comet impacts, surface interactions with the solar wind, and outgassing from the lunar interior (e.g., Anand, 2010; Lucey, 2009; Lucey et al., 2020). It is unclear however, how larger ice deposits can develop from such surface-related processes (Cannon & Britt, 2020). Vertical transport of lunar water down to several meters can occur by impact mixing of the surface regolith. But because this is a highly energetic process, it is likely that released volatile material will be lost by sublimation and can only accumulate via trapping on a colder regolith surface. Subsurface migration therefore mainly happens by diffusion of molecules along particle surfaces or in voids between particles. Molecules trapped at the surface may diffuse to deeper layers of regolith by a mechanism called ice pumping, caused by thermal gradients and thus differences in the sublimation rates over depth. Schorghofer et al. (2014, 2007) have previously studied mechanisms of water transport into the lunar subsurface, driven by either an external supply of water molecules to the surface or an initial ice layer at the surface. They concluded that the efficiency of subsurface water migration is strongly temperature-dependent and ideal for conditions with a surface temperature <105 K on average and >120 K peak. The presence of a protective surface ice layer was found to enhance the efficiency of pumping, but it is generally far less efficient than on Mars, where molecules leaving the surface are not immediately lost but remain in the atmosphere.
While these conclusions suggest a lunar water pumping mechanism is generally feasible, the underlying models have their limitations. Recent model efforts by Schorghofer and Williams (2020) include the depth-dependency of thermal properties, but similar to previous work neglect the time-variability, temperature-variability, and pressure-variability of thermal conductivity and specific heat capacity. Both properties are initialized using an expected temperature depth distribution based on the average surface temperature. Such simplifications make it impossible to assess microscale processes in the lunar subsurface at diurnal timescales. In fact, previous studies focused on geological timescales to assess the general potential of pumping in the polar regions of the Moon. This potential however is not indicative of the time-dependent depth distribution of any subsurface ice. Schorghofer and Taylor (2007) have previously calculated the quantitative amount and mean depth of subsurface ice for a constant supply rate at the surface, but mostly based on a constant soil temperature.
Complementary to this previous work, we present a new numerical model where the heat and mass transport processes in the subsurface are simulated on a smaller scale. The model takes into account the temperature-, pressure-, and depth-dependency of soil properties, such as porosity, tortuosity, particle size, gas density, desorption energy, thermal diffusivity, and gas diffusivity. It is limited to the first meter depth with a comparatively high spatial and temporal resolution. In contrast to averaged and static models, it therefore enables an insight into the highly dynamic heat and mass transfer processes that take place in the lunar subsurface and how these affect the overall pumping potential.
To derive the boundary conditions at the surface, we apply a separate thermal simulation model to compute more realistic heat fluxes in the visible and infrared range, including scattering and shadowing effects on the lunar surface. This allows the investigation of real local surface temperature variations that have a different effect on the efficiency of pumping than previously used simplified sinusoidal heat inputs.
In the following, we describe our theoretical model for subsurface transport of water in lunar soil and the thermal environment at six selected sites of interest at polar and mid-latitudes. We present the results for long-term and short-term transport and storage of subsurface water at these sites, as well as a sensitivity analysis to investigate the influence of selected soil properties on the pumping mechanism.
2 Physical Model
The simulation model is based on the physical formulations described in recent work (Reiss, 2018) for the coupling of temperature-dependent and pressure-dependent heat and mass transfer in lunar soil. It was originally developed to simulate the thermal extraction of water from small lunar samples, but has been modified and enhanced to enable broader thermophysical studies of the lunar subsurface. Compared to previous work, the major changes are the addition of resorption of water within the soil and depth-dependent soil properties, as described in the following.
2.1 Sorption Kinetics
of an adsorbate on a regolith particle can be expressed as the balance of desorption 
rate and adsorption rate 
(e.g., Kolasinski, 2012):
(1)
and 
and the fractional coverage 
, where 
is the coverage at saturation of the surface:
(2)
(3)
(4)
(5)As the Langmuir model only allows adsorption of molecules at free sites, the saturation coverage 
equals monolayer coverage. The desorption rate constant (Equation 4) is described in the form of the Arrhenius equation, using the preexponential factor 
, desorption energy 
, Boltzmann constant 
, and temperature
. Values of 
and 
were determined by Hibbitts et al. (2011) and Poston et al. (2013) for water physisorbed on lunar regolith simulant JSC-1A. These are the default values for the following simulations, but other relevant values will be discussed in the sensitivity analysis.
and the rate of incident molecules per square meter and second in the form of the Hertz-Knudsen equation, where 
is the molecular mass of water. The sticking factor was approximated using a temperature-dependent correlation of empirical data developed by Haynes et al. (1992) for the sticking of water on water ice:
(6)
, 
and 
and 
is the universal gas constant. The classic definition of desorption and adsorption rate as described above uses the dimensionless fractional coverage 
or surface concentration 
in molecules per unit area. Because in the present context the bulk volume of the lunar soil shall be investigated, we prefer a volumetric definition of the adsorbate concentration in the common unit mol/m³. Therefore, the surface concentration needs to be converted into the volumetric adsorbate concentration 
:
(7)Here, 
is the specific surface area of the regolith particles, 
is the particle density, 
is the porosity, and 
is the Avogadro constant. The same relation can be used to convert the surface concentration at saturation 
to a volumetric concentration at saturation 
.
:
(8)
2.2 Depth-Dependent Soil Properties
due to increasing compaction. For the bulk density 
, a relation developed by Vasavada et al. (2012) was used with the scaling parameter 
matching the global (<±60° latitude) lunar average (Hayne et al., 2017):
(10)
(11)For the specific surface area an average value of 0.5 m2/g was used, considered constant over depth in the model. The surface concentration of a monolayer of water is on the order of 1019 molecules per m2 (Schorghofer & Aharonson, 2014). With the aforementioned soil parameters this translates to volumetric monolayer concentrations of 9.0–14.9 mol/m³ from surface to depth, respectively.
2.3 Governing Equations
The model was implemented in COMSOL Multiphysics using the partial differential equation module to define the equations for heat transfer and mass transfer individually. The lunar subsurface was defined as one-dimensional line with a length (depth) of 1 m. A finer mesh was applied to the first 10 cm with 2,000 elements exponentially increasing in size to ensure a robust solution for high temperature and concentration gradients. The depth interval between 10 cm and 1 m was meshed with 500 elements.
and time 
:
(12)
is a combination of the densities and specific heat capacities of the solid and gas phase in the bulk regolith. The effective thermal conductivity 
is a combination of four different terms that account for the conduction via physical contact between solid particles, radiation between particles, conduction via the gas phase in the voids, and the coupling of solid and gas conduction in the vicinity of the contact area. 
is the source term, which is zero over the entire depth but different at the surface boundary to account for absorbed and emitted heat fluxes 
and 
:
(13)
for desorbed water in the gas phase and 
for adsorbed water, the latter being considered immobile (neglecting surface diffusion). The diffusion equation for water in the gas phase, again with all parameters as a function of depth 
and time 
, is:
(14)
is a combination of Knudsen and ordinary diffusion and variable with temperature and pressure. The term 
describes the variation of adsorbate and thus represents a source term for the gas phase, see Equation 9. 
is an additional source term for the surface boundary, accounting for a supply of water 
and the escape of volatile water into the lunar exosphere with the mean thermal velocity 
:
(15)The flux of both phases at the lower boundary is considered zero (
, 
) and there is also no flux at the surface boundary for the adsorbed water (
).
In the model, we define a constant supply rate of volatile water to the lunar surface. This is based on the assumption that volatiles in illuminated or warmer locations are liberated and migrate toward colder locations via ballistic hopping. Such a global transport mechanism for volatile water was observed by the Lunar Exploration Neutron Detector (LEND), concluding that accumulation of up to 0.0125 ± 0.0022 wt% water-equivalent hydrogen within the upper meter of the lunar surface would happen at local dawn (Livengood et al., 2015). Averaged over one lunation this would mean a relatively high supply rate on the order of 10−8 kg/(m2 s). Similar observations of global water transport were made by the Moon Mineralogy Mapper (M3), where the strongest diurnal variations in OH abundance on the order of 0.02% were observed at latitudes between ±30° and ±60° (Li & Milliken, 2017). For our simulations we use a supply rate of 10−15 kg/(m2 s), or 5.55 × 10−14 mol/(m2 s). This value equals the order of magnitude derived by Hayne et al. (2015), based on the sublimation rate at 110 K and consistent with observations made by Diviner and LAMP. Benna et al. (2019) derived similar values based on LADEE data for the combination of water released globally by meteoroid impacts and redeposited on the surface (up to 4 g/s) and water produced in situ by reaction with the solar wind hydrogen (2 g/s). Adding these values returns a surface area specific supply rate on the order of 10−16 kg/(m2 s), which is comparable with our implemented value.
For the initial conditions, we use the last temperature depth profile of a previous temperature initialization run with sufficiently long equilibration time. The initial water concentrations are defined as 
and 
. The latter was chosen small enough to be an insignificant contribution to the physical model (the equivalent gas pressure being on the order of 10−18 Pa, far below the exospheric pressure), but large enough to avoid numerical issues that could lead to negative concentrations.
3 Investigated Sites
We investigated three sites around −83° latitude in the south polar region of the Moon and three sites around −59° in the Clavius region, both depicted in Figure 1. The polar sites P1–P3 were chosen to investigate the pumping potential at sites with generally low temperatures, also partially shadowed, and close to permanently shadowed regions. The sites at Clavius, C1–C3, were chosen to evaluate the possibility of pumping at mid-latitudes and attempt to test a hypothesis for water observations in illuminated nonpolar areas (e.g., Honniball et al., 2020). All six sites have distinct thermal characteristics, representing different boundary conditions for thermal pumping. P1 is located on an equator-facing slope with a relatively high radiation input. P2 and P3 are partially shadowed with maximum diurnal surface temperature variation. They are also located close to larger permanently shadowed regions (PSR), indicated by the outlined areas in Figure 1 (Mazarico et al., 2011). C1 and C2 are located at opposite sides of Clavius D crater with drastically different solar radiation intensities. C3 is located in the flat terrain of the floor of Clavius crater, but due to the increased latitude, solar radiation comes at a high angle of incidence and thus is still moderate.
Investigated sites near the South Pole (P1–P3) and in the Clavius region (C1–C3). Areas with red boundary indicate PSR’s > 10 km2 (Mazarico et al., 2011). The underlying images were generated from LROC WAC mosaic using Lunar QuickMap.
The external thermal radiation on the lunar surface at these sites was calculated with the Oxford 3D thermophysical model (King et al., 2020) for a time period of one Earth year starting in November 2014. This model takes into account direct solar, albedo, and infrared radiation, as well as three-dimensional shadowing and scattering effects on the lunar surface and was validated with LRO Diviner temperature measurements. The diurnal variation of external radiation at these sites is depicted in Figure 2 for a period of 12 lunations (each with a duration of 2,544,601.639 s). Besides the general differences in radiation intensities and the visible seasonal variations, it is noteworthy that site P2 receives direct solar radiation only during five subsequent months of the year.
Absorbed surface heat fluxes over one lunar year at the investigated sites.
4 Temperature Initialization
To derive the initial temperature profile for the subsequent simulations of water migration, the subsurface thermal model was calculated over a duration of 100 years, or 1,200 lunations, with the repeating surface radiation input depicted in Figure 2. For this purpose, the model was solved without applying the mass transfer equations. Table 1 summarizes the final annual subsurface temperature envelope for all sites. After the simulated period, the subsurface temperatures have sufficiently equilibrated with variations of <0.2 K per 12 lunations.
| Site | Latitude (°) | Longitude (°) | Annual thermal skin depth (cm) | Surface temperature (K) | Column-integrated adsorbate concentrationa (mol/m2) | Adsorbate concentration deptha (cm) | ||||
|---|---|---|---|---|---|---|---|---|---|---|
| Min. | Mean | Max. | Range | Peak | Mean | |||||
| P1 | −83.4063 | 27.6563 | 2.0 | 63 | 160 | 312 | 249 | 4.9e-10 | 6.2 | 6.6 |
| P2 | −82.2813 | 23.9063 | 2.0 | 35 | 57 | 220 | 185 | 2.1e-8 | 0.6 | 0.6 |
| P3 | −82.3438 | 30.4688 | 1.8 | 48 | 93 | 273 | 224 | 7.3e-9 | 1.3 | 1.3 |
| C1 | −58.4688 | −12.4375 | 2.3 | 55 | 115 | 225 | 170 | 8.3e-9 | 1.4 | 1.4 |
| C2 | −59.1563 | −12.3125 | 1.9 | 71 | 191 | 372 | 301 | 1.8e-11 | 12.6 | 26.3 |
| C3 | −59.2188 | −15.4375 | 1.9 | 65 | 174 | 326 | 261 | 1.8e-10 | 8.7 | 10.0 |
- a After 300 lunations.
For a perfectly oscillating heat input, the thermal penetration depth or skin depth 
can be used to determine how far heat penetrates into the subsurface (e.g., Grott et al., 2007). However, because in reality the diurnal heat input underlies seasonal and local variations, this parameter changes over time and between different sites. We therefore calculate the annual thermal skin depth on the basis of a period of 12 lunations to account for the recurring pattern of heat input depicted in Figure 2. The values for the investigated sites range from 1.8 to 2.3 cm (Table 1). This is significantly less than the diurnal skin depth of 4.4 cm predicted by the model of Hayne et al. (2017), the 10 cm assumed as a simplified approximation by Schorghofer and Taylor (2007), or the 19 cm described as nominal value over a draconic year by Schorghofer and Williams (2020). However, these models assume thermal properties averaged over larger areas and define a generally higher thermal conductivity than our model with a significantly stronger temperature-dependency. As will be shown in the following, the thermal skin depth alone is not a suitable measure to assess the efficiency of thermal pumping, because this also depends on the absolute amplitude of the temperature change at a certain depth.
5 Water Migration
To investigate the migration of water in the lunar subsurface, we solved the heat and mass transfer equations with the COMSOL model over a duration of 300 lunations, or 25 years. This proved to be sufficiently long to observe long-term trends beyond the initialization phase. The typical migration process of water molecules we observe, common to all investigated scenarios, can be distinguished in the following two phases.
-
Increasing surface temperature (morning): Molecules desorb at the front of the penetrating heat wave, where the desorption energy is reached. They diffuse along the gradient of temperature and concentration and adsorb at the next available site with colder temperature. With the heat wave penetrating further into the subsurface, there is a net downward flux of molecules.
-
Decreasing surface temperature (night): While the temperature drops at the surface, causing the supply of water from the surface to cease, it is still high enough at depth to liberate molecules and drive the downward flux. Free molecules are therefore trapped in the subsurface and accumulate at the temperature minimum, until the temperature difference vanishes. Here, all free molecules finally adsorb and build a distinct peak in the adsorbate concentration.
5.1 Column-Integrated Water Concentration
The first metric to assess the pumping efficiency is the column-integrated adsorbate concentration in the subsurface. As Figure 3 exemplarily shows for site P1, the values vary strongly over the diurnal cycle, mainly due to the capture and release of high water concentrations at the surface. However, the local minima at daytime of the individual cycles increase over time, indicating a net accumulation of adsorbate. In the simulated time period the concentration growth exhibits a logarithmic character, with rapid increase directly after initialization, followed by a constantly decreasing growth rate. To compare the different results, we use the adsorbate concentration after 300 lunations, at the last local minimum before the end of the simulated period. This value ranges between 1.8 × 10−11 and 2.1 × 10−8 mol/m2 among all sites (Table 1), equal to a fraction between 10−7 and 10−4 of the total quantity of water supplied to the surface. To estimate the long-term column-integrated water concentration, we apply a logarithmic growth function to the local minima. Practically, the growth should be limited when saturation is reached. However, during the simulated period the accumulated concentrations are too far from this limit as to be constrained by it. Extrapolating the logarithmic curve fits to a duration of 1 Ga, the column-integrated adsorbate concentration would only reach values on the order of 10−10 to 10−7 mol/m2 among all sites. This equals a total water mass of some nano- to micrograms per m2, or a fraction of 10−12 to 10−9 of the water supplied to the surface over this period.
Column-integrated adsorbate concentration at site P1. The inset shows a detail view of the diurnal variations in concentration over the last simulated cycles.
5.2 Depth of Water Migration
Figure 4 shows the depth profile of the adsorbate concentration that emerges at the local maxima of the diurnal cycle, when the downward flux has ceased. The initial peak accumulates at lower depth and migrates deeper over the subsequent cycles, due to the seasonal change in heat input, which approaches a maximum at perihelion (compare Figure 2). With the surface temperatures decreasing toward the end of the annual cycle, the supplied water migrates less deep, building a second adsorbate peak at a slightly lower depth. In some cases, both peaks merge into a single broader one, but in others they remain separated. In case of site C2, the high temperature variation in the subsurface leads to an enhanced transport of water molecules down below 1 m depth, where they slowly adsorb (The simulated depth in the model was increased to 3 m in this case to avoid boundary effects). However, the quantity of adsorbate at this depth is extremely low.
Adsorbate concentration over depth during the last of 25 simulated lunar years. Gray curves show depth profiles at each diurnal maximum of cumulated subsurface water concentration, the last lunation is highlighted red.
The peak and mean concentration depth at the end of the simulated period are provided in Table 1. As the adsorbate concentration steadily grows, due to the constant supply of water from the surface, it can be expected that the concentration peaks keep accumulating water up to the saturation limit. The depth of water accumulation is therefore an important parameter for the assessment of the efficiency of pumping and the accessibility of subsurface water.
To estimate the long-term trend of the migration depth, we apply a logarithmic growth function to the peak and mean concentration depths over time with a least squares fit. In case of multiple peaks only the largest is taken into account. Although these positions should practically be limited in growth, the investigated timescales are much too small for the water concentration to approach monolayer coverage. In all investigated scenarios the adsorbate coverage at any depth and any time is less than a hundredth of a monolayer. The usage of the Langmuir adsorption model therefore does not constrain the growth of adsorbate and applying a limited growth fit therefore would not be useful for the simulated period. The growth rates we find with the curve fit are extremely small, so that even when extrapolating to a duration of 1 Ga, the average concentration depths for the investigated sites are only 27.8 cm (P1), 1.8 cm (P2), 3.4 cm (P3), 4.8 cm (C1), and 51.7 cm (C3). In case of site C2, the initially established peak remains practically stable on these timescales and only the adsorbate at greater depth continues to accumulate, but at insignificant quantities compared to the other sites.
. However, given the complexity of the underlying model, it is more practical to do this numerically, with the drawback that the entire computational model must be solved. Ideally, a practical measure for the adsorbate peak depth should be related to the surface temperatures, such as it is the case for the thermal skin depth. We therefore attempt to achieve an empirical fit for the peak depth, using the minimum, mean, and maximum values of the temperatures at the surface. They are related via the temperature ratio 
, which describes the relative position of the mean surface temperature to the respective surface temperature extremes (see Table 1):
(16)
strongly correlates with 
in an exponential form:
(17)Equation 17 provides an excellent fit over the entire simulated time period with a coefficient of determination of 0.99. However, as this is an empirical fit based on limited data, it remains to be validated for other scenarios.
5.3 Daytime Water Release
As shown above, thermal pumping enables the long-term accumulation of water and the build-up of adsorbate deposits that remain stable in an undisturbed subsurface. A secondary effect of thermal pumping is the temporary capture and release of water on a diurnal timescale. Trapped water that remains only at shallow depths during night can be released and escape from the surface at the subsequent day. To evaluate the potential for this “breathing” effect, we integrate the boundary daytime flux of water at the surface toward space over time (Figure 5). The results vary strongly depending on the respective thermal conditions and water concentrations present during each lunar day. The average daytime water release is on the order of 10−8 mol/m2 for all investigated sites, similar to the quantity of water supplied over the lunar night (7 × 10−8 mol/m2). On some days the release is even two orders of magnitude higher, meaning that there is a net loss of water to space. During most of the days however, the deposited water is not entirely lost and as shown above, this is sufficient to enable long-term trapping at larger depths. It is noteworthy that the median values are about one order of magnitude higher for the Clavius sites C1 and C3 than for all other sites. Here also the variance of values is lower, so that there is a more homogeneous pumping over the annual cycle.
Histogram for the amount of water released to the exosphere during the lunar day (illuminated period only, compare Figure 2). Median values for each site are provided in the legend.
Concluding on these results, it should in principle be possible to observe temporarily adsorbed water on the lunar surface even in illuminated areas, at least shortly after local sunrise. During 4–12% of the days at the investigated sites a cumulative water mass of >1 µg per m2 is released from the surface. This gives rise to the hypothesis that the release of temporarily trapped water could be one potential contributor to recent observations of surface and near-surface water in sunlit areas (Honniball et al., 2020; Pieters et al., 2009). A second mechanism that could lead to such observations is the release of shallow water accumulations via impacts. As shown above, such impacts would only have to disrupt the first upper centimeters of the surface to expose at least some of the water adsorbed therein.
6 Sensitivity Analysis
The desorption activation energy, gas diffusivity, and thermal diffusivity are key parameters for the heat and mass transfer (compare Equations 12 and 14), but only poorly constraint due little available data. To establish their influence on the results presented above, we performed an additional sensitivity analysis using the example of site P1.
For the desorption energy of water on lunar regolith only a few laboratory studies exist. Poston et al. (2013) for instance determined the desorption energy of water ice on the simulant JSC-1A and found values ranging from 0.45 eV for physisorbed water ice clusters to 1.2 eV for stronger chemisorption bonds. However, more than 90% of the desorption energies in this range were less than 0.9 eV. In another study for the desorption of water or hydroxyl from Apollo 17 sample 72501, desorption energies up to 1.5 eV were found with a peak at 0.7 eV (Poston et al., 2015). Recent studies on water desorption from Apollo samples 14163 and 10084 lead to similar values of 0.57–1.87 and 0.62–1.55 eV with a peak at 0.62 and 0.67 eV, respectively (Jones et al., 2020). Because our simulations presented above used relatively weak bonds with 0.5 eV, we also investigated how increased values of 0.6 and 0.7 eV for adsorbed water affect the results.
As the results in Table 2 show, a higher desorption energy leads to significantly higher accumulation of water, but at a lower depth. The desorption energy has an exponential influence on the desorption rate and governs the change in adsorbate concentration, see Equations 4 and 9. A higher desorption energy means the adsorbed water molecules are less volatile and thus accumulate more easily at shallower depths. While the overall feasibility of thermal pumping remains unchanged by the variation of desorption energy, it must be noted that for a quantitative analysis the actual bonding mechanisms need to be better understood. To resemble an even more realistic scenario, desorption energy should not only depend on the type of lunar minerals involved, but also the degree of surface coverage. In the scenarios investigated here the supplied water and accumulated adsorbate is always at least two orders of magnitude lower than required for monolayer coverage. The formation of multilayers or ice clusters, and thus a significant change in adsorption energy, therefore is not relevant for the investigated conditions.
| Desorption activation energy (eV) | Gas diffusivity factor | Thermal conductivity factor | Column-integrated adsorbate concentrationa (mol/m2) | Adsorbate peak concentration deptha (cm) |
|---|---|---|---|---|
| 0.5 | 1 | 1 | 4.9e-10 | 6.2 |
| 0.6 | 1 | 1 | 2.3e-09 | 3.2 |
| 0.7 | 1 | 1 | 6.0e-09 | 1.8 |
| 0.5 | 0.1 | 1 | 1.1e-10 | 4.6 |
| 0.5 | 10 | 1 | 1.7e-09 | 8.7 |
| 0.5 | 1 | 10 | 1.6e-10 | 12.6 |
| 0.5 | 1 | 100 | 3.9e-11 | 24.5 |
- a After 300 lunations.
The influence of gas diffusivity was investigated by scaling the value used for the previous simulations by a factor of 0.1 and 10. This range broadly covers the uncertainty in physical properties that are used to calculate diffusivity, such as the tortuosity or size of voids. A higher gas diffusivity leads to an improved mass transfer and hence a higher accumulation and deeper migration. This overall positive proportionality to diffusivity is in accordance with earlier studies by Schorghofer and Aharonson (2014).
Thermal diffusivity dominates how fast the heat wave penetrates into the subsurface and hence is another important influence factor for the efficiency of pumping. To evaluate its impact, we scaled the thermal conductivity, which is proportional to thermal diffusivity, by a factor of 10 and 100, representing soil that is either less porous and more dense or much more saturated. The temperature distribution for these scenarios at site P1 changes drastically: The minimum surface temperature increases from 63 to 89 K and 125 K respectively, and the annual thermal skin depth increases from 2.0 to 6.4 cm and 23.7 cm. The effect of an increased thermal conductivity on the accumulation of water is negative, however it is positive for the migration depth.
On the basis of the results summarized in Table 2, we can conclude that the sensitivity of the model on the three investigated parameters generally follows a power law. For the column-integrated adsorbate concentration the correlations are: 
. For the depth of the adsorbate concentration peak the relations are: 
. From these proportionalities, it is evident that desorption energy has the highest influence, but in reality might only vary in a small range. The other two parameters might have stronger deviations in a real lunar scenario, but their impacts compensate each other with respect to the quantity of accumulated adsorbate. However, it needs to be noted that they are not independent of each other, as a soil with high thermal diffusivity is also likely to have a low gas diffusivity, for instance because of a lower porosity.
7 Conclusions
The results of our simulations demonstrate that the rate and depth of water migration strongly depends on the dynamically changing temperature distribution. To enable thermal pumping, the temperature must be low enough at surface and depth during some period of the diurnal cycle to capture and hold volatile water molecules supplied from the lunar exosphere. It must also be sufficiently high to release molecules at the surface and trigger a downward flux during the rest of the diurnal period. Water will accumulate at the depth where the combination of temperature and time prevents further desorption and transport, that is, where the thermal energy input is lower than the desorption energy of the adsorbate.
These results are in principle consistent with previous work on lunar ice pumping (Schorghofer & Aharonson, 2014; Schorghofer & Taylor, 2007). However, as mentioned above, our model differs in some substantial points. The simulation results show that the thermal skin depth is not a reliable indicator for the efficiency of pumping, and that using time-averaged heat fluxes is not sufficient to reveal where pumping might take place in a more irregular manner. Our model enables a more nuanced analysis of the pumping dynamics compared to previous broader classification schemes, which solely take into account surface temperature dependencies. Applying such a scheme developed by Schorghofer and Aharonson (2014) to our six investigated sites on the Moon, we find that only one of them (P3) should enable “strong pumping.” The other sites should all exhibit a net loss of water or no pumping at all. While the trends qualitatively roughly agree with our results on migrated quantity and depth, the simplified classification fails to identify pumping to very shallow depths (P2) and greater depths but lower quantity at sites with relatively high heat input (P1, C1–C3). In contrast to previous work on lunar ice pumping, our model results show that adsorbed water can indeed accumulate even at sites with mean surface temperatures well above 110 K.
In general, it needs to be noted that when assuming a supply rate of 10−15 kg/(m2 s), a value considered reasonable based on previous studies, the quantities of subsurface water achievable even on geologic timescales are not high enough to constitute a relevant resource. For the polar sites, they only reach some µg m−2 for an extrapolation to 1 Ga. This might be different for a higher supply rate or a sudden deposition of higher amounts of water, for example due to an impact, both of which scenarios have not been investigated here.
Different to previous work, we present an analysis of the actual depth distribution of adsorbed water and find that the depth of water migration exponentially depends on the ratios of minimum, mean, and maximum annual surface temperatures, as described by Equation 17. Water accumulations outside permanently shadowed locations predominantly exist in the shallow subsurface, as the depth of water migration is less than or around 10 cm in most of the investigated scenarios after 25 simulated years and only up to around 50 cm for an extrapolation to 1 Ga. We also discuss the influence of main model parameters, such as desorption energy, gas diffusivity and thermal diffusivity, the first having the most significant influence on the quantity and depth of water migration.
In general, the results of our study agree with observations of the diurnal variation in water concentrations and an overall higher water content near the lunar poles compared to mid-latitudes (e.g., Li & Milliken, 2017; Sunshine et al., 2009). Local deviations in recent observations of the presence, depth and quantity of subsurface water might be due to differences in the supply rate, bonding mechanisms, and abundances of water, as well as local influences on subsurface temperature distribution (e.g. regolith composition or shadowing caused by a more rough surface). As it has been demonstrated here, the migration depth of water strongly depends on these parameters. Thus, it is reasonable that observation methods that are sensitive to different depths, such as near-infrared spectroscopy (few millimeters) and neutron spectroscopy (several tens of centimeters) detect different water quantities (e.g., Li et al., 2018; Pieters et al., 2009).
The presented simulations show that pumping is not only a relevant mechanism on larger timescales. Water that is trapped during night at shallow depth can be released from the surface during day, presenting a source of water in illuminated areas. Quantities of water released from the surface can reach several µg m−2 on some lunar days. The release of temporarily stored water could therefore not only be a possible explanation for recent observations made in the Clavius region (Honniball et al., 2020), but also a further relevant contribution to the global lunar water cycle. Temporary capture and storage at shallow depths can act as a buffer for volatile water migrating larger distances above the lunar surface.
In this study, we have focused on vertical transport of adsorbed water molecules, but the same conclusions are applicable to lateral transport in the subsurface. This is mainly of interest for very rough surface terrain, where local shadowing by boulders, craters or steep slopes creates lateral temperature gradients on smaller scale. As temporary shadows caused by terrain are more likely at higher latitudes, lateral pumping in the subsurface might be a relevant mechanism to support migration of water toward the poles. The findings are also relevant for the release of adsorbed water via probes, drills, and similar devices that will be used to explore and extract lunar resources. The main implication here being that a certain fraction of the subsurface water will follow the thermal gradient and move away from the heat source to get trapped in colder areas.
Acknowledgments
We acknowledge the use of imagery from Lunar QuickMap (https://quickmap.lroc.asu.edu), a collaboration between NASA, Arizona State University & Applied Coherent Technology Corp.
Open Research
Data Availability Statement
An online dataset for this research is publicly available at Mendeley Data under the reference Reiss, Philipp (2021), “Dynamics of subsurface migration of water on the Moon”, Mendeley Data, V1, https://doi.org/10.17632/99x7hhsr8g.1
