Introduction
Penitent ice structures, or ‘ice penitentes’ are needle-like structures formed by sublimation and melting of ice or snow due to solar irradiation. Documented by William Scoresby in the early 1800s (Jackson, Reference Jackson2009) and later by Charles Darwin (Darwin, Reference Darwin1889), they are named for their visual similarity to hats worn by penitent monks (Lliboutry, Reference Lliboutry1954). These structures are observed most often on terrestrial snowfields at high altitudes and low latitudes, most notably in the South-American Andes mountains. These locations offer a combination of low humidity, high solar irradiance, and moderate winds with clear weather for week- to month-long durations. All of these conditions are thought to be necessary for large penitente formation (Lliboutry, Reference Lliboutry1954; Betterton, Reference Betterton2001; Warren, Reference Warren2022).
Penitentes form via an ablative process dominated by sublimation at the early stages, and enhanced by melting in the troughs at later stages (Bergeron and others, Reference Bergeron, Berger and Betterton2006; Warren, Reference Warren2022). Incident sunlight warms the ice at or just below the surface, liberating water molecules and causing ablation. As described by Betterton (Reference Betterton2001) and Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015), the net sublimation is modulated by the interaction of incident solar flux, light and thermal diffusion in the bulk ice, and vapor diffusion above the ice surface.
On an undulating surface, self-illumination and surface scattering leads to higher incident heat fluxes at the troughs. This provides a positive feedback mechanism for surface feature growth by heating the troughs more than the peaks and causing locally faster sublimation. Internal diffusion of light and heat in the bulk ice also influence the surface sublimation rates. Ice with greater thermal diffusivity will more effectively carry heat away from the surface and into the bulk ice, thus diminishing the surface temperature rise and consequently decreasing the sublimation rate for a given incident solar flux. Photon penetration depth plays a role as to the vertical distribution of energy deposition in the bulk ice. As a function of wavelength, this can vary from microns (infrared) to several cm (visible, UV), and also varies with ice type.
Water vapor above the ice surface provides a negative feedback mechanism for local sublimation, as diffusion of water molecules away from the surface proceeds more slowly into regions of higher vapor density. In addition, regions of higher vapor density provide a source of molecules for re-deposition, thus lowering the net sublimation rate.
Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) developed a dispersion relation accounting for the relative influence of the above factors for the growth and stability of penitentes, though we note that the Claudin formulation may overestimate light penetration depth by at least an order of magnitude and thus energy deposition depth (Warren, Reference Warren2022). Nevertheless, for a given set of ice and atmospheric parameters, this relation estimates a dominant wavelength for surface undulations that will grow faster than others. This wavelength defines the spacing, and therefore the size, of penitentes.
Penitentes have also been artificially created in laboratory settings. Lliboutry (Reference Lliboutry1954) reports on early experiments performed by Troll (Reference Troll1942), where a patch of natural snow was illuminated with an incandescent light source during a cold night. More recently, Bergeron and others (Reference Bergeron, Berger and Betterton2006) conducted a series of experiments to form penitentes in a laboratory chamber at ambient pressure with controlled humidity. They observed that penitentes form at fine scales and coarsen with time, increasing their spacing until liquid water begins to pool in the troughs at which time coarsening ceases and they grow vertically but not horizontally. Those workers also found that changes in relative humidity up to 70% in the bulk air above the surface had little effect on the formation of penitentes. It should be noted here that they measured the bulk air, but not air in the boundary layer very close to the surface, which may have much higher water vapor content. They did find that penitentes only formed between $-10^\circ$
C and $-20^\circ$
C in their testbed, and attributed their absence at colder temperatures to the low saturation vapor pressure of water, leading to very high local relative humidity.
Bergeron and others (Reference Bergeron, Berger and Betterton2006) also observed that penitente formation ceased when they filtered out near-infrared wavelengths from their halogen lamp source. However, it is not clear whether the total flux reaching the ice surface remained constant during this experiment, as most of the light emitted by a halogen lamp is near-infrared. Figure 1 shows a plot of the spectral reflectance of snow (Berisford and others, Reference Berisford2018), indicating high reflectance at visible wavelengths and strong absorption at long wavelengths. In the near-infrared region between roughly 800–2600 nm, the reflectance is strongly dependent on grain size. This indicates a strong dependence in this region on surface morphology due to multiple scattering of photons in the medium (Warren, Reference Warren1982). For clear ice, visible photons penetrate deeply unless at grazing angles of incidence (Bolsenga, Reference Bolsenga1981; Williams and Ferrigno, Reference Williams and Ferrigno2012).
Reflectance spectra for snow and ice. The coarse and fine- grained snow data were obtained using an ASD Fieldspec 3 in a walk-in freezer, and these are relevant to pulverized or regolith-like surfaces of icy worlds. Glacial ice represents a medium-grained case with impurities, and clear lake ice represents a very large-grain case.
Considering potentially comparable conditions on other worlds, several groups have proposed the existence of penitentes on icy worlds in the solar system, including Europa, Pluto, and Mars (Svitek and Murray, Reference Svitek and Murray1988; Moores and others, Reference Moores, Smith, Toigo and Guzewich2017; Hobley and others, Reference Hobley, Moore, Howard and Umurhan2018; Nguyen and others, Reference Nguyen, Smith, Innanen and Moores2019). Some evidence exists to explain the origin of the bladed terrain on Pluto as long-life penitentes formed in nitrogen-based ices (Moores and others, Reference Moores, Smith, Toigo and Guzewich2017), although their low aspect ratio morphology resembles rolling hills more than sharp penitentes. The atmosphere on Pluto consists mainly of nitrogen with pressure and temperature around 1 Pa and 45 K, yielding mean free paths for molecular collisions of order 1 mm (Gladstone and others, Reference Gladstone2016). This puts Pluto in the regime of continuum fluid mechanics, and the same is true for Mars (Hourdin and others, Reference Hourdin, Le Van, Forget and Talagrand1993). For these worlds, the relations used by Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) remain valid for molecular diffusion and fluid boundary layer theory. Europa, however, is definitively free molecular flow, with mean free paths of order 105 m at the equator (Spencer and others, Reference Spencer, Tamppari, Martin and Travis1999; Murphy and Koop, Reference Murphy and Koop2005). In this regime, the equations for diffusion and boundary layer theory break down, as sublimated water molecules leave the area on ballistic free trajectories. The presence of periodic maxima in vapor density vanish, and with it the dominant wavelength for penitente size distribution. This suggests that they do not form on airless worlds (Hand and others, Reference Hand2019). However, with no observation to confirm or deny their existence, we must attempt to bound their formation conditions using experiments and modeling efforts (Carreon and others, Reference Carreon2023; Macias and others, Reference Macias2023; Macias Canizares and others, Reference Macias Canizares2024).
To date, however, no one has experimentally tested whether penitentes can form at vacuum conditions where a liquid phase of water definitively does not exist. These conditions pertain to icy worlds in the solar system. To address this, we have constructed the Europa Penitent Ice Experiment (EPIX) (Berisford and others, Reference Berisford2018, Reference Berisford2021; Macias and others, Reference Macias2023), to investigate penitente formation at vacuum conditions with ice temperatures and pressures ranging from Earth-like to Europa-like. The 2021 reference describes tests performed at free molecular flow conditions and cryogenic temperatures. Here, we present an initial set of results from the static EPIX testbed at moderate temperatures ($-20^\circ$
C to $-2^\circ$
C) and rough vacuum levels (50–2600 Pa). This range keeps molecular mean free paths short enough to remain in the continuum flow regime in order to apply the relations of Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015), but ensures that no liquid phase of water exists in the chamber, thus all phase changes are sublimation or deposition.
Experimental setup
Berisford and others (Reference Berisford2018) describe the test setup in detail, but we will give an abbreviated description here. The testbed consists of a 0.3 m diameter cylindrical aluminum vacuum chamber with a liquid cooling jacket, as shown in Fig. 2. An external lamp illuminates the ice surface through a quartz window in the chamber lid, and a cold aperture limits the light field of view to the ice surface in order to eliminate direct irradiance of the chamber walls. This aluminum coldstop bolts to an aluminum ring welded to the inner chamber surface near the top of the cooling fluid jacket. The top surface of the aperture is covered with aluminized Mylar Single Layer Insulation (SLI) with approximately 95% effective reflectivity to minimize radiant heat transfer from the lamp to the coldstop. The lower surface is painted black with Avian Black high emissivity paint to present a radiative cold sink to the ice, and to minimize internal reflections inside the chamber. The chamber walls are anodized black for the same purpose, as well as to prevent corrosion to the aluminum surface. The aperture plate contains several small holes to allow instruments and camera access to the ice.
Stockpot schematic diagram and photographs. Motion feedthroughs provide vertical and rotary motion for pressure sensor and cameras, as well as a spring-loaded vertically compliant thermocouple array that can be temporarily moved into position to measure radial surface temperature profiles. Embedded thermocouples are frozen into the ice at 10 mm vertical spacing. Chamber is 0.3 m inner diameter (Berisford and others, Reference Berisford2018).
A recirculating chiller pumps heat transfer fluid (Galden HT-135) through the cooling jacket and cools it to a controllable setpoint temperature between 0$^\circ$
C and $-75^\circ$
C, with accuracy of ±1$^\circ$
C. The chamber fluid jacket contains two baffles to help avoid stagnant areas of heat transfer fluid, as shown in Fig. 2. Fluid enters via a single port at the bottom center and exits through dual ports on opposite sides of the chamber near the top for symmetric flow. This configuration provides a nearly isothermal wall temperature throughout the chamber below the aperture.
The chamber is equipped with instrumentation necessary for monitoring the conditions of the ice sample. The bulk pressure is monitored by two capacitance diaphragm vacuum gauges, with one ranged for 0.5 to 1300 Pa and another for 700 to 130 000 Pa. Two unmodified COTS webcams are mounted inside the chamber to capture time lapse and live observations of the changing ice. The primary camera mounts rigidly to the aperture mounting ring to take time-lapse photography from a static position. The secondary camera mounts to a movable post allowing vertical translation and horizontal rotation, to allow varied perspectives to view the ice surface. The internal ice temperatures are monitored via a near-centrally located vertically spaced array of type T thermocouples, including one just below the ice surface (Fig. 2). The ice surface temperature is periodically measured using a spring loaded thermocouple array. This array is able to pivot horizontally and translate vertically, allowing for a radially spaced surface temperature measurement. When this array is not in use it is rotated out of the light cone to prevent shading. The vacuum source is a hybrid rotary vane and diaphragm pump, capable of handling condensable vapors. For experiments with pressure targets above the sample vapor pressure, dry N2 gas is introduced to the chamber through a needle valve.
Ice samples may be frozen in place from liquid water or created elsewhere and transferred to the chamber. In the latter case, we pre-chill the chamber to roughly −2$^\circ$
C to minimize melting, and purge with dry nitrogen while open to minimize frost buildup.
Some tests include backfilled N2 gas at a controlled leak rate to control total background pressure. Other tests use vacuum pumping only, thereby reaching background pressure very close to the water vapor saturation pressure at the ice surface temperature. It is not feasible to operate at pressures significantly below the saturation curve due to vacuum pumping speed limitations. Gross water vapor flux from sublimation of the ice surface at these temperatures is much greater than maximum pumping speed out of the chamber. Therefore, in the absence of additional N2 gas injection, the chamber pressure settles at a value close to the equilibrium vapor pressure. In this case, much re-deposition of water molecules onto the ice surface occurs as the surface sublimates into a head space very close to 100% relative humidity.
Experimental uncertainties and sources of error
Temperature measurement
Ice surface temperature measurements are inherently challenging, and vapor pressure (and hence, penitente spacing) is strongly sensitive to absolute surface temperature. Thermocouples have manufacturer-quoted uncertainty of ±1$^\circ$
C, but this is mostly a bias error. After calibration against a temperature-sensitive diode in an isothermal water bath, we achieve ±0.2$^\circ$
C for all thermocouples in the test. A more difficult source of error arises from absorption of incident light directly onto a thermocouple. For the surface temperature thermocouple embedded just beneath the ice surface, this effect is most severe, potentially leading to erroneously high temperature measurement. To estimate this effect, consider a cylindrical thermocouple 1 mm diameter × 3 mm long, in perfect contact with ice on all sides, illuminated from one side. This geometry is a close approximation of the exposed (uninsulated) portion of the thermocouples used in our tests. We assume a characteristic length for thermal conduction loss to the ice equal to the length of 3 mm, conducting from the cylindrical surface and tip. An incident light flux of 1600 W m−2 absorbs into the cross sectional area of 3 mm2. Heat loss through the ice via the steady-state Fourier's law of conduction yields a temperature offset between thermocouple and surrounding ice of 0.6$^\circ$
C. We therefore have subtracted this offset from all surface temperature values, and included this amount in the error bars in Fig. 4.
We justify the use of a 3 mm characteristic length for thermal conduction based on the time constant for cooling of the thermocouple when the lamp is turned off. Beginning with steady-state temperature T 1, we switch off the lamp and record T as it cools exponentially to a new steady state T 2. The time constant, τ = 10 s from experimental data, is the time elapsed for the temperature to drop by (1/e)(T 1 - T 2). Using the equation for thermal diffusion time constant, τ = L 2/α, where α is thermal diffusivity α = k/(ρCp). For ice with k = 2 W m−1 K−1, ρ = 917 kg m−3, and Cp = 2030 J kg−1 K−1, we obtain L = 3 mm.
The freestream vapor temperature is not directly measured in our experiments. As an estimate, we use the mean value between chamber wall temperature and lid temperature. However, we cannot account for direct heating due to absorption of incident light by vapor molecules. For future tests, a free-hanging thermocouple in the chamber head space, located just outside of the light cone may be able to provide this data.
Diffusion coefficient
In the subsequent analysis, we assume a constant diffusion coefficient, D, across the boundary layer as used by Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015), who use D = 3 × 10−5 m2 s−1 for diffusion of water vapor into air. This assumption of constant D is valid for the outdoor Earth case, were the local total pressure and hence total gas density is approximately constant across the flowing boundary layer. In the case of our experiments with H2O only with no additional background gas, D can vary slightly as the vapor pressure changes. For example, in Test 35 described below, we measured an ice surface temperature of −12.4$^\circ$
C and we measured a background pressure of 198.92 Pa. We assume that the local pressure at the ice surface is equal to the equilibrium saturation water vapor pressure of 208.12 Pa. Across this range, the diffusivity varies from roughly 7.0 × 10−3 m2 s−1 to 7.4 × 10−3. Therefore we expect to see a nonlinear variation in vapor density across the boundary layer. As an approximation, we use an average constant value for D = 7.2 × 10−3 m2 s−1 and assume a linear profile. This is roughly two orders of magnitude greater than that used by Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) due to the enhanced molecular mobility at the lower pressures used in our tests. Moores and others (Reference Moores, Smith, Toigo and Guzewich2017) use values two orders of magnitude greater, corresponging to the even lower pressures (roughly 1 Pa) and higher molecular mobility on Pluto.
Water is a polar molecule, and therefore may deviate from Chapman-Enskog theory (Poling and others, Reference Poling, Prausnitz and O'connell2001) by 15% or more (Marrero and Mason, Reference Marrero and Mason1972). This error propagates linearly into the calculation for diffusion boundary layer thickness, and may yield a similar scaling uncertainty on the steady-state spacing for penitente formation.
Lamp spectrum and light penetration depth
Most of our experiments use an ice thickness depth of 8 cm to 10 cm. Figure 3 shows the estimated lamp spectrum reaching the ice and solar blackbody spectrum. These numbers are based on lamp and window manufacturer data and account for spectral transmission losses through the window. At the wavelength of predicted peak intensity reaching the ice (~900 nm), the $1/\emph {e}$
penetration depth is approximately 4 cm. This is calculated using data from Warren and Brandt (Reference Warren and Brandt2008), compiled by the University of Waterloo (https://www.npsg.uwaterloo.ca/data/water.php). For shorter wavelengths, the penetration depth becomes longer than the ice thickness. In this case, many of the photons in the visible portion of the spectrum reach the chamber floor and their heating contribution is effectively ‘shorted out’ to the chamber floor and chiller fluid. These photons represent approximately 14% of the total incident flux.
Left Axis: Estimated light spectrum reaching the ice surface in the experimental setup, using manufacturer lamp data and accounting for window transmission. Solar blackbody spectrum shown for reference. Right Axis: Theoretical penetration depth for 1/e attenuation as a function of wavelength for clear ice, based on data from Warren and Brandt (Reference Warren and Brandt2008).
The halogen lamp used for these tests is biased toward infrared wavelengths, compared to the sun. This leads to a shorter mean photon penetration depth, which has the effect of shortening the predicted penitente spacing for the H2O-only case, and lengthening it for the H2O into N2 case. However, increasing or decreasing the penetration depth by one order of magnitude changes the predicted spacing by a maximum of 40%, thus the model shows relatively low sensitivity to this term. More important is the effect of ‘thermally shorting’ shorter wavelengths to the chamber floor if they penetrate more deeply than the ice thickness. In other words, photons that pass through the ice deposit their energy to the chamber floor instead of the ice. This effect is unknown, and could be a notable difference between the experimental apparatus and a semi-infinite planetary surface. The addition of fractures, bubbles, snow grains, or other internal scattering surfaces will shorten the penetration depth in natural applications.
Mass flux
We estimate mass flux of sublimated water vapor from the surface by computing the volume of the void left in the ice by sublimation loss, assuming a density of 917 kg m−3, and dividing by the measurement elapsed time. To obtain the volume of lost ice, we compared two different methods. In the first method, after opening the chamber post-test, we placed a ruler across the top of the depression and used depth-measurement calipers to measure excavated depth between the ruler and ice surface at 1 cm horizontal increments from the depression center to the edge. In tests where penitentes appeared, we measured to the trough surface. Assuming axial symmetry, we calculated the volume of the cylindrical shell at each radial location and summed for a rough integration.
The second method involved a similar calculation, but obtaining the depression depth profile using manual image analysis. This involves simple pixel counting to measure depths/distances compared to the scale ruler embedded in the ice and visible in the image, and correcting for camera geometry based on field of view and distances taken from the CAD model. This enables measurements during the test without opening the chamber. Both methods indicate a nearly parabolic-shaped depression, and assuming this we can estimate the volume by measuring only the center depth and depression diameter and integrating a paraboloid shape. Both measurement methods and the integration method agree within 15% for the first five tests, after which we used only the latter method.
Ridge spacing
At the end of each experiment, we attempt to measure spacing (wavelength) between any ridges formed during the test. As will be seen in the results section, this can vary spatially. For the subsequent analysis, we consider only the central portion and therefore restrict our measurements to within a 5 cm diameter central area. Post-test, we use calipers to manually measure spacing between at least 5 sets of ridges and take the average. Alternatively, we can measure spacing by image pixel counting, similar to the technique described above. Either technique results in an uncertainty dominated by the spatial variability between features of up to 30%, likely due to imperfections in the ice created during the freezing process. Early experiments were repeated at least twice to examine repeatability, which showed that test-to-test variations were small compared to the local spatial variations mentioned above.
Results and analysis
Figure 4 shows the equilibrium water vapor pressure vs. temperature curve, with overlaid symbols corresponding to the conditions of individual experiments. The different symbols correspond to visual results in two distinct categories: no morphology change (i.e., smooth excavation by sublimation), and morphology change with regularly patterned textures (i.e., micro penitentes). Figure 5 shows photographs of representative post-experiment morphology for these categories. Broadly, we find that penitentes form at conditions close to the vapor pressure curve, and they tend to vanish as background pressures increase.
Experimental data points overlaid onto the water ice vapor pressure saturation curve from Murphy and Koop (Reference Murphy and Koop2005). All tests where we see penitente-like structures form are at pressures below the triple point, and are at conditions very close to the saturation curve.
Ice surface before (top) and after experiments showing formation of micropenitentes at low pressures close to the saturation curve (middle), and smooth excavation by sublimation at higher pressures (bottom). The white spots visible in the top image are the result of bubbles leaving the surface during the simultaneous degas/freezing process; we do not see any evidence of penitentes forming preferentially at the bubble locations. We used fixed camera exposure settings, often resulting in poor contrast for images of tests in which the ice albedo decreased, as is the case of the bottom image. This image has been artificially increased in brightness for clarity. The top two are unchanged.
It is unclear whether the structures form purely as an ablative process, or if some redeposition occurs to build up the tips. Overall, we see a reduction in the ice surface height, as the penitentes form concurrently with a bulk depression of the illuminated area. This indicates net material loss from the area, which would imply ablation only. However, we also observe fine-scale ridging in the structures, as shown in Fig. 6. This ridging is typical of frost crystals, as shown in Fig. 51 of LaChapelle (Reference LaChapelle1969). This may be be an indication of deposition and crystal growth, but this will require more refined experiments to determine. Modeling efforts by Macias and others (Reference Macias2023) suggest that redeposition at the tips can occur, although that work assumed free molecular conditions.
Closeup photograph of post-test ice surface showing mm-scale ridges on the penitente structures. The cause of these striations is unclear, but they could be the result of water vapor re-deposition.
Two representative tests of note are summarized in Table 1. These tests were performed at similar conditions with the exception of added background nitrogen gas injected via a leak valve through a sintered diffusion stone near the chamber bottom edge. The addition of the gas suppressed penitente formation, likely due to the difference in diffusion boundary layer thickness, ℓ, which we estimate below.
Test parameters for experiments discussed in detail
.
We note here the distinction between different types of fluid boundary layers. The boundary layer definition used by Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) as a strong parameter for determining penitente spacing is that of a species diffusion boundary layer. This is the distance from the surface at which the water vapor density approximately equals that of the freestream. It is distinct from the momentum-transfer (Blasius) boundary layer often used in aerodynamics. Moores and others (Reference Moores, Smith, Toigo and Guzewich2017) and Nguyen and others (Reference Nguyen, Smith, Innanen and Moores2019) define ℓ as the laminar (viscous) sublayer thickness of a turbulent flow momentum-transfer boundary layer (White, Reference White1991). For a lateral bulk turbulent flow over a surface, as is the case for a planetary wind blowing across a snowfield, turbulent mixing occurs above this layer. Thus, it is reasonable to assume that in the turbulent part of the boundary layer, the vapor density of sublimating species from the surface will be effectively mixed, and the entire turbulent region will have vapor density nearly equal to the freestream value. Hence the viscous sublayer is the only region that can sustain significant vapor density gradients, and is therefore approximately equal to the species diffusion boundary layer thickness, ℓ.
In our experimental setup, however, we have no bulk lateral flow across the surface. We cannot, therefore, use momentum boundary layer relations to estimate our diffusion boundary layer thickness. We instead use Fick's law of diffusion for a one-dimensional water vapor concentration gradient:
where J is the diffusive water vapor mass flux, D is the diffusion coefficient, ρ is the water vapor density, and z is distance from the surface. Assuming constant D gives a linear concentration gradient, and we assume that the vapor density at the ice surface is equal to the equilibrium saturation density at the measured ice surface temperature:
which yields by rearranging:
where ρ o is the freestream vapor density. For the experimental case with no added background gas, we estimate ρ o using the ideal gas relation at the measured chamber lid pressure and measured wall temperature (T wall):
where P o is the freestream (chamber lid) pressure, and $R_{H_2O}$
is the specific gas constant.
We estimate mass flux, J from the volume of the excavated crater as described in the previous section, and convert this into mass flux assuming an ice density of 917 kg m−3, and dividing by the time elapsed before the measurement.
We estimate the diffusion coefficient, D, from Chapman-Enskog theory with Lennard-Jones cross sections (Poling and others, Reference Poling, Prausnitz and O'connell2001). For the case where H2O is the only species, we use the same species for both terms in the equations. This is a rough estimate for self-diffusion, specifically for the case of water because it is a polar molecule and may not behave exactly according to the Chapman-Enskog relations or ideal gas law. Using these values for our experiment, Test 35, yields the numbers in Table 1.
For the case where we inject a small amount of nitrogen gas into the chamber, we use a slightly different method. The diffusion coefficient again comes from Chapman-Enskog theory, this time using tabulated values for H2O diffusing into N2 (Poling and others, Reference Poling, Prausnitz and O'connell2001). To estimate ρ o, we begin with the manufacturer-specified pumping speed for the system pump ($\dot {V}_{pump}$
) of 5.5 m3 h−1, including line conductance. This represents the total volumetric flow rate of N2 + H2O out of the system. This may be a slight underestimation of H2O out and overestimation of N2, because it does not account for cryopumping to the chamber walls. However, we believe this effect to be small due to minimal visible frost buildup compared to the relatively large crater loss volume. We again estimate the volumetric flow rate of H2O from the size of the excavated depression in the ice. The difference gives an approximate value of the N2 flowrate ($\dot {V}_{N_2}$
). If we assume that the partial pressures of the two species in the freestream are proportional to these flowrates, then we can estimate the H2O partial pressure and vapor density as:
Assuming again that the vapor density at the ice surface is at saturation for the surface temperature, we can now estimate the diffusion boundary layer thickness:
Table 1 includes these values for Test 36 with injected nitrogen, and Fig. 7 shows a plot of normalized growth rate vs trough spacing for the two experiments along with earth-like conditions for penitentes observed in nature, following the method of Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) and Moores and others (Reference Moores, Smith, Toigo and Guzewich2017). This shows micropenitentes as the dominant size forming in the H2O-only case, and penitentes larger than the scale of the chamber for the nitrogen-added case.
Predicted growth rate vs size for two experimental conditions and Earth-like conditions, following the methods of Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) and Moores and others (Reference Moores, Smith, Toigo and Guzewich2017). H2O only with no background gas at −13.1$^\circ$
C, 198.9 Pa, ℓ = 0.7 mm. N2 background gas at −16.0$^\circ$
C, 253.3 Pa, ℓ = 19.4 mm. Earth-like snowfield at 1 atmosphere, ℓ = 100 mm shown for reference. The depression edge curve represents the outer portion of depression formed in the H2O-only test where smaller structures are observed (see Discussion and Fig. 8).
Post-test photograph showing decreasing feature spacing with increased radial distance from the center. The incident light flux decreases radially outward from the center, thereby decreasing surface temperature and sublimation flux. The changing values of D, ρ sat, and J all drive ℓ to smaller values (Eqn (3)). White powder has been added post-test to enhance contrast for the photograph.
Discussion
As pressure conditions change from atmospheric towards higher vacuum, the collisional mean free path of a given molecule increases, which increases molecular mobility of water molecules and therefore enhances diffusivity of water vapor away from the surface. This enhanced mobility (increased D and decreased density gradient) shortens the height of the diffusion boundary layer, as defined by Eqn (3) above. A shorter boundary layer can sustain only shorter wavelength vapor density gradients, and is hence the strongest influence on dominant instability wavelength in the Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) analysis. Therefore, as we decrease chamber pressure, we should expect to see penitentes decrease in amplitude and wavelength.
Bergeron and others (Reference Bergeron, Berger and Betterton2006) observed cm-scale penitentes form in a chamber of similar size. However, their test was performed at atmospheric pressure (2 orders of magnitude lower D), with a small lateral airflow across the surface. This airflow generates a Blasius boundary layer, and the viscous sublayer definition for ℓ applies, in this case defined by surface roughness. Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) analyzed this case and found an approximate value for ℓ = 0.1 mm, corresponding to λ = 1 cm. Thus, the Bergeron results and ours are consistent with the theory.
The representative tests shown in detail here with and without nitrogen background gas injection show dramatically different values for the calculated diffusion boundary layer thicknesses and therefore penitente spacing. The model predicts nearly two orders of magnitude larger spacing for the case with injected background gas. Despite the slightly reduced vapor diffusivity for the N2 case, the larger value of ℓ is driven by the much larger difference in $\rho _{sat} - \rho _{o, _{H_2O}}$
. This is likely due to mixing of the two gases, resulting in lower water partial pressure and density.
In many of our experiments, the size of the chamber imposes an upper limit on the size of the possible penitentes. For example, Test 36 with injected N2 gas described above should show a dominant wavelength that is larger than the chamber diameter, and thus we see only smooth excavation instead of penitentes. For intermediate wavelengths of order the size of the chamber, it is likely that edge effects disrupt the formation of the regular instability in the boundary layer and again result in mostly smooth excavation. Testing of the theory to larger wavelengths and therefore larger penitentes requires a larger experimental apparatus, or a collimated light source through a much larger opening to make a more uniform ablation crater area.
The earlier subsection ‘Lamp spectrum and light penetration depth’ mentioned that a 10x change in the light penetration depth term resulted in at most 0.4x change in the predicted penitente spacing. This low sensitivity may help explain the relative accuracy of the model by Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) for predicting the wavelength of terrestrial and Plutonian ridges, despite the penetration depth estimation error pointed out by Warren (Reference Warren2022).
The triple point pressure of water is 612 Pa, marked on Fig. 4. All of our tests that show structure formation fall below this pressure and are therefore in a regime such that no liquid phase of water should exist. Without mm-scale local pressure measurements near the surface, we cannot say for certain that no small pockets of higher local vapor pressure exist e.g. in valleys between penitentes. Sufficiently high local pressures in such areas could lead to melting in addition to sublimation. However the scale of the features and the high diffusion coefficients in this pressure regime make this qualitatively unlikely. We therefore infer that all of the structures observed in the tests are due to solid-vapor phase changes alone.
In these tests, the lamp irradiance is not uniform, delivering higher heat flux to the center of the chamber and weakening radially outward. This creates the dish-shaped sublimation crater that we observe in all tests, with or without penitente formation. This indicates a higher mass flux leaving the surface near the center of the crater and therefore a higher surface temperature at the center. As we move radially outward the changing values of D, ρ sat, and J all drive ℓ to smaller values (Eqn (3)). Figure 7 shows normalized penitente growth rates, including the edge case as the dotted line, assuming half the mass flux and 0.1$^\circ$
C temperature decrease compared to the center. Figure 8 is a post-test photograph, showing larger (approximately 8 mm spacing) structures at the crater center, decreasing in size radially outward, which confirms this trend and strengthens the theory.
In all of our experiments at temperatures above approximately −80$^\circ$
C, the minimum chamber pressure is limited approximately to the vapor pressure associated with the ice surface temperature. This is due to the rapid sublimation rate possible at non-equilibrium conditions at these warmer temperatures, as shown in Fig. 9, which shows gross sublimation rates for a surface sublimating into perfect vacuum. To accommodate net sublimation rates similar to these gross values, we would need prohibitively large pumps and the entire ice sample would sublimate away in a matter of minutes. We therefore have no experimental data points at pressures well below the saturation vapor pressure curve at these temperatures.
Gross sublimation rate into vacuum vs. ice surface temperature (Murphy and Koop, Reference Murphy and Koop2005; Andreas, Reference Andreas2007).
These tests show formation of micro penitentes in rough agreement with size predictions of the Claudin and others (Reference Claudin, Jarry, Vignoles, Plapp and Andreotti2015) model for predicting penitente sizing in a partial vacuum environment. This regime represents conditions intermediate between Earth and hard-vacuum worlds such as Europa and Enceladus, and are representative of the fluid regimes on Mars and Pluto. Ongoing work pushes these experiments to cryogenic temperatures and free-molecular flow pressures relevant to other worlds.
Acknowledgements
This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). We wish to acknowledge the support of the Europa Lander Mission concept pre-project and NASA grants 80NSSC22K1312 and NNH21ZDA001N-PSTAR.
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