Gas flow in Martian spider formation

Martian araneiform terrain, located in the Southern polar regions, consists of features with central pits and radial troughs which are thought to be associated with the solid state greenhouse effect under a CO$_{2}$ ice sheet. Sublimation at the base of this ice leads to gas buildup, fracturing of the ice and the flow of gas and entrained regolith out of vents and onto the surface. There are two possible pathways for the gas: through the gap between the ice slab and the underlying regolith, as proposed by Kieffer et al (2007), or through the pores of a permeable regolith layer, which would imply that regolith properties can control the spacing between adjacent spiders, as suggested by Hao et al. We test this hypothesis quantitatively in order to place constraints on the regolith properties. Based on previously estimated flow rates and thermophysical arguments, we suggest that there is insufficient depth of porous regolith to support the full gas flow through the regolith. By contrast, free gas flow through a regolith--ice gap is capable of supplying the likely flow rates for gap sizes on the order of a centimetre. This size of gap can be opened in the centre of a spider feature by gas pressure bending the overlying ice slab upwards, or by levitating it entirely as suggested in the original Kieffer et al (2007) model. Our calculations therefore support at least some of the gas flowing through a gap opened between the regolith and ice. Regolith properties most likely still play a role in the evolution of spider morphology, by regolith cohesion controlling the erosion of the central pit and troughs, for example.


Introduction
Araneiform terrain consists of groups of so-called spiders; radial, dendritic arrangements of troughs connecting to central pits, found at southern high latitudes of Mars (Kieffer, 2000). Their occurence was thought to be limited to the South Polar Layered Deposits (SPLD). Spiders were also discovered outside the area of the SPLD, although the extent of the SPLD was subsequently re-defined. Tanaka et al. (2014) or Schwamb et al. (2018) provide more insight the geography of spiders and the SPLD. These features morphology was observed by Hao et al. (2020). These recent results highlight that the potential effectiveness of gas flow through the regolith itself -rather than through a regolith-ice gap -has to be addressed because it seems vital for our understanding of spiders.
We approach the problem by providing suitable limits on the effectiveness of the underground flow of gas, based on constraints on regolith properties such as porosity/permeability combined with some known gas physics. Regolith porosity is a function of depth and therefore our treatment of the problem has to involve some stratigraphic considerations. We derive the depth of porous regolith needed to support the magnitudes of flow rate previously calculated in hydrodynamics simulations (Thomas et al., 2011a,b) and energy balance models (Kieffer, 2007;Pilorget et al., 2011;Pilorget and Forget, 2016). We then compare this to the required gap size to support laminar free-flow between the ice and regolith, and to the expected bending of the ice sheet in response to the rise in gas pressure beneath. We present the calculations in Section 2, before discussing the results and their implications in Section 3, and concluding in Section 4.

Modelling
The most common model for spider formation is illustrated in Figure 1; CO 2 slab ice condenses on-top of regolith over the Martian winter. When insolation begins to penetrate this transparent ice layer in spring, CO 2 ice at the base of the slab and top of the regolith starts to sublimate, raising the gas pressure to the point where it exceeds the strength of the ice slab and fracturing occurs. In the model of Kieffer (2007), this gas pressure levitates the slab above the regolith surface by a distance of d gap , estimated at around one or two centimetres. Gas can also diffuse into the permeable part of the regolith (Hao et al., 2019), which extends down to some depth d (also on the order of centimetres (Kieffer, 2007;Portyankina et al., 2010)), below which the regolith is made impermeable by water ice filling the pore spaces. See Mellon et al. (2004) for a detailed discussion of the geography of ground ice on Mars. Irrespective of whether the gas flows above or through the regolith to begin with, once fracturing has opened a vent to the surface, a pressure gradient across the radius of the spider drives gas flow to it, either as free-flow through the gap or viscous flow through the porous regolith. In the latter case, flow is limited by the pressure gradient and the regolith permeability, and we estimate these below.

Gas pressure
The sublimation rate at the bottom of the ice slab controls both the pressure buildup and the flow rate into the spider vent. Upon initial fracturing there may be a very energetic flow, driven by the large initial pressure gradient, followed by a reduction to a steady state balance between gas production and ejection. Both these cases were simulated with a commercial hydrodynamics code in Thomas et al. (2011a) and Thomas et al. (2011b).
The gas production rate is estimated by comparing the amount of energy deposited by insolation at the ice-regolith boundary with the latent heat of sublimation of CO 2 . Kieffer (2007) assumed a perfectly transparent ice slab and calculate an area gas production rate ofṁ = 1.157 × 10 −4 kg m −2 s −1 . Thomas et al. (2011b) meanwhile assume 25% of the z d d gap P P surf r Q Figure 1: Conceptual model for spider formation based on a layer of transparent CO 2 ice of thickness z , overlaying permeable regolith of thickness d, above an impermeable ice-rich regolith layer shaded dark. Gas pressure in the regolith increases from near-atmospheric pressure around the spider vent to a maximum value at the edge of the spider. Some models, such as those by Hao et al. (2019) and Pilorget et al. (2011) assume no gap between ice and regolith, d gap → 0. See section 2.1 for a detailed description of parameters used. solar energy penetrates a z = 20 cm thick slab (based on Pilorget et al. (2011)) and arrive atṁ = 2.21 × 10 −5 kg m −2 s −1 .
This constant gas production during spring-time daylight hours cannot escape and swiftly fills up all available sub-ice volume. Gas pressure will at this point be equal to the cryostatic pressure of P = P surf + ρ ice gz, where g = 3.72 m s −2 is the Martian gravity and ρ ice = 1606 kg m −3 is the solid CO 2 ice density. Taking z as 0.7 m, as in Pilorget et al. (2011), and P surf ≈ 600 Pa, results in sub-ice pressures of P cryo ≈ 4700 Pa. This pressure will continue building until the stress in the ice slab above exceeds its rupture strength and it fractures. During the initial rupture, Thomas et al. (2011a) simulated a total flow rate of Q i = 0.05 − 0.5 kg s −1 draining a 400 square metre area into a central vent. Subsequently, ejection of gas reduces the pressure gradient and, in a steady state, the 400 square metre area will produce a total flow some ∼ 5 times smaller: Q s = 0.00884 kg s −1 forṁ = 2.21 × 10 −5 kg m −2 s −1 . Hydrodynamics simulations of these magnitudes of flow rate produce gas and dust ejection velocities of ∼ 10 ms −1 , consistent with the erosion of regolith and the spread of fine material in fans on the surface (Thomas et al., 2011a,b).
A gas production area of 400 square metres corresponds to a circular gas production radius of only r P = 11.3 m, somewhat smaller than the 55 m separation (spider radius R ∼ 25 m) between densely packed spiders found in Hao et al. (2019). Gas flows here therefore represent lower limits; larger spiders will have larger mass flow rates, but the ice breaking pressure will remain the same.
The critical breaking pressure can be estimated using the fracturing model of Portyankina et al. (2010), which is based on an engineering model of the bending of a thin sheet, as shown in Figure 2. Portyankina et al. (2010) give the maximum stress at the middle of a sheet of radius R, and thickness z, when subjected to a constant pressure P over a radius of r P . Recently, Kaufmann et al. (2020) have measured the yield strength of CO 2 ice at the relevant temperature as σ = 12.3 MPa, and so we can set the stress equal to z R r P P Figure 2: Model for the breaking of a thin plate as in Portyankina et al. (2010). Pressure P builds up in a circular region of radius r P under a plate of thickness z and radius R with simply supported edges.
this value and solve for the critical gas pressure needed to cause fracturing (1) Here, ν I = 0.544 is the Poisson's ratio of CO 2 ice (Yamashita and Kato, 1997). It is assumed that the pressure builds up and is constant over a length-scale r P , comparable to the spider radius R (see Fig. 2). We favour a ratio of r P /R equal to one, in contrast to the value of 0.25 as used by Portyankina et al. (2010). We think this is more realistic because gas pressure should build up over the whole radius of the spider, right up to its edge where the maximum sub-ice pressure exists, and beyond-which pressure begins to decrease again with decreasing distance towards the next spider ( Fig. 1). Nevertheless, we plot P crit for a range of ratios and for two different overall spider sizes in Figure 3. It can be seen from Figure 3 that large pressures, exceeding the cryostatic pressure of P cryo ∼ 5 kPa, can develop before fracturing occurs. This is especially the case for the smaller spiders (small R) and for small gas production radii (small r P /R), both of which can be understood as reducing the total area that the pressure is applied to, meaning more gas pressure must be added to achieve the same total stress in the ice. For the densely packed spider separations observed in Hao et al. (2019) and Hao et al. (2020) and a gas production radius of r P = 0.25R ∼ 6.5 m (even smaller in size than the case simulated by Thomas et al. (2011b) above), a total pressure of P crit ≈ 33 kPa can build up before fracturing occurs. Thus, in this case our maximum pressure gradient driving gas flow at the initial breach would be (P crit − P surf )/r P ≈ 33 kPa/6.5 m.

Permeability
The permeability of a porous medium can be expressed as (see for example Balme and Hagermann (2006) where D is the constituent particle size, φ is porosity and τ is tortuosity. Typical values used for Martian regolith are φ = 40%, τ = 25/12 (Balme and Hagermann, 2006;Demidov et al., 2015), and particle sizes between ∼microns and a millimetre (e.g.  , 2018). Over this range of particle sizes K changes by several orders of magnitude, exceeding any uncertainty in φ and τ . For gasses at a low pressure P such as we are dealing with (rather than liquid flows), K must be corrected by the 'Klinkenberg correction' (Klinkenberg, 1941) where pressure is expressed in psi and permeability in mdarcies. Conversion back to SI units is performed using k = cK and k G = cK G , with c = 0.986923 × 10 −3 µm 2 .
With the above permeability, Darcy's law for the total mass flow, Q, through a crosssectional area A of porous medium, driven by a pressure gradient of dP/dr, is (Ahmed, 2010) with the gas density ρ and viscosity η given by and respectively. Here k B is Boltzmann's constant, m CO2 = 7.3 × 10 −26 kg and d CO2 = 330 × 10 −12 m are the molecular mass and diameter of CO 2 , and the mean free path is given by It is assumed that gas temperature, T , is equal to the sublimation temperature at the sub-ice pressure where we use the constants a = 23.3494 and b = 3182.48, as in Thomas et al. (2011a). Under the 0.7 m deep slab these values are T = 163 K (around 10 K higher than at the surface) ρ = 0.155 kg m −3 , λ = 9.75 × 10 −7 m, and η = 2.12 × 10 −5 Pa s. Assuming a cylindrical geometry with a thickness of permeable regolith d metres, the cross-sectional area through which gas can flow, A, at a distance r from the central vent, is A = 2πrd. The pressure gradient, dP dr , can be approximated as ∆P/r P , where ∆P is the pressure drop from the high-pressure, sub-ice environment to the low-pressure vent over a length-scale r P . Close to the central vent (i.e. r R), almost the entire gas flow, Q from above, must pass through an area of permeable regolith. We therefore set r = 1 m and obtain an expression from Eqn. 4 for the required depth of regolith in terms of this flow as Note that we have assumed a constant permeability for the regolith layer while, in reality, porosity, and therefore permeability, is generally reckoned to decrease with depth (see for example Morgan et al. (2018)). Decreasing permeability will restrict gas flow rates 7 meaning that, again, our estimate here is a 'strongest-case' for flow through porous regolith.
The length-scale r P is the gas buildup radius of above and should be similar in size to the spider radius R. Indeed, Hao et al. (2019) suggest that the distance between adjacent spiders is determined by the pressure fall-off, so that new fractures open (forming a new spider) at a distance from the existing vent where pressure increases to it's maximum sub-ice value, r P ≈ R. In the steady state case, therefore, a further simplification can be made. Total flow into the vent is the sum of the per-metre gas production rate over the area of the spider Q =ṁπR 2 , and, having made the assumption r P = R, Eqn. 1 can be simplified and combined with Eqn. 9 to make the dependence on spider size R explicit. We also make the additional assumption that ∆P ≈ P crit (neglecting the comparatively small surface pressure) so that we are using the maximum pressure gradient possible: the ice breaking pressure, rather than the cryostatic pressure. Then R can be expressed as This is the maximum radius of a cylindrical volume of porous regolith which can be effectively drained into a central vent at a steady state productionṁ before gas pressure at the edge exceeds ice strength and a new vent opens.

Results and discussion
We compute the required regolith depth d from Equation 9 for a range of regolith particle sizes and two scenarios: steady state flow and just after the initial rupture. In the steady state, we use the minimum flow rate from above, Q s = 0.00884 kg s −1 , and cryostatic pressure over a length-scale equal to the smaller spider separations, i.e. ∆P/r P = (P cryo −P surf )/r P ≈ 5kPa/25 m. For the rupture case we use the strongest case of (P crit − P surf )/r P ≈ 33 kPa/6.5 m, from above, as well as the slightly higher Thomas et al. (2011a) flow rate (Q i = 0.05 kg s −1 ). Figure 4 shows the results: in the steady state, a permeable regolith layer of thickness at least d ≈ 13.4 m is needed to support the flow rate for the likely particle size of up to D ≈ 200 µm (Kieffer, 2007). This is rather large, given the presence of ground ice and a decreasing regolith porosity with depth. Even in the case of the strongest possible pressure gradient just after rupture, the required thickness is still d ≈ 5.4 m. Only in the case of very large regolith particles (∼ 1 mm) and the strongest pressure gradients does the required layer thickness fall below one metre. An impermeable layer of water-ice bonded regolith is generally assumed to be present in the Martian polar regions at a depth of between several centimetres Pilorget et al., 2011) and the seasonal skin depth of ∼ 0.8 m (Kieffer, 2007). This is supported by thermal inertia (Bandfield and Feldman, 2008) and gamma-ray and neutron spectrometry measurements (Boynton et al., 2002;Diez et al., 2008), as well as the recent detection of shallow-buried water ice cliffs at mid-latitudes (Dundas et al., 2018).
Given the above, it seems unlikely that the total gas flow into a spider vent can occur purely through a porous regolith. What then is the largest size of a feature than can be supported this way? Figure 5 shows the results of Equation 10, where we assumed a steady state flow with the minimum gas production and a fixed geometry, and solve for 8 R for a set of permeable regolith depths. This is the length-scale over which a spider can effectively drain gas to a central vent before pressure exceeds the overlying ice strength at the edge. Even in the case of large regolith depths, the length-scales (∼ 15 m) are below the observed minimum spider separation distances (∼ 25 m) for a particle size below 200 µm. For more realistic porous layer thicknesses, only areas of a few metres in radius can be drained. Gas flow through porous regolith therefore seems incompatible with the expected polar stratigraphy. Even in the most amenable case of the smallest distances between araneiforms, with the highest pressure gradients and the lowest gas flow rates, there is insufficient depth of permeable regolith to support the flow associated with the whole spider area venting through a single central vent. Small areas of a few metres in radius around local vents (such as the so-called 'baby spiders' identified by Schwamb et al. 2018) may be drained by viscous flow through the pores, but it seems likely that the larger (tens to hundreds of metres across) spiders must involve the original Kieffer (2007) model of ice slab levitation and free gas flow. Slab levitation is likely when considering the large gas pressures, in excess of cryostatic pressure, calculated above, as well as the small gap size needed for relatively large flows. This required gap size d gap can be computed. Assuming laminar Poiseuille flow between two parallel plates (an approximation that likely holds away from the energetic flow at the vent itself) and a pressure gradient of ∆P/R, the mass flow rate per unit circumference is given by q = ρd 3 gap 12η ∆P R . The total mass flow into an annulus at radius r can also be expressed in terms of the steady state area production rate as Q(r) =ṁπ(R 2 − r 2 ), so that per unit of circumference it isṁ 2 (R 2 /r −r). The two can then be equated and solved for d gap , the required distance between the two plates to accommodate the flow, as Figure 6 shows the results of equation 11 for the two spider sizes from above. The required depth is greatest for the large mass flows of the bigger spiders, and increases towards the central vent as the gas is concentrated together in the converging flow (it is undefined at r = 0, the vent itself). Nonetheless, the required gap between the regolith and ice, d gap is much smaller than the depth of permeable regolith d calculated above and rarely exceeds one centimetre. Thus the whole flow can be supported by a gap of a few centimetres, as suggested by Kieffer (2007). Such a regolith-ice gap may occur due to levitation of the ice slab, but also important is the fact that the slab will bend upwards before fracturing, providing an additional gap for gas to flow through. Portyankina et al. (2010) give an expression for the maximum 11 displacement at the centre of the ice sheet (see Fig. 2), which we can evaluate just before rupture by using the pressure P crit calculated above: with the CO 2 ice Young's modulus of E = 11.5 GPa (Yamashita and Kato, 1997). The resulting displacement is shown in Figure 7, for the same spider sizes and r P /R ratios as before. The ice in the centre of 25 m and 150 m radius spiders is bent upwards by ∼ 20 cm and ∼ 10 m, respectively, before fracturing occurs. These values greatly exceed the required gap sizes for the gas flow shown in Fig. 6. Therefore, even though the upwards displacement is smaller towards the edge of the feature, it seems likely that upwards bending of the ice sheet can contribute significantly to the opening of a regolith-ice gap, aiding the flow of gas towards the spider vent. With viscous flow through regolith pores unable to deliver the full gas production of a reasonable sized spider, and large gas pressure enough to bend or levitate the ice slab, it may be that some spiders undergo both types of flow. For example, low volumes of gas moving through a large cross-sectional area of porous regolith at the outskirts of a feature or around multiple vents, transitioning to free-flow nearer the vent where 12 the converging gas has bent or levitated the overlying ice slab upwards, opening up a gap. Meanwhile, flow through and just above a loose regolith will certainly begin to erode material, digging troughs downwards into it. In Hao et al. (2019)'s conceptual model, regolith properties are cited as explaining the different appearances and spacing of spider sub-types. Although the above calculations demonstrate that significant gas flow through the pores is unlikely, regolith properties, such as cohesion, could still have an important effect on morphology. So-called 'fat spiders' could still be examples of lowcohesion, more easily eroded regoliths, as suggested by Hao et al. (2019), for example. Once these troughs and pits are eroded into the regolith, gas will preferentially flow down and further erode them, as described in Hao et al. (2019). This will provide additional flow paths for the gas but, since slab ice is thought to conformally cover the terrain in winter, at least some levitation or bending is required to open up a gap again in the spring. The physics of mixed gas flow through the regolith pores, eroded troughs, and a levitated or bent-open ice gap will be complicated, and may involve turbulent as well as viscous flow. Alternatively, local variations in the depth of the impermeable water-ice layer, below the resolution of orbital observations, may allow increased porous gas flow in some localised areas. It seems unlikely, however, that water-ice depths could vary from ∼cm to ∼metres over a few metres lateral distance in otherwsie identical terrain. Finally, it should also be noted that some recent work (Chinnery et al., 2018) suggests a much lower translucence of CO 2 ice when compared to previous experiments (Hansen, 1999), which would require much thinner ice sheets (on the order of 10 cm) in order to trigger the solid state greenhouse effect. We note that in this case, of a reduced z, cryostatic pressure and the critical breaking pressure would be reduced (see Eqn. 1), further lowering the pressure gradient and making permeable gas flow even less effective, even for small gas flow rates.

Conclusion
In the formation of the so-called araneiform features, seen in Mars' southern polar regions, subsurface flow of CO 2 gas below an ice slab can occur in a gap between ice and regolith, and/or in the substrate itself. The latter possibility has recently received increased attention because spider spatial density scales seem to be related to properties of the underlying regolith. We investigate the effectiveness of subsurface flow of CO 2 and compare it to gas flow through the gap between ice and regolith. Based on previously estimated flow rates and thermophysical arguments, we suggest that there is insufficient depth of porous regolith to support the full gas flow of all but the very smallest observed spiders through the regolith. On the other hand, free gas flow through a regolith-ice gap is capable of supplying the likely flow rates for gap sizes on the order of a centimetre. This size of gap can be opened in the centre of a spider feature by gas pressure bending the overlying ice slab upwards, or by levitating it entirely as suggested in the original Kieffer (2007) model. Our calculations therefore support at least some of the gas flowing through a gap opened between the regolith and ice. Regolith properties most likely still play a role in the evolution of spider morphology, by regolith cohesion controlling the erosion of the central pit and troughs, for example.