Icarus 219 (2012) 25–40
Contents lists available at SciVerse ScienceDirect
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i c a r u s
a
b ,
c
d
b
a
Computer Science, University of Maine, 223 Neville Hall, Orono, ME 04469, USA b
Department of Geological Sciences, Brown University, Providence, RI 02912, USA c
Department of Earth Sciences, Boston University, Boston, MA 02215, USA d Laboratoire de Météorologie Dynamique, Institut Pierre Simon Laplace, Université Paris 6, BP 99, 75252 Paris cedex 05, France a r t i c l e i n f o
Article history:
Received 2 June 2011
Revised 29 December 2011
Accepted 13 February 2012
Available online 24 February 2012
Keywords:
Mars
Mars, climate
Mars, polar caps
Mars, polar geology a b s t r a c t
Currently, and throughout much of the Amazonian, the mean annual surface temperatures of Mars are so cold that basal melting does not occur in ice sheets and glaciers and they are cold-based. The documented evidence for extensive and well-developed eskers (sediment-filled former sub-glacial meltwater channels) in the south circumpolar Dorsa Argentea Formation is an indication that basal melting and wetbased glaciation occurred at the South Pole near the Noachian–Hesperian boundary. We employ glacial accumulation and ice-flow models to distinguish between basal melting from bottom-up heat sources
(elevated geothermal fluxes) and top-down induced basal melting (elevated atmospheric temperatures warming the ice). We show that under mean annual south polar atmospheric temperatures ( 100 ° C) simulated in typical Amazonian climate experiments and typical Noachian–Hesperian geothermal heat fluxes (45–65 mW/m
2
), south polar ice accumulations remain cold-based. In order to produce significant basal melting with these typical geothermal heat fluxes, the mean annual south polar atmospheric temperatures must be raised from today’s temperature at the surface ( 100 ° C) to the range of 50 to 75 ° C.
This mean annual polar surface atmospheric temperature range implies lower latitude mean annual temperatures that are likely to be below the melting point of water, and thus does not favor a ‘‘warm and wet’’ early Mars. Seasonal temperatures at lower latitudes, however, could range above the melting point of water, perhaps explaining the concurrent development of valley networks and open basin lakes in these areas. This treatment provides an independent estimate of the polar (and non-polar) surface temperatures near the Noachian–Hesperian boundary of Mars history and implies a cold and relatively dry
Mars climate, similar to the Antarctic Dry Valleys, where seasonal melting forms transient streams and permanent ice-covered lakes in an otherwise hyperarid, hypothermal climate.
Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction
Current polar layered deposits ( Phillips et al., 2008 ), earlier
Amazonian mid-latitude glacial deposits (
Head et al., 2010
) and tropical mountain glaciers (
Head and Marchant, 2003 ) all were
deposited from (or contain) cold-based ice; none are associated with basal melting except for some tropical mountain glaciers that
were modified by sub-glacial volcanic eruptions ( Shean et al.,
2005a,b; Kadish et al., 2008 ). Some features in southern Argyre
have been interpreted to be eskers indicative of wet-based glaciation as late as the Middle Amazonian (
Kargel and Strom, 1992;
⇑
Corresponding author.
E-mail addresses: fastook@maine.edu
(J.L. Fastook), james_head@brown.edu
(J.W. Head).
0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved.
doi: 10.1016/j.icarus.2012.02.013
Banks et al., 2009 ). Of concern here are the Late Noachian–Early
Hesperian south circumpolar Dorsa Argentea Formation (DAF)
(
Tanaka and Kolb, 2001
) (
Fig. 1
). The DAF contains features interpreted to be indicative of basal melting beneath a regional ice-rich
deposit ( Head and Pratt, 2001; Ghatan and Head, 2002, 2004; Ghatan et al., 2003; Kress et al., 2010 ) at that time ( Plaut et al., 1988 ).
What conditions could have led to the formation and evolution of an ice-rich circumpolar deposit such as the Dorsa Argentea Formation in early Mars history? On Mars, as we know it today, water ice at the surface is expected to be less stable on average in the south polar region than in the north polar region because the global topographic asymmetry tends to favor the transport of water to
the northern hemisphere ( Richardson and Wilson, 2002
) despite the fact that water ice can exist in the near subsurface poleward of 60 °
latitude at both poles ( Mitrofanov et al., 2002; Boynton et al., 2002
).

26 J.L. Fastook et al. / Icarus 219 (2012) 25–40
Fig. 1.
(a) A sketch map with thin arrows within Hd unit showing the location of features interpreted to be eskers formed by water flowing at the base of an ice sheet. Shown are distributions of Api (polar ice deposits), Apl (polar layered deposits), Hd (Hesperian Dorsa Argentea Formation), and HNu (Hesperian–Noachian undivided unit). CA is Cavi
Angusti and CS is Cavi Sisyphi. Heavy arrows outside Hd are channels (1–5: Schmidt, Surius, Dzigai, Doanus, and Palcopas Valles respectively) interpreted to be draining Hd.
(b) Examples of the esker-like ridges located at the thin arrows in (a). Both are from
Head and Pratt (2001)
.
Fig. 2.
Distinguishing between ‘‘top-down induced basal melting’’ and ‘‘bottom-up induced basal melting.’’ An ice sheet at typical Amazonian temperatures is cold-based even with enhanced geothermal heat flux. Climate warming can lead to basal melting without producing melting at the surface of the ice sheet.
Because the current atmosphere is very thin, the dependence of surface temperature and mass balance to altitude is different than on Earth. Indeed, the surface pressure being only of a few hPa on
Mars, contribution of the sensible heat flux to the surface energy budget is almost negligible, and the surface does not follow the adiabatic temperature gradient of the atmosphere (see Section
3.2 of
Spiga, 2011
). Instead, the surface temperature is always close to radiative equilibrium, whatever the altitude. On Earth the

J.L. Fastook et al. / Icarus 219 (2012) 25–40 27
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Fig. 3.
Synthetic climate modified from
Madeleine et al. (2009) . (f) Shows the temperature field taken directly from the GCM results. To this we apply the offsets for the
warmer climates. (b) Shows the basic SMB, with sublimation rates modified from
Madeleine et al. (2009)
by a reduction factor of 50 and a radially symmetric accumulation component added. Panels (a, c, d, and e) are the basic SMB times 0.5, 2.0, 3.0, and 4.0.
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Fig. 4.
The grid for ice sheet reconstruction from the pole at grid (0, 0) to 60 ° S at the edge of the circle. Resolution is non-uniform but is nominally 30 km. To avoid complications associated with the present ice sheet topography, we truncate elevations at 3000 m, shown by the white line. Horizontal units are km, while vertical elevations shown by color contours are m. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) surface temperature tends to follow the atmospheric lapse rate
(4.5
° C/km moist, 9.5
° C/km dry), which results in less ablation by melting as well as less accumulation with increasing elevation.
But while ablation reduces to zero, accumulation never does, so a typical mass balance profile has negative values at low elevations, an equilibrium line at some elevation where accumulation and ablation are equal, a peak mass balance at some elevation above the equilibrium line, and then a declining positive mass balance beyond that. On Mars, ablation by melting is replaced by sublimation, whose rate varies with altitude mainly as a function of surface pressure (lower pressure, more sublimation) and not surface temperature which, as we just said, is close to radiative equilibrium.
Thus on present-day Mars sublimation never goes to zero, but actually increases with elevation. It is thus possible, depending on the vertical structure of the accumulation pattern, to have two equilibrium lines, a lower one and an upper one. It is this sort of mass balance pattern that we suspect produced the flank glaciation on the Tharsis volcanos, a situation we treated in detail in
Fastook et al. (2008b)
. Current climate models suggest that ice would only accumulate at the South Pole when the season of the perihelion favors a southern summer significantly colder than in the north (
Montmessin et al., 2007
). This may explain why the
Amazonian south polar-layered deposits are older than the north polar-layered deposits. If the atmosphere was thicker during the
Noachian and Hesperian eras than today, then conditions at the
South Pole may have been very different. For example, with a denser atmosphere, surface temperature distribution would behave much more like that on Earth and follow the adiabatic cooling of the atmosphere, with high altitude regions significantly colder
than lower plains ( Forget et al., 2004, 2010
). Within this context, it is likely that the high southern latitudes would have become a cold-trap where ice would tend to accumulate and form a large

28 J.L. Fastook et al. / Icarus 219 (2012) 25–40
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Basal temperatures (a and c) and vertical profiles (b and d) for SMB 0.5 scenario with no warming and geothermal flux of 45 (a and b) and 65 (c and d) mW/m
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-110 ice sheet, both because of their latitude and their altitude. It is also possible that the amount of water involved in the atmospheric water cycle was larger than today, and that consequently more water was available to form an ice sheet (
Forget et al., 2004,
2010
). For example, valley network activity, suggestive of flowing
streams and a ‘‘warmer and wetter’’ early Mars ( Carr, 1996; Craddock and Howard, 2002; Head et al., 2003; Kargel, 2004; Fassett and Head, 2008a,b
), characterized the highlands at the end of the
Noachian and into the Hesperian. The active water cycle ( Head et al., 2003 ) implied by these interpretations may have been able
to transport large amounts of water into the southern polar region in the Late Noachian and Hesperian.
Observed in the DAF are features interpreted to be eskers ( Head and Pratt, 2001; Ghatan and Head, 2002, 2004; Ghatan et al., 2003;
Kress et al., 2010
). Examples of eskers and their locations are shown in
Fig. 1
by thin, sinuous arrows within the Hesperian
DAF, labeled as the Hd unit. Api depicts polar ice deposits, Apl polar layered deposits, and HNu the Hesperian–Noachian undivided unit. The bolder sinuous arrows in
Fig. 1
outside the Hd unit are large channels leading away from the margins of the DAF, interpreted to be drainage channels linked to the basal meltwater esker system beneath the paleo ice sheet. Smaller meltwater drainage channels or systems elsewhere around the margins of the DAF have not been observed or documented, suggesting that regional melting of the top of the DAF did not occur.
Eskers form beneath ice sheets when water gathers in basal flow channels kept open by the continuous melting of the ice-cavity wall. Ice melt associated with frictional heat from flowing water tends to balance the overburden-driven creep closure of the cavity
( Rothlisberger, 1972
), permitting continuous water flow. The channels at the bed fill with sediments, and as the ice sheet wanes, the accumulated sediment is left as internally stratified, sinuous ridges. As such, eskers are evidence that basal melting, and possibly regional wet-based glaciation, occurred in the past.
There are three possible sources for the water necessary to create eskers. First, the water could come from meltwater generated at the surface of the glacier by significantly increased atmospheric temperatures at the poles to above the melting point; such surface meltwater would drain off the ice sheet into surrounding areas, forming radial channels in the surroundings, and could also be carried to the glacial bed by moulins (narrow holes that connect the surface to the glacial base). The lack of radial channels surrounding

J.L. Fastook et al. / Icarus 219 (2012) 25–40 29
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Basal temperatures (a and c) and vertical profiles (b and d) for SMB 1.0 scenario with no warming and geothermal flux of 45 (a and b) and 65 (c and d) mW/m
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.
the DAF suggests that this extreme case is not likely to be applicable.
Two less extreme possible heat sources are available to help generate eskers. First, warming (not melting) of ice due to atmospheric heating and second, warming due to elevated geothermal heat flux. We call the first type ‘‘top-down induced basal melting’’ and the second type ‘‘bottom-up induced basal melting.’’ Both of these latter types of melting can occur even if the surface of the ice sheet is well below the melting point. The insulating capability of the ice sheet, coupled with internal shear heating and a geothermal heat flux, can raise the basal temperature to the melting point and water can thus be produced at the bed. We must note here that we are not including the melting point depression associated with
perchlorate salts ( Marion et al., 2010 ) as this would produce only a
very transient melted bed, as the meltwater produced would very quickly dilute the available perchlorate to the point where the melting point would again be close to 0 ° C.
We ask the following two questions: (1) are the observed eskers related to melting from top-down heating (but not surface melting) associated with elevated atmospheric temperatures? Or, alternatively, (2) are the eskers related to bottom-up heat sources linked to local and regional elevated geothermal fluxes relative to today? We first test the hypothesis that they could be related to bottom up melting due to enhanced geothermal fluxes linked to early stages of planetary thermal evolution, and that sufficient ice accumulated to raise the global melting isotherm to the base of the ice sheet (
Fig. 2
, top). We then assess the magnitude of atmospheric surface temperature increase that would be required to en-
sure basal melting from the top down ( Fig. 2 , bottom). The latter
question has important consequences for the equatorial regions, since warming of the poles necessarily warms the equator as well.
For example, models for martian atmospheres with 500 mb CO
2
, both with and without excess H
2
O ( Johnson et al., 2008
) show warming of the poles of as much as +65 ° C and +25 ° C respectively.
Their models indicate a pole-to-equator gradient of approximately
40 ° C.
We begin by describing the basic approaches to these questions using the University of Maine Ice Sheet Model (UMISM) and the
Mars General Circulation Model of the Laboratoire de Météorologie
Dynamique.
2. The University of Maine Ice Sheet Model (UMISM)
The UMISM, adapted for the martian environment (
Fastook et al., 2004, 2005, 2006
), is a terrestrial ice sheet model that has been used successfully for time-dependent reconstructions of Antarctic,
Greenland, and paleo-icesheet evolution in response to changing

30
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Basal temperatures (a and c) and vertical profiles (b and d) for SMB 2.0 scenario with no warming and geothermal flux of 45 (a and b) and 65 (c and d) mW/m
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.
climate on Earth ( Fastook, 1993 ). UMISM uses a thermo-mechanically
coupled Shallow-Ice Approximation (vertically-integrated momentum combined with continuity) where the dominant stress is internal shear and longitudinal stresses are neglected. Primary input to the model is the bed on which the ice sheet is to be reconstructed and the net annual surface mass balance (SMB), or accumulation rate. Secondary input includes the mean-annual surface temperature and the geothermal heat flux, used to calculate internal temperatures from which the mechanical properties of the ice are obtained. In addition, internal temperatures allow for the possibility that the base of the ice reaches the pressure melting point, at which point some sliding criteria can be invoked, a phenomena we have not observed in any modeled martian glaciers of Amazonian time (
Fastook et al., 2008a,b
). This feature will be key to the investigation of conditions that would produce a melted bed for ice associated with the DAF.
The fact that with the exception of bed topography, these inputs are poorly constrained for Mars introduces some uncertainty into the results that will be presented. However, choice of reasonable values for mean annual temperature, geothermal flux, and SMB allow production of results comparable to inferred ice conditions based on geologic observations. Given these uncertainties the choice of a Shallow-Ice Approximation model is also appropriate, since the considerable computational load of a higher-order model would not produce more accurate results. Using data from
McGovern et al. (2004)
and
Solomon et al. (2005)
we adopt a geothermal flux of 45–65 mW/m 2 as a range appropriate for this time period near the Noachian–Hesperian boundary.
3. The Mars GCM of the Laboratoire de Météorologie Dynamique
In order to specify the spatial distribution of the SMB of accumulated ice for a glacial flow model, one can arbitrarily choose values and explore consequences and predictions, or use the results from a Global Circulation Model (GCM) as input. Here we choose the latter and employ results from the Laboratoire de Météorologie
Dynamique GCM (LMD/GCM) ( Forget et al., 1999
). This GCM is able to reproduce the present-day water cycle on Mars with good accu-
racy ( Montmessin et al., 2004
) and has proven robust in our earlier glacial modeling studies of reconstructed ice sheets in the Tharsis region (e.g.,
Forget et al., 2006; Fastook et al., 2004, 2005, 2008b ).

J.L. Fastook et al. / Icarus 219 (2012) 25–40 31
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Fig. 8.
Basal temperatures (a and c) and vertical profiles (b and d) for SMB 3.0 scenario with no warming and geothermal flux of 45 (a and b) and 65 (c and d) mW/m
2
.
0
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As yet, however, the model has not been applied toward predictions of SMB for the Noachian–Hesperian era (
Johnson et al.,
2008; Wordsworth et al., 2011 ) and this requires additional
assumptions as noted below.
4. Model assumptions and input
4.1. Noachian–Hesperian ice accumulation
The goal is to produce an ice sheet whose footprint will approximate the DAF so that we can assess the conditions that will generate melting at the bed under what we assume was a thicker
CO
2 atmosphere. This we do by modifying the GCM climate simulated by
Madeleine et al. (2009)
and obtained using an intermediate obliquity (35 ° ) and a high dust optical depth (2.5). In doing so we understand that this ‘‘synthetic’’ climate is not that which would result from a true GCM for the Noachian–Hesperian era; however, we consider it as a typical paleoclimate simulation that will capture salient features such as the spatial distribution of the temperature field as well as the amount and distribution of potential sublimation that might occur in the DAF region.
SMB from
Madeleine et al. (2009)
predicted extremely large sublimation rates in the DAF region, and to build an ice sheet in such a climate would require extremely large accumulation rates.
We choose to retain the spatial pattern of their sublimation rates, but reduce them by a factor of 50 (11 mm/a maximum sublimation instead of more than 50 cm/a). To this we add a radially symmetric pattern of accumulation, with a maximum value of 6 mm/a centered on the pole, declining with distance such that the region of accumulation has a radius of 2450 km. This, combined with the modified
Madeleine et al. (2009)
sublimation rates is designed to produce an ice sheet footprint in accordance with the DAF.
This basic SMB pattern is shown in
Fig. 3 b. Also in Fig. 3 a and c–e
are scaled versions of this basic SMB pattern (times 0.5, 2.0, 3.0, and 4.0) that we use to test the sensitivity of the ice sheet to our assumed climate. By scaling the basic SMB pattern we obtain the same ice sheet footprint for each, since the equilibrium-line position remains the same and both accumulation and sublimation rates are scaled by the same amount. What is different in these various SMB cases is the flux of ice as it moves from the accumulation zone to the ablation zone. Since internal heat within the body of the ice sheet comes directly from shear heating (stress times strain

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Basal temperatures (a and c) and vertical profiles (b and d) for SMB 4.0 scenario with no warming and geothermal flux of 45 (a and b) and 65 (c and d) mW/m
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1
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Percentage Wet Bed, SMB X 0.5
5 % threshold
warming= 0
warming=10
warming=20
warming=30
warming=40
warming=50
50 55
Geothermal Flux (mW/m^2)
60 65
45
0 5 10 15 20 25 30 35 40 45 50
Temperature Offset (K)
Fig. 10.
(a) Percentage of the bed that is wet for SMB 0.5 scenario as a function of temperature warming and geothermal heat flux. (b) Vertical transects through (a) at temperature warmings of +0, +10, +20, +30, +40, and +50 ° C with the 5% basal melting threshold shown as a horizontal line for reference.

J.L. Fastook et al. / Icarus 219 (2012) 25–40 33
(a) Percent Wet Bed
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5 % threshold
warming= 0
warming=10
warming=20
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warming=40
warming=50
Percentage Wet Bed, SMB X 1.0
50 55
Geothermal Flux (mW/m^2)
60 65
0 5 10 15 20 25 30 35 40 45 50
Temperature Offset (K)
Fig. 11.
(a) Percentage of the bed that is wet for SMB 1.0 scenario as a function of temperature warming and geothermal heat flux. (b) Vertical transects through (a) as in
Fig. 10 .
(a)
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60
55
50
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warming=40
warming=50
45
0 5 10 15 20 25 30 35 40 45 50
Temperature Offset (K)
50 55
Geothermal Flux (mW/m^2)
60 65
Fig. 12.
(a) Percentage of the bed that is wet for SMB 2.0 scenario as a function of temperature warming and geothermal heat flux. (b) Vertical transects through (a) as in
Fig. 10 .
(a)
65
60
55
50
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[SMB*3.0]
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11
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Percentage Wet Bed, SMB X 3.0
5 % threshold
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warming=40
warming=50
50 55
Geothermal Flux (mW/m^2)
60 65
45
0 5 10 15 20 25 30 35 40 45 50
Temperature Offset (K)
Fig. 13.
(a) Percentage of the bed that is wet for SMB 2.0 scenario as a function of temperature warming and geothermal heat flux. (b) Vertical transects through (a) as in
Fig. 10 .

34 J.L. Fastook et al. / Icarus 219 (2012) 25–40
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Percentage Wet Bed, SMB X 4.0
5 % threshold
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warming=40
warming=50
50 55
Geothermal Flux (mW/m^2)
60 65
45
0 5 10 15 20 25 30 35 40 45 50
Temperature Offset (K)
Fig. 14.
(a) Percentage of the bed that is wet for SMB 4.0 scenario as a function of temperature warming and geothermal heat flux. (b) Vertical transects through (a) as in
Fig. 10
.
Table 1
The minimum geothermal fluxes and temperature warmings that achieve 5% basal melting conditions for the 5 SMB scenarios.
At temperature warming (+ ° C) Climate scenario (base SMB )
5% wet bed threshold
0.5
1.0
2.0
3.0
4.0
Lowest geothermal flux (mW/m 2 )
53
49
47
45
45
50
50
50
50
49
Lowest temperature warming at 65 mW/m 2
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Fig. 15.
The ice sheet configuration for SMB 3.0 scenario for +50 ° C warming with the low geothermal heat flux of 45 mW/m 2 , a threshold state for 5% basal melting. (a) The
SMB with peak accumulation of 18 mm/a and sublimation rate of 33 mm/a. (b) Surface elevation (m). (c) Velocity magnitude (mm/a). (d) Basal temperatures ( ° C), white indicates temperatures at the pressure melting point and would be producing melt water. (e) Water amount. (f) Profile along the x -axis though the dome.

MOLA topography (
52
Smith et al., 2001
5. Ice sheet modeling results
) shown in
J.L. Fastook et al. / Icarus 219 (2012) 25–40
5.1. The bottom-up induced basal melting case rate, and strain rates are velocity gradients), the velocities required to pass the different fluxes of the scaled SMB patterns will have the potential to create significantly more or less internal heat, critical in our assessment of the amount of warming necessary to produce the basal melting capable of producing eskers.
Also shown in
Fig. 3
f is the mean annual surface temperature field taken directly from
Madeleine et al. (2009) . Temperatures
range from 95 ° C at 60 ° S to as low at 110 ° C at the pole. It is to this temperature field that we will apply uniform ‘‘offsets’’ to simulate warmer climates.
4.2. Noachian–Hesperian topography
The modeled ice sheet is reconstructed on the bed taken from
Fig. 4
A total of 275 model runs were produced (five climate scenarios, SMB 0.5, 1.0, 2.0, 3.0 and 4.0; five geothermal heat fluxes,
45, 50, 55, 60, and 65 mW/m 2 ; and eleven temperature offsets, 0 to +50 ° C in 5 ° C intervals, each added to the temperature field of
Fig. 3 f). Each scenario was run for 10 Ma, a time sufficient for the
ice sheets to achieve basic equilibrium configurations.
. We use a
52 checkerboard of Finite Element quadrilaterals distorted into a circular grid extending from the pole to 60 ° S that completely
encompasses the DAF ( Fig. 1 ). Resolution in the distorted checker-
board is non-uniform but is nominally 30 km. To avoid complications associated with the present ice sheet topography, we truncate elevations at 3000 m, shown in
Fig. 4
by a solid white line.
35
Using mean annual south polar temperatures typical of Amazonian climate simulations (a 0 remain cold-based.
Figs. 5–9
° C offset or warming) from
Fig. 3 f
(ranging from 95 to 110 ° C,) and the range of Noachian–Hesperian heat fluxes (45–65 mW/m 2 ) described above, south polar ice sheets show calculated basal temperatures
(a and c) and temperature profiles (b and d) for the ice sheets grown in our five different synthetic climates (SMB scaling of 0.5,
1.0, 2.0, 3.0, and 4.0 from
Fig. 3 a–e) for both 45 (a and b) and 65
(c and d) mW/m 2 . Surface elevation contours are shown at 500 m intervals. Even with the higher geothermal flux, all but a few deep holes (for instance, Schmidt Crater at grid-coordinates 1000,
+200 km in
Fig. 6
c where a melted patch emerges) are 15 ° C or more below the melting point. The profiles cross the ice sheet through the dome, roughly along the x -axis of the grid coordinate system.
In the temperature profiles, one can see from the evenly spaced isotherms that the effect of advection of heat by moving material is negligible, to be expected with the low accumulation rates and velocities in the ice sheet. Shear heating is also not contributing significantly. The slope of the linear temperature variation with depth is basically defined by the specified geothermal heat flux
(convertible to a temperature gradient by dividing by the conductivity). The higher accumulation cases on the right produce a warmer bed simply by increasing the insulating capability of the ice sheet through its increased thickness. It is worth noting that equilibrium ice sheet elevations do not depend strongly on the magnitude of the accumulation. An analytic solution for a uniform-accumulation ice sheet on a flat bed yields an elevation that depends on the
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Fig. 16.
The ice sheet configuration for SMB 3.0 scenario for +20 ° C warming with the high geothermal heat flux of 65 mW/m
2
, a threshold state for 5% basal melting. (a–f)
As in
Fig. 15
.

36 J.L. Fastook et al. / Icarus 219 (2012) 25–40
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Fig. 17.
The ice sheet configuration for SMB 3.0 scenario for +25 ° C warming with the high geothermal heat flux of 65 mW/m
2
, 7% basal melting. (a–f) As in
Fig. 15 .
accumulation rate raised only to the 1/8th power. Hence a doubling of the accumulation rate should only yield approximately a
10% thicker ice sheet. This can be seen as one moves from the low-SMB case in
5.2. The top-down induced basal melting case cases
(
Fig. 8 c and d). Because the mechanical properties of ice are an
exponential function of temperature, the higher geothermal flux case yields a much thinner ice sheet (as much as 1000 m thinner) because the warmer, softer ice deforms more easily, allowing for larger velocity gradients, and hence overall faster flow requiring less thickness to pass the same mass flux. This thinner ice provides less insulating thickness counteracting warming due to the higher geothermal flux. In addition, the fact that the thinner ice sheet must pass the same flux of ice through the reduced thickness results in larger velocities and also larger velocity gradients, which then produce more shear heating. Assessment of the magnitude of the competing effects of thinner and less insulating versus faster flowing and greater shear heating can only be effectively assessed with a thermo-mechanically coupled ice sheet model such as
UMISM, the model used in this study. Having determined that simply raising the geothermal flux is insufficient to produce significant basal melting, we can now turn to the question of elevated surface temperatures.
Fig. 5
(0.5 times the base) to the high-SMB cases in
Figs. 7–9
(2.0, 3.0, and 4.0 times the base).
Also worth noting is the contrast between the 45 mW/m 2
(
Fig. 8 a and b) and the higher geothermal flux 65 mW/m
2 cases
Under what conditions can basal melting induced by increased atmospheric surface temperatures create meltwater to produce eskers? The threshold for significant water production for each of the climate scenarios is shown in
Figs. 10–14 . In each figure, panel (a)
shows the percent of the ice sheet bed that reaches the pressure melting point and begins to produce water as a function of the temperature warming (the offset added to the temperature field shown in
Fig. 3 f) along the horizontal axis, and geothermal heat
flux along the vertical. Also shown in panel (b) of each figure are vertical transects through the companion panel (a) at six different temperature offsets. For reference, the 5% melted-bed level is shown. This is the minimum level at which significant quantities of water are produced.
Fig. 10 , corresponding to the lowest SMB
0.5 scenario (peak accumulations of 3 mm/a and maximum sublimation of 5 mm/a) shows no significant melting unless the climate is warmed by
50 ° C and the geothermal heat flux increased to 53 mW/m 2 . Alternatively, one achieves melting with 40 ° C warming by raising the geothermal flux to 65 mW/m 2 . As one progresses through the climate scenarios from lower mass throughput (SMB 0.5) to higher
(SMB 4.0), the onset of significant melting (5%) moves to lower temperatures and smaller geothermal fluxes. For the SMB 1.0 scenario (
Fig. 11
), the smallest geothermal heat flux that reaches the
5% threshold within our modeled temperature range is 49 mW/ m 2 at 50 ° C warming, and the lowest warming at the upper range of geothermal heats is 35 ° C.
Figs. 12–14
for SMB 2.0, 3.0, and 4.0
show similar trends with higher SMB reaching our 5% threshold for progressively lower temperature and smaller geothermal fluxes, with a summary shown in
Table 1
. Only the SMB 4.0 scenario achieves 5% melting with the lowest geothermal flux modeled, and then only with the highest temperature modeled, and none produce 5% without at least 16 ° C warming. The main conclusion to draw from these figures is that lower throughput requires either more temperature warming or a higher geothermal flux to achieve
0
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0
-500
-1000
(c)
1500
1000
500
0
-500
-1000
-1500
VELOCITY (mm/a) [d-50-50]
TIME 10Ma
-1500 -1000
-1500 -1000 -500 0 500 1000 1500
1500
1000
500
0
-500
-1000
-1500
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
37
(d)
TEMPERATURE (degree C) [d-50-50]
TIME 10Ma
-1500 -1000 -500 0 500 1000 1500
-36
-41
-46
-51
-56
-61
-66
-71
-11
-1
-6
(e)
-16
-21
-26
-31 1500
1000
500
0
-76
-81
-500
-86
-91
-1000
-96
-101 -1500
-106
WATER (amount) [d-50-50]
TIME 10Ma
-1500 -1000
-1500 -1000 -500 0 500 1000 1500
1500
1000
500
0
-500
-1000
-1500
0.065
0.060
0.055
0.050
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0.100
0.095
0.090
0.085
0.080
0.075
0.070
(f)
[d-50-50]
[]
1500
1000
500
0
-500
-1000
-1500
-1500 -1000
1500
1000
500
0
-500
-1000
-1500
0
-5
-10
-15
-20
-25
-30
-35
-40
-45
-50
-55
-60
-65
-70
-75
-80
-85
-90
-95
-100
-105
-110
Fig. 18.
The ice sheet configuration for SMB 3.0 scenario for +50 ° C warming with the medium geothermal heat flux of 50 mW/m
2
, 9% basal melting. (a–f) As in
Fig. 15
.
the same level of basal melting, a consequence of reduced shear heating.
Figs. 15 and 16
show two of these ice sheet configurations that just exceeded our 5% threshold, both for the SMB 3.0 scenario
(18 mm/a peak accumulation, 33 mm/a maximum sublimation).
Fig. 15
shows a low geothermal heat flux case (45 mW/m
2
) that requires a full 50 ° C atmospheric surface temperature warming in order to achieve our 5% threshold for significant melting. Shown in this figure are (a) the SMB pattern, (b) the surface elevation, (c) the velocity magnitude in mm/a, (d) basal temperatures ( ° C) with white indicating areas where the bed is at the pressure melting point and melting would be occurring, (e) the water amount (a nominal ‘‘thickness’’ that should not be construed as a water layer), and (f) a profile showing internal temperatures and surface on a transect approximately along the x -axis through the dome.
Fig. 16
shows the same (a–f) for the high geothermal heat flux case
(65 mW/m 2 ), but here only +20 ° C atmospheric surface temperature warming is required to achieve our 5% melted-bed threshold.
This ice sheet, even with the higher geothermal heat flux, behaves as a much colder ice sheet, with much steeper margins, as much as
700 m thicker at the dome, and with much more subdued flow velocities. The pattern of melted bed is, however, very similar, with virtually the same pattern of bed at the pressure melting point and basal melting. We would be unable to distinguish between these two cases from the geologic record alone.
Both
Figs. 15 and 16
are of threshold melting cases that just exceeded an arbitrary 5% melted bed criteria (5% and 6% respectively). We will now turn to some cases with higher percentages of melted beds that may be more representative of ice sheets that would produce larger quantities of basal water.
Fig. 17
shows the SMB 3.0 scenario with a warming of +25 ° C and a high geothermal heat flux of 65 mW/m 2 . This is a relatively cold ice sheet and so is thicker. Melted basal conditions occur at the same positions as in
Figs. 15 and 16
but are slightly larger, and the amount and areal extent of the water at the bed is increased (7%). Note that velocity magnitudes are also higher.
Fig. 18
is also for the SMB 3.0 scenario, now for +50 ° C warming and a medium geothermal flux. Melted basal percentage is 9%. The warmer ice sheet is thinner, velocities are still higher, and definite focused areas of flow are occurring over the melted-bed regions.
Velocities approach 2000 mm/a in the regions of fastest flow. Finally,
Fig. 19
, again for the SMB 3.0 scenario, shows the ice sheet configuration for a slightly smaller warming of +40 ° C with a high geothermal heat flux of 65 mW/m 2 . This is the thinnest of the ice sheet configurations presented here. A summary of the ice sheet configurations shown in
Figs. 15–19
is presented in
Table 2
. Of these,
Fig. 18
showing SMB 3.0 with +50 ° C and 50 mW/m 2 is our favored configuration since it requires the most conservative modifications while producing a melted bed pattern coincident with the observed eskers.
Of note is the fact that one of the major patches of calculated melting occurs close to the esker region shown by arrows in
Fig. 1
( Head and Pratt, 2001
).
Fig. 20 , the basal temperatures from
the case shown in
Fig. 18
, includes an overlay map of site names.
The major patch of melted bed lays directly over the DAF region just south of Joly Crater, a feature visible in the example of eskers shown in
Fig. 1 .
In this exercise we have been guided by the geological observations of the extent of the deposit and the presence and distribution of eskers, indicative of basal melting. An ice sheet approximating

38 J.L. Fastook et al. / Icarus 219 (2012) 25–40
-500
-1000
-1500
1500
1000
500
0
(a) Mass Balance (mm/a) [d-40-65]
TIME 10Ma
-1500 -1000
-1500 -1000 -500 0 500 1000 1500
-500
-1000
-1500
1500
1000
500
0
10
8
6
4
2
0
-4
-8
-12
-16
-20
-24
-28
-32
-36
-40
20
18
16
14
12
(b) Surface [d-40-65]
TIME 10Ma
-1500 -1000
-1500 -1000
10000
9500
9000
8500
8000
7500
7000
6500
6000
5500
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
-500
-1000
(c)
1500
1000
500
0
-500
-1000
-1500
VELOCITY (mm/a) [d-40-65]
TIME 10Ma
-1500 -1000
-1500 -1000 -500 0 500 1000 1500
1500
1000
500
0
-500
-1000
-1500
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
(d)
1500
1000
500
0
-500
-1000
-1500
TEMPERATURE (degree C) [d-40-65]
TIME 10Ma
-1500 -1000
-1500 -1000 -500 0 500 1000 1500
1500
1000
500
0
-500
-1000
-1500
-1 (e)
-46
-51
-56
-61
-66
-71
-76
-81
-6
-11
-16
-21
-26
-31
-36
-41
1500
1000
500
0
-500
WATER (amount) [d-40-65]
TIME 10Ma
-1500 -1000
-86
-91
-1000
-96
-101 -1500
-106
-1500 -1000 -500 0 500 1000 1500
1500
1000
500
0
-500
-1000
-1500
0.100
0.095
0.090
0.085
0.080
0.075
0.070
0.065
0.060
0.055
0.050
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
(f)
[d-40-65]
[]
-20
-25
-30
-35
-40
-45
0
-5
-10
-15
-50
-55
-60
-65
-70
-75
-80
-85
-90
-95
-100
-105
-110
Fig. 19.
The ice sheet configuration for SMB 3.0 scenario for +40 ° C warming with the high geothermal heat flux of 65 mW/m 2 , 11% basal melting. (a–f) As in
Fig. 15
.
Table 2
Summary of ice sheet configurations shown in
Figs. 15–19 .
Figure SMB
15
16
17
18
19
3.0
3.0
3.0
3.0
3.0
Temperature warming (+ ° C)
50
20
25
50
40
Geothermal
(mW/m
2
)
45
65
65
50
65
Melted %
7
9
5
6
11
Average thickness (m)
2215
2961
2669
2135
1936
Volume
(M km
3
)
5.76
7.52
6.79
5.51
5.41
the estimated footprint of the DAF is created with a reasonable synthetic climate linked to our work with GCM results. Systematic exploration of the parameter space spanning the magnitude of the
SMB, the surface temperature, and the geothermal heat flux provide a robust estimate of the conditions under which such an ice sheet could have existed. Thermodynamic calculations of internal temperatures within and at the bed of the ice sheet allow us to identify the conditions under which a melted bed might have occurred, and to assess what the configuration of the ice sheet might have been.
At the high end of the geothermal heat fluxes modeled, 65 mW/ m 2 , the warming required to produce significant basal melting is
+40 ° C for the low throughput case (SMB 0.5) and +20 ° C for the high throughput case (SMB 4.0). Note that this degree of warming would increase maximum temperature for the region to between
70 to 55 ° C, still sufficiently cold that no surface melting would occur. At low heat fluxes, 45 mW/m 2 , more than +50 ° C warming (a maximum temperature for the region of 45 ° C) is necessary to bring the bed to the melting point, even in the highest SMB case
( 4.0), although the trend is for the higher SMB to require less warming. At this point, basal melting will start to take place beneath the ice sheet, and this will produce sufficient melting to account for the extensive esker systems observed in the Dorsa
Argentea Formation ( Head and Pratt, 2001; Kress et al., 2010 ). In
all cases the surface temperatures in the circumpolar regions are still considerably below 0 ° C, and thus independent melting of surface ice is not likely to occur, consistent with geologic observations
( Head and Pratt, 2001 ). GCM results for a suite of obliquities show
similar polar temperature warming (+25 to +50 ° C) as one moves from the current 23 ° obliquity to the extreme of 55 ° predicted by
Laskar et al. (2004)
(see for example
Fig. 6
of
Haberle et al.,
2003 ). Excursions to such extreme obliquities are relatively brief
( 100 Ka) and further would not in themselves build an equilibrium ice sheet with sufficient insulating thickness and shear-heating velocity to produce widespread melting at the bed.

J.L. Fastook et al. / Icarus 219 (2012) 25–40 39
1500
1000
-1000
-1500
TEMPERATURE (degree C) [d-50-50]
TIME 10Ma
-1500 -1000 1000 1500
1500
1000
500
0
-500
-1000
-1500
-1
-6
-11
-16
-21
-26
-31
-36
-41
-46
-76
-81
-86
-91
-96
-101
-106
-51
-56
-61
-66
-71
-1500 -1000 1000 1500
Fig. 20.
Basal temperatures for SMB 3.0, +50 ° C, 50 mW/m
2 case of
Fig. 18
, with overlay showing agreement with the identified eskers shown in
Fig. 1
, adjacent to Joly Crater in the center of the DAF.
6. Discussion and conclusions
We show that under mean annual south polar atmospheric temperatures ( 100 ° C) simulated in typical Amazonian climate experiments and estimated Noachian–Hesperian heat fluxes
(45–65 mW/m 2 ), south polar ice accumulations remain cold-based.
In order to produce significant basal melting at these typical geothermal heat fluxes, the mean annual south polar atmospheric temperatures must be raised from today’s temperature ( 100 ° C) to the range of 50 to 75 ° C. This polar surface temperature increase is insufficient to cause melting at the ice sheet surface, but sufficient to induce basal melting of the ice sheet.
This increase in surface temperatures required to explain the basal melting of a thick, south circumpolar, ancient ice sheet translates into higher temperatures at lower latitudes, but does not appear consistent with a generally ‘‘warm and wet’’ early Mars. Mean annual south polar temperatures in the range of 50 to 75 ° C still imply lower latitude mean annual temperatures generally below the melting point of water (
Forget et al., 2004, 2010 ). On the other
hand, these mean annual lower latitude temperatures, while likely to be below the melting point of water, imply equatorial and midlatitude seasonal temperatures that could be in excess of the melt-
ing point of water ( Forget et al., 2004, 2010
). Thus, the late
Noachian climate of Mars may have been similar to the current hypothermal, hyperarid climate of the Antarctic Dry Valleys on
Earth (
Marchant and Head, 2007 ), where seasonal melting occurs
in microenvironment/microclimate zones similar to latitudedependent zones on Mars. Such seasonal melting in Antarctica leads to the formation of transient streams and ice-covered lakes that have similarities to the valley networks and open basin lakes on Mars (
Fassett and Head, 2008a,b; Marchant and Head, 2007
). In conclusion, this analysis of the Dorsa Argentea Formation provides an independent estimate of elevated surface temperatures in the mid-latitudes and equatorial regions of Mars near the Noachian–
Hesperian boundary of Mars history, but favors a cold climate with seasonal melting, rather than a ‘‘warm, wet’’ climate.
Acknowledgments
We gratefully acknowledge the scientists and engineers contributing to the success of the NASA Mars Observer, Mars Odyssey, and Mars Reconnaissance Orbiter, and the ESA Mars Express missions. J.W.H. gratefully acknowledges support from NASA through the Mars Data Analysis Program and the Mars Express HRSC Team.
J.L.F. gratefully acknowledges support from the NSF for development of the terrestrial version of UMISM through many grants.
Thanks are extended to Anne Côté for assistance in manuscript preparation.
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