Icarus 198 (2008) Contents lists available at ScienceDirect. Icarus.

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1 Icarus 198 (2008) Contents lists available at ScienceDirect Icarus Tropical mountain glaciers on Mars: Altitude-dependence of ice accumulation, accumulation conditions, formation times, glacier dynamics, and implications for planetary spin-axis/orbital history James L. Fastook a, James W. Head b,, David R. Marchant c, Francois Forget d a Climate Change Institute and Department of Computer Science, University of Maine, Orono, ME 04469, USA b Department of Geological Sciences, Brown University, Box 1846, 324 Brook Street, 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, BP99, Paris cedex 05, France article info abstract Article history: Received 11 February 2008 Revised 3 June 2008 Available online 13 September 2008 Keywords: Ices Ices, mechanical properties Mars Mars, atmosphere Mars, climate Geological processes Fan-shaped deposits up to 166,000 km 2 in area are found on the northwest flanks of the huge Tharsis Montes volcanoes in the tropics of Mars. Recent spacecraft data have confirmed earlier hypotheses that these lobate deposits are glacial in origin. Increased knowledge of polar-latitude terrestrial glacial analogs in the Antarctic Dry Valleys has been used to show that the lobate deposits are the remnants of cold-based glaciers that formed in the extremely cold, hyper-arid climate of Mars. Mars atmospheric general circulation models (GCM) show that these glaciers could form during periods of high obliquity when upwelling and adiabatic cooling of moist air favor deposition of snow on the northwest flanks of the Tharsis Montes. We present a simulation of the Tharsis Montes ice sheets produced by a static accumulation pattern based on the GCM results and compare this with the nature and extent of the geologic deposits. We use the fundamental differences between the atmospheric snow accumulation environments (mass balance) on Earth and Mars, geological observations and ice-sheet models to show that two equilibrium lines should characterize ice-sheet mass balance on Mars, and that glacial accumulation should be favored on the flanks of large volcanoes, not on their summits as seen on Earth. Predicted accumulation rates from such a parameterization, together with sample spin-axis obliquity histories, are used to show that obliquity in excess of 45 and multiple 120,000 year obliquity cycles are necessary to produce the observed deposits. Our results indicate that the formation of these deposits required multiple successive stages of advance and retreat before their full extent could be reached, and thus imply that spin-axis obliquity remained at these high values for millions of years during the Late Amazonian period of Mars history. Spin-axis obliquity is one of the main factors in the distribution and intensity of solar insolation, and thus in determining the climate history of Mars. Unfortunately, reconstruction of past climate history is inhibited by the fact that the chaotic nature of the solution makes the calculation of orbital histories unreliable prior to about 20 Ma ago. We show, however, that the geological record, combined with glacial modeling, can be used to provide insight into the nature of the spin-axis/orbital history of Mars in the Late Amazonian, and to begin to establish data points for the geologically based reconstruction of the climate and orbital history of Mars Elsevier Inc. All rights reserved. 1. Introduction The well-known record of recent ice ages on the Earth provides insight into the nature of climate history and the driving forces involved (Imbrie, 1985). The recognition of recent ice ages on Mars (Head et al., 2003), combined with (1) a more in-depth understanding of the spin-axis/orbital parameter driving forces in * Corresponding author. Fax: +1 (401) addresses: fastook@maine.edu (J.L. Fastook), james_head@brown.edu (J.W. Head), marchant@bu.edu (D.R. Marchant), forget@lmd.jussieu.fr (F. Forget). climate history for the last 20 million years (Laskar et al., 2004), and (2) improved atmospheric general circulation models (GCM) (Haberle et al., 2001; Levrard et al., 2004) have permitted a more detailed understanding of the recent climate history of Mars. Reconstruction of spin-axis/orbital parameter climate driving force histories on Mars prior to 20 Ma ago is inhibited by the fact that the chaotic nature of the solution makes the calculation of orbital histories unreliable. Despite these difficulties, the glacial geological record of climate change on Mars, combined with glacial ice-sheet modeling, can be used to provide insight into the ancient climate history of Mars, and to begin to establish data points for the geologically based reconstruction of the orbital history of Mars. In this /$ see front matter 2008 Elsevier Inc. All rights reserved. doi: /j.icarus

2 306 J.L. Fastook et al. / Icarus 198 (2008) contribution, we first review ice-sheet modeling on Earth, assess its application to Mars, and then apply the principles of ice-sheet modeling to tropical mountain glacier deposits on Mars in order to gain insight into the climatic conditions that produced the glaciers, their behavior, and the temporal aspects of their formation and evolution. 2. Ice-sheet modeling on Earth Ice ages on Earth have come and gone over the last 2.5 million years, typically lasting on the order of 100,000 years, with interglacial periods lasting approximately 10,000 years. During these ice ages the average global temperature drops by as much as 10 C, and the volume of water locked up in huge continental ice sheets at the maximum (about km 3 ) lowers sea level by m (Imbrie and Imbrie, 1979). Important insight into the nature of climate change during glaciation can be gained from knowledge of the thickness and geometry of these ancient ice sheets, addressing, for example, the question of how continental ice-sheet size and configuration influence climate (Shinn and Barron, 1989). In lieu of direct observations of an ancient ice sheet, an important contribution of ice-sheet modeling is the reconstruction of these paleo-ice sheets. Geologic evidence helps constrain the spatial distribution of paleo-ice sheets, but tells us almost nothing about their thicknesses and elevations, information that is critical for global circulation models aimed at understanding past climates. Typically, terrestrial ice-sheet models are run for a full glacial cycle, with input parameters derived from climate proxies, such as the oxygen-isotope record from ice cores (Dansgaard et al., 1984) and sediment cores (Prell et al., 1986; Mix, 1987). Oxygen isotope records may span hundreds of thousands of years and can be interpreted as paleo-thermometers, recording long-term climate variations at ice-core sites. Ice-sheet models are typically tuned by forcing them to achieve certain well-documented spatially and temporally constrained ice-sheet margin positions, such as the Last Glacial Maximum or the Younger Dryas re-advance (Hughes et al., 1981; Peltier, 1994). While geologic data provide constraints on ice-sheet models, model results also help geologists interpret field data. The model results provide both temporal and spatial interpolation between incomplete geological data, allowing ages to be assigned to stratigraphic chronologies with much more confidence. The model can also act as an ice-sheet laboratory, allowing one to perform what-if experiments on ice sheets. One can then describe what the spatial and temporal response should be to variations in environmental, climatic, and boundary conditions. Ice-sheet model input parameters most commonly include the bed topography, the geothermal heat flux, the net annual accumulation or ablation rates, and the surface mean annual temperatures. The bed topography is assumed to be fixed and is well known, both on the Earth and on Mars. The exception in both cases is the bed topography beneath existing ice sheets, where ice thicknesses must be determined using indirect geophysical methods such as radar echo or seismic sounding. New developments in orbital radar sounding on Mars are showing great promise in detecting the base and determining the thickness and internal structure of current polar ice deposits (Picardi et al., 2005; Phillips et al., 2007, 2008; Plaut et al., 2007) and paleo-polar deposits (Plaut et al., 2007; Phillips et al., 2008; Farrell et al., 2008). The geothermal heat flux is also fixed in ice-sheet models, with possible positive transients, but is poorly known for large areas of the Earth, and can only be estimated for Mars on the basis of models and flexural histories (Solomon and Head, 1990; Johnson et al., 2000; Zuber, 2001; Phillips et al., 2008). The accumulation rate is the annual snowfall or mass input to the ice sheet, while the ablation rate is the annual loss due to melting and/or sublimation. Both of these are highly variable and non-uniform, and are poorly known for the Earth and Mars. Surface mean annual temperatures, necessary to calculate the mechanical properties of the ice sheet, are also variable and non-uniform, and are also poorly constrained. Nonetheless, the range estimates for these values have proven very useful and productive in terrestrial ice-sheet modeling (Pelto et al., 1990; Fastook and Prentice, 1994). An important application of ice-sheet modeling is to test theories of glacier flow, which is accomplished through comparisons of model results and measurements from existing ice sheets such as Antarctica and Greenland (Rignot and Thomas, 2002). An icesheet model can also be used as a simulator for training researchers; models allow us to ask and perhaps answer such questions as How have glaciers responded in the past to known climate change? and How will they respond in the future? Icesheet models allow for expanded interpretation of geological evidence by reconstruction of paleo-ice sheets based on a geological imprint that can consist of many features and deposits. Some examples include: (1) moraines, which provide information about margin positions and the timing of retreat; (2) till, which defines the areal extent of glaciation and the basal conditions of the ice sheet; (3) striations, which indicate direction and relative timing of flow; and (4) glacial isostatic adjustment, which provides constraints on thickness and the timing of unloading during retreat. In addition, comparison of model results with geological data can provide information about the magnitude and timing of the climate change that produced the transient behavior of the ice sheet evident in the geologic record. An important advantage of ice-sheet modeling on Mars is that the theory of glacier flow is well characterized (Goldsby and Kohlsdedt, 2001; see discussion in Milliken et al., 2003). The extremely cold, hyper-arid environment on Mars means that typical glaciers are cold-based, rather than wet-based, even in martian tropical environments (Head and Marchant, 2003). Thus, there is no need to deal with the difficult and as yet not well-understood sliding mechanism that produces the fast flow of ice streams on the Earth (Bindschadler, 1998). The lack of oceans on Mars at present and in the recent past means that glacial margins are always ablationdominated; thus, there is no need for addressing the effects of calving, pulling power, ice shelves, or any other of the poorly understood issues of marine ice sheets on Earth. Disadvantages of ice-sheet modeling on Mars include the difficulties of making direct measurements on the ground and dating deposits and landforms precisely. We now apply the principles of terrestrial ice-sheet modeling to tropical mountain glacier deposits on Mars in order to gain insight into the climatic conditions that produced the glaciers, the behavior of the glacier, and the temporal aspects of their formation and evolution. 3. Ice-sheet modeling of tropical mountain glaciers on Mars The origin of the fan-shaped deposits on the flanks of the large equatorial Tharsis Montes volcanoes (Arsia, Pavonis, and Ascraeus Mons; Fig. 1A) has been debated for nearly three decades. Suggestions for the origin of part or all of the deposits include landslides, pyroclastic flows, glaciation or combinations of these (see review in Zimbelman and Edgett, 1992, and Shean et al., 2005). Recent spacecraft data have supported earlier suggestions (Williams, 1978; Lucchitta, 1981) that these lobate deposits are glacial in origin. Increased knowledge of polar-latitude terrestrial glacial analogs in the Antarctic Dry Valleys (Marchant and Denton, 1996; Marchant and Head, 2007) has been used to show that the lobate deposits are the remnants of cold-based glacial processes that formed in the extremely cold, hyper-arid climate of Mars (Head and Marchant, 2003). These Amazonian-aged deposits display three distinct facies, ridged, knobby, and smooth (Scott and Zimbelman, 1995), each of

3 DynamicsoftropicalmountainglaciersonMars 307 (A) Fig. 1. (A) Fan-shaped deposits (yellow) on the northwest flanks of the Tharsis Montes interpreted to represent tropical ice-age glaciation (Head and Marchant, 2003) (top, Ascraeus Mons; middle, Pavonis Mons; and bottom, Arsia Mons). Dashed lines show azimuth of trends. Background is MOLA altimetry map with brown being high (the summit of Tharsis, lower right, at about 9 km above mean planetary radius) and green low (the flank of Tharsis at 0 1 km elevation). The summits of the Tharsis Montes (white) extend up to 18 km above mean planetary radius. (B) Geological sketch map of the western Arsia Mons fan-shaped deposit (modified from Zimbelman and Edgett, 1992) superposed on a MOLA topographic gradient map. The fan-shaped deposits include ridged (R), knobby (K) and smooth (S) facies. Other adjacent deposits are shield (SA), degraded western flank (SB), smooth lower western flank (SC), caldera floor (CF), caldera wall (CW), flank vent volcanic fans from Arsia Mons (PF) and undivided Tharsis volcanic plains (P). Images show Viking Orbiter images of the facies of the fan-shaped deposit (top) and USGS aerial photographs of interpreted terrestrial analogs in the Antarctic Dry Valleys (ADV) (bottom). Top left is ridged facies, top middle is knobby facies, and top right is smooth facies. Bottom left is drop moraines in the ADV (TMA 3079/303), bottom middle is sublimation tills in the ADV (TMA ) and bottom right is a rock glacier in the ADV (TMA 3080/275). Modified from Head and Marchant (2003). which can be associated with different glacial processes (Head and Marchant, 2003) (Fig. 1B). The outermost ridged facies is interpreted to be the imprint of ice-margin drop moraines recording the fluctuations of a stable cold-based ice sheet. Inside this, the knobby facies is interpreted as a sublimation till deposited as the ice sheet retreated during a period of major contraction and downwasting. Finally, the smooth facies is concentrated in the proximal parts of the fan-shaped deposit and is interpreted to be similar to debris-covered glaciers found in the Dry Valleys of Antarctica (Marchant et al., 2007). Each of the fan-shaped deposits trend generally to the northwest of the volcanoes (Fig. 1A). Their individual cold-based glacial facies (Fig. 1B) and their configurations have been described elsewhere (Arsia: Head and Marchant, 2003; Shean et al., 2007; Pavonis: Shean et al., 2005; Ascraeus: Parsons and Head, 2004, 2005; Kadish et al., 2007, 2008; Olympus: Milkovich et al., 2006). Taken together, the geological observations suggest that during the Late Amazonian period of Mars history, snow and ice accumulated on the northwest flanks of each of the Tharsis Montes, forming huge glaciers that flowed down the flanks of each edifice. Concentric drop moraines are testimony to step-wise retreat and temporary stabilization of the margins, and superposition relationships suggest major phases of ice-sheet advance and retreat. Extensive development of knobby facies at Arsia Mons suggests that the ice sheet underwent collapse and vertical downwasting, and the distinctive development of lobes of smooth facies suggests that more restricted development of localized debris-covered glaciers has characterized the most recent glacial history of these regions (Shean et al., 2007). These data have permitted the reconstruction of the areal extent of glaciation and evidence for its history in the Tharsis Montes. For example, steady-state flowband profiles of the Tharsis Montes fan-shaped deposits have been shown to produce reasonable ice-sheet configurations of a few hundred to a few thousand meters thickness (Shean et al., 2004, 2005, 2007). However, flowband modeling makes estimates of volumes difficult, and steadystate modeling does not allow for the possibility that these ice sheets never attained full equilibrium configurations during the transient climatic conditions that accompany the large oscillations in obliquity that dominate the climate on Mars (Laskar et al., 2004). Chronologies for martian ice-sheet timing and behavior are also emerging. Impact crater size-frequency distributions on glacial deposits (Shean et al., 2006; Kadish et al., 2008), calibrated to impact flux (Hartmann and Neukum, 2001; Hartmann, 2005) provide estimates of deposit absolute ages and corresponding historical epochs. In addition, model chronologies have arisen from a better understanding of Mars climate change and of the processes driving this change, i.e., spin-axis and orbital variations (Laskar et al., 2004). Furthermore, atmospheric GCMs have improved sufficiently to include high-resolution simulations of circulation under different spin-axis/orbital parameter variations and to include the transport and deposition of water (Richardson and Wilson, 2002; Haberle et al., 2003; Mischna et al., 2004). Consequently, conceptual and quantitative models now show that during periods of high obliquity, strong westerly winds prevail in Tharsis, resulting in upwelling and adiabatic cooling of moist polar air, with water ice and

4 308 J.L. Fastook et al. / Icarus 198 (2008) (B) Fig. 1. (continued) snow deposition on the northwest flanks of the Tharsis Montes to form tropical mountain glaciers (Forget et al., 2006). In summary, ice-sheet models, coupled with reasonable assumptions about the climate, can now obtain estimates for the volumes of these ice sheets, better constraining the water budget for the planet (Fastook et al., 2004; Shean et al., 2006) Modeling Ice-sheet models, such as the University of Maine Ice Sheet Model (UMISM), usually involve an integrated momentum-conservation equation based on the flow law of ice (Glen, 1955), coupled with a mass-conservation, or continuity equation to yield a differential equation for ice extent and thickness as a function of time (Fastook and Prentice, 1994; Huybrechts et al., 1996). Conservation of mass says that the gradient of the flux (the average velocity in a vertical column through the ice sheet, U, times the thickness of that column, H) should equal the source of flux (the net annual accumulation rate, ȧ, which consists of two parts, the annual accumulation and the annual ablation, in m/yr), and that any deviation from this balance should change the ice surface elevation ( h/ t).

5 DynamicsoftropicalmountainglaciersonMars 309 This continuity equation is given by the following: h t = ȧ (UH). The column-average velocity, U, is obtained from the flow law ε = (σ /A) 3, a tensor equation expressing the relationship between velocity gradients or strain rates, ε, and stresses, σ, through a non-linear power law. The shallow-ice approximation neglects all stresses and strain rates except the basal drag (σ xz,astressinthex-direction acting on a surface with normal in the z-direction, and ε xz, the gradient in the z-direction of the x-component of velocity). Conservation of momentum tells us that this stress acting on the bed is given by the following σ xz = ρ I gh h, where ρ I is the density of ice, H is the ice thickness, and h is the surface slope. Assuming a linear variation of stress from its maximum at the bed to zero at the free ice surface, one can integrate the velocity gradient to obtain velocity at any depth. Integrating again through the vertical obtains the column-average velocity needed in the continuity equation. This combination of conservation of mass and integrated momentum yields the fundamental differential equation governing ice-sheet physics. [ { } h 2 ρ 3 ] g h t = ȧ H 5. 5 A This non-linear second-order differential equation can be solved with the finite element method. Such a differential equation requires specification of the source of this mass at each point in the domain, the so-called mass balance or accumulation rate, ȧ. As mentioned, the mass balance consists of two parts, the annual accumulation (ice-equivalent snowfall, in m/yr) and the annual ablation (melting, sublimation, or other erosive processes that remove ice from the ice surface, also in m/yr). The difference between these two is the net mass balance, which is the source (or sink, if negative) in the continuity equation GCM mass balance In order to assess accumulation rates, we use the results from a focused run of an atmospheric general circulation model (GCM) for Mars at high obliquity (Haberle et al., 2004; Forget et al., 2006). The GCM used was the Martian Global Climate Model of the Laboratoire de Meterologie Dynamique (Hourdin et al., 1993; Forget et al., 1999), a well-tested model able to adequately simulate the present martian climate. In particular, it includes a model of the water cycle described in Montmessin et al. (2004), which was shown to provides distributions of atmospheric vapor and clouds in very good agreement with MGS TES spectrometer observations (Smith et al., 2001). The model contains a full description of exchange between surface ice and atmospheric water, transport and turbulent mixing of water in the atmosphere, and simplified microphysics for cloud formation. The radiative effects of water vapor and clouds as well as the exchange of water vapor with the subsurface are not included. The surface albedo is set to 0.4 when an ice layer thicker than 5 μm is present, enabling an ice-albedo feedback process. The surface thermal inertia is not modified, however. In this paper, we based our calculations on the numerical simulations described in Forget et al. (2006) in the case of a water source at the northern pole as on present-day Mars (Figs. 1 3 in Forget et al., 2006). To simulate a typical high obliquity climate, they had performed a simulation similar to the one presented in Fig. 2. General circulation model (GCM) results for ice-equivalent accumulation depth, in mm, as a function of time (expressed as L s, the solar longitude), at several latitudes across Arsia Mons. Montmessin et al. (2004) for present-day Mars, except for the following modifications. First, in order not to favor any hemisphere, they had set the orbit eccentricity to zero and fixed the reference visible dust optical depth to a constant value (0.2). Second, they had increased the obliquity of the planet to 45 (near the most probable value of 41.8 ; Laskar et al., 2004). Last, to better represent the atmosphere topography interaction, they used a higher horizontal resolution of 2 in latitude and longitude. The simulation showed net accumulation of mm/yr on the western flanks of Olympus, Ascraeus, Arsia, and Pavonis Montes, largely due to adiabatically cooled westerlies blowing upslope. The model was relatively insensitive to obliquity, as long as it was greater than 40, but mildly sensitive to atmospheric dust load, an unknown at high obliquity. These predicted accumulation rates were then used to drive the UMISM, and the resulting modeled ice sheets are compared to the geological evidence. This allows us to assess the validity of the GCM results, and also to assess (1) the spatial geometry in terms of areal extent and ice volume, and (2) the temporal response in terms of how long such a high obliquity climate must exist to create ice-sheet imprints that are in agreement with observed landforms (Figs. 1A, 1B). The GCM results yield ice-equivalent depth (IED) as a function of L s (the solar longitude, measured from the Northern hemisphere spring equinox, where L s = 0) (Fig. 2). We then use these estimates to separate positive accumulation from negative sublimation in our quest for the net mass balance at each point. For each point the IED may rise and fall throughout the year. As it rises, we sum up total annual accumulation (Fig. 3a), and as it falls, we sum up total annual sublimation (Fig. 3b). The difference represents the net annual mass balance, which is needed for the source in UMISM (Fig. 3c). By separating these two components of the mass balance, we allow ourselves the opportunity to shift the balance between accumulation and sublimation. For instance, we generated Fig. 3c by scaling sublimation by a factor of two to more closely match the geologic record. This method also allows for the specification of a climate knob that can be connected to signals such as obliquity variations. Also generated by the GCM and required as an input boundary condition for UMISM is the surface temperature at each point in the domain (Fig. 3d). The simulation uses constant, fixed input from the GCM (mass balance and surface temperatures). Fig. 3e shows the ice thicknesses and Fig. 3f the column-averaged flow velocities with surface elevations superimposed as contours at the end of 500 Ka of model time. While artificial in the sense that these fixed climatic conditions likely did not remain steady for more than 100 Ka (Mischna and Richardson, 2005), the resulting pattern of glaciation shows

6 310 J.L. Fastook et al. / Icarus 198 (2008) Fig. 3. Tharsis Montes region. (a) Ice-equivalent accumulation in m/yr. (b) Ice-equivalent sublimation in m/yr. (c) Ice-equivalent net mass balance in m/yr. (d) Surface temperature in Kelvin. (e) Ice thicknesses in meters. (f) Velocities, in mm/yr, with surface elevations after 500 Ka.

7 DynamicsoftropicalmountainglaciersonMars 311 Fig. 4. Ice volume versus time in millions of km 3. Note approach to steady state. striking similarities to the observed deposits. Approach to steady state is indicated in Fig. 4, which illustrates volume as a function of time. Interestingly, the volume attained after 500 Ka is within 15% of the estimate for the volume of the north polar ice cap, which in the GCM is the source of moisture for the Tharsis precipitation. The ice thicknesses and velocities (Figs. 3e and 3f) show broad agreement with the deposit locations and some aspects of their configurations (Figs. 1A and 1B; Head and Marchant, 2003; Shean et al., 2005), although some differences in detail are observed. For example, the Olympus Mons ice sheet (Milkovich et al., 2006) is too large, and extends too far to the south, compared to the actual deposits. The Arsia Mons ice sheet as a whole extends too far up the flanks (although recent results suggest that accumulation occurred in local cold traps high on the flanks; Shean et al., 2007) and at the same time does not go far enough to the west compared to the actual deposits (Head and Marchant, 2003; Fig. 1B). The Pavonis Mons ice sheet engulfs the peak of the volcano, but matches the northern margin well (Shean et al., 2005). The Ascraeus Mons ice sheet, while slightly too large and with some ice on the peak, matches the deposits reasonably well (Parsons and Head, 2004, 2005; Kadish et al., 2008). These results are in remarkably close agreement with the observations considering (1) the extraordinary topographic relief (14 18 km) of the edifices, and (2) the small scale of the fan-shaped deposits relative to global models (the resolution of the simulation is 15 km, while that of the GCM is 120 km at the equator). By way of comparison, a very good Earth-based atmospheric model, Polar MM5 (Box et al., 2004), produces a mass balance distribution that would yield similar inconsistencies with the current Greenland Ice Sheet, especially along its margins. We are thus very encouraged by these results Parameterized mass balance distribution For existing ice sheets on Earth, the mass balance can be measured. For reconstruction of paleo-ice sheets, however, a parameterization in terms of elevation and location on the planet is typically used. The accumulation part of the mass balance is usually taken to be proportional to the saturation vapor pressure, a measure of how much water the atmosphere can hold. The saturation vapor pressure is an exponential function of temperature; a cold atmosphere is capable of holding much less water, and hence will produce much less snow than a relatively warmer atmosphere. The same principle applies to the martian atmosphere (Jakosky and Haberle, 1992; Zurek et al., 1992). On Earth, the ablation component for most glaciers is typically due to surface melting, resulting in liquid runoff from the ice sheet. This is calculated in the model by imposing a seasonal amplitude onto the mean annual temperature at a specific location and counting positive degree-days (for each day, the difference between the day s average temperature and 0 C, summed for the whole year). The melt rate is proportional to the number of positive degree-days. Because the Earth is colder at higher elevations (the so-called lapse rate, which is usually linear) we commonly observe a small positive mass balance at high elevations (cold, little snow, but no melting), a large positive mass balance at mid elevations (warmer, thus more snow, but still little melting), and then a strongly negative mass balance at low elevations (warm, therefore plenty of snow, but much more melting). This leads to highland growth of glaciers and snow-capped mountains, even at low latitudes (Kaser and Osmaston, 2002). The relationships of massbalance as a function of altitude are also the basis of the concept of an equilibrium line (mass balance positive above and negative below) (Fig. 5a). On Mars, the temperature seldom reaches the melting point of water ice, temperature pressure conditions mean that liquid water is highly metastable, the lapse rate is much smaller [4.5 C/km (Beebe, 2008) versus a terrestrial rate of 9.8 C/km], and the dominant mechanism for mass removal is sublimation. Two sublimation mechanisms, buoyancy- and turbulence-driven, are defined (Ingersoll, 1970; Pathare and Paige, 2005). ( ρ E b = 0.17Dρ w (1 r) E t = 0.002wρ w (1 r). ρ a g v 2 ) 1 3, The buoyancy sublimation rate, E b, depends on D, thediffusion coefficient of water vapor in carbon dioxide, ρ a, the density of the total atmosphere, ρ, the difference between ρ a and the density of the saturated gas at the surface, g, the gravitational acceleration, and ν, the kinematic viscosity of carbon dioxide. Both the buoyancy sublimation rate and the turbulent sublimation rate, E t, depend on the relative humidity, r. Turbulent sublimation also depends on ventilation, represented by w, the wind speed 1 m above the surface. The saturation water vapor density, ρ w, that appears in both sublimation mechanisms is given by the following. ( ) e ρ w = ρ a ε m. P Here ε m is the ratio of the molecular mass ratios of water and carbon dioxide, P is the pressure, and e, the saturation vapor pressure is given by the following ( ) bo e = a o exp, T S where a o and b o are constants. Pressure at some elevation, z 2,is given in terms of the pressure at some lower elevation, P(z 1 ), where T 12 is the temperature difference between the two elevations and m is the molecular weight. [ ] mg(z2 z 1 ) P(z 2 ) = P(z 1 ) exp. RT 12 The important point here is that the overall sublimation rate depends on the saturation vapor density and hence has the same exponential dependence on temperature as does the accumulation component of the mass balance. With a lapse rate, declining temperature with elevation leads to reduction of both accumulation and sublimation rates with the same functional form. However, with pressure in the denominator, sublimation declines less rapidly, leading to overall lower, and possibly negative, net mass balance at the higher elevations. Thus on Mars we might expect snow-filled valleys rather than the snow-capped mountains we see on the Earth. Declining relative humidity, increasing ventilation

8 312 J.L. Fastook et al. / Icarus 198 (2008) mass balance as a function of elevation and 0-datum temperature is indicated by hotter colors. Fig. 5b can be seen as a vertical slice through Fig. 5c. Because both components of the mass balance ultimately depend on the base temperature through the lapse rates, an offset, or climate knob, can be added to that base temperature and a variation in climate achieved. We will later attach this climate knob to the obliquity signal as a way of driving ice ages on Mars. The idea is not to replace the GCM results, which of course are the correctly modeled temperatures and accumulation/ablation rates, but instead to generate a parameterization that captures the essential physics, while lumping all that we do not know into a few adjustable parameters, which can then be connected to some climate driver. Since we also do not know the connection between obliquity and this climate parameterization, we will allow for a linear connection between the two, with this adjustable parameter tuned to yield a match between the deposit extents during some period of the 200 million year calculations for obliquity variation (Laskar et al., 2004) Spatial distribution The above considerations produce a mass balance distribution that depends solely on elevation, but the fan-shaped deposits are not distributed uniformly as a function of elevation, but rather are all oriented preferentially on the northwest flanks of the volcanoes (Fig. 1A). This preferred orientation could be due to a snow tail in the lee of the volcanoes, a cloud shadow that reduces ablation, or more likely to orographic effects as moist air masses ascend the volcano slopes (Benson et al., 2003; Elphic et al., 2004; Haberle et al., 2004; Mischna et al., 2004; Forget et al., 2006). Whatever the cause, the climate parameterization needs some way to produce a spatially non-uniform mass balance. One way to do this is to allow accumulation only where the surface slopes in a particular direction (i.e., with the wind blowing upslope). We form the dot product between a unit vector in a specified direction (northwest) with a unit vector in the direction of the surface gradient (the cosine of the angle between these two vectors). Where this exceeds some threshold (0.9 would allow a range of 25 )we allow accumulation, and elsewhere only ablation. Fig. 6a shows the distribution of net mass balance for Tharsis under these circumstances before the ice sheet has formed Simulations Fig. 5. Mass balance distribution as function of elevation for Earth (a) and Mars (b). The net mass balance is the difference between the positive accumulation and the negative ablation, both of which depend on elevation in slightly different ways. Another way of visualizing this is illustrated in (c), mass balance as a function of 0- datum temperature (x-axis) and elevation (y-axis) for a vertical temperature lapse rate of 4 C/km. The region of positive mass balance as a function of elevation and 0-datum temperature is indicated by hotter colors. (b) can be seen as a vertical slice through (c). (both of which affect sublimation), or some melting at lower elevations can lead to net negative mass balance at low elevation. Thus we might have two equilibrium lines (Fig. 5b), one high and one low, with positive mass balance only in between. Another way of visualizing this is illustrated in Fig. 5c, where the region of positive Climate change on Mars is controlled by spin-axis and orbital variations similar to those thought to cause ice ages on Earth (Imbrie, 1982, 1985; Laskar et al., 2004). The primary influence on Mars is believed to be obliquity, which has much greater variability than on Earth due to the absence of a large moon and proximity to Jupiter. Laskar et al. (2004) have calculated obliquity histories and shown a robust solution for the last 20 Ma (Fig. 6b). The pattern shows a general 120 Ka oscillation with an amplitude of 10, and a notable 2.4 Ma modulation. Prior to 20 Ma, the chaotic nature of the solution makes the calculation of orbital histories unreliable, in that the extreme sensitivity to initial conditions (the present, calculating backwards in time) produces radically different histories with very different overall behavior. Laskar et al. (2004) have published several representative obliquity histories back to 250 Ma. We have used these representative scenarios to drive the ice-sheet model for the last 200 Ma for the Arsia Mons ice sheet. Because obliquity is not directly connected to the temperature that serves as a climate knob in our mass balance parameterization, we have chosen to calibrate this connection by requiring that the maximum areal extent obtained during the 200 Ma simulation match the glacial deposits identified in the various reconstructions (Head and Marchant, 2003; Shean et al., 2005;

9 DynamicsoftropicalmountainglaciersonMars 313 (a) (b) Fig. 6. (a) Central Tharsis Montes; mass balance spatial distribution, in mm/yr. (b) Laskar et al. (2004) obliquity signal for the last 20 Ma. Prior to this period, reconstruction of past climate history is inhibited by the fact that the chaotic nature of the solution makes the calculation of orbital histories unreliable. In Fig. 7, the volume (in km 3 )andarea(inkm 2 ) are plotted as a function of time for the period from 200 Ma to the present under various obliquity scenarios of Laskar et al. (2004). Kadish et al., 2008). For this reason the calibration for each of the Laskar et al. (2004) scenarios will be different, and hence the ice sheet response will occur at different times. By comparing the specific response with age estimates for the Arsia deposit (Shean et al., 2006, 2007), we will attempt to distinguish between different obliquity scenarios. Fig. 7 shows five different obliquity scenarios calculated by Laskar (a, P000_N; b, P001_N; c, P022_N; d, N001_N; and e, N006_N) with the left part of each figure showing the full 200 Ma period modeled, and the right part of each figure showing a magnified time period of interest. On each of these are also shown ice volume (lowest blue line) and areal extent (middle green line) produced by the ice sheet model. We can now use estimates of the ages of the deposits to assess these different candidate climate scenarios. Earlier estimates established the broad Late Amazonian age of the fan-shaped deposits (Zimbelman and Edgett, 1992; Scott and Zimbelman, 1995) and more recent a really comprehensive high-resolution image coverage has permitted more detailed age estimates for the Tharsis tropical mountain glaciers (Shean et al., 2006, 2007; Kadish et al., 2008). For example, Shean et al. (2006) used HRSC image data of Arsia Mons to confirm the Late Amazonian age and suggest that the deposit as a whole formed sometime between 60 and 460 Ma ago, most likely a few hundred million years ago. Additional crater counts for a part of the relatively young smooth facies (a debriscovered, glacier-like fill in one of the proximal graben) suggests that it formed in the last 100 Myr (model age of 65 Ma, maximum age of 115 Ma; for both the Hartmann and Neukum production functions) (Shean et al., 2007). On the basis of these age data, we can now examine the different scenarios derived from the Laskar et al. (2004) obliquity scenarios for the last 200 Ma to see if we can distinguish among them. We can initially rule out scenarios such as P022_N (Fig. 7a), because such scenarios only produce glacial deposits in relatively recent periods (<50 Ma). For example, the Arsia deposits as a whole are most likely a few hundred million years old, and the innermost smooth facies deposits around and near the graben have amodelageof 65 Ma (Shean et al., 2007). Scenario P001_N (Fig. 7b) is also unlikely because the configuration best matching the formation of glacial deposits is also too young, less than 50 Ma. It is worth noting that both P022_N and P001_N, whose full ice configurations occur at a time younger than 50 Ma, show some limited ice in the graben right up to the present during the low-obliquity phase of the last 4 Ma. Such late ice is not in good agreement with the observations and crater-count dating (Shean et al., 2007). Scenarios P000_N and N001_N (Figs. 7c and 7d) are in good agreement with the crater-count data (Shean et al., 2007). Both show significant ice sheets before 100 Ma. In P000_N, mean obliquity is in excess of 45 from the beginning of the simulation (200 Ma) to 120 Ma, for a duration of 80 Ma. In N001_N, mean obliquity is in excess of 45 from 160 Ma to 115 Ma, for a duration of 45 Ma. In addition, following the shift to the lower-obliquity phase, both P000_N and N001_N have limited ice, compatible with the young deposits in the Arsia graben, but no ice after 4 Ma. The most compelling differences between P000_N and N001_N are that N001_N (Fig. 7d) is shorter duration and only shows four distinct ice-sheet growth events, whereas P000_N (Fig. 7c) is longer duration, and has many. Scenario N006_N (Fig. 7e) is very different from the previous two, both of which showed episodic glaciation. N006_N, because of its generally high obliquity, has more or less consistent glaciation at close to the maximum extent for most of the time prior to 50 Ma, a duration of 150 Ma. During this period, there are constant variations, manifest as minor advances and retreats of the margin, driven by the higher-frequency 120 Ka obliquity signal, but the ice sheet never fully disappears as it did in P000_N and N001_N (Figs. 7c and 7d). In our simulation beginning at 200 Ma, the ice sheet grows rapidly to a configuration consistent with the deposits, and then over the next 150 Ma retreats very gradually, with a more rapid termination at 50 Ma. After 50 Ma, ice cover is limited and consistent with the limited ice deposits around the graben. After 4 Ma, when the obliquity signal drops to its lower level, no ice accumulation is evident anywhere. Fig. 8a shows the configuration for the N001_N peak at Ma (Fig. 7d), with ice thicknesses displayed as colors and surface elevations indicated by contours. This is the configuration used to calibrate the relationship between obliquity and climate knob, and thus fits the glacial deposits, whose observed configuration is indicated by the dark outline. Also shown in Fig. 8b is a typical late configuration (9.1 Ma) that could be consistent with the relatively young age of features within and around the Arsia graben (Shean et al., 2007). Accumulation of this small amount of

10 314 J.L. Fastook et al. / Icarus 198 (2008) Fig. 7. Volume (blue lines, in km 3 ) and area (green lines, in km 2 ) are plotted as a function of time for the period from 200 Ma to the present under various obliquity scenarios (red lines, in degrees) of Laskar et al. (2004) for the Arsia Mons tropical mountain glacier. The left panel of each plot shows the entire modeled period and the right panel shows a specific 50 Ma subset of the period. (a) Scenario P022_N, showing how the obliquity signal drives resultant ice-sheet volume and area. The left figure shows the entire modeled period, and the right shows the final 50 Ma. (b) Scenario P001_N. (c) Scenario P000_N, which is similar to age estimates of robust Arsia ice sheets early, some ice in the graben area late, but no ice after 4 Ma. The right panel shows the episodic nature of the ice-sheet in detail. (d) Scenario N001_N also displays robust early ice, some ice in the graben area late, and no ice after 4 Ma. In this case there are only four significant growth episodes around 150 Ma (see details in right panel). (e) Scenario N006_N shows a consistent ice sheet at close to its maximum extent for most of the modeled period. The ice sheet experiences rapid retreat after 50 Ma, and there is no ice in the graben area after 4 Ma.The right panel shows the types of small variations in volume and areal extent that could be responsible for the formation of the ridged facies (Fig. 1B). ice disappears after 3.7 Ma, when obliquity has shifted to its lower state. The dynamic nature of the buildup and decay of such scenarios can be seen in the movie in the online supplemental material. Comparison of these obliquity scenarios and model results illustrates the utility of the models in assessing realistic obliquity scenarios, and finding those that are broadly consistent with the geological and chronological analyses. It is clear that this approach can serve to help distinguish among families of obliquity solutions in a time period otherwise unpredictable from the Laskar et al. (2004) analysis. Thus, in a time period prior to the robust solutions of the last 20 Ma, when precise solutions are not possible due to the fact that the chaotic nature of the solution makes the calculation of orbital histories unreliable, geological data linked to spin-axis/orbital parameters causing climate change can be used to refine these predictions, much as cyclic aspects of the geological record on Earth is used in the deconvolution of Earth s climate history (Imbrie and Imbrie, 1979; Lourens et al., 2001; Palike and Shackleton, 2000; Palike et al., 2004). Further analysis of the geological data for the Tharsis tropical mountain glaciers is providing a much more detailed internal stratigraphy and relative chronology for the Arsia Mons tropical mountain glacial deposits that can be used, in concert with impact crater chronologies (Shean et al., 2006, 2007), to link to more detailed and precise obliquity histories (such as the right-hand side of the scenarios in Fig. 7). In summary, the combination of (1) obliquity history predictions, (2) geological analysis and interpretation of deposits containing glacial climate signals, and (3) ice-sheet modeling, is providing a powerful tool for understanding the climate history of Mars and refining our knowledge of the spin-axis/orbital parameters in the past history of the planet. 4. Conclusions 1. The properties of the martian atmosphere make the formation, accumulation, and behavior of snow and ice different from those on the Earth in significant ways (Figs. 5a and 5b). Such distinguishing characteristics include dual equilibrium lines, ice deposition on flanks as opposed to summits of volcanoes, and the dominance of cold-based glaciation.

11 DynamicsoftropicalmountainglaciersonMars 315 Fig. 7. (continued) (a) (b) Fig. 8. (a) The configuration (in this case, the peak in N001_N at Ma) that matches the outermost ridged facies. It is this configuration that serves as the target for calibrating the connection between the obliquity signal and the climate knob. (b) A late configuration (N001_N at 9.1 Ma) generally compatible with the deposits around the graben at Arsia Mons (Shean et al., 2007). 2. Geological analysis of fan-shaped deposits on the northwest flanks of Olympus Mons and the Tharsis Montes volcanoes, together with terrestrial analogs of cold-based glaciers, supports the interpretation that these features represent the deposits of Late Amazonian tropical mountain glaciers (Head and Marchant, 2003; Shean et al., 2005, 2007; Milkovich et al., 2006; Kadish et al., 2008). 3. Atmospheric general circulation models (GCM) show that during periods of high obliquity the upwelling and adiabatic cooling of moist polar air will deposit snow on the northwest flanks of the Tharsis Montes to form tropical mountain glaciers (Forget et al., 2006). 4. An ice sheet driven by the static results of a GCM run for Mars at high-obliquity matches the boundaries of the observed tropical mountain glacier deposits very well considering the limited resolution of the GCM. Equilibrium volumes of the simulated ice-sheet complex are comparable to the current volume of the north polar cap, among the sources of the tropical mountain glaciers. 5. Using these guidelines, models of mass balance and spatial distribution on the western flanks of Tharsis Montes lead to

12 316 J.L. Fastook et al. / Icarus 198 (2008) patterns that are strikingly similar to the geological evidence for ice accumulation and glacial flow (Fig. 8). 6. Five scenarios from the Laskar et al. (2004) calculation of possible past obliquity histories are used to drive an ice-sheet model with a parameterization of climate that allows for variation as the obliquity changes and as the ice sheet grows and shrinks. Two, P022_N and P001_N (Figs. 7a and 7b), are rejected because they do not agree with crater-count dating of the Arsia deposit (being too young; Shean et al., 2006, 2007), and produce too much ice in the most recent times, which is not observed. Two, P000_N (Fig. 7c) and N001_N (Fig. 7d), are acceptable, both in the timing of the maximum ice configurations ( 100 Ma before present) and in the limited configuration of ice during the last 20 Ma. Both of these produce episodic ice-sheet events. One, N006_N (Fig. 7e), is also acceptable but differs from P000_ N and N000_N in that it produces an ice sheet that persists for over 150 Ma with only minor fluctuations in area, a scenario which may well help to explain the formation of the closely spaced drop moraines of the outer ridged facies. 7. Future, more detailed geological and chronological reconstructions of the tropical mountain glacial deposits can be used to link to more detailed and precise obliquity histories (such as those seen on the right-hand side of the scenarios in Fig. 7). 8. A powerful tool for understanding the climate history of Mars and refining our knowledge of the spin-axis/orbital parameters in the past history of the planet is the combination of obliquity history predictions, geological analysis and interpretation of deposits containing glacial climate signals, and ice-sheet modeling. Acknowledgments We gratefully acknowledge the financial support from the National Aeronautics and Space Administration under the Mars Data Analysis Program (NNG05GQ46G), the Mars Fundamental Research Program (GC196412NG-A), and the Mars Express Mission Participating Scientist Program (JPL ) through grants to J.W.H. and the Mars Fundamental Research Program (NNX06AE32G) to D.R.M. A portion of this work was supported by the National Science Foundation, Office of Polar Programs, through grants to J.L.F. (ANT , ANT , and ANT ) and D.R.M. (OPP ). Very special thanks are extended to David E. Shean who helped considerably in discussions during earlier phases of this study in relation to his work on the geology and chronology of the Tharsis tropical mountain glaciers, and in providing thoughts about how these linked to climate cycles and related models. 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