Interior Temperatures of the Northern Polar Cap on Mars

Icarus 144, 456–462 (2000) doi:10.1006/icar.1999.6296, available online at http://www.idealibrary.com on

Interior Temperatures of the Northern Polar Cap on Mars Janus Larsen and Dorthe Dahl-Jensen Department of Geophysics, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen Ø, Denmark E-mail: [email protected] Received February 3, 1999; revised August 26, 1999

Knowledge about the interior temperatures of the polar caps on Mars is crucial for the understanding of stratigraphy and dynamics of the polar caps. The insolation and thereby the temperatures of the surface are strongly influenced by factors including obliquity, conductivity, and albedo. Basal temperatures are mainly determined by surface temperature, cap thickness, thermal conductivity, and areothermal (Areo, Greek for Mars) heat flux. This paper presents a model for the northern polar cap on Mars, where temperature profiles are determined as a function of orbital parameters (obliquity and eccentricity), thermal conductivity, areothermal heat flux, and cap thickness. Furthermore we will look at the thermal behavior of the upper layer (∼100 m) of a martian ice cap. There is strong evidence that the northern polar cap consists mainly of H2 O ice. In the model we will treat the cap as that of H2 O ice with a high dust content and the possibility for solid CO2 or CO2 clathrates. The temperature model shows that the ice is most likely flowing and that the flow rate oscillates with a time lag of 30–70 Kyrs compared to the surface temperature obliquity oscillation. The possibility for measuring the areothermal heat flux by measuring the temperature in the upper 2 m of the ice cap is not very good, but measurements of temperatures through the top 2 m of the ice could provide valuable information on the conductivity of the ice and thereby the constituents of the Mars ice cap. °c 2000 Academic Press Key Words: Mars surface; thermal histories; ices.

INTRODUCTION

In this paper we present a model of temperatures in the northern polar cap on Mars. We assume that the polar remnant cap consists mainly of H2 O ice with impurities (e.g., dust, CO2 clathrates). This is justified by the following observations; high water vapor concentrations over the summer remnant cap (Farmer et al. 1976), summer temperatures greatly exceeding the CO2 frost point (Kieffer et al. 1976), a density of approximately 1 kg/m3 (Malin 1986), a thermal inertia in the range 600–2000 J 1 m−2 s− 2 K−1 (Paige et al. 1994), and an albedo of approximately 0.4 corresponding to dirty water ice (Kieffer et al. 1976). Some work has previously been done trying to determine the basal temperatures of the martian polar caps (e.g., Clifford 1987, Budd et al. 1986), and these have shown that the possibility for basal melting exists. Basal melting can occur when the ice cap

is thick and the conductivity is low. The interior and especially the basal temperature have great influence on the deformation rate of ice. It is essential to determine if the polar cap is flowing or not in order to understand the stratigraphy and to interpret the climate changes that may have been recorded in the polar deposits. The recent MOLA (Mars Orbiter Laser Altimeter) measurements have provided surface elevation data, with a precision far better than any previously obtained. The inclination of the MGS (Mars Global Surveyor) does not normally allow for measurements at latitudes higher than 86.3◦ , but during 2 weeks in June and July 1998 the MGS was tilted approximately 50◦ , allowing the MOLA instrument to cover the north pole (Zuber et al. 1998). These data allow reasonable accurate estimates for the cap thickness, which is one of the key factors in determining the basal temperatures. Since drilling a deep core in the martian polar deposits will not be possible until well into the future, we will focus on what there is to learn by looking at the top layers of the polar cap. It may be possible within the next decade to measure the temperature in the top 2 m during a martian year. The possibility of evaluating areothermal heat flux from temperature measurements of the martin regolith has been modeled by Mellon and Jakosky (1992). In this paper we will evaluate if a 2-m temperature profile of martian polar ice can help to constrain the values of areothermal heat flux and thermal conductivity of the polar ice. To examine these problems we will set up two models: The first calculates the temperature profile through the ice cap at long time scales (Myrs), while the second models the temperature variation at short time scales (yrs) in the upper 100 m of the ice cap. All years in this paper are martian years = 686.98 Earth days. DISCUSSION OF PARAMETERS

Conductivity Impurities in the ice change the thermal conductivity. Even though the conductivity of clay minerals is generally higher than that of ice, the mixture between clay and ice can have a low conductivity. This is due to thin water films on the dust particles. The water films can exist down to temperatures as low as 250 K (Anderson et al. 1967), and might therefore be important in the

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martian polar caps. The dust concentration in the Greenlandic ice sheet is 10 times as high as in the Antarctic ice sheet. This is due to the fact that the continents where dust is produced is closer to the Greenlandic ice sheet than to the Antarctic ice sheet. Considering that the polar caps on Mars are completely surrounded by very dusty continents, the observation of frequent large dust storms on Mars and that the caps show visible dust bands, it is likely that the martian polar caps have a higher dust content than the terrestrial ice sheets. The thermal conductivity of an ice–clay mixture lies between 0.5 and 3.5 W m−1 K−1 (Clifford 1987). At low obliquity climates the temperatures at the poles will be considerably lower than today, and all the CO2 frost will not sublime during the summer. This suggests that the layers deposited under low obliquity climates have high CO2 content. The conductivity of solid CO2 or CO2 clathrates is 0.4 to 0.6 W m−1 K−1 (Mellon 1996). If CO2 is present in the ice this will lower the conductivity considerably. The thermal conductivity also depends on the density of the ice, so that low-density ice or snow will have a lower conductivity. If the martian polar cap has a firn1 layer, like Earth’s ice sheets, it will act as an insulating layer resulting in higher ice temperatures. The firn effect is of the order 2–3 K. Thermal inertia data (Paige et al. 1994) show that the surface of the polar cap is consistent with dense water ice and densification models predicts solid near-surface ice (Arthern et al. 2000). This suggests that the firn effect is negligible. Based on these conductivity considerations we have chosen to examine the temperature profile with conductivities in the interval 0.5 to 2.0 W m−1 K−1 . Cap Thickness Recent MOLA (Mars Orbiter Laser Altimeter) data show that the polar cap rises approximately 3 km over the surrounding terrain. As the bedrock elevation is unknown, it has to be estimated from the surrounding terrain and assumptions of possible isostatic depression. Zuber et al. (1998) estimated the relaxation time for the present cap to be ∼105 years, much shorter than the presumed age of the polar ice cap. This argues in favor of including isostatic depression when estimating the cap thickness. The compensation depends on the rigidity of the lithosphere, with maximum depression in isostatic equilibrium given by the ratio between the densities of ice and bedrock. Zuber et al. (1998) have estimated the depression to be between 500 and 1200 m. To include the possibility that the polar cap is situated on a small bulge, we will examine the temperature distributions using cap thickness in the interval 2.5 to 4 km, with 4 km corresponding to full isostatic depression of a low-rigidity underlying bedrock. Albedo, Surface Temperatures, and Areothermal Heat Flux At the north pole of Mars there is no incoming radiation during winter from the Autumnal equinox (L s = 180) to the Vernal 1

Firn, snow in the intermediate stages of transformation from snow to ice.

equinox (L s = 0). In the other half of the year the incoming radiation is given by a sine function modulated by the eccentricity of the orbit of Mars. There is no daily variation in the incoming radiation. In the autumn when the insolation decreases, high albedo CO2 frost condenses on the surface when the temperature reaches the frost point of CO2 . We assume a surface temperature of 148 K when CO2 frost is present (Paige et al. 1994). When insolation increases during the spring the CO2 frost sublimes and the lower albedo remnant H2 O ice is revealed. The maximum summer surface temperature is around 205 K (Kieffer et al. 1976), and the period of a CO2 frost-free surface is approximately 190 martian days long (James et al. 1992). The albedo of the H2 O-ice is 0.4 (Kieffer et al. 1976), and the albedo of the CO2 frost is assumed to be 0.7. In approximately half of the period where the north pole receives radiation the surface is covered with CO2 frost. This yields an insolation period averaged albedo of 0.55. Estimates of the areothermal heat flux is 3 × 10−2 to 4 × 10−2 W m−2 (Clifford 1987), approximately half of the geothermal heat flux on earth. Where nothing else is written, we have chosen to use 3 × 10−2 W m−2 in our model runs. MODEL 1

A simple conductivity model solved by a Crank–Nicholson finite difference scheme investigates the effect of long time scale temperature variations due to the change in obliquity and eccentricity. The surface temperature is controlled by incoming and outgoing radiation, and at bedrock the heat flow is assumed to be the areothermal heat flux. Heat generated by deformation is assumed to be negligible. This can be justified by estimating the deformation energy as the product between shear stress and shear strain rate (Paterson 1994). Using a surface slope of 1%, an ice thickness of 3 km, and a value for the flow law parameter A of 7.3 × 10−27 s−1 pa−3 the deformation energy becomes less than 1 mW m−2 or a few percent of the areothermal heat flux. The estimation of the deformation energy is very sensitive to the choice of A. The heat equation is k ∂2T ∂T , = ∂t ρc ∂z 2

(1)

where T is the temperature, k the conductivity, ρ the density of the ice, and c the specific heat capacity. At the surface the temperature is given by the radiative energy balance, I¯ in (1 − A) = Iout ,

(2)

where I¯ in in is the incoming short wave radiation, I¯ in (1 − A) is the absorbed incoming radiation, Iout is the outgoing long wave radiation and A is the albedo.

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The annual average incoming radiation is given by (Ward 1974) I¯ in =

SM p sin(θ), π (1 − e2 )

(3)

where SM is the solar constant at Mars, e is the eccentricity, and θ is the obliquity. The obliquity oscillates with a period of 120 Kyr modulated by a 1.2-Myr amplitude change (Ward 1974). The change in incoming radiation caused by variations of the eccentricity is considerably smaller than the change caused by variations of the obliquity. Stefan–Boltzmanns law determines the outgoing radiation, 4 , Iout = εσ Tsurface

(4)

where σ = 5.67 × 10−8 W m−2 k−4 is Stefan–Boltzmanns constant and ε is the emissivity, which is taken to be unity. At the bedrock the boundary condition is ∂T Q =− , ∂z k

(5)

where Q is the areothermal heat flux. Results Model 1 The thermal wave from a short obliquity period (120 Kyr) is damped considerably as it propagates down through the ice. The annual average surface temperature difference between high obliquity (34◦ ) and low obliquity (15◦ ) is 26.6 K. This difference is damped to a 1.5◦ –9 K change at the base of the ice cap depending on the configuration of conductivity and specific heat capacity. The time lag between the surface and basal variations are 30 to 70 Kyr for the 120-Kyr obliquity period. Table I lists the time lag, the maximum temperature difference at the base of the ice cap during a short obliquity period, and the temperature

TABLE I Maximum Temperature Difference at the Base of the Ice Cap, Time Lag, and Damping Factor for a Surface Thermal Wave, with an Amplitude of 26.6 K and a Period of 120 Kyr (a Short Period Obliquity Wave) Specific heat capacity (J kg−1 K−1 )

Time lag (Kyrs)

1T (K)

Damping factor

1

1000 2000

47 66

4.1 1.5

6.5 17.7

1.5

1000 2000

39 54

6.6 2.9

4.0 9.2

2

1000 2000

34 48

8.8 4.1

3.0 6.5

Conductivity (W m−1 K−1 )

Note. Model parameters: cap thickness, 3 km; areothermal heat flux, 0.03 W m−2 ; ice density, 917 kg m−3 .

damping factor for different configurations of conductivity and specific heat capacity. The damping factor listed in Table I is the ratio between the amplitude of the surface temperature oscillation and the amplitude of the basal temperature oscillation during a short obliquity period. Changing the ice density and the ice thickness will affect the time lag and the damping factor of the thermal wave. Since propagation velocity is inversely proportional to the square root of the density, increasing the density by 20% will increase the time lag with approximately 10%. The amplitude, and thereby the damping factor, of the thermal wave decreases exponentially with ice thickness. Figure 1a shows the temperature profile at high and low obliquity and Fig. 1b shows surface and basal temperature as a function of time. In Fig. 2 four contour plots of the basal temperature are shown for different ice cap thicknesses. The contours are plotted as a

FIG. 1. (a) Temperature profiles at high and low obliquity. Note that at high obliquity, where the surface is warmest, the basal temperature is low. The warm thermal wave has not yet propagated to the bed. At the low obliquity the opposite is observed: While the surface is cold the base is still warm. (b) Surface and basal temperatures for the past 2.5 Myrs. The temperature scale on the right is for the surface temperature (thin lines), and the one on the left is for the basal temperature (thick lines). On the figure one can see the lag of the thermal wave, so the basal temperature is out of phase with the surface temperature. Model parameters: ice cap thickness, 3 km; thermal conductivity, 1.5 W m−1 K−1 ; Specific heat capacity, 1500 J kg−1 K−1 ; areothermal heat flux, 0.03 W m−2 .

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FIG. 2. Maximum basal temperatures as a function of cap thickness (H ), areothermal heat flux, and thermal conductivity for the present obliquity.

function of conductivity and areothermal heat flux. The three factors play the dominating role in determining the basal temperatures. White areas, representing basal melting, take up a large part of the figures, revealing that even if the surface temperatures are far below that of Earth and the areothermal heat flux is half of the geothermal heat flux, the possibility for basal melting is still present. This is partly due to the possibility of using low conductivities which yield greater temperature gradients and thus higher basal temperatures. The lack of advection of heat allows greater temperature differences between surface and bedrock as no cold ice is transported downward. Downward advection has a large influence on the temperatures in terrestrial ice caps (e.g., Paterson 1994), but has been ignored in this model due to the very low annual accumulation (Thomas et al. 1992). Neglect of the advective term can be justified by looking at the Peclet number, which measures the relative importance of advection and conduction, Pe = a

H , κ

(6)

where Pe is the Peclet number, a is the accumulation, H the ice

thickness, and κ the thermal diffusivity. Using κ = 30 m2 s−1 and H = 3 km we see that the accumulation cannot be higher than 1 × 10−2 before the advection must be taken into account. This is a good approximation for the present climate, but could be questionable when dealing with different climates where the accumulation is unknown. The melting point temperature in Fig. 2 is set to 273 K. The true melting point is unknown because pressure and the unknown impurity concentration, depress the melting point temperature. The effect of pressure is small (8.7 × 10−4 K m−1 (Paterson 1994)), but the presence of impurities like salt present the possibility of brine pockets forming at low temperatures. MODEL 2

With this model we want to examine the temperature in the upper 100 m of the north polar cap. The same heat equation (1) is used. The surface temperature here is controlled by the heat budget in the top layer, which we assume to be 1.5 cm thick (Toon et al. 1980). The penetration depth of the incoming radiation determines the thickness of the top layer. Furthermore we assume that all the incoming solar radiation is absorbed in this

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top layer and that all the outgoing radiation radiates from this layer. We assume energy balance in which the balancing terms are the incoming radiation, the outgoing radiation, and the heat conduction down through the ice. When the temperature drops below 148 K CO2 condenses on the surface. The surface temperature of 148 K is maintained until all the CO2 is evaporated again. The governing equation for the surface layer is dT dm dT I¯ (1 − A) − εσT 4 − k +L = ρc dz, dz dt dt

(7)

where I¯ is the daily average insolation, A the albedo, ε the emissivity, σ Stefan–Boltzmanns constant, k the conductivity of the ice, L the latent heat of CO2 (590,000 J kg−1 ), m the amount of CO2 frost, ρ the density of the ice, c the specific heat capacity, T the temperature, and dz the thickness of the surface layer. The daily average insolation I¯ is given by (Ward 1974), [1 + e cos(υ)] I¯ = SM [η sin(ε) sin(δ) + sin(η) cos(ε) cos(δ)], π (1 − e2 )2 (8) 2

where δ is the latitude, ε and η are the latitude and the longitude of the sun, and υ is the true anomaly of Mars, which measures the orbital position of Mars from perihelion. At the base the boundary condition is the areothermal heat flux as in model 1. A Crank–Nicholson scheme is used again, with vertical layer thickness increasing downward from 0.015 m to 4 m, and time steps of 0.1 martian day. The calculations are continued for 400 years until the results have stabilized in the top 100 m of interest. Results Model 2 The model reproduces observations of the surface temperature: the surface is covered with CO2 frost approximately half of the polar daylight period and the maximum summer surface temperatures reach 205 K (James et al. 1992). In Fig. 3a temperature profiles for the upper 30 m are plotted, and the figure shows how the annual thermal wave is damped as it propagates down through the ice. Looking at the autumn curve we see that summer warmth has reached 5 m and cold from the previous winter has reached 17 m depth. The upper 2 m of the temperature profile during the martian summer is plotted for different conductivities in Fig. 3b. Lowering the conductivity slows down the propagation of the thermal wave considerably, which has a large impact on the temperature in the upper layers where the thermal gradients are greatest. The surface temperature is dependent on the thickness of the top layer, which was also observed by (Toon et al. 1980). CONCLUSIONS AND DISCUSSIONS

The basal temperature of the northern polar cap on Mars is similar to temperatures in the terrestrial ice caps, i.e., close to

FIG. 3. (a) Temperature profiles in the upper 30 m at different seasons. Model parameters: thermal conductivity, 1.5 W m−1 K−1 ; specific heat capacity, 1500 J kg−1 K−1 ; areothermal heat flux, 0.03 W m−2 . (b) Summer temperature profiles for different conductivities in the upper 2 m. The figure shows that lowering the conductivity makes the thermal wave propagate slower. In 1 m depth the temperature difference between k = 0.5 and k = 2 is more than 15 K. The curve with k = 1.5 W m−1 K−1 is identical to the top 2 m of the summer curve (a). Model parameters: specific heat capacity, 1500 J kg−1 K−1 ; areothermal heat flux, 0.03 W m−2 .

the freezing point. Most of the deformation in an ice cap takes place in the basal layer, where the shear stress has its maximum and temperatures are highest (Budd et al. 1986). Ice thickness of the northern polar cap on Mars is approximately the same as the thickness of the terrestrial ice caps, that is 3–4 km (Greenland, 3 km; Antarctica, 4 km). Resent MOLA data shows that the surface profile is close to that obtained from a plastic flow law (Zuber et al. 1998), as we observe on Earth. These facts lead us to the conclusion that the ice cap is flowing. The fact that we observe a very smooth surface with very few craters (Zuber et al. 1998, Clifford 1987) also favors the interpretation that the cap is flowing. The smoothness of the surface could be accounted for by mechanisms other than flow. High ablation rates or recycling of surface material by sublimation and condensation would obliterate surface features like impact craters. Due to the long-term climate oscillations caused by astronomical effects, the polar caps have been deposited under different climates. This opens the possibility for horizontal layers of different conductivities. If layers of low conductivity exist, e.g., solid CO2 , CO2 clathrates or ice with a high impurity content, then this will create warmer bottom temperatures thereby enhancing the flow. An argument against the conclusion that the ice cap is flowing is the lack of a depositional mechanism. Some modeling on

ICE CAP TEMPERATURES ON MARS

ablation/accumulation on the ice surface has been done (Thomas et al. 1992, Jakosky et al. 1993), and this work revealed that the ice cap probably is ablating under the present climate. An explanation for this could be that different conditions have existed under higher/lower obliquity, so that the ice cap has accumulated under different climatic conditions than we see today. Mass balance estimates range between ablation rates of 8 × 10−2 g cm−2 yr−1 and accumulations rates of 2 × 10−2 g cm−2 yr−1 (Thomas et al. 1992). Even though accumulation rates probably where much higher under past climates, accumulation rates are still much lower than terrestrial values, and from this we must conclude that the flow is very slow compared to the flow in terrestrial ice sheets. The range of possible basal temperatures have been modeled to be between 200 and 273 K. In terrestrial ice caps Glens law expresses the flow, where the flow constant A is dependent on temperature: A = A0 exp(−Q/RT ), where Q (≈ 50 kJ K−1 ) is the activation energy for creep, R the universal gas constant, and T the temperature (Paterson 1994). The deformation rates are thus very dependent on temperature and can vary with a factor of 1000 for the suggested basal temperatures. Even though the modeling indicates that the ice is flowing, the actual deformation rates are still uncertain. The thermal waves created by the change of obliquity propagate down through the ice. When a warm thermal wave from a high obliquity period reaches the basal layers, the deformation rate is increased, but by how much? The ratio of the deformation rate between the coldest and the warmest basal temperatures during a short obliquity cycle is · µ ¶¸ 1 Q 1 ε˙ cold = exp − , ε˙ warm R Twarm Tcold

(9)

where ε˙ is the deformation rate. The difference between Tcold and Twarm will not exceed 10 K as shown in Table I. Using Q = 50 kJ K−1 , a typical terrestrial value, the deformation rate can increase by a factor of 2–3. The time lag is approximately half of that of the short obliquity period, so the maximum deformation would take place when the obliquity is low. As illustrated in Fig. 4, one could imagine an ice cap that under warm, high obliquity conditions has a negative mass balance as more ice sublimates from its surface than what accumulates. The cold from the previous low obliquity period has reached the bottom and is slowing down the flow rate. Half an obliquity cycle later, the surface temperature has decreased, the ice cap is gaining mass and the basal temperature has increased, thereby enhancing the flow. This gives a picture of an oscillating ice cap. Under high obliquity the ice cap decreases its horizontal extent due to low flow rates and mass loss from the surface and under low obliquity the ice cap increases its horizontal extent due to high flow rates and mass gain. In order to ascertain what information can be gained from observations of temperature in the top 2 m, calculations have been

461

FIG. 4. (a) Surface and (b) basal temperatures during a short obliquity period. Possible mass balance for the ice cap is indicated. The mass balance is correlated to the surface temperature, where as the flow rates are proportional to the basal temperature. Because of the time lag between surface and basal temperature, the flow rates will be high when the mass balance is negative.

made for different values of the conductivity and areothermal heat flux. The annual average, at any depth, is dependent on the areothermal heat flux. This effect is small, however, so a change of the areothermal heat flux from 0.02 to 0.04 W m−2 would change the temperature by less that 0.1 K 2 m below the surface. Internal layers of varying conductivity would change the annual average temperature at any depth, so it would be very difficult to estimate the areothermal heat flux from measurements in the upper 2 m. Similar conclusions was reached by Mellon and Jakosky (1992) for martian regolith. Looking at the temperature variation over the year we see a large effect of the conductivity. For low conductivities the warm summer thermal wave propagates slower than that for high conductivities. Changing the conductivity of the ice from 0.5 to 2.0 W m−1 K−1 changes the temperature at 1 m below the surface by more than 15 K in the summer season. This could be used to determine the conductivity of the ice in the upper layers. It should be noted that even though the conductivity can be derived from a single temperature measurement by assuming the last few month’s surface temperature to be known, a thermistor string would have to be buried in the ice for some time in order to reestablish the original temperature profile. Temperature measurements in the top layers would provide valuable information about the conductivity of the ice. Another possible way of obtaining information on conductivity of the Mars polar ice is to use systems that determine conductivity from cooling after an impact. Such systems are mounted on the Deep Space 2 probes (DS2) currently en route to Mars. The two DS2 probes are designed to measure the martian regolith, but similar systems designed for ices would be of great value to the polar research of Mars.

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ACKNOWLEDGMENTS We thank Stephen Clifford and the Lunar and Planetary Institute for support to attend The First International Conference on Mars Polar Science and Exploration, where this work was initiated. We also thank David Fisher and Gary Clow for helpful comments.

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