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[1] If the large, high-albedo surface feature on Titan's leading hemisphere is an elevated “continent” composed mainly of water ice, it will deform under its own weight. We present a model for the axisymmetric spreading of this hypothesized continent based on the similarity solution of Halfar [1983] and the approach of Nye [2000]. We find that the thickness of the model continent is dictated by the spreading time, whereas topographic relief is controlled primarily by isostatic effects. For a mantle density of 1000 kg m^{−3}, the relief of an isostatically supported continent on Titan is not likely to exceed ∼3–7 km.

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[2] Albedo maps of Titan's surface [e.g., Meier et al., 2000; Gibbard et al., 1999; S. G. Gibbard et al., Titan's surface and atmosphere at 2 μm with the W. M. Keck Telescope adaptive optics system, submitted to Geophysical Research Letters, 2004; A. H. Bouchez and C. H. Griffith, A 2.0 micron map of Titan's surface, submitted to Geophysical Research Letters, 2004] show a relatively bright feature with a radius of ∼1000 km centered on the equator on the leading hemisphere. Gibbard et al. [1999] note that the magnitude of its reflectance values and the difference in reflectances at 1.6 and 2.1 μm are consistent with a mixture of water ice and rock. More recent observations by Griffith et al. [2003] show that 0.8–5.0 μm spectra of Titan's leading hemisphere resemble spectra from Ganymede's icy surface. The remainder of Titan's surface is much darker, and is thought to be inundated or blanketed by hydrocarbons.

[3] Several cataclysmic explanations for the exposure of water ice have been proposed, including impact excavation of subsurface ice layers, cryovolcanism, and tectonic rifting [Smith et al., 1996; Meier et al., 2000]. Most of these scenarios have been dismissed as unlikely [Smith et al., 1996]. The most favored explanation is that the bright feature is an area of high topography on which hydrocarbons do not accumulate, perhaps because it is washed clean by precipitation [e.g., Griffith et al., 1991; Lorenz, 1993].

[4] A large region of elevated terrain would have important consequences for atmospheric and surficial processes [e.g., Lorenz, 1993]. In this paper we present an analytic model that relates the topographic relief of a water ice continent on Titan to the age of the continent and the density contrast between the continent and the mantle. Because the solution is not strongly age-dependent, topographic measurements by the Cassini spacecraft could help to constrain the physical properties of Titan's crust and upper mantle.

2. Model

2.1. Similarity Solution to Radial Flow Equations

[5] We model the time evolution of an ice continent on Titan as the viscous spreading of an axisymmetric body of ice with constant volume. Our approach is based on the analysis of Halfar [1983], which was extended in a planetary context by Nye [2000] and Nye et al. [2000]. Here we provide an overview of the governing equations and the approach used to obtain the solution; for a detailed derivation, the reader is referred to Nye [2000]. The continuity equation for incompressible, radial flow is

where q is the radial ice flux per unit circumferential width, r is the radial coordinate, h is the thickness of the ice sheet at radius r, and t is time. This coordinate system is illustrated in Figure 1. Because the thickness of an ice continent is likely to be much smaller than the radius of the bright spot, we neglect the curvature of Titan's surface.

[6] The style of spreading depends on the relative viscosities of the continent and the mantle [e.g., Dorsey and Manga, 1998]. If the continent is much less viscous than the mantle—we will refer to this as the “low viscosity case”—the radial velocity at the base will be close to zero, and spreading will be dominated by shear stresses imposed by the ice surface slope, much like a continental ice sheet on Earth. This is the expected scenario if, for example, Titan's upper mantle contains a significant fraction of methane clathrate hydrates [e.g., Stevenson, 1992], which are much stronger than water ice [Durham et al., 2003]. If the continent is much more viscous than the mantle—the “high viscosity case”—the basal shear stress will be vanishingly small, and spreading will be governed by longitudinal stresses, as in a floating ice shelf. This would be the case if, for example, the ice continent were floating on a liquid H_{2}O-NH_{3} layer [e.g., Grasset et al., 2000; Stevenson, 1992]. We explore both endmember cases below.

[7] In the low viscosity case, the radial velocity u_{r} is related to the radial shear strain rate _{zr} by

where z is depth below the ice surface, A and n are constants, E is the activation energy for the ice deformation mechanism, V is the activation volume, P is hydrostatic pressure, R_{G} is the ideal gas constant (8.3143 J mol^{−1} K^{−1}), T is temperature and τ is the shear stress driving the deformation. (See Section 2.3 and Table 1 for parameter values.) The corresponding expression for the radial extension rate _{rr} in the high viscosity case is

where σ is the effective normal stress driving the deformation. Equations (2) and (3) describe the temperature-dependent, non-Newtonian rheology of ice and are supported by many field and laboratory observations [Paterson, 1994]. The factors to the left of A account for the fact that experimentally determined values for this parameter are derived from uniaxial compression experiments [see Nye, 1953]. Because PV ≪ E near Titan's surface, we ignore the pressure term in subsequent steps.

where g is gravitational acceleration (1.35 m s^{−2}) and ρ is the density of the continent. Note that equation (5) gives the driving stress for a continent of uniform thickness h. Equations (4) and (5) account for isostatic compensation with the parameter f, which is the fraction of the continent's thickness that lies above the base level of the surrounding topography (Figure 1). Assuming Airy isostasy and no flexural effects, f = 1 − ρ/ρ_{m}, where ρ_{m} is the density of Titan's upper mantle.

[9]Equations (2)–(5) allow us to write expressions for the radial ice flux q in equation (1). We solve equation (1) by seeking a similarity solution [Halfar, 1983] in which the aspect ratio of the ice sheet declines but the scaled shape remains the same; that is, h/H is a time-independent function of r/R, with R and H the radius and central thickness, respectively. For the low viscosity case, that scaled shape is described by

[10]Nye [2000] derives an expression for the low viscosity case that can be rearranged to give the present central thickness H_{o} of the ice sheet as a function of its age (i.e., the time since it began spreading) t_{o} and the present radius R_{o}:

The corresponding expression for the high viscosity case is

The topographic relief—the elevation difference between the center and the edge of the continent—is H_{o}f. The decay of the central thickness with time is

where the exponent a is 2/(5n + 3) for the low viscosity case and 1/n for the high viscosity case. Mathematically, the initial condition of this model is a delta function of infinite thickness, which clearly is not representative of the real history of the ice sheet. However, the similarity solution is relatively insensitive to initial conditions and stable with respect to perturbations that preserve the total volume [Halfar, 1983]. Thus, the model should provide a good description of the relationship between radius, thickness and age even if it is a poor representation of the origin and early history of the continent.

2.2. Temperature Gradient

[11] Up to this point we have assumed an ice temperature that is uniform in space and constant with time. However, the viscosity of ice depends strongly on temperature (equations (2) and (3)), and so it is important to consider the effect of a vertical temperature gradient on the flow. Following Nye [2000], we seek an effective temperature that yields the same radial ice flux as a conductive temperature profile with a constant heat flux and surface temperature. We use equations (2) and (4) to write two expressions for the radial flux q, one in terms of a depth-varying temperature T(z) and a second in terms of a depth-independent . Equating the two expressions and solving for gives

for the low viscosity case. Using the same approach with equations (3) and (5) for the high viscosity case gives

[12] The conductive temperature profile T(z) is governed by the heat flux ϕ and the thermal conductivity k(T):

Laboratory experiments have shown that the thermal conductivity of pure ice is well approximated by an empirical function with the form k(T) = B/T + D, with B = 488.19 W m^{−1} and D = 0.4685 W m^{−1} K^{−1} [Hobbs, 1974]. Integrating equation (12) downward from the surface of the ice, where T = T_{s}, gives depth as a function of temperature. The assumption of a conductive temperature profile is valid if flow is slow relative to thermal diffusion, i.e., the Péclet number (≡ρu_{r}hc_{P}/k, c_{P} = 10^{3} J kg^{−1} K^{−1}) is small.

[13] Because equations (10) and (11) have no time dependence, we require a value of h that is representative of the continent's history. It is evident from equation (9) that the spatially averaged thickness at t = t_{o}/2 is a suitable choice for both cases. From equations (6) and (9), these thicknesses are 0.677 · 2^{2/(5n+3)}H_{o} and 2^{1/n}H_{o} for the low and high viscosity cases, respectively. Substituting these values for h in equations (10) and (11) with T = in equations (7) and (8) yields two expressions for the present central thickness, H_{o}. Because H_{o} appears on both sides of these expressions, we use an iterative procedure to find the solutions.

2.3. Parameter Values

[14]Table 1 summarizes the model parameters and the values used in our analysis. Schubert et al. [1986] estimate a radiogenic heat production rate for Titan of 4–5 × 10^{11} W. It is likely that most of the energy of accretion was dissipated rather than retained as primordial heat, but the energy of differentiation, released over 4.5 Gyr, would contribute an additional 2 × 10^{11} W [Schubert et al., 1986]. Tidal dissipation is probably not a major source of heating on Titan [Sohl et al., 1995]. Thus we estimate a surface heat flux ϕ of 5 to 8 mW m^{−2} and calculate solutions for both values.

[15] The rate-controlling ice deformation mechanism depends on temperature, grain size, and driving stress. For the heat flux estimated above, ice thicknesses of 10–100 km and an ice surface slope of 0.001 to 0.01, we expect typical depth-averaged temperatures of 150–200 K and shear stresses of 0.01–1 MPa (equation (4)) or normal stresses of 0.2–2 MPa (equation (5)). Laboratory experiments suggest that at these conditions, and for a grain size of ∼1 mm, the dominant deformation mechanism is grain boundary sliding, with E = 49 kJ mol^{−1}, n = 1.8, and A = 62 MPa^{−1.8} s^{−1} [Goldsby and Kohlstedt, 2001].

[16] The most uncertain parameters are the age of the continent and the density of Titan's upper mantle. Simple consideration of the mass fractions of silicates and ices [Schubert et al., 1986] indicates ρ_{m} < 1300 kg m^{−3} for a fully differentiated Titan with an anhydrous silicate core, and differentiation models [e.g., Stevenson, 1992; Grasset et al., 2000] suggest that the upper mantle density may be considerably lower. We calculate model solutions for mantle densities between 920 and 1300 kg m^{−3} and ages ranging from 10 kyr to 4.5 Gyr.

3. Results and Discussion

[17] Our results are summarized in Figure 2. In general, the predicted relief H_{o}f decreases with age, because the continent has more time to spread, and with decreasing ρ_{m}, because the continent “floats” lower on the mantle. Due to the asymptotic decay of the central thickness with time (equation (9)), the relief is controlled primarily by the isostatic effect, particularly at low to intermediate mantle densities (ρ_{m} < 1100 kg m^{−3}). Increasing the heat flux from 5 mW m^{−2} to 8 mW m^{−2} decreases the relief by a factor of about one third. The low viscosity solutions (Figure 2a) predict slightly less relief than the high viscosity solutions (Figure 2b), but overall the solutions for the two cases are quite similar. Some solutions, particularly in the high viscosity case, are not physically reasonable because the effective or basal temperatures exceed the melting point (see Figure 2).

[18] The calculated relief ranges from zero for ρ = ρ_{m} to 30 km for a continent that began spreading 10 kyr ago with ρ_{m} = 1300 kg m^{−3}. If ρ_{m} ≈ 1000 kg m^{−3}, our model predicts 3–7 km of relief. Ice thickness H_{o} is controlled primarily by the assumed age of the continent. For the low viscosity case, it ranges from 50 to 90 km (35 to 60 km) for ϕ = 5 mW m^{−2} (8 mW m^{−2}); for the high viscosity case, it ranges from 60 to 120 km (40 to 80 km) for ϕ = 5 mW m^{−2} (8 mW m^{−2}).

[19] There are few independent constraints on the relief of the bright spot. Voyager 1 radio occultations suggest that the eastern margin of the feature is at most 1 km higher than the center of the dark region on the opposite hemisphere [Smith et al., 1996]. Moment of inertia effects of an isostatically compensated continent would be small [Smith et al., 1996].

[20] The thickness of an ice continent, and hence its relief, may be limited by basal melting. With a surface temperature of 93 K, a conductive temperature profile, and a melting temperature lowered by 1 K per 13.46 MPa of hydrostatic pressure [Hobbs, 1974], pure ice should melt at depths of 75 km (ϕ = 8 mW m^{−2}) to 115 km (ϕ = 5 mW m^{−2}). Model solutions with effective or basal temperatures that exceed the basal melting temperature are shown in Figure 2. The antifreeze effect introduced by ammonia, if present, could trigger basal melting at even shallower depths.

[21] Several limitations of our model warrant further discussion. Equation (1) includes no accumulation or ablation terms. The lack of atmospheric water vapor suggests that water precipitation would not be a significant mass source. Erosion of ice by chemical dissolution is unlikely due to the low solubility of ice in hydrocarbons, but high elevations might be subject to physical erosion associated with runoff [Griffith et al., 1991; Lorenz, 1993].

[22] Changes in Titan's surface temperature alter the predicted thickness and relief by less than one percent per degree. We do not consider the effects of convection in Titan's ice crust on the temperature profile or the flow. The analysis of Sohl et al. [2003] suggests that the crust should be entirely conductive to the depths considered here, although it may be underlain by a thin, convecting sublayer.

[23] Caution is in order when applying experimentally derived ice flow laws to planetary conditions. Laboratory experiments are performed at relatively high strain rates, and ice rheology depends on ice fabric and chemical composition. We have assumed a relatively large ice grain size (d = 1 mm). The constant A is proportional to d^{−1.4} [Goldsby and Kohlstedt, 2001], and so smaller grains would lower the predicted relief by 12% per ten-fold decrease in grain size. The presence of ammonia hydrates would lower the bulk strength [Durham et al., 1993] and the thermal conductivity [Kargel, 1990] of the continent, resulting in lower effective viscosities and lower relief. In contrast, methane clathrate hydrates are known to be much stronger than water ice [Durham et al., 2003]. Small rock particulate concentrations (<10% by volume) would strengthen the ice in a roughly linear fashion [Durham et al., 1992], but the higher bulk density would outweigh this effect and lower the relief. Because most of these effects would lower the predicted relief, the results in Figure 2 should be taken as upper limits.

4. Conclusions

[24] Using a simple model for the axisymmetric spreading of a water ice continent on Titan, we have shown that the total topographic relief is controlled primarily by isostasy and is relatively insensitive to the age of the continent. The style of spreading, which is governed by the relative viscosities of the continent and upper mantle, does not have a large effect on the results. For a mantle density of 1000 kg m^{−3}, the predicted relief is ∼3–7 km, though less relief is likely if impurities such as ammonia hydrates or silicate particles are present in the ice, the ice grain size is smaller, or melting of the continental root has occurred. In sum, an isostatically compensated ice continent on Titan with a relief of more than a few kilometers seems unlikely. Our approach could be combined with data from the Cassini-Huygens mission to better understand the dynamics of Titan's bright spot, if it is indeed a positive relief feature.

Acknowledgments

[25] We thank K. Cuffey and M. Manga for helpful discussions and J. Mitrovica for constructive comments. Reviews by W. Durham and an anonymous reviewer improved the manuscript. This work was supported in part by NSF grant AST-0205893.