A numerical study of a mantle plume beneath the Tharsis Rise: Reconciling dynamic uplift and lithospheric support models

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2003je002228, 2004 A numerical study of a mantle plume beneath the Tharsis Rise: Reconciling dynamic uplift and lithospheric support models Hannah L. Redmond and Scott D. King Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, Indiana, USA Received 19 December 2003; revised 28 May 2004; accepted 25 June 2004; published 22 September [1] The Tharsis Rise is an area of extensive volcanism containing the most significant long-wavelength topographic and areoid anomalies on Mars. The mechanism for supporting this large topographic expression and correlating areoid remains controversial. The two main competing ideas are dynamic support by a deep, mantle plume and a volcanically constructed lithosphere followed by lithospheric flexure. While both of these models, separately, can account for specific features of Tharsis, neither explain its entirety. Most of the support of the Tharsis topographic anomaly is best explained by a volcanic structure modifying the lithosphere. However, this raises the important question of the origin of the heat source for volcanic construction, leading to possible small-scale convection. This demonstrates that the remaining topography and areoid not accounted for in lower end-members of volcanically constructed surface loads is consistent with a deeper, mantle plume source. Our results show we can produce about 15% and 18% of the total Tharsis long-wavelength areoid and topography anomaly with a plume forming in a strong, temperature-dependent rheology. Additionally, our plume model has a 165 km thick rheological lithosphere (within the range for present-day estimates on Mars) and generates about 1% partial melting, more than sufficient to provide a heat source for recent volcanism. This implies both a model with dynamic support from a plume and a volcanically constructed lithosphere not only can coexist but offer an explanation for both the surface anomalies and heat needed for the minor amount of recent volcanic activity at Tharsis Rise. INDEX TERMS: 6225 Planetology: Solar System Objects: Mars; 5475 Planetology: Solid Surface Planets: Tectonics (8149); KEYWORDS: mantle convection, mantle plume, Tharsis Rise Citation: Redmond, H. L., and S. D. King (2004), A numerical study of a mantle plume beneath the Tharsis Rise: Reconciling dynamic uplift and lithospheric support models, J. Geophys. Res., 109,, doi: /2003je Introduction [2] The currently accepted model of convection on Earth suggests that the pattern of mantle convection takes the form of subducting plates and upwelling, axisymmetric plumes, defined as buoyant columns of hot mantle rising from the core-mantle boundary [e.g., Bercovici et al., 1989; Bunge et al., 2003]. Morgan [1971] proposed that hot spots on Earth, such as Iceland, are a result of mantle plumes. The source of mantle plumes remains controversial and several alternative hypotheses have been suggested [cf. Anderson, 1998]. The most widely accepted mantle plume theory entails a hot thermal plume originating as an instability at a thermal boundary layer in the Earth s mantle [e.g., Griffiths and Campbell, 1990; Davies, 1990]. [3] Because the natural convection pattern in a spherical shell forms cylindrical, plume-like upwellings, it is natural to assume that mantle plumes exist, or once existed, on other terrestrial planets [Schubert et al., 2001]. The radius of Mars is about half that of Earth. It is believed that if plate Copyright 2004 by the American Geophysical Union /04/2003JE tectonics ever existed, it has halted at present [Sleep, 1994]. There are two lines of evidence suggesting this. One is that Mars has a larger surface area to volume ratio than Earth. This allows for more rapid heat loss and cooling of the Martian interior as compared with Earth. The other is crustal differentiation on Mars was more efficient [Spohn, 1991; Schubert et al., 1992]. Thus a large fraction of incompatible, heat-producing elements are thought to be near the surface of Mars where heat loss would be dominated by conduction [Schubert et al., 2001]. Although plate tectonics has ceased, there is still evidence for present-day or geologically recent mantle convection. The major dominant physical feature on Mars associated with past and/or present convective upwelling is the Tharsis Rise. [4] The Tharsis Rise is an elevated, domal-shaped structure composed of relatively young, large volcanoes with significant, long-wavelength topography and areoid anomalies (Figure 1). The associated shield volcanoes are some of the largest in our solar system and are most likely a result of extensive volcanism by a single mantle plume [Zuber, 2001; Schubert et al., 2001] or multiple plumes that feed (or once fed) each volcano [Kiefer, 2003]. Possible theories for the origin of the Tharsis Rise include isostatic uplift followed 1of14

2 Figure 1. Mercator projections of the global topography and areoid of Mars (MOLA data sets ieg0062t.img and jod75d60.img, respectively). by flexural loading [Banerdt et al., 1982, 1992; Banerdt and Golombek, 2000], thick volcanic extrusions through a locally thin lithosphere [Solomon and Head, 1982; Zhong, 2002; Zhong and Roberts, 2003], and crustal thickening by intrusion [Willemann and Turcotte, 1982]. By incorporating an endothermic transition from the gamma phase of olivine to perovskite plus magnesiowustite, Harder and Christensen [1996] developed a global convection model with one dominant and one minor plume. The resulting planform of their model is strikingly similar to the observed pattern of volcanic activity on Mars (i.e., Tharsis (dominant plume) and Elysium (minor plume)); however, their calculation takes longer than the age of the solar system to develop to this point and earlier in their calculation there are multiple plumes. There is apparently no surface expression on Mars for other volcanic centers. In order to support the height of the Tharsis shield volcanoes (up to 24 km), a thick lithosphere is required [Banerdt et al., 1982; Willemann and Turcotte, 1982]. However, large, long-wavelength areoid anomalies that may not be fully explained by lithospheric support alone are observed in models where the isostatic areoid [e.g., Kiefer et al., 1996] has been subtracted from the observed areoid of Mars [e.g., Smith et al., 1999]. If an elastic lithosphere is included, it is possible to explain most, if not all of the long-wavelength anomalies; however, this is not a unique solution [Lowry and Zhong, 2003; Zhong and Roberts, 2003]. Additionally, the lack of craters on the Tharsis Rise implies the volcanoes have had geologically recent flows [Hartmann and Neukum, 2001; Hartmann et al., 1999]. Dynamic models suggesting Tharsis is largely supported by convection [Kiefer et al., 1996; Harder and Christensen, 1996; Harder, 2000; Kiefer, 2001] are consistent with the young age of the shield volcanoes. We are not rejecting the possibility of a region of volcanically thickened crust beneath Tharsis. This must be true, in part. However, in order to generate enough magma for even the small amount of recent volcanism a source of heating is required. An anomalously hot mantle will contribute to the observed topographic uplift [Kiefer, 2003]. Thus there is reason to believe that the mantle beneath Tharsis is still contributing to the overall uplift at present, assuming the core is supplying some significant fraction of the total surface heat flow. If the core only produces a small amount of heat flow, as in some thermal history models [e.g., Hauck and Phillips, 2002], then it may be difficult for a plume to exist at present and the plume mode of convection may have shut off early in the history of Mars. In this paper, we present the results of finite element calculations of Martian mantle convection in a spherical, axisymmetric shell. Our goal is to produce plume calculations with small presentday areoid and topography anomalies that are suggestive of a convectively supported region below Tharsis. We assume that the Tharsis Rise is currently the result of one plume but do not address the possibility that Mars may have (or had) more than one plume. [5] As was recently shown by Zhong and Roberts [2003] and Lowry and Zhong [2003], a significant amount of the support of the Tharsis topographic anomaly is best explained by a volcanic structure modifying the lithosphere. Yet, this raises the important question of the origin of the 2of14

3 Table 1. Scaling Parameters for Mars Definition Value R m radius of Mars 3394 km d depth of convecting layer 1800 km DT vertical temperature contrast 1625 C r m mantle density 3400 kg/m 3 g gravitational acceleration 3.72 m/s 2 a thermal expansion coefficient C 1 k thermal diffusivity m 2 /s heat source that enabled the small amount of recent volcanism. While most of the areoid and topographic anomalies (for example, about 75% of the topographic anomalies and 85% of the areoid anomalies) are likely the result of lithospheric and crustal structure, the thermal anomaly and remaining structure could well be explained by a deeper, mantle plume source. Thus our modeling effort is not to explain the entire areoid and topographic anomalies over Tharsis by a plume convection model. Our goal will be to produce plume calculations that explain of order 15% and 25% of the total present-day areoid and topographic anomalies, respectively. 2. Modeling Procedures [6] We use the finite element code SCAM [Kellogg and King, 1997] to solve the equations governing conservation of mass, momentum, and energy, assuming a creeping viscous fluid. The code uses spherical geometry with axisymmetry about the pole q = 0. We take advantage of the symmetrical shape of plumes using axisymmetry geometry in order to reduce the problem to a two-dimensional computation. An important caveat of axisymmetric calculations is that there are no variations in the f direction (i.e., there are no f derivatives in the governing equations). Thus, as we get further from the pole, the results become increasingly less physical. Downwelling drips off the pole are actually downwelling doughnuts that follow the entire small circle. In a fully 3D calculation, this doughnut feature would in reality be a drip. Therefore we restrict the geometry to the region near the plume. Because we are not modeling the full sphere, we cannot address the reason Mars developed only a single plume. When calculating the areoid and topography, we pad our results to p and use symmetry to 2 fill out the full sphere. [7] A significant advance compared with previous versions of SCAM that enables us to compute the large number of steady state solutions in this work is the use of Picard iteration [Cuvelier et al., 1986]. This enables us to compute steady state solutions in less than 1/100th the time required using the traditional explicit time stepping algorithm in SCAM. Solutions were verified by spot computing several cases with the explicit solve to ensure that true steady state conditions had been achieved. While we highly doubt that the Martian mantle has actually achieved a steady state condition, this allows us to compare all of our models varying a single parameter at a time. Once time dependence enters our calculations is it difficult to quantify only the effect of the parameter under investigation Effects of Grid Resolution [8] Grid resolution tests were performed for a constant viscosity fluid with no internal heating and a Rayleigh number of The radius of Mars is about 3394 km and the core radius ranges from km based on a moment of inertia of [Schubert and Spohn, 1990; Folkner et al., 1997]. Our choice of core radius, 1472 km, is near the lower middle of this range and gives an inner radius (core radius) to outer radius (core + mantle radius) ratio of For computational ease, we use nondimensional values in our calculations. Thus our computation domain in the radial direction, r 0, where r = r 0 * 1800 km, is r and 0 q p in the lateral 4 direction (q). This choice of parameterization gives an aspect ratio of nearly 1.0 and corresponds to the same inner to outer radius ratio (0.45) as calculated before. We only model up to q = p in order to isolate and study the 4 effects from a single plume which, in axisymmetric geometry, occurs at the poles. Other scaling parameters for Mars are listed in Table 1. The value of the temperature change across the mantle, DT, for Mars remains controversial. However, Kiefer [2003] shows that for a T cmb range of 1763 C 1888 C, the corresponding DT ranges from 1525 C to 1650 C. Our value of DT, 1625 C, is based on these data. In our tests, we chose six grid sizes, labeled A through F. The grid parameters and results are presented in Table 2. [9] In Table 2, we can see the effect of increasing the number of elements in the grid on a steady state calculation with Ra = 10 6, no internal heating and constant viscosity. As the number of elements increases, the difference between parameter values on two adjacent grids decreases. For example, the difference between the results on our grid (grid E) and a factor of two higher resolution (grid F) is on the order of ±0.591 meters for the geoid (0.471%) and ±12.16 meters for the topography (0.303%). Thus, on the basis of the grid resolution tests and the amount of time required for these calculations to reach steady state, we chose grid E, which has 192 elements in the radial direction and 150 elements in the lateral direction, as our computational grid. Grids A, B, and C are representative of the grid resolution used in most previous plume studies. The results in Table 2 indicate that the computational errors resulting from coarse grids are on the order of a few percent, but are largest for the geoid. The high-resolution grids become more important for the temperature-dependent viscosity calculations, because we want to avoid large changes in Table 2. Grid Test Parameters and Results a Grid E r E q q B q T V rms Areoid Topography Time A B C D E F a Notes: E r is the number of elements in the radial direction; E q is the number of elements in the theta direction; q B is dimensionless basal heat flux; q T is dimensionless surface heat flux; V rms is dimensionless root mean square velocity; the areoid and topography are in units of meters; and the time is how long it takes to complete 1000 time steps and is in units of hours. 3of14

4 viscosity across a single element. The elements approximate the velocity and temperature fields by piecewise linear functions. Because the viscosity is an exponential function of temperature, even a relatively small change in temperature that is well represented by piecewise-linear functions can lead to a large change in viscosity which would not be well represented by piecewise-linear functions Modeling Parameters [10] In this research, several parameters are examined in depth, including: Rayleigh number, rate of internal heating, and the activation energy in the temperature-dependent viscosity law. We use purely viscous models and do not consider any elastic deformation. We then calculate the areoid anomaly and topographic uplift from these models for comparison with the observed values on Mars, which we use as our constraints. We use steady state calculations so that we can separate time-dependent from parameterdependent effects. Once again, it is unlikely that any Marssized, or larger, planetary body is in steady state. These calculations mainly serve as a guide, allowing us to determine the relationship between surface observations and internal parameters. We will use the insight from these steady state calculations to speculate on the thermal evolution of Mars. [11] The first parameter we examine, one of the most important parameters, is the Rayleigh number, Ra. It is a dimensionless parameter that can be defined by the following equation: Ra ¼ gradtd3 ; ð1þ kh where h is viscosity and the rest of the constants are described in Table 1. The Rayleigh number is the most important factor controlling the vigor of convection because it is a ratio of the buoyancy force to the resisting viscous stresses. In equation (1), we see that the Rayleigh number is inversely related to viscosity; therefore an increase in viscosity will cause the Rayleigh number to decrease, with all other constants fixed. Thus we use constant viscosity models in order to isolate the effect a change in Rayleigh number has on convection, topography and areoid. [12] The rate of internal heating, the second parameter of study, plays a large role in determining whether or not a plume will form. If the mantle is 100% internally heated, no bottom boundary layer will form, and thus a plume will not form [Parmentier et al., 1975]. On the other hand, if there is little to no internal heating, a large thermal boundary layer will form and a strong plume will develop. In order to isolate the effects of adding internal heating into our calculations, we fix the Rayleigh number and then vary the amount of internal heating. We determine the percent internal heating by finding the ratio of the basal heat flux to the surface heat flux observed at the surface. We then can study the effects of internal heating in our plume calculations and on the areoid and topography. [13] Most of the Martian mantle is likely to be in the olivine stability field. Therefore we use the creep properties of olivine [Karato and Wu, 1993] for our rheological law. We use the Arrhenius form of the temperature-dependent part of the viscosity law for the Martian mantle. This is based on the assumption that diffusion creep is the dominating mechanism (i.e., stress depends linearly on strain rate), where the viscosity law depends exponentially on temperature: E* hðþ¼h t o exp t þ T 0 exp E* ; ð2þ 1 þ T 0 where h(t) is the effective viscosity, h o is pre-exponential viscosity, t is the dimensionless temperature (0 t 1.0), T o (=0.2) is the temperature offset to avoid dividing by zero, and E* is the activation energy divided by RDT, where R is the universal gas constant and DT is the temperature drop across the shell. Viscosities exceeding 1000 h 0 are set equal to 1000 h 0 to avoid numerical problems with the matrix solution. [14] In the exporitory Rayleigh number and internal heating calculations, the activation energy is set to zero, yielding an isoviscous rheology. The results of previous studies [King, 1997; Zhong, 2001] indicate that constant viscosity calculations significantly overestimate the topography and areoid. Thus we fix the Rayleigh number and amount of internal heating and incorporate temperaturedependent rheology into the convection simulation. Our value of E* is reduced compared to a wet olivine rheology [Karato and Wu, 1993]; however, Christensen [1984] shows that the dominant effect of stress-dependent viscosity is that the activation energy is reduced by 1 n. Additionally, we are trying to avoid strongly timedependent calculations in order to isolate the effect of an increase in activation energy. Although the study of temperature-dependent viscosity has been examined by others [e.g., King, 1997], our work uses a significantly greater number of elements than used in previous calculations, and we systematically vary the activation energy (E*) over a large range of values, something that has not been done before. While the values we use for the activation energy span a range that includes values unlikely to be representative of mantle minerals, we feel it is illustrative to systematically expand from the wellstudied constant viscosity end-member to the more uncharted territory of temperature-dependent rheology. Our results also allow us to extrapolate beyond the scope of this study to more planet-like values. The use of a finer mesh becomes more important in studying temperature-dependent rheology because we want to minimize the variation of the viscosity from element to element. 3. Results [15] In this section we first study the effects of a change in Rayleigh number, amount of internal heating and activation energy by observing the changes in the temperature field, areoid and topography. Our goal is to find reasonable values for the Rayleigh number, rate of internal heating and activation energy for present-day Mars while creating minimal topographic uplift and areoid anomalies. After attaining a set of parameters, we then perform a depth-dependent viscosity calculation for comparison with our temperaturedependent model to assess the effectiveness of using a temperature-dependent rheology. Finally, we calculate the amount of melt that can be generated from the resulting plumes allowing us to speculate as to whether or not a 4of14

5 Figure 2. Comparison of temperature fields with (a) Ra = 6e4 and (b) Ra = 1e6 to illustrate the effect of the Rayleigh number on the temperature field for spherical, axisymmetric convection. plume can provide enough heat for partial melting of the lithosphere Rayleigh Number [16] The Rayleigh number largely controls the vigor of convection; therefore we should expect a stronger, more buoyant plume with an increase in Rayleigh number. In Figures 2a and 2b (Ra = and Ra = , respectively) we plot the temperature field to illustrate the effect of increasing the Rayleigh number. The higher Rayleigh number calculation produces a thinner thermal boundary layer and a narrower upwelling and downwelling than the lower Rayleigh number calculation. The plume with a higher Rayleigh number is also significantly hotter and more buoyant than the surrounding mantle so it rises with less diffusion into the mantle and, again, results in a thinner upwelling. In Figure 3a we present the results from a wider range of Rayleigh numbers to further illustrate the effect of decreasing plume width. We calculate the plume width as a percentage by determining what fraction of the convecting shell is the plume. We can clearly see, in Figure 3a, that plume width decreases with increasing Rayleigh number. The Rayleigh number also has an effect on the areoid height and the amount of dynamic topography produced due to thermal anomalies in the upwellings, downwellings and thermal boundary layers. In Figures 3b and 3c we plot the peak topography and areoid as a function of the Rayleigh number. We can see that as the Rayleigh number increases, both the topography and areoid decrease. Because the results of higher Rayleigh number calculations are time-dependent and 10 6 is a reasonable value for mantle convection on Mars [Kiefer, 2003], we will use a Ra = 10 6 for all subsequent models Rate of Internal Heating [17] It should be immediately obvious that these models are not representative of planetary interiors because the plume carries more than 50% of the surface heat and there is no top thermal boundary layer. Thus we investigate the effect of including internal heating and then temperaturedependent rheology. The amount of internal heating largely controls plume formation. Using a Rayleigh number of 10 6 and constant viscosity, we observe the effect of increasing the rate of internal heating on mantle convection. In Figure 4a we plot the temperature field of a plume forming in a mantle that is about 10% internally heated. For comparison, we plot a plume forming in a mantle that is about 40% internally heated in Figure 4b. It is clear, from Figure 4, that increasing the amount of internal heating causes the plume to become less pronounced and the temperature contrast across the plume to lessen. While the temperature contrast between the plume and the surrounding mantle is less extreme in this case than the cases shown in Figure 2, the temperature anomaly of the plumes is still comparable to the temperature contrast between the top and the bottom shell. The thermal boundary layer near the surface is still very thin, on the order of 25 km. The result of the decrease in temperature contrast is an increase in plume width as illustrated in Figure 5a. The amount of internal heating also has a significant effect on the areoid and topography (Figures 5b and 5c). Because of the temperature contrast decrease, the plume becomes less 5of14

6 Figure 3. The effect of increasing the Rayleigh number on (a) plume width, (b) peak topography, and (c) peak areoid. The plume width is the fraction of the convecting shell that is occupied by the plume. buoyant, causing less stress on the lithosphere and hence less deformation. In our calculations, we only achieve steady state solutions until about 40% internally heated. However, if we were able to increase the amount of internal heating it would only further reduce the topography and areoid until no plume forms. This is because the temperature contrast between the plume and surrounding mantle would decrease resulting in a decrease in buoyancy. In the next section we study the effects of a plume forming in a temperature-dependent viscosity. We start our calculations with approximately 40% internal heating. However, as temperature dependence increases, the ratio of internal to basal heat observed at the surface increases because there is less of a temperature contrast between the plume and surrounding mantle. This will allow us to achieve more realistic internal heating rates Temperature-Dependent Viscosity [18] In constant viscosity models there is almost no top thermal boundary layer (no lithosphere) and the plumes carry most of the heat observed at the surface. The results of previous studies [King, 1997; Zhong, 2001] indicate that constant viscosity calculations significantly overestimate the topography and areoid. On the basis of a small number of calculations, King [1997] concluded that plumes forming in a temperature-dependent viscosity fluid are reduced by a factor of two in comparison to plumes in a constant viscosity fluid. Thus we modify the above models by incorporating a temperature-dependent viscosity into the convection simulation. [19] For our temperature-dependent viscosity calculations, we use a Rayleigh number of 10 6, a starting internal heating rate of 40% (the amount of internal heating rises with increasing activation energy), and vary the activation energy in the Arrhenius law of equation (2). For E* = 7.5, the corresponding dimensional value is about 120 kj/mole. We realize 120 kj/mole is low for realistic mantle minerals; however, calculations with an activation energy above these Figure 4. Comparison of temperature fields with (a) 10% internal heating and (b) 40% internal heating, illustrating the effect of internal heating on the temperature field. The Rayleigh number in both models is of14

7 Figure 5. The effect of increasing the amount of internal heating on (a) plume width, (b) peak topography, and (c) peak areoid. Plume calculations more than 40% internally heated are time-dependent. values are time-dependent, making them more difficult to quantify. Christensen [1984] has shown that the dominant effect of stress-dependent rheology is the effective decrease of the activation energy by 1 n. In dislocation creep, n is between 3 and 5 for olivine; thus this would decrease our effective activation energy by 3 5, giving us a smaller effective activation energy. Our results show that as the activation energy rises, the lateral variation in viscosity in the near-surface thermal boundary layer greatly increases allowing the thermal boundary to strengthen and thicken (Figures 6a and 6b). Although the plume width is not strongly affected by the increase in viscosity variation, there is an interesting effect on the topographic uplift and areoid anomaly (Figures 7a, 7b, and 7c). For weakly temperaturedependent calculations (E* < 2.0), the topography, areoid and plume width increase. However, beyond E* = 2.0, the plume continues to increase both in temperature and width, but the topography and areoid begin to decrease. In this regime, the increasing stiffness of the surface boundary layer has more of an effect on the topography and areoid than the lateral viscosity variations in the lithosphere and plume conduit. This becomes an important key to understanding convection and/or plume formation on Mars, especially under the Tharsis rise, because it is possible for lateral variations to exist in the near surface due to the crustal dichotomy. Although the dimensionless activation energies examined may be unrealistic, this study serves as a guide to understanding the systematic effect of an increase in activation energy on mantle convection. A caveat for these temperature-dependent calculations is that as temperature dependence increases, the ratio of internal to basal heating observed at the surface increases. For example, there is about 50% internal heating for E* = 1.1, about 80% for E* = 4.0, and 88% for E* = 7.5. In the previous section we only achieved steady state conditions up to 40% internal heating, but, by incorporating a temperaturedependent rheology, we further increase this ratio of internal to basal heating due to the changes in viscosity while Figure 6. Temperature fields of convection models with (a) E* = and 40% internally heated and (b) E* = 7.5 and 88% internally heated. This comparison illustrates the effect of increasing the activation energy (or increasing the amount of temperature dependence) on the temperature field. E* is the dimensionless activation energy (E/RDT). Both models have Ra = of14

8 Figure 7. The effect of varying the dimensionless activation energy on (a) plume width, (b) peak topography, and (c) peak areoid. The decrease in peak topography and areoid after an activation energy of about 2.0 is due to the thick, strong lithosphere that begins to develop. maintaining steady state calculations. This is an important result because if Mars is largely internally heated, it is still possible for a plume to form and contribute to the overall uplift under the Tharsis Rise Rheological Lithosphere [20] We can treat the near-surface thermal boundary layer as a high viscosity lid that stiffens and increases in thickness with increasing activation energy. Estimates for the Martian rheological lithosphere are on the order of km [Grasset and Parmentier, 1998; Schubert and Spohn, 1990], which we can use as a constraint for our lid thickness. By determining the vertical depth where the average viscosity contrast exceeds 100 times the mantle below, we can estimate the thickness of the upper thermal boundary layer or rheological lithosphere. In Figure 8, we plot the rheological lithosphere versus the dimensionless activation energy. At an activation energy of 7.5 the rheological lithospheric thickness is about 165 km, which falls within the range for present-day estimates on Mars. If we increase the activation energy by 50%, we obtain a lithospheric thickness of about 200 km as shown in Figure 8. However, values above E* = 7.5 are time-dependent, making comparisons between models difficult. Nevertheless, the results presented in Figure 8 illustrate that increasing the activation energy causes an increase in the rheological lithosphere. An increase in lithospheric thickness will further reduce the topography and areoid. This merely strengthens our argument that a plume is completely consistent with volcanically constructed surface load models Depth-Dependent Viscosity [21] We next consider a depth-dependent viscosity calculation for comparison with our temperature-dependent results. The purpose of this experiment is to demonstrate that the dominant effect of temperature-dependent rheology is to modify the average radial viscosity structure. The effect on the width of the plume conduit is small. Because the largest temperature differences are in the lithosphere, the largest viscosity differences are also in the lithosphere. In this calculation, we use Ra = 10 6 and about 90% internal heating but no temperature-dependent rheology. Instead, we create a layered viscosity model that parallels the final result of our temperature-dependent calculation for E* = 7.5. The temperature fields of both models are plotted in Figures 9a and 9b. It is not surprising that the depth-dependent calculation is similar to that of the temperature-dependent calculation. In fact, the areoid anomaly and topography closely match the temperature-dependent values (Figure 10). The reason for performing this depth-dependent calculation is not to demonstrate that we can produce a model to emulate our temperature-dependent model, but to illustrate that the main effect of temperature-dependent rheology is the increase in rigidity and production of a viscous lid Melt Generation [22] Thus far we have mantle convection calculations that produce only small areoid and topography values consistent with the idea of a plume as a secondary heat source. In addition to the topography and areoid, another constraint on Tharsis is the amount of volcanic activity. Tharsis was most active during the Noachian [Phillips et al., 2001] but there is Figure 8. The effect of increasing the activation energy on the thickness of the rheological lithosphere. The increase in activation energy correlates to an increase in the temperature dependence of the viscosity. This causes the viscosity variations in the radial direction to increase, producing a thick, strong lithosphere. 8of14

9 Figure 9. Comparison of temperature fields of a temperature-dependent rheology (a) and a depthdepth-dependent rheology (b). Both models use Ra = 10 6 and are about 90% internally heated. The temperature-dependent calculation uses E* = 7.5. The depth-dependent calculation parallels the temperature-dependent model for comparison purposes. evidence for recent volcanism. This suggests that at least a small amount of melting has continued to present-day. Thus we try to generate a small amount of melt with our temperature-dependent plume model (Ra = 10 6, 88% internally heated, E* = 7.5) in order to obtain a possible heat source for volcanically constructed models. This will also aid in supporting the existence of a mantle plume. [23] We use melting relations from McKenzie and Bickle [1988] and calculate the solidus temperature from Hirschmann [2000]. Although this is a fairly straightforward model, it supplies us with an approximate method for studying basic melt production [Hirschmann et al., 1999]. In our convection models the output temperature is in nondimensional form with values ranging from 0 to 1. We use the following equation to convert the nondimensional temperature to a dimensional value: T ¼ T nd DT þ Z T ad ; where T nd is the nondimensional temperature, DT isthe temperature drop across the shell (Table 1), Z is the depth below the surface and T ad is the adiabatic temperature gradient in the mantle. We find T ad from the following equation: ð3þ T ad ¼ dt dy ¼ agdt C p ; ð4þ where a, g and DT are listed in Table 1 and C p is the specific heat at constant pressure, taken to be 1 kj kg 1 K 1. These values are then used to calculate the amount of melt generated from our plume model. Our goal is to produce less than 1 2% partial melting based on the shergottite studies of Norman [1999] and Borg and Draper [2003]. Moving in increments of 25 C, we calculate the minimum temperature it would take for partial melting to begin. Our results are presented in Figure 11 where we plot the thickness of the rheological lithosphere versus temperature required for melt to begin. The amount of melt is sensitive to lithospheric thickness so in order to generate a small amount of melt, it is necessary to increase DT asthe lid stiffens (i.e., the activation energy increases). Previously, we chose E* = 7.5 because it produced a reasonable estimate for the rheological lithosphere. In Figure 11 we see that for a lithospheric thickness of 165 km (i.e., E* = 7.5), a DT of 1625 C is required for partial melting to begin. With DT = 1625 C (a reasonable estimate for the present-day mantle temperature difference on Mars [Kiefer, 2003]), only a small amount of melting occurs (Figure 12a). This has implications toward the amount of melt that could have been generated in the past (i.e., 2 or more billion years ago). In Figure 12b we plot the melt generated by a plume with DT = 1725 C. In this case, a large amount of melt is generated, possibly a source for the extensive volcanism at the time of Tharsis formation. 4. Discussion [24] A minimum of 75% of the areoid and topographic profiles over Tharsis can be explained by crustal thickness and surface volcanic construction [e.g., Zuber, 2001; Solomon and Head, 1982; Zhong, 2002; Lowry and Zhong, 2003; Zhong and Roberts, 2003]. Given the crustal thickness anomalies, some doubt that a mantle plume could be active on Mars presently and speculate that recent volcanism on Tharsis may have formed by a nonplume mechanism [e.g., King and Ritsema, 2000]. However, by systematically studying the effects of Rayleigh number, internal heating, and temperature-dependent viscosity on anomaly profiles in convective calculations designed to model plumes, we show that a thermal plume produces dynamic topography and areoid anomalies that are consistent with the present-day areoid and topography even if 75 85% results from lithospheric structure. 9of14

10 Figure 10. A comparison of the topography and areoid profiles over a plume forming in a temperature-dependent viscosity (solid line) and a plume forming in a depthdependent viscosity (dashed line). [25] The Tharsis Rise dominates the long-wavelength topography and areoid on Mars (Figure 1) and is one of the most prominent surface features. Our goal in this research is to better explain the observations over the Tharsis Rise by suggesting that a mantle plume beneath Tharsis can generate small topography and areoid anomalies and provide a sufficient heat source to explain present volcanism. We begin our plume modeling by studying the Rayleigh number using constant viscosity. At low Rayleigh numbers, a thick, strong plume forms and no upper thermal boundary layer develops. The result is a very large areoid and topography anomaly over the plume. As the Rayleigh number increases the plume width, topography and areoid greatly decrease and a small upper thermal boundary layer begins to form. To obtain small topography and areoid values, as well as select a realistic Rayleigh number for convection on Mars, we use Ra = [26] One thing that should be quite apparent is the calculations in Figure 2 are significantly different from any model planetary interior. In these calculations, all the heat comes from the core. Because the change in surface area is a function of radius, there is a factor of four greater heat flux per unit area at the core-mantle boundary than at the surface. (The integrated heat out of the surface is the same as the integrated heat into the bottom.) As a result the majority of the mantle is cold, there is almost no thermal boundary layer, and the temperature contrast between the plume and surrounding mantle is almost as large as the temperature contrast between the top and the bottom. [27] It is believed that Mars is currently losing heat both by the cooling of the core and by conduction through the loss of radioactive elements. Thus it is important to incorporate internal heating into our calculations. The initial effect of increasing the amount of internal heating is an increase in plume width. This is due to the decrease in temperature contrast between the plume and the rising temperature of the surrounding mantle, which allows more mantle material to be brought up with the plume. The second major effect is further reduction of the topography and areoid. This is because thermal anomalies exist in the upper thermal boundary layers, as well as in the upwellings and downwellings. Therefore the amount of topographic uplift will largely depend on the size of these anomalies. Additionally, density anomalies within the thermal boundary layer generate topography similar to that of Pratt compensation models [Kiefer and Hager, 1992]. Thus, as the boundary layer thickens, the amount of topographic uplift it can support decreases. The decreasing topography implies a decrease in areoid because the areoid is related to density anomalies in the thermal boundary layers as well as topography at the bottom and top of the convection cell. Our models only reach a steady state solution up to 40% internal heating, a good, but moderately low, estimate for Mars [Kiefer, 2003]. However, if one believes that Mars has a higher ratio of internal to basal heating observed at the surface then, as shown in Figure 5, the areoid and topography will decrease even more. This further strengthens our argument that a plume can produce a fraction of the observed areoid and topography on Mars while the rest of the areoid and topography may be explained by the crust and lithosphere. [28] Many plume calculations to date have used an isoviscous rheology and it is partly due to the intuition from these results that a plume under Tharsis is considered inconsistent with the observations. Previously, we noted that constant viscosity calculations greatly overestimate the Figure 11. The effect increasing rheological lithospheric thickness on the minimum temperature (DT) required for a fractional amount of melt to begin. The rheological lithospheric thicknesses correspond to specific activation energies (see Figure 8). 10 of 14

11 Figure 12. Comparison of the amount of melt generated from our preferred plume model with (a) DT= 1625 C and (b) DT = 1725 C. The contours are at an interval of 0.05%. topography and areoid. The reason is quite clear. Without a realistic thermal boundary layer at the surface, the majority of the temperature difference between the surface and core is carried through the plume. This is inconsistent with the view of plumes as a secondary form of heat transfer compared with the plate-slab system on Earth [Davies, 1988]. Additionally, plumes forming in a temperaturedependent rheology are broader, warmer and can pull material from within the mantle into the head of the plume as it rises [Kellogg and King, 1997] which can generate a very different plume structure. Therefore, in order to model realistic mantle conditions, we incorporate a temperaturedependent rheology into our models. We do this by varying the dimensionless activation energy in equation (2). [29] It is clear from Figure 7 that as the activation energy increases up to E* = 2.0, the plume width, topography and areoid increase. This is because the thickening of the upper thermal boundary layer and the lateral variation in viscosity has very little effect on the rising plume. As temperature dependence strengthens, the plume continues to increase in temperature and width because the temperature contrast between the plume and mantle is very small allowing more mantle material to enter. The mantle also increases in temperature with rising activation energy. This, in turn, causes the ratio of internal to basal heating observed at the surface to increase above 40%. However, beyond E* = 2.0, the topography and areoid decrease with increasing activation energy. This is due to the increasing rigidity of the near-surface thermal boundary layer. In this case, the effect of lateral viscosity variations becomes less important than the increasing stiffness of the surface boundary layer. This has important implications for the mantle beneath the Tharsis Rise. If lateral variations, such as those produced in our model, exist in the crust and lithosphere beneath Tharsis due to the crustal dichotomy, then it is possible that a plume cannot produce all of the observed topography and areoid anomalies over Tharsis due to the rigid lithosphere. However, this does not imply that a plume cannot exist. In our model we are still able to produce a small amount of topographic uplift and a small areoid anomaly with a plume. For a nondimensional E* = 7.5, the areoid is about 334 meters and the topographic uplift is about 2560 meters. If the Tharsis Rise is about 10 km above its surroundings and the areoid is over 1 km, then we have produced about 25% of the observed topography and about 30% of the observed areoid. Our dynamic topography percentage is much less than the upper-bound estimate reported by Lowry and Zhong [2003] and the same as the upper-bound estimate of Zhong and Roberts [2003]. Our areoid percentage is only slightly higher than Lowry and Zhong s [2003] upper-bound approximation and doubles that of Zhong and Roberts [2003]. Our main goal was to produce no more than 25% and 15% of the observed topography and areoid anomalies, respectively, which we have not quite done. However, if we incorporate an elastic lithosphere into our modeling we may reduce the areoid anomaly and topographic uplift even more. Zhong [2002] showed that plume calculations using a 150 km thick elastic lithosphere produce topography and geoid anomalies that are about a factor of 1.4 and 2.0 less than purely viscous models. Thus, if we were to incorporate an elastic lithosphere into our models, we should expect the areoid and topography to reduce to 167 meters and 1828 meters, respectively. As a result, the areoid calculated over a plume would only produce about 15% of the 11 of 14

12 negative anomaly deep in the mantle, reducing the areoid anomaly. Using our temperature-dependent viscosity model, as described in section 3.3, we calculate the areoid from the upper 150 km of the mantle and compare it to the areoid calculation over the whole mantle (Figure 13). Clearly, the areoid anomaly from the upper 150 km is much greater (almost a factor of 2) than the areoid anomaly when the whole mantle is included in the calculation. This leads us to believe that the existence of a mantle plume could be the answer. Figure 13. Comparison of the areoid calculated from the whole mantle (solid line) of our preferred model and the areoid calculated from just the upper 150 km of the mantle (dashed line). The areoid from just the upper mantle overestimates the areoid by about 300 m. observed areoid on Mars and the topography would only be about 18% of the observed. Both of these percentages are now much less than the upper-bound estimates of Lowry and Zhong [2003] and similar to those of Zhong and Roberts [2003]. While we agree that the elastic lithosphere will differ at each harmonic, it is important to note that we are not trying to match a specific topographic pattern in our research, only a maximum amplitude. Thus we are able to generate a small topography and areoid anomaly with a mantle plume as well as provide a deep heat source necessary for the formation of the volcanic construction under the Tharsis Rise. [30] We have computed a depth-dependent rheology calculation for comparison with our temperature-dependent model. By systematically increasing the activation energy, we can control the amount of temperature dependence that is added to our system as well as attribute the changes in the topography and areoid to the change in activation energy. While the depth-dependent rheology calculation produces similar results (and is most likely behaving like a temperature-dependent rheology calculation), it is both hard to control and isolate its effects by manually inputting lateral viscosity variations. Furthermore, the increase in activation energy has a more profound effect of reducing the areoid and topography by forming, and increasing the rigidity of, a upper thermal boundary layer, or rheological lithosphere, than the depth-dependent viscosity model. [31] For further evidence toward the existence of a plume, we present the effects of not including a deep mantle plume in our calculations. Despite geologically recent lava flows (e.g., Mya) and large areoid anomalies associated with high topography, many believe a plume beneath Tharsis, if it exists, has little or no contribution to the areoid. However, if there currently is a mantle plume beneath Tharsis, models suggesting the areoid over Tharsis can be fully explained by a volcanically constructed surface load, lithospheric erosion or shallow compensation in the upper mantle (e.g., upper 150 km) will overestimate the areoid, because the density of the hot plume is less than the density of the surrounding, cold material creating a 5. Conclusions [32] Most researchers believe that the Tharsis Rise was formed mainly by volcanic construction and magmatic intrusion followed by lithospheric flexure. If Tharsis were created by volcanic construction then a large volume of magma would be required to produce such massive topography. In order to obtain a large amount of magma, an anomalously hot mantle is required. This suggests that a plume may have existed at the time of Tharsis formation. Many researchers believe the plume mode of convection on Mars subsequently shut off. It is not clear why the volcanic activity would be concentrated in one area. Because models suggesting a volcanically constructed surface load do not address a possible heat source responsible for the late stage volcanic activity, a fortuitous small-scale convection mechanism located beneath Tharsis is required. We approach this problem by asking, if a long-lived plume still exists, how much melt can it generate with present-day mantle temperatures on Mars? If we can generate only a fractional amount of melt, we can provide a heat source for volcanic construction models as well support the existence of a plume beneath Tharsis. This allows us to propose that the formation of the Tharsis Rise is a combination of both models. [33] Because present-day mantle temperatures are significantly cooler than the time of eruptive volcanism but not well constrained, the ability to generate melt is a significant problem in both of the before mentioned models. Using the modeling parameters in Table 1, along with our temperature-dependent rheology plume calculation, a fractional amount of melt can be generated with our mantle plume (Figure 12a). We only wish to produce a small amount of melt (i.e., 1% or less) because a reservoir that behaves elastically will only erupt about a fraction of 1% of its volume [Blake, 1981]. Our plume model in Figure 12a generates a disc-shaped melt region with a volume of about 240,000 km 3 (V = pr 2 h, where r is the radius of the disc and h is the height) with about 1% of this assumed to be melt (about 2400 km 3 ). If the flux of new, unmelted mantle through our melt region is about 1 cm/yr and the average height of our reservoir is about 48 km, we could expect about 2400 km 3 to melt through every 4.8 Ma. Over the lifetime of Tharsis (3 Ga) this would enable a very large amount of magma to be created. We are not suggesting that all of it erupts nor are we stating that there has been a steady output of volcanism through time. However, a likely scenario may be that the magma seeps away from the plume head allowing for smaller amounts of magma to erupt (e.g., through any of the Tharsis volcanoes) and/or becomes underplated followed by isostatic uplift. Scott et al. [1998] propose that a typical Tharsis volcano reservoir may be on 12 of 14