Relative rates of fluvial bedrock incision on Titan and Earth



[1] Observations of likely fluvial channels on the surface of Titan, along with Titan's geologically youthful surface, motivate this study of comparative fluvial erosion rates on Titan and the Earth. The roles of bedload abrasion, suspended load abrasion, plucking, and cavitation are considered. Despite orders of magnitude differences in some of the physical parameters that control fluvial incision on Titan and the Earth, incision rates on Titan are likely to be similar to terrestrial rates, given similar conditions of slope, discharge, and sediment supply.

1. Introduction: Streams on Titan

[2] Saturn's largest moon Titan is the only satellite in the solar system with a significant atmosphere, composed primarily of nitrogen with ∼5% methane. At Titan's surface temperature of ∼95 K and pressure of ∼1.5 bars, it is possible for methane to condense and precipitate from the atmosphere. Dendritic networks of sinuous, steep-sided valleys on the surface of Titan have been recently observed by the descent imager on the Huygens probe [Tomasko et al., 2005] and the radar on the Cassini orbiter [Elachi et al., 2005], while large sinuous low albedo features have been observed by the Imaging Science Subsystem on Cassini [Porco et al., 2005]; all of the aforementioned references have interpreted these features to be fluvial channels. Radar images have also shown triangular, radar-bright regions at the ends of some channels that may be alluvial fans strewn with cobbles and boulders [Lorenz, 2005]. The Huygens probe landed in a flat basin downhill from several channel networks, and imaged a surface covered with gravel and rounded cobbles, made of water ice mixed with an unknown (possibly organic) substance [Tomasko et al., 2005]. Widespread water ice on the surface is also consistent with groundbased spectra of Titan [Griffith et al., 2003]. Optical observations of Titan from the Earth [e.g., Griffith et al., 2000] and the Cassini orbiter [Porco et al., 2005] show that many of the occasional clouds in the atmosphere rapidly appear and disappear, consistent with convective rainstorms [Lorenz et al., 2005]. The paucity of impact craters on Titan's surface [Porco et al., 2005; Elachi et al., 2005] indicates rapid burial or removal of surface topography, which makes the role of fluvial erosion in modifying Titan's surface especially interesting. In summary, the emerging view from the initial exploration of Titan's surface is a world with many Earth-like processes. If we are to understand Titan's geology at more than a superficial level, we must examine the details of these processes and how they may operate in Titan's alien environment.

[3] The goal of this paper is to set some first-order constraints on the rate of fluvial erosion on Titan. Specifically, this paper addresses the scaling of fluvial incision processes that may operate in terrestrial bedrock streams to methane streams cutting into water ice bedrock on the surface of Titan. Fluvial channels on Titan may be carved by bedrock incision if the surface is largely solid ice, or by removal of sediments if the surface is composed of a loose regolith and the methane flow rate exceeds the rate of infiltration into the regolith. Chemical weathering of water ice by liquid methane on Titan's surface is theoretically negligible [Lorenz and Lunine, 1996], so we must consider erosional processes that are mechanical in nature.

[4] Sediment transport by methane streams on Titan is addressed in detail in a separate work (D. M. Burr et al., Sediment transport by liquid overland flow: Application to Titan, submitted to Icarus, 2005, hereinafter referred to as Burr et al., submitted manuscript, 2005), but I will summarize a few key points that relate to the bedrock incision problem. First, sediment falls about 3 to 4 times more slowly on Titan than on the Earth. The lower gravity on Titan's surface (Table 1) accelerates the sediment more slowly, but this is partially offset by the lower viscosity of liquid methane when compared with that of liquid water. In addition, water ice has a specific gravity of 2.06 in liquid methane, while the specific gravity of quartz in water is 2.65, so sediments on Titan are slightly more buoyant. The slower sediment fall velocity on Titan brings up a second important point, which is that for a given shear velocity in a flow, larger sediments can be transported on Titan. For example, on the Earth, fine gravel with a diameter of 4 mm lies at the threshold of bedload transport at a shear velocity of 0.05 m/s, while the same shear velocity on Titan will place very coarse gravel (d = 4 cm) at the threshold of bedload transport and will suspend coarse sand (d = 1 mm) (Burr et al., submitted manuscript, 2005), an order of magnitude difference in the size of mobile sediment between Titan and Earth conditions. Counteracting this ability to move large sediments, Titan's lower gravity requires streams to have steeper slopes and/or greater flow depths to achieve the same shear velocity as a stream on the Earth. Once this is taken into account, the diameter of sediment at the threshold of transport is only a factor of ∼2 higher on Titan than on the Earth, for a given slope or flow depth.

Table 1. Parameters Used for Example Streams in Figure 1
Surface gravity, m/s29.801.35
Density, kg/m31000450
Kinematic viscosity, m2/s1.1 × 10−64.2 × 10−7
SedimentQuartzWater ice
Density, kg/m32600930
Particle diameter, m0.010.01
Supply, particles/s10001000
BedrockSandstoneWater ice
Young's modulus, Pa5 × 10109 × 109
Tensile strength, Pa2 × 1061 × 106
Abrasion resistance parameter1.4 × 1061.9 × 104
Channel width, m1010
Channel slope0.0050.005

2. Bedrock Incision Processes

[5] Understanding sediment transport is only the first step in understanding how stream channels form on Titan. Unless all of the channels are carved from loose regolith, incision into the icy bedrock of Titan is necessary. In this section, I build on the terrestrial reasoning of Whipple et al. [2000] and Sklar and Dietrich [2001, 2004], to consider the relative rates of methane stream incision into water ice bedrock on Titan and water incision into silicate bedrock on Earth under similar stream conditions.

2.1. Abrasion by Saltating Bedload

[6] Saltating grains of sediment may chip off small pieces of bedrock as the grains strike exposed sections of the streambed. We may consider how this process scales from the Earth to Titan thanks to the work of Sklar and Dietrich [2004], who developed a model of the mechanics of bedload abrasion. Sklar and Dietrich model the erosion rate E as the product of three parameters: the fraction of bedrock exposure at the stream bottom Fe, the impact rate of particles on the stream bottom Ir, and the volume of bedrock eroded per impact Vi. The value of Fe depends on the supply of sediment to the stream, and the transport stage of the stream (the ratio of the shear velocity in the flow to the critical shear velocity at the threshold of sediment motion). The value of Ir is a function of the flux of particles, and the saltation hop length. These first two factors are different on Titan due to the ability of Titan streams to move the same size sediments at lower stream velocities. The value of Vi is dependent on the kinetic energy of each particle impact, and the resistance of the bedrock material to abrasion by small impacts. For equal diameter particles falling through fluid from the same height, the kinetic energy delivered by the particle impact is lower on Titan by a factor of ∼50 due to the lower sediment density and slower fall velocity. This factor alone would argue for slower bedload abrasion rates on Titan than the Earth. However, the abrasion resistance of water ice has not been previously measured.

[7] Abrasion resistance can be considered as a dimensionless parameter (kv) that relates kinetic energy put into a target by small impacts to mass loss of the target. Greeley et al. [1982] measured this parameter for different rock types using a carefully controlled sandblasting apparatus in a study of aeolian abrasion rates. Sklar and Dietrich [2004] measured this parameter for a variety of rock types in two ways, by using data from abrasion mills, and by repeatedly dropping a known mass from a known height onto a rock target. The latter method, though it sounds very simple, produces essentially the same results as abrasion mills and sandblasting (though it should be noted when comparing the results that the values of kv reported by Sklar and Dietrich [2004] are a factor of 106 too high, due to an error in nondimensionalization). Drop tests on rocks by Sklar and Dietrich [2004] give a value of kv of 1.4 ± 0.3 × 106.

[8] To determine a value of kv for water ice under Titan conditions, I performed drop tests on clear, frozen disks of deionized water. Tests were performed with a variety of chilled impactors from 2 g to 8 g in mass, dropped from 20 cm to 60 cm in height. Larger masses and drop heights tended to fracture through the ice disks instead of chipping small pieces from the surface. Experiments performed with ice chilled to 200 K in a special freezer and ice chilled to near 100 K with liquid nitrogen showed negligible dependence of kv on ice temperature, as long as the ice was not close to melting. The results of drop tests at 100 K which did not crack through the samples give a value of kv for water ice of 1.9 ± 0.8 × 104, almost two orders of magnitude weaker than rock. The low measured value of kv for ice effectively cancels out the effect of the lower kinetic energy imparted by impacts of sediment grains on Titan.

[9] The competing effects of low kinetic energy and low ice abrasion resistance brings the erosion rate close to Earth-like, with one of the closest Earth analogues being a stream channel carved into sandstone. Figure 1 is an example comparison of similar model streams on Titan and the Earth, showing how erosion rate varies with discharge in a rectangular channel, given a constant sediment supply. The model erosion rate curves were calculated using the method of Sklar and Dietrich [2004], but replacing the Manning equation for mean flow velocity with the Darcy-Weisbach equation to explicitly account for differences in gravity [e.g., Wilson et al., 2004]. Table 1 shows the input parameters for the model. At low discharges, there is no erosion because the sediments are immobile. At high discharges, the erosion rate decreases as the sediment is increasingly suspended in the flow and no longer impacts the bed. Peak erosion rates occur at smaller discharges on Titan than on the Earth, because lower shear velocity is needed to move ice sediment on Titan, and thus the transport stage is always higher on Titan for a given sediment size and a given discharge.

Figure 1.

Example calculation of erosion rates from bedload abrasion, using the parameters in Table 1 as constants in the model of Sklar and Dietrich [2004]. The solid line is the instantaneous erosion rate of water ice bedrock on Titan, while the dashed line is the rate for sandstone bedrock on the Earth in a stream with the same slope, width, and supply of sediment particles.

[10] Further work (e.g. cryogenic abrasion mills) can refine the abrasion resistance of ice beyond the simple drop tests reported here. Further work is also needed on the mechanical properties of ammonia-water ice, if this is a better analogue for bedrock on Titan. Initial measurements of Young's modulus for ammonia-water ice at 100 K show that it has a similar value to the Young's modulus of water ice [Lorenz and Shandera, 2001], so perhaps the uncertainty in ammonia content will not have a large effect on erosion rates.

2.2. Abrasion by Suspended Load

[11] In some high-velocity streams on the Earth, the bedrock may be eroded by the formation of flutes and potholes, often on the downstream surfaces of protrusions into the flow. This erosion may be due to the action of sediments suspended in the flow, which break away from the streamlines and impact rock surfaces where tight vortices form in the flow [Whipple et al., 2000]. Unlike the bedload abrasion problem, no complete physical model of suspended load abrasion has been published that can be scaled to Titan conditions. However, we may consider the relevant physical parameters and how they may enhance or reduce suspended load abrasion on Titan.

[12] Many of the same parameters that influence bedload abrasion would also influence suspended load abrasion. The low resistance of water ice to abrasion wear and the enhanced suspension of sediments on Titan would both tend to speed up suspended load erosion. The low gravity, and thus slower flow velocities, would lower the erosion rate, especially since Whipple et al. [2000] argue that suspended load abrasion rates will scale with flow velocity to the fifth power. To make any detailed predictions about erosion rates due to this process, more work needs to be done to understand how suspended sediments are decoupled from the flow and how much energy they impart to the bedrock surface.

2.3. Plucking

[13] Loosening and plucking of bedrock blocks from stream channels can be an effective incision process. It is most effective in areas where the bedrock is well jointed or layered, so that blocks are detached easily, and the limit on incision may be set by the sediment transport limit of the stream [Whipple et al., 2000].

[14] On Titan, the higher transport stage of equivalent streams would make it easier to transport large blocks, while the low tensile strength of ice (∼1 MPa) promotes the formation and propagation of fractures in the surface that would detach the blocks. It is unknown whether the ice bedrock on Titan is likely to be jointed or layered; while Ockham's razor encourages us to think that the ice is homogeneous, there are several possible volcanic, tectonic, or sedimentary processes that could be acting on Titan to produce joints and/or layers, and the ice bedrock geology could be complex. In the absence of more information about the properties of Titan's bedrock, it is difficult to make any more definitive statements beyond the observation that if plucking is not limited by the production of loose blocks, the transport limit is higher on Titan and plucking would be more effective than on the Earth.

2.4. Cavitation

[15] Cavitation is the formation of small vapor bubbles within a liquid flow, normally caused by a constriction or vortex that causes a pressure drop within the flow. If the local pressure in the flow drops below the vapor pressure of the fluid, vapor bubbles will form. If these bubbles travel to a region of higher pressure in the flow, they will suddenly implode with great force, potentially damaging nearby solid surfaces. In terrestrial rivers, cavitation is theoretically possible at very high flow velocities [Barnes, 1956], but there is a lack of direct field evidence for cavitation or its erosive effects in natural streams [Whipple et al., 2000], even though it is a common problem in engineering.

[16] Cavitation erosion has been proposed as a possible erosive agent on Titan [Lorenz and Mitton, 2002], due to the fact that Titan's surface temperature is close to the boiling point of methane, leading to a high vapor pressure of liquid methane, which would tend to enhance cavitation. The onset of cavitation in a flow is described by the cavitation index

display math

where p0 is the hydrostatic pressure within the flow, pv is the vapor pressure of the liquid, ρ is the density of the liquid, and U is the flow velocity. Cavitation occurs when σ falls below a critical value, which could be between 1 and 4, depending on the particular situation [Arndt, 1981]. Note that the cavitation index relies on the difference between the ambient pressure and the vapor pressure. Even though the vapor pressure of methane on the surface of Titan is about five times higher than the typical vapor pressure of water on the Earth, Titan's dense atmosphere more than compensates with an extra half bar of hydrostatic pressure. The term (p0pv) has a typical value of 98 kPa on the Earth, but is about 137 kPa on Titan. It is interesting to note that the value of this term is less than 1 kPa on Mars, where cavitation may be easily induced in water flows [Baker, 1979]. When examining cavitation at the base of flows, the pressure of the overlying liquid must be added to the atmospheric pressure. Figure 2 shows the relationship between flow velocity and depth for a cavitation index of 2. Above these curves, cavitation is likely to occur in natural streams [Whipple et al., 2000]. Note that the curve for Titan is higher than the curve for the Earth, due to the higher atmospheric pressure on Titan, and it is flatter, due to the low density of liquid methane and the low gravity on Titan. For flow depths <25 m, higher flow velocities are required to induce cavitation at the streambed on Titan than on the Earth. Higher required flow velocities are especially challenging on Titan given the lower gravity.

Figure 2.

Curves showing a cavitation index of 2 for different conditions of flow velocity and depth for streams on Titan (solid line) and the Earth (dashed line). Above the curves, cavitation is likely to occur, while it is unlikely to occur beneath the curves. Mars (thin line) is plotted for comparison with Baker [1979]. Note that for flow depths below 25 m, higher flow velocities are required to induce cavitation on Titan as compared to the Earth.

[17] Cavitation can occur at lower flow velocities with the addition of dissolved gases into the liquid. Nitrogen dissolved in methane streams on Titan could enhance the occurrence of cavitation. However, the addition of dissolved gas also tends to cushion the implosions of the vapor bubbles and inhibit their erosive effects [Barnes, 1956].

3. New Questions on Titan

[18] Despite orders of magnitude differences in some of the physical parameters that control fluvial incision on Titan and the Earth, erosion rates on Titan are likely to be surprisingly similar to terrestrial rates, given similar stream conditions. This begs the question of whether similar stream conditions would be expected on Titan and the Earth. Further detailed analysis of data from the Huygens probe and instruments on the Cassini orbiter will be able to shed some light on the nature of the channels themselves: slopes, depths, widths, and watershed areas. If the channels are fed by rainfall rather than by springs [Tomasko et al., 2005], then it becomes critically important to understand the frequency of rainfall events, and the rate of rainfall during such an event. Current work in this area indicates that rainstorms on Titan are likely to be rare but very intense [Lorenz et al., 2005], so fluvial erosion on Titan may occur rapidly, but in short pulses separated by long dry intervals. It is also important to understand the permeability of Titan's surface to liquid methane, in order to constrain runoff from rainfall events and the possible role of spring-fed streams. With the exception of cavitation, the fluvial incision processes outlined above depend on sediments entrained in the flow to provide abrasive particles (the “tools” effect) [Sklar and Dietrich, 2001], to form cracks by large particle impacts, or to wedge open cracks in the production of blocks for plucking [Whipple et al., 2000]. The presence and production of sediments in high elevation areas on Titan thus becomes an important problem. In the absence of chemical weathering of the bedrock or effective physical weathering by large temperature changes or frost wedging, it is unclear how the supply of ice sediment would be replenished at the headwaters of the streams. Impacts may spread loose ice particles over a wide area, though few impact craters have been observed, or particles may be transported by wind from the sedimentary basins back into the highlands. Particles of organic haze material may also continually fall from the sky, but this is likely to provide only very fine sediments, as the haze particles are generally smaller than 1 μm [Tomasko et al., 2005]. Unless there is a process to aggregate these particles on the ground, they are unlikely to be important erosive tools since they would be easily suspended and faithfully follow the flow streamlines. Finally, to understand fluvial erosion and many other aspects of Titan's geology, we need to better constrain the composition of Titan's bedrock, likely to be a mixture of water ice with something else, and obtain data on the physical properties of likely mixtures.


[19] I wish to thank Alfred McEwen, Zibi Turtle, and Ralph Lorenz for their hospitality and discussions while I spent a sabbatical at the University of Arizona, where much of the background work was performed. Thanks to Theo Collins (no relation) for assisting with the ice abrasion experiments at Wheaton. I also thank Leonard Sklar and Lionel Wilson for their reviews and comments on this work.