On convection in ice I shells of outer Solar System bodies, with detailed application to Callisto
Introduction
It has long been thought that the larger icy satellites of the outer Solar System could contain internal liquid water oceans (Lewis, 1971). Indeed, even smaller midsize (∼500–1500-km diameter) satellites could have had oceans earlier in their histories, if convective transport of internal heat through the ice did not occur (Lewis, 1971, Consolmago and Lewis, 1977). Nevertheless, it is the larger icy satellites (Ganymede, Callisto, Titan, Triton, and Europa) and Pluto that are the best candidates for possessing oceans, because the minimum melting temperature of ice (251 K) is attainable at 0.21 GPa within their icy outer layers, or nearly so in the case of Europa and Pluto (Friedson and Stevenson, 1983, McKinnon et al., 1997, Spohn and Schubert, 2003, Schubert et al., 2004). Dissolved salts, ammonia, or methanol further lower this minimum melting temperature. An icy satellite warming from an initially cold condition thus first melts at depth, and if this melt is denser than the ice I-rich layer above, an internal ocean or aquasphere may open up; conversely if the melt is buoyant, volcanism may result. For a satellite cooling and freezing from a largely molten state (such as due to accretional, tidal, or early radiogenic heating), the residual ocean will concentrate dissolved solutes, enhancing the freezing-point depression (e.g., Kirk and Stevenson, 1987, Grasset and Sotin, 1996). Europa is a special case in that its ice shell is not thick in the above sense (it is unlikely that pressures are high enough for ice II or III at the base of its water/ice shell), but strong tidal heating in the surface ice layer almost certainly maintains a subsurface ocean (Ojakangas and Stevenson, 1989, Greeley et al., 2004).1
The ocean detected by electromagnetic induction within Callisto (Zimmer et al., 2000) was one of the more surprising results of the Galileo mission. It was argued by Ruiz (2001) that the dominant non-Newtonian creep mechanisms of water ice, grain-boundary sliding (Goldsby and Kohlstedt, 2001) and dislocation creep (Durham and Stern, 2001), make the ice shell above the ocean stable against solid-state convective overturn, and stable over geologic time. His argument relied on newly developed scaling and stability relationships for heat transfer by non-Newtonian (or power-law) convection (Solomatov, 1995) that superceded earlier parameterized convection formulations (see McKinnon, 1998, for a review). Moreover, although unstated, his argument applied to all similarly radiogenically heated ice I shells in the outer Solar System, and by extrapolation to midsized icy satellites and large Kuiper Belt objects (in which ice I is stable throughout much or all of their interiors; e.g., Schubert et al., 1986) as well. Conductive heat transport and internal melting are thus predicted to be, or have been, widespread among large and midsize outer planet satellites (cf. Fairén and Ruiz, 2003) and the larger Kuiper Belt objects (McKinnon, 2002). Convection, and the tectonics that result, could then only occur in ice I shells whose viscosities are lowered, e.g., by tidal flexing (McKinnon, 1999).
The arguments of Ruiz (2001) are compromised by a mathematical error (see Appendix A). Nevertheless, if power-law creep is suppressed, then it is important to consider Newtonian alternatives. Indeed, at very low stresses, it can be easily shown that convective instabilities may arise by means of diffusion creep (McKinnon, 2001), specifically lattice diffusion or Nabarro–Herring creep (Poirier, 1985, ch. 7). Lattice diffusion creep has not been observed in water ice (Frost and Ashby, 1982, ch. 16; Goldsby and Kohlstedt, 2001), but I will show that measurements of diffusion coefficients imply Newtonian viscosities low enough that solid-state convection is expected above Callisto's internal liquid layer over most of geologic time for plausible ice grain sizes. Ruiz (2001) offered that a thick, conductive icy shell would explain the lack of active geology on Callisto (e.g., Moore et al., 2004). Even with convection, however, the cold stagnant lid should be quite thick today (about 100 km), and thus also compatible with the lack of active geology. The question of surface geological effects in the past, when the stagnant lid must have been much thinner, is addressed herein.
More recently, Barr et al. (2004) have examined the question of non-Newtonian convective instability when a long wavelength, finite temperature perturbation is applied to basally heated ice shells. Barr and Pappalardo (2005) further compared such non-Newtonian instabilities against the possibility of convection arising through diffusion creep. The present paper addresses the general question of convective instability for temperature perturbations of arbitrarily small amplitude and of any wavelength, and elaborates on the consequences.
I first review the present-day picture for Callisto and the arguments that have been made concerning convection within it (Section 2). This is followed by a detailed examination of the applicability of diffusion creep to ice I (Section 3) and its implications for convection (Section 4). Results are presented in Section 5, first in simple form, and then with increasing levels of detail regarding convective characteristics and heat flow. Inferences are drawn for Callisto's evolutionary history, including the formation of its ocean and whether convection is occurring today. This is followed by an extended discussion (Section 6), which considers the implications of these results for Ganymede, Titan, Triton, Pluto, and other bodies. The question of likely ice grain size is addressed, as are the implications for convective start-up and shut-down for Callisto. The section finishes with an examination of convective stress levels, both in terms of potential tectonic effects and the likely role of multiple deformation mechanisms in icy satellite convection. Section 7 concludes with a brief summary and outline of future work.
Section snippets
The case for Callisto
The post-Galileo view of Callisto's internal structure is rather surprising (Fig. 1). Beneath an ancient, heavily cratered, mass wasted, and rather dirt-rich surface (Moore et al., 2004) lies a relatively clean water-ice I dominated layer, which in turn floats on a aqueous layer (or ocean) whose precise composition is unknown. Below the ocean are layers of higher-pressure ice phases and admixed rock, and possibly, a modest rock (i.e., metal + silicate) core (Schubert et al., 2004). These
Diffusional flow
An important aspect of the convective initiation problem that has been generally left out (at least until recently) is the role of diffusion creep. One reason for its neglect is that it has never been reliably measured in the laboratory for ice (due to experimental difficulties in obtaining measurements on sufficiently fine grained ice at low enough stresses; Frost and Ashby, 1982, Duval et al., 1983, Goldsby and Kohlstedt, 2001). Volume or lattice diffusion is very well understood
Convection on Callisto
Because diffusional flow is temperature-dependent but Newtonian and Callisto's floating shell is bottom heated, the well-understood stagnant lid scaling of Solomatov (1995) can be used with some confidence. Here I adapt the analysis in McKinnon (1999). The temperature drop across the bottom boundary layer, , is of order , where is the ≈adiabatic temperature of the convecting sublayer and is the (pressure-dependent melting) temperature at the base of the shell (Fig. 2). In
Results
The critical basal viscosity for convection to occur by diffusional creep is plotted in Fig. 4 as a function of true shell thickness. Equation (2) is inverted in the manner of Eq. (3), but ΔT is now an explicit function of depth through (Petrenko and Whitworth, 1999, Table 11.1), and α and k are evaluated at . The latter varies linearly between at and at . Density (ρ) is more sensitive to pressure than temperature in this situation, and an
Discussion
Ruiz (2001) argued that Callisto's ice I shell could not convect, citing the theoretical stability criteria for non-Newtonian convection. Although incorrect in detail, the stakes are still high because of the finite amplitude nature of the non-Newtonian instability. The latter applies to all radiogenically heated, separately convecting, ice I shells, not just Callisto's.
Callisto is one of the “big three” icy satellites, with Ganymede and Titan. Furthermore, from Solomatov (1995),
Summary and prospectus
I have shown that Callisto's floating ice I shell, and by implication similar shells in the outer Solar System, are susceptible to convective overturn for sufficiently small grain size. The deformation mechanism responsible is Newtonian, volume diffusion. This is in itself not a remarkable result, and was given in outline form by McKinnon (2001). It relaxes the argument against convection in Callisto of Ruiz (2001), to the extent that it needed relaxing. It complements the numerical results of
Acknowledgements
This research was supported by grants from the NASA Planetary Geology and Geophysics and Jovian System Data Analysis Programs. I thank J.D. Anderson and A.P. Showman for constructive reviews, F. Nimmo for useful comments, and A.C. Barr for discussions.
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2022, IcarusCitation Excerpt :For an ocean to form and remain liquid, the presence of long-lived heat sources is required, of which radioactive and tidal heating are the most significant contributors (McKinnon et al., 1997; Hussmann and Spohn, 2004; Schubert et al., 2010; Robuchon and Nimmo, 2011; Saxena et al., 2018). Consequently, understanding the long-term thermal and tidal evolution of planetary systems is a central tenet in evaluating the possibility for the existence of a present-day subsurface ocean and, in turn, the astrobiological potential of the outer Solar System (McKinnon, 2006; Mottl et al., 2007; Vance et al., 2007). The manuscript is arranged as follows.