Background and Motivation
The cores of terrestrial planets are composed of a mixture that is
predominantly Iron together with light impurities such as Sulphur, Oxygen
or Silicon (see for example Stevenson, 1983). The temperature below which
freezing takes place (the liquidus temperature) is a function of
the concentration of impurity as well as of pressure. As the planet cools,
due to the effect of pressure, the liquidus temperature is reached first
at the centre of the (fluid) core (see for example Loper, 1984). When the
concentration of impurity is small, the solid that freezes is almost pure
Iron, with most of the impurity being left in the remaining fluid. This
segregation process leads to the growth of a dense solid inner core. The
boundary between the solid inner core and the fluid outer core is then
a freezing interface and the segregation of the light impurity into the
fluid means that this interface is a source of compositionally buoyant
fluid (see for example Loper and Roberts, 1981). The latent heat released
by the freezing process also contributes to the buoyancy but is believed
to be a less efficient means of driving convection than the compositional
buoyancy (see for example Gubbins et al, 1979).
Compositionally driven convection is believed to be a (if not the) primary source of energy required to generate the Earth's magnetic field, it may have been responsible for a strong lunar field during the first billion years or so of its history, and it may be maintaining a weak field in Mercury (see for example Merrill and McElhinny, 1983). In the cases of Venus and Mars (which both have extremely weak fields at present), core temperatures (for different reasons) have probably not yet fallen below the liquidus. In the future, further cooling may well result in the nucleation of an inner core and the emergence of a strong magnetic field, generated by a compositionally driven dynamo.
The compositional implications of the inner-core outer-core boundary (IOB) being a freezing interface were first discussed by Braginsky (1963), but it took some time before the importance of compositionally driven convection became widely recognised (see for example Verhoogen, 1980). Even now, though, dynamo theories continue to mainly use thermal buoyancy as the basis for their calculations (see for example Glatzmaier and Roberts, 1995). The justification for this that is usually made is that density differences due to compositional variations obey an equation that is essentially the same as that for density differences due to variations in temperature; the primary difference being that compositional variations typically diffuse very much slower than temperature variations. While this argument is true in the outer core, it completely ignores the role of the bottom boundary. Models of thermally driven dynamos typically consider unstable temperature gradients due to uniform internal heating and/or differential heating between the IOB and the core mantle boundary (CMB). When a strong magnetic field is present, convection is typically large-scale; on a lengthscale comparable with the core radius (see for example Fearn and Proctor 1983). Convection due to compositional buoyancy may be very different because of the source of the buoyancy; in the freezing interface at the IOB.
Compositional convection is a complicated process, and very little is known about how it is affected by rotation and magnetic fields, both possibly important in the application to the core. The discovery of the anisotropy of the inner core (see for example Karato 1993, Clement and Stixrude 1995) may provide some clues.
Our knowledge of the freezing process is based largely on the metallurgical literature (for example Chalmers, 1964; Copley et al, 1970) and more recently from aqueous experiments motivated by geophysical applications (for example Huppert 1990, Tait and Jaupart 1992). It seems certain (Loper and Roberts 1981) that the freezing interface is not flat, instead, freezing taking place in a "mushy zone". Above this zone there is fluid, below it there is solid, and the zone itself is a mixture of fluid and solid. The fluid fills the gaps between the solid which is probably in the form of dendrites. The mass fraction, f,of solid in the mushy zone increases with depth. Formally, the mushy zone probably occupies the entire inner core (with the temperature at the centre of the core greater than the eutectic temperature, see Fearn et al 1981). However, to be consistent with seimic data that show a sharp IOB, the effective depth (where there is a significant fraction of fluid) can only be of the order of a kilometre (see Loper 1983).
Progress in Understanding the Problem
Much of our understanding of the freezing process is based on the aqueous
Ammonium Chloride system first investigated by Copley et al (1970).
In this system, the buoyant fluid generated by the freezing process rises
in narrow plumes that emerge from gaps (called chimneys) in the dendrite
layer. Over most of the rest of the surface of the mushy zone there is
a slow downwelling of fluid from the fluid layer above. The nonlinear processes
in the mushy zone that cause this structure are extremely complicated,
involving the non-equilibrium thermodynamics of a freezing mixture flowing
through an evolving porous medium. Some progress has been made (Hills et
al, 1983; Hills and Roberts, 1987a,b) but we are still a very long
way from understanding this process in detail.
Several complementary approaches have been followed to advance our understanding of the problem of a solidifying alloy, and these have almost exclusively dealt with the laboratory scale problem (so the role of pressure is ignored). This makes sense, particularly since the problem has many practical applications such as the metallic casting problem that motivated Copley et al's (1970) early study. Both theoretical and experimental approaches have been followed. The theoretical subdivide into linear and weakly nonlinear work looking at the onset of convection (see for example Amberg and Homsy 1993, Emms and Fowler 1994, Fowler 1985, Worster, 1992), and nonlinear, dealing with the case where chimneys are established. To make theoretical progress, the former often have to make approximations such as the system being close to the eutectic composition or a steadily advancing mushy layer of constant height, while the latter have in some way to parameterise the chimneys (see for example Worster 1991).
The primary practical applications are in solidifying metallic alloys, but these have their experimental problems. Experimental approaches have therefore used both metallic systems (for example Sarazin and Hellawell 1988) and aqueous systems (for example Chen 1995). In general, the latter may not be a good model for the former, but the NH4Cl-H2O system pioneered by Copley et al (1970) is believed to be a good model for metallic systems. The aqueous models clearly have the advantage of being transparent so that the freezing process can be observed as it progresses. The evolution of the freezing metallic systems can only be inferred from an analysis of the final, completely frozen, ingot.
For application to the Earth we are interested in the nature of compositional convection in the core. The experimental work suggests that the convection may be in the form of narrow chimneys emanating from a mushy zone at the IOB. In the experiments, the chimney flow typically rises to the top of the fluid. In the Earth the effects of rotation and magnetic field are likely to be important; the motivation for the studies of Loper and Moffatt (1993), Moffatt and Loper (1994), but see also St. Pierre (1996). Bergman and Fearn (1994) have questioned whether chimneys do exist in the core. Many factors can prevent the formation of chimneys (see for example Sample and Hellawell 1984, Worster and Kerr 1994), but often these are associated with the suppression of convection in the mushy zone. The effective thinness of the mushy layer at the IOB can only be explained by a vigorously convecting mushy layer.
The nonlinear mechanism causing chimney formation is understood (see for example Tait et al 1992); it is due to the upwelling fluid causing localised melting, reducing the viscous resistance to the flow. A fast narrow upwelling in a chimney devoid of dendrites is balanced by a slow wide downward flow into the densely packed porous medium of the mushy zone. When a sufficiently strong magnetic field is present, it is the Lorentz force that is the dominant resistance to flow. Unlike viscosity this is independent of the lengthscale of the flow (see for example Raptis et al 1982), so the mechanism responsible for chimney formation may no longer dominate the dynamics of convection in the mushy zone. Using what little evidence is known about interdendritic spacing in the core (Esbensen and Buchwald 1982), Bergman and Fearn (1994) have argued that the geomagnetic field may indeed be strong enough to influence the nature of convection in the mushy zone at the IOB.
Tewari et al (1994) have investigated the effect of a magnetic
field experimentally but their field is probably too weak to produce the
predicted effect. Recent work using a higher magnetic field (Bergman et
al 1999) provides evidence that a sufficiently strong magnetic field
may inhibit chimneys in a convecting mushy zone.
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