SEARCH

SEARCH BY CITATION

Keywords:

  • Convection;
  • crystallization;
  • double diffusion;
  • fluid instability

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References

[1] Buoyancy forces derived from either thermal or compositional origins, or both, are the main driving mechanisms for dynamic instabilities within natural fluid systems. Crystallization can contribute to the compositional buoyancy due to density differences between the original bulk fluid and the residual fluid of crystallization. Depending on the density contrast, the compositionally induced buoyancy force may enhance or decrease the thermal buoyancy force established by the prevailing temperature gradient of the environment. The purpose of this paper is to examine the complex interactions between the compositional and the thermal buoyancy forces within a thermal chemical system incorporating the effects of crystallization. Like a thermal chemical system without crystallization (also known as an ordinary double-diffusive system), our results identify a dynamic and an oscillatory instability boundary in a traditional stability diagram. A dynamic instability boundary separates the stability diagram into two regimes. On one side of the boundary, fluid dynamic perturbations will grow indefinitely, leading to rigorous convection eventually. On the other side, perturbations decay in time, and the system will ultimately return to a static condition. In the absence of crystallization the oscillatory instability boundary defines a regime where perturbations lead to oscillatory motions. In the presence of crystallization, however, other investigators have observed the formation of convective layering. Despite these similarities, our results show significant differences in flow characteristic when crystallization is present. When crystallization is taken into consideration, the slope of a dynamic instability boundary is no longer governed by the diffusion coefficient ratio between heat and composition alone. The release of latent heat and the location of crystallization also play some roles. In addition, convective layering (i.e., the oscillatory instability regime) is no longer confined within a domain where fluid composition is stably stratified. It can extend into the compositionally unstable domain as well. Applications of our results to the solidification of magmatic bodies, the formation of columnar joints, and the cooling of the outer liquid core are examined. The existence of oscillatory instability within the compositionally unstable domain is significant because it suggests that the possible existence of convective layering within a cooling tholeiitic magma body cannot be ruled out, in contrast to a conclusion derived from double-diffusion studies in the absence of crystallization. Our analysis also suggests a possible criterion to test if columnar jointing has a double-diffusive origin. If columnar joints were thermal cracks formed along a preexisting double-diffusive fingering pattern as speculated, they should occur only in calc–alkaline basaltic lava, and they are not expected to appear in tholeiitic basalts. Iron–nickel precipitation above the inner–outer core boundary operates in a compositionally unstable domain because the residual liquid is expected to be less dense. Since the condition is unstable both thermally and compositionally at this boundary, solidification should lead to rigorous convection above the inner core. However, the existence of convective layering remains a possibility if the latent heat release is sufficient to cause the solidification process to operate within the newly determined oscillatory instability regime. Although the averaged temperature beneath the core–mantle boundary is believed to exceed the melting temperature of iron and nickel, local solidification is possible, resulting in the production of stabilizing convective layering that can inhibit large-scale convection near the top of the liquid outer core.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References

[2] This paper studies the fluid dynamic instability of a planar, thermal chemical system incorporating the effects of crystallization. Crystallization represents a general cooling of a fluid system within which solid crystals precipitate. Such systems are usually composed of two or more components and operate at temperatures very close to or below the solidification temperature of at least one of the components. Because of the involvement of both heat and multiple chemical constituents, these systems can be considered as a subclass of the general thermal chemical systems. It should be noted that thermal chemical systems are sometimes known as double-diffusive systems because the dynamics of these systems are driven by the diffusion of heat as well as the diffusion of chemical species. In this study we incorporate the effects of crystallization into a simple double-diffusive system and examine its resulting dynamic consequences. The analysis is then applied to investigating the cooling of extrusive lava flows and intrusive magma bodies and the cooling of the Earth's liquid outer core.

[3] Solidification dynamics in geological systems have received much attention during the past few decades. Using a linear convective stability analysis, Bartlett [1969] developed one of the earliest models to calculate temperature distributions within a cooling magma body and to study crystal settling in the presence of convection. Later, it was recognized that convection within cooling geological systems could be driven not only by thermal gradients but by compositional gradients as well. As a result, double-diffusive convection [e.g., Veronis, 1965] was introduced into geological investigations. Turner and Gustafson [1978] reported one of the earliest applications of the double-diffusive phenomenon to explain the formation of geological ore bodies. They reviewed the possible influence of double-diffusive processes in an exhalative massive sulfide ore deposit and other ore bodies. Subsequently, Chen and Turner [1980] conducted a laboratory study using an aqueous solution of sodium carbonate (Na2CO3) to study the effects of crystal precipitation at both the top and the bottom of a fluid layer. They observed that multiple-layered stratifications could form when crystals precipitated at the top. This layered structure is often referred to as double-diffusive layering. It is composed of multiple layers of different thickness, and each layer contains a distinct concentration of sodium carbonate. When crystallization took place at the bottom boundary, Chen and Turner observed that a layer of narrow, elongated fingers was produced. The fingering structure represents a collection of narrow plumes of the expelled less dense residual liquids (i.e., water in this case) after crystallization. These behaviors are in general agreement with the classical results of double-diffusive convection without crystallization [e.g., Veronis, 1965; Baines and Gill, 1969]. Since the study of Chen and Turner [1980] many other laboratory experiments investigating the effects of crystallization within a convective system have also been reported [e.g., Huppert and Turner, 1981; Huppert and Sparks, 1984]. Huppert et al. [1987] studied the effects of cooling at an inclined sidewall. This followed a similar experiment by Huppert and Turner [1978] that was designed to study ocean–iceberg interactions. Huppert et al. [1987] showed that as crystals form on a vertical (or an inclined) wall, a thin layer of residual liquid is produced adjacent to the wall. If the residual liquids are less dense than the bulk fluid, they can move very rapidly toward the top to produce layered stratifications. On the basis of these laboratory investigations, many investigators [e.g., McBirney and Noyes, 1979; Turner, 1980; Irvine, 1980; Sparks and Huppert, 1987] suggested that double-diffusive layering might explain the layered intrusions at Skaergaard, Muskox, and other locations. Later, however, on the basis of a stability analysis of a double-diffusive system by Baines and Gill [1969], McBirney [1985] argued that double diffusion could not be the origin of layered intrusion in a tholeiitic igneous body. Baines and Gill [1969] considered a system where a constant compositional gradient was imposed across the fluid layer. Effects of crystallization were not taken into consideration. At present, the role of double diffusion during the formation of igneous bodies remains a subject of uncertainty [McBirney and Nicolas, 1997].

[4] Besides magma chamber dynamics, solidification may also play important roles in the formation of the inner core and the fluid dynamics of the outer core. The energetics of the core have been studied by many investigators [e.g., Loper, 1991; Jacobs, 1987; Stevenson, 1981; Gubbins et al., 1979; Loper, 1978]. Using an adiabatic assumption, Buffett et al. [1992] presented a theoretical model to study the important parameters that determine the thermal state of the core. They concluded that the thermal state of the inner core is mostly controlled by the heat capacity of the outer core fluid and the heat escaping into the mantle. Considering the heat and mass balance within the outer core, Loper [2000] proposed a model for the dynamical structure of the liquid outer core. Although the energetics of the core have been studied, very little attention has been given to the dynamical effects of core solidification. In recent years, seismological studies have revealed significant information about the structure and the dynamics of the core. For example, Song and Richards [1996] proposed a differential rotation model between the inner core and the Earth's mantle based on their seismological observations. In response to this report, many other investigations have been carried out to understand the anisotropic nature of the inner core [e.g., Cormier et al., 1998; Souriau, 1998; Creager, 2000]. Whether the inner core anisotropy is a natural consequence of core solidification remains a subject to be investigated. It is our hope that our analysis can shed some light in this area of current interest.

[5] At present, most studies of fluid dynamics with crystallization have been carried out on the basis of laboratory experiments [e.g., Huppert and Worster, 1985; Huppert, 1990] or heat budget analyses [e.g., Worster et al., 1993]. In this study we present a more formal theoretical analysis that can be used as the basis for further theoretical and experimental investigations. We emphasize that this paper focuses only on the fluid dynamic aspect of the crystallization process. It can provide information about the possible effects of crystallization upon a convective geological system. The present analysis is unable to provide the finite amplitude solutions needed to address the details of the solidification process, however. For that purpose, numerical simulations are necessary. Although many numerical simulations of solidification processes have been reported in metallurgy literature, they are mostly designed for specific engineering applications [e.g., Prescott and Incropera, 1994] and do not provide the generality needed to address scientific insights of the crystallization process.

2. Ordinary Double-Diffusive Convection: A Summary

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References

[6] Following Baines and Gill [1969], let us consider a planar fluid layer of thickness d and with an infinite horizontal extent. The fluid layer is bounded at the top and the bottom by temperatures T1 and T2, respectively, and by compositional concentrations S1 and S2, respectively, as illustrated in Figure 1. This is a classical setup for a double-diffusive system. Let α be the coefficient of thermal expansion; let β ≡ (1/ρ0)(∂ρ/∂C) be the compositional coefficient that measures density changes as a function of concentration (note that the relationship between C and S is defined after equation (8)); let ν be the kinematic viscosity; and let κT be the thermal diffusivity of the fluid. Dynamic instability of the system is then governed by two parameters. They are the thermal Rayleigh number Rθ and the compositional Rayleigh number RS, as defined in equations (1) and (2), respectively:

  • equation image
  • equation image

In these two equations, ΔTT2T1; ΔSS2S1; and g is the downward gravitational acceleration as illustrated in Figure 1. Notice that both ΔT and ΔS can be either positive or negative depending on the temperature and the composition boundary conditions. Because all other parameters in these expressions are positive, the signs of the thermal and the compositional Rayleigh numbers are strictly dependent on the signs of ΔT and ΔS alone. Within the context of thermal convection, fluid motions are driven by density variations in response to the nonuniform distribution of temperature and composition within the fluid. Generally, when temperature increases, fluid density decreases owing to thermal expansion. The relationship between fluid density and composition is more complex, depending on the solute and the solvent density contrast of the fluid. For some fluid systems, bulk fluid density increases when compositional concentration increases. This occurs when solutes are heavier than the solvents. When solutes are less dense, however, bulk fluid density will decrease when compositional concentration of the solutes increases.

image

Figure 1. A schematic diagram of an infinite fluid layer used for this analysis. Relevant boundary conditions, physical parameters used in the analysis, and the reference coordinate system are illustrated.

Download figure to PowerPoint

[7] Because density changes represent the first-order cause of fluid instability, the stability of a double-diffusive system is best understood through a RθRS, plot as illustrated in Figure 2. This is known as a stability diagram in which Rθ is plotted against RS. The two axes define four quadrants denoted by 1, 2, 3, and 4. A positive Rθ requires a positive ΔT, and it indicates that the bottom temperature of the fluid layer is higher than the top temperature. Such a temperature gradient results in an unstable condition because fluid density near the bottom is less than that immediately beneath the top. Following the same logic, one can show that the negative Rθ suggests a thermally stable situation. Similarly, positive RS represents compositional stability, whereas negative RS suggests that the system is unstable compositionally. On the basis of these qualitative interpretations, quadrant 1 will be stable, and quadrant 3 will be unstable unconditionally. Quadrants 2 and 4 represent mixed conditions. The 45° line (EF) that goes through the origin represents the static stability boundary at which instability caused by compositional effect is exactly balanced by stability produced by the thermal effect, or vice versa.

image

Figure 2. A static stability diagram for a thermal chemical system. Thermal Rayleigh number is plotted against compositional Rayleigh number. The competing stability effects are illustrated. The 45° dashed line represents the static stability boundary. It defines the condition where thermal instability and compositional stability, or vice versa, are exactly balanced statically. To the left of this boundary, a system becomes statically unstable. Conversely, it becomes statically stable when it operates to the right of this boundary.

Download figure to PowerPoint

[8] However, the actual behavior of a fluid system is more complicated than this simple static interpretation because of fluid viscosity and other factors. The dynamical instability boundaries are determined [e.g., Baines and Gill, 1969] to be lines AB and CD, as illustrated in Figure 3. Line AB represents a dynamic stability boundary. To the right of AB, a system is dynamically stable, and it becomes dynamically unstable if a system operates to the left of this boundary. The slope of this boundary is determined by the relative magnitudes between the thermal diffusivity and the compositional diffusivity. If the two diffusivities are identical, line AB will be a 45° line going through point C. Point C falls on the Rθ axis, and it represents the classical critical Rayleigh number of a purely thermal convective system. The stable region to the right of line AB can be subdivided into two regimes by line CD. On the basis of these two dynamic stability boundaries together with the static stability boundary GF, the stability diagram can now be separated into regions I, II, III, and IV (Figure 3). Region I is bounded by lines CB and CD, and it represents a region of unconditional dynamic stability. In other words, for a system operating in this region, all perturbations will eventually dissipate, and the system becomes static in time. Region II is bounded by CA and CD. When a system is operating within this region, the thermal driving force is sufficiently strong that fluid particles tend to rise and sink according to their thermal state relative to their neighbors. However, these particles will not move very far, and the compositional restoring force becomes effective. As a result, fluid particles are returned to their original positions. At that point, thermal instability again takes over, and the cycle of motion repeats itself. Thus this region is known as the region of oscillatory instability. This type of transient behavior in a double-diffusive system has been documented previously [e.g., Veronis, 1965; Huppert and Turner, 1981]. Region III is bounded by lines GA and GF. When a system is operating in this region, it is dynamically unstable unconditionally. All perturbations will lead to convective motions. Region IV is bounded by lines GF and GB. This region represents a condition where a system is dynamically unstable but statically stable. From laboratory experiments this region is found to support fingering formation [Stern, 1960; Shirtcliffe, 1967]. These fingers represent narrow uprising plumes of compositionally unstable fluids through a thermally stable layer.

image

Figure 3. A dynamic stability diagram for an ordinary thermal chemical system in the absence of crystallization. It is plotted according to equations (A17) and (A38). Line ACB derived from (A17) represents the dynamic stability boundary, and line CD derived from (A38) represents the oscillatory instability boundary. Four convective regimes are identified in this diagram as regimes I, II, III, and IV. The dynamic natures of these flow regimes are explained in the text.

Download figure to PowerPoint

[9] Through this brief summary it is clear that when thermal and compositional effects are taken into consideration simultaneously, very complex dynamic phenomena can be produced. Besides the above mentioned ordinary double-diffusive complexities, the process of crystallization introduces additional complications into a solidifying thermal chemical system. In this paper the effects of crystallization are introduced as a perturbation to an ordinary double-diffusive system. Our interests are to examine the first-order fluid dynamic effects that crystallization imposes on an ordinary thermal chemical system.

3. Formulation of Fluid Instability in the Presence of Crystallization

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References

[10] Consider the same fluid layer as sketched in Figure 1. To simplify the analysis, the following assumptions are made. The fluid is first assumed to consist of only two components (e.g., a solute and a solvent) from which solid crystals of one component can precipitate. The saturation temperature of the fluid is assumed to fall between the boundary temperatures. As a result of crystal precipitation, a residual liquid of different concentration is produced, and the latent heat release associated with the crystallization process is also incorporated. In this analysis, effects of crystallization are modeled similar to the process of moisture condensation in cloud formation [Krishnamurti, 1975]. At the onset of crystallization the amount of crystals generated is so small that their presence is assumed to produce negligible effects on the overall flow characteristics. In other words, this paper is not intended to address problems related to flows with suspended crystals or flows within mushy zones.

[11] On the basis of the imposed boundary conditions, the basic static state of the system can be described by the following two equations:

  • equation image
  • equation image

where subscripts 1 and 2 and also d are defined in Figure 1. Here z is the vertical coordinate with its positive direction pointing downward in the direction of gravity. The governing equations for the perturbed state are given below. The Boussinesq approximation is employed in the following formulation such that the perturbed density is important only in the buoyancy term, and density of the fluid is treated as a constant everywhere else.

  • equation image
  • equation image
  • equation image
  • equation image

where equation image is the perturbed velocity vector; ρ0 is the reference density of the fluid; p is the perturbed dynamic pressure of the fluid; α, β, and g are as defined previously; equation image and equation image are the perturbed temperature and concentration, respectively; êk represents a unit vector in the vertically downward direction; κT and κs are the thermal and the compositional diffusivities, respectively; w is the vertical component of the velocity vector; ΔT and ΔS are the temperature and the compositional differences across the fluid layer, as previously defined; cp and is the specific heat under constant pressure. Equation (5) governs the conservation of mass; equation (6) represents the Navier–Stokes equations that govern the conservation of momentum; equation (7) represents the conservation of thermal energy; and equation (8) represents the conservation of compositional species. Because only one solute species exists in our model, there is only one equation for the conservation of compositional species. In more general situations, there should be a conservation equation for each species. ET and ES are the rates of heat and compositional changes due to crystallization, respectively. Following Krishnamurti [1975], ET and ES can be described by the following equations:

  • equation image
  • equation image

where h is the latent heat of crystallization per unit volume; Tsat is the saturation temperature (or the solidification temperature); δ(TTsat) is a Dirac delta function; Csat = Csat (p,T) and represents the saturation concentration, which is a function of temperature and pressure in most general applications. According to the chain rule, dCsat/dt can be rewritten in the following form:

  • equation image

As a result, the change of saturation composition as a function of time can be expressed in terms of the change of temperature with respect to time, and dCsat/dT represents a material property of the fluid.

[12] Although this analysis is also intended to apply to the solidification process within the outer liquid core, the Coriolis and the Lorentz forces are not included in the governing equations for the following reason. As indicated by Loper [2000], both the thermally and the compositionally induced buoyancy forces are believed to be the dominating forces that drive fluid motions within the outer core. Thus, to the first-order approximation and for the purpose of isolating the effects of crystallization, we choose to assume that the effects of these forces are indeed small and negligible.

[13] Following a common practice, equations (5)(8) are nondimensionalized according to a length scale d; a timescale d2/κT; a temperature scale ΔT; a concentration scale ΔS; and a pressure scale ρ0κTν/d2. The resultant linearized, nondimensional governing equations become the following:

  • equation image
  • equation image
  • equation image
  • equation image

where Prν/κT is the Prandtl number; Rθ and RS are the thermal and the compositional Rayleigh numbers, respectively, as defined in equations (1) and (2); mT is defined by (h/ρ0cp)(dCsat/dT), mS is defined by (ΔTS)(dCsat/dT), and τ is the ratio between the compositional diffusion coefficient and the thermal diffusion coefficient. Further, mT represents a ratio between the latent heat and the slope of solid–liquid transition of the mixture. If mT is large, it suggests that either the latent heat is large or the saturation concentration is very sensitive to temperature. Here mS simply represents a nondimensional slope of the solid–liquid transition of the mixture.

[14] Equations (12)(15) have also been simplified by introducing a nondimensional scale height z0 at which T = Tsat, or z0 ≡ (TsatT1)/(T2T1). As a result, the Dirac delta functions in (14) and (15) are now expressed in terms of z0 instead of Tsat. It is further assumed that z0 is very close to either the top or the bottom boundary so that the effects of a possible crystal mushy zone can be neglected.

[15] For igneous systems the viscosity of magma near the solidification temperature is typically of the order of 102 poise and higher [Murase and McBirney, 1973]. Consequently, Prandtl numbers are likely to exceed 104. Similarly, according to parameter values reported by Anderson [1995], the Prandtl number at the inner–outer core boundary can be estimated to be ∼20 and higher. The viscosity values given above are derived when the materials are in a totally molten state. It is conceivable that when a system is operating near the state of solidification, its viscosity can increase by many orders of magnitude. Thus, to further simplify the analysis, an infinite Prandtl number approximation is used.

[16] Before the system of governing equations can be solved, mechanical boundary conditions must also be specified. Normally, one would use a no-slip boundary condition for geological applications because geologic systems are likely bounded by solid surfaces. However, for mathematical simplicity and in order to obtain an analytic solution, a stress-free boundary condition is used instead. Experience indicates that flow structures are not very sensitive to the mechanical boundary conditions as long as the ratio of the Rayleigh number to the corresponding critical Rayleigh number remains the same. For example, if a flow of a Rayleigh number that is 5 times the no-slip critical value under a no-slip boundary condition yields a hexagonal cell structure, the same hexagonal cell structure is expected under a stress-free boundary condition if the Rayleigh number is also 5 times the stress-free critical value. In addition, the nature of the stability boundaries is insensitive to the mechanical boundary conditions as well. Therefore a stress-free boundary is considered appropriate for this analysis.

[17] Equations (12)(15) together with the constant temperature, constant concentration, and stress free boundary conditions form a well-posed problem mathematically. It can be solved for the field variables of temperature, concentration, pressure, and velocity components. A small perturbation analytical procedure is used to seek solutions for the problem. Details of the solution method are given in Appendix A.

4. Results and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References

4.1. Effects of mT and mS on Stability Diagrams

[18] Equation (A22) represents the dynamic stability boundary for nonoscillatory motions, and equation (A41) is the boundary for oscillatory instability. These two equations are reproduced below:

  • equation image
  • equation image

Notice that (16) is identical to (A17) if mT = mS = 0. In other words, in the absence of crystallization, solutions of the system should become that of an ordinary double-diffusive system. Hence terms that contain the mT and the mS parameters represent the contributions from crystallization. Similarly, (17) shows the same characteristics relative to (A38). We also note that in most natural systems, crystallization produces a latent heat release, and the saturation concentration of the solute usually increases as temperature increases. These suggest that mT and mS are positive quantities in most applications. In such a case, the critical thermal Rayleigh number at RS = 0 becomes larger in the presence of crystallization. This is because latent heat release retards cooling of the system, resulting in the delay of fluid instabilities. In other words, it requires a higher thermal potential to overcome the stabilizing effect of crystallization to produce fluid motions. It is also immediately apparent from (16) and (17) that the dynamic instability boundary and the oscillatory instability boundary no longer intersect at the Rθ axis because of crystallization.

[19] To illustrate the effects of these stability boundaries, let us consider a situation where crystallization is assumed to take place very close to the top and the bottom horizontal boundaries such that the following relationship holds:

  • equation image

By making such an assumption, the actual magnitude of mS and mT become less important. One can always choose z0 to be very close to 0 or 1, which implies solidification immediately next to the boundaries, to satisfy the above assumption. Substituting (18), (16) and (17) become the following:

  • equation image
  • equation image

These two stability boundaries are plotted in a traditional RθRS diagram (Figure 4). Line AD represents (19), whereas line EF represents (20). Point B is the intersection between AD and the Rθ axis, and point C is the intersection between lines AD and the extension of EF. An enlargement of the area surrounding these intersections is given in the inset. Statically, positive RS suggests compositional stability, and positive Rθ indicates thermal instabilities, as discussed earlier for Figure 2. The 45° line (i.e., OH) represents the static stability boundary. As indicated in Figure 2, a system operating to the left of OH is statically unstable, and it becomes statically stable when operating to the right of this boundary. On the basis of both the dynamic and the static stability boundaries, the RθRS plot can be separated into four different stability domains, identified by the Roman numerals I, II, III, and IV, similar to that of Figure 3. Region I is bounded on the left by lines FC and CD and is the region where stability exists both dynamically and statically, indicating that all perturbations will decay in time. Region II, bounded by lines AC and CF, admits oscillatory instability. Despite its name, this is the regime where stably stratified layering was observed [e.g., Chen and Turner, 1980]. The area to the left of lines AG and GH (III) represents a regime where it is unstable both dynamically and statically. Perturbations within this region will grow in time and turn into dynamic flows. Region IV is the triangular area bounded by GH and GD. This is an area where the system is dynamically unstable but statically stable. Laboratory experiments [Chen and Turner, 1980] showed that elongated fingering is produced under this condition. Although these four stability regimes are qualitatively similar to that determined under ordinary double-diffusive conditions (e.g., Figure 3), significant differences exist. First, the slope of the dynamic stability boundary is no longer dependent only on the ratio between the thermal diffusion coefficient and the compositional diffusion coefficient. It also depends on mS and mT, which are related to the solid–liquid transition slope of the system and the latent heat of solidification among some other parameters. Under an ordinary double-diffusive condition the coefficient of compositional diffusion is usually small compared to the thermal diffusion coefficient. As a result, the slope of a dynamic stability boundary is usually quite steep. When crystallization is taken into consideration, the resultant slope of the dynamic stability boundary can be modified because both mS and mT can be of order one. In addition, the intersection between the dynamic stability boundary and the oscillatory instability boundary no longer occurs at the Rθ axis, as pointed out earlier. Consequently, oscillatory instability can now extend into the negative RS domain. This effect is illustrated in the enlarged inset of Figure 4. The existence of oscillatory instability in the negative RS domain might have important implications for the formation of igneous layering, to be discussed below.

image

Figure 4. A stability diagram for a thermal chemical convective system in the presence of crystallization. Line ABCD derived from (19) is the dynamic stability boundary, and line CEF derived from (20) is the oscillatory instability boundary. Line OGH is the static stability boundary. The inset is an enlarged picture of the area near points B, C, and E. The dynamic natures of regimes I, III, and IV are similar to those in Figure 3. Laboratory observations showed that double-diffusive layers are found in regime II.

Download figure to PowerPoint

4.2. Application to Magma Dynamics

[20] Igneous magmas possess a very broad spectrum of physical properties. To the dynamic process that is of interest here, the most important property is the density change associated with magma differentiation. Tholeiitic and calc–alkaline magmas behave differently during differentiation [e.g., McBirney, 1985]. Crystallization of tholeiitic magma can produce a wide range of residual liquid densities [Huppert and Sparks, 1980]. Generally, during the early stage of differentiation, olivine precipitation takes place and generates a less dense residual liquid. After the initial precipitation of olivine crystals, tholeiitic magma differentiation gradually becomes dominated by plagioclase precipitation, resulting in the production of denser residual liquids. Following McBirney [1985], it is assumed here that differentiation of tholeiitic magma results in a residual liquid that contains a higher concentration of FeO. Consequently, the resultant residual liquid has a higher density than the original bulk fluid. At the roof of a tholeiitic magma body, therefore, solidification can produce a compositionally unstable situation because the residual liquid immediately beneath the ceiling is heavier than the bulk fluid underneath. On the other hand, at the floor of a tholeiitic magma body a compositionally stable condition can result. From the thermal point of view the top of a magma chamber is under an unstable condition because the interior of a magma body is at a higher temperature than the roof wall. At the bottom of a magma body, however, the thermal gradient is stabilizing because a lower temperature exists within the country rocks sitting beneath the magma body. Combining both the thermal and the compositional effects, the following stability conditions can be derived qualitatively. At the roof of a magma chamber, tholeiitic magma solidification results in a thermally and compositionally unstable situation. On the other hand, solidification of tholeiitic magma at the floor creates a stabilizing condition both thermally and compositionally.

[21] Calc–alkaline magma bodies show opposite density trends after differentiation [McBirney, 1980]. Their residual liquids tend to be less dense than the original bulk fluid. Following the same qualitative reasoning as discussed for the tholeiitic magma, a thermally unstable but compositionally stable condition exists at the roof of a calc–alkaline body. At the floor of the same magma body, however, it is unstable compositionally but stable thermally.

[22] A stability diagram for the roof and the floor of these two different types of magma body in terms of Rθ and RS is given in Figure 5. In this diagram the thermal and the compositional conditions resulting from solidification of the two different types of magma at two different locations are illustrated by the four quadrants distinguished with different colors. A similar diagram had been reported by McBirney [1985]. Although these colored quadrants illustrate the qualitative operating domains for each case, they do not provide any information about the dynamical instability of the system under these conditions. Thus the stability boundaries illustrated in Figure 4 must be superimposed to understand the dynamic behaviors of magma solidification. For illustration purposes, the parameters used previously are adopted here. Because our model assumes a two-component system, it is undoubtedly a highly simplified model for any natural magmatic system. Nevertheless, the parameters chosen are appropriate for the initial stage of magma solidification. As a result, our model should be adequate to illustrate the general dynamical behaviors within solidifying magmatic bodies.

image

Figure 5. Same as Figure 4, except that the possible geological applications to the roof and the floor of a tholeiitic and a calc–alkaline magma body are superimposed. The top right-hand quadrant is applicable to the roof of a calc–alkaline magma body, and the top left-hand quadrant is applicable to the top of a tholeiitic magma body. The bottom right-hand quadrant represents the floor of a tholeiitic body, whereas the lower left-hand quadrant represents the bottom of a calc–alkaline body.

Download figure to PowerPoint

[23] On the basis of Figure 5, convective layering remains possible within tholeiitic bodies if these bodies operate within the orange triangular area bounded by BCE as illustrated in the inset. Notice that this area represents a condition that is suitable for the cooling at the roof of a tholeiitic magma body, as indicated by the pink color (i.e., the top left-hand) quadrant. We must reiterate that double-diffusive layers are observed within the oscillatory instability regime II [Chen and Turner, 1980]. Therefore oscillatory instability implies the existence of step-like layered stratification within a crystallizing double-diffusive system. As to the solidification of calc–alkaline magmatic bodies, occurrence of double-diffusive layering should be more likely according to this analysis. Vertical zonation of major element geochemistry within calc–alkaline magma bodies has been well documented. McBirney [1980] provided a good description of this remarkable characteristic. He further suggested that double-diffusive layers might be the explanation for the rectilinear trend of calc–alkaline magma differentiation both in terms of the major and the trace element geochemistry. However, cooling at the top of a calc–alkaline magma body can also operate within the blue regime bounded by line AB and the Rθ axis. In that case, thermal buoyancy dominates, and the system becomes unstable dynamically. As a result, no convective layers are expected. Although the size of this dynamic regime is relatively small in this diagram, the actual extent of this unstable area is dependent on several factors, including the latent heat release and the ratio between the thermal and the compositional diffusivities, for example. In order to evaluate if double-diffusive convection can exist within a specific magmatic body, a more complete, nonlinear analysis using appropriate parameter values is necessary.

4.3. Application to Columnar Joints

[24] Another unique dynamic feature of double-diffusive convection is the formation of fingering. Within many cooling basaltic lava flows, and recorded within some eroded intrusive basaltic plugs, elongated fracture patterns with hexagonal or rectangular cross sections called columnar joints are often observed. These fracture patterns have been explained as thermal cracks within a cooling basaltic body [Ryan and Sammis, 1978; Aydin and Degraff, 1988]. Although it is agreed that these fractures are initiated by thermal contraction during the cooling of a freshly emplaced basaltic body, a hypothesis has been suggested that once initiated, these thermal cracks propagate along a preexisting finger-like pattern established by the cooling of a double-diffusive system near the floor of the cooling magma bodies [Kantha, 1980, 1981; Hsui, 1989]. If the double-diffusive hypothesis is correct, it follows that it is highly unlikely for tholeiitic lava to show columnar jointing because a fingering regime does not exist on the floor of a tholeiitic magma body, as illustrated in Figure 5. On the other hand, the floor of a calc–alkaline basaltic lava flow is more likely to produce double-diffusive fingering. This analysis therefore suggests a criterion to discriminate if columnar jointing has a double-diffusive origin by testing the chemical composition of the basaltic lava flows where columnar joints are found. If columnar joints are observed only in tholeiitic basalts, double-diffusive processes probably do not play any role in columnar joint formation. However, if columnar joints are found largely in calc–alkaline basalts, the role of double-diffusive convection must be examined in more detail.

4.4. Application to the Core

[25] As indicated previously, this analysis can also be applied to the solidification process within the outer core. On the basis of thermodynamic considerations, the thermal gradient within the core is always unstable regardless of whether it is at the top or the bottom of the liquid outer core. As a result, solidification beneath the core–mantle boundary and above the inner–outer core boundary must operate in the positive thermal Rayleigh number (i.e., Rθ > 0) regime. Current thinking suggests [e.g., Anderson, 1995] that solidification within the core represents the removal of the heavy elements (i.e., the iron and nickel components) from the molten liquid mixture. Therefore the residual fluid will be less dense. This results in a compositionally stable condition beneath the core–mantle boundary and becomes unstable above the inner–outer core boundary. A question one needs to address before going any further is whether iron–nickel precipitation is possible beneath the core–mantle boundary. Most of the classical textbooks indicate that the laterally averaged temperature of the outer core exceeds the melting temperature of iron and nickel. This is to ensure that the outer core is in a molten state as constrained by the lack of seismic shear waves in this region. However, one must remember that these textbook plots are one-dimensional plots. In a real three-dimensional situation, core convection will take place with upwelling and downgoing currents. These currents will produce lateral temperature variations from the averaged value. Bloxham [2000a] also suggested the possible existence of lateral temperature variations at the core–mantle boundary on the basis of an analysis of paleomagnetic secular variations. Generally, at a location where an upwelling current impinges, the local temperature can reach a value higher than the average. Conversely, at a location where a downgoing current initiates, the local temperature can be lower than the average value. Consequently, at the top of a downgoing current, temperature can be below the melting temperatures of iron and nickel. This would be especially true in areas beneath the lower mantle where cold subducted surface materials accumulate. Solidification beneath the core–mantle boundary therefore is possible at least locally. The actual magnitudes of the lateral temperature variations are uncertain, however. On the basis of an analysis of the geomagnetic axial dipole, Bloxham [2000b] estimated that lateral heat flow variations at the core–mantle boundary could vary from 5 to 23% of the adiabatic value.

[26] The stability regimes where the solidification processes operate beneath the core–mantle boundary and above the inner–outer core boundary are illustrated in Figure 6. Dynamic stability boundaries have also been superimposed in this diagram. On the basis of this illustration it is concluded that double-diffusive layers are likely to occur beneath the core–mantle boundary at least locally. At present, however, other than the D″ layer at the base of the mantle, there is no observation indicating the existence of any large-scale layered structure beneath the core–mantle boundary. Although our analysis suggests that double-diffusive layering can occur above the inner–outer core, it must take place under the condition that the system is operating within the triangular oscillatory instability regime defined by BCE in Figure 6. Not knowing the actual physical parameters at the inner–outer core boundary, it is not clear whether double-diffusive layering can be used to explain the recent seismic observations that suggested the possible existence of spherical layering within the inner core [Cormier et al., 1998; Creager, 2000].

image

Figure 6. Same as Figure 4 with the applications to the liquid outer core superimposed. The top right-hand quadrant is applicable to the region immediately beneath the core–mantle boundary, and the top left-hand quadrant is applicable to the region immediately above the inner–outer core boundary.

Download figure to PowerPoint

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References

[27] In summary, a theoretical model for the thermal chemical system incorporating the effects of crystallization has been developed. The analysis is based on a small perturbation theory applied to an ordinary double-diffusive system [e.g., Baines and Gill, 1969]. Results show that owing to latent heat release from crystallization, the slope of the dynamic stability boundary is no longer controlled by the ratio between the thermal and the compositional diffusion coefficients alone. Both the amount of latent heat release and the location of crystallization can make significant contributions to the determination of the slope of this boundary. Unlike the results of an ordinary double-diffusive system, the domain of oscillatory instability can extend into the regime where it is unstable both thermally and compositionally. Because of crystallization, our results suggest that double-diffusive layering remains a plausible mechanism for tholeiitic differentiation suites from a dynamical point of view, given favorable conditions for the system to operate in the oscillatory instability regime. This is in contrast with the conclusion derived on the basis of a double-diffusive analysis without the consideration of crystallization [e.g., McBirney, 1985]. In addition, our analysis also suggests a criterion to test if columnar joints can have a double-diffusive origin by examining the chemical compositions of basaltic lava flows where columnar joints are observed. If the formation of columnar jointing requires a preexisting fingering structure, only calc–alkaline lava flows may show columnar jointing because double-diffusive fingering can be produced only at the floor of this type of cooling lava. Double-diffusive fingers are unlikely to occur within a tholeiitic lava flow because cooling at its base is stabilizing and it cannot produce any dynamic fingering feature. As to the solidification above the solid inner core, double-diffusive layers can occur only if the conditions are such that the cooling process operates within the oscillatory instability regime (i.e., the triangular area BCE as illustrated in Figure 6). Whether double-diffusive layers can be used to explain the recently reported layered structure within the inner core [Cormier et al., 1998; Creager, 2000] remains to be investigated. On the other hand, local solidification of iron and nickel beneath the core–mantle boundary may produce a stabilizing condition compositionally, which in turn may prohibit large-scale convective motions immediately beneath the core–mantle boundary. This is consistent with the conclusion derived by Loper [2000]. Although our results indicate that local layering structures might exist beneath the core–mantle boundary, there is no observation to support this result at present. In order to examine the finite amplitude solutions for a more realistic investigation of these solidifying systems, numerical approaches are necessary. This study represents only a preliminary investigation of the dynamical behaviors induced by crystallization within a thermal chemical system.

Appendix A.: Details of Mathematical Analysis

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References

[28] The purpose of this appendix is to provide the mathematical details for solving the nondimensional governing equations (12)(15). First, it is recognized that a divergent free velocity field is used to describe the conservation of mass (i.e., equation (12)). For a divergent free velocity field, velocity vectors can be represented by the following representation [Chandrasekhar, 1961], where ϕ and ψ are the poloidal and the toroidal scalar velocity potentials, respectively:

  • equation image

Taking the vertical component of the curl of the curl of (13) and substituting (A1) into (14) and (15), we have

  • equation image
  • equation image
  • equation image

where equation image is the horizontal Laplacian and τ is the ratio of compositional diffusion coefficient to thermal diffusion coefficient. In most cases this is a small number, possibly of the order of 0.0001 and smaller. However, in some systems it can be as large as order 0.01 [Lundstrom, 2000]. Equations (A2)(A4) can now be solved subjecting to a stress-free, nondeformable, constant temperature and constant concentration boundary condition as described below.

  • equation image

[29] Solutions for equations (A2)(A5) are then sought in terms of an infinite series in the horizontal directions. This is a very standard solution method for flows with infinite horizontal extents. They are taken to have the following forms:

  • equation image

where equation image is the position vector, Cn are the harmonic coefficients, and equation image are the horizontal wave number vectors. Further, Cn and equation image must also satisfy the following conditions:

  • equation image

where the superscript * represents the complex conjugate and α is a scalar representing the magnitude of wave numbers. Substituting (A6) and (A7) into (A2)(A5), we have

  • equation image
  • equation image
  • equation image
  • equation image

Notice that (A8)(A11) are identical to that of the classical double-diffusive convection if mT = mS = 0. Thus the terms containing mT and mS represent the contributions from crystallization. For our stability analysis, magnitude of the crystallization effect is assumed to be small. Consequently, both mT and mS are chosen to be of O(ɛ). Here ɛ is a small parameter related to the magnitude of crystallization. As a result, solutions of the following form are sought:

  • equation image

Quantities with subscript 0 are solutions in the absence of crystallization, and quantities with subscript 1 are the leading terms of solutions that include the effects of crystallization. These leading terms are of O(ɛ). Additionally, it should also be noted that the initial compositional Rayleigh number is altered as crystallization is taking place. This is necessary because crystallization will change the total solute concentration of the fluid. To account for this effect, therefore, RS should be expanded in terms of ɛ similar to (A12) as well. Thus

  • equation image

where RS1 is of O(ɛ). This is a fairly standard approach to seeking solutions for this type of equation. It is known as the method of successive approximation [e.g., Busse and Riahi, 1980].

A1. Exchange of Stability

[30] According to Chandrasekhar [1961], the condition of stability exchange for a monotonic (i.e., nonoscillatory) stability can be determined by setting ∂/∂t = 0. In the absence of crystallization (i.e., for the zeroth order solution), equations (A8)(A11) become the following:

  • equation image

Solutions for (A14) are given below:

  • equation image

Substituting (A15) into the first equation of (A14), we have

  • equation image

Equation (A16) defines the stability boundary in a RθRS space for various wave numbers represented by α. At the minimum critical wave number (i.e., equation image), (A16) becomes

  • equation image

This relationship is identical to that obtained by Baines and Gill [1969] for an ordinary double-diffusive system. This should be the case since the effect of crystallization has not yet been incorporated. To study the effect of crystallization, we need to examine the first-order governing equations. To the order of ɛ, the governing system obeys the following equations:

  • equation image

To obtain the correction of stability boundary due to crystallization, it is necessary to find the solubility condition for (A18). Following Schlüter et al. [1965], the solubility condition can be derived by multiplying the first equation in (A18) by G0, the second by −RθH0/α2, and the third by (RS0F0/α2). Then, sum the three equations and integrate over the entire fluid layer utilizing the given boundary conditions. The resultant equation is given below:

  • equation image

Since G0, F0, and H0 are given in (A15), the three integrals in (A19) take on the following forms:

  • equation image

Substituting (A20) into (A19) and rearranging, we have the following:

  • equation image

The boundary of exchange of stability that includes the effects of crystallization can be obtained by combining (A21), (A17), and (A13). The result is given below:

  • equation image

Since (A21) does not contain wave number, the critical minimum wave number remains unchanged.

A2. Transient Analysis of Oscillatory Motion

[31] The nature of transient oscillatory motion in a double-diffusive system has been discussed in section 2. For this analysis the Prandtl number is retained initially. Solutions for an infinite Prandtl number represent a limiting case of the general solutions. Again, the linear system in the absence of crystallization is given by the following equations:

  • equation image

A set of solution for G0, H0, and F0 can be constructed by taking

  • equation image

where ω0 is the angular frequency. Substituting (A24) into the second and third equations of (A23), we obtain the following:

  • equation image
  • equation image

It is recognized that G0, H0, and F0 as given in equation (A24)(A26) satisfy the boundary conditions specified in (A23). Substituting (A24)(A26) into the first equation of (A23), and after some algebraic manipulations, the value of ω0 corresponding to a purely oscillatory motion and the associated stability boundary are found to be the following:

  • equation image
  • equation image

Again, (A27) and (A28) are identical to that obtained by using a more traditional eigenvalue problem approach [e.g., Veronis, 1965]. To the first order of crystallization, the governing equations for oscillatory motion become the following:

  • equation image

It should be noted that unlike the equations in (A23), the equations in (A29) represent a set of inhomogeneous equations. To determine the existence condition for this type of equations, a method of adjoint solutions [Riahi, 1984] is adopted here. Special solutions of the adjoint linear system (i.e., adjoint to the system represented by the equations in (A23)) corresponding to the solutions (A24)(A26) are the following:

  • equation image

It should also be noted that the equations in (A23) possess another set of possible solutions as given below:

  • equation image

Special solutions for the linear adjoint system corresponding to (A31) can then be constructed.

  • equation image

Substituting (A24)(A26) into (A29), we have the following:

  • equation image

In order to obtain the solubility condition of this set of equations, multiply the first equation of (A33) by Ĝ0, the second equation by −(Rθ/α2)Ĥ0, and the third equation by (Rs0/α2)equation image. Sum the resultant equations and integrate across the layer utilizing the given boundary conditions. Thereafter, average the equation in time over a period of 2π/ω0. The following equation is obtained:

  • equation image

Similarly, one can substitute (A31) into (A29) and form a set of equations similar to (A33). Again, to obtain a solubility condition, it is necessary to multiply the first equation by equation image, the second equation by −(Rθ/α2)equation image, and the third equation by (Rs0/α2)equation image. Add the three equations. Integrate across the fluid layer and average over a time period of 2π/ω0. The resultant equation is given below:

  • equation image

Equations (A34) and (A35) form two equations for two unknowns, RS1 and ω1. These two equations can be rewritten explicitly in terms of the two unknowns as in the following:

  • equation image
  • equation image

Equations (A36) and (A37) can be solved for RS1 and ω1 uniquely. To the limits of a large Prandtl number and a small diffusivity ratio (i.e., Pr[RIGHTWARDS ARROW]∞ and τ≪1), (A27) and (A28) become the following:

  • equation image

Utilizing (A38), it follows that ω1 and RS1 can be expressed by the following equations:

  • equation image
  • equation image

Since RSRS0 + RS1, and based on (A38) and (A40), we have the following relationship:

  • equation image

Equation (A41) defines the boundary of oscillatory instability. Together with (A22), these two equations specify the different modes of instability in the RθRS space.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References

[32] A.T.H. is grateful for many helpful discussions with Craig Lundstrom and Chu-Yung Chen. Reviews by Peter van Keken (the associate editor), Alison Leitch, and two other anonymous reviewers have greatly improved the manuscript. Their efforts are very much appreciated. Any mistakes, however, remain ours. This research was initiated through a grant from the Research Board of the University of Illinois at Urbana-Champaign.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Ordinary Double-Diffusive Convection: A Summary
  5. 3. Formulation of Fluid Instability in the Presence of Crystallization
  6. 4. Results and Discussion
  7. 5. Conclusions
  8. Appendix A.: Details of Mathematical Analysis
  9. Acknowledgments
  10. References
  • Anderson, O. L. (1995), Mineral physics of iron and of the core, Rev. Geophys., 33, supplement, 429441.
  • Aydin, A., and J. M. Degraff (1988), Evolution of polygonal fracture patterns in lava flows, Science, 239, 471476.
  • Baines, P. G., and A. E. Gill (1969), On thermohaline convection with linear gradients, J. Fluid Mech., Part 2, 37, 289306.
  • Bartlett, R. W. (1969), Magma convection, temperature distribution, and differentiation, Am. J. Sci., 267, 10671082.
  • Bloxham, J. (2000a), The effect of thermal core–mantle interactions on the palaeomagnetic secular variation, in Geomagnetic Polarity Reversals and Long-Term Secular Variation, edited by D. Gubbins, D. V. Kent, and C. Laj, Philos. Trans. R. Soc. Math. Phys. Eng. Sci., 358(1768), 11711179.
  • Bloxham, J. (2000b), Sensitivity of the geomagnetic axial dipole to thermal core–mantle interactions, Nature, 405(6782), 6365.
  • Buffett, B. A., H. E. Huppert, J. R. Lister, and A. W. Woods (1992), Analytical model for solidification of the Earth's core, Nature, 356, 329331.
  • Busse, F. H., and N. Riahi (1980), Nonlinear convection in a layer with nearly insulating boundaries, J. Fluid Mech., 96, 243256.
  • Chandrasekhar, S. (1961), Hydrodynamic and Hydromagnetic Stability, 654 pp., Oxford Univ. Press, New York.
  • Chen, C. F., and J. S. Turner (1980), Crystallization in a double-diffusive system, J. Geophys. Res., 85(B5), 25732593.
  • Cormier, V. F., X. Li, and G. L. Choy (1998), Seismic attenuation of the inner core: Viscoelastic or stratigraphic? Geophys. Res. Lett., 59(21), 40194022.
  • Creager, K. C. (2000), Inner core anisotropy and rotation, in Earth's Deep Interior: Mineral Physics and Tomography From the Atomic to the Global Scale, Geophys. Monogr. Ser., vol. 117, pp. 89114, AGU, Washington, D. C.
  • Gubbins, D., T. G. Masters, and J. A. Jacobs (1979), Thermal evolution of the Earth's core, Geophys. J. R. Astron. Soc., 59, 5799.
  • Hsui, A. T. (1989), Possible consequences of a fluid dynamic origin for columnar joints in basalts, Arch. Sci. Genève, 42, 425435.
  • Huppert, H. E. (1990), The fluid dynamics of solidification, J. Fluid Mech., 212, 209240.
  • Huppert, H. E., and R. S. J. Sparks (1980), The fluid dynamics of a basaltic magma chamber replenished by influx of hot, dense ultrabasic magma, Contrib. Mineral. Petrol., 75, 279289.
  • Huppert, H. E., and R. S. J. Sparks (1984), Double-diffusive convection due to crystallization in magmas, Annu. Rev. Earth Planet. Sci., 12, 1137.
  • Huppert, H. E., and J. S. Turner (1978), On melting icebergs, Nature, 271, 4648.
  • Huppert, H. E., and J. S. Turner (1981), Double-diffusive convection, J. Fluid Mech., 106, 299329.
  • Huppert, H. E., and M. G. Worster (1985), Dynamic solidification of a binary melt, Nature, 314, 703707.
  • Huppert, H. E., R. S. J. Sparks, J. R. Wilson, M. A. Hallworth, and A. M. Leitch (1987), Laboratory experiments with aqueous solutions modeling magma chamber processes, II, Cooling and crystallization along inclined planes, in Origins of Igneous Layering, edited by I. Parsons, pp. 539539, D. Reidel, Norwell, Mass.
  • Irvine, T. N. (1980), Magmatic infiltration metasomatism, double-diffusive fractional crystallization, and adcumulus growth in the Muskox intrusion and other layered intrusions, in Physics of Magmatic Processes, edited by R. B. Hargraves, pp. 325383, Princeton Univ. Press, Princeton, N.J.
  • Jacobs, J.A. (1987), The Earth's Core, 2nd ed., 417 pp., Academic, San Diego, Calif.
  • Kantha, L. H. (1980), A note on the effect of viscosity on double diffusive processes, J. Geophys. Res., 85, 43984404.
  • Kantha, L. H. (1981), ‘Basalt fingers’—Origin of columnar joints? Geol. Mag., 118(3), 251264.
  • Krishnamurti, R. (1975), On the cellular cloud patterns, part 3, Applicability of the mathematical and laboratory models, J. Atmos. Sci., 32, 13731383.
  • Loper, D. E. (1978), The gravitationally powered dynamo, Geophys. J. R. Astron. Soc., 54, 389404.
  • Loper, D. E. (1991), The nature and consequences of thermal interactions twixt core and mantle, J. Geomagn. Geoelectr., 43, 7991.
  • Loper, D. E. (2000), A model of the dynamical structure of Earth's outer core, Phys. Earth Planet. Inter., 117, 179196.
  • Lundstrom, C. C. (2000), Rapid diffusive infiltration of sodium into partially molten peridotite, Nature, 403, 527530.
  • McBirney, A. R. (1980), Mixing and unmixing of magmas, J. Volcanol. Geotherm. Res., 7, 357371.
  • McBirney, A. R. (1985), Further considerations of double-diffusive stratification and layering in the Skaergaard Intrusion, J. Petrol., Part 4, 26, 9931001.
  • McBirney, A. R., and A. Nicolas (1997), The Skaergaard layered series, part II, Magmatic flow and dynamic layering, J. Petrol., 38(5), 569580.
  • McBirney, A. R., and R. M. Noyes (1979), Crystallization and layering of the Skaergaard intrusion, J. Petrol., 20, 487554.
  • Murase, T., and A. R. McBirney (1973), Properties of some common igneous rocks and their melts at high temperature, Geol. Soc. Am. Bull., 84, 35633592.
  • Prescott, P. J., and F. P. Incropera (1994), Convective transport phenomena and macrosegregation during solidification of a binary metal alloy, I, Numerical predictions, J. Heat Transfer, 116, 735741.
  • Riahi, N. (1984), On a nonlinear overstable convection roll in a rotating system, J. Aust. Math. Soc., Ser. B, 25, 406418.
  • Ryan, M. P., and C. G. Sammis (1978), Cyclic fracture mechanisms in cooling basalt, Geol. Soc. Am. Bull., 89, 12951308.
  • Schlüter, A., D. Lortz, and F. H. Busse (1965), On the stability of steady finite amplitude convection, J. Fluid Mech., 23, 129144.
  • Shirtcliffe, T. G. L. (1967), Thermaosolutal convection: Observation of an overstable mode, Nature, 213, 489490.
  • Song, X., and P. G. Richards (1996), Seismological evidence for the rotation of the Earth's inner core, Nature, 382, 221224.
  • Souriau, A. (1998), Is the rotation real? Science, 281, 5556.
  • Sparks, R. S. J., and H. E. Huppert (1987), Laboratory experiments with aqueous solutions modeling magma chamber processes, I, Discussion of their validity and geological application, in Origins of Igneous Layering, edited by I. Parsons, pp. 527538, D. Reidel, Norwell, Mass.
  • Stern, M. E. (1960), The salt fountain and thermohaline convection, Tellus, 12, 172175.
  • Stevenson, D. J. (1981), Models of the Earth's core, Science, 214, 611619.
  • Turner, J. S. (1980), A fluid dynamical model of differentiation and layering in magma chambers, Nature, 285, 213215.
  • Turner, J. S., and L. B. Gustafson (1978), The flow at hot saline solutions from vents in the sea floor—Some implications for exhalative massive sulfide and other ore deposits, Econ. Geol., 73, 10821100.
  • Veronis, G. (1965), On finite amplitude instability in thermohaline convection, J. Mar. Res., 23, 117.
  • Worster, M. G., H. E. Huppert, and R. S. J. Sparks (1993), The crystallization of lava lakes, J. Geophys. Res., 98(B9), 1589115901.