Abstract: Untersuchung der Konvektion im Erdinneren und dafür wichtiger Materialparameter unter hohem Druck
U. Walzer. Untersuchung der Konvektion im Erdinneren und dafür wichtiger Materialparameter unter hohem Druck. Veröff. Zentralinst. Physik d. Erde, 75:1-259, 1981.
Untersuchung der Konvektion im Erdinneren und dafür wichtiger Materialparameter unter hohem Druck
U. Walzer
Abstract
The present paper deals primarily with convection in the Earth's mantle. But it includes also considerations about the possibility of thermal convection in the outer core of the Earth and about the pressure dependence of significant material parameters, with a close relationship between the subjects considered becoming apparent.
In Section 1, we give a substantiation why convection is the most important motor of the geotectonic processes. Other geotectonic theories are considered, too. In Section 2, we deduce the system of equations of mantle convection. The content of Section 3 will be discussed after that of 4. Starting from geochemical results, the author assumes in Section 4 that the lower mantle has a lower amount of radioactive heat produced per unit volume per unit time than the upper mantle. Consequently, the thermoconvective driving energy in the lower mantle is too weak to p e r m a n e n t l y maintain convection there. At the same time, however, the lattice and radiative heat conductivies are low there, so that the heat generated is dissipated only to a minor extent. Therefore, temperature increases. As a result, due to the well-known temperature dependence of the effective viscosity, the Rayleigh number increases until the critical Rayleigh number is surpassed and convection also starts in the lower mantle. Through the currents, heat is dissipated, causing an increase in the permanently existing upper-mantle convection, in magmatism and orogenesis as well as Earth-wide transgressions. As a result of the heat dissipation, the Rayleigh number again drops beyond the critical value, i, e., due to the low internal heat source density lower-mantle convection dies down until the next convection interval is started after a long time span of heat accumulation. The differential equations of the problem have been reduced to a system of equations with a Hammerstein integral equation and solved numerically. Four convection episodes resulted which agree, in respect of time, with the four highest maxima of Gastil's curve of magmatic activity: These four overturns are found 2820, 3633, 4128 and 4496 m. y. after the accretion of the Earth, an age of the Earth of 4600 m. y. being assumed. A comparison of empirical curves showed that these times also correspond to earthwide transgressions and that the latter are found to precisely lie in the periods in the Phanerozoic in which the geomagnetic dipole field only rarely reversed polarity. The latter most probably has to do with the fact that the lower mantle determines part of the boundary conditions of the hydromagnetic convection in the Earth's outer core. In Section 3, it is shown that there cannot be convection intervals without internal heating.
This leads us to Section 5 where the conditions have been considered under which the geodynamo can be driven in the outer core by thermal convection. It is found that the core paradox is completely conditional on the assumed dependence of the melting point, Tm, on pressure or on the atomic volume, v. This dependence is inferred in Section 6 in two different ways from different premises. The first deduction is based on an interatomic pair potential and on a dislocation model, the second deduction is based on the equation of state by Ullmann and Pan'kov, on the Vashchenko-Zubarev relation and on Lindemann's law. The equation of state can also be deduced from the lattice theory alone. From this, the exponents of the Tm(v)-equations are independently determined. They are found to be in agreement. Reversing the sequence of conclusions, Lindemann's law can be inferred on the basis of the other conditions mentioned. However, this necessarily applies only to high coordination numbers or high pressures. The relations of the melting temperature to macroscopic physical quantities known as a function of depth for the outer core have been examined. A new dimensionless quantity, Q0, not containing the Grüneisen parameter is found to be suited for serving in future studies as an auxiliary quantity for the determination of the melting temperature in the Earth's outer core. The pressure dependence of even more general dimensionless quantities, Qn, has been determined analytically and, for chemical elements, numerically, too. Various interesting dimensionless quantities have been shown in the Periodic Table and compared.
To be able to study the question of the pressure dependence of melting temperature and, thus, the problem of the possibility of thermal convection in the outer core in an even more realistic manner in the future, a new equation of state for metals has been inferred in Section 7. The starting point of the deduction is a realistic expression for Gibbs' free energy, where not only the lattice terms, but also the Fermi energy, the exchange and correlation energies of the electrons are taken into account. This expression is extended by means of three free parameters. The author is successful in transforming the new equation of state so that exclusively macrophysical material parameters occur in it. Consequently, the equation becomes applicable both to experimental high-pressure physics and to geophysics, however, it has the advantage that the bonding structure has been better taken into account in it than in other equations of state. The new equation, together with two other equations of state that have already proven well, has been tried out using isothermal and Hugoniot data of 40 substances. Not only for metallic elements but also for halides and some geophysically relevant oxides, we obtained a good agreement with the measured data which is better for the most substances than that of the other two equations of state.
In Section 8, the hydrodynamic stability problem of a horizontal, plane, viscous fluid layer with internal heat sources is solved both for the case of free and for the case of rigid boundaries. The new aspect in the present study is that the following quantities and their temperature dependencies are taken into account in the convection problem in a manner adapted to the conditions of the Earth's mantle: Lattice and radiative heat conductivity, specific heat and heat expansion coefficient. While only insignificant neglects were made, it was possible to determine the analytical solutions, all pertinent constants and the secular equation.
Whereas the previous sections were primarily devoted to an examination of the influence of the temperature and pressure dependence of the material parameters on convection, in particular, on the time behaviour of convection, Section 9 concerns the question of the geometry of the currents. Starting from a system of structurally simple postulates, the kinematics of mantle convection is inferred. This theory uses no prerequisites with respect to the constitutive equations valid in the mantle and the energy sources of convection. The currents are assumed to be rolls. From the possible current types, a theoretical topography is inferred which is in quantitative agreement with the topography observed. The distribution of the seismic discontinuities, too, indicates the correctness of the postulates.
In Section 10, the following hydrodynamic stability problem is solved: A horizontal, plane layer consisting of a Newtonian fluid is maintained at the lower and upper boundary planes at constant temperatures of T0 and T1, respectively, with T0 > T1. In addition, a heat-producing plane that is parallel to the boundary planes exists inside the fluid. According to our assumption, this plane is not displaced at the onset of current, but it also does not mechanically obstruct the current. In these conditions, convection rolls develop under and above the heat-producing plane, which are tangent to one another at this plane. The theory developed here shows an analogy to the kinematic theory of Section 9.
In Sections 11 and 12, the hydrodynamic stability problem of a horizontal layer has been computed as a model for the Bénard convection in the partially (1 to 2 %) molten asthenosphere. In Section 11, the layer consists of an incompressible micropolar fluid, in Section 12, of a compressible micropolar fluid with microstretch. The field equations for the velocity vector, microrotation vector, microstretch, microinertia, density, temperature and pressure are a system of eleven partial differential equations for the determination of eleven scalar functions. The author succeeds in decoupling the system and reducing the problem to a common differential equation. The analytical solution can be given for the special case of a micropolar Oberbeck-Boussinesq fluid.
In Section 13, the stability problem of a non-Newtonian fluid has been considered, with a constitutive equation corresponding to the solid creep in olivine. In Sections 1 and 14, the differentiation of the Earth's core that may have played a role at the beginning of the evolution of the Earth is considered. Here, the question is considered in which way the potential energy is transformed into heat and influences the convection due to gravitational differentiation. Therefore, in Section 14, the problem of the Rayleigh-Taylor instability with viscous dissipation is solved.
Key words: mantle convection, Earth, mantle, convection, tectonics, viscosity, Rayleigh number, transgression, outer core, core paradox, equation of state, geophysics, hydrodynamic stability, micropolar fluid, melting temperature, Earth's mantle, continent, spherical harmonics