Do you realize that tectonics does not host a single physical principle that we can find in a physics book? {Not one} There are no mathematical formulations!
The earth's mantle in the numerical model is treated as an irrotational, infinite Prandtl number, anelastic Newtonian fluid within a spherical shell with isothermal, undeformable, traction-free boundaries. Under these conditions the following equations describe the local fluid behavior:
Here p denotes pressure, r density, g gravitational acceleration, t deviatoric stress, u fluid velocity, T absolute temperature, g the Grueneisen parameter, k thermal conductivity, H volume heat production rate, cv specific heat at constant volume, m dynamic shear viscosity, K the isothermal bulk modulus, and a the volume coefficient of thermal expansion. The quantities pr, rr, and Tr are, respectively, the radially varying pressure, density, and temperature of the reference state used for the mantle. I is the identity tensor. The superscript T in (4) denotes the tensor transpose. Equation (1) expresses the conservation of momentum in the infinite Prandtl number limit. In this limit, the deformational term is so large that the inertial terms (as well as the rotational terms) may be completely ignored. The resulting equation (1) then represents the balance among forces arising from pressure gradients, buoyancy, and deformation. Equation (2) expresses the conservation of mass under the anelastic approximation. The anelastic approximation ignores the partial derivative of density with respect to time in the dynamics and thereby eliminates fast local density oscillations. It allows the computational time step to be dictated by the much slower deformational dynamics. Equation (3) expresses the conservation of energy in terms of absolute temperature. It includes effects of transport of heat by the flowing material, compressional heating and expansion cooling, thermal conduction, shear or deformational heating, and local volume (e.g., radiogenic) heating.
The expression for the deviatoric stress given by equation (4) assumes a viscosity m that is dependent on the radial temperature and pressure distribution but independent of the strain rate. The stress therefore is linear with respect to velocity and represents the customary description for the deformation of a Newtonian fluid. This rheological law applies to the type of deformation in solids known as diffusion creep that is believed to occur in the mantle under conditions of extremely small strain rate. Equation (5) represents density variations as linearly proportional to pressure and temperature variations relative to a reference state. The compressible reference state is chosen to match observational data for the earth to a high degree of precision. It includes the density jumps associated with mineralogical phase changes. In the numerical model the set of equations (1)-(5) is solved for each grid point in the computational domain during each time step.
The Morse equation of state [2], derived from an atomic potential model of a crystalline lattice, is employed for the density dependence and given as follows:
p1(r)=[3Ko/(Ko' - 1)] (r/ro)2/3 E (E - 1)
E=exp{ (Ko' - 1) [1 - (r/ro)-1/3] }
Here ro is the uncompressed zero-temperature density, Ko is the uncompressed zero-temperature isothermal bulk modulus, and Ko' is the derivative of Ko with respect to pressure. These three material parameters specify the pressure-density relationship for a given mineral assemblage. By choosing appropriate values for the upper mantle, the transition zone between 410 and 660 km depth, and the lower mantle, one can match the density profile given by the seismic models quite closely.
Depth variation in the dynamic shear viscosity is modeled using a temperature and pressure dependent relationship of the form
m=mo exp[ -(E* + prV*)/RTr]
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I suggest you do your homework :x
