GeoModBox.jl

Geodynamic Modelling ToolBox is a julia package mainly used for teaching purposes. The package provides different finite differences, fully staggered, discretization schemes for the governing equations to describes a geodynamic problem. The geoverning equations are the conservation equations of energy, mass, momentum, and compositon.

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Staggered Finite Difference

• brief general description of a staggered finite difference scheme
• Energy equation
• Momentum equation

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Energy equation

  The conservation of energy is a fundamental principle in physics and defines that the loss and generation of energy needs to be equal. In terms of a geodynamical problem, energy can be described by temperature, which is transported mainly through conductive and convective processes, such that a general energy equation is defined as followed (assuming only radioactive heat sources):

$$ (\frac{\partial E}{\partial t} + \overrightarrow{v} \cdot \nabla E) + \frac{\partial q_{i}}{\partial x_{i}} = \rho H, \tag{} $$

where the energy is described as $E=c_{p} \rho T$, and c_p is the specific heat capacity [J/kg/K], ρ is a reference density [kg/m³], T is the temperature [K], t is the time [s], $\overrightarrow{v}$ is the velocity vector [m/s], q_i is the heat flux in direction of i [W/m²], ∂/∂xi is a directional derivative in direction of i, and H the heat production rate per mass [W/kg]. The repeated index means a summation of derivatives. This conservation law contains the variation of the heat flux in a certain direction, where the heat flux is defined by the Fourier’s law as followed:

$$ \overrightarrow{q} = - k \nabla T, \tag{} $$

where k is the thermal conductivity [W/m/K]. The heat flux is the amount of heat that passes through a unit surface area, per unit time and is positive in the direction of decreasing temperature, that is in the case when the temperature gradient is negative. The temperature conservation equation in an Eulerian form can then be written as:

$$ \rho c_p (\frac{\partial T}{\partial t} + \overrightarrow{v} \cdot \nabla T) = -\frac{\partial q_i}{\partial x_i} + \rho H. \tag{} $$

• General formulation of the energy equation (advection + diffusion term + radiogenic heating)
• Operator splitting
• Solution of the conductive part

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Heat Diffusion

• Conducitve part of the energy equation + radiogenic heating + ?
• Constant thermal parameters
• Variable thermal paramters

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Numerical Schemes

All numerical schemes methods can be used in the thermal convection code and the Blankenbach Benchmark and are generally available to chose in the code. A more detailed analysis on the accuracy of each discretization scheme and the effect of the grid resolution is given in the Gaussian Diffusion Benchmark.

Explicit, FTCS

Implicit, FTCS

Cranck-Nicolson Approach (CNA)

Alternating Direction Implicit (ADI)

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Thermal Boundary Conditions

1. Dirichlet
2. Neumann

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Steady State Solution

Poisson solution, constant k

Poisson solution, variable k

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Advection Equation

Discretization Schemes

Upwind

Staggered Leap Frog (SLF)

Semi-Lagrangian

Passive Tracers

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Stokes Equation

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Benchmarks

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Examples

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