Thermal Convection Models in UnderWorld

What Kind of Thermal Convection Systems can be Modelled using UnderWorld?

The following thermal convection systems can be modelled using UnderWorld:

Temperature-dependent only

• Arrhenius
• Frank Kamenetskii

Temperature and stress-dependent

• one rheology - Non-Newtonian
• multiple rheologies - Multi-Component

Anisotropic rheology

• Anisotropic

Plastic rheology

• MobileLid

Radiogenic heating

• InternalHeating

Depth dependent viscosity

• DepthDependentViscosity

Multi thermal diffusivity

• MultiThermalDiffusivity

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General Equations

We solve the equations of thermal convection for a creeping fluid with the Bousinnesq approximation and infinite Prandtl number,

where x_i are the spatial coordinates, u_i is the velocity, T is the temperature, is the thermal expansivity, is the fluid density, g is the gravitational acceleration, is the unit vector in the direction of gravity and is the coeffecient of thermal diffusion. Gravity acts in the y direction.

The deviatoric stress tensor, is given by,

where is the viscosity. This is a constititive relationship. Please refer to description of each case for which viscosity laws are used (e.g. Arrhenius, FrankKamenetskii, NonNewtonian, Depth Dependent Viscosity).

The surface temperature is T₀ = 0. and the bottom temperature is T_l = . All boundaries are free-slip (rather than no-slip).

Choosing the following scales

results in a nondimensional form of the equations, given as,

With constant viscosity, convection is fully described by the non-dimensional Rayleigh number,

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Base Application (BaseApp)

Most thermal convection template InputFiles include the ThermalConvection.xml Base Application (view this BaseApp in the latest "Bleeding Edge" repository or the Underworld-1.1.0 repository), except for Anisotropic and MultiThermalDiffusivity, which use the ThermalConvectionPIC.xml Base Application (view this BaseApp in the latest "Bleeding Edge" repository or the Underworld-1.1.0 repository).

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Boundary Conditions

All thermal convection template InputFiles include the free-slip boundary condition, and all except MultiThermalDiffusivity include the temperature boundary condition.

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Initial Conditions

All thermal convection template InputFiles except MultiThermalDiffusivity include a sinusoidal perturbation initial condition to initiate convection. See MultiThermalDiffusivity for alternative initial conditions used in that model.

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-- WendyMason - 29 Jun 2007

-- RobertTurnbull - 12 Jan 2006

-- JustinFreeman - 19 Aug 2003 (equations)
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