Heat Conduction Equation Boundary Conditions

Earthbox uses the approach of Gerya (2010) for specifying boundary conditions for the heat-conduction equations on a basic grid whereby simple coefficients are used that control type of boundary condition.

Heat Boundary Conditions: Top and Bottom Boundaries

Boundary conditions for the heat conduction equation are implemented along the top and bottom boundaries using the following equations:

eq:heat-bc-top-bottom

\[\begin{split} T_{(1,j)_b} & = T_{top} + f_T^{top}T_{(2,j)_b} \\ T_{(N_{y,b},j)_b} & = T_{bot} + f_T^{bot}T_{(N_{y,b}-1,j)_b} \\ \end{split}\]

where $j$ ranges from $1$ to $N_{x,b}$, $f_T^{top}$ and $f_T^{bot}$ are the boundary condition coefficients for the top boundary and bottom boundaries, respectively, and $T_{top}$ and $T_{bot}$ are the prescribed temperature or gradient values at nodes along the top and bottom boundaries, respectively. The coefficients $f_T^{top}$ and $f_T^{bot}$ control how the temperature at boundary nodes is related to the internal temperature and can have a value of -1, 0 or 1 depending on the type of boundary condition.

Heat Boundary Conditions: Left and Right Boundaries

Boundary conditions for the heat conduction equation are implemented along the left and right boundaries using the following equations:

eq:heat-bc-left-right

\[\begin{split} T_{(i,1)_b} & = T_{left} + f_T^{left}T_{(i,2)_b} \\ T_{(i,N_{x,b})_b} & = T_{right} + f_T^{right}T_{(i,N_{x,b}-1)_b} \\ \end{split}\]

where $i$ ranges from $1$ to $N_{y,b}$, $f_T^{left}$ and $f_T^{right}$ are the boundary condition coefficients for the left boundary and right boundaries, respectively, and $T_{left}$ and $T_{right}$ are the prescribed temperature or gradient values at nodes along the left and right boundaries, respectively. The coefficients $f_T^{left}$ and $f_T^{right}$ control how the temperature at boundary nodes is related to the internal temperature and can have a value of 0 or 1 depending on the type of boundary condition.

Boundary nodes have a large matrix coefficient of 1 for the unknown temperature $T$ and a right hand side value equal to the prescribed temperature value $T_{top}$, $T_{bot}$, $T_{left}$ or $T_{right}$ for the top, bottom, left and right boundaries, respectively. The internal nodes used in equations Eq. and Eq. are associated with unknowns that have coefficients equal to $-f_T^{top}$, $-f_T^{bot}$, $-f_T^{left}$ and $-f_T^{right}$ for the top, bottom, left and right boundaries, respectively, after rearranging unknowns to the left hand side of equations Eq. and Eq..

Constant Temperature Boundary Conditions

Constant temperature boundary conditions are implemented by setting $f_T^{top}$, $f_T^{bot}$, $f_T^{left}$ and $f_T^{right}$ to 0 and setting $T_{top}$, $T_{bot}$, $T_{left}$ and $T_{right}$ to the prescribed temperature values at the top, bottom, left and right boundaries, respectively. Constant gradient boundary conditions are implemented by setting $f_T^{top}$, $f_T^{bot}$, $f_T^{left}$ and $f_T^{right}$ to 1 and setting $T_{top}$, $T_{bot}$, $T_{left}$ and $T_{right}$ to the prescribed temperature gradient values at the top, bottom, left and right boundaries, respectively.

Zero Flux Boundary Conditions

Zero heat flux boundary conditions are implemented by setting $f_T^{top}$, $f_T^{bot}$, $f_T^{left}$ and $f_T^{right}$ to 1 and setting $T_{top}$, $T_{bot}$, $T_{left}$ and $T_{right}$ to zero at the top, bottom, left and right boundaries, respectively.

Heat Boundary Conditions for Extensional Models

For extensional models constant temperature boundary conditions are used along the top and bottom boundaries and zero heat flux boundary conditions are used along the left and right boundaries as described in the following equations:

eq:heat-bc-extension-models

\[\begin{split} T_{(1,j)_b} & = T_{top} \\ T_{(N_{y,b},j)_b} & = T_{bot} \\ T_{(i,1)_b} & = T_{(i,2)_b} \\ T_{(i,N_{x,b})_b} & = T_{(i,N_{x,b}-1)_b} \text{.} \\ \end{split}\]