Stokes-continuity Boundary Conditions

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

Stokes Boundary Conditions: Top and Bottom Boundaries

Boundary conditions for the x-component of velocity are implemented along the top and bottom boundaries using the following equations:

eq:vx-bc-top-bottom

\[\begin{split} v_{x(1,j)_{v_x}} & = v_{x}^{top} + f_{v_x}^{top}v_{x(2,j)_{v_x}} \\ v_{x(N_{y,b}+1,j)_{v_x}} & = v_{x}^{bot} + f_{v_x}^{bot}v_{x(N_{y,b},j)_{v_x}} \\ \end{split}\]

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

Boundary conditions for the y-component of velocity are implemented along the top and bottom boundaries using the following equations:

eq:vy-bc-top-bottom

\[\begin{split} v_{y(1,j)_{v_y}} & = v_{y}^{top} + f_{v_y}^{top}v_{y(2,j)_{v_y}} \\ v_{y(N_{y,b},j)_{v_y}} & = v_{y}^{bot} + f_{v_y}^{bot}v_{y(N_{y,b}-1,j)_{v_y}} \\ \end{split}\]

where $j$ ranges from $1$ to $N_{x,b}+1$, $f_{v_y}^{top}$ and $f_{v_y}^{bot}$ are the boundary condition coefficients for the top boundary and bottom boundaries, respectively, and $v_{y}^{top}$ and $v_{y}^{bot}$ are the prescribed velocity components at boundary unknown nodes along the top and bottom boundaries, respectively. The coefficients $f_{v_y}^{top}$ and $f_{v_y}^{bot}$ control how the y-component of velocity at boundary unknown nodes is related to the internal velocity and can have a value of -1, 0 or 1 depending on the type of boundary condition.

Stokes Boundary Conditions: Left and Right Boundaries

Boundary conditions for the x-component of velocity are implemented along the left and right boundaries using the following equations:

eq:vx-bc-left-right

\[\begin{split} v_{x(i,1)_{v_x}} & = v_{x}^{left} + f_{v_x}^{left}v_{x(i,2)_{v_x}} \\ v_{x(i,N_{x,b})_{v_x}} & = v_{x}^{right} + f_{v_x}^{right}v_{x(i,N_{x,b}-1)_{v_x}} \\ \end{split}\]

where $i$ ranges from $1$ to $N_{y,b}+1$, $f_{v_x}^{left}$ and $f_{v_x}^{right}$ are the boundary condition coefficients for the left boundary and right boundaries, respectively, $v_{x}^{left}$ and $v_{x}^{right}$ are the prescribed velocity components at boundary unknown nodes along the left and right boundaries, respectively. The coefficients $f_{v_x}^{left}$ and $f_{v_x}^{right}$ control how the x-component of velocity at boundary unknown nodes is related to the internal velocity and can have a value of -1, 0 or 1 depending on the type of boundary condition.

Boundary conditions for the y-component of velocity are implemented along the left and right boundaries using the following equations:

eq:vy-bc-left-right

\[\begin{split} v_{y(i,1)_{v_y}} & = v_{y}^{left} + f_{v_y}^{left}v_{y(i,2)_{v_y}} \\ v_{y(i,N_{x,b}+1)_{v_y}} & = v_{y}^{right} + f_{v_y}^{right}v_{y(i,N_{x,b})_{v_y}} \\ \end{split}\]

where $i$ ranges from $1$ to $N_{y,b}$, $f_{v_y}^{left}$ and $f_{v_y}^{right}$ are the boundary condition coefficients for the left boundary and right boundaries, respectively, and $v_{y}^{left}$ and $v_{y}^{right}$ are the prescribed velocity components at boundary unknown nodes along the left and right boundaries. The coefficients $f_{v_y}^{left}$ and $f_{v_y}^{right}$ control how the y-component of velocity at boundary unknown nodes is related to the internal velocity and can have a value of -1, 0 or 1 depending on the type of boundary condition.

No-slip Boundary Conditions

For no-slip boundary conditions along all boundaries, boundary coefficients $f_{v_x}^{top}$, $f_{v_x}^{bot}$, $f_{v_y}^{left}$ and $f_{v_y}^{right}$ are set equal to -1, and boundary coefficients $f_{v_y}^{top}$, $f_{v_y}^{bot}$, $f_{v_x}^{left}$ and $f_{v_x}^{right}$ are set equal to 0. Additionally, prescribed velocities $v_{x}^{top}$, $v_{x}^{bot}$, $v_{y}^{top}$, $v_{y}^{bot}$, $v_{x}^{left}$, $v_{x}^{right}$, $v_{y}^{left}$ and $v_{y}^{right}$ are set to zero. These conditions ensure that velocity at ghost nodes has the opposite sign of the internal velocity along the boundary, making the velocity at the model boundary zero as described in the following equations:

eq:no-slip-bc

\[\begin{split} v_{x(1,j)_{v_x}} & = -v_{x(2,j)_{v_x}} \\ v_{y(1,j)_{v_y}} & = 0 \\ v_{x(N_{y,b}+1,j)_{v_x}} & = -v_{x(N_{y,b},j)_{v_x}} \\ v_{y(N_{y,b},j)_{v_y}} & = 0 \\ v_{x(i,1)_{v_x}} & = 0 \\ v_{y(i,1)_{v_y}} & = -v_{y(i,2)_{v_y}} \\ v_{x(i,N_{x,b})_{v_x}} & = 0 \\ v_{y(i,N_{x,b}+1)_{v_y}} & = -v_{y(i,N_{x,b})_{v_y}} \\ \end{split}\]

Free-slip Boundary Conditions with Inflow/Outflow

For free slip with inflow/outflow boundary conditions, boundary coefficients $f_{v_x}^{top}$, $f_{v_x}^{bot}$, $f_{v_y}^{left}$ and $f_{v_y}^{right}$ are set equal to 1, and boundary coefficients $f_{v_y}^{top}$, $f_{v_y}^{bot}$, $f_{v_x}^{left}$ and $f_{v_x}^{right}$ are set equal to 0. Additionally, prescribed velocities $v_{x}^{top}$, $v_{x}^{bot}$, $v_{y}^{left}$ and $v_{y}^{right}$ are set to zero, and prescribed velocities $v_{y}^{top}$, $v_{y}^{bot}$, $v_{x}^{left}$ and $v_{x}^{right}$ are set to the inflow/outflow velocity. These conditions ensure that velocity parallel to the boundary at ghost nodes is equal to the internal velocity, and velocity perpendicular to the boundary at boundary nodes is equal to the inflow/outflow velocity as described in the following equations:

eq:free-slip-bc-inflow-outflow

\[\begin{split} v_{x(1,j)_{v_x}} & = v_{x(2,j)_{v_x}} \\ v_{y(1,j)_{v_y}} & = v_{y}^{top} \\ v_{x(N_{y,b}+1,j)_{v_x}} & = v_{x(N_{y,b},j)_{v_x}} \\ v_{y(N_{y,b},j)_{v_y}} & = v_{y}^{bot} \\ v_{x(i,1)_{v_x}} & = v_{x}^{left} \\ v_{y(i,1)_{v_y}} & = v_{y(i,2)_{v_y}} \\ v_{x(i,N_{x,b})_{v_x}} & = v_{x}^{right} \\ v_{y(i,N_{x,b}+1)_{v_y}} & = v_{y(i,N_{x,b})_{v_y}} \\ \end{split}\]

Boundary Conditions for Extensional Models with Fixed Boundaries

Extensional models with fixed boundaries use free-slip boundary conditions with inflow/outflow along all boundaries as described in Eq.. Prescribed outflow velocity along the side boundaries is set to $0.5v_{ext}$ where $v_{ext}$ is the full extension velocity. Prescribed inflow velocity along the top and bottom boundaries are defined so that the net inflow/outflow volume is zero over the entire model domain and the net inflow/outflow volume is zero within the sticky layer. The prescribed inflow velocity along the top boundary $v_y^{top}$ is calculated as:

eq:inflow-velocity-top

\[v_y^{top} = \frac{v_{ext}\left(H_{sticky,l} + H_{sticky,r}\right)}{x_{size}}\]

where $H_{sticky,l}$ and $H_{sticky,r}$ are the thickness of the sticky layer on the left and right sides of the model domain, and $x_{size}$ is the width of the model domain. The prescribed inflow velocity along the bottom boundary $v_y^{bot}$ is calculated as:

eq:inflow-velocity-bottom

\[v_{y}^{bot} = v_y^{top} -2v_{ext}\frac{y_{size}}{x_{size}}\]

where $y_{size}$ is the height of the model domain. Inflow velocities $v_{y}^{top}$ and $v_{y}^{bot}$ are recalculated at each time step to account for changes in the sticky layer thickness due to crustal thickness changes and isostatic adjustment.