# Calving front stress boundary condition¶

## Notation¶

Variable

Meaning

$$h$$

ice top surface elevation

$$b$$

ice bottom surface elevation

$$H = h - b$$

ice thickness

$$g$$

acceleration due to gravity

$$\viscosity$$

vertically-averaged viscosity of ice

$$\n$$

normal vector

$$B(T)$$

ice hardness

$$D$$

strain rate tensor

$$d_{e}$$

effective strain rate

$$t$$

Cauchy stress tensor

$$t^{D}$$

deviatoric stress tensor; note $$\td{ij} = t_{ij} + p \delta_{ij}$$

## Calving front stress boundary condition¶

In the 3D case the calving front stress boundary condition ([56], equation (6.19)) reads

$\left.t\right|_{\text{cf}} \cdot \n = -\psw \n.$

Expanded in component form, and evaluating the left-hand side at the calving front and assuming that the calving front face is vertical ($$\nz = 0$$), this gives

$\begin{split}\begin{array}{rcrcl} (\td{xx} - p) \nx &+& \td{xy} \ny &=& -\psw \nx,\\ \td{xy} \nx &+& (\td{yy} - p) \ny &=& -\psw \ny,\\ \td{xz} \nx &+& \td{yz} \ny &=& 0. \end{array}\end{split}$

Because we seek boundary conditions for the SSA stress balance, in which the vertically-integrated forms of the stresses $$\td{xx},\td{xy},\td{yy}$$ are balanced separately from the shear stresses $$\td{xz},\td{yz}$$, the third of the above equations can be omitted from the remaining analysis.

Let $$\pice=\rhoi g (h-z)$$. In the hydrostatic approximation, $$t_{zz}=-\pice$$ ([56], equation (5.59)). Next, since $$\td{}$$ has trace zero,

\begin{align}\begin{aligned}p &= p - \td{xx} - \td{yy} - \td{zz}\\&= - t_{zz} - \td{xx} - \td{yy}\\&= \pice - \td{xx} - \td{yy}.\end{aligned}\end{align}

Thus

(50)$\begin{split}\begin{array}{rcrcl} (2\td{xx} + \td{yy}) \nx &+& \td{xy} \ny &=& (\pice - \psw) \nx,\\ \td{xy} \nx &+& (2\td{yy} + \td{xx}) \ny &=& (\pice - \psw) \ny.\\ \end{array}\end{split}$

Now, using the “viscosity form” of the flow law

\begin{align}\begin{aligned}\td{} &= 2\viscosity\, D,\\\viscosity &= \frac 12 B(T) d_{e}^{1/n-1}\end{aligned}\end{align}

and the fact that in the shallow shelf approximation $$u$$ and $$v$$ are depth-independent, we can define the membrane stress $$N$$ as the vertically-integrated deviatoric stress

$N_{ij} = \int_{b}^{h} t^{D}_{ij} dz = 2\, \bar \viscosity\, H\, D_{ij}.$

Here $$\bar \viscosity$$ is the vertically-averaged effective viscosity.

Integrating (50) with respect to $$z$$, we get

(51)$\begin{split}\begin{array}{rcrcl} (2N_{xx} + N_{yy}) \nx &+& N_{xy} \ny &=& \displaystyle \int_{b}^{h}(\pice - \psw) dz\, \nx,\\ N_{xy} \nx &+& (2N_{yy} + N_{xx}) \ny &=& \displaystyle \int_{b}^{h}(\pice - \psw) dz\, \ny. \end{array}\end{split}$

## Shallow shelf approximation¶

The shallow shelf approximation written in terms of $$N_{ij}$$ is

(52)$\begin{split}\begin{array}{rcrcl} \D \diff{}{x}(2N_{xx} + N_{yy}) &+& \D \diff{N_{xy}}{y} &=& \D \rho g H \diff{h}{x},\\ \D \diff{N_{xy}}{x} &+& \D \diff{}{y}(2N_{yy} + N_{xx}) &=& \D \rho g H \diff{h}{y}. \end{array}\end{split}$

## Implementing the calving front boundary condition¶

We use centered finite difference approximations in the discretization of the SSA (52). Consider the first equation:

(53)$\diff{}{x}(2N_{xx} + N_{yy}) + \D \diff{N_{xy}}{y} = \D \rho g H \diff{h}{x}.$

Let $$\tilde F$$ be an approximation of $$F$$ using a finite-difference scheme. Then the first SSA equation is approximated by

$\frac1{\dx}\left(\partI_{i+\frac12,j} - \partI_{i-\frac12,j}\right) + \frac1{\dy}\left(\partII_{i,j+\frac12} - \partII_{i,j-\frac12}\right) = \rho g H \frac{h_{i+\frac12,j} - h_{i-\frac12,j}}{\dx}.$

Now, assume that the cell boundary (face) at $$i+\frac12,j$$ is at the calving front. Then $$\n = (1,0)$$ and from (51) we have

(54)$2N_{xx} + N_{yy} = \int_{b}^{h}(\pice - \psw) dz.$

We call the right-hand side of (54) the “pressure difference term.”

In forming the matrix approximation of the SSA [29], [37], instead of assembling a matrix row corresponding to the generic equation (53) we use

$\frac1{\dx}\left(\left[\int_{b}^{h}(\pice - \psw) dz\right]_{i+\frac12,j} - \partI_{i-\frac12,j}\right) + \frac1{\dy}\left(\partII_{i,j+\frac12} - \partII_{i,j-\frac12}\right) = \rho g H \frac{h_{i+\frac12,j} - h_{i-\frac12,j}}{\dx}.$

Moving terms that do not depend on the velocity field to the right-hand side, we get

$\frac1{\dx}\left(- \partI_{i-\frac12,j}\right) + \frac1{\dy}\left(\partII_{i,j+\frac12} - \partII_{i,j-\frac12}\right) = \rho g H \frac{h_{i+\frac12,j} - h_{i-\frac12,j}}{\dx} + \left[\frac{\int_{b}^{h}(\psw - \pice) dz}{\dx}\right]_{i+\frac12,j}.$

The second equation and other cases ($$\n = (-1,0)$$, etc) are treated similarly.

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