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
$\nu$	vertically-averaged viscosity of ice
$\mathit{n}$	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 $t_{ij}^D=t_{ij}+p{\delta }_{ij}$

Calving front stress boundary condition

In the 3D case the calving front stress boundary condition ([55], equation (6.19)) reads

$${t|}_{\text{cf}}\cdot \mathit{n}=-p_{\text{water}}\mathit{n}.$$

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

$$\begin{array}{r}\begin{array}{rcrcl}(t_{xx}^D-p){\mathit{n}}_x& +& t_{xy}^D{\mathit{n}}_y& =& -p_{\text{water}}{\mathit{n}}_x,\\ t_{xy}^D{\mathit{n}}_x& +& (t_{yy}^D-p){\mathit{n}}_y& =& -p_{\text{water}}{\mathit{n}}_y,\\ t_{xz}^D{\mathit{n}}_x& +& t_{yz}^D{\mathit{n}}_y& =& 0.\end{array}\end{array}$$

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

Let $p_{\text{ice}}={\rho }_{\text{ice}}g(h-z)$. In the hydrostatic approximation, $t_{zz}=-p_{\text{ice}}$ ([55], equation (5.59)). Next, since $t_{}^D$ has trace zero,

$$\begin{array}{r}\begin{array}{rl}p& =p-t_{xx}^D-t_{yy}^D-t_{zz}^D\\ & =-t_{zz}-t_{xx}^D-t_{yy}^D\\ & =p_{\text{ice}}-t_{xx}^D-t_{yy}^D.\end{array}\end{array}$$

Thus

(60)$$\begin{array}{r}\begin{array}{rcrcl}(2t_{xx}^D+t_{yy}^D){\mathit{n}}_x& +& t_{xy}^D{\mathit{n}}_y& =& (p_{\text{ice}}-p_{\text{water}}){\mathit{n}}_x,\\ t_{xy}^D{\mathit{n}}_x& +& (2t_{yy}^D+t_{xx}^D){\mathit{n}}_y& =& (p_{\text{ice}}-p_{\text{water}}){\mathit{n}}_y.\end{array}\end{array}$$

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

$$\begin{array}{r}\begin{array}{rl}{t}_{}^D& =2\nu D,\\ \nu & =\frac{1}{2}B(T)d_e^{1/n-1}\end{array}\end{array}$$

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

$$N_{ij}={\int }_b^h{t}_{ij}^{D}dz=2\overline{\nu }H{D}_{ij}.$$

Here $\overline{\nu }$ is the vertically-averaged effective viscosity.

Integrating (60) with respect to $z$, we get

(61)$$\begin{array}{r}\begin{array}{rcrcl}(2N_{xx}+N_{yy}){\mathit{n}}_x& +& N_{xy}{\mathit{n}}_y& =& {\displaystyle {\int }_b^h(p_{\text{ice}}-p_{\text{water}})dz{\mathit{n}}_x,}\\ N_{xy}{\mathit{n}}_x& +& (2N_{yy}+N_{xx}){\mathit{n}}_y& =& {\displaystyle {\int }_b^h(p_{\text{ice}}-p_{\text{water}})dz{\mathit{n}}_y.}\end{array}\end{array}$$

Shallow shelf approximation

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

(62)$$\begin{array}{r}\begin{array}{rcrcl}{\displaystyle \frac{\partial }{\partial x}(2N_{xx}+N_{yy})}& +& {\displaystyle \frac{\partial N_{xy}}{\partial y}}& =& {\displaystyle \rho gH\frac{\partial h}{\partial x},}\\ {\displaystyle \frac{\partial N_{xy}}{\partial x}}& +& {\displaystyle \frac{\partial }{\partial y}(2N_{yy}+N_{xx})}& =& {\displaystyle \rho gH\frac{\partial h}{\partial y}.}\end{array}\end{array}$$

Implementing the calving front boundary condition

We use centered finite difference approximations in the discretization of the SSA (62). Consider the first equation:

(63)$$\frac{\partial }{\partial x}(2N_{xx}+N_{yy})+{\displaystyle \frac{\partial N_{xy}}{\partial y}={\displaystyle \rho gH\frac{\partial h}{\partial x}.}}$$

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

$$\frac{1}{\mathrm{\Delta }x}((2{\tilde{N}}_{xx}+{\tilde{N}}_{yy}{)}_{i+\frac{1}{2},j}-(2{\tilde{N}}_{xx}+{\tilde{N}}_{yy}{)}_{i-\frac{1}{2},j})+\frac{1}{\mathrm{\Delta }y}(({\tilde{N}}_{xy}{)}_{i,j+\frac{1}{2}}-({\tilde{N}}_{xy}{)}_{i,j-\frac{1}{2}})=\rho gH\frac{h_{i+\frac{1}{2},j}-h_{i-\frac{1}{2},j}}{\mathrm{\Delta }x}.$$

Now, assume that the cell boundary (face) at $i+\frac{1}{2},j$ is at the calving front. Then $\mathit{n}=(1,0)$ and from (61) we have

(64)$$2N_{xx}+N_{yy}={\int }_b^h(p_{\text{ice}}-p_{\text{water}})dz.$$

We call the right-hand side of (64) the “pressure difference term.”

In forming the matrix approximation of the SSA [10], [17], instead of assembling a matrix row corresponding to the generic equation (63) we use

$$\frac{1}{\mathrm{\Delta }x}({[{\int }_b^h(p_{\text{ice}}-p_{\text{water}})dz]}_{i+\frac{1}{2},j}-(2{\tilde{N}}_{xx}+{\tilde{N}}_{yy}{)}_{i-\frac{1}{2},j})+\frac{1}{\mathrm{\Delta }y}(({\tilde{N}}_{xy}{)}_{i,j+\frac{1}{2}}-({\tilde{N}}_{xy}{)}_{i,j-\frac{1}{2}})=\rho gH\frac{h_{i+\frac{1}{2},j}-h_{i-\frac{1}{2},j}}{\mathrm{\Delta }x}.$$

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

$$\frac{1}{\mathrm{\Delta }x}(-(2{\tilde{N}}_{xx}+{\tilde{N}}_{yy}{)}_{i-\frac{1}{2},j})+\frac{1}{\mathrm{\Delta }y}(({\tilde{N}}_{xy}{)}_{i,j+\frac{1}{2}}-({\tilde{N}}_{xy}{)}_{i,j-\frac{1}{2}})=\rho gH\frac{h_{i+\frac{1}{2},j}-h_{i-\frac{1}{2},j}}{\mathrm{\Delta }x}+{\left[\frac{{\int }_b^h(p_{\text{water}}-p_{\text{ice}})dz}{\mathrm{\Delta }x}\right]}_{i+\frac{1}{2},j}.$$

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

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