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 ([55], 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, \(t_{zz}=-\pice\) ([55], 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

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

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 (59) with respect to \(z\), we get

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

Shallow shelf approximation

The shallow shelf approximation written in terms of \(N_{ij}\) is

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

Implementing the calving front boundary condition

We use centered finite difference approximations in the discretization of the SSA (61). Consider the first equation:

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

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 (60) we have

(63)\[2N_{xx} + N_{yy} = \int_{b}^{h}(\pice - \psw) dz.\]

We call the right-hand side of (63) 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 (62) 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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