# Calving front stress boundary condition¶

Contents

## 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

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

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,

Thus

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

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

Here \(\bar \viscosity\) is the vertically-averaged effective viscosity.

Integrating (49) with respect to \(z\), we get

## Shallow shelf approximation¶

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

## Implementing the calving front boundary condition¶

We use centered finite difference approximations in the discretization of the SSA (51). Consider the first equation:

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

Now, assume that the cell boundary (face) at \(i+\frac12,j\) is at the calving front. Then \(\n = (1,0)\) and from (50) we have

We call the right-hand side of (53) 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 (52) we use

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

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

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