# Grounding Line Migration

## Overview

Presently CISM uses the same grounding line retreat and advance mechanism used in Glimmer. This is based up the flotation criteria combined with a calving rate. Algorithmically

At a cell $i,j$ at the grounding line, if the continuity equation predicts that at time $t+1$ $H_{i,j}^{t+1} > -\frac{\rho_w}{\rho_i} z_b f$
,

then the cell becomes occupied with ice, thickness $H_{i,j}^{t+1}$, otherwise the cell has zero ice thickness. Here $f$ is the fraction of the ice that is assumed to calve away.

While this is a fairly typical treatment, we hope to do better.

### van der Veen's Method

Following Fundamentals of Glacier Dynamics (2002) page 396, from continuity, the evolution of thickness at the grounding line obeys $\frac{dH}{dt} = -\nabla H\mathbf{u} + M = -\mathbf{u} \cdot \nabla H - H \nabla \cdot \mathbf{u} +M.$

The term $H \nabla \cdot \mathbf{u}$ is our primary concern, as it is the stretching of the ice as it moves onto an ice shelf. It is simpler to consider the one dimensional problem of the ice spreading outwards, normal to the shelf. The one dimensional continuity equation is $\frac{dH}{dt} = -u \frac{dH}{dx} - H \frac{du}{dx} +M.$ $\frac{du}{dx}$ is the longitudinal stretching of the ice shelf, $\dot \epsilon_{xx}$. A one dimensional analytical expression of this can be derived from the constituative relation for ice $\dot \epsilon_{xx} = \theta A \sigma_{xx}^{'n},$

and by noting that hydrostatic equilibrium at the ice/ocean interface implies $\sigma'_{xx} = \frac{1}{2} \rho_i g \left(1-\frac{\rho_i}{\rho_w}\right) H.$

The balance of forces in an ice shelf is $\tau_d = \frac{\partial }{\partial x} H \sigma_{xx}' + \frac{H\tau_s}{W}$

where the lateral drag on the ice shelf is $\tau_s$ and the ice shelf's half width is $W$.

Substitution of previous expressions for $\sigma_{xx}$ and recalling that an ice shelf at floatation has driving stress $\tau_d = \rho_i g H \frac{dh}{dx} = \rho_i g H \frac{d}{dx} \left(1-\frac{\rho_i}{\rho_w}\right) H$ gives $\frac{\partial H \sigma_{xx}'}{\partial x} =\frac{1}{2} \rho_i g \left(1-\frac{\rho_i}{\rho_w}\right) \frac{\partial H^2}{\partial x} - \frac{H\tau_s}{W}$

After integration $\dot \epsilon_{xx} = \theta A \left( \frac{1}{2} \rho_i g \left(1-\frac{\rho_i}{\rho_w}\right) H - \sigma_b\right)^n$

With the 'back pressure' equal to $\sigma_b = \frac{1}{H} \int^L_x \frac{H\tau_s}{W} dx.$

This derivation neglects the back pressure that may arise from ice rises. As we will ultimately seek a parametrization of $\sigma_{b}$, rather than doing an actual integral, it is alright to lump that contribution into $\sigma_b$.

#### Estimating $\sigma_b$

With out being able to evaluate $\sigma_b$ by direct means, we hypothesis that it is a function of 1) geometry of the embayment, 2)temperature of the ocean and atmosphere, and 3) position of the calving front. Each is now treated in turn.

##### geometry of embayment

The geometry can be crudely measured in terms of the aspect ratio of the embayment. If the ratio is length in transverse divided by length in longitudinal flow direction, then larger ratios produce larger resistances. $\sigma_b \propto \frac{l_t}{l_l}$

##### temperatures of ocean and atmosphere

The temperature and position dependence would lead to a time delay for the temperature sensitivity that is asymmetric. The delay between cold temperatures and ice shelf advance is longer than the delay between warm temperatures and ice shelf retreat.