Damage Evolution

Physical basis

Damage is a state variable introduced to account for the influence of fractures on ice flow, while maintaining a continuum representation of the ice domain. For purely viscous ice flow modeling, damage is linked to flow enhancement—specifically the increase in strain rate—due to a fracture or a multitude of fractures in the ice.

Inferring damage from remote sensing data

Remote sensing data can be used to calculate damage from the static stress balance in the ice. At present, this is only implemented in two dimensions for the SSA approximations to ice flow. Damage can be inferred in one of two ways:

Inverting for damage directly
Inverting for ice rigidity $Equation 1$ and then post-processing to determine damage (and optionally backstress)

Make sure that you are using the matdamageice class for md.materials. You can do that conversion using:

md.materials = matdamageice(md.materials);

Inverting for damage directly

For the SSA equations, the damage-dependent ice viscosity ( $Equation 2$ ) is:

$Equation 3$

where:

$Equation 4$ is damage
$Equation 5$ is the ice rigidity
$Equation 6$ is the effective strain rate
$Equation 7$ is the flow law exponent

Damage can be calculated using an inverse control method in the same manner as an inversion for the ice rigidity $Equation 8$ . Simply specify the following field in md.inversion:

md.inversion.control_parameters = {'DamageDbar'} (MATLAB)
md.inversion.control_parameters = ['DamageDbar'] (Python)

The remainder of the inversion procedure is described on the on the ‘Advanced Features’ → ‘Inversions’ page. This was the procedure followed by [Borstad2012] in determining the damage for the Larsen B ice shelf prior to its collapse (see the ‘Publications’ page for a link to the paper).

Post-processing to determine damage

Damage can also be calculated from the results of an inverse method solution for ice rigidity $Equation 9$ . This procedure uses the analytical solution for the strain rate of a damaged ice shelf, derived by [Borstad2013]:

$Equation 10$

where:

$Equation 11$ is the longitudinal strain rate
$Equation 12$ accounts for the lateral and shear strain rate terms
$Equation 14$ and $Equation 13$ are the densities of ice and seawater, respectively
$Equation 15$ is gravitational acceleration
$Equation 16$ is the ice thickness
$Equation 17$ is the backstress resisting the flow
$Equation 18$ is the damage
$Equation 19$ is the ice rigidity
$Equation 20$ is the flow law exponent

To determine damage, an inverse control method solution for ice rigidity $Equation 22$ is first carried out. The initial guess $Equation 21$ for the control method (contained in md.materials.rheology_B) is assumed to be based on a temperature parameterization, given a reasonable estimate of the depth-averaged temperature of the ice. Damage is then calculated in locations where the inverse solution for $Equation 23$ is less than the ice rigidity appropriate for the local temperature of the ice. A post-processing function carries out this calculation directly:

>> D=damagefrominversion(md);

Additionally, the scalar backstress can be calculated from the inversion results:

>> backstress = backstressfrominversion(md);

This procedure for calculating damage and backstress was used in [Borstad2013] for the Larsen C ice shelf (see the ‘Publications’ page for a link to the paper).