## Rheology

There are various ice rheology implemented in Elmer/Ice.

### Glen's flow law

This is a SIF entry for Glen's flow law (after: Paterson, W. S. B. 1994. `The Physics of Glaciers.` Pergamon Press, Oxford etc., 3rd edt.) using the built-in Elmer viscosity law (recommended, as it is evaluated at Gauss-points):

```! Glen's flow law (using Glen)
!----------------
! viscosity stuff
!----------------
Viscosity Model = String "Glen"
! Viscosity has to be set to a dummy value
! to avoid warning output from Elmer
Viscosity = Real 1.0
Glen Exponent = Real 3.0
Critical Shear Rate = Real 1.0e-10
! Rate factors (Paterson value in MPa^-3a^-1)
Rate Factor 1 = Real 1.258e13
Rate Factor 2 = Real 6.046e28
! these are in SI units - no problem, as long as
! the gas constant also is
Activation Energy 1 = Real 60e3
Activation Energy 2 = Real 139e3
Glen Enhancement Factor = Real 1.0
! the variable taken to evaluate the Arrhenius law
! in general this should be the temperature relative
! to pressure melting point. The suggestion below plugs
! in the correct value obtained with TemperateIceSolver
Temperature Field Variable = String "Temp Homologous"
! the temperature to switch between the
! two regimes in the flow law
Limit Temperature = Real -10.0
! In case there is no temperature variable
!Constant Temperature = Real -10.0```

With the values of the activation energies above, the gas constant has to be given in SI units, i.e., 8.314 J/(mol K). If you do not provide the following section

```Constants
Gas Constant = Real 8.314 !Joule/mol x  K
End```

the suggested SI default value is used automatically.

This Material section gives the law with a fixed rate factor:

```! Glen's flow law (using Glen)
!-----------------
! viscosity stuff
!----------------
Viscosity Model = String "Glen"
Viscosity = Real 1.0 ! To avoid warning output
Glen Exponent = Real 3.0
Critical Shear Rate = Real 1.0e-10
! gives a fixed value in MPa^-3a^-1
Set Arrhenius Factor = Logical True
Arrhenius Factor = Real \$1.0E-16 * 1.0E18
Glen Enhancement Factor = Real 1.0```

This is a SIF entry for Glen's flow law (after: Paterson, W. S. B. 1994. `The Physics of Glaciers.` Pergamon Press, Oxford etc., 3rd edt.) using the old power law (MATC function):

```!! Glen's flow law (using power law)
!-----------------
\$ function glen(Th) {\
EF = 1.0;\
AF = getArrheniusFactor(Th);\
_glen = (2.0*EF*AF)^(-1.0/3.0);\
}

!! Arrhenius factor needed by glen
!! (in SI units)
!---------------------------------
\$ function getArrheniusFactor(Th){ \
if (Th<-10) {_getArrheniusFactor=3.985E-13 * exp( -60.0E03/(8.314 * (273.15 + Th)));}\
else {\
if (Th>0) _getArrheniusFactor=1.916E03 * exp( -139.0E03/(8.314 *  (273.15)));\
else _getArrheniusFactor=1.916E03 * exp( -139.0E03/(8.314 *  (273.15 + Th)));}\
}```

Its call within the Material section looks as follows:

```!! call in SI units
Viscosity = Variable Temperature
Real MATC "glen(tx)"
Critical Shear Rate = 1.0E-09

!! call in scaled units (m-MPa-years)
Viscosity = Variable Temperature
Real MATC "glen(tx)*31556926.0^(-1.0/3.0)*1.0E-06"
Critical Shear Rate = \$1.0E-09 * 31556926.0

!! this holds for both unit systems
Viscosity Model = String "power law"
Viscosity Exponent = \$1.0/3.0```

Strictly speaking the homologous temperature should be used as input to the Glen function above, but if homologous temperature is not readily available then using temperature (in Celsius) is a good approximation (which deteriorates for thicker glaciers/ice sheets).

Be very careful in choosing the correct value of the critical shear rate. A too high value leads to a way too soft ice at low shear rates, a too low value can have consequences on the numerical stability (singularity of shear thinning fluid at zero shear).

An example using Glen's flow law can be found in `[ELMER_TRUNK]/elmerice/examples/Test_Glen_2D`..

### Anisotropic Ice

The flow of anisotropic ice can be modelled using the General Orthotropic Flow Law (GOLF) from Gillet-Chaulet et al. (2005) implemented in the AIFlow Solver or the Continuum-mechanical Anisotropic Flow model based on an anisotropic Flow Enhancement factor (CAFFE, Seddik et al., 2008) implemented in the User Function CAFFE. The evolution of the fabric as a function of stress and velocity gradient for both anisotropic models can be computed using the Fabric Solver.

### Firn and Snow Rheology

The rheology of porous ice, namely firn and snow, is represented using the porous law proposed by Gagliardini and Meyssonnier (1997). This law is implemented in Elmer/Ice in the Porous Solver. Density evolution can be computed from the mass conservation equation.

### Damage

Damage is accounted for through the enhancement factor. Damage evolution is modelled following the approach in Krug et al. (2014). More information can be found here. 