<br />
<b>Warning</b>:  Undefined array key 1 in <b>/home/np29546/public_html/elmerice/wiki/inc/auth.php</b> on line <b>78</b><br />
<br />
<b>Warning</b>:  Cannot modify header information - headers already sent by (output started at /home/np29546/public_html/elmerice/wiki/inc/auth.php:78) in <b>/home/np29546/public_html/elmerice/wiki/inc/auth.php</b> on line <b>431</b><br />
<br />
<b>Warning</b>:  Cannot modify header information - headers already sent by (output started at /home/np29546/public_html/elmerice/wiki/inc/auth.php:78) in <b>/home/np29546/public_html/elmerice/wiki/inc/Action/Export.php</b> on line <b>104</b><br />
<br />
<b>Warning</b>:  Cannot modify header information - headers already sent by (output started at /home/np29546/public_html/elmerice/wiki/inc/auth.php:78) in <b>/home/np29546/public_html/elmerice/wiki/inc/Action/Export.php</b> on line <b>104</b><br />
<br />
<b>Warning</b>:  Cannot modify header information - headers already sent by (output started at /home/np29546/public_html/elmerice/wiki/inc/auth.php:78) in <b>/home/np29546/public_html/elmerice/wiki/inc/Action/Export.php</b> on line <b>104</b><br />
===== 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: Cuffey and Paterson, 2010. ''The Physics of Glaciers.'' Pergamon Press, Oxford etc., 4th edt.) using the built-in Elmer viscosity law (recommended, as it is evaluated at Gauss-points):
<code>
! Define some constant at the top of the SIF file (using LUA)
! Define the parameter in MPa - a - m
#yearinsec = 365.25*24*60*60
#rhoi = 900.0/(1.0e6*yearinsec^2)
#rhow = 1000.0/(1.0e6*yearinsec^2)
#gravity = -9.81*yearinsec^2

! Prefactor from Cuffey and Paterson (2010) in MPa^{-3} a^{-1}
#A1 = 2.89165e-13*yearinsec*1.0e18
#A2 = 2.42736e-02*yearinsec*1.0e18
#Q1 = 60.0e3
#Q2 = 115.0e3


Material 1
  Density = Real #rhoi
  
  ! Glen's flow law (using Glen)
  !----------------
  ! viscosity stuff
  !----------------

  Viscosity Model = String "glen"
  Viscosity = 1.0 ! Dummy but avoid warning output
  Glen Exponent = Real 3.0

  Limit Temperature = Real -10.0
  Rate Factor 1 = Real #A1
  Rate Factor 2 = Real #A2
  Activation Energy 1 = Real #Q1
  Activation Energy 2 = Real #Q2
  Glen Enhancement Factor = Real 1.0
  Critical Shear Rate = Real 1.0e-10

  Constant Temperature = Real -1.0 
End
</code>

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
<code>
Constants
  Gas Constant = Real 8.314 !Joule/mol x  K 
End
</code>
the suggested SI default value is used automatically.

This Material section gives the law with a fixed rate factor:
<code>
Material 1
! 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
End
</code>

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) (Not recommended, use build-in implementation of Glen's flow law - first solution on this page):
<code>
!! 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)));}\
}
</code>
Its call within the Material section looks as follows:
<code>
!! 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
</code>

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 [[solvers:aiflow|AIFlow Solver]] or the Continuum-mechanical Anisotropic Flow model based on an anisotropic Flow Enhancement factor (CAFFE, Seddik et al., 2008) implemented in the [[userfunctions:caffe|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 [[solvers:fabric|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 [[solvers:porous|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 [[userfunctions:damage|here]].