**USF Fortran File:**`USF_Sliding.f90`

**USF Name:**`Friction_Coulomb`

**Required Input Variable(s):**A`Flow Solution`

in`Flow Solution Name`

,`Normal Vector`

,`Stress`

or the`Effective Pressure`

variable.

The file `USF_Sliding.f90`

contains three user functions to apply non-linear friction at the base of glacier.

The first user function (`Sliding_Weertman`

) is a non-linear Weertman-type friction law and is described here. The second user function (`Friction_Coulomb`

) is a non-linear water pressure dependant friction law, as proposed by Schoof (2005) and Gagliardini et al. (2007), and is presented in this page. The third user function (`Sliding_Budd`

) is described here and is from Budd et al 1984 (Annals of Glaciology 5, page 29-36).

The friction law in *Friction_Coulomb* is of the form:

where

and

The *Slip Coefficient* in Elmer is then given as

When , in the previous equation is replaced by .

The parameters to be given are:

`Friction Law Sliding Coefficient`

→`Friction Law Post-Peak Exponent`

→`Friction Law Maximum Value`

→ ~ max bed slope`Friction Law Exponent`

→ m = (n Glen's law)`Friction Law Linear Velocity`

→

The effective pressure is defined as , where is the normal Cauchy stress and the water pressure. If a variable `Effective Pressure`

exists, it is used to evaluate directly . Else, the normal Cauchy stress is estimated from the stress computed at previous timestep. The water pressure is prescribed as an `External Pressure`

(Negative - Compressive convention, and therefore 'External Pressure' should be equal to the opposite of the water pressure in the sif).

The required keywords in the SIF file for this user function are:

!!! Bedrock Boundary Condition Boundary Condition 1 Target Boundaries = 1 Normal-Tangential Velocity = Logical True Flow Force BC = Logical True !! Water pressure given through the Stokes 'External Pressure' parameter !! (Negative = Compressive) External Pressure = Equals Water Pressure Velocity 1 = Real 0.0 Slip Coefficient 2 = Variable Coordinate 1 Real Procedure "ElmerIceUSF" "Friction_Coulomb" Slip Coefficient 3 = Variable Coordinate 1 Real Procedure "ElmerIceUSF" "Friction_Coulomb" !! Parameters needed for the Coulomb Friction Law Friction Law Sliding Coefficient = Real 4.1613e5 Friction Law Post-Peak Exponent = Real 1.0 !(q=1) Friction Law Maximum Value = Real 1.0 !(C=1) Friction Law PowerLaw Exponent = Real 3.0 !(m = n = 3 Glen's law) Friction Law Linear Velocity = Real 0.01 End

The Coulomb friction law is tested in `[ELMER_TRUNK]/elmerice/Tests/Friction_Coulomb`

with a direct input of the effective pressure and `[ELMER_TRUNK]/elmerice/Tests/Friction_Coulomb_Pw`

with the effective pressure computed from the stress and a prescribed water pressure.

When this friction law is used, it can be cited using the following reference:

Gagliardini O., D. Cohen, P. Råback and T. Zwinger, 2007. Finite-Element Modeling of Subglacial Cavities and Related Friction Law. J. of Geophys. Res., Earth Surface, 112, F02027.