**Solver Fortran Files:**`GlaDSCoupledSolver.F90`

and`GlaDSchannelSolver.F90`

**Solver Names:**`GlaDSCoupledSolver`

,`GlaDSsheetThickDummy`

and`GlaDSchannelOut`

**Required Output Variable(s):**`Hydraulic Potential`

,`Sheet Thickness`

and`Channel Area`

**Required Input Variable(s):**None**Optional Output Variable(s):**`Vclose`

,`Wopen`

,`Water Pressure`

,`Effective Pressure`

,`Sheet Discharge`

,`Sheet Storage`

,`Flux from Moulins`

(nodal variables) and`Channel Flux`

(edge variable, to be exported in`GlaDSchannelOut`

).**Optional Input Variable(s):**`Zb`

If in parallel a moulin belongs to two partitions, the flux from the moulin is taken into account twice. This is may be not a bug, but the partitioning should avoid to have moulins at the boundary of partitions. The python tool `makemoulin.py`

(in `[ELMER_TRUNK]/elmerice/Meshers`

) take care to have no duplicated moulins on the boundaries of partitions.

If running the solver on a 3d internally extruded mesh, one should specify `Preserve Edges = True`

and `Preserve Baseline = True`

in the Simulation section.

If running on a “true” 3d mesh, the `GlaDSchannelsSolver`

has to be executed on the 3d body (not the bottom surface body as for the two other GlaDS solvers).

The complete description of the equations solved by the GlaDS solver can be found in Werder et al. (2013). The implementation follows exactly these equations, except that optionally the hydraulic potential can be computed at the top of the water sheet instead than at the bed (keyword: `Neglect Sheet Thickness in Potential`

).

The GlaDS solver solves for the hydraulic potential, the water sheet thickness and the cross-sectional area of the channels. Whereas the two first variables are nodal variable and define continuous fields, the Channel area is a discrete field only defined on the edge of the elements.

The GlaDS model is composed of three solvers:

`GlaDSCoupledSolver`

is the main solver and couple the solve of the 3 main variables`Hydraulic Potential`

,`Sheet Thickness`

and`Channel Area`

. Detail on the keywords for this solver are given below.`GlaDSsheetThickDummy`

is just a solver to declare the`Sheet Thickness`

variable as a primary variable.`GlaDSchannelOut`

has two functions: declare that the`Channel Area`

variable is an edge variable (`Element = “n:0 e:1”`

) and create output vtu files for edge variables.

Since version Version 8.3 (Rev: b213b0c8), `GlaDSchannelOut`

works for parallel simulation (no more vtk or acscii output, only vtu). These solvers only work in transient. They can be executed either on a 2d plane view mesh defining the bedrock or on the boundary of a 3d mesh. If using internal extrusion within Elmer see the structured mesh page for essential keywords to preserve baseline and edges. More details about the specificity of the solvers are given below.

The SIF examples given here are from the tests SHMIP. The name of the variables as well as some constants have to be defined in the `Constants`

section:

Constants Latent Heat = Real $Lw Gravity Norm = Real $gravity Water Density = Real $rhow Ice Density = Real $rhoi Sheet Thickness Variable Name = String "Sheet Thickness" Hydraulic Potential Variable Name = String "Hydraulic Potential" Channel Area Variable Name = String "Channel Area" Bedrock Variable Name = String "Zb" End

The GlaDS solvers depend on a lot of physical parameters. The main parameters to be defined in the `Material`

section are:

! For the sheet Sheet Conductivity = Real $Ks Sheet flow exponent alpha = Real $alphas Sheet flow exponent beta = Real $betas Englacial Void Ratio = Real $ev Bedrock Bump Length = Real $lr Bedrock Bump High = Real $hr Sheet Closure Coefficient = Real $Ar ! For the Channels Channel Conductivity = Real $Kc Channel flow exponent alpha = Real $alphac Channel flow exponent beta = Real $betac Channel Closure Coefficient = Real $Ac Sheet Width Over Channel = Real $lc Pressure Melting Coefficient = Real $Ct Water Heat Capacity = Real $Cw ! Coupling with ice flow and glacier geometry Sliding Velocity = Real $ub Ice Normal Stress = Variable Coordinate 1 Real MATC "rhoi*gravity*H(tx)"

In the `Body Force`

section, one can set a water input source:

Body Force 1 Hydraulic Potential Volume Source = Real $Source End

`GlaDSCoupledSolver`

solves for the three variables `Hydraulic Potential`

, `Sheet Thickness`

and `Channel Area`

in a coupled way inside the solver itself (`Coupled Max Iterations`

and `Coupled Convergence Tolerance`

). Equations for the `Hydraulic Potential`

and `Channel Area`

are non linear. Only the equation for the `Hydraulic Potential`

needs to solve a system. The two others are local and can be solved either explicitely, implcitely or using the Crank-Nicholson method.

Solver 1 Equation = "GlaDS Coupled sheet" Procedure = "ElmerIceSolvers" "GlaDSCoupledSolver" Variable = -dofs 1 "Hydraulic Potential" ! activate or not the development of channels Activate Channels = Logical True ! activate or not the growth of channels by melt Activate Melt from Channels = Logical True ! compute the hydraulic potential at the top of the water sheet (''False'') or at the bed (''True'') Neglect sheet Thickness in Potential = Logical True ! choices are EXPLICT, CRANK-NICOLSON, IMPLICIT Channels Integration method = String "Crank-Nicolson" Sheet Integration method = String "Implicit" ! define exported variables for visualization Exported Variable 1 = -dofs 1 "Vclose" ! closure velocity of the water sheet layer Exported Variable 2 = -dofs 1 "Wopen" ! opening velocity of the water sheet layer Exported Variable 3 = -dofs 1 "Water Pressure" ! water pressure at the base Exported Variable 4 = -dofs 1 "Effective Pressure" ! effective pressure at the base Exported Variable 5 = -dofs 2 "Sheet Discharge" ! water discharge (vector) in the water sheet layer Exported Variable 6 = -dofs 1 "Sheet Storage" ! storage in the water sheet layer Exported Variable 8 = -dofs 1 "Zs" Exported Variable 9 = -dofs 1 "Zb" Exported Variable 10 = -dofs 1 "Flux from Moulins" ! flux of water from the moulins (Qm) Linear System Solver = Direct Linear System Direct Method = umfpack Nonlinear System Max Iterations = 30 Nonlinear System Convergence Tolerance = 1.0e-6 Nonlinear System Relaxation Factor = 1.00 Coupled Max Iterations = Integer 10 Coupled Convergence Tolerance = Real 1.0e-3 Steady State Convergence Tolerance = 1.0e-03 End

`GlaDSsheetThickDummy`

is just here to declare the variable `Sheet Thickness`

Solver 2 Equation = "GlaDS Thickness sheet" Procedure = "ElmerIceSolvers" "GlaDSsheetThickDummy" Variable = -dofs 1 "Sheet Thickness" End

`GlaDSchannelOut`

allows to declare the `Channel Area`

variable as an edge variable and to save edge variables. Output file formats is VTU.

Solver 3 Exec Solver = After Saving Equation = "GlaDS Channel OutPut" Procedure = "ElmerIceSolvers" "GlaDSchannelOut" Variable = -dofs 1 "Channel Area" ! the variable is define on the edges only Element = "n:0 e:1" Exported Variable 1 = -dofs 1 "Channel Flux" VTU OutPutFile = Logical False ! set to TRUE to have VTU output VTU BinaryFile = Logical False Channels OutPut Directory Name = String "results" Channels OutPut File Name = String "$namerun"_channels" End

The possible boundary conditions are:

- channels not allowed to growth (recommanded on all the domain boundary)

Boundary Condition 1 Target Boundaries(2) = 1 3 No Channel BC = Logical True End

- fixed value of the
`Hydraulic Potential`

at some specific outlet nodes:

Boundary Condition 2 Name = "point front" Target Coordinates(1,2) = Real 0.0 0.0 Hydraulic Potential = Variable Zb Real MATC "rhow*gravity*tx" End

- the possibility to impose water flux at some nodes in the domain (moulins type inflow). The nodes have to be declared as node element (101) in the mesh (
`mesh.header`

and`mesh.boundary`

files have to be modified by hand for that, or using the python tool`makemoulin.py`

- see below).

Boundary Condition 4 Name = "moulins" Target Boundaries(20) = 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Moulin Storage = Logical True Moulin Area = Real $Am Moulin Flux = Real $4.5*yearinsec End

Moulins are 101 boundary elements. ElmerGrid does not export correctly 101 boundary elements from gmsh or when partitioning a mesh. To add 101 boundary elements to an existing mesh, thanks to Mondher Chekki (IGE), one can use the python tool `makemoulin.py`

(in `[ELMER_TRUNK]/elmerice/Meshers`

).

Usage:

python makemoulin.py --meshdir mesh_dir --moulin moulin_file --partition number_of_partition

where `moulin_file`

is an ascii file which contains the (x,y) coordinates of the moulins. The same file has to be used in gmsh so that nodes with the moulin coordinates already exist.

Examples using the *GlaDS* Solver can be found in `[ELMER_TRUNK]/elmerice/Tests/`

.

The description of the GlaDS model is in:

- Werder M.A., I.J. Hewitt, C.G. Schoof and G.E. Flowers, 2013. Modeling channelized and distributed subglacial drainage in two dimensions. Journal of Geophysical Research: Earth Surface, 118(4), 2140-2158.

The implementation of the GlaDS model in Elmer/Ice is described here:

- Gagliardini O. and M. Werder, 2018. Influence of increasing surface melt over decadal timescales on land-terminating Greenland-type outlet glaciers, Journal of Glaciology, 64(247), 700-710, doi:10.1017/jog.2018.59

Results of the SHMIP experiments using Elmer/Ice are discussed in the SHMIP paper:

- De Fleurian, B., M. Werder, S. Beyer, D. Brinkerhoff, I. Delaney, C. Dow, C., J. Dows, O. Gagliardini, M.J. Hoffman, R. LeB Hooke, J. Seguinot, A.N. Sommers, 2018. SHMIP The subglacial hydrology model intercomparison Project. Journal of Glaciology, 1-20. doi:10.1017/jog.2018.78