• Solver Fortran File: GlaDSCoupledSolver.F90 and GlaDSchannelSolver.F90
• Solver Name: 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 and Channel Flux
• Optional Input Variable(s): Zb

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 vtk files for edge variables.

Currently (June 2017), GlaDSchannelOut doesn't support parallel simulation. 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. More details about the specificity of the solvers are given below.

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 SIF example given here is from the test A1 of SHMIP. 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-NICHOLSON, IMPLICIT
  Channels Integration method = String "Crank-Nicholson"
  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 4 = -dofs 1 "Water Pressure"       ! water pressure at the base 
  Exported Variable 5 = -dofs 1 "Effective Pressure"   ! effective pressure at the base
  Exported Variable 6 = -dofs 2 "Sheet Discharge"      ! water discharge (vector) in the water sheet layer 
  Exported Variable 7 = -dofs 1 "Sheet Storage"        ! storage in the water sheet layer

  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 are vtk or ascii.

Solver 3
  Exec Solver = After Saving
  Equation = "GlaDS Channel OutPut"
  Procedure = "ElmerIceSolvers" "GlaDSchannelOut"
  Variable = -dofs 1 "Channel Area"
! Define that the variable is define on the edges only
  Element = "n:0 e:1"

  Exported Variable 1 = -dofs 1 "Channel Flux"

  VTK OutPutFile = Logical True
  ASCII OutPutFile = Logical False
  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):

 
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

An example using the GlaDS Solver can be found in [ELMER_TRUNK]/elmerice/examples/GlaDS.

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.