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solvers:enthalpy [Elmer/Ice Wiki]

## Enthalpy Solver

### General Informations

• Solver Fortran File: EnthalpySolver.f90
• Solver Name: EnthalpySolver
• Required Output Variable(s): Enthalpy_h, Phase Change Enthalpy, Temperature and Water Content
• Required Input Variable(s): a velocity field
• Optional Output Variable(s): None
• Optional Input Variable(s): None

### General Description

Solves the enthalpy equation: where

• is the enthalpy variable
• is the ice density
• is the ice velocity vector
• is the enthalpy diffusivity
• the strain heating
• a complementary source term accounting for melt water refreezing

Enthalpy is defined as a function of the water content and the temperature , such that:  where

• is the enthalpy of fusion, defined from the fusion temperature according to the pressure dependent Clausius-Clapeyron relationship.
• is the temperature dependant heat capacity, defined as • is the latent heat of fusion

For the boundary conditions, a flux (Enthalpy Heat Flux) has the same meaning than for the temperature solver (W/m2). For a Dirichlet boundary condition on the enthalpy variable, the same definition as in the solver has to be used, i.e. . See example below.

### SIF contents

In this example, ice velocity are in m/s and pressure en MPa.

Solver 2
Equation = String "Enthalpy Equation"
Procedure = File "ElmerIceSolvers" "EnthalpySolver"
Variable = String "Enthalpy_h"
Linear System Solver = "Iterative"
Linear System Iterative Method = "BiCGStab"
Linear System Max Iterations = 500
Linear System Convergence Tolerance = 1.0E-07
Linear System Abort Not Converged = True
Linear System Preconditioning = "ILU0"
Linear System Residual Output = 1
Steady State Convergence Tolerance = 1.0E-04
Nonlinear System Convergence Tolerance = 1.0E-03
Nonlinear System Max Iterations = 10
Nonlinear System Relaxation Factor = Real 1.0

Apply Dirichlet = Logical True
Stabilize = True

Exported Variable 1 = String "Phase Change Enthalpy" ! (J kg-1)
Exported Variable 1 DOFs = 1

Exported Variable 2 = String "Water Content" ! (%)
Exported Variable 2 DOFs = 1

Exported Variable 3 = String "temperature" ! (°C)
Exported Variable 3 DOFs = 1
End

Constants
T_ref_enthalpy = real 200.0 !(J kg-1)
L_heat = real 334000.0 !(J kg-1)
! Cp(T) = A*T + B
Enthalpy Heat Capacity A = real 7.253 !(J kg-1 K-2)
Enthalpy Heat Capacity B = real 146.3 !(J kg-1 K-1)
P_triple = real 0.061173 !Triple point pressure for water (MPa)
P_surf = real 0.1013 ! Surface atmospheric pressure(MPa)
beta_clapeyron = real 0.0974 ! clausus clapeyron relationship (K MPa-1)
End

Body Force 1
Heat Source = real 0.0
End

Material 1
Enthalpy Density = real 917.0 !(kg m-3)
Enthalpy Heat Diffusivity = Real \$2.1/2050.0 ! = k / Cp (kg m-1 s-1)
Enthalpy Water Diffusivity = real 1.045e-4 ! (kg m-1 s-1)
End

! bed rock interface
Boundary Condition 1
Target Boundaries = 1
Velocity 1 = Real 0.0
Velocity 2 = Real 0.0
Velocity 3 = Real 0.0

Enthalpy Heat Flux BC = logical True
Enthalpy Heat Flux = real 0.02 !(W m-2)
End

! Upper Surface
Boundary Condition 2
Target Boundaries = 2
Enthalpy_h = variable coordinate 3
real MATC "25000.0/150.0*(tx-3250)+140000.0" ! (J kg-1)
End
End

### Examples

An example solving for the enthalpy within the Tete Rousse glacier assuming an elevation dependent enthalpy at the upper surface can be found in [ELMER_TRUNK]/elmerice/Tests/Enthalpy.

Gilbert, A., O. Gagliardini, C. Vincent, and P. Wagnon, 2014. A 3-D thermal regime model suitable for cold accumulation zones of polythermal mountain glaciers, J. Geophys. Res. Earth Surf., 119, doi:10.1002/2014JF003199. 