~~NOTOC~~
===== 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:
rho {{\partial H}/{\partial t}} + rho u .grad H = div(k grad H) + tr (sigma epsilon) + Q_lat
where
* H is the enthalpy variable
* rho is the ice density
* u is the ice velocity vector
* k is the enthalpy diffusivity
* tr (sigma epsilon) the strain heating
* Q_lat a complementary source term accounting for melt water refreezing
Enthalpy is defined as a function of the water content omega and the temperature T, such that:
If H < H_f, then H(T, omega) = int_T0^T C_p (T) dT
If H > H_f, then H(T, omega) = int_T0^Tm C_p (T) dT + omega L
where
* H_f is the enthalpy of fusion, defined from the fusion temperature according to the pressure dependent Clausius-Clapeyron relationship.
* C_p is the temperature dependant heat capacity, defined as C_p = AT+B
* L 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. H(T, omega) = int_T0^T C_p (T) dT. 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''.
==== References ====
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.