## Solving the age equation

### General Description

This page explains how to use the general AdvectionReactionSolver from the Elmer distribution to get Age/Depth relation: The AdvectionReactionSolver solves the general equation In the particular case of the age equation, one has and . Obviously for an incompressible flow. For a compressible flow (snow/firn rheology), the trace of the strain-rate tensor should be evaluated using the ComputeStrainRate Solver or directly from the Porous Solver.

### SIF contents

The Solver section looks as

Solver 8
Exec Solver = "After Timestep"
! this is the DG variable, which is not part of the output
Variable =  -nooutput "DGAge"
! this tells that the solver is run on DG mesh
Discontinuous Galerkin = Logical True
! the solver can account for upper and lower limits of the variable
! imposed by formulation of an variational inequality (VI)
! next line switches the VI to be accounted for
Limit Solution = Logical True

Linear System Solver = Iterative
Linear System Iterative Method = BiCGStab
Linear System Max Iterations  = 1000
Linear System Preconditioning = ILU1
Linear System Convergence Tolerance = 1.0e-06
! Variational inequality makes it a non-linear problem
Nonlinear System Max Iterations = 40
Nonlinear System Min Iterations = 2
Nonlinear System Convergence Tolerance = 1.0e-04

! This is the variable that is used to interpolate
! the DG solution to the regular FEM mesh in order
! to get a correct output
Exported Variable 1 = Age
Exported Variable 1 DOFS = 1
End

The source in case of the age/depth equation is 1 (da/dt + u da/dx + … = 1)

Body Force 1
...
DGAge Source = Real 1.0  ! result in years
End

Initial and boundary conditions are then being set for the DG variable and not the exported one!

Initial Condition 1
...
DGAge = Real 0.0
End

! only Dirichlet BC can be set
! the solver automatically uses this
! condition only on inflow boundaries
! outflow boundaries are ignored
Boundary Condition 2
Name = "surf"
Target Boundaries = 2
Body ID = 2
...
! fresh snow or ice
DGAge = Real 0.0
End

! a suggestion for a trick at no-slip BC's to
! keep the age in bounds
!----------------------------------------------
Boundary Condition 1
Name = "bedrock"
Normal-Tangential Velocity = True
! normal outflow velocity set to a very small value
! in order to avoid infinitely large age values
Velocity 1 = Real 1.0e-06 (a micro-meter per year)
! no-slip tangential
!-------------------
Velocity 2 = Real 0.0
Velocity 3 = Real 0.0
.
.
.
End

Just make sure that a free surface can have both, inflow and outflow. NB: If you have a model for re-freezing at the bedrock, you also should set a DGAge = Real 0.0 on this boundary.

The Material section contains the reaction rate as well as the upper/lower limit for the DG variable

Material 1
..
! this has to be sometimes set in diagnostic
! runs to avoid infinite [actually O(1/machine precision)]
! ages at stagnation points in the flow field
DGAge Upper Limit = Real 100000

! well, no ice from the future
! anyway, if you have a well behaving velocity field
! you should not be in need of that constraint
DGAge Lower Limit = Real 0

!this would be a reaction rate,
! is equal to -tr(Eij) (minus trace of the strain-rate)
! in the case of an incompressible material (ice), it is then 0
DGAge Gamma = Real 0.0
End

### Example

An example using the AdvectionReaction solver to solve the age equation can be found in [ELMER_TRUNK]/elmerice/examples/Test_Dating. For a steady 1D flow, the dating equation simplifies to dA/dx = 1/u and solutions are of the form A = ln(Kv) with u = v/v'.
Two tests can be also found in [ELMER_TRUNK]/elmerice/Tests/Dating and [ELMER_TRUNK]/elmerice/Tests/Density. 