AdjointThickness_ThicknessSolver.md

AdjointThickness : Direct Solver {#thickness_direct_solver}

Module name: AdjointThickness_ThicknessSolver.F90 Module subroutines: AdjointThickness_ThicknessSolver Module authors: Fabien Gillet-Chaulet (IGE-Grenoble) Document authors: Fabien Gillet-Chaulet Document edited: 10/12/2020

Solver Variable:

• H : thickness

Introduction

This solver solves the steady-state Thickness evolution equation: $$div(uH) = Ms + Mb$$ where:

• $u$ is the mean horizontal velocity

• $Ms$ and $Mb$ are the surface and bottom mass balance (>0 for accumulation)

The convection velocity can be provided as a variable using the keyword Flow Solution Name in the Solver Section, or alternatively the velocity componenents can be read in the body force section.

In general $Ms$ and $Mb$ will be the apparent mass basalance, i.e. they will include a correction with the observed thickness rate of change.

This solver has been derived from the legacy Elmer/Ice Thickness solver. It has been simplified and separated from the legacy solver to derive the adjoint code (AdjointThickness : Gradient Solver) and make sure that they are consistent.

In particular, this solver allows to use only the stabilised method (i.e. a StreamLine Upwind Petrov-Galerkin (SUPG) method) and does not include the limiters.

It can only be used in a cartesian reference frame and in steady-state.

Note that in the absence of limiters this equation is linear and does not require non-linear iterations.

Keywords

Solver Section:

  Solver *id*
   Equation = "Thickness"

  !# The solver variable 
   Variable = "H"

   Procedure = "ElmerIceSolvers" "AdjointThickness_ThicknessSolver"

 !# Keywords related to linear system
  Linear System Solver = ...

 !#  the convection velocity $u$
  Flow Solution Name = String  ...

 !# if the convection velocity is read from the body force, 
 !#  provide the diemnsion
 Convection Dimension = Integer
End

Body Forces:

!# Ms and Mb
Body Force *id*
  Top Surface Accumulation = Real ...
  Bottom Surface Accumulation = Real ...

  !# If <Flow Solution Name> is not provided in the solver section
  !#  the convection velocity can be read here:
  ! x-componenent
  Convection Velocity 1 = Real ...
  ! y-componenent if Convection Dimension = 2
  Convection Velocity 1 = Real ...
End

Boundary Conditions:

Requires Dirichlet boundary conditions at inflow boundaries.

Boundary Condition 1 
! Dirichlet condition
  H = Real ...
End

Tests and Examples

• See examples for the Mass Conservation methods