===== Retrieve Emergence Velocity =====
==== General Information ====

  * **Solver Fortran File:** ''Emergence.F90''
  * **Solver Name:** ''GetEmergenceVelocity''
  * **Required Output Variable(s):** 
    * (1) ''EmergenceVelocity'' 
  * **Required Input Variable(s):** 
    * (1) ''Normal Vector''
    * (2) ''Flow Solution'' 
  * **Optional Output Variable(s):** None  
  * **Optional Input Variable(s):** None

==== General Description ====

This is a pseudo solver (i.e., it only composes a new variable from given, not solving a matrix) to retrieve the emergence velocity from a given surface normal vector and a (ice) velocity field. It uses the fact that the scalar-product between surface velocity and surface normal gives (apart from a factor that is very close to unity) the emergence velocity

<m>
v_{em} =  - u  {{\partial h}/{\partial x}} - v  {{\partial h}/{\partial y}} + w,
</m>

as given in the kinematic boundary condition of the free surface.

<m>
{{\partial h}/{\partial t}} - v_{em} = a ||grad F_h||,
</m>

where <m>h</m> is the <m>z</m>-coordinate of the free surface, <m>(u,v,w)</m> the components of the ice velocity vector <m>\vec{u}</m>, <m>a</m> the net normal accumulation/ablation. The gradient of the implicit free surface function <m>F_h = z - h</m>

<m>
||grad F_h|| = \sqrt{({{\partial h}/{\partial x}})^2 + ({{\partial h}/{\partial y}})^2 + 1},
</m>

usually is approximated by unity, as the derivatives of the free surface equation are of the order of the aspect ratio (usually small). Consequently, the surface normal is given by

<m>
\vec{n} = {{grad F_h}/{||grad F_h||}} \approx{ {grad F_h}} 
</m>

and hence
<m>
\vec{n} =( -{{\partial h}/{\partial x}}, -{{\partial h}/{\partial y}}, 1 ), 
</m>

and the emergence velocity can be approximated by

<m>
v_{em} = \vec{u} . \vec{n}
</m>
==== SIF Contents ====
The following SIF excerpt additionally contains solvers needed for the surface ''Normal Vector'' (Solver 2) and the ''Flow Solution'' (Solver 3). 

<code>
!==============================================================================
! /// Compute Normals ///
!==============================================================================
Solver 2
   Exec Solver = "Before Simulation"
   !Exec Solver = Never
   Equation = "NormalVector"
   Procedure = "ElmerIceSolvers" "ComputeNormalSolver"
   Variable = String "Normal Vector"
   Variable DOFs = 3
   Optimize Bandwidth = Logical False
   ComputeAll = Logical False
End
!==============================================================================
! /// Stokes Equation ///
!==============================================================================
Solver 3
  Equation = String "Navier-Stokes"
  Flow Model = "Stokes"
  Stabilization Method = Stabilized
  Optimize Bandwidth = Logical True
  Steady State Convergence Tolerance = 1.0E-03
  Linear System Solver = Direct
  Linear System Direct Method = umfpack
  Nonlinear System Convergence Tolerance = 1.0E-03
  Nonlinear System Max Iterations = 50
  Nonlinear System Min Iterations = 10
  Nonlinear System Newton After Iterations = 50
  Nonlinear System Newton After Tolerance =  1.0E-01
  Nonlinear System Reset Newton = Logical True
End
!==============================================================================
! /// Computing emergence velocity ///
!==============================================================================
Solver 4
  Equation = "SMB"
  Procedure = "ElmerIceSolvers" "GetEmergenceVelocity"
  Variable = -dofs 1 EmergenceVelocity
End
</code>
''GetEmergenceVelocity'' (Solver 4) is - in contrary to the other Solvers - executed only on the body declared at the free surface boundary, where ''ComputeNormal = Logical True'' has to be set. The corresponding ''Equation 2'' has to contain a keyword ''Convection = "Computed"'' as well as the name of the variable contining the velocities (usually ''Flow Solution'')

<code>
Equation 2
  Name = "Surface Equations"
  Active Solvers(1) = 4 
  Convection = String "Computed"
  Flow Solution Name = String "Flow Solution"
End
</code>
==== Example ====
An example solving the emergence velocity for a surface velocity and shape distribution computed on a Bueler-profile can be found in ''[ELMER_TRUNK]/elmerice/Tests/Emergence''.

==== References ====

Välisuo, I., T. Zwinger, and J. Kohler, 2017. Inverse solution of surface mass balance of Midtre Lovénbreen,
Svalbard, accepted Journal of GLaciology