**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

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

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

where is the -coordinate of the free surface, the components of the ice velocity vector , the net normal accumulation/ablation. The gradient of the implicit free surface function

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

and hence

and the emergence velocity can be approximated by

The following SIF excerpt additionally contains solvers needed for the surface `Normal Vector`

(Solver 2) and the `Flow Solution`

(Solver 3).

!============================================================================== ! /// 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

`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`

)

Equation 2 Name = "Surface Equations" Active Solvers(1) = 4 Convection = String "Computed" Flow Solution Name = String "Flow Solution" End

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`

.

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