Using a free surface#

This section was contributed by Ian Rose.

Free surfaces are numerically challenging but can be useful for self consistently tracking dynamic topography and may be quite important as a boundary condition for tectonic processes like subduction. The parameter file cookbooks/free_surface/free_surface.prm provides a simple example of how to set up a model with a free surface, as well as demonstrates some of the challenges associated with doing so.

supports models with a free surface using an Arbitrary Lagrangian-Eulerian framework (see Free surface calculations). Most of this is done internally, so you do not need to worry about the details to run this cookbook. Here we demonstrate the evolution of surface topography that results when a blob of hot material rises in the mantle, pushing up the free surface as it does. Usually the amplitude of free surface topography will be small enough that it is difficult to see with the naked eye in visualizations, but the topography postprocessor can help by outputting the maximum and minimum topography on the free surface at every time step.

The bulk of the parameter file for this cookbook is similar to previous ones in this manual. We use initial temperature conditions that set up a hot blob of rock in the center of the domain.

The main addition is the Mesh deformation subsection. In this subsection you need to give a comma separated list of the boundary indicators where the ‘free surface’ deformation should be applied. In this case, we are dealing with the ‘top’ boundary of a box in 2D. There is another significant parameter that needs to be set here: the value for the stabilization parameter “theta”. If this parameter is zero, then there is no stabilization, and you are likely to see instabilities develop in the free surface. If this parameter is one then it will do a good job of stabilizing the free surface, but it may overly damp its motions. The default value is 0.5.

Also worth mentioning is the change to the initial time step size. Stability concerns typically mean that when making a model with a free surface you will want to take smaller time steps. In general just how much smaller will depend on the problem at hand as well as the desired accuracy. Because this model has a very smooth time evolution it is sufficient to reduce the time step size of the first few time steps.

Following are the sections in the input file specific to this testcase. The full parameter file may be found at cookbooks/free_surface/free_surface.prm.

set Maximum first time step                = 1e3
set Maximum relative increase in time step = 30
set Pressure normalization                 = no

subsection Mesh deformation
  set Mesh deformation boundary indicators = top: free surface

  subsection Free surface
    set Free surface stabilization theta = 0.5
  end
end

subsection Boundary velocity model
  set Tangential velocity boundary indicators = left, right, bottom
end

subsection Boundary temperature model
  set Fixed temperature boundary indicators = left, right, bottom, top
  set List of model names = constant
  subsection Constant
    set Boundary indicator to temperature mappings = left:0, right:0, bottom:0, top:0
  end
end

subsection Initial temperature model
  set Model name = function
  subsection Function
    set Variable names      = x,y
    set Function expression = sqrt((x-250.e3)^2 + (y-100.e3)^2) < 25.e3 ? 200.0 : 0.0
  end
end

subsection Postprocess
  set List of postprocessors = visualization, velocity statistics, topography

  subsection Visualization
    set Time between graphical output = 1.e6
  end
end

Running this input file will produce results like those in Fig. 45. The model starts with a single hot blob of rock which rises in the domain. As it rises, it pushes up the free surface in the middle, creating a topographic high there. This is similar to the kind of dynamic topography that you might see above a mantle plume on Earth. As the blob rises and diffuses, it loses the buoyancy to push up the boundary, and the surface begins to relax.

After running the cookbook, you may modify it in a number of ways:

  • Add a more complicated initial temperature field to see how that affects topography.

  • Add a high-viscosity lithosphere to the top using a compositional field to tamp down on topography.

  • Explore different values for the stabilization theta and the CFL number to understand the nature of when and why stabilization is necessary.

  • Try a model in a different geometry, such as spherical shells.

Figure

Fig. 45 Evolution of surface topography due to a rising blob. On the left is a snapshot of the model setup. The right shows the value of the highest topography in the domain over 18 Myr of model time. The topography peaks at 167 meters after 5.5 Myr. This cookbook may be run with the cookbooks/free_surface/free_surface.prm input file.#