Simple convection in a quarter of a 2d annulus#

Let us start this sequence of cookbooks using a simpler situation: convection in a quarter of a 2d shell. We choose this setup because 2d domains allow for much faster computations (in turn allowing for more experimentation) and because using a quarter of a shell avoids a pitfall with boundary conditions we will discuss in the next section. Because it’s simpler to explain what we want to describe in pictures than in words, Fig. 73 shows the domain and the temperature field at a few time steps. In addition, you can find a movie of how the temperature evolves over this time period at http://www.youtube.com/watch?v=d4AS1FmdarU.1

figure

Fig. 73 Simple convection in a quarter of an annulus: Snapshots of the temperature field at times \(t=0\), \(t=1.2\times 10^7\) years (time step 2135), and \(t=10^9\) years (time step 25,662). The bottom right part of each figure shows an overlay of the mesh used during that time step.#

Let us just start by showing the input file (which you can find in cookbooks/shell_simple_2d/shell_simple_2d.prm):

# A simple setup for convection in a quarter of a 2d shell. See the
# manual for more information.


set Dimension                              = 2
set Use years in output instead of seconds = true
set End time                               = 1.5e9
set Output directory                       = output-shell_simple_2d


subsection Material model
  set Model name = simple

  subsection Simple model
    set Thermal expansion coefficient = 4e-5
    set Viscosity                     = 1e22
  end
end


subsection Geometry model
  set Model name = spherical shell

  subsection Spherical shell
    set Inner radius  = 3481000
    set Outer radius  = 6336000
    set Opening angle = 90
  end
end


subsection Boundary velocity model
  set Zero velocity boundary indicators       = inner
  set Tangential velocity boundary indicators = top, left, right
end


subsection Heating model
  set List of model names =  shear heating
end


subsection Boundary temperature model
  set Fixed temperature boundary indicators = top, bottom
  set List of model names = spherical constant

  subsection Spherical constant
    set Inner temperature = 4273
    set Outer temperature = 973
  end
end


subsection Initial temperature model
  set Model name = spherical hexagonal perturbation
end


subsection Gravity model
  set Model name = ascii data
end


subsection Mesh refinement
  set Initial global refinement          = 5
  set Initial adaptive refinement        = 4
  set Strategy                           = temperature
  set Time steps between mesh refinement = 15
end


subsection Postprocess
  set List of postprocessors = visualization, velocity statistics, temperature statistics, heat flux statistics, depth average

  subsection Visualization
    set Output format                 = vtu
    set Time between graphical output = 1e6
    set Number of grouped files       = 0
  end

  subsection Depth average
    set Time between graphical output = 1e6
  end
end

In the following, let us pick apart this input file:

  1. Lines 1–4 are just global parameters. Since we are interested in geophysically realistic simulations, we will use material parameters that lead to flows so slow that we need to measure time in years, and we will set the end time to 1.5 billion years – enough to see a significant amount of motion.

  2. The next block (lines 7–14) describes the material that is convecting (for historical reasons, the remainder of the parameters that describe the equations is in a different section, see the fourth point below). We choose the simplest material model has to offer where the viscosity is constant (here, we set it to \(\eta=10^{22} \text{ Pa . s}\)) and so are all other parameters except for the density which we choose to be \(\rho(T)=\rho_0(1-\alpha (T-T_{\text{ref}}))\) with \(\rho_0=3300 \text{ kg}\;\text{ m}^{-3}\), \(\alpha=4\times 10^{-5} \text{ K}^{-1}\) and \(T_{\text{ref}}=293 \text{ K}\). The remaining material parameters remain at their default values and you can find their values described in the documentation of the simple material model in Material model and Subsection: Material model / Simple model.

  3. Lines 17–25 then describe the geometry. In this simple case, we will take a quarter of a 2d shell (recall that the dimension had previously been set as a global parameter) with inner and outer radii matching those of a spherical approximation of Earth.

  4. The second part of the model description and boundary values follows in lines 28–42. The boundary conditions require us to look up how the geometry model we chose (the spherical shell model) assigns boundary indicators to the four sides of the domain. This is described in Geometry model where the model announces that boundary indicator zero is the inner boundary of the domain, boundary indicator one is the outer boundary, and the left and right boundaries for a 2d model with opening angle of 90 degrees as chosen here get boundary indicators 2 and 3, respectively. In other words, the settings in the input file correspond to a zero velocity at the inner boundary and tangential flow at all other boundaries. We know that this is not realistic at the bottom, but for now there are of course many other parts of the model that are not realistic either and that we will have to address in subsequent cookbooks. Furthermore, the temperature is fixed at the inner and outer boundaries (with the left and right boundaries then chosen so that no heat flows across them, emulating symmetry boundary conditions) and, further down, set to values of 700 and 4000 degrees Celsius – roughly realistic for the bottom of the crust and the core-mantle boundary.

  5. Lines 45–47 describe that we want a model where equation (3) contains the shear heating term \(2\eta \varepsilon(\mathbf u):\varepsilon(\mathbf u)\) (noting that the default is to use an incompressible model for which the term \(\frac{1}{3}(\nabla \cdot \mathbf u)\mathbf 1\) in the shear heating contribution is zero). Considering a reasonable choice of heating terms is not the focus of this simple cookbook, therefore we will leave a discussion of possible and reasonable heating terms to another cookbook.

  6. The description of what we want to model is complete by specifying that the initial temperature is a perturbation with hexagonal symmetry from a linear interpolation between inner and outer temperatures (see Initial composition model), and what kind of gravity model we want to choose (one reminiscent of the one inside the Earth mantle, see Gravity model).

  7. The remainder of the input file consists of a description of how to choose the initial mesh and how to adapt it (lines 60–65) and what to do at the end of each time step with the solution that computes for us (lines 68–81). Here, we ask for a variety of statistical quantities and for graphical output in VTU format every million years.

Note

Having described everything to ASPECT, you may want to view the video linked to above again and compare what you see with what you expect. In fact, this is what one should always do having just run a model: compare it with expectations to make sure that we have not overlooked anything when setting up the model or that the code has produced something that doesn’t match what we thought we should get. Any such mismatch between expectation and observed result is typically a learning opportunity: it either points to a bug in our input file, or it provides us with insight about an aspect of reality that we had not foreseen. Either way, accepting results uncritically is, more often than not, a way to scientifically invalid results.

The model we have chosen has a number of inadequacies that make it not very realistic (some of those happened more as an accident while playing with the input file and weren’t a purposeful experiment, but we left them in because they make for good examples to discuss below). Let us discuss these issues in the following.

Dimension.#

This is a cheap shot but it is nevertheless true that the world is three-dimensional whereas the simulation here is 2d. We will address this in the next section.

Incompressibility, adiabaticity and the initial conditions.#

This one requires a bit more discussion. In the model selected above, we have chosen a model that is incompressible in the sense that the density does not depend on the pressure and only very slightly depends on the temperature. In such models, material that rises up does not cool down due to expansion resulting from the pressure dropping, and material that is transported down does not adiabatically heat up. Consequently, the adiabatic temperature profile would be constant with depth, and a well-mixed model with hot inner and cold outer boundary would have a constant temperature with thin boundary layers at the bottom and top of the mantle. In contrast to this, our initial temperature field was a perturbation of a linear temperature profile.

There are multiple implications of this. First, the temperature difference between outer and inner boundary of 3300 K we have chosen in the input file is much too large. The temperature difference that drives the convection, is the difference in addition to the temperature increase a volume of material would experience if it were to be transported adiabatically from the surface to the core-mantle boundary. This difference is much smaller than 3300 K in reality, and we can expect convection to be significantly less vigorous than in the simulation here. Indeed, using the values in the input file shown above, we can compute the Rayleigh number for the current case to be2

\[\textrm{Ra} = \frac{g\, \alpha \Delta T \rho L^3}{\kappa\eta} = \frac{10\, \text{ m}\,\text{ s}^{-2} \times 4\times 10^{-5}\, \text{ K}^{-1} \times 3300\, \text{ K} \times 3300\, \text{ kg}\,\text{ m}^{-3} \times (2.86\times 10^6 \, \text{ m})^3}{10^{-6}\, \text{ m}^2\,\text{ s}^{-1}\times 10^{22}\, \text{ kg}\,\text{ m}^{-1}\,\text{ s}^{-1}}.\]

Second, the initial temperature profile we chose is not realistic – in fact, it is a completely unstable one: there is hot material underlying cold one, and this is not just the result of boundary layers. Consequently, what happens in the simulation is that we first overturn the entire temperature field with the hot material in the lower half of the domain swapping places with the colder material in the top, to achieve a stable layering except for the boundary layers. After this, hot blobs rise from the bottom boundary layer into the cold layer at the bottom of the mantle, and cold blobs sink from the top, but their motion is impeded about half-way through the mantle once they reach material that has roughly the same temperature as the plume material. This impedes convection until we reach a state where these plumes have sufficiently mixed the mantle to achieve a roughly constant temperature profile.

This effect is visible in the movie linked to above where convection does not penetrate the entire depth of the mantle for the first 20 seconds (corresponding to roughly the first 800 million years). We can also see this effect by plotting the root mean square velocity, see the left panel of Fig. 74. There, we can see how the average velocity picks up once the stable layering of material that resulted from the initial overturning has been mixed sufficiently to allow plumes to rise or sink through the entire depth of the mantle.

figure

Fig. 74 Simple convection in a quarter of an annulus. Left: Root mean square values of the velocity field. The initial spike (off the scale) is due to the overturning of the unstable layering of the temperature. Convection is suppressed for the first 800 million years due to the stable layering that results from it. The maximal velocity encountered follows generally the same trend and is in the range of 2-3 cm/year between 100 and 800 million years, and 4-8 cm/year following that. Right: Average temperature at various depths for \(t=0\), \(t=800,000\) years, \(t=5\times 10^{8}\) years, and \(t=10^9\) years.#

The right panel of Fig. 74 shows a different way of visualizing this, using the average temperature at various depths of the model (this is what the depth average postprocessor computes). The figure shows how the initially linear unstable layering almost immediately reverts completely, and then slowly equilibrates towards a temperature profile that is constant throughout the mantle (which in the incompressible model chosen here equates to an adiabatic layering) except for the boundary layers at the inner and outer boundaries. (The end points of these temperature profiles do not exactly match the boundary values specified in the input file because we average temperatures over shells of finite width.)

A conclusion of this discussion is that if we want to evaluate the statistical properties of the flow field, e.g., the number of plumes, average velocities or maximal velocities, then we need to restrict our efforts to times after approximately 800 million years in this simulation to avoid the effects of our inappropriately chosen initial conditions. Likewise, we may actually want to choose initial conditions more like what we see in the model for later times, i.e., constant in depth with the exception of thin boundary layers, if we want to stick to incompressible models.

Material model.#

The model we use here involves viscosity, density, and thermal property functions that do not depend on the pressure, and only the density varies (slightly) with the temperature. We know that this is not the case in nature.

Shear heating.#

When we set up the input file, we started with a model that includes the shear heating term \(2\eta \varepsilon(\mathbf u):\varepsilon(\mathbf u)\) in (3). In hindsight, this may have been the wrong decision, but it provides an opportunity to investigate whether we think that the results of our computations can possibly be correct.

We first realized the issue when looking at the heat flux that the heat flux statistics postprocessor computes. This is shown in the left panel of Fig. 75.3 There are two issues one should notice here. The more obvious one is that the flux from the mantle to the air is consistently higher than the heat flux from core to mantle. Since we have no radiogenic heating model selected (see the List of model names parameter in the Heating model section of the input file; see also Heating model), in the long run the heat output of the mantle must equal the input, unless is cools. Our misconception was that after the 800 million year transition, we believed that we had reached a steady state where the average temperature remains constant and convection simply moves heat from the core-mantle boundary the surface. One could also be tempted to believe this from the right panel in Fig. 74 where it looks like the average temperature does at least not change dramatically. But, it is easy to convince oneself that that is not the case: the temperature statistics postprocessor we had previously selected also outputs data about the mean temperature in the model, and it looks like shown in the left panel of Fig. 76. Indeed, the average temperature drops over the course of the 1.2 billion years shown here. We could now convince ourselves that indeed the loss of thermal energy in the mantle due to the drop in average temperature is exactly what fuels the persistently imbalanced energy outflow. In essence, what this would show is that if we kept the temperature at the boundaries constant, we would have chosen a mantle that was initially too hot on average to be sustained by the boundary values and that will cool until it will be in energetic balance and on longer time scales, in- and outflow of thermal energy would balance each other.

figure

Fig. 75 Simple convection in a quarter of an annulus. Left: Heat flux through the core-mantle and mantle-air boundaries of the domain for the model with shear heating. Right: Same for a model without shear heating.#

figure

Fig. 76 Simple convection in a quarter of an annulus. Left: Average temperature throughout the model for the model with shear heating. Right: Same for a model without shear heating.#

However, there is a bigger problem. Fig. 75 shows that at the very beginning, there is a spike in energy flux through the outer boundary. We can explain this away with the imbalanced initial temperature field that leads to an overturning and, thus, a lot of hot material rising close to the surface that will then lead to a high energy flux towards the cold upper boundary. But, worse, there is initially a negative heat flux into the mantle from the core – in other words, the mantle is losing energy to the core. How is this possible? After all, the hottest part of the mantle in our initial temperature field is at the core-mantle boundary, no thermal energy should be flowing from the colder overlying material towards the hotter material at the boundary! A glimpse of the solution can be found in looking at the average temperature in Fig. 76: At the beginning, the average temperature rises, and apparently there are parts of the mantle that become hotter than the 4273 K we have given the core, leading to a downward heat flux. This heating can of course only come from the shear heating term we have accidentally left in the model: at the beginning, the unstable layering leads to very large velocities, and large velocities lead to large velocity gradients that in turn lead to a lot of shear heating! Once the initial overturning has subsided, after say 100 million years (see the mean velocity in Fig. 74), the shear heating becomes largely irrelevant and the cooling of the mantle indeed begins.

Whether this is really the case is of course easily verified: The right panels of Fig. 75 and Fig. 76 show heat fluxes and average temperatures for a model where we have switched off the shear heating by setting

subsection Heating model
  set List of model names =
end

Indeed, doing so leads to a model where the heat flux from core to mantle is always positive, and where the average temperature strictly drops!

Summary.#

As mentioned, we will address some of the issues we have identified as unrealistic in the following sections. However, despite all of this, some things are at least at the right order of magnitude, confirming that what is computing is reasonable. For example, the maximal velocities encountered in our model (after the 800 million year boundary) are in the range of 6–7cm per year, with occasional excursions up to 11cm. Clearly, something is going in the right direction.


1

In YouTube, click on the gear symbol at the bottom right of the player window to select the highest resolution to see all the details of this video.

2

Note that the density in 2d has units \(\text{ kg}\,\text{ m}^{-2}\)

3

The heat flux statistics postprocessor computes heat fluxes through parts of the boundary in outward direction, i.e., from the mantle to the air and to the core. However, we are typically interested in the flux from the core into the mantle, so the figure plots the negative of the computed quantity.