Geophysical setups

Geophysical setups#

Having gone through the ways in which one can set up problems in rectangular geometries, let us now move on to situations that are directed more towards the kinds of things we want to use ASPECT for: the simulation of convection in the rocky mantles of planets or other celestial bodies.

To this end, we need to go through the list of issues that have to be described and that were outlined in How to set up computations, and address them one by one:

  • What internal forces act on the medium (the equation)? This may in fact be the most difficult to answer part of it all. The real material in Earth’s mantle is certainly no Newtonian fluid where the stress is a linear function of the strain with a proportionality constant (the viscosity) \(\eta\) that only depends on the temperature. Rather, the real viscosity almost surely also depends on the pressure and the strain rate. Because the issue is complicated and the exact material model not entirely clear, for the next few subsections we will therefore ignore the issue and start with just using the “simple” material model where the viscosity is constant and most other coefficients depend at most on the temperature.

  • What external forces do we have (the right hand side) There are of course other issues: for example, should the model include terms that describe shear heating? Should it be compressible? Adiabatic heating due to compression? Most of the terms that pertain to these questions appear on the right hand sides of the equations, though some (such as the compressibility) also affect the differential operators on the left. Either way, for the moment, let us just go with the simplest models and come back to the more advanced questions in later examples.

    One right hand side that will certainly be there is that due to gravitational acceleration. To first order, within the mantle gravity points radially inward and has a roughly constant magnitude. In reality, of course, the strength and direction of gravity depends on the distribution and density of materials in Earth – and, consequently, on the solution of the model at every time step. We will discuss some of the associated issues in the examples below.

  • What is the domain (geometry)? This question is easier to answer. To first order, the domains we want to simulate are spherical shells, and to second order ellipsoid shells that can be obtained by considering the isopotential surface of the gravity field of a homogeneous, rotating fluid. A more accurate description is of course the geoid for which several parameterizations are available. A complication arises if we ask whether we want to include the mostly rigid crust in the domain and simply assume that it is part of the convecting mantle, albeit a rather viscous part due to its low temperature and the low pressure there, or whether we want to truncate the computation at the asthenosphere.

  • What happens at the boundary for each variable involved (boundary conditions)? The mantle has two boundaries: at the bottom where it contacts the outer core and at the top where it either touches the air or, depending on the outcome of the discussion of the previous question, where it contacts the lithospheric crust. At the bottom, a very good approximation of what is happening is certainly to assume that the velocity field is tangential (i.e., horizontal) and without friction forces due to the very low viscosity of the liquid metal in the outer core. Similarly, we can assume that the outer core is well mixed and at a constant temperature. At the top boundary, the situation is slightly more complex because in reality the boundary is not fixed but also allows vertical movement. If we ignore this, we can assume free tangential flow at the surface or, if we want, prescribe the tangential velocity as inferred from plate motion models. ASPECT has a plugin that allows to query this kind of information from the GPlates program.

  • How did it look at the beginning (initial conditions)? This is of course a trick question. Convection in the mantle of earth-like planets did not start with a concrete initial temperature distribution when the mantle was already fully formed. Rather, convection already happened when primordial material was still separating into mantle and core. As a consequence, for models that only simulate convection using mantle-like geometries and materials, no physically reasonable initial conditions are possible that date back to the beginning of Earth. On the other hand, recall that we only need initial conditions for the temperature (and, if necessary, compositional fields). Thus, if we have a temperature profile at a given time, for example one inferred from seismic data at the current time, then we can use these as the starting point of a simulation.

This discussion shows that there are in fact many pieces with which one can play and for which the answers are in fact not always clear. We will address some of them in the cookbooks below. Recall in the descriptions we use in the input files that ASPECT uses physical units, rather than non-dimensionalizing everything. The advantage, of course, is that we can immediately compare outputs with actual measurements. The disadvantage is that we need to work a bit when asked for, say, the Rayleigh number of a simulation.