“Anti-Squeeze” for Mantle Convection Simulations in 2-D Spherical Geometries

Hi all,

I have been reading this poster presented by Paul Tackley at EGU 2020 on the importance of
“Anti-Squeeze” for Mantle Convection Simulations in 2-D Spherical Geometries.

Does ASPECT have such a feature, I can’t see something like this in the manual?

Thanks,
Elodie

Hello Elodie,

This is an interesting question to discuss here. My understanding is that the 2D “sphere” or 2D “chunk” geometry is actually a cylindrical geometry, not a sphere. I think this has something to do with how this geometry is set-up in deal.II and Aspect inherits this. Maybe someone else can confirm.

So, I think the squeezing issue is not actually an issue in Aspect. However, there is another issue using cylindrical geometry, which is that surface area of an element at the core mantle boundary is larger than it would be for a spherical geometry, so if heat flux from the core is important to your problem, you also need to make adjustment for the radius of the core when using a cylindrical geometry.

I think the best combination is a 2D spherical annulus with the “anti-squeeze” for the stresses. I would very interested in discussion of what would be involved to make that possible in Aspect?

Magali

Dear Elodie,

ASPECT doesn’t have 2D spherical geometries, so it also has no need for “anti-squeeze”. See this lengthy exchange from 2018: [aspect-devel] Cylindrical coordinates

Here is the conclusion from Wolfgang:

This is not a frequently requested feature (I’m sure this has come up a couple of times over the ~8 years of ASPECT, but not much more often), and so it is not implemented. It is simply a question of who takes the time to implement it and find all of the places where one needs to change things…

P.S. I’m not sure whether there’s been a study on the differences between 2D cylindrical and 3D spherical.

  • The van Keken (2001) paper deals only with the difference between 3D axisymmetric and 2D cylindrical.
  • The Hernlund and Tackley (2008) paper only compared full 3D spherical, 2D spherical annulus and 2D rescaled cylindrical geometries (and their new 2D spherical annulus was without anti-squeeze).

Anti-squeeze also has implications for heat flow. It’s not entirely clear to me where the biggest benefits of 2D spherical annulus+anti-squeeze lie relative to 2D cylindrical.

Bob already helpfully pointed out the right thread from several years ago. The important point is that there are many different ways in which one can think about what exactly a 2d simulation should actually represent. The world is three-dimensional, and what ASPECT assumes is that in a 2d x-y (or r-phi) simulation, the velocity in the third Cartesian dimension (z) is zero. An alternative point of view would assume that the velocity in the second angle variable (psi) is zero. These are not equivalent. The point is simply that different people have different views/different situations require different solutions for how the 3d world should be represented in 2d simulations. ASPECT implements one in which the geometric shortening is not relevant.

A bit about the choice of coordinates can be found in this mail in the thread already pointed out: [aspect-devel] Cylindrical coordinates

Best
W.

Oh I see! Thanks for all your helpful comments! :slight_smile:

There are a couple conceptual issues involved that need to be separated:

(1) Basic Compression (Pressure): All features that sink/rise in a spherical planet are subject to increased/decreased (respectively) lateral compression (from both lateral directions) and radial compression (bearing the weight of the overlying matter), this is clear. The way that pressure/density/temperature scales isentropically in a spherical annulus is identical to the full 3D spherical planet. On the other hand, the way that these quantities scale in a cylindrical model is not realistic, the basic (reference) state equations in 1D (radius) aren’t the same for a cylinder as they are for a sphere. This is relevant to “compressible” models or in other scenarios where an accurate representation of the reference state is important.

(2) Deviatoric Stress: Assumptions about the form of 3D downwelling carries implications for the kinds of deviatoric stresses that are induced by sinking/rising motions alone (i.e., those that arise purely from radial motion by itself without shear coupling to the surroundings). This is what Paul’s “anti-squeeze” is concerned with, the elimination of induced deviatoric stresses upon compression for certain kinds of motions/planforms. It may be important if one is implementing stress-dependent rheologies that manifest over a wide depth range in the mantle (his group has been working on these kinds of problems). I’m not yet convinced that it is important for any other reason, but there may be some cases where this arises.

There is also a philosophical issue, which is that one has to consider how much effort they will put into modestly important high-order corrections for models having significant low-order uncertainties. For example, we know very little about deep mantle rheology, so in the big picture does it really matter if something far less important is handled with great precision? On the other hand, if the way the model is constructed requires certain accommodations in order to avoid introducing unwanted artifacts (Tackley’s proper anti-squeeze motive), then it is important to take care of it. This is the “art” of modeling.

  • The Hernlund and Tackley (2008) paper only compared full 3D spherical, 2D spherical annulus and 2D rescaled cylindrical geometries (and their new 2D spherical annulus was without anti-squeeze).

We also included 2D axisymmetric cases, like the old school Zebib, Schubert, et al. models.

PS: I came up with the spherical annulus as an undergrad, presented it to the UCLA geodynamics group during my PhD, and most of them hated it…but with >100 citations on the paper the idea must be useful to some people.

Hi John,

Thanks for elaborating!

I wanted to particularly thank you for pointing out the issue with compressible cylindrical models. Highly viscous inclusions in a truly cylindrical compressible simulation should also get “stuck” at a particular depth, because they have to undergo deviatoric strain in order to compress or expand. However, in ASPECT, an “anti-stretch correction” is already done (see Section 2.1.3 of the manual). Essentially, this correction relaxes the mass-conservation requirement in favour of zeroing the ?z-components of the stress tensor (this is a bad explanation… I’ll come up with a better one at some point). I think we could probably make this clearer in the text. What do the other developers think?

In writing the above, I thought of a possible alternative to the spherical annulus that might be more straightforward to implement in ASPECT. I haven’t worked through any of the implications, so please shoot me down if I missed something! The spherical annulus + anti-squeeze is designed to correct for the fact that in 3D spherical geometries, we have:
dVs/dr = 4 pi r^2
whereas in 3D cylindrical geometries, we have
dVc/dr = 2 pi r * a

Perhaps a “negative compressibility” term could be added to the mass conservation equation to account for this? Then downwellings will expand in the 2D plane. I think the term would look like this:
drho / dr = 1 * rho / r (we’d need to double check the constant)
This term would only be used in the mass conservation equation; it wouldn’t actually be used to modify the density. ASPECT’s “anti-stretch” should already deal with the stress (but we should be more explicit about what discarding the edot_zz term means for mass conservation).