From: Wolfgang Bangerth Date: Sat, 12 Sep 2009 04:41:02 +0000 (+0000) Subject: Talk about what one would have to change to get the pressure right. X-Git-Tag: v8.0.0~7130 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=8f2430cc020be64b12709fc96dc73bf00d8eb80a;p=dealii.git Talk about what one would have to change to get the pressure right. git-svn-id: https://svn.dealii.org/trunk@19443 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-32/doc/results.dox b/deal.II/examples/step-32/doc/results.dox index dce0e7882e..032eebe8ab 100644 --- a/deal.II/examples/step-32/doc/results.dox +++ b/deal.II/examples/step-32/doc/results.dox @@ -344,4 +344,37 @@ in the @ref step_31 "step-31", here are a few more ideas: (downward) it will be compressed and because in that case $\mathbf u \cdot \mathbf g > 0$ we get a positive heat source. Conversely, the fluid will cool down if it moves against the direction of gravity. + + Implementing this requires one additional step, however. As mentioned in the + introduction, we use a rather simplified model for gravity in which the + gravity force diminishes with depth, as if Earth had a homogenous + density. That isn't the case, however: the earth core is much denser than + the mantle, and gravity actually tops out at the core + mantle boundary. A consequence of this is that our computations predict + pressures around 80 GPa at the core mantle boundary, whereas in reality the + value is closer to 140 GPa. With a pressure wrong by so much, we can't + expect compression heating to be accurate. + + A more realistic model for the gravity vector in the program would take the + spatially variable density into account, and that wouldn't actually be + terribly complicated: by integrating the PDE for the gravity potential under + the assumption that $\rho(\mathbf x)=\rho(r)$, we get + @f[ + \varphi(r) = 4\pi G \int_0^r \frac 1{t^2} \int_0^t s^2 \rho(s) \; ds \; dt, + @f] + and consequently for the gravity vector + @f[ + \mathbf g = - 4\pi G \frac 1{r^2} \left( + \int_0^r s^2 \rho(s) \; ds \right) + \frac{\mathbf x}{\|\mathbf x\|}. + @f] + This expression reduces to the one we use for the case that the density is + constant, but a more complete model would, for example, assume that the + density varies with the radius (in the simplest case it could be constant in + various layers). In either case, it can relatively easily be evaluated if + for non-trivial models of $\rho(r)$. Of course, a really complete model + would consider that $\rho$ can also vary in the tangential direction, for + example in a time dependent way as a consequence of the thermal expansion of + rocks as a result of the convection. Taking into account this self + gravitational effect of convection would be much harder, however.