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Projects
Scalable Modeling of Mantle Dynamics
ProblemMantle dynamics is the only theory that can elucidate the earliest origins of our planet, describing the motion of tectonic plates, origin of mountain belts and rift zones, and spectacular physics at subduction zones, such as great earthquakes and volcanism. High resolution simulations combined with the explosion of observational data is creating an enormous opportunity for scientific advancement.
ApproachMany papers from the late 1990s and early 2000s show that there exist efficient multigrid smoothers for the Stokes equation. However, these have fallen out of favor, I think largely due to insufficient software support in that linear algebra packages do not integrate topological information. In addition, there has been much more work on block Schur complement methods for nonlinear Stokes problems. We will test Vanka and Braess-Sarazin smoothers against Schur complement methods for the variable-viscosity Stokes problem arising from mantle dynamics.
ImpactWe are seeking to create a large, scale thermal and stress map of the Earth's mantle, which has implications for geochemistry, volcanology, petrology, and economic geophysics. The coupling of regional and global scales, through inverse problems and sensitivity analysis, will enable scientists to look precisely at places on earth, rather than cartoons which lack the geometric and physical detail to trace develop over geologic time.
PlanWe have already constructed the infrastructure for linear Vanka and Braess-Sarazin preconditioners for the Stoeks equations. Going forward, we will scale these implementations to very large problems, as well as extend the implementations to nonlinear problems.
- Inversion for Temperature in Mantle Dynamics
The new thing here for DD could be the multilevel solution on the control problem. We could do the level solves using the DD that Xiao-Chuan has shown. The adjoint equation is linear, but we would need nonlinear FIELDSPLIT in order to smooth the three fields independently.
Start with TAO ex1, which is working. Get the proposed smoother working there.
Create another simple example with a nonlinearity, such as nonlinear conductivity..
- Sensitvity Analysis in Mantle Dynamics
A beginning problem is to look at the effect of introducing a subducted Yakatat plateau on the predicted upthrust of the Chugash? mountains. In earlier work, Jadamec[?] accurately predicted uplift in the Chugash? Range as well as subsidence of the Cook Inlet Basin based upon dynamic topography calculations from regional mantle convection simulations. However, the extent of subsidence was overpredicted, and this could be related to the absence of a Yakutat plateau in the model. This problem could, of course, be attacked by running yet more simulations. However, we will try and disentangle it by using sensitivity analysis. We will look at the sensitivity of the integrated normal traction on a section of the boundary to a change in the density of a section of our regional volume.
Work Steps:
- Margarete: Debug mantle rheology in SNES ex69
- Matt: Calculate normal traction over a labeled section of the boundary
This can be verified by computing the analytical traction from a known solution. This seems easiest for a solution in SNES ex62.
- Margarete/Matt: Add density parameter to the code
The right way to do this is probably to use the Boussinesq approximation, so that density difference result only in a change in bouyancy, and meaning that the field we want is likely \(\Delta\rho\). This means that the density parameter field would be used only in an \(f_0\) function
- Matt: Get forward sensitivity of all tractions to scalar density parameter
The forward sensitvity of our equation \( F(u, u_x, t; p) = 0 \) is given by \( 0 &= \frac{dF}{dp} \\ &= \frac{\partial F}{\partial u} \frac{du}{dp} + \frac{\partial F}{\partial p} \\ \frac{du}{dp} &= -\frac{\partial F}{\partial u}^{-1} \frac{\partial F}{\partial p} \) The direct dependence on the parameter comes only from the bouyancy term, so this amount to a Jacobian solve at the current solution using the bouyancy (without \(\Delta\rho\)) as the driving term. Since traction is a linear functional, the sensitvity of the traction is just \(\psi\left(\frac{du}{dp}\right\).
We could get the sensitivity of the traction on each face instead of the integrated traction just by choosing \(n\) functionals, but that does not seem to give us much more information.
- Matt: Get adjoint sensitivity of integrated traction to all density parameters
We can go back to the defintiion of the sensitivity \( \frac{d\psi}{dp} &= \frac{d\psi}{du} \frac{du}{dp} \\ &= -\frac{d\psi}{du} \frac{\partial F}{\partial u}^{-1} \frac{\partial F}{\partial p} \\ \) Now, instead of solving the Jacobian system for each \(p\), we could instead associate left, and look at the adjoint equation \( \frac{\partial F}{\partial u}^T \lambda = -\frac{d\psi}{du} \) so that \( \frac{d\psi}{dp} = \lambda^T \frac{\partial F}{\partial p} \\ \) Thus we can do one solve using the adoint operator, and find the sensitvity to many parameters, but we have to do that for each functional. If the functional is linear, the value \(\psi(u) = v^T u\) for some vector \(v\), which is the rhs.
Resources
- Short note on sensitivity analysis by Linda Petzold.
- Impact of Large Earthquakes on Mantle Flow
Setup mantle flow in Japan, Alaska, Chile, or Cascadia. Run with long term continental drift, and run with earthquakes added in (from catalogue or jsut double-couple sources). Look at flow differences.
Maxwell times for mantle can be 10,000 years, so the effect should not relax out.
- How did Plate Tectonics begin and what are the organizing principles?
I think that it is possible that biological systems has has a large effect on lithospheric evolution, by organizing the process of plate tectonics. For instance, could chemotaxic bacteria living at depth have altered chemical gradients on a large scale which resulted in certain patterns of spreading, subduction, and plate boundary evolution.
A particular thing I was thinking of is the salinity instability near a nid-ocean ridge that Rich was simulating. Can we include a biological component in the simulation as another influence on the chemical concentration?