1. Magma Migration Applied to Oceanic Ridges Geophysical Porous Media Workshop Project Josh Taron - Penn State Danica Dralus - UW-Madison Selene Solorza- UABC - Mexico Jola Lewandowska - UJF France Angel Acosta-Colon - Purdue 2 M.I.A.s Advisors: Scott King & Marc Spiegelman Core Crust and Lithosphere (~100km) (~3000km) Magma Migration Applied to Oceanic Ridges

2. 1.Plate Tectonics Intro (Angel) 2.Magma Migration (Danica) 3.Solitary Waves (Selene) 4.Modeling Results (Josh) Outline

3. Magma Migration Applied to Oceanic Ridges Earthquakes

4. Magma Migration Applied to Oceanic Ridges Volcanoes

5. Magma Migration Applied to Oceanic Ridges Plate Tectonics Boundaries

6. Magma Migration Applied to Oceanic Ridges Types of Boundaries

7. Magma Migration Applied to Oceanic Ridges Plate Tectonics Boundaries Earth is divided into dynamics rigid plates. The plates are continuously created and “recycled”. Magma migration affects the plates evolution. In ocean ridges, the magma will control the geochemical evolution of the planet and fundamentals of the plate tectonics dynamics.

8. Magma Migration Applied to Oceanic Ridges Core Crust and Lithosphere (~100km) (~3000km)

9. Magma Migration Applied to Oceanic Ridges So, what makes magma migration strange? Localized Flow

10. Magma Migration Applied to Oceanic Ridges Supporting Evidence (an example) MORBs are typically undersaturated in OPX. OPX is plentiful in the mantle and dissolves quickly in undersaturated mantle melts. Observations suggest MORBs travel through at least the top 30 km of oceanic crust without equilibrating with residual mantle peridotite. MORBs are also not in equilibrium with other trace elements.

11. Magma Migration Applied to Oceanic Ridges Implications? USGS National Geographic BBC News

12. Magma Migration Applied to Oceanic Ridges What do we need for a working theory? At least 2 phases (melt and solid) Allow mass-transfer between phases (melting/reaction/crystallization) System must be permeable at some scale System must be deformable (consistency with mantle convection) Chemical Transport in open systems

13. Magma Migration Applied to Oceanic Ridges Governing Equations

14. Magma Migration Applied to Oceanic Ridges Compressible Flow Equations (No Shear, No Melting)

15. Magma Migration Applied to Oceanic Ridges Dimensionless Compressible Flow Equations That is, porosity only changes by dilation/compaction. The compaction rate is controlled by the divergence of the melt flux and the viscous resistance of the matrix to volume changes.

16. Magma Migration Applied to Oceanic Ridges Solitary Waves

17. Magma Migration Applied to Oceanic Ridges History On August 1834 the Scottish engineer John Scott Russell (1808-1882) made a remarkable scientific discovery: The solitary wave. Russell observed a solitary wave in the Union Canal, then he reproduced the phenomenon in a wave tank, and named it the “Wave of Translation.

18. Magma Migration Applied to Oceanic Ridges History Drazin and Johnson (1989) describe solitary wave as solutions of nonlinear Ordinary Differential Equations which: 1.Represent waves of permanent form; 2.Are localized, so that they decay or approach a constant at infinity; 3.Can interact with other solitary waves, but they emerge from the collision unchanged apart from a phase shift.

19. Magma Migration Applied to Oceanic Ridges Then substituting eq. (1) into (2), we have 1-D Magmatic Solitary Wave where is porosity and C is the compaction rate.

20. Magma Migration Applied to Oceanic Ridges By the chain rule Where is the distance coordinate in a frame moving at constant speed c.  Assuming a solution of the form 1-D Magmatic Solitary Wave

21. Magma Migration Applied to Oceanic Ridges From eq. (1) and (5), the compaction rate satisfies Thus, eqs. (1) and (2) are transformed into the non-linear ODE 1-D Magmatic Solitary Wave

22. Magma Migration Applied to Oceanic Ridges Animation of the collision of the solitary wave (From Spiegelman) 1-D Magmatic Solitary Wave For n=3, using the second order Runge-Kutta numerical method to solve the 1-D magmatic solitary wave eq. (4) for periodic boundary conditions and initial conditions:

23. Magma Migration Applied to Oceanic Ridges Modeling Results

24. Magma Migration Applied to Oceanic Ridges How do behaviors vary? The simplest case: –Convection/Conduction transport – No mechanical considerations (uncoupled) Coupled examples: –Elastic systems:The Mendel-Cryer effect –Viscous systems: The solitary wave Fluid-Mechanical Coupling

25. Magma Migration Applied to Oceanic Ridges Convection/Conduction Transport Homogeneous porosity No mechanical considerations

26. Magma Migration Applied to Oceanic Ridges Velocity Field Low Porosity Region Convection/Conduction Transport in Heterogeneous Media A bit more exciting No mechanical considerations

27. Magma Migration Applied to Oceanic Ridges Darcy flow with Convection/Conduction to track magma location Level Set Media Magma Smoothing Function Coupling: Convection Velocity = Darcy Velocity Why COMSOL? Starting from scratch…time constraints A bit about the method so far…

28. Magma Migration Applied to Oceanic Ridges What about mechanical coupling? Does it dramatically change the system? 1.The elastic scenario (near surface) 2.The viscous scenario (way down there)

29. Magma Migration Applied to Oceanic Ridges Elastic Systems: The Mendel-Cryer Effect Images from Abousleiman et al., (1996). Mandel’s Problem Revisited. Géotechnique, 46(2): 187-195. Mandel, J. (1953). Consolidation des sols (étude mathématique). Géotechnique, 3: 287-299. Skempton, A.W. (1954). The pore pressure coefficients A and B. Géotechnique, 4: 143-147. Described by Biot Theory (Linear Poroelasticity) Verified in laboratory and at field scale Is well defined (unlike for a viscous medium) and pressure effects of a similar response will alter behavior of fluid transport (coupled system)

30. Magma Migration Applied to Oceanic Ridges And the viscous scenario… Recall the derivation for coupled flow and deformation in a viscous porous medium No need for level-set What are the mechanical effects? –Remember the solitary wave Neglects melting (reaction)

31. Magma Migration Applied to Oceanic Ridges Fluid-Mechanical in a Viscous Medium: Solitary Wave The mathematics are well posed. Does this actually occur?? In the second video, the matrix is allotted a downward velocity. Watch for the phase shift.

32. Magma Migration Applied to Oceanic Ridges 3D Solitary Waves From Wiggings & Spiegelman, 1994, GRL

33. Magma Migration Applied to Oceanic Ridges What would we like to do? Couple the reaction equation (mass transfer)… …to the fluid-mechanical viscous medium derivation System mimics the “salt on beads” interaction Da(R) = Damkohler Number (relation of reaction speed to velocity of flow) A = Area of Dissolving phase (matrix) available to reaction c f eq -c f = Distance of reacting solubility (i.e. melting solid fraction in molten flow) from equilibrium

34. Magma Migration Applied to Oceanic Ridges What would we like for that to look like?

35. Magma Migration Applied to Oceanic Ridges What does it look like? What do we need to make it work? 1.Time 2. Bigger computer 3. Sanity 4. Siesta 5. Beer The backup plan…

36. Magma Migration Applied to Oceanic Ridges Applying the level set method from before… Adding reaction (melting) the result becomes

37. Magma Migration Applied to Oceanic Ridges Concluding Remarks (in picture form) Fluid only Fluid only Fluid/Mechanical Fluid/Reactive (melt)

38. Magma Migration Applied to Oceanic Ridges The End… Questions?