Tuesday, March 26, 2013

Geophysics Problem of the Day: Mantle Couette Flow



Geophysics Problem of the day: one-dimensional flow with varying viscosity and shear stress

I still have vivid and occasionally lucid dreams even 25+ years after my college roommate and I stopped telling each other our dreams every morning. In a recent dream, I was trying to set up this problem on the blackboard, but my legs kept buckling out from under me.

Here’s the picture I was trying to draw (in my dream).
Couette Flow Cartoon
 
Here’s today’s Geophysics Problem of the Day:
Part 1. Remind yourself how to derive Couette Flow (1-dimensional flow between plates). For the first part, assume that the viscosity of the fluid is constant. Wikepedia is a good help on this. Derive formulas for velocity as a function of depth and shear stress as a function of depth.

Part 2. Using your derived relationships, determine velocity and shear stress as a function of depth for the Earth’s upper mantle, assuming that the lower mantle is fixed. Also calculate accumulated strain over a ~100 million year timescale as a function of depth.Use typical plate velocities (1-10 cm/yr) and mantle viscosities (~1019-1020 Pa s)

In real materials, viscosity is strongly dependent on temperature, with the following general relationship:
where THom is the “homologous” temperature: the ratio of the actual temperature to the melting point. Using the homologous temperature is important here, because is shows that for many solids the viscosity—resistance to flow—decreases as the melting temperature is approached. Below the lithosphere, the temperature increases with depth through the mantle. But likely the melting temperature also increases with depth. So calculating homologous temperature as a function of depth is easy to say, but hard to do.

Part 3. Rederive Couette-style flow using a depth-dependent viscosity, and use these relationships to calculate velocity, shear stress, and accumulated strain as a function of depth for Earth’s upper mantle and transition zone using a depth-dependent viscosity. Try a depth-dependent viscosity model such as Peltier, Science 1996.

Part 3. Rederive Couette-style flow using a depth-dependent viscosity, and use these relationships to calculate velocity, shear stress, and accumulated strain as a function of depth for Earth’s upper mantle and transition zone using a depth-dependent viscosity. Try a depth-dependent viscosity model such as Peltier, Science 1996. 

Mantle Couette Cartoon.

Part 4. Viscosity describes a material's ability to support a shear stress over a long timescale. During a solid-state phase transformation, the ability of a material to support a shear stress is expected to approach zero at equilibrium (=long time scales). In the Earth, the top and bottom of the transition zone correspond to major mineralogical phase transitions. What if these transformations significantly reduce the viscosity in thin shells at the top and bottom of the transition zones? To examine the potential effects of this, calculate the depth-dependent velocity, shear stress, and accumulated strain for a constant viscosity mantle with two sharp lower-viscosity zones, corresponding to the top and bottom of the transition zone, as shown in the picture at left. You might want to do this for different values of low-viscosities. How high does it need to be to avoid an effective "free-slip" boundary condition, thereby ruining the flow for everything underneath?

Please send me your solutions!

Note 1: If this problem is too boring, then try it in a spherical coordinate system, with the outside surface of the sphere forced to rotate with an angular velocity, and a fixed small interior volume.

Note 2: If you need additional information or values such as constants or material properties don’t be afraid to look them up and/or make them up.

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