Introduction: Plate tectonics has been regarded as a scientific revolution. If so one might argue that it is incomplete. While we have a detailed understanding of how the plates behave, the relationship between plate motions and convection in the mantle is still vigorously debated. This is evident in the ongoing discussion about mantle plumes and hot spot trails. New information from seismic tomography and modeling capacities promise to provide insight in the near future. As we expand our knowledge in science we also expand the boundaries of the unknown. Convection is also a fundamentally important process that controls much of the dynamics of atmospheric, ocean, mantle and outer core systems.

Image to right depicting top to bottom mantle convection from http://geomag.usgs.gov/about.php .

Two basic simple end-member models:

• coupled: currents in the mantle raft the overlying plates around. Traction stresses at the base of the plates would be critical. This could be akin to sea ice being moved by underlying currents in the sea water.
• decoupled: plates move due to internal body forces, and influence the shallow convection current pattern in the mantle.
• locally exclusive, but not globally.

Schematic diagram of coupled (left) and decoupled (right) models. In the first case the motion in the asthenosphere rafts the plate above along. The plate would move slower than the underlying asthenosphere, and the relative sense of motion between the two would be as the blue arrows show. In the other case some other force such as trench pull would pull the lithosphere and drag would induce movement in the asthenosphere below, which would diminish downwards, and the relative sense of motion between the two would be the opposite. The second possibility ignores the possibility that some other motion could exist in the asthenosphere due to mantle convection, and treats the asthenosphere as passive. In other words, the arrows in the decoupled case only show the movement component in the asthenosphere that would be created by drag as the over riding plate moves and not movements due to other causes.

Possible driving forces for plate tectonics:

• bottom lithosphere tractions by convection currents.
• trench pull (covered earlier).
• ridge push (sliding off a high, crust in compression).
• trench suck (rollback).
• global expanding or contracting forces.
• membrane forces on spinning ellipsoid (e.g. variants of polar fleeing forces).

Lava lake videos that demonstrate possibilities on a small scale:

• http://www.youtube.com/watch?v=PZjPGdXMMso.
• http://www.youtube.com/watch?v=PbG-8URKnm8 .
• http://www.youtube.com/watch?v=KARzrNa7y_4 .

Conduction vs. convection vs. advection. Advection is often assumed to be absent in the mantle, but is it (partial melt migration)? We won't have time to really address that here. Realize that the term advection is used in different ways.

Buoyancy is driven by gravity acting on density contrasts caused by thermal differences and by phase changes. Both are important in earth's mantle.

Importance of Clapeyron slope: For most mineral transformations the transformation pressure increases with increasing T (i.e. a positive Clapeyron slope). This means that in a colder mantle region the transformation can occur at a shallower level and in a hotter region it occurs at a deeper level. We already considered this in the context of the olivine to spinel transition in a subducting slab and the mechanism of trench pull. We can also consider thermal plumes. If there is a density increase in a colder region of convective downwelling, the phase transition will occur at a elevated level producing a negative buoyancy and enhancement of downwelling would be expected. By the same line of reasoning a hot spot a positive bouynacy force would exit. In this case a plume may be self perpetuating once it forms. Which is consistent with the long history of some mantle plumes (see below). In fact you could ask why would a mantle plume ever die? What is the slope for the 670 km, spinel to perovskite boundary?

Mantle viscosity: a most crucial parameter!!

viscosity

• resistance to flow.
• technically it is ratio of deviatoric stress (measure of size of shear stress) over shear strain. There are different types of viscosity defined in slightly different ways (see Davies for an explanation). There are also different stress-strain viscosity relationships - linear (Newtonian) vs. power law.
• higher the number the more viscous a material is.
• typical units are Pascals per second.
• for kinematic viscosity units are meters squared per second.

Shallow mantle viscosity dominated by olivine rheology.

Mantle material is of varying viscosity: lithosphere vs. asthenosphere, and upper vs. lower mantle. Is there a viscosity contrast across the 370 km phase change boundary?

Estimations of mantle viscosity from glacio-isostatic rebound:

• rebound history takes an exponential form.
• deformation due to ice load will penetrate to depth comparable to its radius (Davies).
• some larger loadings (bigger ice sheets) reach into lower mantle.
• lower mantle viscosity of 6*E+21Pa-s, upper mantle viscosity of 3*E+20 Pa-s. This is a large viscosity contrast.

Significance of the Rayleigh number in convection behavior:

• balance between buoyancy and viscous forces in a simple layer of thickness D. Dimensionless number.
• input parameters:
  • coefficient of thermal expansion.
  • temperature gradient in excess of that associated with increasing pressure (superadiabatic component).
  • gravitational acceleration.
  • thermal diffusivity (how fast heat moves by conduction)
  • thickness of convecting fluid = d.
  • kinematic viscosity
• Ra = gravitational acceleration * density * volume coefficient of thermal expansion * temperature of interior fluid * depth of layer cubed all divided by the thermal diffusivity * viscosity (Davies)
• note that as viscosity increases Ra decreases and convection less likely. As thermal diffusivity increases Ra decreases and convection less likely.
• for a given geometry there is a number that needs to be exceeded for convection to occur. For the spherical mantle this value is 2380 (Keary & Vine). As R increases there are also different styles/geometries of convection.
• estimates for even the lower mantle yield a Ra value of circa 3*E6. In other words buoyancy forces far overpower viscous retarding forces and convection should be vigorous. See Turcotte or Davies for a fuller explanation.

Geometries of convection:

• Image to right from http://pubs.usgs.gov/gip/dynamic/unanswered.html, shows a very simple large scale convection cells. How is this simple depiction certainly wrong in detail?? What doesn't it take into account?
• parallel elongate cylindrical cells (sheet upwell and downwelling).
• hexagonal patterns
• spokes, plumes.
• bimodal patter of perpendicular cells.
• stable vs. unstable (turbulent) patterns.
• pattern a function of the Rayleigh number.
• very tall or very wide circulation cells are not stable (excepting plumes). Often a width to depth ratio of 2 for cells for earthly conditions.
• below is an experimentally produced convection pattern. How would you describe the pattern? Image is from http://www.etl.noaa.gov/about/eo/science/convection/Table.html .
• 

Hot spots:

• well known examples: Hawaiian island chain, Iceland, Yellowstone.
• J Tuzo Wilson initial proponent.
• can persist for over 100 Ma.
• oceanic island related hotspots have a different trace element chemistry than ridge basalts, one less evolved.
• continental hotspot volcanism can be more complex.
• depth of origin?
• birth in an LIP - plume head arrival.
• minority view consider it as a propagating crack, and not driven by a deep plume.
• good place to enter debate: http://www.mantleplumes.org/index.html .
• image to right from http://education.usgs.gov/common/resources/mapcatalog/earthquakes.html, shows the emergent part of the Hawaiian island chain. Note how the earthquakes are associated preferentially with the southern end, reflecting the younger more active portion.
• Below is an image below the Hawaiian hotspot suggesting the plume extends to substantial depth. Image source: http://www.iris.edu/hq/gallery/photo/9228, Credit: Cecily Wolfe, University of Hawaii, Sean Solomon, Carnegie Institution of Washington, Gavi Laske, Scripps Institution of Oceanography, Robert S. Detrick, Woods Hole Oceanographic Institution, John A. Orcutt, Scripps Institution of Oceanography, David Berc
• 
• image below from http://pubs.usgs.gov/gip/dynamic/world_map.html . Shows distribution of world's hot spots at present. Are they evenly spread? Which plates show the largest concentration of hot spots?
• 
• Diagram below from http://oceanexplorer.noaa.gov/explorations/05galapagos/background/mission_intro/media/GSC3D_600.html shows the Galapagos hot spot. Note how the plume is shown to partly feed spreading ridge. As depicted one might be forgiven for wondering why such a broad plume has such a focused volcanic expression.
• 
• Image of Mons Olympus, the largest basaltic volcanic construct in the Solar System, found on the Tharsis Bulge of Mars.
• 
• Site describing Tharsis crustal dichotomy, with animation that shows global scale asymmetry convection model.