***Under Construction***
Mantle Background
Properties of the Mantle
Key properties of the mantle are summarized in the table below. These properties respond to driving forces on the mantle and control its overall behavior.
*Figure of seismic profile: http://www.uni-muenster.de/Physik.GP/Seismologie/Projekte/ss-pp-precursors.html
Methods of Studying the Mantle
- Telesiesmic waves (P, S, etc), which travel at different speeds in materials of different densities
- waves are produced by large seismic actions (earthquake or artificial production)
- waves travel faster in higher density material
- Assumption: change in density is primarily a function of temperature.
- Relationship for converting S wave velocities to density ρ = 620xVs +560
- For a review of seismic waves and how we use them to determine characteristics of the interior of the earth, see http://eqseis.geosc.psu.edu/~cammon/HTML/Classes/IntroQuakes/Notes/waves_and_interior.html
- Shear wave splitting:
- Other Telesiesmic shear waves (like SKS PKS SKKS) can be used to find the anisotropy of mantle fabric. SKS provide estimates of the two main splitting parameters: the azimuth of the fast wave polarization and the delay time between fast and slow waves, respectively related to the direction and strength of anisotropy.
- see description and figure at http://www.awi.de/en/research/research_divisions/geosciences/geophysics/research_themes/seismology/shear_wave_splitting/See
- an example of measurements on foliated rocks http://earthquake.usgs.gov/research/physics/lab/shearwave.php
- Other methods include
- Numerical Modelling (check out our numerical model!)
- Composition of basalts from which parameters such as temperature (e.g. Herzberg 2011) and depth of mantle flow (e.g. Camp 2013) can be inferred.
- Composition of mantle xenoliths, see this website for a brief introductory description http://earthandsolarsystem.wordpress.com/2012/11/24/mantle-xenoliths-a-window-into-the-interior-of-the-earth/
- Inferring past mantle dynamics from the timing of volcanism and/or uplift induced erosion (Faccenna et al. (2013) and references therein)
Describing Mantle Convection with Math
*figure from http://www.studyblue.com/notes/note/n/how-the-earth-works-final/deck/1690726
What conditions, forces, factors control mantle convection? Convection occurs in predictable ways, so we can describe it with mathematics. Breaking down the math that has been developed helps us understand what factors are important to mantle convection; we shall see that flow is primarily driven by temperature (density) anomalies.
The Navier-Stokes equation describes flow; it does not necessarily need to be convective flow. The full form is: 
Where (for the magma parcel under consideration)
𝜌 = density
v = velocity
t = time
g = acceleration due to gravity
p = pressure
µ = viscosity
What each of the parts of the equation mean:
- 𝜌(dv/dt): From our basic F = ma, mass is in the density term and is acceleration.
- g: Again, density and acceleration, but the is the difference in density between the parcel and its surroundings. The acceleration term is due to gravity, so if the parcel is more dense than its surroundings, it will sink and vice versa.
- ∇p: A pressure gradient, for use in cases, for example, in ground water flow where variation in water table height sets up a pressure gradient so that water flows from high to low head.
- µ ∇2v: A packets’ velocity is resisted by viscosity and the higher the velocity, the greater this drag force.
Essentially,
(Density)(acceleration) = (density)(acceleration from gravity) + (external pressure) + (drag)
Typically the pressure term is negligible or nonexistent, so we can simplify to:
Hence, the key parameters are
- the difference in densities between the parcel and surroundings
- Driven by gravity
- Resisted by viscosity (increasingly with greater velocity)
We can also describe mantle convection with the Rayleigh number. 
Where
g = acceleration due to gravity
ρ = density
ΔT = the temperature difference between the bottom and top of the convection cell
α = the volume coefficient of thermal expansion
d = the height of the convection cell
μ = viscosity
κ = thermal diffusivity
There is a critical Rayleigh number that depends on the size of the convection cell and some constants where:
Ra < Racr convection will decay
Ra > Racr convection will grow
So the most key factors for maintaining convection are
- temperature differences between the base and top of the convection cell
- size of the convection cell
In short, temperature differences lead to density differences which move, driven by gravity and resisted by viscosity.
Since convection in the mantle is mainly dependent on temperature and mantle flow is dependent on density and viscosity which are inversely proportional to temperature, it is also important that we describe how heat evolves through time with this heat equation:
where
T = temperature
t = time
κ = thermal diffusivity
x = space
v = velocity
A = a term which encompasses the production of heat, in the case of the earth from radioactive decay
In the mantle, the production term is small compared to the crust which contains more radioactive elements. Advection comes from convection of the mantle and is the primary way heat is transferred in the mantle, though conduction still occurs and is particularly useful to consider in the case of a cool down-going slab as it conductively warms during its descent.
Recent Literature on the Mantle
What is the mantle doing? How does it interact with the other parts of the earth – the lower and upper crust and the surface? We review for you here some of the recent literature that addresses these questions.
Background for the tectonic systems studied
Mediterranean
- A mosaic of micro plates between the Nubian (African) and Eurasian plates move and deform independently of overall plate convergence.
- N-S convergence of Nubai and Eurasia plate (.5-1cm/y).
- Extension in the Western Med. And Tyrrian Sea (2-3cm/y to 10cm/y)
- What was once a clean subduction zone is now broken up into a complex system of tectonic boundaries.
From Faccenna et al. 2013. Orange arrow shows general direction of both mantle flow and surficial plate motion. “Red indicates active volcanic fields (undistinguished); blue indicates deep basins; red lines indicate active subduction zone; black lines indicate collisional thrust system; yellowish background area indicate regions under influence of Afar plume”
Figure from Jolivet et al. (2009) demonstrating the complicated tectonics of the Mediterranean with various microplates.
Himalaya
Location and basic tectonics:
*Figure from http://earthobservatory.nasa.gov/Features/KashmirEarthquake/
Simplified cross section of Himalayan tectonics:
*Figure from Keary et al. 2009
Alaska
A map of Alaskan earthquakes demonstrating the Pacific Plate subducting below the North American Plate
*Figure from http://www.alaska.net/~logjam/earthquake.html
3D depiction of the complex shape of the Pacific Plate subducting near and to the east of Anchorage. North to the right.
*Figure from Jadamec & Billen 2010
The Literature
The following sections summarize some of the current research that supports the hypothesis that mantle flow interacts with the crust and is expressed in surface plate motion, geology and topography.
- Subducting slabs are the major control on convection. Large scale convection is driven by lower mantle anomalies and hot upper mantle anomalies finer tune mantle convection and therefore plate kinematics. This is the conclusion of Faccena et al. (2013) from their model of the Mediterranean which is density (temperature) driven; models best matched observed plate kinematics when they incorporated both large and small convection cells.
*Figure from Faccena et al. (2013) demonstrating correlation between their model results of plate motion (white arrows) and measured geodesy (orange arrows). Background color is dynamic topography estimates from their model.
*Figure from Faccenna and Becker 2010
- The mantle and crust are well coupled, flowing in the same direction. Jolivet et al. 2009 found a correlation between mantle anisotropy, fabric of metamorphic core complexes and current direction of plate flow as determined by geodesy. They recognize that while mantle and crust are flowing in the same direction, the mantle could be flowing faster than the crust. Also, they note some decoupling between the lower and upper crust by shear zones at the frictional-to-viscous transition due to the differing patterns of deformation in the lower and upper crust.
Figures from Jolivet et al. (2009). A map showing measured strike of stretching lineation in the metamorphic core complexes which correlates well with the direction of mantle flow.
- There is some degree of decoupling at a slab. Jadamec & Billen 2010 model the Alaska subduction and find that in the vicinity of a subducting slab, the mantle flow is complex and can be ten times faster than surface plate motion, hence some decoupling between the two. They explain this as a result of non-Newtonian viscosity in which the mantle becomes easier to deform as strain increases.
*Figure from Jadamec & Billen 2010 demonstrating the complex mantle flow around a slab.
- Slab and mantle dynamics affect surface deformation. Tibet switched from thickening to normal faulting and thinning around 13.5 Ma; this change in deformation style is proposed by Hatzfeld and Molnar (2010) among others to be initiated by slab breakoff below Tibet around that time, causing a change in mantle flow and thus deformation of the crust. This attests to both control of the subducting slab on mantle flow and coupling between the behavior of the mantle and crust.
*Figure from Guillot et al (2003). Even though there is still convergence in the Himalaya, there has more recently been normal faulting.
- Dynamic topography is that which is due to flow in the mantle as opposed to that which is due to isostasy. Dynamic topography helps explain transient topography (millions to tens of millions of years) and must be taken into account when developing past sea levels. See Flament et al. (2013) for a good recent review.
*figure from Flament et al. (2013). Models of dynamic topography for the present (left) and 100Ma (right). The dynamic topography of North America has evolved much in part with the behavior of the Farallon slab which subducted under western North America.
Summary Statement
- Subducting slabs are a major control on convection.
- Overall, the mantle and crust are well coupled.
- Slab and mantle dynamics affect surface deformation.
Limitations on mantle knowledge
How well do we really know what is going on in the mantle? These papers note that we have limits on what we know about the mantle.
- Matching fabrics in the crust and mantle are often attributed to coupling between mantle and crust, fabric development in the crust driven from below. Jolivet et al. (2009) recognize that the same pattern could be driven from the top down by topography.
- Jolivet et al. (2009) conclude that mantle fabrics could be reset in as little as 5-8 Myr in particular settings such as a warm backarc basin. When we compare crustal and mantle fabrics to discuss coupling, we need to consider if both fabrics are of comparable age or if one (especially the crust) could be preserving fabric from a past tectonic regime.
- Seismic data provides much of what we know about the mantle. Jolivet et al. (2009) recognize the that when interpreting shear wave splitting (see Methods, this webpage), we assume that the majority of anisotropy is due to olivine in the mantle rather than the crust. Also they mention that this method has poor vertical resolution (~300km), making it difficult to determine whether the anisotropy resides in the lithosphere or asthenosphere.
Mantle Model
***Under Construction***
References
- Camp, V. E. (2013) Origin of Columbia River Basalt: Passive rise of shallow mantle, or active upwelling of a deep-mantle plume? The Geological Society of America Special Paper 497.
- Faccena, C., Becker, T. W., Jolivet, L. and Keskin, M. (2013) Mantle convection in the Middle East: Reconciling Afar upwelling, Arabia indentation and Aegean trench rollback. Earth and Planetary Science Letters. v. 375; p. 254-269.
- Faccenna, C. and Becker, T. W. (2010) Shaping mobile belts by small-scale convection. Nature. v. 465; doi:10.1038/nature09064.
- Flament, N., Gurnis, M., and Müller, R. D. (2013). A review of observations and models of dynamic topography. Lithosphere. v. 5, no. 2; p. 189-210.
- Hatzfeld, D., and Molnar, P. (2010) Comparisons of the Kinematic and Deep structures of the Zagros and Himalaya and of the Iranian and Tibetan Plateaus and geodynamic implications. Reviews of Geophysics. v. 48; RG2005, doi:10.1029/2009RG000304
- Herzberg, C. (2011) Basalts as temperature probes of Earth’s Mantle. Geology. v. 39, no. 12; p. 1179-1180.
- Jolivet, L., Faccenna, C. and Piromallo, C. (2009) From mantle to crust: Stretching the Mediterranean. Earth and Planetary Science Letters. v. 285; p. 198-209.
- Jadamec, M. A. and Billen, M. I. (2010) Reconciling surface plate motions with rapid three-dimensional mantle flow around a slab edge. Nature. v. 465; doi:10.1038/nature09053.
- Keary, P., Klepeis, K. A., and Vine, F. J. (2009) Global Tectonics, 3rd ed. Wiley-Blackwell. West Sussex, UK.
Testing The Si- Earth Hypothesis in COMSOL
-COMSOL is a powerful tool that allows us to test certain physical parameters in a controlled environment. COMSOL is perfect for tweaking a specific variable (like viscosity or density) and seeing how that change effects the system.
-However, one must always keep in mind that any computer model today is not a perfect clone of the real-earth scenario. Many variables (like unconformities within the mantle, or accurate temperature gradients throughout the mantle) are very difficult or impossible to reproduce within the limitations of COMSOL.
Using COMSOL to study mantle/crustal interactions
We have used COMSOL to test under what conditions the SI -Earth hypothesis remains true. If the SI hypothesis is a realistic explanation we should be able to see a direct interaction between Mantle forces and the lower crustal boundary, resulting in a topographical change on the surface.
Once we have shown that, we will test various realistic conditions to see if/when the SI- Earth hypothesis fails. In other words can we create a realistic scenario where there is no interaction between the mantle and the crust.
Parameters constant in all models unless otherwise specified
Density:
Crust: 2700 kg/m^3 Upper Mantle: 3200 kg/m^3 Lithosphere: 3300 kg/m^3 Oceanic crust: 3000 kg/m^3
Viscosity:
Mantle: 1e^21 Pa*s Lithosphere: 1e^22 Pa*s
Angle of Subduction: 30deg
Thermal Conductivity: k = 3 W/m*K
-A standard subduction model 
Here is a basic subduction model run in COMSOL. Note how the crust thickens above the subducting slab, and how the crust thins in the areas around it. Also note right underneath the lithosphere we see a change in Y strain, driven by mantle convection, supporting the hypothesis that mantle processes (in this case the convection cell driven by the subducting slab) have a direct effect on the crustal layers.
So if this is true we should see a change in topography as well.
Here is a plot of the surface topography as the slab is subducting. Even in this very simple model we can see how a subducting slab can create a positive increase in topography.
Changing the Density
How does the system behave if the density of the subducting lithosphere is altered? 
In this model the density of the subducting lithoshpere has been changed to 3225 kg/m^3, so only slightly more dense than the surrounding mantle material. As you can see the system nearly grinds to a halt. Density is clearly the primary driving force in this system, and without a density difference convection is impossible.
As you can see the topographic change in this model is minimal compared to the standard.
Looking at different mantle senarios
Of course not every real earth subduction zone is structured like the previous models. For example there is evidence of ‘free floating’ slabs underneath the Mediterranean. In other words, sinking lithosphere that has broken off of from the subducting slab. Would a scenario like this still have an effect on surface topography?
We can see that the ‘free floating’ slab creates its own unique Y strain on the lithosphere. But it is not easy to see so let us take this to the extreme.
Here we can see that a standard subduction zone is not required to create topographic changes. Additional models support the conclusion that a slab of any realistic size and shape can drive a mantle convection cell, which in turn effects the lithosphere creating topographical changes.
Conclusion
While COMSOL is not a perfect representation of a real-earth subduction zone it does allow us to make a number of important conclusion in terms of the SI Earth hypothesis.
-Density differences between the subducting object and the surrounding mantle material is the primary driving force behind mantle convection.
-This mantle convection creates a direct force (as shown as Y- direction strain in COMSOL) upon the lithosphere and crustal layers.
-This force is translated into topographic change on the surface (ie. mountain building)
-While changing specific variables (like density, viscosity, subduction angle, slab shape) can have an effect on the magnitude or rate of topographic change, it is very difficult to create a scenario, with a subducting object of any sort, to not have a resulting effect on topography.
-In conclusion COMSOL models support the overall SI earth hypothesis. It also supports that the SI Earth hypothesis is a very robust explanation of mantle/ crustal interactions and can be used to explain and model many real earth scenarios.
UMaineGeodynamics2014 