Geodynamics

Geodynamics is what we often call “modelling”.

We use what we know about the physics of how materials behave and interpret our observations in ways that conform to that physics.

In this way we can use primary observations to make assertions about how the Earth works beyond just how material properties are distributed.

In the following we will touch briefly on Isostacy, then talk about rock deformation and fluid flow. This will allow us to make some conclusions about the nature of convection in the mantle, and it’s viscosity.

Isostacy

We discussed isostacy already in connection with Gravity. Recall the Pratt vs Airy models for local compensation.

But in many instances neither of those models works well; as discovered by Vening Meinesz. He proposed “regional compensation”.

Vening Meinesz worked in subs in 1920’s and proposed plate bending in 1931. His conclusions were prescient of plate tectonics.

The K-XVIII sails from Nieuwediep (Netherlands) to Soerabaja with Dr. F.A. Vening Meinesz on board to make gravity measurements.

Post Glacial Rebound

A consequence of

Isostatic Adjustment – we will return to this observation to infer the

viscosity of the mantle.

Rheology

Rheology is the science of the deformation and flow of solids. Or – how a material reacts to stress (what kind of strain and what are the rules governing stress-strain relations?)

We already discussed the elastic case in seismology ad nauseum. With elasticity, all deformation caused by stress is recoverable once the stress is removed:

klijklij c

If you go beyond the elasic limit (or yield stress), permanent deformation results. Two main types:

Brittle: the material physically breaks or ruptures. e.g., Earthquakes

Ductile: the material flows.

The kind of deformation that rocks experience will depend on:

1. Temperature: low -> brittle; high -> ductile (cf. candy bar in summer).

2. Strain rate -> high -> brittle; low -> ductile (cf. Bubble gum)

3. Confining pressumre -> low brittle, high -> ductile.

Because of P-T dependance, rocks tend to be brittle at shallow depths and ductile at deeper depths (transition is generally about 15 km or so). BUT, again, strain rates can change this. Consider the depth of the lithosphere determined seismically vs geologically (loading).

Viscosity

In the case of laminar flow, a fluid will have an internal friction due to particles migrating perpendicular to the flow direction.

In a class of fluids known as Newtonian fluids the stress is proportional to the strain rate.

Recall that

dz

duxxz

The constant of proportionality is called the viscosity

Note that with low viscosity, a small stress can give a high gradient (easy flow).

dz

dv

dt

du

dz

d

dz

du

dt

d

dt

d xxxxz

dz

dv

dt

d xxzxz

Viscoelastic Flow in Solids

Some materials, when the yield stress is exceeded, deform indefintiely (keep straining) with no further increase in stress. This is called perfectly plastic deformation.

Rocks behave like fluids with very high viscosities, and show a combined elastic and viscous behavior called visco-elastic. In this case

We define a characteristic time called the retardation time:

dt

dE

E

Dividing by the Young’s modulus:

m is a type of elastic strain.

mdt

d

dt

d

EE

m

dt

d

The solution to the above is

/1 tm e

so the strain asymptotically approaches m.

Creep.

Most solids will deform even at low stresses due to some fraction of atoms in a lattice having enough energy to jump into vacancies. (Maxwell – Bolzman law).

The distribution function f(E) is the probability that a particle is in energy state E.

Note that M is the molar mass and that the gas constant R is used in the expression. If the mass m of an individual molecule were used instead, the expression would be the same except that Boltzmann's constant k would be used instead of the molar gas constant R.

The idea is that some subset of atoms will have sufficient energy to jump out of their lattice position. If they fill a vacancy, you could think of vacancies “jumping” to where the atoms left.

The creep flow history in rocks is illustrated in figure to the right.

Note that primary creep is just visco-elastic, while secondary is purely viscous. The tertiary stage leads to failure.

There are different types of creep, but all have to do with the movement of vacancies and imperfections through a rock.

Example of Dislocation Glide

Example of Screw Dislocation Glide

The most important types of creep are Plastic Flow, Dislocation or Power Law Creep and Diffusion creep.

The regimes depend mostly on temperature, and in particular the fraction of the melting temperature (the homologous temperature).

Plastic Flow takes place at low temperatures and is most important in the lower crust. Large strains possible, but large differential stresses are required as well.

Dislocation Creep is important at temperatures between 0.55 Tm and 0.85 Tm, which is the most of the mantle. It is the most pervasive and is the mechanism of convection.

Dislocation creep is also called Power Law Creep because of the dependance of strain rate on a power (usually = 3) of the stress:

Where Ea is called the activation energy, and k is Boltzman’s constant. Note the strong dependance on Temperature in the exponential term that comes from the Maxwell-Boltzmann relationship. Again, n is typically 3 in this equation.

kTE

n

aeAdt

d /

At temperatures T > 0.85 Tm, Diffusion Creep takes over, which involves migration of defects long grain interiors (Nabarro-Herring creep) or along grain boundaries (Coble creep).

Coble creep brings us back to Newtonian flow (n = 1). This type of creep important in the asthenosphere.

RIGIDITY OF THE LITHOSPHERE

To a good approximation, we can think of the lithosphere as a thin elastic sheet. We can use characterizations of such representations, like the flexural rigidity,

32 )1(12

hE

D

Where E is Young’s modulus, v is Poisson’s ratio, and h is the thickness of the place. D is a measure of how difficult it is to bend a plate. Big D means the plate is stiffer.

We solve the above for the shape of the plate when loaded by islands or bending to subduct into the lithosphere.

To determine the strength of the plate. It is instructive to see how the plate responds to loads.

We consider a surface load L(x,y) on a plate of thickness h. A balance of the load by the elastic forces within the plate and the bouancy force due to density contrast gives a formula for the plate deflection w:

),()(24

4

22

4

4

4

yxLgwdy

wd

dydx

wd

dx

wdD lm

Example of detemining “D” for a “point load” produced by a sea mount.

Example of detemining rheology by fitting the profiles of subducting lithosphere.

Note that the lithosphere looks like an elastic plate in many cases, but the elastic limit can be reached at the edges because we exceed the yield stress and in this case we get an elastic-perfectly plastic behavior.

The effects of strain rate are quite evident when comparing geologic vs seismic strain rates.

Mantle Viscosity

We can estimate the viscosity of the mantle by observing how it responds to changing loads, such as the removal of ice sheets following the ice age.

A model that works well for response to load removal is:

/)( toewtw

gm

4

Where is the wavelength of the depression. We can therefore use the relaxation rate to estimate the viscosity of the mantle.

We can compare the effects of assuming different channel depths by varying that parameter in the model and seeing how it affects the uplift profile.

The bigger radius the load, the deeper into the mantle is the effect. To look at what happens deep in the mantle, we can apply the same analysis to uplift of North America – a very large radius load!

We can also look at the change of the position of the rotation pole due to shifts mass within the earth. The rate of the shift is a reflection of the rate of mass movement (readjustment) in the Earth, and this in turn is a function of the viscosity of the mantle.

Plate Dynamics

Mantle Convection

Flow is usefully described in the form of several dimensionless constants.

We look at the balance between pressure gradients and buoyancy, which drive flow, and viscosity and inertia, which resist it.

For example, t he relative importance of viscosity to inertia is given by the Prandtl number, a ratio of viscosity to thermal diffusivity:

Which is really big in the earth, meaning we don’t worry about inertial forces.

Pr

For convection, the Rayleigh number is the ratio of buoyancy forces (thermal expansion and gravity) to viscosity. There are two kinds to worry about. One is due to the superadiabatic temperature gradient :

4Dg

RaT

5Dk

QgRaQ

Note the strong dependence on the physical dimension of the system (D). A big Rayleigh number means convection is likely. Under almost any conditions, the Rayleigh number is very big in the Earth, meaning convection is virtually certain.

The other is due to radiogenic heat production Q:

At the same time, the flow is laminar (not turbulent) as indicated by the Reynolds number which is a ratio of the momentum to the viscosity.

This is a small number in the mantle (as you might expect; hard to imagine what a turbulent mantle would be like!).

vD

Re

A long standing question in Geophysics is the scale of convection: is it layered or whole mantle?

Note that if layered all heat must pass through 660 km by conduction.

Recent tomography results are in favor of whole-mantle convection.

A recent idea of how mass tranfers in the mantle is shown below. Note the complexity at the the CMB – plumes originate and slabs founder. Plumes appear at the surface as hot spots, which we noticed at some time ago.

The deep origin of plumes is strongly suggested by the correlation of plume activity with very long wavelength characteristics of the Geoid, as shown below. There seems to be some deep seated origin of low density material responsible for the plumes.

Plumes seem to rise up through the mantle independent of the lithospheric plate motions, and have been suggested as a way to determine absolute plate speeds. The best evidence for this idea comes from the history of eruptions at Hawaii (Yellowstone shows this as well).

A final note about the Forces on plates: we understand the sources of these forces, but which are important?

A comparison of force magnitudes on different plats shows that slab pull and trench suction tend to be larger than the rest, but there is clearly no one single force responsible.

The relative lack of importance of convective drag may be a bit surprising.