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 εσ =

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 =ε

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 ===ε

dz

dv

dt

d xxzxz ηεησ ==

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:

εηεσdt

dE +=

E

ητ =

Dividing by the Young’s modulus:

εm is a type of elastic strain.

mdt

d

dt

d

EEε

ετε

εηε

σ=+=+=

τεε

τε m

dt

d=+

The solution to the above is

[ ]τεε /1 t

m e−−=

so the strain asymptotically approaches εm.

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 /−

=

µσε

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,

3

2 )1(12h

ED

ν−=

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 =−+

++ ρρ

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:

τ/)( t

oewtw −=

ηλρ

πτ

gm

4=

Where λ 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.

κη

=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 θ:

4Dg

RaT κηαθ

=

5Dk

QgRaQ κη

α=

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!).

ηρvD

=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.