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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 KXVIII 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 stressstrain 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 PT 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 viscoelastic. 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 tm 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
viscoelastic, 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 MaxwellBoltzmann
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
(NabarroHerring 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 elasticperfectly
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/)( toewtw−=
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 wholemantle
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.