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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


    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


    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


    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).












    d xxxxz ===ε




    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


    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:


    dE +=


    ητ =

  • Dividing by the Young’s modulus:

    εm is a type of elastic strain.










    τε m



    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


    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


    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


    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.




    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


    Coble creep brings us back to Newtonian flow (n = 1). This type

    of creep important in the asthenosphere.



    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



    2 )1(12h



    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:












    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


  • 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


  • 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


  • A model that works well for

    response to load removal is:

    τ/)( toewtw−=





    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


    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


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

    inertial forces.



  • 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:


    RaT κηαθ



    QgRaQ κη


    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


    This is a small number in the mantle (as you might expect; hard

    to imagine what a turbulent mantle would be like!).



  • 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


    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


  • 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


    The relative lack of importance of

    convective drag may be a bit