Download EARTH SYSTEM SCIENCE II

EARTH SYSTEM SCIENCE II
EES 717 – SPRING 2009
Buonaiuto & Salmun
Mantle Convection & Plate Tectonics
Bercovici, D., Ricard Y., Richards M.A., 2000. The Relation Between Mantle Dynamics and
Plate Tectonics: A Primer. In The History and Dynamics of Global Plate Motions,
GEOPHYSICAL MONOGRAPH 121.
1. Abstract/Introduction
a. Plates are the top thermal boundary layer of mantle convection, which is still widely
believed to be the engine of surface motions.
b. The direct energy source for plate tectonics (mountain building, earthquakes, volcanoes,
etc.) is the release of the mantle’s gravitational potential energy through convective
overturn, (radiogenic heating and core cooling continue to replenish the mantle’s
gravitational potential energy)
c. It is becoming more widely accepted that the plates are an integral part of mantle
convection, or more to the point they are mantle convection.
d. Many unanswered questions remain. Perhaps most difficult of these questions concerns
the property of the mantle-lithosphere system which permits a form of convective flow
that is unlike most forms of thermal convection in fluids, i.e., a convection which looks
like plate tectonics at the surface.
2. Basic Thermal Convection
Subsection 2.1 – Define the Rayleigh number Ra; understand its physical meaning; onset of
convection and the critical Ra (Racr); Racr and convective instability
Convective self-organization: non-linear system that develops an organized (polygonal)
circulation of hot and cold thermals.
Rayleigh’s stability analysis of convection predicted the conditions necessary for the onset of
convection, as well as the expected size of convection cells relative to the layer’s thickness.
Depending on the nature of the top and bottom plate the convection cells typically develop
widths approximately equal to the depth of the convecting fluid. This is to say that the aspect
ratio (H/L, where H and L measure the characteristic vertical and horizontal dimensions,
respectively) is of order unity (O(1)).

The Rayleigh Number:
ΔT > μκ/ρgαd3
ΔT: temperature difference between plates (top/bottom isothermal boundary of
convecting fluid)
1
μ: dynamic viscosity
κ:
ρ:
g:
α:
thermal diffusivity: ratio of thermal conductivity to volumetric heat capacity (m2/s)
fluid density: mass per unit volume
gravitational field strength
thermal expansivity (also known as coefficient of thermal expansion): is the tendency
of matter to change in volume in response to a change in temperature.
d: thickness of layer (conducting fluid, distance between plates)
If ΔT exceeds this parameter convection will commence. Re-arranging this relationship
we generate the Rayleigh number (dimensionless):
Ra = ρgαΔTd3/μκ

The critical Rayleigh number, Racr, corresponds to the least stable mode or smallest
perturbation that will initiate convection ~1000. Once convection has begun the
difference between the Rayleigh number and the critical Rayleigh number is an
indication of convective vigor. Mantle convective Rayleigh number is 107-109
 viscosity: is a measure of resistance of a fluid to being deformed by either shear stress
or extensional stress. Dynamic viscosity: determines the dynamics of an
incompressible Newtonian Fluid.
 Newtonian Fluid: stress versus rate of strain curve is linear and passes through origin.
(stress = force/area; strain = geometrical expression of deformation as a result of
stress), This is expressed mathematically as (for a one dimensional flow, u, in the xdirection but changing in magnitude in the y-direction) – air, water, honey.
τ = μ*du/dy
 non-Newtonian Fluid: viscosity changes with applied rate of strain – mayonnaise,
latex paint, molten plastics, lava, foams.
Vertical Structure of Simple Convection
Symmetry & Figure 1 – Once steady state/final stage convection is achieved the top boundary
is cooling the fluid at the same rate the bottom boundary is heating the fluid (Figure 1).
After this subsection should be able to explain this figure conceptually and in detail.
Thermal Boundary Layers (bl) – The Earth’s lithosphere and tectonic plates are essentially the
horizontal thermal boundary layer along the top surface of the Earth’s convecting mantle.

Relatively thin region on the top (cold) and bottom (hot) of the convecting fluid where
temperatures transition from the bulk/average homogeneous interior. The thermal
boundary layers are gravitationally unstable. (Figure 2). Identify the thickness, δ, of these
layers in the figure, i.e., give an estimate in % of distance separating plates.
2

Thermal Conduction: transfers heat into (influx, bottom) and out of (efflux, top) the
thermal boundary layer. Note that the heat transfer in the bl is via conduction (as
opposed to heat transfer via convection).

The stronger the temperature gradient in the thermal boundary the greater the rate of heat
transfer, the more vigorous the convection in the interior and the thinner the thermal
boundary layer. Hence δ is a function of intensity of convection (a good measure of this
would be the speed of convective currents) and temperature gradients (∆T/∆h).

Gravitational instability in the thermal boundary layers leads to downwelling/sinking in
the cold boundary and upwelling in warm boundary that feed the convection currents.
The feeding of currents induces motion in and thickening of the boundary toward the
regions of downwelling (cold) and upwelling (warm). Think of convergence
zones/divergence zones in this context and the thinning and thickening of the bl.
Size of Convection Cells – One of the things determining the lateral extent of a single
convection cell is the length of the horizontal boundary layer currents. Depending on material
properties and temperature gradients at the top of the thermal layer, material can only travel so
far before sinking/downwelling into the mantle, thus controlling the size.
Thermal Boundary Layer Forces: horizontal pressure gradients are the primary driving force
for lateral flow in thermal boundary layers.
It should be clearly understood that buoyancy does not drive the boundary layer currents directly;
buoyancy only acts vertically while boundary layer currents move horizontally (buoyancy or
gravity does eventually deflect these currents back into the convecting layer, but it cannot drive
their lateral flow).
Be sure to understand the fundamental concepts (they are fundamental to all geophysical fluid
flow!) in the following passage of the paper:
When hot upwelling fluid impinges on the top surface, it is forced to move
horizontally away from a high pressure region which is centered above the
upwelling itself; the high pressure region results from a force exerted by the
surface on the fluid to stop the vertical motion of the upwelling thermal. As the
top cold thermal boundary layer moves away from an upwelling to its own
downwelling it thickens, gets heavier and acts to pull away from the surface; this
induces a suction effect and thus, because of the boundary layer’s growing
weight, increasingly lower pressures in the direction of motion, eventually
culminating in a concentrated low pressure zone where the downwelling separates
from the surface. Thus the horizontal boundary layer current flows from the
induced high pressure over the upwelling to the low pressure over the
downwelling, i.e., it flows down the pressure gradient. Invariably plate driving
forces are related to these pressure highs and lows, i.e., pressure gradient effects.
Now explain in your own words the generation of areas of low and high pressure that lead to
horizontal pressure differences (pressure gradients) that ultimately constitute the forces that drive
plate motion.
3
The three main forces driving the movement of plates are slab pull, ridge push and basal
tractions (see handout on forces acting on the plates)


Ridge Push: gradual pressure gradient from the ridge high outward
Slab Pull: pressure low created from slabs pulling away from surface
Patterns of convection – Varied patterns, relation to Rayleigh Number (Figure 3). Purely
basally-heated plane-layer isoviscous convection is naturally a very symmetric system; this
requires geometries that naturally fit together such as long rolls, rectangles, squares, hexagons
and triangles. This symmetrical planforms can breakdown internally due to internal heating and
variable viscosity.
Influence of Internal Heating: mantle is radiogenically heated from the decay of uranium,
thorium and potassium.

In a simple internally heated convection cell only a top (cool) thermal boundary layer is
created, in which the cooling and downwelling balances the internal heat source. Slow
broad/diffuse upwelling takes place to balance the mass movement however it is
considered a passive current. Whereas the downwelling slab/current is active. Please note
the boundary conditions imposed on the top and bottom boundaries: top is kept
isothermal (temperature is a constant but there can be heat flux); bottom is thermally
insulated which does not allow for temperature gradients to develop (there cannot be loss
or gain of heat there, the flux is zero). In all cases of convecting systems the boundary
conditions play an important role on the outcome.

Having a system with both basal and internal heating leads to regions with more intense
downwelling and weaker upwelling. The Earth system is thought to have approximately
80% of the heat source through radiogenic heating. So the large scale circulation is driven
by downwellings (slabs) fed by an intense thermal boundary layer (upper, cool; in the
case of the earth the lithosphere and plates) while the active upwelling (mantle plumes)
are relatively weak (see Figure 4).
Influence of temperature-dependant viscosity

A few hundred degree change in temperature can cause many orders of magnitude
changes in viscosity. A strongly temperature dependent viscosity can break the symmetry
between upwellings and downwellings in much the same way as internal heating.

Viscosity may change by as much as 7 orders of magnitude in the top 200 km of the
mantle. The effect of temperature-dependent viscosity on mantle convection is to make
the top colder thermal boundary layer (the lithosphere) much stronger than the rest of the
mantle. Interestingly, this helps make thermal convection in the mantle plate-like at the
surface in some respects, but it can also make convection less plate-like in other respects.

T dependent viscosity can create asymmetries in upwelling and downwelling because the
stronger (cold slabs) act as a heat plug, which increases the temperature gradient between
the upper thermal boundary and the mantle, and reduces the temperature gradient
4
between the mantle and core.

Result in large aspect ratios between lateral extent of convection cells and depth of
convecting fluid because it takes a great deal of time to cool and become negatively
buoyant and sink through the cold stiff surroundings. Large aspect ratio convection cells
are considered to occur in the Earth, for example it is possible that the Pacific plate and
its subduction zones reflect the dominant convection cell in the mantle.

If viscosity is very strongly temperature dependent, the top thermal boundary layer can
also become completely immobile and the large aspect ratio effect vanishes. The
immobile boundary layer happens simply because it is so strong that it cannot move. This
is like imposing a rigid lid on the rest of the underlying fluid, which then convects much
as if it were nearly in isoviscous convection with a no-slip top boundary condition. This
would yield convection cells of aspect ratio equal to 1.
Summary of convection regimes is shown in Figure 6: 1) low T-depend viscosity, unit aspect
ratio, isoviscous regimes, 2) higher T-dependency, high aspect ratios, 3) high T-dependence back
to unit ratios as no slip prevails.
Sphericity: the spherical shape of Earth makes the bottom thermal boundary less surface area
than the top thermal boundary, which in theory would lead to stronger upwelling movements and
weaker downwelling motion. However since Earth is the opposite (stronger downwelling,
weaker upwelling), this points to the greater importance of internal heating and T-dependent
viscosity.
Poloidal and Toroidal Flow – Figure 7. Poloidal: cycle of upwelling and downwelling
(ridges/rises and trenches) currents and associated divergent and convergent zones, respectively.
Toroidal: associated with rotation, strike-slip motion and spin of plates, i.e., San Andreas Fault.
In isoviscous flow toroidal motion does not occur naturally so it must be excited from top or
bottom boundary layers. This can also be achieved through lateral changes in viscosity.
A few important concepts:

Incompressible mantle (can be said of any fluid if it applies): a fluid with constant density

Boussinesq fluid: density is a function of temperature (and thus actually not constant), the
density fluctuations are so small that the fluid is still essentially incompressible except
when the density fluctuations are acted on by gravity. Thus the fluid acts incompressible,
but can still be driven by buoyancy.
Both conditions (or valid assumptions if not exact conditions) require that the rate at which mass
is injected into a fixed volume must equal the rate at which the mass is ejected, since no mass
can be compressed into the volume if it is incompressible; i.e., what goes in must equal what
goes out. This is another way of stating the conservation of mass principle.
The important point to get out of this subsection is that the poloidal motion is directly driven by
buoyancy forces (and can be shown from basic principles) while there are no internal driving
forces for toroidal flow (also derived from basic dynamics).
5
3. Success of Basic Convection Theory in Explaining Plate Tectonics
Convective forces and plate driving forces –
I. Slab Pull: as important driving mechanism for plate driving motion, and the associated
convective downwelling. This can be illustrated with simple dimensional analysis to compute
slab velocities (~10 cm/yr) which roughly match the observed plate motions.
Recall that the pull of a slab on a plate is in fact a horizontal pressure gradient acting in
the cold upper thermal boundary layer and caused by the low pressure associated with a
slab pulling away from the surface.
Estimation of the force and velocity of a cold downwelling using simple scaling analysis –
this approach uses basic relationships and fundamental physical dimensions of physical
quantities that are known to relate to a physical problem, it is widely used in geophysics
(geophysical fluid mechanics, for example) and earth sciences, as well as in all engineering
fields.
Follow the cartoon of Figure 8 illustrating the slab-pull force problem and consider a cold top
thermal boundary layer which is L long (from its creation at a divergent zone to its destruction at
a convergent zone) and W wide. As it moves from the divergent to convergent zone, the
boundary layer thickens by vertical heat loss which, in the horizontally moving boundary layer,
is only due to thermal diffusion (no mechanical effects here, such as those due to the ‘bodily’
transport of heat by a current) .
To estimate the boundary layer thickness δ(t) we apply dimensional homogeneity ≡ physical
dimensions on both sides of an equality should be the same. Assuming that only thermal
diffusivity κ (with units of m2/s) and age t (where t=0 at the divergent zone where the boundary
layer is initiated) control the growth of the boundary layer, then we can write
 (t )  A a t b
where A is a dimensionless constant (which has to be dimensionless since we assume no other
dimensional properties of the system have any influence on the cooling process). The constants a
and b are then determined to match the dimensions of either side of the equation for δ. Doing the
algebra leads to a=b=1/2. This is very much what one would get for MANY ‘decay’ time of
problem, which is typical of problems where one starts (initial condition) with a certain amount
of “something” (pollution point source, a high concentration of plankton in one small space, and
such) concentrated in space and then the “something” diffuses away (that’s the key word: a
6
diffusion process which is always represented as divergence of the flux of the “something”
mathematically, in words we say that “stuff” diffuses ‘down the gradient’). When we use that
t=L/v then we get the thickness of the boundary layer  (t )  L / v . Note here that v is not
known at this point, to obtain that we need to consider a balance of forces that leads to the
motion. The two forces acting on this system are the buoyancy force FB and the drag force due
to viscosity (resistance to motion, viscosity provides the stress which is force per unit area) FD.
The buoyancy force of the downwelling current as follows:
Considerations made: the current (with speed v) goes down to a depth D, the temperature
contrast (T anomaly) between bl and interior is ∆T (see figure), we use ONLY what we have
at hand to obtain an expression of dimensional units “F=ma”. In the formula, the first factor
in the square brackets represents the density anomaly of the downwelling, and the second
factor in brackets is the volume of the downwelling.
The drag force on the downwelling current is the viscous stress
Considerations made: stress is approximated by viscosity times the rate of strain on the fluid,
as it is sheared vertically between the downwelling side of ‘things’ and the stagnant fluid in
the interior. Again we need a “F=ma” expression but recall that “stress=F/Area” so F=stress
x Area (also recall the expression for the stress in a Newtonian Fluid at the beginning of
these notes).
Now FB – FD = 0 yields:
and when using the relation between δ and v found earlier:
Now we can use this formula to estimate plate motion and see that what we get is in good
agreement with the observations. Take that L≈D (aspect ratio of O(1)) and using the followin
values (viscosity value is for for a ‘lower-mantle dominated’ mantle)
we obtain v ≈ 10 cm/yr. Can play a little with parameters such as viscosity, L and D but we will
get the same order of magnitude estimate for velocity.
Summary: this section shows, using basic scale analysis, that the velocity of downwelling
currents is indeed the correct order of magnitude for velocities of active (i.e., slab-connected)
7
plates, strongly suggesting that the plate force deemed “slab pull” is well modeled by a simple
cold downwelling current in basic convection.
II. Ridge Push: convective pressure gradient force distributed throughout the slab above the
compensation depth. It is the next most important force behind slab pull, although it has been
estimated that it constitutes only 5-10% the driving force due to subducting slabs. It is now
recognized that it is a force distributed across the area of a plate (note that area of plate does not
enter in estimation of slab pull from a mantle convection model approach). It is not so clearly
related to convection, as slab pull is.
The second paragraph in this subsection summarizes the mechanical (as opposed to ‘thermal’, as
that due to buoyancy effects in a stratified fluid) concepts related to hydrostatic pressure
(vertical) and isostatic compensation depth, and how to create the differences in pressure along
the horizontal dimension that will ultimately provide for this force called ridge push. Yet, the
third and final paragraph in this subsection (nicely written again!) states in relation to the role of
convection in the ridge-force pressure gradient:
“The ridge-push pressure gradient is essentially indistinguishable from the
pressure gradient in a convective thermal boundary layer.”
And you will read that the authors arrive to this conclusion by using the same concepts as before:
(i) horizontal pressure gradient in the bl drives the lateral motion of the bl itself
(ii) with a deformable upper surface, H pressure are at divergent zones and push up the surface
(iii) L pressure lows are at convergent zones and pull down on the surface
(iv) the manifestation of this pressure gradient in the bl is surface subsidence from a divergent to
convergent zone
Structure of Ocean Basins: some of the basic structures can be explained through simple
convection theory including trenches, to first order heat flow and sea floor subsidence. The
dimensional analysis of the previous section led to an expression for how the boundary layer
grows in time due to diffusion, namely that  ~ t . In addition, as the thermal boundary layer
thickens it weighs more and thereby pulls down on the surface with increasing force; if the
surface is deformable, it will be deflected downward (until isostasy is established). The surface
subsidence thus mirrors the increasing weight of the boundary layer; the weight increases only
because δ grows, and thus surface subsidence goes as t . Sea floor subsidence is predicted by
convective theory (more precisely convective boundary layer theory) to increase as the squareroot of lithospheric age; this is called the (age)1/2 law.
Moreover, this type of boundary layer theory predicts that the heatflow out of the ocean floor
should obey a 1/(age)1/2 law, which can be inferred by the fact that the heatflow across the
boundary layer is mostly conductive and thus goes as T /  (where к is thermal conductivity
and ΔT is the temperature drop across the thickening thermal boundary layer). Measurements of
sea floor bathymetry and heatflow versus age are presented in Fig. 9 and show that, to first order,
the theoretical curve fit the observations quite well lending support to the idea that sea floor
subsidence and heatflow do indeed follow simple results from boundary layer theory, (age)1/2
and 1/(age)1/2, respectively.
8
Some discrepancy results from additional cooling at the ridge/rise from hydrothermal vent
activity and additional heating far away from the ridge axis as a result of secondary heating small
scale convection cells, mantle plumes etc.
Slab-like downwellings are characteristic of 3D convection – The point of this section is that
although a pattern of sheets of downwellings does not occur always it is nonetheless a very
common feature of simple, purely basally heated plane-layer convection. In that sense, the
occurrence of slab-like downwellings in mantle convection is characteristic of basic thermal
convection and again, these patterns are consistent with plate tectonics. It is also stressed that
cylindrical upwellings and mantle plumes are also characteristic of 3D convection. Cylindrical
upwelling plumes in the Earth’s mantle are indirectly inferred from hotspot volcanism,
geochemical analyses and perhaps most directly in seismic analysis. It is worth noting here that
mantle plumes and hotspots are not part of plate tectonics. They still play an important role in
our ability to measure absolute plate motions (e.g., because of assumed hotspot fixity), and in
understanding the nature of mantle convection and the relative sizes of its thermal boundary
layers.
Relative fluxes of plumes and slabs – The heat flux transported by mantle plumes, relative to
that transported by the cooling lithosphere and slabs, is well predicted by convection models
using the Earth’s presumed proportion of internal heating. The net plume heat flux is
approximately the same as the heating injected through the bottom boundary; i.e., plumes
essentially carry only heat input from below. From considerations about the Earth’s magnetic
field it is estimated that the heat flux out of the core (core-mantle boundary, this heat is then
input into mantle from below) is approximately 10% of the net terrestrial flux. Calculations from
model output suggest that plumes transport roughly 10% of the net heat flow (of Earth’s interior)
which is very consistent with the picture of basic, predominantly internally heated convection.
Mantle heterogeneity and history of subduction – The main point from this subsection is that
the expected relationships among downwelling flow, mantle density heterogeneity, and plate
motion (via subduction) have been confirmed observationally, so that this constitutes a clear
success of concepts derived from standard convection theory applied to plate motions.
Two things to keep in mind:

This does not address, even less answer, the question of how the plates themselves are
generated in the Earth’s system.

On a long-time average over the past 120-150 million years, global plate motions are well
predicted using results from mantle dynamics theories and models. But plate motions,
and hence subduction zones, are ephemeral on timescales of tens to hundreds of million
years, leading to a complicated picture when looking at correspondence between presentday subduction zones and mantle downwellings. (e.g., Intermittent Plate Tectonics?).
Temperature-dependent viscosity, internal resistive boundaries and the aspect ratio of
convective cells – The Pacific plate, with most of the kinetic energy and subduction zones of the
plate tectonic system, can in many ways be considered the top of the Earth’s dominant
convection cell; a very large convection cell with a large aspect ratio, much longer than it is
9
possibly deep. Since large aspect-ratio or long-wavelength convection cells in the mantle are also
inferred from seismic tomography, mantle convection in the Earth is thought to be characterized
by very large aspect ratio convection cells.
It was also discussed that convection with temperature-dependent viscosity can allow for large
aspect ratio cells, in particular in the sluggish convection regime (moderately temperaturedependent viscosity and a sluggish but mobile upper thermal boundary layer). This effect can
theoretically allow for a Pacific-sized convection cell; however, if the Earth’s mantle were only a
temperature-dependent-viscosity fluid, estimates of the its convective regime places it more in
the stagnant lid regime where the top thermal boundary layer is essentially frozen and thus
would be more characterized by unit aspect ratio convection cells. Other relatively simple effects
may also cause long wavelength convection cells; these effects primarily involve an internal
resistive boundary which, while not a feature of the fundamental fluid dynamics of convection,
can be readily added to simple convection models to improve their match with the Earth.
The main paragraph of this section focuses on the nature of the well-known seismic discontinuity
that separates the upper and lower mantle at a depth of 660 km. The discontinuity involves both
phase change and a viscosity jump with depth by most likely a factor of 30-100. The kind of
convective mass transfer at the discontinuity is debatable but there exists evidence from seismic
tomography for transfer of both slabs and plumes across the phase change. Despite the transfer,
it is argued (sound physical arguments) that the phase change and the viscosity jump can still
provide resistance to downwellings as they attempt to penetrate into the lower mantle. The result
of the resistance is that convection tends to develop large-volume downwellings (basically: the
thermal boundary layer/slab needs to get ‘fat and heavy’ enough to go down) – and thus large
wavelength convection cells – in order to gather enough negative buoyancy (again here slab
needs to become heavy, so its buoyancy is decreasing) to penetrate a moderately resistive
boundary. This tendency to create large aspect ratio (nearly-Pacific size cells) has been well
documented in three-dimensional models of spherical convection at reasonably high Rayleigh
number for both the phase change and the viscosity-jump effects. Here again we see that
convection theory can be successful at explaining known features of the Earth as well as
processes that are known or believed to be at work in plate tectonics.
The bottom line: temperature-dependent viscosity, or an internal boundary resistive to
downwellings, basic convection can generate Earth-like, large-wavelength (or long aspect ratio)
plate-sized convection cells. However, while these effects move us closer to obtaining plate-like
scales in convection models, none actually generate plates.
**************************** O *****************************
The last part of this review article is the big
WHERE DOES BASIC CONVECTION THEORY FAIL IN EXPLAINING PLATE
TECTONICS, AND WHAT ARE WE DOING (OR MIGHT WE BE DOING) TO FIX IT?
Here is where the authors go over all the little (and not so little) nagging details of the aspects
that we discussed above in Section 3, aspects that although to quite a large extent were
10
reasonable explained by convection theory, there were never really – REALLY – explained to
completion, and in some cases far from that. Item re-visited and discussed are:







Plate forces not well explained by basic convection
Structure of ocean basins (deviations from the (age)1/2 law)
Dynamic topography
Changes in plate motion
Plate-like strength distributions: weak boundaries and strong interiors
Convergent margins: initiation and asymmetry of subduction
Divergent margins: ridges, and narrow, passive upwellings
And a final sub-section, Section 4.8, where the focus is the generation of toroidal motion and the
necessity of this component of the motion to provide a complete theoretical frame for the platetectonics form of convection seen on the Earth.
Section 4 of this paper is well presented and although the details are beyond the scope of the
purpose of our course, we would like to suggest (an encourage you!) that you read to get the
general ideas. It will provide context and contrast to the detail discussion of Section 3, and it
might provide a good qualitative answer to a general PhD exam question!
**************************** O *****************************
11