Geodynamics
Lecture 10
The forces driving plate tectonics
Lecturer: David Whipp!
david.whipp@helsinki.fi!
!
2.10.2014
Geodynamics
www.helsinki.fi/yliopisto
1
Goals of this lecture
•
Describe how thermal convection operates according to
basic physical principles!
!
•
Quantify the basic plate-driving forces!
•
•
•
Slab pull
Ridge push
Drag force
2
NEWS & VIEWS
Mantle convection
Compositional stratification
around 660 km
Secondary
plumes
Mid-ocean
ridge
Primary
plume
Deflected
slab
Penetrating
slab
Enriched
piles
Continent
Slab
grave yards
660 km
discontinuity
•
•
Core–mantle
boundary
Tackley, 2008
Plate tectonics is ultimately driven by convection in the
Figure 1 Leaky layers. The emerging model of mantle convection suggests that some relatively cool subducting
mantle
reflecting
produced
and(dashed
lostblack
as line)
thewhereas
Earth
slabs
of oceanic plate
(blue) are deflheat
ected at the
660 km discontinuity
otherscools!
penetrate
all the way to the core–mantle boundary (solid black line), forming slab graveyards. Piles of material that are
enriched in incompatible elements compared with the expected mantle average (orange) are pushed around at the
core–mantle boundary by incoming slab material, and plumes form from their edges and tops (red). Some plumes
penetrate below 660 km, whereas others are deflected and may produce secondary upper-mantle plumes. An
average compositional stratification exists either side of 660 km.
How convection operates in the mantle is somewhat
controversial, but there is clear evidence of fossil plates in the
upper and lower mantle
most of the mantle could be similar to the
depleted upper mantle, with only small
sense, this does not have much effect
on the evolution of the mantle and
3
mate
trace
at ho
stud
both
melt
Univ
thou
from
this
abov
Th
mixt
conv
inter
a glo
strat
of ph
subs
man
mate
grav
perm
disti
leng
may
may
in tr
of tr
Layered versus whole-mantle convection
•
Layered and whole-mantle
convection are endmember models of mantle
convection!
!
•
Layered convection is
based on the strong
seismic discontinuity at
670 km, among other lines
of evidence, with the idea
that this discontinuity
separates upper and lower
mantle circulation
4
http://instruct.uwo.ca
Thermal convection
A Rayleigh-Taylor instability
occurs when a denser fluid
overlies a less dense fluid,
which is gravitationally!
unstable (the denser fluid
wants to sink, and the less
dense fluid wants to rise due
to its buoyancy)
Fig. 6.21, Turcotte and Schubert, 2014
•
A fluid heated from beneath will undergo thermal expansion
and may become less dense than the cooler overlying fluid!
•
•
This scenario is gravitationally unstable!
If the denser fluid is able to sink into the less dense fluid,
thermal convection will occur
5
Thermal convection
Fig. 6.21, Turcotte and Schubert, 2014
•
What is going to happen to these two fluids with time?
6
Thermal convection
Time evolution of a Rayleigh-Taylor instability
7
Thermal convection
•
Thermal convection in the Earth results from buoyancy forces
owing to thermal expansion of mantle rocks!
•
For an incompressible viscous fluid, the force balance of
pressure, gravity and viscous forces in 2D is
✓ 2
◆
2
@p
@ v
@ v
0=
+ ⇢g + ⌘
+ 2
2
@y
@x
@y
where ! is pressure, " is density, # is viscosity and $ is velocity
in the % direction!
•
To account for the buoyancy forces from thermal expansion,
the density term must be variable
⇢ = ⇢0 + ⇢0
where "0 is a reference density and "ʹ is a density perturbation
much smaller than "0
8
Thermal convection
•
•
If the variable density is substituted into the 2D force balance,
and the hydrostatic pressure based on the reference density is
eliminated by using ' = ! - "0(% we find
✓ 2
◆
2
@P
@ v
@ v
0
0=
+⇢g+⌘
+ 2
2
@y
@x
@y
!
A change in density as a result of thermal expansion is
⇢0 =
⇢0 ↵v (T
T0 )
where )$ is the volumetric coefficient of thermal expansion
and *0 is a reference temperature corresponding to the
reference density "0!
•
The vertical force balance
including
thermal
buoyancy
is
thus
✓ 2
◆
2
@P
@ v
@ v
0=
+⌘
+ 2
g⇢0 ↵v (T T0 )
2
@y
@x
@y
9
Thermal convection
•
In order to determine the thermal buoyancy, we require a 2D
equation for heat transfer via conduction and convection!
•
Turcotte and Schubert give a detailed derivation, arriving at the
following relationship for heat conduction and convection in
✓ 2
◆
2D
2
@T
@T
@T
@
T
@
T
+u
+v
=
+
2
2
@t
@x
@y
@x
@y
where + is the velocity in the , direction and - is the thermal
diffusivity
10
Thermal convection
•
In order to determine the thermal buoyancy, we require a 2D
equation for heat transfer via conduction and convection!
•
Turcotte and Schubert give a detailed derivation, arriving at the
following relationship for heat conduction and convection in
✓ 2
◆
2D
2
@T
@T
@T
@
T
@
T
+u
+v
=
+
2
2
@t
@x
@y
@x
@y
Time dependence
Convection
Conduction
where + is the velocity in the , direction and - is the thermal
diffusivity
11
Onset of thermal convection
•
For a fluid heated to temperature *1 at its
base and cooled to temperature *0 at its
upper surface thermal buoyancy will drive
convection if the viscous resistance to fluid
flow is overcome!
•
When the temperature difference *1 - *0 is
small, convection will not occur and the fluid
velocities will be + = $ = 0, and temperature
will not change with time (././ = 0) or along
the , axis (./., = 0), reducing the heat
transfer equation to
Fig. 6.38, Turcotte and Schubert, 2014
d2 Tc
=0
2
dy
where 0 indicates conductive heat transfer
12
Onset of thermal convection
•
For the thermal boundary conditions * = *0
at % = -1/2 and * = *1 at % = 1/2, the
solution to the heat transfer equation is
T1 + T0
(T1 T0 )
Tc =
+
y
2
b
where 1 is the thickness of the fluid layer!
Fig. 6.38, Turcotte and Schubert, 2014
•
Convection will begin when the temperature
* just begins to exceed the conductive
temperature *0!
•
In this case, the temperature difference *ʹ is
extremely small
T0 ⌘ T
Tc
13
Onset of thermal convection
•
The temperature difference *ʹ can be
inserted into the heat transfer and force
balance equations for a 2D incompressible
viscous fluid to determine the conditions
under which convection will occur!
•
From this, a dimensionless number known as
the Rayleigh number can be defined
Fig. 6.38, Turcotte and Schubert, 2014
⇢0 g↵v (T1 T0 )b3
Ra =
⌘
14
Onset of thermal convection
Fig. 6.39, Turcotte and Schubert, 2014
•
Instability and convection will occur
when
2 2
4⇡
2
(⇡ + 2b )3
Ra >
4⇡ 2 b2
2
where 2 is the wavelength of
convection!
Minimum at
p
= 2 2b
•
The flow is thus stable when Ra is
less than the right side of the
equation above!
•
The critical Rayleigh number
when convection begins is thus
2
Ra ⌘ Racr =
(⇡ +
4⇡ 2 b2 3
2
)
4⇡ 2 b2
2
15
Onset of thermal convection
•
For a fluid cooled from above, heated from within and with no
heat flux across its base the Rayleigh number equation is
2
5
↵
⇢
gHb
v 0
RaH =
k⌘
where 3 is the heat production per unit mass and 4 is the
thermal conductivity!
•
The equation above is appropriate for the Earth’s mantle
16
Onset of thermal convection
!
!
RaH
↵v ⇢20 gHb5
=
k⌘
•
The critical Rayleigh number for an internally heated fluid is
Racr = 2772!
•
What is the Rayleigh number for the upper mantle?!
•
•
Should it convect?!
What about the whole mantle?!
!
•
Assume )$ = 3 ⨉ 10-5 K-1, "0 = 4000 kg m-3, 3 = 9 ⨉ 10-12 W
kg-1, 4 = 4 W m-1 K-1, # = 1 ⨉ 1021 Pa s, and - = 1 mm2 s-1
17
What drives tectonic plate motions?
The driving forces of plate motion
Forsyth and Uyeda, 1975
Driving forces
Resisting forces
FDF
FRP
FTF
FCR
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
FCD
165
FSP
FDF
FSR
18
What drives tectonic plate motions?
The driving forces of plate motion
Forsyth and Uyeda, 1975
Driving forces
Resisting forces
FDF
FRP
FTF
FCR
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
FCD
165
FSP
FDF
FSR
19
Slab (or trench) pull
FSP
Forsyth and Uyeda, 1975
•
Slab pull results from the gravitational body force acting on the dense, sinking
oceanic lithosphere.!
•
It can be divided into two components:!
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
•
•
Fb1, the force resulting from the slab being colder than the surrounding mantle!
Fb2, the force resulting from the elevation of the olivine→spinel phase change!
Mathematically, we can say
FSP = Fb1 + Fb2
20
Slab (or trench) pull
FSP
•
Forsyth and Uyeda, 1975
For one half of a sinking mantle plume, Turcotte and Schubert show
✓
◆1/2
u0

Fb = 2⇢0 g↵v b(Tc T0 )
v0 2⇡u0
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
We can simplify this relationship slightly for a sinking slab to estimate the slab
pull force resulting from the relatively cold temperature of the slab
"
Reference mantle density
*
Temperature at LAB
( Gravitational acceleration
+
Horiz. velocity of upper layer
)
Coeff. of thermal expansion
$
Average vertical velocity
1
Convecting layer thickness
- Thermal diffusivity
*
Temperature at conv. center
2 Width of 2 convection cells
LAB = Lithosphere-asthenosphere boundary
21
Slab (or trench) pull
FSP
Forsyth and Uyeda, 1975
6.21 A Steady-State Boundary-Layer Theory
509
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
Here is the simplified
convection cell geometry
used for calculating the
force acting on a sinking
mantle plume!
!
•
Note that areas in each
triangle are equal, so mass is
Turcotte and Schubert, 2002
conserved, or
1 Linear velocity profiles used to model the core flow in a conv0
= u0 b
l. The areas under the triangles are equal to conserve fluid.
2
22
Slab (or trench) pull
FSP
Forsyth and Uyeda, 1975
•
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
Because a sinking slab is rigid, we can assume that +06=6$0, so the force
resulting from the slab being colder than the surrounding mantle is
✓
◆1/2

Fb1 = 2⇢0 g↵v b(Tc T0 )
2⇡u0
If we use typical values for these variables, we see that 8116is6~3×1013 N m-1!
•
To be clear, this value is the force per meter of trench length
23
Slab (or trench) pull
FSP
Heat Transfer
Forsyth and Uyeda, 1975
Fig. 4.59, Turcotte and Schubert, 2014
6.22 The Forces that Drive Plate Tectonics
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
Theortransition
fromseparating
olivine to
e Clapeyron
equilibrium curve
twospinel
phases of
al.
in the mantle generally occurs at
400-500 km depth, but the transition
e 4–58. Since
the subducted
depends
on bothlithosphere
pressure was
andformed on the
thermal structure upon subduction is given by Equation
temperature!
ndence of temperature upon depth prior to subduction is
erm given in Figure 4–56. As the subducted lithosphere
Because the descending slab is
mantle, frictional heating occurs at its upper boundary.
relatively cold, the olivine-spinel phase
ional heating were studied in Section 4–26. As discussed
transition
at lower
pressure
ture distribution
dueoccurs
to frictional
heating
– Equation (4–
(shallower
depth)
erimposed
on the initial
temperature
distribution
to give
Figure
6.43 Elevation
he slab. The result is shown in Figure 4–58.
•
Turcotte and Schubert, 2002
24 in
of the olivine–spinel phase change
Slab (or trench) pull
FSP
Heat Transfer
Forsyth and Uyeda, 1975
6.22 The Forces that Drive Plate Tectonics
Warm slab
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
Fig. 4.59, Turcotte and Schubert, 2014
Cold slab
•
Theortransition
fromseparating
olivine to
e Clapeyron
equilibrium curve
twospinel
phases of
al.
in the mantle generally occurs at
400-500 km depth, but the transition
e 4–58. Since
the subducted
depends
on bothlithosphere
pressure was
andformed on the
thermal structure upon subduction is given by Equation
temperature!
ndence of temperature upon depth prior to subduction is
erm given in Figure 4–56. As the subducted lithosphere
Because the descending slab is
mantle, frictional heating occurs at its upper boundary.
relatively cold, the olivine-spinel phase
ional heating were studied in Section 4–26. As discussed
transition
at lower
pressure
ture distribution
dueoccurs
to frictional
heating
– Equation (4–
(shallower
depth)
erimposed
on the initial
temperature
distribution
to give
Figure
6.43 Elevation
he slab. The result is shown in Figure 4–58.
•
Turcotte and Schubert, 2002
25 in
of the olivine–spinel phase change
Slab (or trench) pull
FSP
Heat Transfer
Forsyth and Uyeda, 1975
6.22 The Forces that Drive Plate Tectonics
Warm slab
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
Fig. 4.59, Turcotte and Schubert, 2014
Cold slab
•
Theortransition
fromseparating
olivine to
e Clapeyron
equilibrium curve
twospinel
phases of
al.
in the mantle generally occurs at
400-500 km depth, but the transition
e 4–58. Since
the subducted
depends
on bothlithosphere
pressure was
andformed on the
thermal structure upon subduction is given by Equation
temperature!
ndence of temperature upon depth prior to subduction is
erm given in Figure 4–56. As the subducted lithosphere
Because the descending slab is
mantle, frictional heating occurs at its upper boundary.
relatively cold, the olivine-spinel phase
ional heating were studied in Section 4–26. As discussed
transition
at lower
pressure
ture distribution
dueoccurs
to frictional
heating
– Equation (4–
(shallower
depth)
erimposed
on the initial
temperature
distribution
to give
Figure
6.43 Elevation
he slab. The result is shown in Figure 4–58.
•
Fig. 6.43, Turcotte and Schubert, 2014
26 in
of the olivine–spinel phase change
Slab (or trench) pull
FSP
Forsyth and Uyeda, 1975
•
6.22 The Forces that Drive Plate Tectonics
Thus, the driving force from the phase
transition depends on the location of
the phase transition isotherm *os in
the sinking slab!
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
•
The driving force from the phase
transition is
✓
◆1/2
2(Tc T0 ) ⇢os

Fb2 =
⇢0
2⇡u0
The olivine-spinel phase transition
increases density by ~270 kg m-3
@
A"
Slope of Clapeyron curve
Fig. 6.43, Turcotte and Schubert, 2014
27 in
Gravitational acceleration
Figure 6.43 Elevation of the olivine–spinel phase change
Slab (or trench) pull
FSP
Forsyth and Uyeda, 1975
Using typical values for a sinking slab,
the gravitational body force due to
the olivine-spinel phase transition in
the slab 812 is ~1.5×1013 N m-1, or
about half of that from the difference
in temperature alone!
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
6.22 The Forces that Drive Plate Tectonics
!
•
Going back to our original equation
for the slab pull force,
F = Fb1 + Fb2
SP
we find a total slab pull force of
8E'6=6~4.5×1013 N m-1
Fig. 6.43, Turcotte and Schubert, 2014
28 in
Figure 6.43 Elevation of the olivine–spinel phase change
Ridge push
FRP
Forsyth and Uyeda, 1975
•
Ridge push results from the elevation of oceanic ridges relative to the
seafloor!
•
The difference in elevation results in a pressure head that drives the plate away
from the ridge!
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
This motion may also be viewed as gravitational sliding!
!
•
To calculate the ridge push force 8G' we must consider the forces acting on
the top (81), bottom (82) and side (83) of the oceanic lithospheric plate:
FRP = F1
F2
F3
29
Ridge push
FRP
6.22 The Forces that Drive
Plate
Tectonics
Forsyth
and Uyeda,
1975
With this force balance in mind, we
can see!
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
Fig. 6.44, Turcotte and Schubert, 2014
1. The horizontal force on the base
of the plate must be equal to the
integrated lithostatic pressure in
the mantle along RD!
2. The horizontal force on the top of
the plate must be equal to the integrated hydrostatic pressure along AB!
Figure 6.44 Horizontal forces acting on a section of the ocean, li
and mantle at an ocean ridge.
3. The horizontal force acting on the lithospheric section BC is equal to the
integrated pressure in the lithosphere!
•
The integrated pressure force on the upper surface of the lith
Note that this pressure
that resulting
frombecause
the overlying
equal toshould
F4 , theinclude
net pressure
force on AB,
the section o
oceanic water must be in equilibrium. Thus we can integrate the hydrostat
30
the water to obtain
!
Ridge push
FRP
6.22 The Forces that Drive
Plate
Tectonics
Forsyth
and Uyeda,
1975
Fig. 6.44, Turcotte and Schubert, 2014
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
!
•
At a constant depth H
P = ⇢gz
assuming constant density of the overlying material!
•
Integrated over a
Pint =
Z
Figure 6.44 Horizontal forces acting on a section of the ocean, li
depthand
range
H1 to
mantle
at Han
2
ocean ridge.
z2
The integrated pressure force on the upper surface of the lith
⇢gzdz
equal to F4 , the net pressure force on AB, because the section o
z1
must be in equilibrium. Thus we can integrate the hydrostat
31
the water to obtain
!
Ridge push
FRP
6.22 The Forces that Drive
Plate
Tectonics
Forsyth
and Uyeda,
1975
•
Going back to the original force
balance equation for ridge push,
we see
FRP = F1
•
F2
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
Fig. 6.44, Turcotte and Schubert, 2014
F3
After some mathematical substitutions
and integrations we find
F
RP = g⇢m ↵v (T1
"
"
*

2 ⇢m ↵v (T1 T0 )
Figure
T0 ) 6.44
1 +Horizontal forces acting on a section
t of the ocean, li
and mantle at an
⇡ ocean
(⇢ridge.
⇢w )
m
Temperature
surface
The integrated*pressure
force onat
theplate
upper
surface of the lith
force onplate
AB, because the section o
Water density equal to F4 , the net
Age of oceanic
/ pressure
must be in equilibrium. Thus we can integrate the hydrostat
32
Mantle temperature
the water to obtain
Mantle density
!
Ridge push
FRP
6.22 The Forces that Drive
Plate
Tectonics
Forsyth
and Uyeda,
1975
Fig. 6.44, Turcotte and Schubert, 2014
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
Using typical values, we find the ridge
push force is
8G'6=6~4×1012 N m-1!
•
Note that this is about an order
Figure
6.44
Horizontal forces acting on a section of the ocean, li
of magnitude smaller than
the
slab
and mantle at an ocean ridge.
pull force
The integrated pressure force on the upper surface of the lith
equal to F4 , the net pressure force on AB, because the section o
must be in equilibrium. Thus we can integrate the hydrostat
33
the water to obtain
!
Drag force
FDF
•
Fluid
Mechanics
Forsyth
and Uyeda, 1975
The drag force on the base of the oceanic
lithosphere can both drive and resist plate
tectonics, depending on the relative motion
between the plate and the underlying mantle!
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
416
!
•
If we assume that the underlying mantle
Fig. 6.2, Turcotte and Schubert, 2014
resists or drives plate motion by viscous
flow across a fixed-thickness layer, the drag force on the plate is simply
FDF = ⌘as
#
Viscosity of asthenosphere
A+ Velocity difference
u
L
h
ℎ Thickness of viscous layer
J
Length of the plate
34
Figure 6.2 One-dimensional channel flows of a constan
Drag force
FDF
416
Fluid
Mechanics
Forsyth
and Uyeda, 1975
Downloaded from http://gji.oxfordjournals.org/ at Dalhousie University on September 1, 2013
•
Again using typical values, we find the drag
force is
8K86=6~1×1013 N m-1!
•
Fig. 6.2, Turcotte and Schubert, 2014
Note that this value is similar in magnitude to the slab pull force
35
Figure 6.2 One-dimensional channel flows of a constan
Recap
•
Thermal convection in the mantle results from buoyancy
forces as the mantle is heated and undergoes thermal
expansion!
!
•
Slab pull and the drag force are similar and about an order of
magnitude larger than the ridge push force
36
References
Forsyth, D., & Uyeda, S. (1975). On the Relative Importance of the Driving Forces of Plate Motion*.
Geophysical Journal International, 43(1), 163–200. doi:10.1111/j.1365-246X.1975.tb00631.x
37