Kinematics II: Euler’s poles and triple junctions
Important
: This chapter
follows mainly on chapter 2
And section 4.2.8 in
Fowler’s textbook.
Plate tectonics on a spherical earth: Rotation axes and rotation poles
Euler’s fixed point theorem
: “Every displacement from one position to
another on the surface earth can be regarded as a rotation about a
suitably chosen axis passing through the center of the earth.”
The
axis of rotation
is the suitably chosen axis passing through the
center of the earth.
The
poles of rotation
or the
Euler’s poles
are the two points where the
axis of rotation cuts through the earth surface.
Plate tectonics on a spherical earth: Angular velocity and relative
velocity
•
The relative velocity,
, of a certain point on the earth surface is a
function of the angular velocity,
, according to:
where R is the earth radius and
is the angular distance between the
pole of rotation and point in question.
•
Thus, the relative velocity is equal to zero at the poles, where
=0
degrees, and is a maximum at the equator, where
=90 degrees.
•
The relative velocity is constant along small circles defined by
=constant.
•
Note that large angular velocity does not mean large relative velocity.
Plate tectonics on a spherical earth: Determination of rotation poles and
rotation axes for plates on either side of mid

ocean ridges
Transform faults are arcs of small circles
about a rotation pole (why?). The rotation
pole, therefore, must lie somewhere on a
great circle that is perpendicular to that
small circle. So if two transform faults are
available, the intersection of the great
circles marks the position of the rotation
pole.
Plate tectonics on a spherical earth: Determination of rotation poles and
rotation axes for plates on either side of mid

ocean ridges
The spreading rate of mid ocean
ridge changes as a function of
sine the angular distance,
, from
the rotation pole. Thus if the
spreading rates at various points
along the plate boundary can be
measured, the rotation pole may
be estimated.
Plate A
Plate B
Because the angular velocities are in fact vectors pointing at the
direction of the rotation axes, they can be written as:
where k is a unit vector in the direction of the rotation axis and
is the
(scalar) angular velocity. In analogy to plate kinematics on flat earth (a
few slides back), if
C
B
and
B
A
are known, one can use vector
summation to compute
C
A
as:
Alternatively one can use (why?):
Plate tectonics on a spherical earth: Determination of rotation poles and
rotation axes for plates using vector summation
Plate boundaries can change with time: The Farallon plate
The formation of new plates and the destruction of existing plate are the
most obvious reasons why plate boundaries and relative motion
change.
For example, the Farallon plate
subducted underneath north
America 20

30 million years ago.
Plate boundaries can change with time: The Farallon plate
Animation from: http://www.seismo.unr.edu/ftp/pub/louie/class/333/atwater
Plate boundaries can change with time: The Farallon plate
Seismologist think that they can
still see the remaining of the
Farallon plate underneath N.
America.
The blue (i.e., faster)
anomaly.
Triple junctions: Examples
Figure from NASA Goddard Space Flight Center
The topography of the Atlantic Ocean bottom is shown as if the ocean had been removed. The blue
lines indicate the edges of the ocean tectonic plates. The yellow dots indicate locations of
earthquakes that have occurred in the period 1960 to 1985. The red triangles are the locations of
volcanic eruptions that have occurred in the period 1980 to 1995. Both the earthquakes and the
volcanic eruptions follow the plate boundaries.
Cape
mendocino
Afar
Azores
Triple junctions: Examples
Figure from
Hugh Rance site:
geowords.com
The Afar rift
Triple junctions: Examples
The Azores triple junction
Figure from: www.ija.csic.es/gt/ivone/research_AFEU.html
Triple junctions: Stability issues
•
Triple junction is a point at which three plates meet.
•
A triple junction is stable if the relative motion of the three plates and
the azimuth of their boundaries do not change in time.
•
An unstable triple junction exist only momentarily before evolving to a
different geometry.
For example, triple
junction between
three ridges is always
stable

why?
Triple junctions: Stability issues
Geometry and stability of all
possible triple junctions (after
McKenzie and Morgan,
1969)
R=ridge
T=trench
F=transform
Fault plane solution
Most plate boundaries are offshore and are thus inaccessible to
geologist. How then the relative plate motion along these boundaries
can be inferred?
The seismograms produced at a given location are a function of the
medium along the ray path, the response of the recording instrument
and the source mechanism, i.e. thrust versus strike

slip:
By studying the polarity of the first waves arriving at different stations,
one can determine both the fault mechanism and the orientation of the
fault plane.
Fault plane solution
Imagine a strike

slip fault within a flat earth and stations A through E
distributed on the ground surface at different distances from the fault
trace. The first P

wave arrival at each of the stations will be either
compressional or dilatational. In this example the distribution of the
compressional and dilatational P

waves falls into 4 quadrants.
Note that station D receives no P

wave energy.
Fault plane solution
Because earth is a sphere, one must work in spherical coordinates.
•
We imagine a sphere centered on and surrounding the focus of an
earthquake, this imaginary sphere is referred to as the
focal sphere
.
•
The rays traveling from the source to the station intersect the lower
hemisphere of the focal sphere at an angle I of the vertical and an
azimuth A.
•
If the P

wave velocity structure is
known, the rays arriving to each station
can be traced back to the and the angle
of departure can be obtained.
Fault plane solution
•
The azimuth is easily measured geographically.
•
The lower focal hemisphere is then projected onto a horizontal
plane.
•
The polarity of the first motion at each seismogram is then
plotted on the projection.
Fault plane solution
•
The first arriving P

wave for a seismograph close to the station travels
almost horizontally, thus i=90 degrees (red path).
•
The first arriving P

wave for a seismograph at the opposite side of the
earth travels almost vertically from source, thus i=0 degrees (blue path).
•
Thus, nearby stations plot close to
the edge of the projection, and
distant station plot near the
projection center.
Fault plane solution
•
The four quadrants are separated by two orthogonal planes, or
focal
planes
.
•
One nodal plane represents the fault plane, the other is an
auxiliary
plane
.
•
From the fault plane solution alone one cannot determine which plane
is which.
•
The radiation pattern to the right can
either represent a EW trending vertical
right lateral strike slip fault, or a NS
trending vertical left lateral strike slip fault.
Fault plane solution
The previous example corresponds to the rare situation in which the
fault plane is exactly vertical. If the fault is not vertical, the fault plane
solution of a strike

slip fault still has four quadrants, but the nodal planes
do not pass through the origin. Instead they plot as orthogonal great
circles offset from the origin by 90
0

, with
being the fault dip.
Fault plane solution
•
Fault plane solutions for a
normal fault (left) and a
reverse faults (right).
•
In these cases the
ambiguity is only with
respect to the fault dip, but
not with respect to the
strike.
Fault plane solution
Rarely do the fault mechanisms fall into the category of pure strike or
dip slips. The examples below show fault plane solution of predominant
normal slip (left) and a predominant reverse slip (right). In both case
there is only a strike

slip component in addition to the dip

slip
component.
Fault plane solution
The examples below are for explosion (left) and implosion (right):
In either case, the fault plane solution possess only one nodal plane.
Fault plane solution
The pattern of seismic waves from some earthquakes cannot be
produced by slip along a planar fault surface. The focal plane
solution of such earthquakes is referred to as
non

double

couple
.
The simplest explanation for such earthquake mechanisms is that
they are complex, with slip occurring on two or more non

parallel
fault surfaces.
Here are a few examples for non

double

couple solutions from a
volcanic complex in Iceland (from
Julian et al., 1997).
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