Kinematics II: Euler’s poles and triple junctions

Important: This chapter follows mainly on chapter 2And 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.

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v = ωRsinθ,

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 APlate 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 CB and BA are known, one can use vector summation to compute CA as:

Alternatively one can use (why?):

Plate tectonics on a spherical earth: Determination of rotation poles and rotation axes for plates using vector summation

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C

r ω A =C

r ω B +B

r ω A .

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C

r ω A +A

r ω B +B

r ω C = 0.

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r = ˆ k ,

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.

Capemendocino

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=ridgeT=trenchF=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.€

seismogram = medium⊗ instrumental response ⊗ mechanism

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 900-, 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).