STRUCTURAL GEOLOGY EXERCISES
with Glaciotectonic Examples – Part IV

James S. Aber, Professor Emeritus
Emporia State University, Kansas

10. FAULT GEOMETRY

Basic geometry of faults

Nearly all types of rocks, from granite to chalk to glacier ice, contain fractures of some kind. These fractures range from mm-size features to breaks that may extend for 100s km. A fault is simply a fracture along which visible movement or displacement of the opposing rocks has taken place. The stresses that cause faulting are generated in a great many ways: plate movement, meteorite impact, glacier pushing, volcanic activity, soft sediment collapse, salt-dome uplift, etc. Most faults are strictly crustal structures of limited depth, because under high pressure and temperature rocks deform by plastic flow rather than by fracturing. The deepest known faults are associated with subduction zones extending several 100 km into the mantle.

In many situations, faults may be considered as roughly planar features. Thus, the orientation of a fault is given by strike and dip. Based on the fault's dip and the relative movement of blocks along the fault, four basic kinds of faults are recognized.

The displacements of these faults are relative movements only. It is usually impossible to tell which side of a fault actually moved, or if both sides moved. Normal, reverse, and thrust faults move primarily parallel to dip of the fault plane, so they are called dip-slip faults, in contrast to strike-slip faults. Many faults represent a combination of dip-slip and strike-slip movement, and some faults have undergone rotational slip, so that displacement varies along the fault.

Slip is the actual measurement of displacement between two originally adjacent points in the fault plane—see Fig. 10-1. The net slip or total slip is the direct distance between the two points measured in whatever units are convenient: feet, m, miles, or km. Net slip is made up of two components: strike slip and dip slip, which are measured parallel to the strike and dip of the fault plane. The angle formed between the net-slip vector and strike of the fault is called rake.

Figure 10-1. Block diagram illustrating slip and rake for a normal fault. Points A and B were originally adjacent.

Although it is often possible to recognize distinctive strata offset along a fault, it may be impossible to identify unique points from which to measure slip. Where slip proves impossible to determine, other measurements of fault displacement, such as throw and heave, could be made using offset strata—see Fig. 10-2. Throw is the vertical distance of separation between matching offset beds, and heave is the horizontal distance of separation. Both measurements are made in a vertical cross section which is perpendicular to strike of the fault.

Figure 10-2. Cross section of a thrust fault illustrating measurement of throw and heave on the top of a limestone bed. Cross section is vertical and oriented perpendicular to the fault strike.

Fault Patterns

Faults are often found in crossing sets which form distinctive patterns. The displacement of a single fault may be small, but the combined displacements of many faults within a well organized set may be substantial. The conjugate fault pattern is a common one in which two sets of similar faults cross each other at a consistent angle—see Fig. 10-3.

Figure 10-3. Cross sections showing idealized normal and thrust conjugate fault patterns. Large arrows show overall sense of horizontal displacement.

Conjugate normal-fault sets dip in opposite directions, and the overall result is thinning and lengthening of the rock mass. Conversely, conjugate thrust-fault sets produce overall thickening and shortening of the rock mass. Both normal and thrust conjugate patterns represent shear fractures (compare with Fig. 9-4), the difference being that õ1 is vertical for normal faults and horizontal for thrust faults.

Another category of fault patterns involves the rotation of fault blocks during displacement. A single set of normal fault blocks which undergo rotational tilting results in overall thinning and lengthening of a rock mass—see Fig. 10-4. Tilted fault blocks produce crustal extension, so they often accompany conjugate normal faulting. Stacked or imbricated thrusts also undergo rotational tilting during thrusting—see Fig. 10-5. This style of faulting may accompany conjugate thrusts and results in shortening and thickening of the rock mass.

Figure 10-4. Cross sections illustrating development of idealized tilted normal fault blocks. Large arrows indicate overall sense of horizontal displacement.
Figure 10-5. Cross sections illustrating development of idealized imbricated thrust blocks. Large arrow shows overall sense of horizontal displacement.

Recognition of faults

Several kinds of evidence may indicate the presence of a fault. However, a single indicator may not be conclusive, so additional features should be examined to verify the existence of a fault. To begin with, every fault represents a structural break or discontinuity within a body of rock. A discontinuity is usually shown by an abrupt change in lithology or rock age, especially where the break cuts across stratification or some other primary structure in the rock. Such a discontinuity could represent several possible structures—fault, unconformity, intrusion, or sedimentary channel filling. Consideration of the general geological setting and reference to other kinds of evidence should indicate whether the discontinuity is truely a fault.

Faults and fractures.

In a stratified sequence, faulting may create the repetition or omission of particular strata. Thrusting causes the stacking and repeating of strata, whereas normal faulting may "cut out" strata. Such stratigraphic anomalies are the easiest way to recognize subsurface faulting in well logs or seismic sections. Of course, the regional stratigraphy must be known and must be reasonably consistent in order to recognize that strata are either repeated or omitted at a particular location. An unconformity could explain local omission of strata, and an overturned fold would produce an inverted repetition of strata.

The fault zone itself may possess one or more features, such as slickensides and striations, resulting from the grinding and abrasion of rock during movement. Fault gouge is a finely powdered rock debris; breccia consists of jumbled, angular stones broken from the fault walls. Under conditions of high pressure and temperature, a banded metamorphic rock called mylonite may develop. Mylonite consists of crushed rock debris strongly welded together with a fault-parallel foliation. Shearing in strata adjacent to the fault often results in small folds, called drag folds, whose orientation indicates the direction of slip.

Reverse fault with drag folding in Permian red beds with gypstone layers. Overview (left) and close-up shot (right); cliff is about 80 feet (~25 m) tall. Lea Lake, Bottomless Lakes State Park, New Mexico.

Fault zones provide easy avenues for movement of groundwater, hydrothermal fluid, or magma. Consequently, secondary mineral deposits, such as pegmatites, and quartz or calcite veins, commonly form along faults. Important economic deposits of gold, silver, copper, uranium, and unusual minerals are often concentrated in fault zones. Faults are frequently marked at the surface by springs and distinctive vegetation.

Thrust fault on Mount Maxwell, Culebra Range
 Sangre de Cristo Mountians, southern Colorado
Left: brecciated zone with contorted blocks of displaced sandstone. Right: highly polished slickensides on fault surface.
Highly fractured sandstone with quartz veins. Such fracturing and veins are quite common for faults in hard rock. Comb for scale is 5 inches long.
Left: copper-bearing mineral (malachite?) filling veins within sandstone is evidence for hydrothermal activity. Right: chattermarks on fault surface.

Problem

The island of Møn, southeastern Denmark, is undoubtedly among the most famous glaciotectonic locales in the world—see title image. The island is underlain by soft, white, Cretaceous chalk. The chalk along with overlying drift were severely disturbed by ice pushing during multiple advances by the late Weichselain glaciation (~32,000 to 13,000 years ago). Enormous masses of chalk were ripped up from Fakse Bugt and Hjelm Bugt and thrust up as hills on Møn—see Fig. 10-6.

Figure 10-6. Locality map, Møn, southeastern Denmark. S = City of Stege and SN = Stege Nor, a lagoon. Dashed line shows position of ice-shoved hill in which Hvideklint is cut. Adapted from Aber (1979).

The major structures are large thrust faults beneath uplifted and displaced chalk masses. Smaller, complex folds and faults accompany the thrusts. The main structural disturbances on Møn were created by late Weichselian glaciations from two general directions (Aber and Ber 2007).

  1. Main advance coming from the northeast, the Swedish ice stream, which may have experienced repeated small advances and retreats during the overall northeasterly overspreading of Møn (Berthelsen et al. 1977).

  2. Southerly to southeasterly advances by the subsequent Baltic ice lobe, which caused the primary deformations at Møns Klint (Pedersen and Graversen 2021). This ice lobe was responsible for the curved shape of southern Møn along Hjelm Bugt.

Hvideklint is a cliff over 1 km long and 20 m high eroded through multiple upthrust chalk bodies on the southern coast of Møn—see Fig. 10-7. The cliff cuts through an ice-shoved hill that exceeds 35 m height inland from the cliff. More than a dozen thrust-fault blocks of chalk and drift underlie Hvideklint to a depth of 70-80 m below sea level (Pedersen and Graversen 2021). Additional exposures of deformed glacial strata are found to the northeast at Vagtbo Bakke and to the southwest along Hjelm Nakke.

Undulating terrain of ice-shoved hill inland from Hvideklint. Hill tops exceed 35 m (115 feet) above sea level. Kite aerial photograph looking southward by J.S. and S.E.W. Aber (2023).

Figure 10-7. Measured profile of Hvideklint section as it appeared in 1979. Note 2X vertical exaggeration; scale in m; adapted from Aber et al. (1989).

Cliff exposures at Hvideklint, Møn
 As photographed in 1979
Overview of a large chalk raft (left) that is pushed over deformed drift and chalk along a major thrust fault next to the scale pole. Between meters 830 and 900 of the measured section. Chalk resting on smeared glacial sediment (right). Note thin slivers of glacial sediment sheared into base of chalk mass. Fault #5 in Table 10-2.
Highly folded, faulted, and contorted till, sand, and gravel are "sandwiched" between chalk bodies (left). Between meters 860 and 900 of the measured section. Scale pole at center is marked in 20-cm stripes. Western end of Hvideklint (right) displays extremely stretched and folded chalk-till mélange.

Cliff exposures along Hjelm Nakke, Møn
 As photographed in 2023
Folded sand within till (left). Knitting needle indicates the fold axis. Overturned fold of sand (right) with a till core and expanded upper limb in silty fine sand (swallow nests).
Isoclinal, nearly recumbent fold of sand and till. Overview (left) and close-up shot (right). Knitting needle (^) indicates the fold axis. Fold #19 in Table 10-1.

The following structural data represent ice-pushed folds and faults found in glacial strata and chalk at Hvideklint, Vagtbo Bakke, and Hjelm Nakke, southern Møn, as measured in 1979 (Aber 1979) with additional measurements collected by the author in 2023—see Tables 10-1 and 10-2.

Table 10-1. Fold-axis data from Hvideklint,
Vagtbo Bakke, and Hjelm Nakke.
Folds Trend/Plunge
1. Pinched syncline
overturned to SW 		130/10
2. Isoclinal fold 120/06
3. Small recumbent fold 162/10
4. Overturned tight synform 118/06
5. Tight synform 128/12
6. Tight fold in clay bed 110/00
7. Isoclinal fold in sand 164/00
8. Isoclinal fold in sand 138/06
9. Isoclinal fold in sand 342/40
10. Small open fold parallel to large overturned fold 290/08
11. Recumbent isoclinal
fold of till in sand 		300/08
12. Kink fold on limb of overturned fold 125/10
13. Large overturned fold 116/06
14. Overturned fold in chalk-banded till 265/40
15. Small overturned and recumbent fold 330/20
16. Small overturned and recumbent fold 255/10
17. Small overturned fold in sand 155/10
18. Small overturned fold in sand 155/15
19. Large isoclinal fold in sand 85/06
20. Large overturned fold with bulbous limb 135/10
21. Large overturned fold in flint breccia 135/14
22. Drag fold in till along chalk/till fault 260/26

Table 10-2. Fault and fold-limb data from Hvideklint and Hjelm Nakke. Strike/dip values given according to the right-hand rule.
Faults and fold limb Strike/dip
1. Small normal fault 118/62
2. Normal fault 118/68
3. Small normal fault 100/58
4. Small thrust fault 296/16
5. Banded thrust zone below large chalk mass 342/32
6. Limb of large syncline in chalk 110/40
7. Reverse fault between chalk and till 130/75

  1. Plot all data on an equal-area stereonet. Fold axes plot as points; fault planes and fold limbs plot as arcs. Stereonet 11 is recommended given the large number of data entries. Enter values for two datasets: a) linear for fold axes, and b) planar for faults and fold limb. You may put these onto the same display for visual inspection.

  2. Assume that faults strike and folds trend perpendicular to ice movement and that thrust faults dip upice. Is there general agreement between orientations of folds and faults? Explain your interpretation.

  3. From your stereonet plot, determine the general direction of ice advance. Do you find any evidence for more than one direction of ice pushing at Hvideklint, Vagtbo Bakke and Hjelm Nakke? Explain your interpretation.

  4. From which direction and what location were the chalk masses now at Hvideklint probably derived?

  5. Which glacial advance caused the primary disturbances at Hvideklint? Is there evidence for more than one episode of glaciotectonism along the southern coast of Møn? Explain your answer.

References

11. GEOMETRY OF PLATE MOTION

Plate rotation

The movement of rigid lithospheric plates over the surface of the Earth is properly understood in terms of spherical geometry. The Earth may be treated as a perfect sphere with a circumference of roughly 40,000 km for purposes of plate-motion calculations. The relative motion of any two plates which meet along a common boundary—trench, mid-ocean ridge, or transform fault—represents a rotation of the plates about an imaginary axis running through the center of the Earth.

The example shows two plates (A and B) separated by spreading ridges and transform faults, and rotating about an axis running through the center of the Earth—see Fig. 11-1. The rotation axis intersects the surface at two points called poles of rotation on opposite sides of the globe. The positions of the poles of rotation are specified by latitude and longitude coordinates. As plates may move in any direction, rotation axes may occur in all orientations. Hence, rotation poles may be located anywhere on the surface of the Earth.

Figure 11-1. Geometric elements of plate motion. Spreading ridges shown by double lines; transform boundaries are single lines. Lr = latitude of rotation; Mr = longitude (meridian) of rotation; C = center of Earth; P = pole of rotation.

A great circle is one whose center coincides with the center of the Earth; longitude (meridian) lines and the equator are examples of great circles, whose radii and circumferences are all the same and equal to those of the spherical Earth. Any other circle drawn on the surface of the Earth, whose center does not coincide with the Earth's center, is called a small circle. Latitude lines north and south of the equator are small circles, whose radii and circumferences are less than for a great circle.

A series of great circles drawn through the rotation poles for any pair of plates form so-called meridians or longitudes of rotation (fig. 11-1). Likewise, a series of concentric small circles and an equator constructed around the rotation poles are called circles or latitudes of rotation. Latitudes and longitudes of rotation are useful for analyzing plate motion, but they should not be confused with the standard latitude and longitude lines shown on maps. The latitudes and longitudes of rotation are related solely to the poles of rotation and, thus, may be positioned in any orientation on the Earth's surface.

Linear and angular velocity

Velocity of plate movement may be described in two manners: (1) linear or surface velocity at a point and (2) angular velocity of rotation about the axis. Linear velocity is the more familiar, such as miles or km per hour, with plate velocities usually given in cm per year. However, linear velocity is not constant for a plate, being less near the poles of rotation and greater near the equator of rotation. For this reason, it is preferable to measure plate velocity as an angular movement about the rotation axis in degrees or radians per million years. Angular velocity is constant throughout the plate.

Angular velocity is typically symbolized as wAB, where the subscripts (A and B) refer to the two plates involved, with the first one assumed to be fixed in position for purposes of calculations. WAB reads literally, the angular velocity with respect to A of B. To determine the sense of rotation—clockwise or counterclockwise—the so-called right-hand rule is used. The fingers of an imaginary right hand positioned at the center of the Earth with the thumb pointing northward along the axis of rotation would curl in a positive or clockwise direction.

In the example (fig. 11-1), plate A is assumed to be stationary, while plate B is moving toward the southeast; according to the right-hand rule plate B is moving clockwise. For example, wAmAf, the angular velocity with respect to America of Africa, is +0.37° per million years (LePichon 1968).

The nature of the boundary between two plates depends on the position of the boundary relative to the poles of rotation for the pair of plates. Where the boundary is parallel to latitudes of rotation, a transform fault is formed (fig. 11-1), and the plates slide past each other side by side. Two plates which are diverging develop a ridge or spreading boundary which is parallel to a longitude of rotation.

The positions of the rotation poles for a pair of plates are easily found by reference to spreading ridge segments or transform faults developed along their common boundary. Consider, for example, that portion of the Mid-Atlantic Ridge separating North America and Africa. Great circles drawn parallel to ridge segments and perpendicular to transform faults all intersect at the poles of rotation. In this case, the poles of rotation are located at 58° N, 37° W and 58° S, 143° E (LePichon 1968).

Age of North Atlantic oceanic crust. NOAA
image obtained from Wikimedia Commons.

An ocean trench marking a convergent plate boundary may assume any orientation intermediate between latitudes and longitudes of rotation. For this situation, the poles of rotation could be located only by reference to a third plate which meets the other two at a triple junction and which has ridge or transform boundaries with the other two plates (LePichon 1968).

Given w and the poles of rotation for a pair of plates, one of which is assumed fixed in position, the linear velocity at any point on the mobile plate may be found. The maximum linear velocity occurs on the equator of rotation, which represents a great circle of 40,000 km circumference. One degree of a great circle equals approximately 111 km. Continuing our example, wAmAf = 0.37° or 41 km per million years (4.1 cm per year) on the equator of rotation. Other points on the plate move at slower linear velocities depending on their latitudes of rotation.

Figure 11-2. Angular and linear velocity of a point (Z) on a plate (B). Cr = colatitude of rotation; R = radius of Earth.

The second example shows two continents (A and B) separated by a spreading ocean basin—see Fig. 11-2. Linear velocity of the point (Z) marked on continent B depends on wAB, which is constant for the plate, and on the angular distance from the nearest pole of rotation to the point. The angular distance between the pole and the point is called the colatitude of rotation (Cr). It is measured along the longitude of rotation which connects the pole and the point. Colatitude is the complement of the latitude of rotation. The circumference of any latitude or colatitude of rotation is given by:

6.282R • cos(Lr), or 6.282R • sin(Cr)

where:

R = radius of the Earth, approximately 6366 km.
Lr = latitude of rotation.
Cr = colatitude of rotation (Cr + Lr = 90).

For instance, a point on the 35° latitude of rotation (Cr = 55°) would move at 82 percent of the equatorial linear velocity; at 65° (Cr = 25°), only 42 percent of the equatorial velocity occurs. Linear velocity is zero at the poles of rotation, which simply turn in a circle without moving in position.

Haversine function

In order to make the conversion between angular and linear velocities for any given point on a plate, it is necessary to know the angular distance (colatitude) between the point and the nearest pole of rotation. This difficult problem in spherical trigonometry is simplified by using the haversine (meaning half of the versed sine) function.

haversine(A) = hav(A) = [1-cos(A)]/2

The haversine value increases from 0 to 1 for angles of 0° to 180°; it is always positive and never greater than 1. Table 11-1 gives natural haversine values for 0° to 180°.

Table 11-1. Haversine
values for 0° to 180°.

To calculate the angular distance between two points, the map (globe) latitude and longitude coordinates of each must first be determined. By convention, northern latitudes are positive, and southern latitudes negative; longitude is given from 0° to 360° measuring eastward (clockwise) from the Prime Meridian at Greenwich, England. The following formula is used (Bradley 1942).

hav(D) = [cos(L1) • cos(L2) • hav(Md)] + hav(L1 - L2)

where:

D = angular distance between points 1 and 2 (= Cr).
L = latitudes of points 1 and 2.
Md = difference in longitudes between points 1 and 2 (using the lesser of M1 - M2 or M2 - M1).

The angular distance (D) between any point on the plate and the nearest pole of rotation is equivalent to the colatitude of rotation (Cr). Thus, a general formula for relating linear and angular velocities of a plate is:

V = 11.1w • sin(Cr)

where:

V = linear velocity of a given point in cm per year.
w = angular velocity in degrees per million years.
Cr = colatitude of rotation for the point (= D, between the point and the nearest pole of rotation).

Problem

The Mid-Atlantic ridge in the vicinity of Iceland (see above) separates the North American and European plates. The American-European pole of rotation is located at 78° N, 102° E (LePichon 1968). Active seafloor spreading is taking place along the Reykjanes ridge—see Fig. 11-3. A hot spot is presently located beneath east-central Iceland at about 65° N, 17° W, as shown by high levels of volcanic and seismic activity and by a pronounced gravity anomaly.

Figure 11-3. Map of the Iceland-Faroe region of the North Atlantic. Position of the North American-European plate boundary shown by Reykjanes Ridge; approximate position of Iceland hot spot shown by asterisk. Depth contours on seafloor in fathoms (100 f = 183 m).

Volcanic features in Iceland
Eldgjá scoria deposit (left) in south-central Iceland. The scoria was erupted from a fissure vent about 1000 years ago. Laki lava flows (right), southeastern Iceland. The eruption at Lakagígar in 1783-84 was the largest such historical event in the world. The rough lava flows are now covered by soft moss, but the terrain remains largely impassable today.
Relatively old Neogene basalt flows (>3 million years old) in southeastern Iceland. Water fall at Fagifoss (left), and cliff behind the Smyrlabjörg farm (right). Photographs © J.S. Aber.

Volcanoes of Iceland.

The Faroe Islands, located at 62° N, 7° W, are composed largely of basalt flows and pyroclastic materials with an exposed thickness above sea level totaling nearly 3 km. These strata were deposited during three cycles of eruptions followed by dike intrusions over what is now the Iceland hot spot during the Paleocene and Eocene (~62 to 54 million years ago). The Faroe platform was originally slightly larger than Iceland, but later erosion and subsidence have reduced the islands greatly to just 1400 km2 land area (Rasmussen 1982; Humlum et al. 2023).

Faroe Islands geology
Cliff section at Gjógv at the northern end of Streymoy Island. Overview (left); note people at lower right for scale. Closer shot (right) showing a vertical dike (*) that cuts through layers of basalt and pyroclastic strata of the Malinstindur Formation (middle basalt series).
Villingadalsfjall mountain and village of Vidareidi on the island of Vidoy, northernmost Faroe Islands. Overview (left); mountain peak is 841 m above sea level. Closer shot (right) showing basalt lava flows on coastal cliff and steep hill side. Enni Formation (upper basalt series).
Cliff, cleft, sea caves, and isolated pillar (left) on the eastern coast of Vágar. Close-up view of pillar (right) shows well-developed columnar jointing in thick basalt flows of the Malinstindur Formation (middle basalt series). Photographs © J.S. Aber.

Geology of the Faroe Islands.

For purposes of calculation, assume an average age of 60 million years for the Faroe Islands. The North American, European, and African plates all meet on the Mid-Atlantic Ridge at a triple junction (29° N, 30° W) near the Azores Islands.

Calculations

  1. Calculate the average linear velocity of seafloor spreading between the Iceland hot spot and the Faroes. Round off degrees to nearest tenth during your calculations.

  2. What is the average linear velocity of seafloor spreading between Greenland (North America) and the Faroes (Europe)?

  3. Calculate wAmEu – the average angular velocity of seafloor spreading with respect to North America (Greenland) of Europe (Faroes).

  4. What is the average linear velocity of seafloor spreading at the triple junction near the Azores Islands?

  5. How much seafloor spreading took place in the Azores' portion of the Atlantic during the past 60 million years?

References

12. POLAR WANDERING

Natural remnant magnetism

All iron-bearing rocks may acquire at the time of their formation or sometime later a magnetic field, called natural remnant magnetism (NRM), which is roughly parallel with the Earth's magnetic field at that time. Thus, ancient rocks may serve as a kind of "fossil compass." This is of particular usefulness in plate tectonics for establishing the past positions, movements, and configurations of continental masses. Rocks may acquire NRM in several manners.

Not all rocks, of course, retain a useable NRM. In some cases, the age of the rock may be known, but the time when NRM was acquired may be indefinite. Later metamorphism and diagenesis could overprint a secondary NRM. Weathering and lightning strikes may also alter the fossil NRM of surface rocks. In most paleomagnetic laboratories, NRM is measured using a spinner magnetometer. Samples are subjected to stepwise demagnetization in order to determine the consistency and reliability of NRM measurements from particular rocks.

Earth's magnetic field

The Earth behaves basically as a large bar-magnet generating a dipole field—see Fig. 12-1. The orientation of the Earth's magnetic field at any point is determined by two measurements: declination and inclination. Declination (= trend) is defined as the angle between magnetic north and true north. The modern magnetic north pole is located in Arctic Canada, and its position drifts slowly from year to year. This magnetic drift is insignificant over geological time spans, and the average position of the magnetic pole is thought to coincide with the geographic pole.

Figure 12-1. Dipole magnetic field of the Earth with magnetic inclination shown by arrows.

Inclination (= plunge) is the angle between the magnetic field and the horizon, that is the angle of tilting. The sense of inclination is indicated as follows: downward inclination, as in the northern hemisphere today, is positive; upward inclination, as in the southern hemisphere, is negative. Generally, the Earth's magnetic field is strongest near the poles, and a magnetically quiet zone exists near the equator. However, strong magnetic anomalies exist, as near shallow iron-ore or magma bodies, that could produce large deflections in the local magnetic field.

At high latitudes, the magnetic field has a steep inclination, and at low latitudes it has a shallow inclination (Fig. 12-1). In fact, there is a strict relationship between latitude and inclination—see Fig. 12-2.

tan(I) = 2tan(L), or cot(I) = 2cot(C)

where:

L = latitude of site relative to modern pole.
C = colatitude of site relative to modern pole.
M = meridian (longitude: 0° to 360°) of site.
I = inclination of magnetic field at site.
D = declination of magnetic field at site.

Figure 12-2. Geometric elements of modern geomagnetic field. At the site indicated by the small box, I = +49°. GM = Greenwich Meridian.

The Earth's magnetic field is known to undergo periodic flips in polarity. During episodes of magnetic reversal, a compass needle points south, and the sense of magnetic inclination is the opposite. Although the sense of inclination switches, the angle of inclination remains the same for any given latitude. Such reversals have occurred on average every half million years during the Cenozoic and have resulted in magnetic anomalies on the seafloor parallel to mid-ocean ridges. The age and spacing of these seafloor anomalies may be used to calculate the rate of seafloor spreading, in a manner like the previous exercise. The last major reversal in polarity happened about 720,000 years ago.

Polar wandering

The inclination of NRM in an ancient rock is a direct indicator of the paleolatitude at which the rock originated. This is the case assuming the original horizontal position of the rock may be ascertained and the NRM is found to be stable to a high level of demagnetization. In itself, determination of paleolatitude is of great usefulness for paleogeographic reconstructions. For example, paleomagnetic evidence indicates that North America was astride the equator during most of the Paleozoic (Irving 1964). This is consistent with widespread Paleozoic coal beds, carbonate reefs, evaporites, and other low-latitude indicators in the sedimentary record of the continent. NRM data may be further used to calculate the positions of paleopoles.

The NRM of Cenozoic rocks is generally parallel to the modern (or reversed) geomagnetic field. However, older rocks have NRM significantly different in orientation from the modern field. For rocks of a given age and continent, the apparent position of the paleopole may be calculated, and when paleopoles representing each geologic period are plotted, a progressive movement or wandering of the pole through time may be seen. The path of apparent pole movement is called a polar wandering curve.

The term polar wandering is, unfortunately, somewhat misleading. It does not mean that the Earth's magnetic field has actually wandered, for the polar wandering curves of each continent are different. In fact, the Earth's magnetic field has remained more-or-less constant, aside from reversals and minor magnetic drift, while the continents have migrated independently of each other through time. Thus, polar wandering curves actually represent the paths of continental drifting.

The position of the magnetic paleopole may be determined from NRM declination and inclination of samples from a specific site—see Fig. 12-3. Declination is assumed to be approximately parallel to the paleomeridian, and inclination is used to calculate the paleolatitude (or paleocolatitude). The various geometric elements necessary to solve for paleopoles are:

M = meridian of sample site (longitude: 0° to 360°).
L = latitude of sample site.
C = colatitude of sample site.
PL = paleolatitude: tan(I) = 2tan(PL).
PC = paleocolatitude: cot(I) = 2cot(PC).
Lp = latitude of paleopole (modern coordinates).
Mp = meridian of paleopole (modern coordinates).
I = inclination of NRM of sample rocks.
D = declination of NRM of sample rocks.

Figure 12-3. Geometric elements of paleomagnetic field and location of paleopole at point P. The dotted line from the North Pole to the paleopole shows the polar wandering curve. GM = Greenwich Meridian; sample site shown by small box.

The diagonal arc passing through the sample site toward the northeast in the example (Fig. 12-3) is a great circle which forms a 40° angle with the meridian of the sample site. This great circle represents the paleomeridian with D = 40° relative to modern north. The paleopole is located somewhere along this paleomeridian at a point determined by paleocolatitude; in this case, I = +59°, so PC = 50°. The following formulas are used to calculate the exact location of the paleopole in terms of modern latitude and longitude coordinates (based on LePichon et al. 1976).

sin(Lp) = [sin(L) • cos(PC)] + [cos(L) • sin(PC) • cos(D)]

sin(Mp - M) = [sin(PC) • sin(D)] ÷ cos(Lp)

In this example:

sin(Lp) = (sin30 • cos50) + (cos30 • sin50 x cos40)
Lp = arc sin (.8296) = 56° N.

sin(Mp - M) = (sin50 • sin40) ÷ cos56
Mp - M = arc sin (.88056)
Mp = 62 + 330 = 392°, or 32° E.

During the time since the NRM of rocks at the sample site formed, the pole appears to have wandered a considerable distance from the North Pole toward the south (Fig. 12-3). The rate of apparent polar wandering depends on length of the polar wandering curve (distance from paleopole to modern pole) versus age of the sample rocks. The shortest possible polar wandering curve is along a meridian (Fig. 12-3), and its distance is simply the colatitude of the paleopole times 111 km per degree; for this example: 90 - 56 = 34°, or 3774 km. This assumes the paleopole has shifted position N-S along a meridian, which is probably not the case, so the actual polar wandering curve is likely longer.

If the sample rocks in the example are, say, Late Eocene (38 million BP) in age, then the average minimum rate of polar wandering would be: 3774 km/38 million years, or approximately 10 cm per year. In reality, the pole has not wandered, rather the sample rocks have been displaced from their site of origin. However, this displacement may involve both rotational and translational movements, and so it is not always possible to determine the actual distance of rock movement without additional paleomagnetic information.

Problem

Table 12-1 presents paleomagnetic data for rocks of six ages from the contiguous United States—see Fig. 12-4 (below). From this information, answer the following questions.

  1. Determine the paleolatitude and paleocolatitude (PL and PC) for each rock, and then calculate the latitude and longitude coordinates (Lp and Mp) for each paleopole.

    Note: the paleopole equations do not distinguish magnetic polarity. Be sure to pay attention to positive and negative values. If PL is negative, then PC also is negative, such that PL + PC = -90°. Your resulting answers may turn out in the northern hemisphere (positive Lp) or in the southern hemisphere (negative Lp). Round off PL, PC, Lp, and Mp values to nearest whole degree.

  2. Plot the locations of the paleopoles on the polar-projection graph (below). This graph represents a polar view of the northern hemisphere, with the North Pole at the graph center. Meridians (longitude lines) radiate away from the center, and latitudes form concentric circles around the center. Longitude is measured eastward (right on graph) from the Prime Meridian (zero longitude) at Greenwich, England.

    Some calculations in question 1 produce southern hemisphere paleopoles. These must be converted into equivalent northern hemisphere poles to plot on the graph. The conversion procedure is simple: (1) drop the negative sign from latitude and (2) subtract (or add) 180° to Mp.

    Connect paleopoles with a smooth line reaching up to the North Pole to construct the North American polar-wandering curve. To orient yourself to the present position of the United States on this graph, plot the location of Denver, Colorado: 40° N, 105° W.

  3. Discuss the possible past positions of North America which would explain the pattern of your polar wandering curve.

Table 12-1. Paleomagnetic data for the United States.
Taken from Irving (1964).
Formation Age L M D I
1. Green River Formation Eocene 40 252 345 65
2. Granite, Sierra Nevada Cretaceous 38 240 335 61
3. Supai Formation Permian 35 250 150 11
4. Barnett Formation Carboniferous 31 261 322 -5
5. Clinton Iron Ore Silurian 34 273 143 19
6. Juniata Formation Ordovician 40 281 131 26

Figure 12-4. Map of the contiguous United States showing locations of samples sites for paleomagnetic data (Table 12-1).

Polar-projection graph. The center of the graph represents the geographic North Pole; Greenwich Meridian is the “0” radial line. Plot Lp and Mp coordinates for paleopoles. Print at full size.

References

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Notice: This course was prepared for the use and benefit of students enrolled at Emporia State University. Others are welcome to view the course webpages. Any other use of text, imagery or curriculum materials is prohibited without permission. © J.S. Aber (2023).