An estimate of motion between the spin axis and the hotspots over the past century

Authors


Abstract

[1] Estimates of the motion of the spin axis over the past century either neglect plate motion or are relative to the mean lithosphere. To understand the response of the spin axis to mass changes, it is more useful to estimate spin axis motion relative to the hotspots because the hotspots well represent the solid earth. The hotspots originate in the lower mantle, which comprises 2/3 of the earth's volume. Relative to the lower mantle the plates and the upper mantle move in opposite directions, to a large extent canceling each other out. The mean lithosphere moves quickly relative to the hotspots and poorly represents the solid earth. The spin axis has moved relative to the hotspots over the past century along the 68.4°W meridian at 4.03 milliarcsecond/yr. This motion is 11° counterclockwise of, 0.5 milliarcsecond/yr faster than, and 25% different than spin axis motion relative to the mean lithosphere.

1. Introduction

[2] Polar motion, the motion of the spin axis relative to the solid earth, consists mostly of 4 parts: (1) a forced wobble with an amplitude of 0.1 arcsecond (3 m) and a period of 1 year, (2) a free Chandler wobble with an amplitude varying from 0.1 to 0.2 arcsecond (3 to 6 m) and a period of 433 days, (3) the Markowitz wobble with an amplitude of 0.03 arcsecond (1 m) and a period of about 30 years, and (4) a velocity of 0.00351 arcsecond/yr (11 m/100 yr) along the 79°W meridian. Part 4, the mean velocity of the spin axis relative to the solid earth over the past century, is the subject of this study.

[3] Gross and Vondrák [1999] determine three estimates of the velocity of the spin axis. Their estimate from solely (ILS) International Latitude Service data, which is determined from observations from 1900 to 1978 of the latitude of 7 sites along the 39°8′N parallel, neglects plate motion (Figure 1, Table 1). Their estimate from the Hipparcos star catalog, which is determined from the ILS observations and other astronomical observations of latitude, accounts for plate motion and is relative to the mean lithosphere (Figure 1). The mean lithosphere is the reference frame yielding no net rotation of the plates [Argus and Gordon, 1991]. Gross and Vondrák's [1999] estimate from an inversion of observations from 1976 to 1997 from four space geodetic techniques is also relative to the mean lithosphere. Because the time span that the velocity is determined over is shorter than the period of the Markowitz wobble, this space geodetic estimate does not reflect the mean velocity of the spin axis over the past century.

Figure 1.

Estimates of the motion of the spin axis over the past century: ILS, Gross and Vondrák's [1999] ILS estimate not corrected for plate motion; Hipparcos NNR, Gross and Vondrák's [1999] Hipparcos estimate relative to the mean lithosphere; ILS NNR, this study's ILS estimate relative to the mean lithosphere; ILS HS, this study's ILS estimate relative to the hotspots; Hipparcos HS, this study's Hipparcos estimate relative to the hotspots; 1, this study's estimate relative to mean mantle frame 1 [Steinberger, 2000]; 2, this study's estimate relative to mean mantle frame 2 [Steinberger, 2000]; 3, this study's estimate relative to mean mantle frame 3 [Steinberger, 2000].

Table 1. Estimates of the Motion of the Spin Axis Over the Past Century
Speed mas/yrMeridian Lon.°EFixedModel
ILS
   3.81−75.5uncorrected for plate motion 
   3.53−79.9mean lithosphereNNR-NUVEL1A
   4.04−69.1hotspotsHS3-NUVEL1A
Hipparcos
   3.51−79.2mean lithosphereNNR-NUVEL1A
   4.03−68.4hotspotsHS3-NUVEL1A
   3.75−73.2mean mantle 1Steinberger [2000]
   3.72−73.1mean mantle 2Steinberger [2000]
   3.60−73.0mean mantle 3Steinberger [2000]

[4] To understand the response of the spin axis to changes in distribution of mass, we seek to estimate the velocity of the spin axis relative to the mean solid earth. That parts of the solid earth move relative to one another complicates this estimation. The mean lithosphere, the reference frame relative to which the velocity of the spin axis is estimated in prior studies, moves quickly relative to the mean solid earth. We next present an argument that the hotspots, the hot plumes cutting through the upper mantle that create lines of volcanic islands as plates pass over them, well represent the mean solid earth:

[5] 1. The solid earth by volume is 6% lithosphere, 27% upper mantle, 66% lower mantle, and 1% inner core. Thus the mantle comprises 93% of the solid earth's volume.

[6] 2. The plates, which constitute the lithosphere, move relative to one another at speeds as fast as 150 mm/yr. The lithosphere and upper mantle are convecting. Because the thickness across which flow in the direction of plate motion occurs is less than the thickness across which return flow occurs, motion among the plates is believed to be faster than motion among parts of the upper mantle [Steinberger and O'Connell, 1998]. Motions among parts of the lower mantle, the layer of the earth that is most nearly rigid, are very slow.

[7] 3. Hotspots originate in and move slowly relative to the lower mantle [Steinberger and O'Connell, 1998; Steinberger, 2000]. Motion of the upper mantle relative to the lower mantle is in the opposite direction as motion of the plates, resulting in very slow net motion of these two layers relative to the lower mantle. Over the past 5 Myr the mean mantle in Steinberger's [2000] Models 1, 2, and 3 has moved relative to the hotspot-fixed frame in Steinberger's [2000] Model 0 at 0.083°/Myr, 0.083°/Myr, and 0.055°/Myr, respectively, yielding relative motion between the frames of at most 9 mm/yr, 9 mm/yr, and 6 mm/yr.

[8] 4. Hotspots move slowly relative to one another. Using volcanic propagation rates and volcanic island trends, Gripp and Gordon [2002] estimate motion between any 2 of their 10 hotspots over the past 5 Myr to differ insignificantly from zero, though their 95% confidence limits on relative hotspot motions are typically ±20 mm/yr to ±40 mm/yr. In Steinberger's [2000] three models, motions at the surface of 44 hotspots relative to the lower mantle since 40 Ma are typically 5 mm/yr to 20 mm/yr. Molnar and Stock [1987] estimate motion of hotspots in the Pacific Ocean relative to those in the Atlantic and Indian Oceans since 65 Ma to be 10 mm/yr to 20 mm/yr.

[9] The mean lithosphere moves quickly at as fast as 48 mm/yr relative to the hotspots (Figure 2). The motion of the mean lithosphere relative to the hotspots is described by a rigid-body right-handed rotation of 0.436°/Myr about a pole at 55.9°S, 66.9°E [Gripp and Gordon, 2002]. This angular velocity is the difference between model NNR-NUVEL1A [Argus and Gordon, 1991; DeMets et al., 1994] and model HS3-NUVEL1A [Gripp and Gordon, 2002]. The speed of motion of the mean lithosphere relative to the hotspots ranges from 0 mm/yr at the rotation pole to 48.5 mm/yr at places 90° from the rotation pole. At Hawaii the mean lithosphere moves relative to the hotspots at 36.8 mm/yr toward N58.4°W. At the North pole the mean lithosphere moves relative to the hotspots at 27.2 mm/yr along the 20.1°W meridian.

Figure 2.

Estimates of the angular velocity relative to the Pacific plate of various reference frames: HS3, hotspot frame HS3-NUVEL1A; NNR, mean lithosphere frame NNR-NUVEL1A; NA, North American plate NUVEL1A; 0, hotspot frame 0 [Steinberger, 2000]; 1, mean mantle frame 1 [Steinberger, 2000]; 2, mean mantle frame 2 [Steinberger, 2000]; 3, mean mantle frame 3 [Steinberger, 2000]. At top are poles of rotation. At bottom left are cross sections of angular velocities from east to west. At bottom right are cross sections of angular velocities from south to north.

2. Method and Result

2.1. Correcting for Plate Motion

[10] We correct Gross and Vondrák's [1999] ILS estimate of the spin axis velocity for plate motion using the method of Dickman [1977]. The north component of velocity relative to the mean lithosphere of each of the 5 ILS sites with observations over most of the twentieth century are first computed (Table 2) from model NNR-NUVEL1A [Argus and Gordon, 1991; DeMets et al., 1990, 1994]. The 2 parameters specifying the spin axis velocity that best fit the 5 apparent latitude change rates are next estimated. The ILS estimate of the velocity of the spin axis relative to the mean lithosphere (Figure 1, Table 1) is then computed to be the sum of Gross and Vondrák's [1999] ILS estimate and this spin axis velocity correction for plate motion relative to the mean lithosphere. The ILS estimate relative to the mean lithosphere we thus compute nearly equals Gross and Vondrák's [1999] Hipparcos estimate relative to the mean lithosphere. Correcting the ILS estimate for plate motion reduces the difference between the ILS and Hipparcos estimates from 12 to 2 mm/yr.

Table 2. North Components of Velocity of ILS Sites Relative to the No Net Rotation and Hotspot Reference Framesa
 Plate VelocityDeform North mm/yrTotal North mm/yr
Speed mm/yrAzimuth °CW of NNorth mm/yr
  • a

    Velocities relative to the no net rotation frame are computed using NNR-NUVEL1A [Argus and Gordon, 1991; DeMets et al., 1990, 1994]. Velocities relative to the hotspot frame are computing using HS3-NUVEL1A [Gripp and Gordon, 2002; DeMets et al., 1990, 1994]. Kitab and Carloforte are taken to be on the Eurasian plate. Gaithersburg is taken to be on the North American plate. The north component of velocity of Mizusawa relative to the Eurasian plate is taken to be 3 mm/yr, the same as that of VLBI sites Kashima and Tsukuba. The north component of velocity of Ukiah relative to the North American plate is taken to be 23 mm/yr, the same as that of GPS site HOPB. Caloforte, on the island of Sardinia, lies along the margin of the Eurasian plate, near SLR and GPS site Caglia, also on Sardinia. In geodetic inversions we do not assume Caglia to be on the Eurasian plate, but find its motion relative to the plate to be 0 mm/yr.

Mizusawa, Japan 39.1°N 141.1°E (Eurasia)
NNR24.3130.5−15.83−12.8
HS320.7−61.49.9312.9
 
Kitab, Russia 39.1°N 66.9°E Eurasia
NNR26.089.50.2  
HS322.3−93.2−1.2  
 
Carloforte, Italy 39.1°N 8.3°E Eurasia
NNR25.255.814.2  
HS320.8−117.8−9.7  
 
Gaithersburg, Maryland 39.1°N 77.2°W North America
NNR15.5−77.03.5  
HS333.8−109.5−11.2  
 
Ukiah, California 39.1°N 123.2°W (North America)
NNR18.6−138.5−13.9239.1
HS327.9−106.1−7.72315.3

[11] We similarly correct Gross and Vondrák's [1999] ILS estimate of the spin axis velocity for plate motion relative to the hotspots. The north velocity component of each of the 5 ILS sites is computed (Table 2) from model HS3-NUVEL1A [Gripp and Gordon, 2002; DeMets et al., 1990, 1994], the spin axis velocity that best fit the 5 apparent latitude change rates is estimated, and the ILS estimate of the velocity of the spin axis relative to the hotspots (Figure 1) is computed to be the sum of Gross and Vondrák's [1999] ILS estimate and this spin axis velocity correction for plate motion relative to the hotspots.

2.2. Transforming to the Hotspot Reference Frame

[12] We transform the Hipparcos estimate of the spin axis velocity from the mean lithosphere reference frame to the hotspot reference frame as follows. The Hipparcos estimate relative to the hotspots (Figure 1) is computed to be the sum of Gross and Vondrák's [1999] Hipparcos estimate relative to the mean lithosphere and the velocity of the mean lithosphere relative to the hotspots at the North pole. The latter term, 27.2 mm/yr along the 20.1°W meridian, is computed from the angular velocity of the mean lithosphere (NNR-NUVEL1A) relative to the hotspots (HS3-NUVEL1A), a right-hand rotation of 0.436°/Myr about 55.9°S, 69.9°E [Gripp and Gordon, 2002]. The spin axis velocity relative to the hotspots is 4.03 milliarcsecond/yr along the 68.4°W meridian. This motion is 11° counterclockwise of, 0.5 milliarcsecond/yr faster than, and 25% different than spin axis motion relative to the mean lithosphere.

[13] To show the two methods we use to be consistent, we similarly transform the ILS estimate of the spin axis velocity from the mean lithosphere reference frame to the hotspot reference frame. The ILS estimate relative to the hotspots is computed to be the sum of the ILS estimate relative to the mean lithosphere we compute above and the velocity of the mean lithosphere relative to the hotspots at the North pole. The ILS estimate relative to the hotspots we thus compute equals the ILS estimate relative to the hotspots we compute above. The two methods we use are therefore consistent.

3. Discussion

3.1. Uncertainty in Plate-Hotspot Motion and Consequence of Motion Among the Hotspots

[14] The results and statements in this article rely on the assumptions that hotspots move slowly relative to one another and that model HS3-NUVEL1A approximately describes motion of the plates relative to the hotspots. The speed at which hotspots move relative to one another is hotly debated. Müller et al. [1993] find that motion among hotspots in the Atlantic and Indian Oceans has been slow (no faster than several mm/yr) since 84 Ma, but O'Neill et al. [2003] find motion relative to the mean mantle of hotspots in the Indian Ocean to be in some places faster than 10 mm/yr. Plate reconstructions [Raymond et al., 2000] and paleomagnetic observations of latitude [Tarduno et al., 2003] suggest that motion of the Hawaiian hotspot relative to Atlantic and Indian Ocean hotspots has been slow (about 5 mm/yr) since 47 Ma but fast (about 40 mm/yr) between 81 and 47 Ma.

[15] Model HS3-NUVEL1A is a carefully constructed and widely accepted model that averages motion over just the past 6 Myr. Because the speed at which plates in the Atlantic and Indian Oceans move across the hotspots is in most places too slow to make a meaningful trend over the past 6 Myr, the 2 volcanic propagation rates and 9 of the 11 volcanic trends used in HS3-NUVEL1A come from plates in the Pacific Ocean. What if we were to instead use a model determined more from Atlantic and Indian Ocean hotspot volcanic trends over the past few tens of millions of year? Using the angular velocity of the African plate relative to the mean mantle in Steinberger's [2000] three models yield estimates of motion of the spin axis that are 6° clockwise of that estimated using HS3-NUVEL1A (Figure 1). The big difference between the two estimates reflects uncertainty in plate-hotspot motion and uncertainty about the hotspots being fixed relative to one another.

3.2. Implications for Geodynamics and the International Terrestrial Reference Frame

[16] The spin axis is believed to have moved over the past century mostly in response to the present-day isostatic adjustment of the solid earth in viscous response to the unloading of the ice sheets over the past 20 Kyr [Nakiboglu and Lambeck, 1980; Peltier, 1996; Peltier and Jiang, 1996; Vermeersen et al., 1997; Mitrovica and Milne, 1998; Nakada and Okuno, 2003]. Thus the spin axis is presently moving toward the center of rebound, along a meridian between the late Pleistocene Laurentide and Fennoscandian ice sheets. Substituting the spin axis velocity relative to the hotspots for spin axis velocity relative to the mean lithosphere changes the meridian along which the center of rebound lies by 11° so that it's nearer Fennoscandia.

[17] The (ITRF) International Terrestrial Reference Frame 2000 [Altamimi et al., 2002] is the reference frame relative to which geodetic site velocities are presently estimated. The position of the spin axis as a function of time is also estimated [International Earth Rotation and Reference Systems Service, 2003] in ITRF2000. The angular velocity of ITRF2000 is defined using the mean lithosphere. To determine site and spin axis motions relative to the mean solid earth, it would be better to define the angular velocity of the terrestrial reference frame to be fixed to the hotspots. If hotspots were substituted for the mean lithosphere, horizontal site velocities would change by as much as 48.5 mm/yr and spin axis velocities would change by 27.2 mm/yr. In practice inconsistencies would arise between positions relative to the mean lithosphere estimated before the substitution and positions relative to the hotspots estimated after the substitution. It may be best to over the next several years continue to estimate positions in the ITRF 2000, but to then transform horizontal site velocities and spin axis velocities from the mean lithosphere frame to the hotspots frame simply using the angular velocity describing the difference between the two frames.

Acknowledgments

[18] This research was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We are grateful to an anonymous reviewer for constructive suggestions.

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