Research Article Superrotation of Earth s Inner Core

Research ArticleSuperrotation of Earthrsquos Inner Core Extraterrestrial Impactsand the Effective Viscosity of Outer Core

Pirooz Mohazzabi1 and John D Skalbeck2

1Department of Mathematics and Physics University of Wisconsin-Parkside Kenosha WI 53141 USA2Department of Geosciences University of Wisconsin-Parkside Kenosha WI 53141 USA

Correspondence should be addressed to Pirooz Mohazzabi mohazzabuwpedu

Received 13 October 2014 Revised 12 March 2015 Accepted 6 April 2015

Academic Editor Petr Vanıcek

Copyright copy 2015 P Mohazzabi and J D Skalbeck This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The recently verified superrotation of Earthrsquos inner core is examined and a new model is presented which is based on the tidaldespinning of the mantle and the viscosity of the outer core The model also takes into account other damping mechanisms arisingfrom the inner core superrotation such as magnetic and gravitational coupling as well as contribution from eddy viscosity in theouter coreThe effective viscosity obtained in this model confirms a previously well constrained value of about 103 Pa s In additionthe model shows that the currently measured superrotation of the inner core must be almost exactly equal to its asymptotic orsteady-state value The effect of extraterrestrial impacts is also investigated and it is shown that perturbations due to such impactscan only persist over a short geological time

1 Introduction

In 1996 Song and Richards analyzed two types of seismicwaves traveling through Earthrsquos solid and fluid cores respec-tively [1] They discovered a differential travel time betweenthe two waves which increased systematically by about 03 sfrom the year 1967 to the year 1995 This temporal changewas later attributed to a shift of the lateral velocity gradientin the inner core caused by the inner core rotation [2 3]Subsequent studies provided further support for the fasterrotation of Earthrsquos solid inner core compared to the rest ofthe planet [4ndash7]

Because of the anisotropy due to wood-like grain struc-ture of the crystalline iron in the inner core the speedof seismic waves is different in different directions [8] Ifthe inner core rotates faster than the rest of Earth seismicwaves that are generated at the same place on Earth butdecades apart will have different travel times passing throughthe inner core and detected diametrically across the globeAccurate recent investigations using high-quality waveformdoublets generated in the South Sandwich Islands region and

detected in and near Alaska have revealed definite temporalchanges over a period of up to 35 years [9]These experimentsconfirm that Earthrsquos inner core rotates faster than the mantleand crust (hereafter simply referred to as mantle) with asuperrotation of about 027 to 053 degyr

Estimates of the inner core differential rotation ratesrange from low values of 02 to 03 degyr [2] to high valuesof 3 degyr [10] Deuss [11] presents a compilation of studieson inner core differential rotation rates as a function ofpublication year that suggests more recent data supportslower inner core differential rotation rates however the datais quite scattered The author cites a recent study [12] thatreports no inner core rotation and another study [13] thatshows subrotation but themajority of the studies report innercore superrotation with respect to the mantle

Several Earth-based mechanisms have been suggestedfor Earthrsquos inner-core differential rotation Gubbins [14]attributes the phenomenon to the existence of a large toroidalmagnetic field inside Earthrsquos core which together with thedipole components produces a magnetic torque on theinner core that tends to accelerate it relative to the mantle

Hindawi Publishing CorporationInternational Journal of GeophysicsVolume 2015 Article ID 763716 8 pageshttpdxdoiorg1011552015763716

2 International Journal of Geophysics

He then assumes that some equilibrium is reached so thatthe inner core rotates with constant angular velocity andexperiences zero net torque Glatzmaier and Roberts [15]numerically solve the self-consistent magnetohydrodynamicequations that describe thermal convection and magneticfield generation in a rapidly rotating spherical fluid shellwith a solid conducting inner core Their solution whichserves as an analog for the geodynamo shows that viscousand magnetic coupling of the outer core with the innercore and the mantle causes time-dependent variations intheir respective rotation rates the inner core usually rotatesfaster than the mantle Buffett and Glatzmaier [16] allowedgravitational coupling between the inner core and the mantleby incorporating viscous deformation of the inner coreinto their numerical simulations of the geodynamo Theydiscovered that differential rotation between the inner coreand the mantle is permitted by allowing the inner core todeform Numerical calculations by Aurnou et al [17] showedexcess temperature inside the tangent cylinder surroundingthe inner core which generates a prograde thermal windand a strong azimuthal magnetic field inside the tangentcylinder They conclude that the electromagnetic torque onthe inner core resulting from induced azimuthal magneticfields and the ambient poloidal field equilibrate when certainconditions are met Dumberry [18] studied the steady andtime-dependent rates of inner core rotation based on angularmomentum balance between the inner core fluid core andmantle He concluded that the rotational rate of an oscillatinginner core is constrained by the changes in mantle rotationinduced by gravitational coupling

Tkalcic and others [19] analyzed earthquake doubles toconstruct a model for inner core differential rotation ratesof 025 to 048 degyr with decadal fluctuations around themean of 1 degyr The authors suggest that these decadalfluctuations can account for discrepancies between previouscore rotation models and agree with recent geodynamosimulations A three-dimensional model by Livermore andothers [20] suggests that axial electromagnetic torque is thedominant influence for inner core differential rotation andthat decadal variations of the magnetic field may drive thequasioscillatory nature of the inner core differential rotation

Magnetic coupling between the inner and outer coresseems to play a role not only in the superrotation of the innercore but also in the generation of the Earthrsquosmagnetic field Suet al [10] reported an anomalous variation in the inner coreorientation that temporally coincided with the geomagneticldquojerkrdquo (a sudden change in the strength of Earthrsquos magneticfield of 1969-1970)This suggests a correlation between Earthrsquosmagnetic field and the inner core superrotation Glatzmaierand Roberts [21] have suggested that the inner core rotatesin response to the magnetic torque Γ

119861and the viscous torque

Γ] to which it is subjected with Γ119861+ Γ] = 0 The magnetic

torque drags the inner core eastward and the viscous torqueacts westward Nevertheless they state that even though thefluid viscosity in their model is several orders of magnitudegreater than is likely for real Earth the viscous torque on theinner core has little effect

Although each of these theories provides a reasonableexplanation for the differential rotation of the inner core

there is no experimental evidence supporting one over theothers and they are all based on processes that are assumedto be taking place inside Earth Furthermore about a decadebefore the most recent data of 027 to 053 degyr werepublished calculations based on magnetic coupling betweenthe inner and outer cores suggested superrotations that wereabout an order of magnitude higher than these values [10 21]

Perhaps the most significant and tangible external factorresponsible for the inner core superrotation is the tidal forces[22 23] Dissipation of tidal energy in oceans and transferof angular momentum between Earth Moon and the Sunresults in torques on the mantle causing it to dispin [24]which gradually increases the length of the day continuouslyThis despinning of the mantle leaves the inner core witha small excess eastward rotational velocity relative to themantle Su et al [10] used the known tidal increase in thelength of day of approximately 2ms per century to extrapolatebackward and conclude that the inner core was rotating withthe same period as the mantle about 105 years agoThe recentconfirmation of superrotation of Earthrsquos inner core by Zhangand others [9] prompted us to examine this phenomenon inthe context of tidal effects from a phenomenological point ofview

In addition to the oceanic tidal effects yet anotherexternal factor could contribute to the differential rotation ofthe Earthrsquos inner core which has not been addressed in theliterature Earth impacts Since the formation of our SolarSystem some 46 billion years ago collisions and impacts haveplayed a fundamental role in establishing its characteristicsranging from the accretion of planetesimals and the earlyformation of planets [25 26] to the recent series of impactson Jupiter by the fragments of Shoemaker-Levy 9 comet inJuly 1994 [27] These collisions and impacts have affected thedynamics of various components of the Solar System Forexample most planets have obliquity or axial tilt with respectto their orbital planes about the Sun Earth has an obliquity ofabout 235∘ while Uranus the third largest planet in the SolarSystem has an obliquity of about 97∘ In other words Uranusis tilted on its side so that its rotation axis is nearly in its orbitalplane about the Sun Yet giant planets are believed to formwith nearly zero obliquity [28] The axial tilts of planets arebelieved to have been caused by major impacts [29 30]

The above discussion is the motivation for examining asecond question in this paper Because Earthrsquos solid inner corerotates inside the fluid outer core is it possible for bolideimpacts to alter the angular velocity of Earthrsquos mantle relativeto the inner core resulting in a superrotation a subrotation ora transition from one to the other If so is it possible for theseimpacts to result in differential rotations that are comparableto the experimentally observed superrotation values andhowlong would it take for such perturbations to damp out

The model presented here could equally be applied forconditions that would generate subrotation or no differentialrotation depending on the size and angle of bolide impactExamination of the probability of direction of differentialrotation is beyond the scope of this paper The focus hereis to examine the potential for bolide impact contribution ifsuperrotation of the inner core is present

International Journal of Geophysics 3

2 Tidal Despinning of the Mantle andViscosity of the Outer Core

As stated earlier dissipation of tidal energy in the oceansresults in gradual despinning of the mantle which leaves theinner core with a small excess relative eastward differentialrotation relative to the mantle called superrotation [24] Thisdifferential rotation is communicated between the mantleand the inner core though a Couette flow in the fluid outercore which tends to damp the superrotation

Consider two concentric solid spheres separated by a fluidlayer between them with a coefficient of viscosity 120578 Let theradius of the inner sphere be 119903

1and the inside radius of the

outer sphere be 1199032 Then if the inner sphere rotates with

an angular velocity 120596 with respect to the outer sphere thedamping torque Γ

120578on it due to the fluid viscosity is given by

[31]

Γ120578= 8120587120578(

11990331119903

32

11990332 minus 119903

31)120596 (1)

Although Dai and others [32] have presented seismic reflec-tion data suggesting significant topography on the inner coreboundary this model can be applied to Earthrsquos inner core andmantle with the fluid between them being the outer coreHere the effect of the topography is absorbed in the estimateof effective viscosity

Using (1) the rotational equation of motion of the innercore

Γnet = 119868119889120596

119889119905(2)

becomes

119868120572 minus 8120587120578(11990331119903

32

11990332 minus 119903

31)120596 = 119868

119889120596

119889119905 (3)

where 119868 is the rotational inertia of the solid inner sphereThe left hand side in this equation is the net torque on theinner sphere The first term represents the torque due to thetidal effects which tends to increase the eastward rotationalvelocity 120596 of the inner core relative to the mantle 120572 is theconstant rate at which the rotational speed of themantle slowsdown due to the tidal effects It has a value of 23 millisecondsper day per century [33] or 345 times 10minus5 degyr2 The secondtermon the left hand side of (3) is the viscous damping torque

Writing 119868 in terms of the inner core density 1205881and the

inner core radius 1199031 reduces (3) to

119889120596

119889119905= minus

151205781205881119903

21(

11990332

11990332 minus 119903

31)120596+120572 (4)

Furthermore defining the characteristic time 120591 by

120591 =1205881119903

21

15120578[1minus(1199031

1199032)

3] (5)

reduces (4) to

119889120596

119889119905= minus

120596

120591+120572 (6)

Integration of this equation using the initial condition120596(0) =0 gives

120596 = 120572120591 (1minus 119890minus119905120591) (7)

From this equation we see that 120591 is in fact the relaxation timefor the process With the value of 120591 obtained from (5) thisequation gives the superrotation of the inner core 120596 at anytime 119905 after the dissipation of tidal energy began about 4 times109 years ago since oceans existed as early as the relativelystable Earth The value of 120572 as mentioned earlier is 345 times10minus5 degyr2

Earthrsquos inner core has a radius of 1220 times 106m and adensity of 13000 kgmminus3 The liquid outer core has a radius of3473 times 106m [34]The viscosity of the outer core however ishighly uncertain depending on its method of determinationIn fact its estimated values from various sources span over 14orders of magnitude from 10minus3 to 1011 Pa s [35] For examplede Wijs et al [36] report a value of 15 times 10minus2 Pa s withan uncertainty of a factor of three through dynamical firstprinciples simulations of liquid iron Similarly Rutter et al[37] report a value of 16 times 10minus2 Pa s using experimental highpressure study of liquid Fe-S systemOn the other hand usingamplitude of forced nutation Molodenskiy [38] finds a valueof about 1 times 106 Pa s

Although the viscosity of the Earthrsquos outer core is noto-riously uncertain it is plausibly considered to be boundedby 102 Pa s le 120578 le 1011 Pa s [32 34] Estimates of the fluidouter cores viscosity show variation from 102 Pa s at the topof the outer core to 1011 Pa s at the bottom near the innercore boundary using Arrhenius extrapolation of pressuredependencies for laboratory measurements on liquid iron[39 40] A viscosity value of 122 times 1011 Pa s near the solidinner corewas found by Smylie [41] using Ekman layer theoryto estimate viscous drag forces from Coriolis splitting of thetwo equatorial translational models of oscillation of the solidinner core Palmer and Smylie [42] found a viscosity valueof 615 Pa s near the top of the outer core from the free decayof free core nutations Smylie et al [43] present a viscosityestimate of 2371 times 103 Pa s for the top of the outer core whichrelies on the decay of free core nutations method and anestimate for the bottom of the outer core of 1247 times 1011 Pa sfrom the method of Smylie [41] and accounting for pressuredependence of the activation volume The authors found theviscosity profile fits a nearly log-linear trend across the outercore but values still vary by 9 orders of magnitude Smylie[44] presents this log-linear viscosity profile for the outercore based on the Arrhenius description of temperature andpressure of the viscosity by Brazhkin [39] and also reports amean viscosity value of 3124 Pa s at the top of the outer core

Therefore instead of using a highly variable value of theviscosity to calculate the superrotation of the inner core wesolve the inverse problem We use the measured value of thesuperrotation 120596 to find the viscosity of the outer core To doso we first solve the transcendental equation (7) numericallyfor the relaxation time 120591 using 120596 = 04 degyr (mean valueof the measured superrotation) 120572 = 345 times 10minus5 degyr2 and119905 = 4 times 109 yr (age of oceans) This gives a value 120591 = 11600 yr


(or 366 times 1011 s) Then we use (5) and solve for the viscosityof the outer core 120578 We find 120578 = 337 times 103 Pa s This numberis in incredible agreement with the value obtained by Bills[24] and by Smylie [44] Using a well described differentmodel Bills [24] reports an outer core viscosity of the order of103 Pa s assuming a value of superrotation of about 1 degyrIt is interesting to note that if we use 120596 = 1 degyr (insteadof 04 degyr) in our calculations we find 120578 = 135 times 103 Pa sOur calculated viscosity does not agree with the study bySu et al [10] that implied a viscosity of 10minus4 Pa s by ignoringelectromagnetic forces and assuming differential rotation of3 degyr

Admittedly our values of the viscosity are calculated atthe inner core boundary as our theoretical model considersrotation of the inner core However the extremely highviscosity of 1011 Pa s near the inner core strongly couples thelower zone of the outer core to the inner core Thus thedifferential rotation effectively takes place further out in theouter core where the viscosity is considerably lower Sincewe are solving the inverse problem our calculated viscositycorresponds to this region where the differential rotationeffectively takes place Ironically due to the functional formof(5) the values of the viscosity calculated for a superrotation of04 degyr with 120588

1= 12000ndash13000 kgm3 and with 119903

1ranging

from the inner core radius all the way to 98 of the outer coreradius vary in a narrow range of 103-104 Pa s

Let us now return to the notorious uncertainty in thereported values of outer core viscosity Generally these num-bers fall into two distinct categoriesThehigh values are basedon seismologic geodetic and geomagnetic observations ofEarth [32 34 41] whereas the low values are based on theoryand laboratory investigations of liquid metals [24 45 46]It has been suggested that this dichotomy is possibly dueto contribution of eddy viscosity caused by fluid motionwhereas the liquid metal investigations only account forintrinsic or molecular viscosity [24] In addition it is possiblethat electromagnetic and gravitational couplings also playroles Since all these effects arise from the differential rotationof the inner core therefore they should each be a functionof 120596 Furthermore according to Lenzrsquo Law and LeChatelierPrinciple they should all act in such a way to reduce oreliminate the cause which is the inner core superrotation

Let the combined torque resulting from all these dampingeffects which for the sake of simplicity from now on we referto as core coupling be Γ

119888= Γ119888(120596) Expanding this function in

a Taylor series we have

Γ119888 (120596) = Γ

119888 (0) + Γ1015840

119888(0) 120596 + 1

2Γ10158401015840

119888(0) 1205962

+ sdot sdot sdot (8)

Because 120596 is small we can neglect the second- and higher-order terms in the expansion In addition since wemust haveΓ119888(0) = 0 we obtain

Γ119888(120596) = Γ

1015840

119888(0) 120596 (9)

Adding this retarding torque to the left hand side of (3) weobtain

119868120572 minus 8120587120578(11990331119903

32

11990332 minus 119903

31)120596minusΓ

1015840

119888(0) 120596 = 119868

119889120596

119889119905 (10)

Repeating the calculations as before instead of (6) we get

119889120596

119889119905= minus(

1120591+1120583)120596+120572 (11)

where 120583 is a constant with the dimension of time Integrationof this equation with the initial condition 120596(0) = 0 gives

120596 = 120572120581 (1minus 119890minus119905120581) (12)

where 120581 is defined by

1120581=1120591+1120583

(13)

which is the new relaxation time taking into account both theviscosity of the outer core as well as all other damping effectswhich we called core coupling

Equation (12) has exactly the same functional form as(7) except that 120591 is replaced by 120581 Therefore if we solve thisequation for 120581 using the known value of 120596 we find the samevalue for 120581 that we found for 120591 before that is 120581 = 11600 yrThus the relaxation time of 11600 yr corresponds not only tothe viscosity of the outer core but also to all other dampingeffects as well Using this value in (5) and solving for 120578therefore gives an effective value for the outer core viscositythat could be greater than the actual value It is interesting tonote that since the relaxation time 120581 is 11600 yr the value of120596 today is so close to its saturation value 120572120581 that it would beimpossible to measure the difference

3 Extraterrestrial Impacts

Because geomagnetism and its reversals are associated withthe differential rotation of Earthrsquos inner core therefore anychanges in the Earthrsquos magnetic field can be attributed to achange in the inner corersquos superrotation Muller and Morris[47] have suggested that the impact of a large extraterrestrialobject on Earth can produce a geomagnetic reversal throughamechanism involving a sequence of events If so it would bepossible for such an impact to alter the superrotation of theinner core In what follows we study the dynamics of a bolideimpact and calculate the change in the steady-state value of120572120581 of the differential rotation of Earthrsquos inner core resultingfrom such impacts We then investigate the time scale overwhich such a perturbation in the inner core superrotationwould damp out

Consider an asteroid or comet of mass 119898 and velocity vrelative to the center of Earth just before impact as shownin Figure 1 For simplicity we assume that the velocity vectorof the asteroid or comet is in the equatorial plane of EarthThe center of mass of the bolide-Earth system is located at adistance of

119889cm =119898119877

119898 +119872lt119898

119872119877 (14)

from Earthrsquos center where119872 is the mass of Earth and 119877 is itsradius For a typical bolide119898 ≪ 119872 and therefore 119889cm is very


Earth

v

West

East

mR

120579

120596

Figure 1 A bolide of mass 119898 and velocity k relative to Earthrsquoscenter impacts Earth with an angle 120579 from the eastward directionThe velocity vector of the bolide just before the impact lies in theequatorial plane

small For example for a bolide of radius 119903 = 20 km and adensity comparable to that of Earth we have

119898

119872= (

119903

119877)

3= (

200006378 times 106

)

3= 308times 10minus8 (15)

Then with 119877 = 6378 times 106m we find 119889cm lt 02m which isless than 20 cm from Earthrsquos center Therefore we can safelytake the center of mass of the bolide-Earth system at themoment of impact to be simply Earthrsquos center

The time rate of change of the total angularmomentumofa system of particles is equal to the net external torque on thesystem all measured with respect to some inertial coordinatesystem

119889

119889119905= Γnet (16)

However if (16) is written relative to the center of mass of thesystem then it is valid even if the center ofmass is accelerating[48]

For the bolide-Earth system the net external torque aboutEarthrsquos center is zero and the angular momentum of thesystem about that point is conserved during the impactTherefore we have

119868Ω+119898V119877 cos 120579 = (119868 + 119889119868) (Ω+119889Ω) (17)

where V is the speed (magnitude of the velocity vector) of thebolide and 120579 is the angle between the bolidersquos velocity vectorand the tangent to Earth in the direction of Earthrsquos rotationas shown in Figure 1The first term on the left hand side is theangular momentum of Earth before the impact in which 119868 isthe rotational inertia of Earth andΩ is its angular velocityThesecond term is the angular momentum of the bolide beforethe impact The term on the right is the angular momentumof the combination after the impact in which 119889119868 and 119889Ω arevery small compared to 119868 andΩ respectively

Expanding the right hand side of (17) neglecting thesecond-order infinitesimal term 119889119868 119889Ω and rearranging theremaining terms we find

119898V119877 cos 120579 = 119868119889Ω+Ω119889119868 (18)

which may also be written as

119889 (119868Ω) = 119898V119877 cos 120579 (19)

Equation (19) simply states that the change of the angularmomentum of Earth is that imparted to it by the bolide

Dividing both sides of (18) 119868Ω we obtain

119889Ω

Ω=119898V119877 cos 120579

119868Ωminus119889119868

119868 (20)

Now 119889119868 = 1198981198772 because the bolide lodges itself near Earthrsquos

surface Furthermore assuming Earth to be a solid sphere ofuniform density for the moment (we address this point lateron) we have 119868 = (25)119872119877

2 Substituting in (20) we obtain

119889Ω

Ω=52119898

119872(V cos 120579119877Ω

minus 1) (21)

Finally using

119898

119872=120588119887

120588(119903

119877)

3 (22)

where 120588119887and 120588 are themean densities of the bolide and Earth

respectively we obtain

119889Ω

Ω=52120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (23)

Equation (23) measures the fractional change in Earthrsquosangular velocity as a result of a bolide impact This equationassumes that the velocity vector of the bolide is in theequatorial plane of Earth and that Earth is a solid sphere ofuniform density However since the outer core is fluid it doesnot immediately transfer the angular momentum resultingfrom the bolide impact to the solid inner core Thereforewe need to only consider the rotation of the mantle in ourcalculations The moment of inertia of the mantle 119868

119898 is the

difference between the moment of inertia of the entire Earthand that of the inner and outer cores

119868119898= 119868 minus 119868

119888= (1minus

119868119888

119868) 119868 = (1minus

1198721198881198772119888

1198721198772 ) 119868

= [1minus(120588119888

120588)(

119877119888

119877)

5] 119868

(24)

where 119877119888is the radius of the outer core 120588

119888is the combined

mean density of the inner and outer cores and 120588 is the meandensity of the entire Earth If (24) is written as

119868119898= 120574119868 (25)


where

120574 = 1minus(120588119888

120588)(

119877119888

119877)

5 (26)

(23) becomes

119889Ω

Ω=

52120574

120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (27)

where now Ω and 119889Ω refer respectively to the angularvelocity and the change in angular velocity of the mantle

Because Earthrsquos mass and mean radius are respectively119872 = 5976 times 1024 kg and 119877 = 6371 times 106m we calculate amean density of 120588 = 5517 kgmminus3 for Earth Furthermore theradius of the outer core and the combinedmean density of theinner and outer cores are respectively 119877

119888= 3473 times 106mand

120588119888= 11770 kgmminus3 [34] These data together with (26) give

120574 = 08973 Thus the correction to the fractional change inthe angular velocity of themantle as compared to entire Earthis only about 103

The change in the angular velocity of themantle as a resultof asteroid or comet impact can be positive negative or zerodepending on the parameters involved in (27) There are tensof thousands of asteroids in the Solar System and about 220have diameters greater than 100 km Ceres the largest knownasteroid has a diameter of about 950 km [49] The orbit ofsome of these asteroids is such that they could potentially beperturbed into a collision course with Earth

An object attracted by Earthrsquos gravitational field fromlarge distances will impact Earth with a minimum speed ofabout 11 km sminus1 A more likely collision however is whenEarth intercepts an object that is being attracted by the Sunwhich has amuchmore powerful gravitational attraction andthe impact speed will be much higher In fact it has beenestimated that if the bolide responsible for the Chicxulubcrater was a comet it must have struck Earth with a speedas high as 70 or 80 km sminus1 [50]

To obtain a numerical estimate for the change of theangular velocity of the mantle as a result of an asteroid orcomet impact we consider a bolide of radius 15 km and adensity comparable to that of Earth impacting Earth with aspeed of 40 km sminus1 The angle of impact 120579 can be anywherefrom 0 to 180∘ as shown in Figure 1 For 120579 between 0 and 90∘the impact speeds up the rotation of the mantle relative to theinner core and for 120579 between 90∘ and 180∘ the impact slows itdown Let us consider the special cases of near glancing angles120579 = 0 and 120579 = 180∘ Because Earthrsquos angular velocity is

Ω =2120587

24 times 3600= 72722times 10minus5 rads (28)

from (27) with 120579 = 0 we find 119889Ω = 2257 times 10minus10 rads =041 degyrOn the other hand if the impact is in the direction120579 = 180∘ we find a change in the mantlersquos angular velocity of119889Ω = minus042 degyr Because the change of the differentialrotation of the inner core relative to the mantle as a resultof an impact is 120575120596

119894= minus119889Ω the impact in the former case

decreases the superrotation of the inner core by 041 degyr

Bolide diameter (km)

00

01

02

03

04

05

06

07

08

minus120575120596i

(deg

yr)

25kms

20kms

15kms

10kms

5kms

10 15 20 25 30 35

Figure 2 Magnitude of the impact-induced superrotation of theEarthrsquos inner core (minus120575120596

119894) as a function of bolide diameter and speed

The horizontal dashed lines indicate the range of superrotationreported by Zhang et al [9]

and that in the latter case increases it by 042 degyr A strongenough impact in the direction 120579 = 0 can impart a change in120596 that exceeds the value of 120572120581 resulting in a subrotation ofthe inner core

Figure 2 shows plots of the change in superrotation ofEarthrsquos inner core 120575120596

119894 as a result of bolide impacts of various

diameters and speeds Bolide densities are assumed to becomparable to that of Earth and the impacts are in thedirection 120579= 180∘The limits of themeasured values of Earthrsquosinner core superrotation are also shown in the same figure forcomparison

Immediately after impact the value of the differentialrotation of the inner core relative to the mantle changes to120572120581+120575120596

119894 To find out how this new value decays we solve (11)

using the initial condition 120596(0) = 120572120581 + 120575120596119894 This gives

120596 = 120572120581+ 120575120596119894119890minus119905120581

(29)

Because 120581 = 11600 yr (29) shows that in a time scale of a fewtens of thousand years the effect of bolide impact damps outTherefore impacts only contribute to geologically short-termperturbations in the steady-state value 120572120581 of the inner-coresuperrotation

4 Discussion and Conclusion

There are a number of existing theories explaining thesuperrotation of Earthrsquos inner core Most of these theoriesare based on internal Earth processes Although each theoryprovides a reasonable account for the phenomenon there isnot sufficient experimental evidence to support one theoryover the others The external tidal effect on the other handappears to be one of the most reasonable explanations which


is also supported by experimental evidence and measure-ments

The frictional torques generated by the tidal effects tendto slow down the rotational motion of the mantle therebyincreasing the superrotation of the inner core This effectis counterbalanced by the viscous forces of the Earthrsquos fluidouter core as well as other effects such as the electromagneticand gravitational couplings and eddy viscosity caused by fluidmotion in the outer coreThe combined effect gives rise to aneffective viscosity of the outer core on the order of 103 Pa sin agreement with a previously reported value based on adifferent model [24] The viscosity of the molten materialof the outer core without these additional contributionshowever could be considerably less Because the relaxationtime for the inner core superrotation is of the order of 104 yrthe superrotation today is almost exactly equal to its saturatedor steady-state value

Bolide impacts yet another external factor are possiblyresponsible for sudden changes in the differential rotation ofEarthrsquos inner core from time to time and can render changescomparable in magnitude to the presently observed valuesAlthough the existing geological evidence indicates fewmajorbolide impacts with Earth in recent geological times the largenumber of craters on the Moon strongly suggests that Earthwith a considerably larger cross-sectional area has also beensubject to a large number of impacts since its formationMost of the resulting craters however have been hiddenor destroyed by Earthrsquos weathering and tectonic processesTherefore the number of bolide impacts on Earth is notlimited to the number of existing craters but rather is muchhigher Furthermore as our science and technology advanceswe discover more impact craters on Earth of which we werepreviously unaware For example it was originally believedthat the Vredefort dome in South Africa was formed by avolcanic explosion But in the mid-1990s evidence revealedthat it was in fact the site of a huge bolide impact abouttwo billion years ago Evidence of four impacts older thanVedefort that occurred between 32 to 35 billion years ago hasalso been found in South AfricaThese sites however are noteasily recognizable as impact structures on Earthrsquos surface

Asteroids approaching Earth and the Moon have anaverage speed of about 17 km sminus1 and depending on theircomposition can have densities ranging in the extreme fromabout 3 kgmminus3 (rock) to about 8 kgmminus3 (iron)The diameterscan be as high as tens of kilometers As shown in Figure 2there is a wide range of diameters and velocities for animpacter which can cause superrotations of Earthrsquos innercore comparable to the observed values In fact severalasteroids among the near Earth objects fall in this categoryFor example 1036 Ganymed the largest asteroid in this classis about 32 km across According to Figure 2 with an impactspeed of about 10 km sminus1 an asteroid of this size can causea superrotation in Earthrsquos inner core that is in the range ofthe observed values Furthermore cumulative effects fromsmaller impacts may also result in such superrotations

Nonetheless as stated earlier these impacts that canchange the steady-state differential rotation of the Earthrsquosinner core can be viewed as short lived perturbations in

geological terms with decaying times of the order of a fewtens of thousands of years The discrepancy between ourdamping time scale of a few tens of thousands of years andthose obtained by Tkalcic et al [19] and by Livermore et al[20] of the order of decades is the result of invoking differentmechanisms and models for the inner-core superrotation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] X D Song and P G Richards ldquoSeismological evidence fordifferential rotation of the earthrsquos inner corerdquo Nature vol 382no 6588 pp 221ndash224 1996

[2] K C Creager ldquoInner core rotation rate from small-scaleheterogeneity and time-varying travel timesrdquo Science vol 278no 5341 pp 1284ndash1288 1997

[3] X D Song ldquoJoint inversion for inner core rotation inner coreanisotropy and mantle heterogeneityrdquo Journal of GeophysicalResearch B Solid Earth vol 105 no 4 pp 7931ndash7943 2000

[4] J E Vldale D A Dodge and P S Earle ldquoSlow differentialrotation of the Earthrsquos inner core indicated by temporal changesin scatteringrdquo Nature vol 405 no 6785 pp 445ndash448 2000

[5] X D Song ldquoTime dependence of PKP(BC)-PKP(DF) timescould this be an artifact of systematic earthquakemislocationsrdquoPhysics of the Earth and Planetary Interiors vol 122 no 3-4 pp221ndash228 2000

[6] X D Song and A Y Li ldquoSupport for differential inner coresuperrotation from earthquakes in Alaska recorded at SouthPole stationrdquo Journal of Geophysical Research B Solid Earth vol105 no 1 pp 623ndash630 2000

[7] J D Collier and G Helffrich ldquoEstimate of inner core rotationrate from United Kingdom regional seismic network data andconsequences for inner core dynamical behaviourrdquo Earth andPlanetary Science Letters vol 193 no 3-4 pp 523ndash537 2001

[8] R A Kerr ldquoEarthrsquos inner core is running a tad faster than therest of the planetrdquo Science vol 309 no 5739 p 1313 2005

[9] J Zhang X Song Y Li P C Richards X Sun and FWaldhauser ldquoInner core differential motion confirmed byearthquake waveform doubletsrdquo Science vol 309 no 5739 pp1357ndash1360 2005

[10] W-J Su A M Dziewonski and R Jeanloz ldquoPlanet within aplanet rotation of the inner core of earthrdquo Science vol 274 no5294 pp 1883ndash1887 1996

[11] A Deuss ldquoHeterogeneity and anisotropy of earthrsquos inner corerdquoAnnual Review of Earth and Planetary Sciences vol 42 no 1 pp103ndash126 2014

[12] AMMakinen andADeuss ldquoGlobal seismic body-wave obser-vations of temporal variations in the Earthrsquos inner core andimplications for its differential rotationrdquo Geophysical JournalInternational vol 187 no 1 pp 355ndash370 2011

[13] A Souriau ldquoNew seismological constraints on differentialrotation of the inner core fromNovaya Zemlya events recordedat DRV Antarcticardquo Geophysical Journal International vol 134no 2 pp F1ndashF5 1998

[14] D Gubbins ldquoRotation of the inner corerdquo Journal of GeophysicalResearch vol 86 no 12 pp 11695ndash11699 1981


[15] G A Glatzmaier and P H Roberts ldquoA three-dimensional con-vective dynamo solution with rotating and finitely conductinginner core and mantlerdquo Physics of the Earth and PlanetaryInteriors vol 91 no 1-3 pp 63ndash75 1995

[16] B A Buffett and G A Glatzmaier ldquoGravitational braking ofinner-core rotation in geodynamo simulationsrdquo GeophysicalResearch Letters vol 27 no 19 pp 3125ndash3128 2000

[17] JM Aurnou D Brito and P L Olson ldquoMechanics of inner coresuper-rotationrdquoGeophysical Research Letters vol 23 no 23 pp3401ndash3404 1996

[18] M Dumberry ldquoGeodynamic constraints on the steady andtime-dependent inner core axial rotationrdquo Geophysical JournalInternational vol 170 no 2 pp 886ndash895 2007

[19] H Tkalcic M Young T Bodin S Ngo and M SambridgeldquoThe shuffling rotation of the Earthrsquos inner core revealed byearthquake doubletsrdquo Nature Geoscience vol 6 no 6 pp 497ndash502 2013

[20] P W Livermore R Hollerbach and A Jackson ldquoElectromag-netically driven westward drift and inner-core superrotation inEarthrsquos corerdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 110 no 40 pp 15914ndash159182013

[21] G A Glatzmaier and P H Roberts ldquoRotation and magnetismof Earthrsquos inner corerdquo Science vol 274 no 5294 pp 1887ndash18911996

[22] K Lambeck The Earthrsquos Variable Rotation Geophysical Causesand Consequences Cambridge University Press New York NYUSA 1980

[23] K LambeckGeophysical GeodesyThe Slow Deformations of theEarth Oxford University Press New York NY USA 1988

[24] B G Bills ldquoTidal despinning of the mantle inner core superro-tation and outer core effective viscosityrdquo Journal of GeophysicalResearch B Solid Earth vol 104 no 2 pp 2653ndash2666 1999

[25] G W Wetherill ldquoThe formation of the Earth from planetesi-malsrdquo Scientific American vol 244 no 6 pp 162ndash174 1981

[26] A H Marcus ldquoFormation of the planets by accretion ofplanetesimals some statistical problemsrdquo Icarus vol 7 no 1ndash3pp 283ndash296 1967

[27] K Zahnle and M-M M Low ldquoThe collision of jupiter andcomet shoemaker-levy 9rdquo Icarus vol 108 no 1 pp 1ndash17 1994

[28] G Boue and J Laskar ldquoA collisionless scenario for uranustiltingrdquoTheAstrophysical Journal Letters vol 712 no 1 pp L44ndashL47 2010

[29] A G W Cameron ldquoCosmological considerations regardingUranusrdquo Icarus vol 24 no 3 pp 280ndash284 1975

[30] S S Sheppard D Jewitt and J Kleyna ldquoAn ultradeep surveyfor irregular satellites of uranus limits to completenessrdquo TheAstronomical Journal vol 129 no 1 pp 518ndash525 2005

[31] H Lamb Hydrodynamics Dover Publications New York NYUSA 1945

[32] Z Dai W Wang and L Wen ldquoIrregular topography atthe Earthrsquos inner core boundaryrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 109 no20 pp 7654ndash7658 2012

[33] Ocean Tides and the Earthrsquos Rotation 2012 httpbowiegsfcnasagovggfctidesintrohtml

[34] J Verhoogen F J Turner L E Weiss et al The Earth HoltRinehart and Winston New York NY USA 1970

[35] Y D Fomin V N Ryzhov and V V Brazhkin ldquoProperties ofliquid iron along the melting line up to Earth-core pressuresrdquo

Journal of Physics Condensed Matter vol 25 no 28 Article ID285104 pp 1ndash5 2013

[36] G A de Wijs G Kresse L Vocadlo et al ldquoThe viscosity ofliquid iron at the physical conditions of the Earthrsquos corerdquoNaturevol 392 no 6678 pp 805ndash807 1998

[37] M D Rutter R A Secco T Uchida et al ldquoTowards evaluatingthe viscosity of the Earthrsquos outer core an experimental highpressure study of liquid Fe-S (85 wt S)rdquoGeophysical ResearchLetters vol 29 pp 1217ndash1221 2002

[38] S M Molodenskiy ldquoUpper viscosity boundary of the Earthrsquoscorerdquo Iznestiya Physics of the Solid Earth vol 17 pp 903ndash9091981

[39] V V Brazhkin ldquoInvestigation of the crystallization of liquidiron under pressure extrapolation of the melt viscosity into themegabar rangerdquo Journal of Experimental andTheoretical PhysicsLetters vol 68 no 6 pp 502ndash508 1998

[40] V V Brazhkin and A G Lyapin ldquoUniversal viscosity growthin metallic melts at megabar pressures the vitreous state of theEarthrsquos inner corerdquo Physics-Uspekhi vol 43 no 5 pp 493ndash5082000

[41] D E Smylie ldquoViscosity near earthrsquos solid inner corerdquo Sciencevol 284 no 5413 pp 461ndash463 1999

[42] A Palmer and D E Smylie ldquoVLBI observations of free corenutations and viscosity at the top of the corerdquo Physics of theEarth and Planetary Interiors vol 148 no 2ndash4 pp 285ndash3012005

[43] D E Smylie V V Brazhkin and A Palmer ldquoDirect observa-tions of the viscosity of Earthrsquos outer core and extrapolation ofmeasurements of the viscosity of liquid ironrdquo Physics-Uspekhivol 52 no 1 pp 79ndash92 2009

[44] D E Smylie Earth Dynamics Deformations and Oscillations ofthe Rotating Earth Cambridge University Press 2013

[45] L I Lumb and K D Aldridge ldquoOn viscosity estimates for theEarthrsquos fluid outer core and core- mantle couplingrdquo Journal ofGeomagnetism amp Geoelectricity vol 43 no 2 pp 93ndash110 1991

[46] R A Secco ldquoViscosity of the outer corerdquo inMineral Physics andCrystallography AHandbook of Physical Constants T J AhrensEd vol 2 pp 218ndash226 AGU Washington DC USA 1995

[47] R A Muller and D E Morris ldquoGeomagnetic reversals fromimpacts on the Earthrdquo Geophysical Research Letters vol 13 no11 pp 1177ndash1180 1986

[48] G R Fowles andG LCassidayAnalyticalMechanics SaundersNew York NY USA 6th edition 1999

[49] J W Parker S A Stern P CThomas et al ldquoAnalysis of the firstdisk-resolved images of ceres fromultraviolet observations withthe Hubble Space Telescoperdquo Astronomical Journal vol 123 no1 pp 549ndash557 2002

[50] W AlvarezT Rex and the Crater of Doom PrincetonUniversityPress Princeton NJ USA 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of


EarthquakesJournal of


Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining


Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

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OceanographyInternational Journal of


Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in


MineralogyInternational Journal of


MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in


He then assumes that some equilibrium is reached so thatthe inner core rotates with constant angular velocity andexperiences zero net torque Glatzmaier and Roberts [15]numerically solve the self-consistent magnetohydrodynamicequations that describe thermal convection and magneticfield generation in a rapidly rotating spherical fluid shellwith a solid conducting inner core Their solution whichserves as an analog for the geodynamo shows that viscousand magnetic coupling of the outer core with the innercore and the mantle causes time-dependent variations intheir respective rotation rates the inner core usually rotatesfaster than the mantle Buffett and Glatzmaier [16] allowedgravitational coupling between the inner core and the mantleby incorporating viscous deformation of the inner coreinto their numerical simulations of the geodynamo Theydiscovered that differential rotation between the inner coreand the mantle is permitted by allowing the inner core todeform Numerical calculations by Aurnou et al [17] showedexcess temperature inside the tangent cylinder surroundingthe inner core which generates a prograde thermal windand a strong azimuthal magnetic field inside the tangentcylinder They conclude that the electromagnetic torque onthe inner core resulting from induced azimuthal magneticfields and the ambient poloidal field equilibrate when certainconditions are met Dumberry [18] studied the steady andtime-dependent rates of inner core rotation based on angularmomentum balance between the inner core fluid core andmantle He concluded that the rotational rate of an oscillatinginner core is constrained by the changes in mantle rotationinduced by gravitational coupling

Tkalcic and others [19] analyzed earthquake doubles toconstruct a model for inner core differential rotation ratesof 025 to 048 degyr with decadal fluctuations around themean of 1 degyr The authors suggest that these decadalfluctuations can account for discrepancies between previouscore rotation models and agree with recent geodynamosimulations A three-dimensional model by Livermore andothers [20] suggests that axial electromagnetic torque is thedominant influence for inner core differential rotation andthat decadal variations of the magnetic field may drive thequasioscillatory nature of the inner core differential rotation

Magnetic coupling between the inner and outer coresseems to play a role not only in the superrotation of the innercore but also in the generation of the Earthrsquosmagnetic field Suet al [10] reported an anomalous variation in the inner coreorientation that temporally coincided with the geomagneticldquojerkrdquo (a sudden change in the strength of Earthrsquos magneticfield of 1969-1970)This suggests a correlation between Earthrsquosmagnetic field and the inner core superrotation Glatzmaierand Roberts [21] have suggested that the inner core rotatesin response to the magnetic torque Γ

119861and the viscous torque

Γ] to which it is subjected with Γ119861+ Γ] = 0 The magnetic

torque drags the inner core eastward and the viscous torqueacts westward Nevertheless they state that even though thefluid viscosity in their model is several orders of magnitudegreater than is likely for real Earth the viscous torque on theinner core has little effect

Although each of these theories provides a reasonableexplanation for the differential rotation of the inner core

there is no experimental evidence supporting one over theothers and they are all based on processes that are assumedto be taking place inside Earth Furthermore about a decadebefore the most recent data of 027 to 053 degyr werepublished calculations based on magnetic coupling betweenthe inner and outer cores suggested superrotations that wereabout an order of magnitude higher than these values [10 21]

Perhaps the most significant and tangible external factorresponsible for the inner core superrotation is the tidal forces[22 23] Dissipation of tidal energy in oceans and transferof angular momentum between Earth Moon and the Sunresults in torques on the mantle causing it to dispin [24]which gradually increases the length of the day continuouslyThis despinning of the mantle leaves the inner core witha small excess eastward rotational velocity relative to themantle Su et al [10] used the known tidal increase in thelength of day of approximately 2ms per century to extrapolatebackward and conclude that the inner core was rotating withthe same period as the mantle about 105 years agoThe recentconfirmation of superrotation of Earthrsquos inner core by Zhangand others [9] prompted us to examine this phenomenon inthe context of tidal effects from a phenomenological point ofview

In addition to the oceanic tidal effects yet anotherexternal factor could contribute to the differential rotation ofthe Earthrsquos inner core which has not been addressed in theliterature Earth impacts Since the formation of our SolarSystem some 46 billion years ago collisions and impacts haveplayed a fundamental role in establishing its characteristicsranging from the accretion of planetesimals and the earlyformation of planets [25 26] to the recent series of impactson Jupiter by the fragments of Shoemaker-Levy 9 comet inJuly 1994 [27] These collisions and impacts have affected thedynamics of various components of the Solar System Forexample most planets have obliquity or axial tilt with respectto their orbital planes about the Sun Earth has an obliquity ofabout 235∘ while Uranus the third largest planet in the SolarSystem has an obliquity of about 97∘ In other words Uranusis tilted on its side so that its rotation axis is nearly in its orbitalplane about the Sun Yet giant planets are believed to formwith nearly zero obliquity [28] The axial tilts of planets arebelieved to have been caused by major impacts [29 30]

The above discussion is the motivation for examining asecond question in this paper Because Earthrsquos solid inner corerotates inside the fluid outer core is it possible for bolideimpacts to alter the angular velocity of Earthrsquos mantle relativeto the inner core resulting in a superrotation a subrotation ora transition from one to the other If so is it possible for theseimpacts to result in differential rotations that are comparableto the experimentally observed superrotation values andhowlong would it take for such perturbations to damp out

The model presented here could equally be applied forconditions that would generate subrotation or no differentialrotation depending on the size and angle of bolide impactExamination of the probability of direction of differentialrotation is beyond the scope of this paper The focus hereis to examine the potential for bolide impact contribution ifsuperrotation of the inner core is present









[31]

Γ120578= 8120587120578(

11990331119903

32

11990332 minus 119903

31)120596 (1)



Γnet = 119868119889120596

119889119905(2)

becomes

119868120572 minus 8120587120578(11990331119903

32

11990332 minus 119903

31)120596 = 119868

119889120596

119889119905 (3)




119889120596

119889119905= minus

151205781205881119903

21(

11990332

11990332 minus 119903

31)120596+120572 (4)


120591 =1205881119903

21

15120578[1minus(1199031

1199032)

3] (5)

reduces (4) to

119889120596

119889119905= minus

120596

120591+120572 (6)


120596 = 120572120591 (1minus 119890minus119905120591) (7)









1ranging






Γ119888 (120596) = Γ

119888 (0) + Γ1015840

119888(0) 120596 + 1

2Γ10158401015840

119888(0) 1205962



Γ119888(120596) = Γ

1015840

119888(0) 120596 (9)


119868120572 minus 8120587120578(11990331119903

32

11990332 minus 119903

31)120596minusΓ

1015840

119888(0) 120596 = 119868

119889120596

119889119905 (10)


119889120596

119889119905= minus(

1120591+1120583)120596+120572 (11)


120596 = 120572120581 (1minus 119890minus119905120581) (12)


1120581=1120591+1120583

(13)






119889cm =119898119877

119898 +119872lt119898

119872119877 (14)



Earth

v

West

East

mR

120579

120596



119898

119872= (

119903

119877)

3= (

200006378 times 106

)




119889

119889119905= Γnet (16)



119868Ω+119898V119877 cos 120579 = (119868 + 119889119868) (Ω+119889Ω) (17)



119898V119877 cos 120579 = 119868119889Ω+Ω119889119868 (18)


119889 (119868Ω) = 119898V119877 cos 120579 (19)



119889Ω

Ω=119898V119877 cos 120579

119868Ωminus119889119868

119868 (20)




119889Ω

Ω=52119898

119872(V cos 120579119877Ω

minus 1) (21)

Finally using

119898

119872=120588119887

120588(119903

119877)

3 (22)



119889Ω

Ω=52120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (23)


119898 is the


119868119898= 119868 minus 119868

119888= (1minus

119868119888

119868) 119868 = (1minus

1198721198881198772119888

1198721198772 ) 119868

= [1minus(120588119888

120588)(

119877119888

119877)

5] 119868

(24)




119868119898= 120574119868 (25)


where

120574 = 1minus(120588119888

120588)(

119877119888

119877)

5 (26)

(23) becomes

119889Ω

Ω=

52120574

120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (27)



119888= 3473 times 106mand






Ω =2120587






00

01

02

03

04

05

06

07

08

minus120575120596i

(deg

yr)

25kms

20kms

15kms

10kms

5kms

10 15 20 25 30 35











120596 = 120572120581+ 120575120596119894119890minus119905120581

(29)













References






























































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in









[31]

Γ120578= 8120587120578(

11990331119903

32

11990332 minus 119903

31)120596 (1)



Γnet = 119868119889120596

119889119905(2)

becomes

119868120572 minus 8120587120578(11990331119903

32

11990332 minus 119903

31)120596 = 119868

119889120596

119889119905 (3)




119889120596

119889119905= minus

151205781205881119903

21(

11990332

11990332 minus 119903

31)120596+120572 (4)


120591 =1205881119903

21

15120578[1minus(1199031

1199032)

3] (5)

reduces (4) to

119889120596

119889119905= minus

120596

120591+120572 (6)


120596 = 120572120591 (1minus 119890minus119905120591) (7)









1ranging






Γ119888 (120596) = Γ

119888 (0) + Γ1015840

119888(0) 120596 + 1

2Γ10158401015840

119888(0) 1205962



Γ119888(120596) = Γ

1015840

119888(0) 120596 (9)


119868120572 minus 8120587120578(11990331119903

32

11990332 minus 119903

31)120596minusΓ

1015840

119888(0) 120596 = 119868

119889120596

119889119905 (10)


119889120596

119889119905= minus(

1120591+1120583)120596+120572 (11)


120596 = 120572120581 (1minus 119890minus119905120581) (12)


1120581=1120591+1120583

(13)






119889cm =119898119877

119898 +119872lt119898

119872119877 (14)



Earth

v

West

East

mR

120579

120596



119898

119872= (

119903

119877)

3= (

200006378 times 106

)




119889

119889119905= Γnet (16)



119868Ω+119898V119877 cos 120579 = (119868 + 119889119868) (Ω+119889Ω) (17)



119898V119877 cos 120579 = 119868119889Ω+Ω119889119868 (18)


119889 (119868Ω) = 119898V119877 cos 120579 (19)



119889Ω

Ω=119898V119877 cos 120579

119868Ωminus119889119868

119868 (20)




119889Ω

Ω=52119898

119872(V cos 120579119877Ω

minus 1) (21)

Finally using

119898

119872=120588119887

120588(119903

119877)

3 (22)



119889Ω

Ω=52120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (23)


119898 is the


119868119898= 119868 minus 119868

119888= (1minus

119868119888

119868) 119868 = (1minus

1198721198881198772119888

1198721198772 ) 119868

= [1minus(120588119888

120588)(

119877119888

119877)

5] 119868

(24)




119868119898= 120574119868 (25)


where

120574 = 1minus(120588119888

120588)(

119877119888

119877)

5 (26)

(23) becomes

119889Ω

Ω=

52120574

120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (27)



119888= 3473 times 106mand






Ω =2120587






00

01

02

03

04

05

06

07

08

minus120575120596i

(deg

yr)

25kms

20kms

15kms

10kms

5kms

10 15 20 25 30 35











120596 = 120572120581+ 120575120596119894119890minus119905120581

(29)













References






























































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in





1ranging






Γ119888 (120596) = Γ

119888 (0) + Γ1015840

119888(0) 120596 + 1

2Γ10158401015840

119888(0) 1205962



Γ119888(120596) = Γ

1015840

119888(0) 120596 (9)


119868120572 minus 8120587120578(11990331119903

32

11990332 minus 119903

31)120596minusΓ

1015840

119888(0) 120596 = 119868

119889120596

119889119905 (10)


119889120596

119889119905= minus(

1120591+1120583)120596+120572 (11)


120596 = 120572120581 (1minus 119890minus119905120581) (12)


1120581=1120591+1120583

(13)






119889cm =119898119877

119898 +119872lt119898

119872119877 (14)



Earth

v

West

East

mR

120579

120596



119898

119872= (

119903

119877)

3= (

200006378 times 106

)




119889

119889119905= Γnet (16)



119868Ω+119898V119877 cos 120579 = (119868 + 119889119868) (Ω+119889Ω) (17)



119898V119877 cos 120579 = 119868119889Ω+Ω119889119868 (18)


119889 (119868Ω) = 119898V119877 cos 120579 (19)



119889Ω

Ω=119898V119877 cos 120579

119868Ωminus119889119868

119868 (20)




119889Ω

Ω=52119898

119872(V cos 120579119877Ω

minus 1) (21)

Finally using

119898

119872=120588119887

120588(119903

119877)

3 (22)



119889Ω

Ω=52120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (23)


119898 is the


119868119898= 119868 minus 119868

119888= (1minus

119868119888

119868) 119868 = (1minus

1198721198881198772119888

1198721198772 ) 119868

= [1minus(120588119888

120588)(

119877119888

119877)

5] 119868

(24)




119868119898= 120574119868 (25)


where

120574 = 1minus(120588119888

120588)(

119877119888

119877)

5 (26)

(23) becomes

119889Ω

Ω=

52120574

120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (27)



119888= 3473 times 106mand






Ω =2120587






00

01

02

03

04

05

06

07

08

minus120575120596i

(deg

yr)

25kms

20kms

15kms

10kms

5kms

10 15 20 25 30 35











120596 = 120572120581+ 120575120596119894119890minus119905120581

(29)













References






























































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


Earth

v

West

East

mR

120579

120596



119898

119872= (

119903

119877)

3= (

200006378 times 106

)




119889

119889119905= Γnet (16)



119868Ω+119898V119877 cos 120579 = (119868 + 119889119868) (Ω+119889Ω) (17)



119898V119877 cos 120579 = 119868119889Ω+Ω119889119868 (18)


119889 (119868Ω) = 119898V119877 cos 120579 (19)



119889Ω

Ω=119898V119877 cos 120579

119868Ωminus119889119868

119868 (20)




119889Ω

Ω=52119898

119872(V cos 120579119877Ω

minus 1) (21)

Finally using

119898

119872=120588119887

120588(119903

119877)

3 (22)



119889Ω

Ω=52120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (23)


119898 is the


119868119898= 119868 minus 119868

119888= (1minus

119868119888

119868) 119868 = (1minus

1198721198881198772119888

1198721198772 ) 119868

= [1minus(120588119888

120588)(

119877119888

119877)

5] 119868

(24)




119868119898= 120574119868 (25)


where

120574 = 1minus(120588119888

120588)(

119877119888

119877)

5 (26)

(23) becomes

119889Ω

Ω=

52120574

120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (27)



119888= 3473 times 106mand






Ω =2120587






00

01

02

03

04

05

06

07

08

minus120575120596i

(deg

yr)

25kms

20kms

15kms

10kms

5kms

10 15 20 25 30 35











120596 = 120572120581+ 120575120596119894119890minus119905120581

(29)













References






























































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


where

120574 = 1minus(120588119888

120588)(

119877119888

119877)

5 (26)

(23) becomes

119889Ω

Ω=

52120574

120588119887

120588(119903

119877)

3(V cos 120579119877Ω

minus 1) (27)



119888= 3473 times 106mand






Ω =2120587






00

01

02

03

04

05

06

07

08

minus120575120596i

(deg

yr)

25kms

20kms

15kms

10kms

5kms

10 15 20 25 30 35











120596 = 120572120581+ 120575120596119894119890minus119905120581

(29)













References






























































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in










References






























































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in
















































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in










Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

Documents

Research Article Superrotation of Earth s Inner Core