International Journal of Geophysics

Volume 2015, Article ID 763716, 8 pages

http://dx.doi.org/10.1155/2015/763716

## Superrotation of Earth’s Inner Core, Extraterrestrial Impacts, and the Effective Viscosity of Outer Core

^{1}Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, WI 53141, USA^{2}Department of Geosciences, University of Wisconsin-Parkside, Kenosha, WI 53141, USA

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

Academic Editor: Petr Vaníček

Copyright © 2015 Pirooz Mohazzabi and John D. Skalbeck. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The recently verified superrotation of Earth’s inner core is examined and a new model is presented which is based on the tidal despinning of the mantle and the viscosity of the outer core. The model also takes into account other damping mechanisms arising from the inner core superrotation such as magnetic and gravitational coupling as well as contribution from eddy viscosity in the outer core. The effective viscosity obtained in this model confirms a previously well constrained value of about 10^{3} Pa s. In addition, the model shows that the currently measured superrotation of the inner core must be almost exactly equal to its asymptotic or steady-state value. The effect of extraterrestrial impacts is also investigated, and it is shown that perturbations due to such impacts can only persist over a short geological time.

#### 1. Introduction

In 1996, Song and Richards analyzed two types of seismic waves traveling through Earth’s solid and fluid cores, respectively [1]. They discovered a differential travel time between the two waves, which increased systematically by about 0.3 s from the year 1967 to the year 1995. This temporal change was later attributed to a shift of the lateral velocity gradient in the inner core caused by the inner core rotation [2, 3]. Subsequent studies provided further support for the faster rotation of Earth’s solid inner core compared to the rest of the planet [4–7].

Because of the anisotropy due to wood-like grain structure of the crystalline iron in the inner core, the speed of seismic waves is different in different directions [8]. If the inner core rotates faster than the rest of Earth, seismic waves that are generated at the same place on Earth but decades apart will have different travel times passing through the inner core and detected diametrically across the globe. Accurate recent investigations, using high-quality waveform doublets generated in the South Sandwich Islands region and detected in and near Alaska, have revealed definite temporal changes over a period of up to 35 years [9]. These experiments confirm that Earth’s inner core rotates faster than the mantle and crust (hereafter simply referred to as mantle), with a superrotation of about 0.27 to 0.53 deg/yr.

Estimates of the inner core differential rotation rates range from low values of 0.2 to 0.3 deg/yr [2] to high values of 3 deg/yr [10]. Deuss [11] presents a compilation of studies on inner core differential rotation rates as a function of publication year that suggests more recent data supports lower inner core differential rotation rates; however, the data is quite scattered. The author cites a recent study [12] that reports no inner core rotation and another study [13] that shows subrotation but the majority of the studies report inner core superrotation with respect to the mantle.

Several Earth-based mechanisms have been suggested for Earth’s inner-core differential rotation. Gubbins [14] attributes the phenomenon to the existence of a large toroidal magnetic field inside Earth’s core which, together with the dipole components, produces a magnetic torque on the inner core that tends to accelerate it relative to the mantle. He then assumes that some equilibrium is reached so that the inner core rotates with constant angular velocity and experiences zero net torque. Glatzmaier and Roberts [15] numerically solve the self-consistent magnetohydrodynamic equations that describe thermal convection and magnetic field generation in a rapidly rotating spherical fluid shell with a solid conducting inner core. Their solution, which serves as an analog for the geodynamo, shows that viscous and magnetic coupling of the outer core with the inner core and the mantle causes time-dependent variations in their respective rotation rates; the inner core usually rotates faster than the mantle. Buffett and Glatzmaier [16] allowed gravitational coupling between the inner core and the mantle by incorporating viscous deformation of the inner core into their numerical simulations of the geodynamo. They discovered that differential rotation between the inner core and the mantle is permitted by allowing the inner core to deform. Numerical calculations by Aurnou et al. [17] showed excess temperature inside the tangent cylinder surrounding the inner core which generates a prograde thermal wind and a strong azimuthal magnetic field inside the tangent cylinder. They conclude that the electromagnetic torque on the inner core resulting from induced azimuthal magnetic fields and the ambient poloidal field equilibrate when certain conditions are met. Dumberry [18] studied the steady and time-dependent rates of inner core rotation based on angular momentum balance between the inner core, fluid core, and mantle. He concluded that the rotational rate of an oscillating inner core is constrained by the changes in mantle rotation induced by gravitational coupling.

Tkalčić and others [19] analyzed earthquake doubles to construct a model for inner core differential rotation rates of 0.25 to 0.48 deg/yr with decadal fluctuations around the mean of 1 deg/yr. The authors suggest that these decadal fluctuations can account for discrepancies between previous core rotation models and agree with recent geodynamo simulations. A three-dimensional model by Livermore and others [20] suggests that axial electromagnetic torque is the dominant influence for inner core differential rotation and that decadal variations of the magnetic field may drive the quasioscillatory nature of the inner core differential rotation.

Magnetic coupling between the inner and outer cores seems to play a role not only in the superrotation of the inner core but also in the generation of the Earth’s magnetic field. Su et al. [10] reported an anomalous variation in the inner core orientation that temporally coincided with the geomagnetic “jerk” (a sudden change in the strength of Earth’s magnetic field of 1969-1970). This suggests a correlation between Earth’s magnetic field and the inner core superrotation. Glatzmaier and Roberts [21] have suggested that the inner core rotates in response to the magnetic torque and the viscous torque to which it is subjected, with + = 0. The magnetic torque drags the inner core eastward and the viscous torque acts westward. Nevertheless, they state that even though the fluid viscosity in their model is several orders of magnitude greater than is likely for real Earth, the viscous torque on the inner core has little effect.

Although each of these theories provides a reasonable explanation for the differential rotation of the inner core, there is no experimental evidence supporting one over the others, and they are all based on processes that are assumed to be taking place inside Earth. Furthermore, about a decade before the most recent data of 0.27 to 0.53 deg/yr were published, calculations based on magnetic coupling between the inner and outer cores suggested superrotations that were about an order of magnitude higher than these values [10, 21].

Perhaps the most significant and tangible external factor responsible for the inner core superrotation is the tidal forces [22, 23]. Dissipation of tidal energy in oceans and transfer of angular momentum between Earth, Moon, and the Sun results in torques on the mantle causing it to dispin [24], which gradually increases the length of the day continuously. This despinning of the mantle leaves the inner core with a small excess eastward rotational velocity relative to the mantle. Su et al. [10] used the known tidal increase in the length of day of approximately 2 ms per century to extrapolate backward and conclude that the inner core was rotating with the same period as the mantle about 10^{5} years ago. The recent confirmation of superrotation of Earth’s inner core by Zhang and others [9] prompted us to examine this phenomenon in the context of tidal effects from a phenomenological point of view.

In addition to the oceanic tidal effects, yet another external factor could contribute to the differential rotation of the Earth’s inner core which has not been addressed in the literature, Earth impacts. Since the formation of our Solar System some 4.6 billion years ago, collisions and impacts have played a fundamental role in establishing its characteristics, ranging from the accretion of planetesimals and the early formation of planets [25, 26] to the recent series of impacts on Jupiter by the fragments of Shoemaker-Levy 9 comet in July 1994 [27]. These collisions and impacts have affected the dynamics of various components of the Solar System. For example, most planets have obliquity or axial tilt with respect to their orbital planes about the Sun. Earth has an obliquity of about 23.5° while Uranus, the third largest planet in the Solar System, has an obliquity of about 97°. In other words, Uranus is tilted on its side so that its rotation axis is nearly in its orbital plane about the Sun. Yet, giant planets are believed to form with nearly zero obliquity [28]. The axial tilts of planets are believed to have been caused by major impacts [29, 30].

The above discussion is the motivation for examining a second question in this paper. Because Earth’s solid inner core rotates inside the fluid outer core, is it possible for bolide impacts to alter the angular velocity of Earth’s mantle relative to the inner core, resulting in a superrotation, a subrotation or a transition from one to the other? If so, is it possible for these impacts to result in differential rotations that are comparable to the experimentally observed superrotation values, and how long would it take for such perturbations to damp out?

The model presented here could equally be applied for conditions that would generate subrotation or no differential rotation depending on the size and angle of bolide impact. Examination of the probability of direction of differential rotation is beyond the scope of this paper. The focus here is to examine the potential for bolide impact contribution if superrotation of the inner core is present.

#### 2. Tidal Despinning of the Mantle and Viscosity of the Outer Core

As stated earlier, dissipation of tidal energy in the oceans results in gradual despinning of the mantle, which leaves the inner core with a small excess relative eastward differential rotation relative to the mantle called superrotation [24]. This differential rotation is communicated between the mantle and the inner core though a Couette flow in the fluid outer core which tends to damp the superrotation.

Consider two concentric solid spheres separated by a fluid layer between them with a coefficient of viscosity . Let the radius of the inner sphere be and the inside radius of the outer sphere be . Then, if the inner sphere rotates with an angular velocity with respect to the outer sphere, the damping torque on it due to the fluid viscosity is given by [31]Although Dai and others [32] have presented seismic reflection data suggesting significant topography on the inner core boundary, this model can be applied to Earth’s inner core and mantle, with the fluid between them being the outer core. Here the effect of the topography is absorbed in the estimate of effective viscosity.

Using (1), the rotational equation of motion of the inner core,becomeswhere is the rotational inertia of the solid inner sphere. The left hand side in this equation is the net torque on the inner sphere. The first term represents the torque due to the tidal effects, which tends to increase the eastward rotational velocity of the inner core relative to the mantle. is the constant rate at which the rotational speed of the mantle slows down due to the tidal effects. It has a value of 2.3 milliseconds per day per century [33] or 3.45 × 10^{−5 }deg/yr^{2}. The second term on the left hand side of (3) is the viscous damping torque.

Writing in terms of the inner core density and the inner core radius , reduces (3) toFurthermore, defining the characteristic time byreduces (4) toIntegration of this equation using the initial condition givesFrom this equation, we see that is in fact the relaxation time for the process. With the value of obtained from (5), this equation gives the superrotation of the inner core at any time after the dissipation of tidal energy began, about 4 × 10^{9} years ago, since oceans existed as early as the relatively stable Earth. The value of , as mentioned earlier, is 3.45 × 10^{−5 }deg/yr^{2}.

Earth’s inner core has a radius of 1.220 × 10^{6} m and a density of 13000 kg m^{−3}. The liquid outer core has a radius of 3.473 × 10^{6} m [34]. The viscosity of the outer core, however, is highly uncertain depending on its method of determination. In fact, its estimated values from various sources span over 14 orders of magnitude from 10^{−3} to 10^{11} Pa s [35]. For example, de Wijs et al. [36] report a value of 1.5 × 10^{−2} Pa s with an uncertainty of a factor of three through dynamical first principles simulations of liquid iron. Similarly, Rutter et al. [37] report a value of 1.6 × 10^{−2} Pa s using experimental high pressure study of liquid Fe-S system. On the other hand, using amplitude of forced nutation, Molodenskiy [38] finds a value of about 1 × 10^{6} Pa s.

Although the viscosity of the Earth’s outer core is notoriously uncertain, it is plausibly considered to be bounded by 10^{2} Pa s ≤ ≤ 10^{11} Pa s [32, 34]. Estimates of the fluid outer cores viscosity show variation from 10^{2} Pa s at the top of the outer core to 10^{11} Pa s at the bottom near the inner core boundary using Arrhenius extrapolation of pressure dependencies for laboratory measurements on liquid iron [39, 40]. A viscosity value of 1.22 × 10^{11} Pa s near the solid inner core was found by Smylie [41] using Ekman layer theory to estimate viscous drag forces from Coriolis splitting of the two equatorial translational models of oscillation of the solid inner core. Palmer and Smylie [42] found a viscosity value of 615 Pa s near the top of the outer core from the free decay of free core nutations. Smylie et al. [43] present a viscosity estimate of 2.371 × 10^{3} Pa s for the top of the outer core which relies on the decay of free core nutations method and an estimate for the bottom of the outer core of 1.247 × 10^{11} Pa s from the method of Smylie [41] and accounting for pressure dependence of the activation volume. The authors found the viscosity profile fits a nearly log-linear trend across the outer core but values still vary by 9 orders of magnitude. Smylie [44] presents this log-linear viscosity profile for the outer core based on the Arrhenius description of temperature and pressure of the viscosity by Brazhkin [39] and also reports a mean viscosity value of 3124 Pa s at the top of the outer core.

Therefore, instead of using a highly variable value of the viscosity to calculate the superrotation of the inner core, we solve the inverse problem. We use the measured value of the superrotation to find the viscosity of the outer core. To do so, we first solve the transcendental equation (7) numerically for the relaxation time , using = 0.4 deg/yr (mean value of the measured superrotation), = 3.45 × 10^{−5 }deg/yr^{2}, and = 4 × 10^{9} yr (age of oceans). This gives a value = 11600 yr (or 3.66 × 10^{11} s). Then, we use (5) and solve for the viscosity of the outer core . We find = 3.37 × 10^{3} Pa s. This number is in incredible agreement with the value obtained by Bills [24] and by Smylie [44]. Using a well described different model, Bills [24] reports an outer core viscosity of the order of 10^{3} Pa s, assuming a value of superrotation of about 1 deg/yr. It is interesting to note that if we use = 1 deg/yr (instead of 0.4 deg/yr) in our calculations, we find = 1.35 × 10^{3} Pa s. Our calculated viscosity does not agree with the study by Su et al. [10] that implied a viscosity of 10^{−4} Pa s by ignoring electromagnetic forces and assuming differential rotation of 3 deg/yr.

Admittedly, our values of the viscosity are calculated at the inner core boundary as our theoretical model considers rotation of the inner core. However, the extremely high viscosity of 10^{11} Pa s near the inner core strongly couples the lower zone of the outer core to the inner core. Thus, the differential rotation effectively takes place further out in the outer core where the viscosity is considerably lower. Since we are solving the inverse problem, our calculated viscosity corresponds to this region where the differential rotation effectively takes place. Ironically, due to the functional form of (5), the values of the viscosity calculated for a superrotation of 0.4 deg/yr with = 12000–13000 kg/m^{3} and with ranging from the inner core radius all the way to 98% of the outer core radius vary in a narrow range of 10^{3}-10^{4} Pa s.

Let us now return to the notorious uncertainty in the reported values of outer core viscosity. Generally these numbers fall into two distinct categories. The high values are based on seismologic, geodetic, and geomagnetic observations of Earth [32, 34, 41], whereas the low values are based on theory and laboratory investigations of liquid metals [24, 45, 46]. It has been suggested that this dichotomy is possibly due to contribution of eddy viscosity caused by fluid motion, whereas the liquid metal investigations only account for intrinsic or molecular viscosity [24]. In addition, it is possible that electromagnetic and gravitational couplings also play roles. Since all these effects arise from the differential rotation of the inner core; therefore, they should each be a function of . Furthermore, according to Lenz’ Law and LeChatelier Principle, they should all act in such a way to reduce or eliminate the cause, which is the inner core superrotation.

Let the combined torque resulting from all these damping effects, which for the sake of simplicity from now on we refer to as core coupling, be = . Expanding this function in a Taylor series, we haveBecause is small, we can neglect the second- and higher-order terms in the expansion. In addition, since we must have = 0, we obtainAdding this retarding torque to the left hand side of (3), we obtainRepeating the calculations as before, instead of (6) we getwhere is a constant with the dimension of time. Integration of this equation with the initial condition = 0 giveswhere is defined bywhich is the new relaxation time, taking into account both the viscosity of the outer core as well as all other damping effects which we called core coupling.

Equation (12) has exactly the same functional form as (7) except that is replaced by . Therefore, if we solve this equation for using the known value of , we find the same value for that we found for before; that is, = 11600 yr. Thus, the relaxation time of 11600 yr corresponds not only to the viscosity of the outer core, but also to all other damping effects as well. Using this value in (5) and solving for ; therefore, gives an effective value for the outer core viscosity that could be greater than the actual value. It is interesting to note that since the relaxation time is 11600 yr, the value of today is so close to its saturation value that it would be impossible to measure the difference.

#### 3. Extraterrestrial Impacts

Because geomagnetism and its reversals are associated with the differential rotation of Earth’s inner core, therefore, any changes in the Earth’s magnetic field can be attributed to a change in the inner core’s superrotation. Muller and Morris [47] have suggested that the impact of a large extraterrestrial object on Earth can produce a geomagnetic reversal through a mechanism involving a sequence of events. If so, it would be possible for such an impact to alter the superrotation of the inner core. In what follows, we study the dynamics of a bolide impact and calculate the change in the steady-state value of of the differential rotation of Earth’s inner core resulting from such impacts. We then investigate the time scale over which such a perturbation in the inner core superrotation would damp out.

Consider an asteroid or comet of mass and velocity** v** relative to the center of Earth just before impact, as shown in Figure 1. For simplicity, we assume that the velocity vector of the asteroid or comet is in the equatorial plane of Earth. The center of mass of the bolide-Earth system is located at a distance offrom Earth’s center, where is the mass of Earth and is its radius. For a typical bolide, and therefore is very small. For example, for a bolide of radius km and a density comparable to that of Earth, we haveThen, with = 6.378 × 10^{6} m, we find < 0.2 m, which is less than 20 cm from Earth’s center. Therefore, we can safely take the center of mass of the bolide-Earth system at the moment of impact to be simply Earth’s center.