EARTH ROTATION AND DEFORMATION SIGNALS CAUSED BY DEEP EARTH PROCESSES
Andrew Watkins
A Thesis
Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
December 2017
Committee:
Yuning Fu, Advisor
Richard Gross
Marco Nardone
Margaret Yacobucci
ii ABSTRACT
Yuning Fu, Advisor
The length of a day on Earth (abbreviated LOD) is not exactly 24 hours. There is a small excess LOD that varies on timescales ranging from a few days to thousands of years, generally on the order of milliseconds. One characteristic of LOD variations is a sinusoidal component with a period of ~6 years. The cause of the ~6-year signal is unknown, but is generally suspected to be exchanges of angular momentum between the mantle and the core. This study aimed to test the hypothesis that the ~6-year LOD signal is due to coupling between the mantle and fluid outer core. The flow of the core’s fluid deforms the base of the mantle, leading to redistribution of
Earth’s mass (causing changes in the gravitational field) and deformation of the overlying crust.
Surface deformation data from a global network of high-precision Global Positioning System
(GPS) stations was analyzed, and the component that acts on the ~6-year timescale was isolated and inverted for the core’s flow. Resulting angular momentum changes were computed for the outer core and compared to the LOD signal to search for evidence of core-mantle coupling.
Outer core angular momentum changes obtained from GPS deformation data exhibit evidence of the suspected core-mantle coupling, but this result is sensitive to inversion parameters. Changes in the gravitational field were also modeled and found to be smaller than the errors in the currently available data.
iii ACKNOWLEDGEMENTS
I would like to thank Yuning Fu, Richard Gross (JPL), Peg Yacobucci, and Marco
Nardone for their support and guidance during this project, and for their willingness to serve as members of my thesis committee. I would also like to thank Mike Heflin (JPL), Mike Chin
(JPL), and Ming Fang (MIT) for their helpful comments. Finally, this thesis would not be possible if not for the work of previous researchers in this field, and of those who have contributed to the production of datasets used in this project.
iv TABLE OF CONTENTS
Page
1. INTRODUCTION .…………………………………………………………………...... 1
2. BACKGROUND AND METHODS ...... 4
2.1. LOD Signal Isolation ...... 4
2.2. Deformation Signal Isolation ...... 7
2.3. Inversion for the Outer Core’s Flow ...... 10
2.3.1. Physical Model and Discretization ...... 10
2.3.2. Staggered Inversion Approach ...... 12
2.3.3. Geostrophic Flow Solutions ...... 15
2.3.4. Angular Momentum Solutions ...... 18
2.4. Gravitational Field Changes ...... 21
2.5. Error Estimation ...... 24
2.6. Robustness Test ...... 26
2.7. Values of Physical Parameters ...... 27
3. RESULTS ...... 29
3.1. LOD Signal and Solid Earth Angular Momentum ...... 29
3.2. Deformation Signal ...... 31
3.3. Outer Core Pressure and Flow ...... 36
3.4. Outer Core Angular Momentum ...... 37
3.5. Modeled Gravitational Field Changes ...... 40
4. DISCUSSION AND CONCLUSION ...... 42
REFERENCES ...... 47 v LIST OF FIGURES
Figure Page
1.1 Decadal LOD Signals From the Outer Core ...... 2
2.1. Spectral Properties of Loading Removal ...... 10
2.2. Spatial Distribution of GPS Stations I ...... 13
2.3. Spatial Distribution of Grid Cells ...... 15
2.4. Geometry of the Direction Vector ...... 17
2.5. Geometry of Taylor’s Constraint ...... 18
2.6. Geometry of a Cylindrical Annulus ...... 21
3.1. Surface Fluid Effects on LOD ...... 29
3.2. Spectral Properties of LOD Signal Isolation ...... 30
3.3. Rotation Signals of the Mantle and Crust ...... 30
3.4. Spectral Properties of Deformation Signal Isolation I ...... 31
3.5. Spectral Properties of Deformation Signal Isolation II ...... 32
3.6. Spectral Properties of Deformation Signal Isolation III ...... 32
3.7. Spatial Distribution of GPS Stations II ...... 33
3.8. Spatial Distribution of GPS Stations III...... 34
3.9. Spatial Distribution of GPS Stations IV ...... 34
3.10. Deformation Signal Size ...... 35
3.11. Circulating Flow ...... 36
3.12. Latitudinal Flow ...... 37
3.13. Interannual Angular Momentum Signals I ...... 38
3.14. Interannual Angular Momentum Signals II ...... 39 vi 3.15. Interannual Angular Momentum Signals III ...... 39
3.16. Modeled Gravitational Field Changes I ...... 40
3.17. Modeled Gravitational Field Changes II ...... 41
4.1. Columnar Flow ...... 44
vii LIST OF TABLES
Table Page
2.1. Inversion Acronyms ...... 27
2.2. Values of Physical Parameters ...... 27
1 1. INTRODUCTION
Earth is a complex system with many related components. One feature of this system is that components directly accessible at the surface are affected by inaccessible components deep below the crust. These effects are opportunities, as they provide a means to investigate deep
Earth processes which are otherwise elusive. The fluid outer core is an illustrative example of such an inaccessible component. Clues about the core are available in the form of the geomagnetic field, gravitational field variations, surface deformation, and length of day (LOD) variations. This study made use of the latter three observations to investigate the outer core’s behavior on sub-decadal timescales.
Measurements of LOD are made at Earth’s surface and are therefore related to the angular velocity of the solid Earth (crust and mantle). In general, the angular momentum L = Iω of a body rotating about a fixed axis is conserved, where ω is the body’s angular velocity and I, called the moment of inertia, is related to the body’s mass distribution (Barger and Olsson,
1994). The principle of conservation of angular momentum is therefore a valuable means of investigating Earth’s rotation, and requires either mass redistribution or some external torque to explain LOD variations.
Previous investigations have established the outer core as one important source of torque on the solid Earth (Gross 2015). In these investigations, variations in the geomagnetic field were used to determine changes in the outer core’s flow and angular momentum L%&. Researchers found that a torque coupling the ∆L%& to the solid Earth would cause LOD changes that agree well with measured ∆LOD on decadal timescales (Figure 1.1). 2
Figure 1.1. Decadal LOD Signals From the Outer Core. Image from Gross (2015) showing the agreement between measured ∆LOD (Stephenson and Morrison, 1984; McCarthy and Babcock, 1986; Gross 2001) and modeled ∆LOD (Jackson et al., 1993; Jackson 1997; Pais and Hulot, 2000) computed from core flow solutions derived from magnetic field data.
There is an unexplained ~6-year LOD signal that is smaller in magnitude than the decadal variations. Both the fluid outer core (Holme and De Viron, 2013) and the solid inner core
(Davies et al., 2014) have been suggested as causes for the signal. This study aimed to test the hypothesis that the ~6-year LOD signal is a result of angular momentum exchange between the solid Earth and outer core.
If the outer core is responsible for the ~6-year LOD signal, there should be a ~6-year signal in the outer core’s motion compensating for the motion of the solid Earth. This study assumed that the flow in the outermost region of the core is geostrophic (Le Mouël et al., 1985), implying that a ~6-year signal in the outer core’s motion would be associated with pressure fluctuations just below the core-mantle boundary (CMB). Fang et al. (1996) discussed how such pressure fluctuations would deform the overlying mantle and crust, forming an important piece of the mathematics used in this study. 3 The number of high-precision Global Positioning System (GPS) stations has increased in recent years, providing valuable data about crustal deformation. This study analyzed a global network of GPS data, inverting a ~6-year deformation component for the CMB pressure variations and for the associated fluid flow at the edge of the core. Additional constraints on the organization of the outer core were then used to solve for the flow of the bulk of the outer core’s fluid. Modeled ∆L%& were then derived analytically from the flow field.
If the ~6-year LOD signal is due to angular momentum exchanges between the solid
Earth and fluid outer core, a comparison of their angular momenta should reflect the conservation principle. If the components of L%& and the solid Earth’s angular momentum (L(&) due to all external effects have been removed, this takes the form of L( + L%& = 0, where the dot accent represents a derivative with respect to time. Integrating and removing the constant values, this leads to the expectation:
∆L%& = −∆L(& [Eq. 1.1]
The ∆L%& solutions were compared to the time series ∆L(& computed directly from the
~6-year LOD signal. The results are interpreted in terms of their relevance to the hypothesis that the outer core is responsible for the ~6-year LOD signal. Modeled gravitational field changes are also presented. 4 2. BACKGROUND AND METHODS
2.1. LOD Signal Isolation
The LOD dataset used in this study was COMB2015 Noon (Ratcliff and Gross, 2016), a daily time series of excess LOD (in milliseconds) at noon for a ~54-year period beginning in
1962. The data is made publicly available (Ratcliff n.d.).
Tidal forces from the Moon, Sun, and other planets result in deformation of Earth, which redistributes its mass and changes the moment of inertia about the rotation axis. This in turn causes LOD variations that have been modeled (Yoder et al., 1981; Kantha et al., 1998). These modeled LOD variations were removed from COMB2015 Noon prior to publication (Ratcliff and Gross, 2016). LOD variations due to tidal mass redistribution therefore do not influence the results of this study.
Annual and higher-frequency variations due to the atmosphere are a prominent feature of non-tidal LOD variations (Aoyama and Naito, 2000; Eubanks et al., 1985; Zhou et al., 2008).
The atmosphere exchanges angular momentum with the crust due to the winds (the “motion” term) and the redistribution of atmospheric mass (the “mass” term) (Zhou et al., 2006). The significance of the atmosphere to many scientific fields led The National Center for
Environmental Protection (NCEP) and The National Center for Atmospheric Research (NCAR) to collaborate on a Reanalysis Project that assimilates global atmospheric data (Kalnay et al.,
1996). This data has been commonly used to produce time series of the Atmosphere’s Angular
Momentum (AAM) (Salstein et al., 1993; Salstein and Rosen, 1997; Salstein et al., 2005; Zhou et al., 2006). The Special Bureau for the Atmosphere (SBA) of the International Earth Rotation and
Reference Systems Service (IERS) makes this AAM data publicly available in 6-hour intervals
(Special Bureau for the Atmosphere n.d.).
5 In this study, the 6-hourly axial AAM values from the SBA (for both the motion and mass terms) were combined into daily averages. For each day, 5 samples were considered: midnight, 06:00, 12:00, 18:00, and midnight on the following day. These values were added with weights: 1/8, 1/4, 1/4, 1/4, and 1/8 (respectively) to center the daily average at noon, following the convention of Gross et al. (2004). Two mass terms are given by the SBA, one with the inverted barometer (IB) correction applied (considers the ocean’s response to atmospheric pressure, but idealistically), and one without (considers the ocean as rigid) (Chao and Yan,
2010). The IB-corrected mass term was used, and multiplied by a factor of 0.7 to account for the elastic response of Earth to atmospheric pressure (Chao and Yan, 2010). Computed ∆AAM has been found to underestimate the observed ∆LOD at annual and shorter periods (Chao and Yan,
2010) by 10–20%, so the sum of the atmospheric motion and mass terms was multiplied by a factor of 1.15. The effects of AAM changes on LOD were then computed according to (Chao and Yan, 2010):
86400 s ∆LOD = ∆AAM [Eq. 2.1] I/,12ω4 where ω4 is Earth’s mean rotation rate and I/,12 is the axial moment of inertia of the mantle and crust (replacing their usage of the axial moment of inertia of the mantle only).
Oceanic Angular Momentum (OAM) variations also influence LOD (Gross et al., 2004;
Marcus et al., 1998). The Estimating the Circulation and Climate of the Ocean (ECCO)
Consortium developed a general circulation model of the ocean (Wunsch et al., 2009), which is used by the IERS’ Special Bureau for the Ocean (SBO) to produce datasets that directly provide the effects of OAM on LOD (Gross et al., 2005). The data is made publicly available online
(Special Bureau for the Oceans n.d.). This study used the (daily) ECCO_kf080h.chi dataset for
6 1993/01/02 and onward, and linearly interpolated the ECCO_50yr.chi dataset (10-day sampling interval) to daily for prior dates to determine the effect of OAM on LOD.
The effects of OAM (mass and motion) and AAM (mass and motion) were then subtracted from COMB2015 Noon. Variations in continental water storage, also known as the mass term of Hydrological Angular Momentum (HAM), also contribute noticeably to LOD.
These effects were not accounted for in this study as they are not known as well as AAM and
OAM, and appear to be dominantly annual (Chen et al., 2000). A Savitzky-Golay filter was then applied to smooth the dataset and remove any remaining sub-annual variations. Elaborating for clarity, this means at each data point, a polynomial of a certain order (3 in this case) was defined by a least-squares fit to a certain number of samples symmetrically surrounding that data point
(equal to 1095 samples ≈ 3 years in this case, with smaller windows as necessary towards the endpoints). The polynomial values were then taken as the new data values (Savitzky and Golay,
1964). The frequency content of the resulting LOD was analyzed by looking at its Power
Spectral Density (PSD) as a function of frequency. An un-windowed periodogram (used for all
PSD estimates in this study) was used to estimate the PSD (Stoica and Moses, 2005). The data was zero-padded out to 10 times its original length to obtain an estimate of the period of the
AB desired LOD signal. A curve of the form λ cos( t + φ) was then fit by least squares to the C smoothed LOD to extract the desired ~6-year signal, where T is the chosen period in days.
The ∆LHI time series was then computed by rearranging Eq. 2.1, using the extracted signal for ∆LOD and replacing ∆AAM with −∆LHI. The negative sign was introduced because the equation in this case describes the relationship of a body’s angular momentum with its own rotation rate (rather than with the rotation rate of an external object to which the body is coupled, as is the case in Eq. 2.1).
7 2.2. Deformation Signal Isolation
The deformation data used in this study was the Jet Propulsion Laboratory’s (JPL) daily position solutions for ~3000 GPS stations worldwide, made publicly available online (Heflin n.d.). The JPL residual time series were used, which have had a linear trend, abrupt jumps (due to events such as earthquakes), annual terms, semiannual terms, and tidal ocean loading removed
(Mike Heflin, personal communication 2017). This study focused solely on the radial component of the deformation data.
GPS deformation data has seen wide applications because it is sensitive to a wide range of factors. Local effects have significant influence on the deformation. These characteristics present serious challenges to the analysis of a globally distributed set of GPS deformation time series. Due to these challenges, this study focused on the frequency domain to extract the desired
~6-year component. The average PSD (denoted PSD) at a frequency f was used in this study for the analysis of the aggregate periodicity of a set time series. The desired component of the GPS data was that which satisfies the following conditions:
PSDPQR ST ≈ PSDRUP ST [Eq. 2.2]
PSDPQR SV ≈ 0 [Eq. 2.3]
PSDRUPXPQR SV ≈ PSDRUP SV [Eq. 2.4]
PSDRUPXPQR ST ≈ 0 [Eq. 2.5] where the subscript SIG indicates the set of time series of the extracted components, the subscript
GPS indicates the set of time series of the GPS data from which SIG was extracted, the subscript
GPS − SIG indicates the set of time series of the differences, ST is the frequency corresponding to the desired ~6-year period, and SV represents all other frequencies. The first three conditions (Eq.
2.2-2.4) correspond to the expectation that the desired signal (and only the desired signal) is
8 isolated, the last condition (Eq. 2.5) ensures the correct phase of the isolated signal, and the use of “approximately equal to” signs reflects the complexities associated with analyzing the aggregate properties of a set which varies significantly from one element to the next.
The GPS time series all contain some missing points. In this study, any GPS time series with data missing for more than 30% of the days in the time period to be analyzed was discarded.
The time series with less than 30% of the data missing were linearly interpolated to fill in the gaps, and the interpolated points are assigned a standard error estimate (1-σ) of 1 cm (about 3–4 times as large as the typical values).
All inversions of GPS deformation data first computed weighted monthly averages of the deformation. The weighted averages were calculated according to:
^ w^x^ x = [Eq. 2.6] ^ w^ where x is the weighted average, the x^ are the 31 deformation samples included in the average,
XA and the w^ are weights given by w^ = σ_` , where σ_` is the standard error estimate of x^.
The following methods were optional efforts to isolate the desired component and were not all used in every inversion: 1) A 12-sample (roughly 1 year) moving average was used to smooth the time series. The endpoints where the 12-sample window could not be defined symmetrically were discarded. 2) A sum of 8 sine terms was fit by least squares to the time series. The terms outside of the period range 5–7 years were dropped. 3) A curve of the form
AB λ cos( t + φ) was fit by least squares to the time series, where T = 2191.5 days = 6 years. C
Start points were provided (λf = 1 mm, φf = 0) for the curve fits. Following these processes, the mean was removed.
9 A deformation signal isolated from GPS data is not desirable if it is known to be due to some cause other than CMB pressure variations. A variety of potential causes of deformation signals exist at Earth’s surface, including but not limited to: non-tidal atmospheric loading
(NTAL), non-tidal oceanic loading (NTOL), and a hydrological loading (HYDL) contribution that tends to be the largest. The former two terms are the effects of mass loads applied from surface fluids, while the latter is the effect of changing water mass distribution associated with the hydrologic cycle.
The German Research Centre for Geosciences (GFZ) provides publicly available data
(GFZ n.d.) about these loading contributions on a 0.5° x 0.5° global grid based on global atmospheric data for NTAL, and models for NTOL and HYDL contributions (Dill and Dobslaw,
2013). These data were bi-linearly interpolated to each GPS station location (except for 55 GPS stations whose locations were detected to be in the ocean by GFZ’s 0.5° x 0.5° mask file, and thus did not have trustworthy loading data). The NTAL and NTOL time series data were provided at a 3-hour sampling interval (HYDL contributions were provided as daily), and were converted to daily using a simple arithmetic mean. A linear trend was fit by least squares to the total loading contribution (NTAL + NTOL + HYDL) and removed, as was a seasonal estimate of
AB mB the form λ cos t + φ + λ cos t + φ + ε, so that the loading was directly i jkl.Al i A jkl.Al A comparable to the GPS residuals used in this study. A global 6-year signal is present in GPS data
(vertical component) and is not accounted for by surface loading data (Figure 2.1), giving further justification for the use of the frequency domain to extract the deformation component.
10
Figure 2.1. Spectral Properties of Loading Removal. The set of GPS stations used (444 total) was those with trustworthy loading data and no gaps longer than 180 days for the period between 2002/01/01 and 2014/01/01. Surface loading does not account for the 6-year component of the deformation (Equations 2.2 and 2.5 are not satisfied).
2.3. Inversion for the Outer Core’s Flow
2.3.1. Physical Model and Discretization
The inversion of the surface deformation field for the CMB pressure fluctuations was based on the work of Fang et al. (1996), who provided theoretical relationships between pressure anomalies at the CMB and surface deformation in the radial and horizontal directions. The radial deformation Dr is given in the following form:
Dr θ, ϕ = K(θ, ϕ, θr, ϕr)p(θr, ϕr)dQ [Eq. 2.7] u where θ is the colatitude, ϕ is the longitude, primed coordinates represent points at the CMB, p is a pressure anomaly, dQ = sin θ dθdϕ (Ming Fang, personal communication 2017), and the integral is taken over the pressure distribution on a unit sphere. The K term is given by:
11 r r 3 K θ, ϕ, θ , ϕ = h|P| cos α [Eq. 2.8] 4πgρ4 | where g is the gravitational acceleration at Earth’s surface, ρ4 is the average density of Earth, h| is the Love number h of degree n for a pressure load from the core acting on the surface, α is the
r r arc length between the points θ, ϕ and θ , ϕ on a unit sphere, and P| is the Legendre polynomial of degree n given by (Suetin n.d.):
1 d| P x = xA − 1 | [Eq. 2.9] | n! 2| dx|
Surface deformation data comes from a finite number of GPS stations, so the continuous form of Eq. 2.7 could not be directly implemented. Instead Earth’s surface and the CMB surface were sampled with a finite number of grid cells. Each grid cell was demarcated by two lines of latitude and two lines of longitude. A center was assigned to each grid cell, where the latitude and longitude coordinates were given by the mean of the boundary latitudes and longitudes, respectively. The only exception was that of polar grid cells, which simply covered a polar cap extending to a boundary latitude. The centers of polar cells were assigned to be the pole itself.
Each grid cell’s deformation value Dr was given by the weighted average (see Eq. 2.6) of the isolated deformation components for the GPS stations within its boundaries.
Equation 2.7 gives each deformation value as being the result of contributions from an infinite number of pressure elements. The discretization used in this study considered each deformation value Dr to be the result of a finite number of contributions (one from each grid cell on the CMB). Accordingly, the integral in Eq. 2.7 was replaced with a finite sum:
∆r = KipidQi + KApAdQA + ⋯ + K|Xip|XidQ|Xi + K|p|dQ| [Eq. 2.10]
Considering a total of m grid cells on Earth’s surface that are each the result of contributions of n grid cells on the CMB, a linear system arises:
12 ∆ri = KiipidQi + KiApAdQA + ⋯ + Ki,|Xip|XidQ|Xi + Ki|p|dQ|
∆rA = KAipidQi + KAApAdQA + ⋯ + KA,|Xip|XidQ|Xi + KA|p|dQ|
. . .
∆r1Xi = K1Xi,ipidQi + K1Xi,ApAdQA + ⋯ + K1Xi,|Xip|XidQ|Xi + K1Xi,|p|dQ|
∆r1 = K1ipidQi + K1ApAdQA + ⋯ + K1,|Xip|XidQ|Xi + K1|p|dQ| or simply:
∆r1_i = C1_|p|_i [Eq. 2.11] where C^ = K^dQ^ .
The deformation and pressure grid cell elements ∆r and p were given a single set of coordinates each to describe their locations (the latitude and longitude of their centers). Thus the
K^ operate in much the same way as in the continuous case, with the input α being the arc length
(on the surface of a unit sphere) between the center of the ith deformation grid cell and the center of the jth pressure grid cell, evaluated using Vincenty’s formula (Vincenty 1975). When calculating the K^, the summation index was cut off at n = 10, about the degree at which the
Love numbers h| become negligibly small (Fang et al., 1996). The dQ were then obtained by directly evaluating sin θ dθdϕ over the boundaries of the ith grid cell. The estimation of the
XÖ XÖ pressure field pÇÉÑ was obtained according to pÇÉÑ = C ∆r, where C was the exact inverse
CXi if it exists and if not, the Moore-Penrose pseudoinverse was used (Barata and Hussein,
2012).
2.3.2. Staggered Inversion Approach
A unique challenge of this study was the limitation on spatial resolution due to the locations of GPS stations around the globe. Most of Earth’s surface is excluded by necessity
13 from having any samples (the oceans), and many areas (particularly in the southern hemisphere) have a sparse distribution of GPS stations (Figure 2.2). Thus, a grid layout with a high spatial resolution (small grid cells) would have some cells with missing data. The effect of this can be
XÖ understood by considering the resolution matrix R1 = C C. The resolution matrix gives the relationship between the estimation pÇÉÑ of the pressure field and the pressure field p|_i that
XÖ XÖ would exactly explain the data, since pÇÉÑ = C ∆r = C Cp|_i = R1p|_i. Each missing grid cell would eliminate one row of the linear system in Eq. 2.11, causing R1 to deviate further from the identity matrix. The estimation pÇÉÑ of the pressure elements would then be confounded and would deviate further from the ideal p|_i.
Figure 2.2. Spatial Distribution of GPS Stations I. The set of GPS stations (499 total) with coverage between 2004/01/01 and 2016/01/01 and no gaps more than 180 days for that period. Shown here in the Hammer projection (Weisstein n.d.) along with the continents.
On the other hand, a system of grid cells on Earth’s surface that is fully determined (each grid cell containing at least 1 GPS station) would only capture very large scale features of the deformation, and therefore would only be able to model very large scale features of the pressure
14 and flow at the edge of the core. Such a coarse spatial resolution would likely miss features of the core’s behavior that affect the angular momentum solutions.
This study solved this dilemma by using multiple spatial designs. For each point in time, the deformation on Earth’s surface was sampled and inverted multiple times using grid cell layouts that are slightly offset from each other. Each inversion had a sparse spatial resolution, and the grid cell layout on the CMB was the same as the layout on the surface. The pressure samples from each inversion were then combined onto the same sphere, considerably improving the spatial resolution of the modeled pressure field while ensuring the linear systems (Eq. 2.11) were fully determined.
The grid cells for each inversion were laid out in a systematic manner. First, a base number of latitude bands bàâ was specified, along with the desired number of longitude bands bàf and desired number of longitude layouts äàf. Latitude bands were then defined, each with a π uniform width ( radians). Within each latitude band, there were then bàf equally spaced bàâ grid cell centers assigned beginning at -p and moving eastward (each separated by
2π radians), with the edges of the cells placed equidistant from each adjacent grid cell center. bàf
Another layout was then defined by placing the centers of latitude bands on the edges of the bands for the previous inversion (for a total of bàâ + 1 latitude bands). For each latitude band, there were then bàf grid cell centers defined with the same longitude coordinates of the previous inversion (except for samples at the poles, which were not divided into multiple longitude bands). This formed a set of related spatial layouts, and a total of äàf sets were defined in the same manner (except the longitude coordinate of the grid cell centers was assigned beginning
15 2π radians to the east of the starting point for the previous set). Figure 2.3 shows an bàfäàf example of the result.
Figure 2.3. Spatial Distribution of Grid Cells. Locations of grid cell centers are shown in relation to the coastlines for 6 inversions (bàâ = bàf = äàf = 3). Each pole has multiple samples from the inversions with an alternate layout of latitude bands (those that end in “b”).
When there were multiple pressure values sampled at the poles, a weighted average was used (as in Eq. 2.6) to assign the polar grid cell a single pressure value.
2.3.3. Geostrophic Flow Solutions
Flow solutions were derived from the pressure field based on the assumption of tangentially geostrophic flow (where Coriolis and pressure gradient forces dominate) in the outermost region of the core. The equation governing this assumption was given by Le Mouël et al. (1985):
å×∇ p ã = è [Eq. 2.12] 2ρêIω4 cos θ
16 where ã is the flow vector, å is the unit normal vector, ∇è is the horizontal gradient operator, p is the pressure field, and ρêI is the density in the outer core.
The operator ∇è in Eq. 2.12 is typically an analytical operator. In these typical cases, the gradient of the entire pressure field is taken, and the normal component removed. However, the description of the pressure field in this study was limited to the finite set of nodes on the CMB surface. Thus, a numerical substitute for the operator ∇è was used. The formation of this substitute was based on the physical meaning of the gradient of a scalar field. This gradient is a vector that points in the direction of the greatest increase per unit of spatial separation, and its magnitude corresponds to the magnitude of that increase per unit of spatial separation (Stewart
2011).
A pressure gradient vector was determined for each pressure sample p within the geostrophic region of the CMB. The direction was determined by looping through the set of adjacent samples p^, and finding the sample that maximizes the scalar gradient:
∆p^ p^ − p = [Eq. 2.13] α^ α^ where α^ is the arc length on the CMB between the pressure samples p and p^. The gradient vector ∇èp was first assigned a direction vector s in Cartesian coordinates. The coordinate system was chosen so that the origin is the center of a sphere with radius rIHë (the radius of the
CMB), and the z-axis is Earth’s rotation axis. Let the Cartesian vectors p1 and p2 give the positions of the CMB pressure sample point p and the position of the adjacent CMB pressure sample point p^ that maximizes Eq. 2.13, respectively. The first condition for the vector s was that it lie on the plane tangent to the sphere at p1. This is equivalent to forcing the dot product í ∙
îï to be zero. The second condition is that í must point in the direction of greatest increase. That
17 is, í must “point” towards p2. Consider the line in ℝó that intersects both the origin and the point p2. This line will also intersect the plane tangent to the sphere at îï at another point kp2
(whenever îï ∙ îò > 0), where k is some constant. Mathematically, this condition was implemented by requiring that í = kîò − îï. Combining these conditions gives the direction vector í of the gradient vector ∇èp :
îï A í = kîò − îï = îò − îï [Eq. 2.14] îò ∙ îï
The geometry of this formation is shown in Figure 2.4.
The direction vector í was then rescaled to unit magnitude to form a vector í. The unit direction vector í was then rescaled by the magnitude of the scalar gradient in Eq. 2.13 to form the gradient vector ∇èp. Equation 2.12 was then used to form the flow vector ã. That vector is not defined at the equator, since there is a factor of cos θ in the denominator. As will be discussed in Section 2.3.4, the flow vectors were associated with cylindrical annuli parallel to the rotation axis. The flow vector was calculated for equatorial samples by moving the coordinate θ northward by õ , which is halfway to the northern edge of the equatorial annulus. 8bàâ
Figure 2.4. Geometry of the Direction Vector. The direction of the gradient ∇èp at a pressure sample point îï, where p2 is the adjacent pressure sample point that maximizes Eq. 2.13. The position vectors are shown as large dots, while the direction vector is shown as an arrow.
18 2.3.4. Angular Momentum Solutions
Implicit in the implementation of the geostrophic flow equation (Eq. 2.12) is the assumption of constant density (the mean density of the outer core was used in this study for this constant). Combined with the geostrophic assumption, the constant density assumption leads to the conclusion that the flow ã does not vary in the direction parallel to the rotation axis (Roberts and Aurnou, 2012). This conclusion, known as the Taylor-Proudman Theorem, is important for studies of the outer core, including this one, where it was used to obtain the flow of the core’s fluid at depth from the surface samples ã. A related result, known as Taylor’s constraint, describes the outer core’s flow as rigid rotations of nested liquid cylinders parallel to the rotation axis. A good visualization of this constraint was given by Livermore et al. (2008), presented here in Figure 2.5.
Figure 2.5. Geometry of Taylor’s Constraint. The boundaries of nested cylindrical annuli are shown, formed from the portion of cylindrical shells contained inside the sphere. Image from Livermore et al. (2008).
19 A finite number of cylindrical annuli were defined by considering their intersection with the CMB. Since the annuli have a non-zero width, they intersect the CMB at two latitudes (a polar edge and an equatorial edge) in both the northern and southern hemispheres (with the exception of an equatorial annulus, which has just one edge in each hemisphere). The polar edge of the first annulus was taken to be the boundary latitude of the cylinder tangent to the inner core
úùûü (which is arccos ( ), where rQIë is the radius of the inner core). The remaining edges were úû†ü placed halfway between the latitude bands of flow samples. Each flow sample rests on the surface of one of these annuli. An angular velocity vector ° was associated with each flow vector ã by forming a direction vector ° and scaling by the appropriate magnitude:
¢×ã ° = [Eq. 2.15] |¢×ã|
|ã| ° = ° [Eq. 2.16] |¢| where r is the position vector of the flow sample ã. For each annulus, a weighted average (again as in Eq. 2.6) of the z-components ω/ was taken, and the entire annulus was assumed to be moving with this angular velocity.
The axial moment of inertia I/ was formed for a general cylindrical annulus of outer core
úùûü fluid that intersects the CMB at latitudes of ±ψf (See Figure 2.6), where ψf ≤ arccos : úû†ü
A A I/ = dI/ = x + y dm [Eq. 2.17] u u
A A A A I/ = ( x + y ) ρdV = ρêI (x + y )dxdydz [Eq. 2.18] u u
20 where Q describes the geometry of the annulus. The transformation to spherical coordinates with radius r, latitude, and longitude was then used:
x = r cos ψ cos ϕ [Eq. 2.19]
y = r cos ψ sin ϕ [Eq. 2.20]
z = r sin ψ [Eq. 2.21]
The Jacobian matrix J1 of this transformation is then:
∂x ∂x ∂x ∂r ∂ψ ∂ϕ ∂y ∂y ∂y J = [Eq. 2.22] 1 ∂r ∂ψ ∂ϕ ∂z ∂z ∂z ∂r ∂ψ ∂ϕ
The appropriate substitutions were made and the integrand was multiplied by det J1 =
A −õ õ r cos ψ (cos ψ is nonnegative whenever ψ ∈ 2 , 2 ) to complete the change of variables
(Stewart 2011):
A A A A A A A I/ = ρêI ( r cos ψ cos ϕ + r cos ψ sin ϕ)r cos ψ drdψdϕ [Eq. 2.23] u
The limits of integration for Q are clear upon examining Figure 2.6:
AB ƨ≠ úû†ü m j I/ = ρêI r cos ψ drdψdϕ [Eq. 2.24] Ø X¨ 2fɨ ≠ ú ≠ û†ü 2fɨ
ƨ 4πρ rl sinj ψ ≠ I (ψ ) = êI IHë sin ψ − − cosl ψ tan ψ [Eq. 2.25] / f 5 3 f X¨≠ where the vertical bar implies the antiderivative is evaluated at −ψf, and that value is subtracted from the antiderivative evaluated at +ψf. Performing this operation, and making use of the oddness of the sine function and the evenness of the cosine function, we have:
21 4πρ rl sinA ψ I (ψ ) = êI IHë sin ψ 1 − f − cosm ψ [Eq. 2.26] / f 5 f 3 f
The axial moment of inertia for each annulus was determined by evaluating Eq. 2.26 at the annulus’ equatorial edge, and subtracting that value from Eq. 2.26 evaluated at the annulus’ polar edge. For the equatorial annulus, only one evaluation is necessary.
The axial angular momentum was formed for each annulus according to: L/ = I/ω/. The outer core’s angular momentum variations (∆LêI) were determined by summing over all the annuli. The mean was then removed from the ∆LêI time series.
Figure 2.6. Geometry of a Cylindrical Annulus. The axial moments of inertia are defined for cylindrical annuli (shaded region) that lie outside a single cylinder.
2.4. Gravitational Field Changes
The crustal deformation from CMB pressure variations is an extension of the mantle deformation. Another consequence of CMB pressure variations is the redistribution of Earth’s mass (through deformation of its components), which alters the gravitational field (Fang et al.,
22 1996). Any object outside Earth’s surface has some gravitational potential energy. The gravitational potential U rf, θ, ϕ describes this potential energy per unit mass at a given location outside Earth, where rf is the distance from Earth’s center of mass. The gravitational potential changes at Earth’s surface that would arise from CMB pressure anomalies were given by Fang et al. (1996) in a form similar to Eq. 2.7:
r r r r DU θ, ϕ = KA(θ, ϕ, θ , ϕ )p(θ , ϕ )dQ [Eq. 2.27] u where KA is given by:
r r 3 KA θ, ϕ, θ , ϕ = k|P| cos α [Eq. 2.28] 4πρ4 | where k| is the Love number k of degree n.
The CMB was divided into a set of grid cells based on the combined set of pressure samples. Demarcating lines of latitude and longitude were placed halfway between each adjacent sample. Equations 2.27 and 2.28 were then discretized in a manner similar to Eq. 2.7 and 2.8.
Considering a set of deformation samples on Earth’s surface with the same coordinates as the pressure samples gives rise to a linear system similar to Eq. 2.11:
∆U|_i = CA,|_|p|_i Eq. 2.29
Since both CA,|_| and p|_i were known in this case, no inversion was necessary and the ∆U|_i were forwardly modeled.
Gravitational potential changes are often described by geodesists in terms of an expansion in the following form (Chao and Gross, 1987):
≥ ± ± GM4 r4 U rf, θf, ϕf = P±1(cos θf) C±1 cos mϕf + S±1 sin mϕf [Eq. 2.30] rf rf ±≤Ø 1≤Ø
23 where the C±1 and S±1 are unitless “Stokes coefficients,” M4 is the mass of Earth, r4 is Earth’s radius, G is the universal gravitational constant, and the P±1 are the normalized associated
Legendre functions of degree ä and order m given by:
2 − δ 2ä + 1 ä − m ! P µ = P µ 1Ø [Eq. 2.31] ±1 ±1 ä + m !
where δ is the Kronecker delta function (δ_∂ = 1 if x = y, and δ_∂ = 0 otherwise) and the P±1 are the unnormalized associated Legendre functions given by:
1 − µA 1 A d±Æ1 P µ = µA + 1 ± [Eq. 2.32] ±1 2±ä! dµ±Æ1
Changes in the gravitational field are thus described by changes in the Stokes coefficients. Of particular importance to the mass distribution of Earth are the degree-2 terms
(Gross 2015; Fang et al. 1996). This study focused on the CAØ term, related to Earth’s equatorial bulge (also known as Earth’s oblateness or “flattening”). The CAØ term was related to the gravitational potential changes by first considering the gravitational potential in Eq. 2.30 at a
Æ general point exactly at Earth’s surface (rf = r4), and assuming S±1∑ = C±1∑ = 0 for all m >
0, giving:
≥ GM4 U r4, Ω = C±ØP±Ø(cos θ) [Eq. 2.33] r4 ±≤Ø where Ω is an abbreviation for the angles θ and ϕ. The surface spherical harmonic functions Y±1 were then considered (Chao and Gross, 1987):
2ä + 1 ä − m ! Y Ω = (−1)1P cos θ e1º [Eq. 2.34] ±1 ±1 4π ä + m !
Using Equations 2.31-2.34, the surface gravitational potential can be expressed as:
24 ≥ 2GM4 π U r4, Ω = C±ØY±Ø Ω [Eq. 2.35] r4 ±≤Ø
∗ Multiplying by the complex conjugate YAØ of the degree-2 spherical harmonic YAØ (the asterisk denotes complex conjugation), and integrating with respect to Ω gives:
≥ ∗ 2GM4 π ∗ U r4, Ω YAØ Ω dΩ = C±Ø Y±Ø Ω YAØdΩ [Eq. 2.36] r4 u ±≤Ø u where Q gives all possible values of Ω on a sphere and dΩ = sin θ dθdϕ (Richard Gross, personal communication 2017). The summation on the right-hand side of Eq. 2.36 is simply equal to CAØ due to the normalization of the spherical harmonics (Chao and Gross, 1987). With
∗ an evaluation of YAØ directly from Eq. 2.34 and elementary algebra, Eq. 2.36 leads to:
r4 5 A CAØ = U r4, θ, ϕ (3 cos θ − 1) sin θ dθdϕ [Eq. 2.37] 8GM4π u
The gravitational potential U was given for a finite number of grids on Earth’s surface (Eq. 2.29), so the integration in Eq. 2.37 was considered as the sum of the integrations over each of these grids, and U was considered constant within the grids:
ºø¿ æø¿ r4 5 A CAØ = U (3 cos θ − 1) sin θ dθdϕ [Eq. 2.38] 8GM4π ºø¡ æø¡
r4 5 j j CAØ = U ϕi − ϕA cos θA − cos θA − cos θi + cos θi [Eq. 2.39] 8GM4π
th where the ϕ^ and θ^ are the appropriate boundaries for the i grid cell.
2.5. Error Estimation
25 Error estimates from GPS data were propagated through the operations described in
Sections 2.2–2.4 using a general formula that describes the error estimate σ¬ of a function f of a set of variables β (Tellinghuisen 2001):
C σ¬ = g Vg [Eq. 2.40]
th ∂f where g is a column vector whose i element is given by g = , and the matrix V is the ∂β covariance matrix (the sample covariance was used in this study): V^ = cov β, β^ . When averaging different values of the same GPS time series (as was necessary when taking monthly averages and moving averages), V was the sample autocovariance matrix, with V^ being the sample autocovariance at lag i − j. For vector quantities, error was propagated through the magnitude only, and the vector cross product in Eq. 2.12 was considered equivalent to scalar multiplication when computing the g. The standard error estimates for the 6-year curves fitted to
GPS time series were obtained by considering the version of Eq. 2.40 where the β are uncorrelated:
σ = gAσA [Eq. 2.41] ¬ ∆ø
AB The form of the fit is again: λ cos( t + φ), where T is exactly 6 years. The parameters are then C
λ and φ. The errors σ« and σ» are then taken to be the half-width of the 68% confidence interval
(corresponding to the definition of a 1-σ standard error on a normal distribution) for the appropriate parameter, and Eq. 2.41 becomes:
2π 2π σ t = σAcosA t + φ + λAσA sinA t + φ [Eq. 2.42] ¬ « T » T
26 Although Eq. 2.40 is based on linear functions of the β, Monte-Carlo simulations have shown
that these estimates are reasonable for some nonlinear functions when the parameter errors σ∆ø are sufficiently small (Tellinghuisen 2001). In the case of fitting sum-of-sine models to GPS time series, the reported confidence intervals were too wide for the error estimate from Eq. 2.41 to remain reasonable, so these error estimates remained unchanged from the GPS time series to the fitted curves.
2.6. Robustness Test
The sensitivity of the LêI solutions was examined by independently varying 4 parameters between 2 possible values each: 1) The time period examined (2004/01/01 to 2016/01/01 or
2000/06/01 to 2012/06/01), 2) The maximum gap size during the time period in question allowed for any given GPS station to be included (90 days or 180 days), 3) Smoothing of the GPS time series prior to curve fitting (no smoothing or 12-sample moving average), and 4) The type of curve fit used (the 6-year curve fit or the fit of a sum of 8 sines model, with only the curves at periods 5–7 years being kept). The spatial layout in Figure 2.3 was used for each inversion. Thus, a total of 2m = 16 inversions were run. Acronyms are used to refer to individual inversions (see
Table 2.1). Each inversion was examined for its isolation of the 6-year deformation signal in each quadrant of the globe so that regional differences could be seen. The estimations of the
CMB pressure field, outer core flow and angular momentum, as well as forward-modeled changes in the CAØ Stokes coefficient were also examined. The time series ∆LêI and ∆LHI were compared with the expectation in Eq. 1.1 as a means of investigating the idea that the outer core is responsible for the ~6-year LOD signal.
27 Table 2.1. Inversion Acronyms. Sets of inversions are referred to by replacing one or more of the letters with X (for instance, EXXX refers to all inversions for the earlier time period). Time period: Time period: Time period: Time period: 2000/06/01 to 2000/06/01 to 2004/01/01 to 2004/01/01 to 2012/06/01 (E), 2012/06/01 (E), 2016/01/01 (L), 2016/01/01 (L), maximum gap maximum gap maximum gap maximum gap size: 90 days (N) size: 180 days size: 90 days (N) size: 180 days (O) (O) Isolation by sum ENUE EOUE LNUE LOUE of sine expansion (UE) Isolation by ENSE EOSE LNSE LOSE smoothing and sine expansion (SE) Isolation by 6- ENUF EOUF LNUF LOUF year curve fit (UF) Isolation by ENSF EOSF LNSF LOSF smoothing and 6-year curve fit (SF)
2.7. Values of Physical Parameters
Table 2.2. Values of Physical Parameters. Parameter Value Source ρ (outer core’s mean kg Dziewonski and Anderson êI 1.12393×10m density) mj (1981) give the density in the outer core as a function of radius. This function was integrated over the range of radius values for the outer core, and the result divided by the width of the domain for the radial coordinate to obtain the mean value. ρ (mean density of Earth) kg Mussett and Khan (2000) 4 5.5×10j mj m g (gravitational acceleration 9.81 Mussett and Khan (2000) at Earth’s surface) sA
28 h| , n = 1, … ,10 (load Love -1.42523, 0.65898, 0.35221, Fang et al. (2013) numbers for a pressure load 0.19433, 0.11313, 0.06797, from the core acting on the 0.04135, 0.02519, 0.01530, surface) 0.00924 (respectively)
k| , n = 1, … ,10 (load Love 0.22776, 0.31735, 0.10924, Fang et al. (2013) numbers for a pressure load 0.04381, 0.01990, 0.00976, from the core acting on the 0.00500, 0.00263, 0.00140, surface) 0.00075 (respectively) k rQIë (radius of the inner core) 1.2215×10 m Dziewonski and Anderson (1981) k rIHë (radius of the outer core 3.48×10 m Dziewonski and Anderson (1981) k r4 (radius of Earth) 6.731×10 m Dziewonski and Anderson (1981) ω (Earth’s mean rotation rad Groten (2004) 4 7.292115×10Xl rate) sA
Am M4 (mass of Earth) 5.974×10 kg Dziewonski and Anderson (1981) G (universal gravitational m3 Groten (2004) 6.67259×10X11 constant) s2kg
37 A I/,12 (axial moment of inertia 7.1236×10 kg m Matthews et al. (1991) of the mantle and crust)
29 3. RESULTS
3.1. LOD Signal and Solid Earth Angular Momentum
AAM and OAM effectively accounted for the seasonal LOD variations, as shown in
Figure 3.1. Noticeable non-periodic variability remained at sub-annual timescales, possibly due to AAM or OAM fluctuations not accounted for in the NCEP/NCAR Reanalysis project or the
ECCO model.
Figure 3.1. Surface Fluid Effects on LOD. Surface fluids account for the seasonal signals in COMB2015 Noon.
The Savitzky-Golay filter successfully removed this remaining variability. The smoothed residual LOD includes a ~6-year signal but is dominated by the longer-period variations due to the outer core. The power spectrum displays a peak at 5.85 years that is successfully accounted for by the curve fit, as shown in Figure 3.2. The LOD signal is plotted along with the equivalent
L"# signal for the past ~20 years in Figure 3.3. 30
Figure 3.2. Spectral Properties of LOD Signal Isolation. The fit of a single 5.85-year curve accounts well for the LOD signal, satisfying the equivalent for LOD of the conditions given in Equations 2.2–2.5.
Figure 3.3. Rotation Signals of the Mantle and Crust. The fitted LOD signal generally agrees in phase with the isolation of Holme and De Viron (2013) since 2000, but has a slightly shorter period (5.85 years as opposed to their 5.9-year fit) and a slightly larger amplitude (0.154 ms as opposed to their result of 0.127 ms). 31 3.2. Deformation Signal
In 5 of the 16 inversions, the 6-year deformation signal was considered not sufficiently isolated, because at least one of the conditions in Equations 2.2–2.5 was violated for at least one quadrant of the globe. An illustrative example is shown in Figure 3.4. A sum of 8 sines model was used as the isolation method in each of these cases. The XXXE inversions that successfully satisfied Eq. 2.2–2.5 were LNSE, ENSE, and EOSE. The characteristics of quadrant-by-quadrant average power spectra for all inversions that did satisfy Equations 2.2–2.5 are similar by definition, so only 2 of these results are presented (Figures 3.5 and 3.6).
Figure 3.4. Spectral Properties of Deformation Signal Isolation I. The LOSE inversion did not properly isolate the desired GPS component, as the condition in Eq. 2.5 is violated in the SW quadrant. 32
Figure 3.5. Spectral Properties of Deformation Signal Isolation II. The EOSF inversion properly isolated the desired GPS component.
Figure 3.6. Spectral Properties of Deformation Signal Isolation III. The LOSF inversion properly isolated the desired GPS component.
33 The set of GPS stations used was dependent upon the time period being analyzed and the maximum gap size allowed, so the spatial distribution of each of these sets is presented. The spatial distribution of the set for the LOXX inversions was shown in Figure 2.2, so the remaining
3 sets are displayed in Figures 3.7–3.9. The inversion parameters used in this study were chosen intentionally so that each inversion would involve a fully determined linear system (no empty grid cells on Earth’s surface). However, for the sets in Figures 3.7–3.9, there are some instances where adjacent grid cells, while having sampled different areas and being assigned different centers, have included the exact same set of GPS stations. This is expected for polar grid cells, but for others (where flow vectors are computed) is undesirable, since they imply a greater spatial resolution than is justified by the data.
Figure 3.7. Spatial Distribution of GPS Stations II. The locations of GPS stations (400 total) for the LNXX inversions. Two pairs of nonpolar grid cells have shared sets of stations. One pair has centers at (60°W,60°S) and (100°W,60°S) and another has centers at (20°E,60°S) and (20°E,30°S).
34
Figure 3.8. Spatial Distribution of GPS Stations III. The locations of GPS stations (353 total) for the EOXX inversions. One pair of nonpolar grid cells has a shared set of stations, with centers at (60°W,60°S) and (100°W,60°S).
Figure 3.9. Spatial Distribution of GPS Stations IV. The locations of GPS stations (293 total) for the ENXX inversions. Two pairs of nonpolar grid cells have shared sets of stations. One pair has centers at (60°W,60°S) and (100°W,60°S) and another has centers at (20°E,60°S) and (20°E,30°S). 35
For the LOXX inversions, all nonpolar grid cells sampled unique sets of GPS stations. Of these inversions, only the LOXF instances successfully isolated the desired GPS component. The
Signal-to-Noise Ratios (SNRs), defined as the ratio of the amplitude of the fitted curve to the maximum assigned error estimate (see Eq. 2.42), were examined. Unsurprisingly, the LOSF inversion had higher SNRs (mean = 3.62) than the LOUF inversion (mean = 2.15), so LOSF was considered the preferred inversion.
The amplitudes of the fitted curves in the preferred solution had a mean value of 0.89 mm, and a median value of 0.75 mm. There was no clear pattern in the spatial distribution of the amplitudes, as shown in Figure 3.10.
Figure 3.10. Deformation Signal Size. The amplitude of the ~6-year component of GPS data is shown as a function of location for the preferred inversion. For all inversions that successfully isolated the ~6-year component of GPS data, the mean amplitude ranged from 0.82–1.01 mm, and the median ranged from 0.71–0.92 mm.
36 3.3. Outer Core Pressure and Flow
The estimated pressure field just below the CMB consisted of anomalies on the order of
101 Pa for all inversions, a small fraction of the mean pressure in that region. The geostrophic
23 m flow vectors associated with these anomalies are generally on the order of 10 s . The flow solutions at the outer core’s edge can be characterized by three broad categories: 1) Large-scale circulations of the core’s fluid, 2) Uniform flow within latitude bands, and 3) Chaotic. Categories
1) and 2) are much more common, generally symmetric about the equator, and generally westward. The onset and the fading of these organized flows sometimes exhibits a lag between the northern and southern hemispheres. The chaotic flows are very short-lived and tend to occur as a transition stage between the two organized categories. Examples of the organized flows 1) and 2) from the preferred inversion are given in Figures 3.11 and 3.12.
Figure 3.11. Circulating Flow. From the preferred (LOSF) solution. Centers of the vortices correspond to pressure peaks and are generally located at latitudes ±60°. Flow vectors at the equator have been scaled down to ½ of their original length for display.
37
Figure 3.12. Latitudinal Flow. From the preferred (LOSF) solution. The strongest flows are uniform within latitude bands at ±60° and the less organized flow elsewhere is generally of a smaller magnitude. Flow vectors at the equator have been scaled down to ½ of their original length for display.
3.4. Outer Core Angular Momentum
13 In every inversion, the magnitudes of the L6# and L"# variations are similar – about 10
J s. However, the phase and period of the ∆L6# solutions are not consistent. Of the 3 XXXE inversions that properly isolated the deformation signal, 2 (ENSE and EOSE) gave ∆L6# solutions that are dominantly periodic at ~6 years and completely out of phase with the prediction given by Equation 1.1, and 1 (LNSE) gave a ∆L6# solution that was dominantly periodic at ~3 years. Of the ∆L6# solutions from the XXXF inversions, the results varied with respect to the time period analyzed. The EXXF inversions gave results either periodic at ~3 years or out of phase with the prediction from Eq. 1.1, and the LXXF inversions (including the preferred solution) gave results dominantly periodic at ~6 years that agreed in phase with the 38 prediction from Eq. 1.1. The phase agreement of the LXXF solutions was verified using the sample cross-covariance function between ∆L6# and −∆L"# (a peak was located within 1.5 years of zero lag). Representative examples of these results are shown in Figures 3.13–3.15.
Figure 3.13. Interannual Angular Momentum Signals I. From the ENSE inversion. The L6# and L"# time series are similar (Pearson’s r = 0.77). The shaded area represents the 1-σ standard error estimates.
39
Figure 3.14. Interannual Angular Momentum Signals II. From the EOSF inversion. The L6# and L"# time series display different periodicities. The shaded area represents the 1-σ standard error estimates.
Figure 3.15. Interannual Angular Momentum Signals III. From the preferred (LOSF) inversion. The L6# and L"# time series are anticorrelated (Pearson’s r = −0.59). The shaded area represents the 1-σ standard error estimates. 40
3.5. Modeled Gravitational Field Changes
The magnitude of the forward-modeled C1> Stokes coefficient changes was similar for all inversions. Maximum changes were between 102?1 and 4×102?1, roughly one order of magnitude smaller than the 1-σ errors in the currently available C1> data from Satellite Laser
Ranging (SLR) measurements (Cheng et al., 2013). The ∆C1> solutions from the inversions that did not properly isolate the deformation signal tend to exhibit time-dependent amplitudes, while the solutions from inversions that properly isolated the signal are all periodic at ~6 years. Of the periodic solutions, the LXXX solutions had a trough roughly at 2007/01/01 (except for the LNSE solution, which had an offset from this phase), and the EXXX solutions had a trough roughly at
2005/01/01. Representative examples are shown in Figures 3.16 and 3.17.
Figure 3.16. Modeled Gravitational Field Changes I. From the EOSF inversion. The shaded area represents the 1-σ standard error estimates.
41
Figure 3.17. Modeled Gravitational Field Changes II. From the preferred (LOSF) inversion. The shaded area represents the 1-σ standard error estimates.
42 4. DISCUSSION AND CONCLUSION
The results of the LOD signal isolation in this study primarily reaffirm the conclusions of previous researchers that have identified a ~6-year LOD signal (Vondrak 1997; Abarca del Rio et al., 2000; Mound and Buffett, 2006; Holme and De Viron, 2013; Duan et al., 2015).
The fact that 6-year curves were fit with a higher SNR to smoothed GPS time series than to unsmoothed GPS time series is unsurprising, since there is higher frequency variability of a larger amplitude than the fits themselves. The fact that the SNRs in the curve fits are generally well above 1.0 suggests that the curve fits are trustworthy. However, the error estimates in Eq.
2.42 may be overly optimistic. Sine functions are nonlinear, violating the premise on which the error estimates (Eq. 2.40) are based. Another difficulty is that the deformation signal is small – the amplitudes are about half of the size of the typical standard errors in the monthly averages
(smoothed or not). This implies that even the typical variability within a month is of a larger magnitude than the desired 6-year deformation signal. Weighting the averages was unable to significantly reduce propagated errors, due to the autocovariance of GPS time series. It was through spatial averages that the propagated error was significantly reduced. This was evident in the XXXE inversions, where the ~1.3 mm error estimates in the monthly time series were kept after the curve fit, and reduced by about a factor of 3.5 by the weighted spatial averages. Further, nonstationarity (due to time-dependent autocovariance) of GPS residual time series complicates the use of the PSD, which is only strictly defined for stationary processes. Nonstationarity also undermines the use of a single autocovariance matrix in Eq. 2.40, since the diagonal (lag 0) elements are known to be dependent on time by observing the reported position error estimates.
Finally, time-dependent autocovariance for other lags is evident in the average power spectra. 43 For instance, a close examination of the GPS time series PSD (Figures 3.5 and 3.6) reveals that the interannual characteristics of the spectrum seem to have changed over time.
The geostrophic flow solutions exhibit interesting similarities with previous inversions from the magnetic field. The “westward drift” of the magnetic field corresponds to an equatorially symmetric westward flow of the bulk of the core’s fluid, with fluctuations that explain the several-decade LOD signals (Jault et al., 1988). The latitudinal flows at ±60° shown in Figure 3.12 are consistent with Taylor’s constraint. The circulating flow solutions resemble similar features in some westward drift solutions (an example is shown in Figure 4.1). These circulations have been interpreted as the twisting motion of a column of core fluid, although the larger spatial scale of the vortices in this study complicates this interpretation. Taken with the westward and equatorially symmetric nature of the flow solutions in this study, these features suggest that the flows could be associated with the same process that causes the longer-period
LOD signals, only the scale here is about one to two orders of magnitude smaller (Le Mouël et al., 1985; Holme and Olsen, 2006). In fact, the view of the ~6-year LOD signal as resulting from a smaller-scale instance of these longer-period outer core processes has been suggested by previous research (Gillet et al., 2010). Although these similarities are interesting, the assumption made in this study of exactly geostrophic flows just below the CMB is a drastic one. This is an extension of another assumption (for which there is little direct evidence) that Coriolis forces are much stronger than Lorentz forces in that region. There is reason to believe that the outer core’s net angular momentum arises from geostrophic part of its flow (Roberts and Aurnou, 2012), but non-geostrophic flows would also be associated with the pressure at the edge of the core, and hence would influence the surface deformation data. 44
Figure 4.1. Columnar Flow. Image from Holme and Olsen (2006). A circular flow feature near the projection of Asia onto the CMB is accompanied by a similar feature of opposite rotational sense in the Southern hemisphere, consistent with columnar flow. Compare with Figure 3.11.
The −∆L+, from Earth rotation data and ∆L-, from surface deformation data in some solutions show surprisingly high similarity considering that they were computed from independent datasets. This similarity agrees with Eq. 1.1. and therefore supports the idea that coupling between the solid Earth and outer core is responsible for the ~6-year LOD signal.
However, the breakdown of this phase agreement with even one pair of non-unique deformation samples on Earth’s surface suggests there is more to the story than was captured by the methodology in this study. The finer details of the ∆L-, results reveal abruptness (Figure 3.15) that betrays the discrete nature of this study. Thus, the finer details in the computed ∆L-, are less important.
The question of whether the results of this study support the idea of outer core-mantle coupling at ~6-years or not is an important one. For the moment though, the separate (but 45 related) question of the validity of the preferred results is considered. Is the component of GPS deformation data isolated in this study really due to CMB pressure variations? Do the flow and
∆L-, solutions really describe the motion (and angular momentum) of the core’s fluid? Is
Earth’s oblateness really changing according to Figure 3.15? Factors that support an answer of
“yes” to these questions include: 1) The agreement of the flow solutions with the expectation that they be symmetric about the equator, 2) The features in the flow solutions that are similar to features from magnetic field inversions, and 3) The agreement of the ∆L-, result with LOD data and the principle of conservation of angular momentum, which by itself is not a reason to believe the result, but should be considered. Factors challenging the validity of the preferred solution include 1) The sparse spatial distribution of the deformation data, 2) The use of a non-standard staggered inversion approach, which has not been explored on more theoretical grounds, 3) The low amplitude of the isolated ~6-year GPS components relative to the standard error measurements of the time series from which they are derived, 4) The nonstationarity of GPS time series and lack of a clear or consistent PSD peak at 6 years in several of the quadrants considered, 5) The lack of rigor in the error estimates due to the use of nonlinear functions and false assumptions of stationarity, 6) The use of assumptions to obtain flow solutions that lack strong tangible evidence and were made largely out of convenience, and 7) The fact that the result is not robust. The hypothesis of outer core-mantle coupling at ~6-years is clearly supported by the preferred solution, so the conclusions that can be drawn from this study depend largely on the relative weight of these factors relevant to its validity.
In theory, some of these factors could be addressed directly. A Monte-Carlo approach would more provide appropriate error estimates. If in the future more GPS stations are placed around the globe, particularly in the southern hemisphere, it could allow for a more conventional 46 sampling of the deformation data, and would allow for additional inversions that have uniquely sampled deformation grid cells. If in the future C/0 data becomes available with decreased errors, the computed ∆C/0 could be compared to the data. Finally, when the GPS deformation data has a longer time span of suitable data to analyze, a clearer picture will arise, since only 2 cycles (at most) of the 6-year signal were analyzed in this study.
The results of this study have implications for a variety of fields beyond Earth’s rotation, so implementation of the above suggestions to obtain clearer results would certainly be worthwhile. Stokes coefficients for the gravitational field modeled here are also used to study processes at the surface, such as hydrological mass redistribution due to climate change. Surface deformation data is used in a wide sense to study the climate (Davis et al., 2004; Fu and
Freymueller, 2012; Fu et al., 2013). The existence of outer core-mantle coupling at ~6-years would further aid the understanding of the dynamics of deep Earth processes and of planetary interiors in general. Improved understanding of Earth’s rotation would aid the exploration of space, because changes in Earth’s orientation are a major source of error in spacecraft tracking and navigation (Estefan and Folkner, 1995).
Angular momentum conservation on the ~6-year timescale is clearly exhibited between measurements for the solid Earth and computations for the liquid outer core, with only moderate support given to the hypothesis of coupling due to methodological limitations. This study identified specific areas for potential improvements and laid a foundation for their implementation, while pioneering the use of GPS data as a means of directly investigating deep
Earth processes for their relevance to Earth’s rotation.
47
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