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PUBLICATIONS Earth and Space Science RESEARCH ARTICLE 10.1002/2016EA000198

Coherence between sea level oscillations and orbital perturbations

Special Section: The Arctic: An AGU Joint Special Collection

Ramy Mawad1

Key Points: • Earth’s orbital perturbation • Earth’s mass inflow • Earth’s mass outflow

Abstract Sea level rising and oscillations indicate global Earth’s mass variability. I propose inflow of mass to the Earth from space through coronal mass ejections, solar wind, small comets, and meteoroids that brings water to Earth’s atmosphere. Outflow mass can also escape plasma into space. Sea level rising is highly correlated with Earth’s orbital perturbations. Monthly and annual periodicities are found in the two variables. Monthly periodicities are attributed to the oscillations of center of gravity of the Earth-Moon system as the Moon revolves around the Earth. Annual periodicities are found with minimum in February and maximum in November, pointing to mass variability.

Supporting Information: • Table S1 Correspondence to: R. Mawad [email protected]

Citation: Mawad, R. (2017), Coherence between sea level oscillations and orbital perturbations, Earth and Space Science, 4, 138–146, doi:10.1002/2016EA000198. Received 19 JUL 2016 Accepted 20 FEB 2017 Accepted article online 24 FEB 2017 Published online 28 MAR 2017 Corrected 17 APR 2017 This article was corrected on 17 APR 2017. See the end of the full text for details.

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Astronomy and Meteorology Department, Faculty of Science, Al-Azhar University, Cairo, Egypt

1. Introduction The Earth travels around the Sun in an ellipse. Variation of orbital parameters can affect climate [Sharaf and Budnikova, 1969]. Perturbations simulated include nonspherical terms in the Earth’s geopotential field, lunar and solar gravity, and solar radiation pressure [Friesen et al., 1992]. The Earth is not isolated from space. Solar variability induces variations in space weather which has impacts on Earth’s climate and weather. It causes periodic changes in the amount of solar irradiance experienced on Earth. Zhao et al. [2004] used continuous wavelet transform; they examined the relationship between solar activity and the annual precipitation in the Beijing area. Their results indicate that the annual precipitation is closely related to the variation of sunspot numbers, and that solar activity probably plays an important role in influencing the precipitation on land. On the side of the Earth facing the Moon, lunar gravitational force is applied to water particles toward the Moon. This force produces a lunar bulge in the layer of ocean water. At the same time, the centrifugal force of the Earth-Moon system acting on the water particles at Earth’s surface opposite the Moon creates a second bulge. Two lunar bulges on opposite sides of Earth are thus created on a planet covered by a uniformly deep ocean. The bulges represent the crests of the two tidal waves (high tide), directly opposite each other, and the low water areas are the two troughs (low tide). The equilibrium tidal theory predicts tides that are semidiurnal, which means two high and two low tides each day. Tides are usually the largest source of short-term sea level fluctuations [Reddy and Affholder, 2002]. Changes in sea level are mainly caused by two components: Tides and weather effects. The dominant forcing of the tides is the variation in the gravitational field on the surface of the Earth due to regular movements of the Earth-Moon and Earth-Sun systems. Weather effects changes in sea level are mainly caused by changes in atmospheric pressure and wind [Reddy and Affholder, 2002]. From 1950 to 2009, measurements show an average annual rise in sea level of 1.7 ± 0.3 mm per year, with satellite data showing a rise of 3.3 ± 0.4 mm per year from 1993 to 2009 [Nicholls and Cazenave, 2010]. The reason for recent increase is unclear.

©2017. The Authors. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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Sea level can be raised by adding water mass. This comes primarily from melting land ice, i.e., from mountain glaciers, small ice caps, and the big ice sheets on Greenland and Antarctica. Melting sea ice hardly affects sea level, since it already floats on the sea displacing water corresponding to its weight (Archimedes’ principle). Mass addition (or removal) can affect to some extent, also come from water stored on land in liquid form, e.g., from storage of water in human-built reservoirs, which corresponds to about 3 cm worth of sea level [Chao et al., 2008]. Scientists have come to the consensus opinion that an increase in atmospheric carbon dioxide has resulted in a lower percentage of reradiated radiant heat escaping into space. The result is that the world is warming up,

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trending toward higher equilibrium temperatures, which will continue to increase if the carbon dioxide level does not stabilize. The raised temperature is responsible for two major kinds of contributions to changes [NOVA, Science in the News, 2004]. A possible approximate approach to this problem is to group glaciers into climatic regions, assuming glaciers in the same region to have a similar specific mass balance. With this method, we need to know only the specific mass balance for a typical glacier in each region [Kuhn et al., 1999]. The source of Earth’s volatiles, especially water, has been a subject of debate. The similarity of Earth’s bulk composition to that of meteorites known as enstatite chondrites suggests a dry proto-Earth with subsequent delivery of volatiles by neither local accretion or impacts of asteroids or comets [Javoy et al., 2010]. The measurements of D/H ratio of some comets from the Oort cloud yielded asteroids were the main source of Earth’s water with ≤10% being delivered by comets [Morbidelli et al., 2000]. Solar wind is the other source of the water into our solar system; lunar water may continuously produce in situ by the hydrogen ions (protons) of the solar wind affecting oxygen-bearing minerals (NASA-Lunar Prospector). It has been theorized that the latter may occur when hydrogen ions in the solar wind chemically combine with the oxygen atoms present in the lunar minerals (oxides, silicates, etc.) to produce small amounts of water trapped in the minerals’ crystal lattices or as hydroxyl groups, potential water precursors [Teodoro et al., 2009].

2. Approach The Earth’s orbit around the Sun is slightly elliptical; therefore, the distance between the Earth and the Sun varies throughout the year. Theoretically, every celestial body inside the solar system produces a nonzero force that causes oscillation around the Earth’s elliptical orbit called perturbation. The Earth is not a sphere; it is an oblate spheroid. There are also regions of higher mass and higher density called mascons that also introduce irregularities; this is another reason for orbital perturbation. Theoretically, these reasons are considered by numerous studies. The previous studies did not found that the perturbation of Earth’s orbit is restricted to forces from everybody inside the solar system and Earth’s shape. Neither empirical nor theoretical studies proved this. In addition, previous studies assumed that the Earth’s mass is constant ignoring tiny rising in the Earth’s mass. The aim of present paper is to study the impact of orbital perturbations of the Earth taking into account sea level variations. Previous studies assumed that the main source of sea level rising is the melting of polar ice caps and glaciers and that the total Earth’s mass is constant according to mass balance assumption (global warming depends on mass balance assumption) [Intergovernmental Panel on Climate Change, 2001]. However, the present study assumes that the Earth is interacting with space which leads to changes in the sea level. I also study other reasons for the rising of the sea level. When rising tides occur in an ocean (sublunar and antipodal points) that leads to flood tide in another location(vertical diameter on line between sublunar and antipodal points), which cause a zero sea level rise according to equilibrium theory of tides [Michelson, 1971], hence sea level is a global estimation. The correlation between orbital perturbations and sea level oscillations was not recognized in the past. However, recently it is found that the atmosphere reacts with solar wind and coronal mass ejections. Some of the solar wind or coronal mass ejections can pass through the Earth’s atmosphere causing aurora and some other phenomena. In addition, some studies assumed that the solar wind is a source of water in some planets that have no atmosphere such as the Moon [Teodoro et al., 2009]. I have neglected the tiny variability, as quick passing of debris or particles through Earth; I only considered the daily variations or greater in our consideration because data sources resolution is greater than 1 day.

3. Data Sources Geocentric ephemeris data during the period 1995–2006 of the Sun is used in this paper from Astronomical Ephemeris Data from NASA data center (resolution is 2 days): http://eclipse.gsfc.nasa.gov/ TYPE/ephemeris.html.

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Sea level data is obtained from TOPEX/Poseidon satellite altimeters based level series (joint mission of NASA and CNES). Satellite data were adjusted to give the same average level as the tide gauges, and it is downloaded in the same period 1995–2006 (resolution is ten days) from http://sealevel.colorado.edu/ current/sl_ib_ns_global.txt.

4. Algorithm I need to compare the perturbations in Earth’s orbit (Distance Shift) with the oscillation in sea level (Sea Level Shift). This can be done in the following steps.

Figure 1. Time series of sea level variations.

4.1. Estimating the Sea Level Shift 1. Calculate the linear fitting of sea level during the total period from a given satellite data. I found that h ¼ 6445:7 þ 3:2305 y; RL ¼ 0:97595;

(1)

where y is the year including months and days as fraction in the year, h is sea level in millimeter, and RL is the correlation coefficient of linear fitting. The correlation coefficient of linear fitting RL is determined as N ∑xy ∑x ∑y RL ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 N ∑x ð∑x Þ2 N ∑y 2 ð∑y Þ2 The sea level rise during the period 1995–2006 is 3.23 mm per year. 1. Complete the date to become daily by using Lagrange’s Interpolation Method. 2. Subtract the sea levels from linear fitting (equation (1)) to get the perturbation only “Sea Level Shift” by the following formula: Δh ¼ H h;

(2)

where H is the observed value of sea level and h is the calculated value of linear fitting of the sea levels with time as in equation (1). 4.2. Estimating the Shift of Sun’s Distance 1. Read the Sun’s distance R from geocentric ephemeris data and completing it to become daily by using Lagrange’s Interpolation Method. 2. Estimate the distance in elliptical orbit of the Earth around the Sun by using Kepler’s equation [Meeus, 1998] as follows: T 1 ¼ 367*Y 7 * ðY þ ðM þ 9Þ=12Þ = 4 þ 275*M=9 þ D 730530 T ¼ intervalðT 1 Þ þ UT=24:0;

(3)

where T is time scale and Y is year, M is month and D is day. The eccentricity of the Earth (0 = circle, 0–1 = ellipse, 1 = parabola) e ¼ 0:016709 1:151109 T:

(4)

The mean anomaly of the Sun M (0 at perihelion; increases uniformly with time) M ¼ 356:0470 þ 0:9856002585 T:

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(5)

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Figure 2. Comparison between orbital perturbation and sea level shift from linear fit shown in Figure 1, showing long and short periods as indicated by arrows (all period 1995–2006).

The eccentric anomaly E E ¼ M þ esinMð1 þ ecosMÞ: Then the elliptical distance r can be calculated from rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 2 1 e sinE : r ¼ ðcosE eÞ þ

(6)

(7)

3. Now, we can get the distance shift Δr between the distance recorded in geocentric ephemeris data RL and elliptical orbital distance r as follows: Δr ¼ r RL :

(8)

4.3. Sea Level Shift-Distance Shift Comparisons 1. List the combination of the sea level shifts and distance shifts in the same table. 2. Plot the time series for distance shift and sea level shift, as well as the smoothing curves. 3. Plot the cross correlation between distance shift and sea level shift time series.

5. Results and Discussions Figure 1 shows that the sea level is rising with time in linear fitting as given by equation (1). When I plot the perturbation of the Earth’s orbit with oscillation in the sea level with time as shown in Figure 2, I found that

Figure 3. Comparison between orbital perturbation and sea level shift from linear fit shown in Figure 1, showing long and short periods as indicated by arrows (1995–1998).

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Figure 4. Comparison between orbital perturbation and sea level shift from linear fit shown in Figure 1, showing long and short periods (1998–2001).

the behavior of both curves are very alike in many parts. The long periodic of both oscillations (orbital perturbation shifts and sea level shifts) are similar and coincident as shown by the smooth curves. Figures 3–6 are extensions of Figure 2. Each figure represents 3 years interval in order to show finer details. The short periodic oscillations in Earth’s orbit: such short regular period oscillations in Earth’s orbit are very distinguished as seen in Figure 3. They are definitely attributed to Moon’s revolution around the Earth. The center of mass of the Earth-Moon system moves toward and away from the Sun at the beginning and full moon phases causing cyclic perturbation in Earth’s orbit. In addition, short periodic oscillations in orbital perturbations and sea level shifts alternate in phase and out of phase. The center of revolution of this motion of the Earth and Moon around their common center of mass lies at a point approximately 1068 miles beneath the Earth’s surface, on the side toward the Moon, and along a line connecting the individual centers of mass of the Earth and Moon. The center of mass of the Earth describes an orbit around the center of mass of the Earth-Moon system (G) just as the center of mass of the Moon describes its own monthly orbit around this same point (as shown in Figure 7). We can show from Figures 1–6 that the amplitude of the perturbation of Earth’s orbit is increasing with the rising of the sea level. In other words, the perturbation of Earth’s orbit is correlated with the sea level shifts. Figure 8 shows the relation between amplitudes of the peaks of long periodic orbital perturbations and the corresponding sea level linear fitting values given by equation (1). We found a strong correlation coefficient between them, as shown in the following fitting formula: H ¼ 6:124 þ 0:35655 A; RL ¼ 0:77521;

(9)


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where H is the fitting value of the sea level, A is the amplitude of the long periodic peaks of orbital perturbation of the Earth, and RL is the correlation coefficient of linear fitting (equation (1a)). From Figure 5 and other figures (3 and 6), it is noticed that there is a wave in the orbital perturbation (twice long periodic peaks) of duration 1 year. The minimum occurs in February trough-peak and the maximum in November crest peak. This means that Earth’s mass becomes greater in November and smaller in February according to equation (11). Notice also that the minimum in Sun-Earth distance shift of February coincides with a peak in sea level. The 1997 peak occurred at the start of the weak solar cycle number 23 that caused solar induced climate change. In 1997, the rift valley lakes in Western Africa also suddenly showed abrupt rise. For example, a sudden rise of Lake Victoria from 11.32 m on 1997.769 to 12.672 m on 1998.308 occurred. In other words, the lake rose by 1.55 m within 0.539 years. These sudden rises were due to El Niño event that caused excessive rain in the area [Elfaki et al., 2014]. El Niño is a hot water current leading to thermal expansion of water thus to sea level rise. Those three years 1997, 2002, and 2006 were El Niño events. The cross correlation R at delay d between two series in Figure 9 is determined as ∑½ðx ðiÞ mx Þ*ðy ði dÞ my Þ Rðd Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ∑ðx ðiÞ mx Þ2 ∑ðy ði d Þ my Þ2 where x(i) and y(i) are two series, i = 0,1,2…N 1. The cross correlation R at delay d, and mx and my are the means of the corresponding series. In my case I assumed d = 3 months, x is Earth’s orbital perturbation, and y is sea level oscillations. Also shown in Figure 9 is a depression in 1999. This depression can be attributed to La Niña event which is a cold water current in the Pacific Ocean that caused a contraction of sea water volume thus lowered the elevation of the sea level. The Earth is moving around the Sun in an elliptical orbit with the Sun being located at one of the ellipse foci. The Sun-Earth distance depends on the Sun and the Earth masses according to Newton’s law of gravitation r2 ¼

Figure 7. Sketch of orbital perturbation caused by center of mass of SunEarth system.

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G m⊙ mE ; F

(10)

where r is the Sun-Earth distance, G is the gravitational constant, F is the gravitational force of attraction, mE is Earth’s 143


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mass, and m⊙ is solar mass. Assuming a constant solar mass, we can say, r 2 ∝mE :

Figure 8. The relationship between amplitude of Earth’s orbital perturbation and fit value of sea level.

(11)

This means that the square of the Sun-Earth distance is related to Earth’s mass variability, i.e., the Earth’s mass variation causes orbital perturbation according to Newton’s law. The Earth’s mass enclosed within the top of the atmosphere (magnetosphere) is equal to the summation of Earth’s components (land, water, and atmosphere). Consequently, the change of the mass of any portion of the Earth leads to a change in the total Earth’s mass. The Earth’s mass correlation with orbital perturbation points to mass variability according to Newton’s law.

Sea level oscillations lead to motion of the Earth’s central mass of gravity, but total Earth’s mass is still constant. The mass of Earth’s water is variable because sea level oscillation is correlated to Earth’s orbital perturbation according to Newton’s law. The Earth’s mass (including ocean mass) exchanges its mass with outside matter. Thus, I suggest a space source increasing “inflow” and decreasing “outflow” of Earth’s mass. The air including water vapor in the atmosphere may escape from atmosphere to space outflow and the solar wind precipitation enter onto the upper atmospheres inflow. Both cases have been detected in many solar system planets and compatible previous researches. Stenberg et al. [2014] used the Ion Mass Analyser (IMA) to investigate both the atmospheric escape outflow from planets (Mars and Venus) and the solar wind precipitation onto the upper atmospheres inflow. They found that on Venus they move mainly antisunward and on Mars toward the tail center. Studying the inflow I conclude that on Mars I regularly observe precipitating solar wind ions (H (+) and He (2+) ) inside the induced magnetosphere boundary (IMB), while on Venus no precipitating alpha particles have been detected and only a few cases of solar wind proton precipitation. Other earlier studies indicate oscillation in the altitude of magnetopause attributed to changes in the solar wind pressure. This is an indication of inflow process in the Earth’s atmosphere [Mawad et al., 2011]. Small comets and asteroids are still carrying water to the Earth’s upper atmosphere, Sun, and all solar system planets [Javoy et al., 2010]. Very hot water has been detected in the solar sunspots regions [Tennyson and Polyansky, 1998]. The Sun ejects plasma to space “solar wind” which have water composition and may have water composition ionized hydrogen (electrons and protons) and oxygen [Teodoro et al., 2009].

Figure 9. Cross correlation between Earth’s orbital perturbation and sea level oscillations. Peaks occurred in 1997, 2002, and 2006 (y axis is the correlation at zero lag of year (x axis) for the period of 3 months).

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Table 1. Summary Action Inflow Outflow

Matters

Indications

solar wind, coronal mass ejections, comets, and meteoroids Plasma ejection from upper atmosphere including dissociated water molecules (hydrogen and oxygen)

One long periodic peak at around November for Earth’s orbital perturbation, sea level rising, coherent with El Niño, and magnetosphere’s altitude oscillation One long periodic peak around February for Earth’s orbital perturbation, sea level flood, coherent with La Niña, and magnetosphere’s altitude oscillation

The Earth’s atmospheric mass changes due to the leak of some of the solar wind and coronal mass ejections into the magnetosphere. The aurora is the signature of arrival of coronal mass ejections into the Earth’s upper atmosphere. Unfortunately, there are no accurate measurements or estimations for atmospheric mass, and it is so difficult to trace the change in the mass or motion of the atmosphere. The annual Earth’s precipitation is closely related to the variation of sunspot numbers, and that solar activity probably plays an important role in influencing the precipitation on land [Zhao et al., 2004]. The peak 1997–1998 sea level rise was contemporary with an El Niño event that also caused dramatic sudden rise of Lake Victoria and other African equatorial lakes. El Niño also induces global warming [Meehl et al., 2006]. El Niño events in the Pacific and Indian Oceans are warm water currents. They cause expansion of water and rise of sea level. This leads to a shift of center of mass of the Earth. El Niño are solar induced phenomena [Yousef, 2006]. They are initiated by coronal mass ejections hitting the polar atmosphere and thus adding extra mass to the Earth’s atmosphere. The motion of Earth’s central mass caused by El Niño enhances the correlation between sea level oscillations and orbital perturbation. El Niño occurred 3 times during the studied period (1997, 2002, and 2006) as shown in Figure 9. Following drought conditions in African equatorial lakes by the end of cycle 23 around 2008 ± 2 years, Yousef and Amer [2003] successfully predicted cyclic rises and falls of lakes level in coherence with the weak solar cycle 24. The following factors affect the correlation between level oscillations and orbital perturbation: (1) Data of the sea level and Sun’s distance is not daily; it depends on interpolations to predict the unavailable values; (2) the daily maximum values are as amplitude of the peaks, and it is not an accurate method; and (3) estimating the eccentric anomaly simple formula without iteration.

6. Conclusion It is found that during our period of study 1995–2006, the sea level rise was 3.23 mm per year. This value is to be compared with 3.3 ± 0.4 mm per year from 1993 to 2009. Since there was a steady increase of sea level since 1870, I suggest that there is a steady flow of deep underground water from the 40,000 km long continuous system of mid-ocean ridges on the floors of all the Earth’s oceans coming out with the magma. Mainly, the orbital perturbation of the Earth is correlated with sea level oscillation, as an additional factor besides gravity of celestial bodies and shape of the Earth. The sea level rising leads to motion of the central mass of the Earth. According to equation (11), Earth’s orbital perturbation is correlated with a changeable mass of the Earth. This is an indication to inflow of matter into the upper atmosphere and outflow of matter from Earth’s atmosphere to space. The inflow matters include solar wind, coronal mass ejections, comets, and meteoroids. They include water molecules and ionized hydrogen and oxygen atoms. Outflow matter includes plasma from upper atmosphere and may include water vapor ionized by solar UV to hydrogen and oxygen. The amplitude of Earth’s orbital perturbation increases with sea level rising. The perturbation of Earth’s orbit is highly correlated with the sea level rising during the period 1995–2006 with correlation coefficient R = 0.77521 as given by equation (9). This indicates that the inflow that contain water molecule increases sea level. El Niño warm currents in the Ocean cause thermal expansion of water that adds up to the rising of Sea level. Such cases are found for the El Niño events of 1997, 2002, and 2006. On the other hand, La Niña cold currents in the ocean causes shrinking in water volume, thus lead to drop in sea level as noticed in the case of the 1999 La Niña.

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The reason of the rising in the sea level is not restricted only to global warming and melting of icebergs according to mass balance, but the outer space is an important factor for this rising. Therefore, we can conclude that an important source of the sea level rising is the space inflow. The altitude of magnetosphere is variable according to solar wind hits, and inflow and outflow in upper atmosphere with space. The behavior of the perturbation of the Earth’s orbit and oscillation in the sea level long periodic curves is very alike in many parts. The long periodic of both oscillations (orbital perturbations and sea level shifts) is similar and coincident. The short periodic oscillations (in orbital perturbations and sea level shifts) alternate in phase or out of phase. The short periodic monthly oscillations are due to oscillation of center of gravity of the Earth-Moon system as the Moon revolves around the Earth. Earth’s mass is greater in November (long periodic trough-peak of Earth’s orbital perturbation), while it is smaller in February (short periodic trough-peak of Earth’s orbital perturbation). The Earth ejects outflow matter through its magnetotail through its flight around the Sun. It is similar to comets collect matter from space and eject it when arrives nearby to Sun through its tail. Finally, we can summarize my conclusion in Table 1. Acknowledgments I thank the NASA data center and TOPEX/Poseidon for providing me the data (section 3).

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Erratum In the originally published version of this article, the image in Figure 7 and its caption were incorrect. The figure and caption have since been updated, and this version may be considered the authoritative version of record.

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