Revisiting the Excitation of Free Core Nutation

Earth possesses a Poincaré mode called Free Core Nutation (FCN) due to the misalignment of the rotation axes of the mantle and fluid outer core. FCN is the primary signal in the observations of Celestial Pole Offsets (CPO) and maintained by geophysical mechanisms that are yet to be understood. Earlier studies suggested an origin in Atmospheric Angular Momentum (AAM)—and to a lesser degree Oceanic Angular Momentum (OAM)—but discrepancies between these geophysical excitations and the geodetic (CPO‐based) excitation were too large to reach definite conclusions. Here we use newly calculated, 3‐hourly AAM and OAM series for the 1994–2022 period, in conjunction with the latest CPO series from the International Earth Rotation and Reference Systems Service (IERS 20 C04 series), to demonstrate a markedly lower power ratio ( ∼ 4.6) of geophysical over geodetic excitation at the FCN frequency compared to previous works (ratio ∼ 10). Among all excitation sources, the AAM pressure term exhibits the highest coherence (0.56) and correlation (0.48) with the geodetic excitation, whereas the coherence with OAM is smaller by a factor of 3. Similar analyses using existing angular momentum series give comparable, albeit smaller coherence and correlation results. We attribute the relevant AAM pressure term signal to Northern Hemispheric landmasses and further show consistent temporal variations in the amplitude of geophysical and geodetic excitations around the FCN band. Our results thus corroborate evidence for large‐scale atmospheric mass redistribution to be the main cause of continuous FCN excitation.


Introduction
The motion of the Earth's spin axis in space-comprising precession and nutation-is caused mainly by the gravitational impacts of Sun and Moon (Gross, 2015).The primary effect is due to the gravitational torques, which are modeled with the laws of rotating fluid dynamics given the presence of fluids inside the Earth (Wahr, 1981).In addition, the gravitational effect of the Sun and Moon also cause body and ocean tides that further influence nutation.These tides are modeled together with the primary effect in a unified framework for forced nutations (Mathews et al., 1991).However, the framework does not account for more irregular nutations that are thought to be driven by atmospheric and non-tidal oceanic mass redistributions.The dominant signal of interest in this regard is the so-called Free Core Nutation (FCN), a nutation with typical amplitudes of less than 1 milliarcsecond (mas, see e.g., Brzeziński et al., 2002) and a celestial period of ∼ 431 days (Gross, 2015).The FCN owes its existence to the misalignment between the rotation axes of the fluid core and mantle.Due to the ellipticity of the core-mantle boundary and Earth's elliptical stratification, gravitational and pressure forces are generated that cause oscillatory motions between core and mantle (Wahr, 1988).With the advent of Very Long Baseline Interferometry (VLBI, Sovers et al., 1998), nutations could be measured with sufficient accuracy and compared to predictions from theory (Gwinn et al., 1986;Herring et al., 1986Herring et al., , 1991)).The small angle differences making up the residual in such comparisons are called Celestial Pole Offsets (CPO), in which the prominent signal is the FCN.
The existence of FCN has long been known, at least since the work of Hough (1895).Sasao et al. (1980) developed a rigorous dynamical theory for FCN based on the rotational fluid dynamics for a stratified core and Poincaré modes.FCN is thus considered to be a Poincaré mode (a form of spin-over modes) of the Earth (see, e.g., Seyed-Mahmoud & Rogister, 2021;Rekier, 2022).The theory of Sasao et al. (1980) was subsequently used in Sasao and Wahr (1981) to suggest that the atmosphere and ocean are capable of exciting and maintaining FCN.When considered in a terrestrial reference frame, to affect nutation, geophysical excitations must have a retrograde diurnal period.The relevant phenomenon in this case is the radiational S 1 tide in the atmosphere and ocean (Brzeziński et al., 2004;Schindelegger et al., 2016), and more precisely for the FCN, temporal variations of S 1 with nearly semiannual period (Bizouard et al., 1998;Ray et al., 2021).These large-scale mass redistributions imply changes in both Atmospheric and Oceanic Angular Momentum (AAM and OAM) that are conveniently expressed as Effective Angular Momentum functions (EAM, Barnes et al., 1983).
The dynamical theory of Sasao et al. (1980) and then available EAM series enabled Brzeziński (1994) to formulate and test the dynamical relation between CPO and effective AAM and OAM, called broad-band Liouville equation in analogy to the Liouville equation (Munk & MacDonald, 1960).In the broad-band Liouville equation, nutations are taken as retrograde diurnal polar motion, in keeping with the conventional polar motion and nutation frequency bands (see, e.g., Rekier et al., 2022).The so-called complex demodulation procedure is applied to the sub-daily-sampled EAM functions to transfer the excitations from the terrestrial frame to the celestial frame, thus leading to the Celestial EAM (CEAM, Brzeziński & Capitaine, 1993).CEAM can be subsequently filtered near the FCN frequency band (approximately periods between 440 and 420 days) and compared with the geodetic excitation derived from the VLBI CPO series.This approach, referred to as deconvolution, has been suggested to be the most appropriate way of studying the excitation sources for Earth's rotational normal modes (Chao, 2017).By contrast, in the convolution approach, one would numerically integrate the broad-band Liouville equation to convert the geophysical excitations to nutation values and then compare with VLBI CPO series (Brzeziński, 1994), as, for example, in Vondrák and Ron (2017).In addition, there are alternative formulations of the Liouville equation for FCN, such as the one presented in Chao and Hsieh (2015), which is very similar in form to the classical Liouville equation for polar motion.Nevertheless, the conclusions of the numerical application of these alternatives are essentially the same, that is, FCN is most probably excited by mass redistributions in geophysical fluids (as expressed by EAM or CEAM).
As the length and accuracy of the EAM series and CPO observations increased, studies followed on the work of Brzeziński (1994).Bizouard et al. (1998) gave evidence for the atmospheric mass (i.e., pressure) term to be likely the most important driver for variability of FCN, consistent with results in Gegout et al. (1998) and later in Brzeziński et al. (2004).However, even though the coherence between the geodetic and geophysical excitations was as large as 0.8, power estimates for the geophysical excitation near the FCN frequency were more than an order of magnitude larger than that of the geodetic excitation, thus prohibiting robust conclusions (Brzeziński & Bolotin, 2006;Brzeziński et al., 2004).Later, a digital filter was developed by Brzeziński (2007) based on the assumption that only the pressure term of EAM contributes to the FCN excitation.This filter was used in conjunction with newer EAM data sets (Brzeziński et al., 2014) to more reliably compare the geodetic and geophysical excitations.Notwithstanding the improvements in modeling and data, the power of geophysical excitation was still almost 10 times that of geodetic excitation in the FCN band.Most recently, Kiani Shahvandi et al. (2024) showed that CPO can be well modeled and predicted, if augmented with the broad-band Liouville equation and sub-daily AAM and OAM data.An important insight from this study was that a part of the temporal changes in the FCN period were correlated with changes in EAM, thus indicating that with newer, more accurate angular momentum estimates it might be possible to explain the origin of the FCN excitation mechanism.
With the release of the latest CPO series of the International Earth Rotation and Reference Systems Service (IERS), namely IERS 20 C04, and the promising results in Kiani Shahvandi et al. (2024), we are motivated to revisit the excitation of FCN and take a fresh look at the potential atmospheric and oceanic drivers.To this end, we analyze newly generated AAM and OAM series and follow the deconvolution approach.In Section 2 we outline the methodology in detail, before elaborating on the IERS 20 C04 CPO data and our new AAM and OAM series in Section 3. We present the results and discussions in Section 4, complemented by conclusions and a short outlook in Section 5.

Methodology
Following the recommendations of Chao (2017), we mainly use the deconvolution approach.For comparison purposes, however, we also briefly examine results obtained from the convolution method.Both approaches are described below.

Complex Demodulation
Our inputs to calculate the geophysical excitations of FCN are AAM and OAM (as effective angular momentum functions) in the terrestrial frame.Each of these has two equatorial components in the direction of 0°and 90°E longitude, denoted by χ A 1 (t), χ A 2 (t) for AAM and χ O 1 (t), χ O 2 (t) for OAM, where t is time in days.Calculations are generally performed using complex-valued quantities (i ≡ ̅̅̅̅̅̅ 1 √ ) defined as follows, Equation 1.
Brzeziński ( 2007) suggested that it is sufficient to consider only the pressure (mass) terms in the excitation of FCN, while contribution of motion (wind) terms to FCN variations are negligible.We have been able to confirm this suggestion with our own analyses and therefore present results only based on mass terms.
CEAM, denoted by χ′(t), is calculated from χA (t) and χO (t) using the complex demodulation procedure at the retrograde diurnal frequency (Brzeziński, 1994).The demodulation is a simple linear transformation based on the Greenwich Apparent Sidereal Time (GAST, computed based on IERS conventions, Petit & Luzum, 2010) by which nearly diurnal and higher frequencies in the terrestrial frame are transformed to long-period variations in the celestial frame.The Celestial AAM and OAM (CAAM and COAM), which are denoted by χ′ A (t) and χ′ O (t) respectively, are defined as in Equation 2.
Note that for the highest accuracy, we use GAST, but various simpler quantities such as Greenwich Mean Sidereal Time (GMST) or even Ω ⋅ t (Ω = 7.292115 ⋅ 10 5 rad sec 1 is the mean rotation rate of the Earth) have been applied in the literature (Brzeziński, 1994;Chao & Hsieh, 2015).Nevertheless, our analyses show that the use of GAST can be more accurate by up to 10 µas (microarcseconds) in comparisons of geodetic and geophysical excitations.

Deconvolution
A central element of this work is the comparison of the geophysical excitation-as represented by CEAM or its components (CAAM, COAM)-with the geodetic excitation of nutation.The latter can be computed using the digital filter developed by Brzeziński (2007), which is ultimately based on the broad-band Liouville equation (Brzeziński, 1994).We invoke the following simplifying assumptions during deconvolution: (a) only the pressure terms of CAAM and COAM contribute to the excitation of FCN, and (b) the effect of Chandler resonance is taken as iΩ.
The inputs to calculate the geodetic excitation are the dX and dY components of CPO series, combined into the single complex variable P(t) = dX(t) + idY(t).The procedure is iterative and summarized in Equation 3.
a p = 9.2 × 10 2 , (3d) χ′ G (0) = 0, (3h) Similar to Brzeziński et al. (2014), we remove the seasonal, semi-annual, annual, 9.3-year and 18.6-year oscillations, both prograde and retrograde, from P by fitting simple harmonic functions.Subsequently, we smooth the series by a running mean before converting them to the geodetic excitation.Tests have shown that the removal of these harmonics and the smoothing have only a minor influence (<10 μas) on the derived excitation from IERS 20 C04 series and do not alter our conclusions.
In Equation 3, δt is the sampling rate of geodetic excitation, which we choose as 1 day corresponding to the temporal resolution of the CPO series (Brzeziński, 2007 discussed that, even for smaller sampling rates, the method could accurately derive the geodetic excitation).σ′ f (rad day 1 ) is the FCN frequency in the celestial frame, while σ c (rad day 1 ) is the Chandler frequency in the terrestrial frame.These frequencies are defined based on the period and quality factor of FCN (T FCN , Q FCN ) and Chandler wobble (T CW , Q CW ).We choose T FCN = 431 days, T CW = 433 days, Q FCN = 20000 and Q CW = 179 by testing a large number of candidates (around 2 × 10 5 ) in a comprehensive grid search.Our criteria were (a) maximizing the correlation and (b) minimizing the root mean square difference between geodetic and geophysical excitations.The selected values are comparable to the ones suggested and used by other authors (Brzeziński, 2007;Brzeziński et al., 2014;Vondrák & Ron, 2017).a p is a dimensionless quantity that depends on the structure of the adopted Earth model and the ratio in terms of equatorial components of inertia between the fluid core and the whole Earth.We use a p = 9.2 × 10 2 , consistent with prior work (Brzeziński et al., 2004;Kiani Shahvandi et al., 2024;Koot & de Viron, 2011;Vondrák & Ron, 2019), and an initial value of χ′ G (0) = 0 for the iteration (Brzeziński, 2007).The parameters β, κ p , and ϕ are the constants that appear naturally when deriving the aforementioned digital filter based on trapezoidal numerical integration of the convolution problem (cf.Brzeziński, 1994).Note also that Equation 3 only yields one possible solution for the geodetic excitation, because in principle the problem of deriving excitation from CPO cannot be solved uniquely.However, as discussed in Brzeziński (2007), the described scheme is useful for practical implementations and gives reasonable results.

Filters and Metrics for Comparisons
By computing χ′ G (t) and χ′(t) and invoking the principle of conservation of angular momentum, comparisons can be made between geodetic and geophysical excitations.For this purpose, we employ the measures of linear correlation (at the 5% significance level) and coherence, as in previous FCN studies (Brzeziński et al., 2014).The coherence is computed using a windowing approach with a window length of 7 years to allow for separation of FCN and retrograde annual signals, which have a beat period of around 7 years.Similar to Schindelegger et al. (2013), we also compute the 95% confidence level for coherence using the method in Jenkins and Watts (1968), resulting in a value of around 0.14 in our analyses.
Given our focus on the FCN band, we evaluate the correlation and coherence between χ′ G (t) and χ′(t) after removing from both series all frequencies with absolute values higher than 1/ 400 days 1 , using a low-pass Journal of Geophysical Research: Solid Earth 10.1029/2024JB029583 filtering approach.The chosen cutoff at 1/ 400 days 1 sits between the FCN and retrograde annual signals, thus suppressing much of the latter while conserving spectral content near the former.In supplemental analysis, we varied this threshold between 1/420 and 1/370 days 1 .We deduced that removing too few or too many frequencies would obscure the agreement between geodetic and geophysical excitations.It is important to mention that filtering is integral to the analysis (cf.Brzeziński, 1994), since a significant part of the frequencies in the demodulated series falls outside the nutation frequency band.For instance, Brzeziński et al. (2004) applied a Gaussian smoothing filter with a full width at half-maximum equivalent to 10 days, set to 2 months in a later study (Brzeziński et al., 2014).Vondrák and Ron (2019) applied a different filter, called Vondrák filter (Vondrák, 1977), which is rather comparable to splines (Wahba, 1990).In general, the choice of filter and its window length can be flexible as long as the required information on FCN excitation in the series is preserved.
In addition, we perform wavelet analyses to examine the amplitude variations of the observed and modeled FCN excitation in time.We compute these patterns for the periods in the range 500 to 300 days (with 1-day steps) using the unfiltered excitation series to cover the FCN and nearby retrograde annual signals.Specifically, we employ the Normal Morlet Wavelet Transform (NMWT), which allows one to retrieve correct amplitudes (and phases) across the specified frequencies (Liu et al., 2007).Note also that the NMWT spectra are derived for geodetic and geophysical excitations separately, but can be compared afterward.We also compute the 95% confidence levels for the derived amplitudes based on the method presented in Torrence and Compo (1998).

Convolution Approach
In the convolution approach, the broad-band Liouville equation is integrated using initial CPO and geophysical excitation values ( P(t 0 ), χ′ (t 0 ) respectively) at a reference time t 0 .The original solution of Brzeziński (1994) was modified in Brzeziński (2007) based on practical considerations, from which the geodetic excitation approach mentioned above was derived.The idea is to use the pressure term of χ′(t) and convert it to a CPO series, which can be subsequently compared with VLBI CPO observations.Given the caveats with this approach for studies of Earth's rotational normal modes (Chao, 2017), we only compare the mean and standard deviation of the convolved CEAM and VLBI series.The standard deviations in particular can shed light on the efficacy of different forcings (i.e., CAAM, COAM) in causing FCN variability.The convolution approach is performed as in Equation 4. (4)

CPO Series
For the computation of geodetic excitation χ′ G (t) we use the IERS 20 C04 series and thus take advantage of the most recent realization of the International Terrestrial Reference Frame (ITRF2020, Altamimi et al., 2023).In ITRF2020, the determination of Earth Orientation Parameters (EOPs) is rigorous, since the procedure is consistent with the determination of other geodetic parameters, such as station positions and velocities, using various space-geodetic techniques in a unified framework.Kiani Shahvandi et al. (2023) showed that this approach significantly reduced some remaining biases in the EOPs compared to the IERS 14 C04 series (Bizouard et al., 2019).This was followed by a similar confirmation for CPO components (Kiani Shahvandi et al., 2024), thus testifying to the accuracy and robustness of the IERS 20 C04 solution.Among the technique centers that provide EOPs related to the ITRF2020, only the International VLBI Service for Geodesy and Astrometry (IVS, Nothnagel et al., 2017) disseminates the CPO series.IVS receives its inputs from various analysis centers.The IERS 20 C04 series is a combination of all these individual solutions.Even though we mainly focus on IERS 20 C04, we also present the statistics of agreement between geodetic and geophysical excitations for eight different CPO series in the Appendix.We perform our analyses of both the VLBI and geophysical excitation series from the beginning of 1994 to the end of 2022.We focus on the data after 1994 because of the insufficient quality of prior data, as per recommendations of Brzeziński (2007) and Chao and Hsieh (2015).Figure 1 illustrates the IERS 20 C04 CPO series.

AAM and OAM Series
As in Kiani Shahvandi et al. ( 2024), we use AAM and OAM series provided by German Research Center for Geosciences (GFZ, Dobslaw et al., 2010).
The AAM series was computed from ECMWF (European Center for Medium-Range Weather Forecasts) reanalysis fields and, from 1 January 2007 onward, from operational analysis data.The associated OAM estimates are based on a free simulation with a baroclinic ocean model, the Max-Planck-Institute for Meteorology Ocean Model (MPIOM, Jungclaus et al., 2013), forced with the respective ECMWF data (wind stress, pressure loading, heat fluxes, etc.).The GFZ AAM and OAM series are provided with a temporal resolution of 3 hr and henceforth denoted as ECMWF.
In DEBOT has been previously used in quantifying oceanic contributions to nutation, particularly those associated with the radiational S 1 tide (Schindelegger et al., 2016).The setup considered in the present work is similar to that in Harker et al. (2021) but employs 3-hourly MERRA-2 sea level pressures and wind stress vectors as forcing fields.The model solves the shallow water equations on a 1/3°latitude-longitude grid, extending to latitudes of ±86°.Energy losses are parameterized by quadratic bottom friction and a linear topographic wave drag, following the scheme suggested by Carrère and Lyard (2003).We also switch on the model's time step-wise handling of self-attraction and loading effects, realized through a Load love number formalism on ocean bottom pressure anomalies p b (cf.Vinogradova et al., 2015).The rationale for this choice is that the ocean's dynamic response to self-attraction and loading is relatively pronounced at diurnal frequencies in the terrestrial reference frame (Schindelegger et al., 2016).
The mass terms of AAM and OAM, or more precisely the atmospheric and oceanic EAM functions, are calculated as where a is a mean Earth radius, g is the acceleration due to gravity, C A is the difference between polar and mean equatorial moments of inertia, and (ϕ,λ) are geographical coordinates.Note that the local bottom pressure p b = p b (ϕ,λ) represents the mass change of the water column after dynamic adjustment to the atmospheric forces, that is, it does not contain contributions from barometric pressure.Instead, and identical to the GFZ approach for computing EAM, barometric pressure variations over the ocean are replaced by their time-varying spatial mean before evaluating the AAM integral (cf.also Harker et al., 2021).In Figure 2, we show the MERRA-2-inferred CAAM and COAM estimates, that is, the excitation functions after complex demodulation and low-pass filtering as described in Section 2.3.For comparison, the original AAM and OAM determinations in the terrestrial frame, as well as the corresponding complex-demodulated series are shown in the Appendix (Figures A1 and A2).
The real and imaginary components of these unfiltered excitations (Figures A1 and A2) are visually rather similar due to the nature of the complex demodulation.In fact, it is only after the application of the mentioned filters (Section 2.3) that we could clearly observe differences between χ′ 1 and χ′ 2 , both for the atmospheric and oceanic component.At the frequencies under consideration (≤ 1/400 days 1 ), the oceanic excitation varies nearly lock-step with the atmospheric excitation but is a factor of ∼3 smaller in amplitude.

Results and Discussion
We present the cross-spectral coherence between geodetic and geophysical excitations, as well as the respective power spectra over period in days in Figure 3.The results can be taken in together with a time-domain comparison between geodetic and geophysical excitations for both sources (MERRA-2, ECMWF) in Figure 4 and statistics of coherence and correlation in Table 1.Note that since we use complex variables in our analysis, the values presented are the absolute values of the complex coherence and correlation.It is evident that the MERRA-2 excitation series exhibit higher correlation with the geodetic excitation than the ECMWF-based series.Inferred coherence values are also slightly larger, suggesting that MERRA-2 might be more accurate in the retrograde diurnal band than ECMWF reanalysis (cf.Schindelegger et al., 2016) and thus more suitable in the study of FCN excitation.There are nevertheless discrepancies between geodetic and geophysical excitations.Potential reasons include data noise in all three components (AAM, OAM, and CPO), numerical model imperfections, as well as the deconvolution approach itself.As for the latter, deconvolution is sensitive to the value of σ′ f , which is considered fixed in the current approach.However, as shown in studies such as Kiani Shahvandi et al. ( 2024), σ′ f is variable in time.Future research might reconsider the modeling of FCN based on variable σ′ f derived from independent observations, including those of superconducting gravimeters (Cui et al., 2018).
Focusing on the MERRA-2 excitation series, the coherence with χ′ G (t) in the FCN band amounts to 0.56 and 0.16 for AAM and OAM, respectively.This implies that even though OAM contributes to the excitation of FCN, its role is at least three times smaller, confirming that the ocean is not capable of maintaining FCN per se (Bizouard et al., 1998;Brzeziński et al., 2014).The similar conclusion can be drawn by analyzing the amplitudes near FCN frequency band: the amplitude of geodetic excitation is ∼0.08 mas, while the respective values for geophysical excitations from atmosphere and ocean are ∼0.14 and ∼0.03 mas (and ∼0.16 and ∼0.04 based on ECMWF).A total coherence of 0.72 is very much comparable to results in Brzeziński and Bolotin (2006), where a coherence of 0.7-0.8 was reported.However, the key insight from our analysis is that according to Figure 3 the ratio in power of geophysical over geodetic excitation is around 4.6, based on MERRA-2 data (and around 6.4 based on ECMWF data).The MERRA-2 number-while still pointing to a considerable power excess in χ′(t)-is considerably smaller than the value of 10-30 reported in Brzeziński and Bolotin (2006) or ∼10 in Brzeziński et al. (2014).Furthermore, all correlations are positive and larger than the maximum (0.33) obtained by Brzeziński et al. (2014), implying that the observed and geophysically modeled excitations are in phase.These results add confidence to the notion that changes in AAM, and to a lesser degree OAM, are indeed responsible for the excitation of FCN.
Two additional notes are warranted.First, Figure 4 and Table 1 also contain contributions from positive frequencies in the range 420-440 days, which are consistent with those observed in Brzeziński et al. (2014).They may reflect temporal modulations in the radiational S 1 tide and, in the case of χ′ G (t), noise carried over from the space geodetic analysis or small residual ocean tide effects (Desai & Sibois, 2016;Huda et al., 2021).Second, gives a value of ∼0.08 mas at the FCN frequency.This is comparable but smaller than the value of 0.10 mas reported in Brzeziński et al. (2014).One reason for the difference in excitation amplitude might be the decrease in FCN amplitude evident in CPO data in recent years (starting from around 2020), see Figure 1.However, we stress that this behavior is observed in the CPO, but the FCN excitation series depends not only on CPO itself, but also on its first and second derivatives, owing to the nature of the broad-band Liouville equation (Brzeziński, 1994).Hence, diminished FCN amplitudes over extended time periods are not necessarily translated to the excitation series.
To analyze whether variations in the FCN excitation from geodetic and geophysical approaches can be reconciled, we use the NMWT method mentioned in Section 2. The results are presented in Figure 5, for the frequencies in the range 500 to 300 days.According to this figure and within the 95% confidence level, the geodetic and geophysical excitations exhibit very similar temporal modulations from 1994 to 2022, particularly in the FCN frequency band.We again find evidence for slightly improved consistency between the geodetic excitation and MERRA-2, in terms of the structure of spectral peaks around the FCN period, the magnitude of the excitation itself, and the location of statistically significant values.In particular, the factor ∼2 mismatch in amplitude between Figures 5a and 5b is commensurate with the afore-noted excess of power in the geophysical estimates over the geodetic excitation.
The mentioned consistency bears implications.According to our analyses, the temporal variation of the FCN amplitude can be largely reconciled with angular momentum changes in geophysical fluids, and more specifically, the effect of atmospheric mass redistribution.This finding contrasts with claims   .Shown are the (a) real and (b) imaginary components of the geophysical excitation series for MERRA-2 and ECMWF after low-pass filtering (cf.Section 2.3).The unit is mas.
that recent anomalous changes in amplitude and phase in the CPO series, especially the post-2020 low excitation of FCN, are caused by Geomagnetic Jerks (GMJ, Malkin et al., 2022).These claims build on an earlier conjecture by Shirai et al. (2005), who suggested that epochs of GMJ and rapid FCN variations coincide.GMJ are related to the rapid variations in the core field and are generally attributed to the core-mantle boundary (Mandea et al., 2010) and the presence of various torques, including magnetic, viscous, and topographic.However, the magnetic torque at the core-mantle boundary is too weak to solely excite FCN and probably only the viscous torque can cause the variations (Deleplace & Cardin, 2006).Then again, if the viscous torques are to be considered as the cause of the mentioned variations, a specific range of viscosity should be prescribed (Triana et al., 2021), which may be at odds with other constraints and observations pertaining to the core-mantle boundary.Furthermore, the effect of topographic coupling is highly uncertain but supposed to happen on decadal-and not diurnal-time scales in the terrestrial frame (Hide et al., 1996).Given that we have used a rigorous dynamical theory and state-of-the-art geodetic and geophysical fluid data, there is strong basis to suggest that the main driver for FCN variations are likely changes in AAM.As further evidence, we analyzed several other CPO series (see Appendix A) and concluded that the patterns of temporal variations of geodetic and geophysical excitations are consistent.A small, residual contribution from GMJ cannot be excluded, as shown by Cui et al. (2020) and based on correlation with the epochs of GMJ.However, until the arrival of a dynamical theory that could connect GMJ to Earth rotation, such inferences should be treated with caution.
Further evidence for the origin of the temporal FCN variations in the atmospheric pressure term can be obtained by complex demodulation at higher orders (Brzeziński et al., 2004).This involves multiplying the geophysical excitations in the terrestrial frame by factors exp(n iGAST) as in Equation 2, where n = 2,3,… is the order.Given the 3-hourly sampling of EAM series in the terrestrial frame, we can perform the demodulation up to the order n = 4, corresponding to the series' Nyquist period of 6 hr.We report that by applying the complex demodulation at orders n = 2,3,4 and after removing frequencies with absolute value larger than 1/400 days 1 , the main remaining signal is a retrograde oscillation with a period of around 2341 days ( 6.41 years) and typical amplitudes between 0.01 mas and 0.02 mas.This is very close to the beat frequency of the retrograde annual and FCN signals, reproduced here simply with AAM determinations.Hence, the atmosphere is effective in exciting both oscillations, consistent with previous studies that have examined either signal (e.g., Bizouard et al., 1998;Brzeziński et al., 2004;Brzeziński et al., 2014;Koot & de Viron, 2011).We also observe a band of prograde frequencies between 1000 and 2000 days, which potentially points toward the so-called Free Inner Core Nutation (FICN, de Vries & Wahr, 1991).However, due to the limited accuracy and length of the EAM data we can neither precisely resolve these frequencies nor robustly analyze their causes.Future improvements in sub-daily EAM estimates will help draw a more comprehensive picture of the long-period, small-amplitude nutations such as FICN.
The 3-hourly MERRA-2 surface pressure data also allow us to locate, at least approximately, the region(s) most relevant in exciting FCN.Specifically, we have repeated the computation of the AAM pressure term (Equation 5) over four spatial domains, comprising land, ocean, land in the Northern Hemisphere, and land in the Southern Hemisphere (Figure A3 in the Appendix).We perform complex demodulation and comparisons with χ′ G (t) for each of these four excitation series and summarize the resulting coherence and correlation statistics in Table 2.This partitioning, when compared with the statistics for the full (i.e., global) atmospheric excitation in Table 1, reveals a dominant role for AAM contributions over land and particularly the Northern Hemisphere.Accordingly, atmospheric pressure over land is the main contributor to FCN excitation, while barometric pressure changes over oceans (spatially uniform, see Section 3.2) have only a minor influence.
We complement the above analysis with a brief discussion of the convolution problem.To avoid overemphasis of results from this approach, we only present the mean and standard deviation (std) of the involved series, listed in Table 3.The mean and std of IERS 20 C04 CPO series are 0.04 mas and 0.19 mas, respectively.Corresponding values for P(t) obtained by integrating the MERRA-2 geophysical excitation series (4) are 0.05 mas and 0.15 mas, with largest contributions coming again from the atmosphere.The CPO and MERRA-2-based statistics are quite close to each other, implying that the geophysical excitations analyzed in this study also work well in the convolution approach.Comparable results can be obtained using ECMWF atmosphere-ocean excitation (std = 0.15 mas), with the difference being that the mean of series (0.07 mas) is larger than for MERRA-2 (0.05 mas).This in turn points to a potential bias incurred during the integration according to Equation 4. Overall, Table 3 further lends credence to our MERRA-2 geophysical excitation series for FCN analyses.

Conclusions and Outlook
We have revisited the long-standing problem of finding the origin of FCN excitation.To this end, we computed AAM and OAM series from MERRA-2 reanalysis data and barotropic ocean model simulations, and followed the deconvolution approach based on dynamical theories to derive geophysical and geodetic excitations.Comparisons between these independent excitations reveal high correlation and coherence values (0.72 and 0.59, respectively) and consistent temporal variations in the FCN band.Among the considered geophysical fluidsatmosphere and ocean-the former has by far the largest impact on the excitation of FCN, albeit the latter also has a positive contribution.Our results therefore corroborate the understanding that the atmosphere is the most probable cause of continuous FCN excitation, including its temporal variations.We have also shown that the main contribution to this excitation arises from atmospheric surface pressure changes over extended landmasses in the Northern Hemisphere.
Even though we have obtained large and statistically significant coherence and correlation values in our analysis, the ratio of power of geophysical to geodetic excitation is still around 5. This ratio embodies the current state-ofthe-art in comparisons of geodetic and geophysical excitations, representing an improvement of at least 50% compared to previous studies, where values of over 10 were reported.As such, the results derived from MERRA-2 reanalysis data (see in particular Figures 3 and 5) hold promise for future studies on the nutation of the Earth.Nonetheless, errors in both geodetic and atmospheric data, as well as numerical model imperfections, can affect the analyses.Such errors are likely pronounced at the FCN frequency, that is, the retrograde diurnal band in the terrestrial frame.Resolving geophysical processes in this band requires continuous VLBI experiments, highfrequency meteorological observations over large spatial scales, apt atmospheric data assimilation strategies, and complementary ocean model simulations.The arrival of new, more accurate AAM and OAM series in the future may further reduce the discrepancies between geodetic and geophysical excitations.In particular, they may be leveraged for higher-resolution CEAM by complex demodulation at higher orders, thus extending and increasing confidence in the results obtained here.Likewise, improvements in the determination of CPO series, including regular provision of high-quality CPO estimates from 24-hr VLBI sessions, can bring benefits to the analyses.• Goddard Space Flight Center (GSFC) • IVS (quarterly series) • Jet Propulsion Laboratory (JPL) • Technische Universität Wien (TUW) • Bundesamt für Kartographie und Geodäsie (BKG) Note that other available series such as those of GFZ and ASI Matera are not included because the signalto-noise ratio is very low.We present the coherence (near FCN) and correlation between the geophysical and geodetic excitation (computed from the eight mentioned series) in Tables B1 and B2, respectively.Evidently, the three series IERS 20 C04, IERS 14 C04, and SYRTE exhibit the largest coherence and correlation with geophysical excitations.This finding is not surprising, as the IERS 20 C04 series is an improved and internally more consistent version of IERS 14 C04, while SYRTE is the rapid form of IERS 14 C04.The next best series, offering fair agreement with geophysical excitation (e.g., coherence ∼0.5), is GSFC, as used in earlier studies (Brzeziński et al., 2014).The other series yield slightly smaller coherence and correlation results than GSFC.This is probably due to the fact that they are solutions by a single institute, rendering them somewhat less suited for geophysical inference and interpretation (Gattano et al., 2017).Nevertheless, and similar to IERS 20 C04, the temporal variations of FCN derived from each of these series are consistent with the ones derived from MERRA-2 geophysical excitation (and to a lesser degree to that of ECMWF).
addition to the GFZ products, we also analyze newly computed mass term estimates for the period 1994-2022 from MERRA-2 (Modern Era-Retrospective Analysis for Research and Applications, Version 2,Gelaro et al., 2017) atmospheric data and our own ocean model simulations.MERRA-2 is a modern global atmospheric reanalysis, produced by sequentially adjusting the state of an atmospheric circulation model (the Goddard Earth Observing System, Version 5,Molod et al., 2015) to fit-in a weighted least squares sense-most available meteorological data in predefined analysis windows.The model solves the governing equations for atmospheric flow and thermodynamics on a cubed-sphere horizontal grid with a nominal cell width of 0.5°× 0.625°.Vertical discretization is in terms of 72 model levels from the surface to 0.01 hPa.The MERRA-2 gridded output comprises 3-hourly analyses (so-called "assimilated states" at 00 UTC, 03 UTC, … ) of both observable parameters and diagnostics.For the calculation of the AAM mass term (see below), only surface pressures, p s , are required.We complement the MERRA-2 AAM estimates with 3-hourly OAM series from a barotropic (2D) time-stepping model referred to as DEBOT (David Einšpigel's Barotropic Ocean Tide model,Einšpigel & Martinec, 2017).

Figure 1 .
Figure 1.IERS 20 C04 CPO series in the range 1 January 1994 to 31 December 2022.The raw series are shown in light cyan, with the slightly smoothed series in black.Panels (a) and (b) show the dX and dY series, respectively.The unit is mas.

Figure 2 .
Figure 2. The pressure term of the MERRA-2 CAAM ( χ′ A ), COAM ( χ′ O ), and their sum χ′) in the range 1 January 1994 to 31 December 2022.Panels (a) and (b) show the first and second equatorial components (χ′ 1 and χ′ 2 , respectively).The unit is mas.Note that these excitations have been low-pass filtered as mentioned in Section 2.3.

Figure 3 .
Figure 3. Cross-spectral coherence between geodetic and geophysical (i.e., atmosphere-ocean) excitation from MERRA-2 and ECMWF for (a) frequencies between 500 and 300 days and (b) frequencies between 300 and 500 days.The 95% confidence level is also shown.Power spectrum of geodetic and the two geophysical excitations over period (in days) for frequencies in the range (c) 500 to 300 days and (d) 300-500 days.The unit here is mas 2 .The narrow vertical shaded areas indicates the frequency band of interest around FCN.

Figure 4 .
Figure 4. Geodetic excitation χ′G (t) versus geophysical excitation χ′(t) in the time domain.Shown are the (a) real and (b) imaginary components of the geophysical excitation series for MERRA-2 and ECMWF after low-pass filtering (cf.Section 2.3).The unit is mas.

Figure 5 .
Figure5.Amplitude of the NMWT spectra for observed and modeled excitation of nutation around the FCN and retrograde annual band, discretized in 1-day steps for periods between 500 and 300 days.Shown are magnitudes (i.e., absolute values of the complex-valued NMWT output) in unit of mas.Panels (a), (b), and (c) are for geodetic, geophysical MERRA-2, and geophysical ECMWF excitations, respectively.Note the difference in scale among the panels and that the spectra were derived from excitations before filtering.The horizontal dashed lines bracket the FCN frequency band.Black contours identify regions with statistically significant magnitudes, whereas white contours highlight statistically insignificant magnitudes, both evaluated at the 95% confidence level.

Figure A2 .
Figure A2.The pressure term of the MERRA-2 OAM in the range 1 January 1994 to 31 December 2022.Panels (a) and (b) show the first and second equatorial components (χ O 1 and χ O 2 , respectively).The corresponding complex-demodulated series (χ ′ O 1 and χ ′ O 2 , respectively) are shown in panels (c) and (d).The unit is mas.

Table 1
Coherence Between Geodetic and Geophysical Excitation at the FCN Frequency, Along With the Correlation of the Corresponding TimeSeries,  Separated Into Atmospheric and Oceanic Contributions, As Well As Their  Sum (i.e., the Total CEAM)

Table 3
Statistics for the Convolution Approach (Mean and Standard Deviation, Std, in Mas)Note.For CPO, mean and std have values of 0.04 and 0.19, respectively.

Table 2
Coherence and Correlation Statistics as in Table 1 but for the MERRA-2 Atmospheric Excitation Evaluated for Different Spatial Domains

Table B1
Statistics of Coherence Between Geophysical and Geodetic Excitations Near the FCN Frequency Band