OAM signal content
Before turning to excitation budgets, we highlight some characteristics of the modeled OAM functions, displayed as amplitude spectra of mass and motion components in Fig. 2. Axial terms allow for little discrimination among the five models and are omitted. Different estimates of \(\chi ^m_{x,y}\) show a high degree of consistency and point to greater levels of excitation in the y component, rather than x, for periods longer than 10 days (cf. Gross et al 2003). A faint cusp of mass term variability at \(\sim \)5 days is likely the angular momentum signature of the ocean’s dynamic response to barometric pressure and wind stress fluctuations associated with the gravest symmetric mode of the Rossby–Haurwitz waves (Madden 2019), see also Ponte and Ali (2002). All models suggest pronounced bottom pressure effects in \(\chi ^m_{y}\) around \(T=20\) days, a peculiarity previously noted by Bizouard and Seoane (2010). We have performed complementary checks of \(p_b'\) fields in that band (18–22 days) and found evidence for an out-of-phase relationship between the Indian and Pacific sectors of the Southern Ocean. Such geometry is generally conducive to a strong OAM signal in the y direction (cf. Sect. 4.3).
Examination of the LLC amplitude spectra and related RMS (root-mean-square) values at \(T<\) 120 days, also included in Fig. 2, reveals that the LLC540 simulation yields weaker OAM variability than LLC90, especially at periods below 7 days. This is an interesting result, as the amount of damping conveyed by bottom friction and viscosity schemes (Sect. 3.3) is, to first order, identical between the two runs (D. Menemenlis, 2021, personal communication). Higher dissipation in LLC540 must stem from the sixfold increase in horizontal resolution, which can have various effects. First, grid spacing of
\(^\circ \) is sufficiently small to resolve the first baroclinic Rossby radius in the deep ocean within latitudes \(|\phi |< 40^\circ \); cf. Figure 1 in Hallberg (2013). Transient eddy features are therefore admitted in the LLC540 simulation and tend to drain energy from the large-scale fields important to OAM quantities. Second, higher resolution necessarily results in sharper topographic gradients, which can enhance scattering of barotropic motions into baroclinic waves. In fact, localized interactions with topography have been shown to be relevant to the dynamic response of a stratified ocean to barometric pressure (Ponte and Vinogradov 2007), a process most active on sub-weekly time scale where LLC540 variability in \(\chi ^{m,v}_{x,y}\) is weak.
A paramount feature in Fig. 2, particularly clear in the motion terms, are power deficits in the OAM functions from DEBOT. Assuming depth-independence of the flow field at large scales is generally justified on physical grounds (Willebrand et al 1980), but dynamics at a given latitude become more baroclinic with increasing period (Bingham and Hughes 2008). Hence, we expect a drop in coherence between DEBOT OAM estimates and rotation parameters toward the lower end of the frequencies considered. Most glaring though in Fig. 2 are large amplitudes in the MPIOM-based equatorial motion terms at periods of less than \(\sim \)2 weeks and 2–4 mas in excess relative to other models. Below, we offer more thoughts on these anomalies and their ramifications for modeling rapid polar motion.
Excitation budgets
Table 1 Excitation budget for sub-seasonal Earth rotation changes over 2007–2011
Table 1 presents the paper’s main results in terms of the percentage of sub-seasonal excitation variance accounted for by geophysical excitation processes. For the ocean, these broadband statistics can be taken in together with a decomposition into bottom pressure and currents effects in Table 2 and a plot of PVE by four ocean models in \({\hat{p}}\) and \({\Delta \Lambda }\) changes over period, shown in Fig. 3. Note that the residual polar motion curves in Fig. 3a tend to follow the transfer function’s ascending slope (\(\sigma \rightarrow 2\pi T_c^{-1}\), Eq. 2) and should be interpreted cautiously in quantitative terms. The baseline for assessing model “skill” is defined by residual geodetic excitation (or residual polar motion) with atmospheric contributions removed, i.e., \(\chi _j^G - \chi _j^A\). Over 2007–2011, atmospheric processes—mostly tropospheric winds (Gross et al 2004)—explain as much as 93.7% of the observed non-tidal \({\Delta \Lambda }\) variance. They are less effective in exciting sub-seasonal wobbles (PVE of 54.2% in \({\hat{\chi }}\)), consistent with Table 7 in Gross et al (2003).
Our assessment of oceanic effects contains several findings of value. First, geodesy’s leading OAM series given by MPIOM do not adequately reduce variance of the observed sub-seasonal polar motion excitation. Table 2 and Fig. 3a suggest that the deficiency has its source in the motion terms and at high frequencies, in line with the spectral characteristics of \(\chi _{x,y}^v\) in Fig. 2. These short period fluctuations are absent from \(\chi _z^v\) and inherently emphasized in evaluations of polar motion excitation based on deconvolution (Eq. 1, Chao 1985).
As a consistency check (not shown), we have computed proxy equatorial OAM motion terms from daily-averaged MPIOM bottom pressure fields (Dobslaw et al 2017). The analysis is based on time integration of relevant torques on the ocean in the frequency domain, similar to what we have previously tested for the atmosphere; cf. Schindelegger et al (2013). Benchmarks of the method with DEBOT along with results in Fujita et al (2002) indicate that a combination of ellipticity and seafloor topographic torques—both estimated from \(p_b'\)—account for most of the variance (PVE \(=86\)%) in the oceanic motion term. This first-order budget constraint is, however, poorly fulfilled in MPIOM (PVE \(=44\)%), implying the presence of a non-standard angular momentum source. Direct analysis of the model’s depth-integrated velocities could shed light upon the issue, as would an examination of equatorial friction torques. The latter are generally thought to be small (Fujita et al 2002) but might be inflated in MPIOM depending on the exact nature of dissipative closures and the offline bulk parameterization for air–sea momentum flux (Dobslaw and Dill 2019). While the origin of the high-frequency energy excess in MPIOM remains arcane, it is clear that these OAM series are not a good choice for evaluating the quality of geodetic polar motion determinations at sub-weekly periods (Dill et al 2020).
Table 2 PVE by oceanic excitation in atmosphere-corrected geodetic excitation (\(T<\) 120 days, 2007–2011), split up into mass and motion terms for each \(\hbox {model}\)
None of the other statistics in Tables 1 and 2 suggests sizeable modeling issues. PVE calculated from ECCOv4, LLC90, and DEBOT fall in a narrow range (52.0–54.3% in \({\hat{\chi }}\), 56.8–59.9% in \(\chi _z\)), even though frequencies of closest correspondence to observations vary from model to model (Fig. 3). In particular, limitations of DEBOT’s constant-density formulation are evident outside the sub-monthly band, and data-constrained optimization in ECCOv4 provides for a better agreement with atmosphere-corrected rotation signals the greater the period. Improvements to OAM quantities with adjusted atmospheric forcing are most obvious in \({\Delta \Lambda }\), including a band (70–120 days) where the \({\Delta \Lambda } - \chi _z^A\) residual has been shown to be coherent with west equatorial Pacific wind stress variability (see Figure 3 in Marcus et al 2001). The potential role of optimization at these time scales could be clarified by dedicated analyses of changes in momentum flux, wind stress torque, bottom pressure, and depth-averaged horizontal currents relative to LLC90 or, better, an unconstrained ECCOv4 integration (cf. Ponte et al 2001).
In most, if not all of our broadband comparisons (Table 1), high PVE with LLC540 stand out. A residual RMS of 20.3 \(\upmu \)s in \({\Delta \Lambda }\) (both \(\chi _z^A\) and \(\chi _z^O\) removed) is marginally better (p-value \(\sim 0.15\)) than what MPIOM can give. More drastically, LLC540 accounts for 65.7% of the non-atmospheric polar motion excitation variance, with particular progress in modeling evident at periods below 50 days (cf. PVE of polar motion in Fig. 3a). DEBOT performs similarly for bands < 20 days, a result pointing to the benefits of higher horizontal resolution or enhanced dissipation, or both. As suggested in earlier work (Nastula and Ponte 1999; Ponte and Ali 2002), weak OAM variability, as long as with the right phases, is most commensurate with atmosphere-corrected polar motion excitation at rapid time scales. For ease of comparison with a similar analysis in Zhou et al (2005), we have recomputed similarity measures for the 4–20-day band. In this case, PVE by LLC540 (DEBOT) in \({\hat{\chi }}^G - {\hat{\chi }}^A\) is 72.7% (68.8%), compared to 51.6% from a barotropic ocean model in Zhou et al (2005).
Secondary excitation processes warrant a brief note. Most of the variability in HAM is compensated by barystatic effects (SLAM), so we evaluate their sum against \(\chi _j^G - \chi _j^A - \chi _j^O\) using the ESMGFZ series. Additional OAM contributions in ECCOv4 due to net freshwater flux are readily computed as difference between the “yesFWF” and “noFWF” OAM functions available at the SBO website. Variance ratios of these terms relative to \(\chi _j^O\) are around 1% for HAM/SLAM and \(\ll \) 0.1% (1.6% in \(\chi _z\)) for ECCOv4 freshwater loads. Partly because of their smallness, neither component can account for appreciable variance in the residual geodetic excitation, see values in Table 1. Last, we note that only minor quantitative, but not qualitative, changes to our conclusions in this section were observed when another Earth rotation solution (IERS 14C04, Bizouard et al 2019)—or a different excitation formalism (Chen et al 2013)—was adopted for the comparison.
Insights from bottom pressure
Results above for the LLC simulations can be given additional context by mapping their differences in terms of \(\chi \) functions in space and identifying areas that give the largest contribution to the global signal. Such analysis typically draws on gridded \(\chi \) values for mass and velocity before their being summed up (e.g., Salstein and Rosen 1989; Nastula et al 2003, 2012). Here, we proceed along similar lines but (i) restrict ourselves to mass effects in the equatorial component, and (ii) work in units of bottom pressure instead of excitation functions. Specifically, we take differences of LLC540 relative to LLC90 in dynamic bottom pressure (\({\Delta }p'_b\)) and deduce complex-valued, spatially weighted bottom pressure differences
$$\begin{aligned} {\Delta }{\hat{p}}_w\left( \phi ,\lambda \right) = {\Delta }p'_b\left( \phi ,\lambda \right) \sin \phi \cos ^2\phi \mathrm {e}^{\mathrm {i}\lambda } \end{aligned}$$
(7)
by applying trigonometric weighting functions implicit to the products of inertia \({\hat{c}} = c_{xz} + \mathrm {i} c_{yz}\), multiplied with \(\cos \phi \) from the grid point’s area element (cf. Nastula et al 2012). Prior to computing \({\Delta }p'_b\), different grid resolutions of the dynamic LLC fields were conformed using harmonic expansion to degree 179 and projection of the coefficients to a 1\(^\circ \) target grid. With Eq. (7), the differential equatorial mass term (LLC540 minus LLC90) reads
$$\begin{aligned} {\Delta }{\hat{\chi }}^m = -a^4 \rho {\Omega } \sum _{\phi , \lambda } {\Delta }{\hat{p}}_w\left( \phi ,\lambda \right) \mathrm {d}\phi \mathrm {d}\lambda \end{aligned}$$
(8)
where a is Earth’s mean radius and \(\rho \) is a reference density for seawater.
A generalized form of principal component (PC) analysis for vector quantities (Hardy and Walton 1978; Nastula et al 2003) was performed on filtered \({\Delta }{\hat{p}}_w\) (\(T<\) 120 days) to separate these anomalies into complex-valued spatial modes (Empirical Orthogonal Functions, EOFs) and their time-dependent modifications (PCs). We rank the so derived modes by their variance in globally integrated excitation functions (Eq. 8), upon a synthesis step involving both the EOFs and the corresponding PCs. Summing up all modes according to Eq. (8) gives the original \({\Delta }{\hat{\chi }}^m\) time series, testifying to the correctness of our method.
Panels a and b in Fig. 4 display the spatial pattern of the leading mode in the re-ranked EOF spectrum, which accounts for 26% (2% in x, 49% in y) of the variance in \({\hat{\chi }}^m\) differences between LLC540 and LLC90 (panels c and d). The mode’s energy is concentrated in the imaginary part, resulting in a relatively large standard deviation of 3.9 mas in \(\chi _y^m\) since the associated PC has a negligible imaginary component. That this differential excitation signal is important for improving the agreement with rotation data is highlighted in Table 3, where we re-evaluate the LLC90 polar motion excitation budget with the model’s original mass terms corrected such that they include the contributions shown as blue lines in Fig. 4c, d. The correction reduces the RMS of \(\chi _y\) residuals from 15.1 mas to 13.9 mas; cf. 12.1 mas with LLC540 in Table 1. Differences in the x OAM functions contain contributions from several other EOFs and are not discussed here.
Table 3 Extension of Table 1 for LLC90 with modified mass \(\hbox {terms}\) Can we make physical assertions based on the spatial pattern depicted in Fig. 4b? To some extent. The EOF’s negative maxima in the Southern Ocean coincide with the centers of the Australian-Antarctic Basin (A) and the Chile Rise (B) in the Bellingshausen Basin, i.e., areas known for their high levels of barotropic variability on intraseasonal time scales (Fukumori et al 1998; Fu 2003). These deep basins are encircled by closed, or almost closed, contours of potential vorticity
, where f is the Coriolis parameter and H the local water depth. The specific
distribution facilitates a near-resonant response to wind stress curl with characteristic decay time scales of about 4 days (Weijer 2010, 2015). At constricted topographic features, energy of the mode is expended to a residual flow that may dissipate elsewhere, e.g., the greater South Indian Ocean surrounding area A. Figure 4b, paired with Fig. 6 in “Appendix,” indicates that the trapped modal circulation and its leakage to areas outside the Australian-Antarctic basin are of too large magnitude in LLC90. The most consistent explanation of this picture is that the model’s 1\(^\circ \) horizontal grid inhibits a proper representation of topographic effects that set the near-resonant response and its energetics.
Figure 5 underpins the point in the frequency domain and across all models analyzed. We show amplitude spectra of area-averaged \(p_b'\) series in region A, along with similar estimates from daily GRACE solutions at periods from 4 to 60 days. All bottom pressure series imply enhanced variability in the same bands, most notably near 10 days (cf. Fukumori et al 1998). DEBOT and LLC540 are tightly aligned with spectral characteristics in GRACE, a result also borne out by their low RMS differences with satellite-based \(p_b'\) fluctuations (statistics are included in Fig. 5). In contrast, all 1\(^\circ \) models—especially LLC90—are subject to a systematic excess in power from 10 to 16 days, consistent with some of the main periodicities seen in the LLC mass term differences (\(\chi _y^m\), Fig. 4d). Although rather local in character, these model comparisons and insights from satellite gravimetry provide, at the very least, valuable hints on where to improve bathymetry in LLC90 (and thus ECCOv4) for better agreement with observed polar motion excitation. Evidently, a more complete analysis of ocean model differences and regional sources of excitation calls for scrutiny of volume transports and possible cancellation between mass and motion effects in the global OAM integrals.