## Abstract

Diurnal S\(_1\) tidal oscillations in the coupled atmosphere–ocean system induce small perturbations of Earth’s prograde annual nutation, but matching geophysical model estimates of this Sun-synchronous rotation signal with the observed effect in geodetic Very Long Baseline Interferometry (VLBI) data has thus far been elusive. The present study assesses the problem from a geophysical model perspective, using four modern-day atmospheric assimilation systems and a consistently forced barotropic ocean model that dissipates its energy excess in the global abyssal ocean through a parameterized tidal conversion scheme. The use of contemporary meteorological data does, however, not guarantee accurate nutation estimates per se; two of the probed datasets produce atmosphere–ocean-driven S\(_1\) terms that deviate by more than 30 \(\upmu \)as (microarcseconds) from the VLBI-observed harmonic of \(-16.2+i113.4\) \(\upmu \)as. Partial deficiencies of these models in the diurnal band are also borne out by a validation of the air pressure tide against barometric in situ estimates as well as comparisons of simulated sea surface elevations with a global network of S\(_1\) tide gauge determinations. Credence is lent to the global S\(_1\) tide derived from the Modern-Era Retrospective Analysis for Research and Applications (MERRA) and the operational model of the European Centre for Medium-Range Weather Forecasts (ECMWF). When averaged over a temporal range of 2004 to 2013, their nutation contributions are estimated to be \(-8.0+i106.0\) \(\upmu \)as (MERRA) and \(-9.4+i121.8\) \(\upmu \)as (ECMWF operational), thus being virtually equivalent with the VLBI estimate. This remarkably close agreement will likely aid forthcoming nutation theories in their unambiguous a priori account of Earth’s prograde annual celestial motion.

## Keywords

Earth rotation variations Nutation Geophysical excitation Atmospheric tides Ocean tides## 1 Introduction

Describing variations of our planet’s orientation in space is a multidisciplinary subject matter that has occupied the attention of mathematicians, astronomers, and geophysicists alike. Nutations, that is, periodic motions of a predefined physical or conventional reference axis with respect to an inertial system, have been classically modeled in a Lagrangian or Hamiltonian framework (Woolard 1953; Kinoshita 1977) as the rigid Earth response to gravitational lunisolar torques. The estimates of these standard treatments are accurate to a few tenths of mas (milliarcseconds), but the advent of precise observational data as well as the pursuit of insights into Earth’s internal constitution has stimulated the development of non-rigid nutation theories for a realistic Earth (Jeffreys and Vicente 1957; Molodensky 1961; Sasao et al. 1980). In 1980, the International Astronomical Union (IAU) adopted theoretical values of the lunisolar forced nutations for an elliptical, oceanless, elastic Earth with a fluid outer core and solid inner core (Wahr 1981), though the geophysical approximations and omissions within that model were soon to become larger than the requirements posed by space geodetic techniques such as VLBI (Very Long Baseline Interferometry). A timely formation of a new nutation series, which has been the reference model since its endorsement by the IAU in 2000, is documented in Mathews et al. (2002) (MHB for short) and explicitly allows for mantle anelasticity, inner core dynamics, and non-hydrostatic equilibrium effects. Basic Earth parameters that govern the nutation response to lunisolar and planetary torques are constrained to their “best estimates” from a least-squares fit of the theoretical nutation expressions to VLBI results. This semi-analytical approach to modeling Earth’s nutation leaves residuals with observational data below 0.1 mas.

Some effort has been devoted by MHB to properly account for the nutation perturbations associated with the Earth’s fluid layers. These contributions range from a few tens of \(\upmu \)as (microarcseconds) to 1 mas in amplitude and can be understood as the manifestations of (quasi-)diurnal atmosphere–ocean dynamics in the terrestrial frame. Their underlying excitation mechanisms are twofold, comprising (1) the daily cycle of solar heating and (2) the differential gravitational forces that directly act upon the atmosphere and ocean and produce global-scale waves known as tides. At the major diurnal tidal frequencies, the oceanic variability is almost exclusively driven by the gravitational influence with minute modulations related to the hydrodynamic response to atmospheric forcing. Satellite altimetry provides an accurate global record of these signals in the modern ocean and is also typically used to infer the associated oceanic angular momentum (OAM) variations of the largest diurnal tides, K\(_1\), P\(_1\), O\(_1\), and Q\(_1\). Early OAM determinations for these four waves (Chao et al. 1996) were adopted by MHB to predict the full OAM spectrum across the diurnal band and subsequently correct the equations of motion and anelasticity in the nutation theory.

Tides in the atmosphere around the central diurnal period of S\(_1\) are capable of exciting small additional, seasonal nutation waves that exceed the statistical uncertainties of VLBI-based parameters on the order of 10 \(\upmu \)as (Dehant et al. 2003). The gravitational components of these oscillations are—to the extent they have not been implicitly accounted for in the MHB model—negligibly small (Bizouard and Lambert 2002) and in fact overshadowed by thermal tides due to periodic radiation and absorption processes (Chapman and Lindzen 1970). Nutation contributions of some minor radiational constituents, such as the P\(_1\) tide, have been thoroughly addressed by MHB, yet the main S\(_1\) wave proved to pose some sort of conundrum to the authors. Whereas the VLBI data (1979.8–1999.11) distinctly testified to the existence of an S\(_1\) influence in the form of a prograde annual nutation residual, available geophysical model estimates (Bizouard et al. 1998; Yseboodt et al. 2002) were deemed unreliable, and thus no “theoretical” account of the effect was incorporated into the dynamical equations. To compensate for this mismatch, MHB subtracted from the VLBI spectrum an a priori S\(_1\) harmonic of somewhat more than 100 \(\upmu \)as and superimposed the very same signal as a post-fit correction term to the final nutation series.

Studies of this subject matter are required to accommodate not only the atmospheric portion of the tide but also the substantial 24-h oceanic mass redistributions driven by S\(_1(p)\), the diurnal pressure variations at the sea surface. Hence, both atmospheric and oceanic oscillations are closely interrelated aspects of the same “global S\(_1\) tide,” labeled as such by Ray and Egbert (2004) as well as in the context of the present work. Owing to its radiational origin, the oceanic S\(_1\) variability might be also perceived as an indirect influence of the atmosphere on the rotation of the solid Earth, and it has therefore been occasionally classified as a “non-tidal” phenomenon (Brzeziński et al. 2004; de Viron et al. 2004)—a terminology that shall, however, not be used in the following.

Investigations of the S\(_1\) effect in nutation on the basis of dynamically coupled atmosphere–ocean models have been pursued primarily by A. Brzeziński and collaborators; cf. Brzeziński et al. (2004), Brzeziński (2011) and references therein. These studies employed different atmospheric analyses and ocean models both in a full 3D baroclinic formulation as well as 2D (constant density) barotropic versions that efficiently capture short-period hydrodynamic processes. The inferred atmosphere–ocean excitation terms of the prograde annual nutation vary substantially, though, both among each other and from the geodetic VLBI value, with deviations usually being larger than 50 \(\upmu \)as. While the VLBI estimate itself is possibly perturbed by other, imperfectly modeled seasonal effects, we must proceed on the assumption that the probed general circulation models were not appropriately designed for S\(_1\)-related investigations. Another though brief assessment of the radiational tidal influence on nutation is given in de Viron et al. (2004) and their seemingly close agreement (<15 μas) with the VLBI value has been acknowledged by Dehant and Mathews (2009) (Sect. 10.11, *ibid.*). We think, however, that the results of de Viron et al. (2004) are questionable and actually affected by an incorrect conversion of excitation values to periodic nutation terms. This deficiency is particularly evident for their tabulated atmospheric contributions (P\(_1\), S\(_1\), and \(\psi _1\)), which are inconsistent with what has been documented for the very same atmospheric dataset by Koot and de Viron (2011) and Bizouard et al. (1998), even if one makes allowance for differences in the utilized transfer functions and the analyzed time spans. Note, e.g., that over the period 1991–2002, Fig. 2 of Koot and de Viron (2011) suggests \(\sim \)75 \(\upmu \)as for the S\(_1\) out-of-phase term, while de Viron et al. (2004) specify a value of 38 \(\upmu \)as. Without further insight into the actual (corrected) S\(_1\) nutation predictions of these authors, we will employ Brzeziński (2011) as a reference study by which our results can be measured.

Building on the elucidations of these pilot investigations, the key objective of the present work is to provide an up-to-date treatment of the global S\(_1\) tidal effect in nutation and ultimately arrive at an explanation of MHB’s empirical prograde annual nutation term. The modern-day aspect of our effort resides in the use of four of the currently most advanced atmospheric assimilation systems, comprising three constant-model, retrospective analyses (so-called reanalyses) for a principal time span from 1994 to 2013, and a shorter, operational dataset (2004–2013) that stems from the near real-time weather analysis with a steadily improving model. This atmospheric portion of our study can be rightly understood as a continuation of similar earlier assessments (Bizouard et al. 1998; Yseboodt et al. 2002), and as such, it is partly motivated by the good agreement of atmospheric nutation estimates from reanalyses that are essentially the precursors of the presently tested models (Koot and de Viron 2011). To ensure conformity with these investigations, nutation values for the minor solar constituents (\(\psi _1\), P\(_1\), \(\pi _1\), and \(\phi _1\)) are tabulated, even though our prime focus is on the S\(_1\) tide throughout.

A second key theme of this work is to numerically model the dynamic ocean response to diurnal atmospheric pressure forcing, which has been a fruitful geophysical industry over the last decade; cf. Ray and Egbert (2004), Dobslaw and Thomas (2005), Ponte and Vinogradov (2007), or Carrère et al. (2012). Mere superpositions of these modeling results to our atmospheric excitation terms are invalid, though, and a rigorous treatment of the geophysically driven prograde annual nutation requires deducing hydrodynamic S\(_1\) solutions and respective OAM values that are consistent with the utilized atmospheric datasets. We adopt the recent barotropic time-stepping model of Einšpigel and Martinec (2015), designated as DEBOT (Barotropic Ocean Tide model developed by D. Einšpigel), and implement the necessary modifications for the problem in hand. Specifically, to obtain S\(_1\) tidal solutions that are on par with Ray and Egbert (2004), ocean self-attraction and loading effects (SAL, Ray 1998b) are accounted for in an iterative fashion and the overestimation of sea surface elevations in deep water is mitigated by a parameterized expression for the barotropic-to-baroclinic energy conversion over abyssal hills (Bell 1975; Jayne and St Laurent 2001). Moreover, forcing the same hydrodynamic model with different pressure tide solutions should go some way to reveal the dependence of the global/regional character of S\(_1\) (and its OAM values) on variations in the barometric input data. This is a subtle issue that has yet not been addressed by the oceanographic community.

Our approach is a climatological one inasmuch as the present formulation of DEBOT only allows for a strictly harmonic pressure loading by the S\(_1\) air tide, even though the temporal variability of this forcing can be large (Ray 1998a). Seasonal modulations of S\(_1(p)\) by 1 cpy (cycle per year) correspond to the P\(_1\) and K\(_1\) constituents, inducing small radiational ocean tides that are automatically included in altimetric solutions of P\(_1\) and K\(_1\) and are thus of no practical significance. Variations on inter-annual timescales (e.g., Vial et al. 1994) pose, however, a more delicate challenge, which can be partly resolved by working with decadal-scale S\(_1\) averages for the coupled atmosphere–ocean system. To determine a favorable, i.e., inter-annually “quiet” averaging period, we assess the S\(_1\) variability both in the integrated atmospheric nutation values and in surface pressure. Specifically, the analysis of S\(_1(p)\) is conceived as a validation of model pressure tides against “ground truth” estimates from 50 island- and buoy-based barometers. This comparison is in fact a vital (though limited) measure in deciphering the fine margins in quality among the different models regarding the diurnal cycle. Further observational constraints on our model-based investigations are supplied by S\(_1\) determinations at 56 coastal tide gauges, which, to some extent, echo the varying degree of reliability of the simulated tidal heights from each atmospheric dataset.

The paper is organized as follows. Section 2 places the present and previous nutation studies in the context of an evolving collection of meteorological assimilation systems and describes the main characteristics of the four models utilized herein. Atmospheric excitation time series are computed and mapped to nutation amplitudes in Sect. 3, complemented by the validation of model pressure tides against in situ estimates from pelagic barometers. The ocean model and its hydrodynamic configuration are thoroughly discussed in Sect. 4, and we assess the quality of our forward simulations both from an angular momentum perspective as well as in a comparison to coastal tide gauges. Section 5 finally synthesizes atmospheric, oceanic, and VLBI nutation results to address the mismatch between theory and observation in the S\(_1\) band.

## 2 Meteorological Data for Nutation Studies

Overview of current atmospheric reanalyses as operated by various meteorological agencies^{a}

Name | Source | GI | Model vintage | Resolution (km) | Assimilation |
---|---|---|---|---|---|

NCEP R1 | NCEP | 1 | 1995 | 210 | 3DVar |

NCEP R2 | NCEP | 1 | 1995 | 210 | 3DVar |

ERA-40 | ECMWF | 2 | 2001 | 125 | 3DVar |

JRA-25 | JMA | 2 | 2002 | 120 | 3DVar |

MERRA | NASA GMAO | 3 | 2004 | 60 | IAU |

CFSR | NCEP | 3 | 2004 | 40 | 3DVar |

ERA-Interim | ECMWF | 3 | 2006 | 80 | 4DVar |

JRA-55 | JMA | 3 | 2009 | 55 | 4DVar |

Previous assessments of atmosphere-driven nutations have relied heavily on NCEP’s first-generation reanalysis R1, whose physical formulation and relatively coarse resolution (2°–2.5° for surface and vertical parameters) date back to 1995. Bizouard et al. (1998), Yseboodt et al. (2002), and Brzeziński et al. (2004) derived R1-related estimates at tidal frequencies for particular reanalysis periods, while Koot and de Viron (2011) additionally analyzed NCEP R2 and the second-generation ERA-40 model over a common time span from 1979 to 2002. As a sole modern-day reanalysis, ERA-Interim (henceforth ERA) has been subject to an evaluation of nutation signals by Brzeziński (2011). Ignoring differences due to varying analysis periods, the S\(_1\) estimates of these studies exhibit a fair agreement, roughly at 20–30 \(\upmu \)as, though individual outliers exist. In an attempt to document the same level of agreement or even further convergence for third-generation reanalyses, the present work derives nutation values for ERA and its contemporaries MERRA (Rienecker et al. 2011) and CFSR (Saha et al. 2010).

Grid specifications of the utilized atmospheric assimilation models

Data stream | Time span | \(\Delta {t}\) (h) | Horizontal resolution | |
---|---|---|---|---|

Surface | Pressure level | |||

MERRA, assimilated state | 1994–2013 | 3 | \(1.25^{\circ }\) | \(1.25^{\circ }\) |

CFSR, analysis & 3-h forecast | 1994–2010 | 3 | \(0.5^{\circ }\) | \(2.5^{\circ }\) |

ERA-Interim, analysis | 1994–2013 | 6 | \(0.5^{\circ }\) | \(2.0^{\circ }\) |

ECMWF, operational analysis | 2004–2013 | 6 | \(1.0^{\circ }\) | \(1.0^{\circ }\) |

Table 2 summarizes the main specifications of the globally gridded datasets from MERRA, CFSR, ERA, and the ECMWF operational model, denoted as EC-OP in the following. The analysis was initially conceived for the period of 2004–2013 (Schindelegger et al. 2015) but extended in retrospect to a 20-year window (1994–2013) for the three reanalyses. CFSR constitutes an exception, having been produced as a genuine, constant-model reanalysis until the end of 2010 with subsequent operational extensions that were, however, disregarded in the frame of the present work. We extracted standard 6-h analysis fields from the respective data archives for both ECMWF models as well as CFSR, whereas 3-h analysis/forecast combinations, designated as *assimilated states*, were utilized for MERRA. This high-resolution dataset has been a source of continued Earth rotation research at TU Wien, comprising also investigations of the semidiurnal atmospheric tide S\(_2\), which is not properly resolved by four-times-daily analysis products. To prepare for further subsequent work on S\(_2\), we also interlaced 3-h forecast data to the CFSR analysis fields after verifying that their inclusion had no evident effect on the S\(_1\) signature in surface pressure and nutation estimates.

Atmospheric excitation is classically inferred from the two components of AAM (atmospheric angular momentum), comprising effects both due to particle movement (wind or motion term) and redistribution of matter (pressure or mass term). The evaluation of the wind term involves vertical integration over an appropriate number (>15) of isobaric levels and is thus computationally intensive. Yet, the higher-altitude horizontal convection associated with S\(_1\) constitutes a robust, large-scale signal that is integrated with sufficient accuracy from comparatively coarse mesh sizes of about 2°; cf. Table 2. Given the pronounced small-scale (and possibly subgrid-scale) characteristics of the diurnal surface pressure variability over landmasses (e.g., Li et al. 2009), the mass component of AAM is preferably deduced from better resolved surface grids, although limitations are imposed by the intrinsic model resolution and our hardware resources. MERRA’s assimilated states of surface pressure are distributed at 1.25° in latitude and longitude, whereas 0.5° grids could be utilized for CFSR and ERA. Note that the native 80-km spacing of ERA (Table 1) is somewhat coarser than 0.5°, suggesting that these data were interpolated during the assignment process. By contrast, the chosen 1° mesh for the operational data is a largely downsampled version of the model’s fine intrinsic discretization, having improved in resolution from 40 km in 2004 to 16 km at the end of 2013.

*s*(Chapman and Lindzen 1970) and retaining only the main Sun-synchronous, migrating S\(_1^1\) tide (\(s=1\)). Forced by tropospheric absorption processes, S\(_1^1\) corresponds to a longitudinally uniform wave that is clearly evident in Fig. 2 over oceanic areas. Latitudinal profiles of this migrating tide from MERRA and CFSR exhibit a fair agreement (not shown), although equatorial peak amplitudes differ slightly (64.1 Pa for MERRA, 67.2 Pa for CFSR) and an overestimation of about 10 Pa at latitudes close to 60\(^\circ \) S can be observed for the CFSR solution. These discrepancies in pressure are likely to have a bearing on the simulation of the oceanic S\(_1\) tide.

## 3 Atmospheric Contributions to Nutation

### 3.1 Implementation

Mode | \({\tilde{\sigma}} \) (cpsd) | \({\tilde{N}}^p \times 10^3\) | \({\tilde{N}}^w \times 10^3\) | |||
---|---|---|---|---|---|---|

Real | Imag | Real | Imag | Real | Imag | |

CW | 0.00251794 | −0.00000564 | 2.561471 | −0.004259 | 3.702480 | \(-\)0.000004 |

FCN | −1.00232436 | 0.00002539 | 0.235658 | 0.000612 | 0.000973 | \(-\)0.000009 |

*X*and

*Y*in response to changes of AAM at some Earth-referred, retrograde diurnal frequency \(\sigma \) (in cycles per sidereal day, cpsd) are modeled as

*A*and

*C*denote the equatorial and polar principal moments of inertia of an axisymmetric solid Earth (i.e., mantle and crust), and \(\tilde{T}^{p,w}\left( \sigma \right) \) are transfer functions describing Earth’s nutation response to atmospheric forcing as conveyed by the periodic terms \(\tilde{H}'^{p,w}\left( \sigma \right) \), which are defined below. The transfer functions read

Multipliers of the fundamental arguments of nutation terms that apply to atmospheric tidal lines^{a}

Term | Fundamental arguments | Period | Phase | ||||
---|---|---|---|---|---|---|---|

| \(l'\) | | | \(\Omega \) | (solar days) | (°) | |

S\(_1\) | 0 | 1 | 0 | 0 | 0 | 365.260 | 357.529 |

\(\psi _1\) | 0 | −1 | 0 | 0 | 0 | −365.260 | −357.529 |

P\(_1\) | 0 | 0 | 2 | −2 | 2 | 182.621 | −159.067 |

\(\phi _1\) | 0 | 0 | −2 | 2 | −2 | −182.621 | 159.067 |

\(\pi _1\) | 0 | 1 | 2 | −2 | 2 | 121.749 | 198.462 |

The harmonic decomposition in Eq. (4) exactly follows the model of Koot and de Viron (2011) except for our inclusion of the \(\phi _1\) component at \(-2\) cpy that suggests some inter-annual modulation of the thermal S\(_1\) tide. While the existence of such a modulation is debatable (Bizouard et al. 1998), its impact on nutation is at the level of 10 \(\upmu \)as and thus comparable to the usually modeled \(\pi _1\) term. Moreover, for numerical reasons, we performed the least-squares fit on the basis of prescaled AAM time series \(\tilde{H}'\left( t\right) /\left( \Omega \left( C-A\right) \right) \) (Eq. 1) in units of \(\upmu \)as, similar to the classical “celestial effective angular momentum functions” of Brzeziński (1994). The excitation scheme of this author was tested briefly and found to yield nutation results well within 5 % of the estimates from the above formalism.

### 3.2 Results

By contrast, a signal with a possibly physical origin is evident for MERRA during 1997–2001, coinciding in time with a peak El Niño event in 1997/1998 and subsequent cold La Niña conditions up to 2001; cf., e.g., the Oceanic Niño Index tabulated at http://www.cpc.noaa.gov/products/analysis_monitoring/ensostuff/ensoyears.shtml (accessed 29 September 2015). The warm phase of ENSO (El Niño–Southern Oscillation) has been previously suggested to alter the radiative forcing of the solar tide in the troposphere (Lieberman et al. 2007), thereby providing significant enhancement to diurnal pressure oscillations across the Pacific (Vial et al. 1994). The response of the climate system to ENSO events is, however, not restricted to the Tropics but can entail atmospheric circulation changes in higher latitudes that may ultimately couple to nutation; see similar conjectures in Yseboodt et al. (2002). Assessing whether the irregular nutation changes from MERRA in Fig. 3 are linked to ENSO or merely represent spurious variabilities in the wake of observing system changes (Robertson et al. 2011) is beyond the scope of this study, though. We will in fact avoid these signals in our selection of the mean analysis window below.

Periodic atmospheric contributions to nutation (\(\upmu \)as, with 1\(\sigma \) errors) computed from the model-specific pressure and wind terms over an analysis period of 2004–2013^{a}

Term | Model | Pressure | Wind | Total | |||
---|---|---|---|---|---|---|---|

ip | op | ip | op | ip | op | ||

S\(_1\) (\(+1\) year) | MERRA | \(-25.9\pm 0.4\) | \(18.1\pm 0.4\) | \(4.0\pm 0.2\) | \(27.6\pm 0.2\) | \(-21.9\pm 0.5\) | \(45.8\pm 0.5\) |

MERRA | \(-30.8\pm 0.5\) | \(23.0\pm 0.5\) | \(5.0\pm 0.2\) | \(26.6\pm 0.2\) | \(-25.8\pm 0.6\) | \(49.7\pm 0.6\) | |

CFSR | \(-32.3\pm 0.6\) | \(57.5\pm 0.6\) | \(-2.0\pm 0.2\) | \(30.0\pm 0.2\) | \(-34.3\pm 0.6\) | \(87.6\pm 0.6\) | |

ERA | \(-31.8\pm 0.5\) | \(31.7\pm 0.5\) | \(-4.4\pm 0.2\) | \(25.8\pm 0.2\) | \(-36.1\pm 0.5\) | \(57.5\pm 0.5\) | |

EC-OP | \(-21.9\pm 0.5\) | \(43.8\pm 0.5\) | \(-0.5\pm 0.2\) | \(23.6\pm 0.2\) | \(-22.4\pm 0.5\) | \(67.5\pm 0.5\) | |

\(\psi _1\) (\(-1\) year) | MERRA | \(-30.1\pm 5.9\) | \(-44.5\pm 5.9\) | \(-2.5\pm 0.3\) | \(-3.0\pm 0.3\) | \(-32.6\pm 5.9\) | \(-47.5\pm 5.9\) |

MERRA | \(-45.1\pm 6.5\) | \(-48.4\pm 6.5\) | \(-2.7\pm 0.4\) | \(-3.6\pm 0.4\) | \(-47.9\pm 6.5\) | \(-52.0\pm 6.5\) | |

CFSR | \(-58.2\pm 7.3\) | \(0.4\pm 7.3\) | \(0.6\pm 0.3\) | \(-3.2\pm 0.3\) | \(-57.5\pm 7.3\) | \(-2.9\pm 7.3\) | |

ERA | \(-79.3\pm 6.1\) | \(22.0\pm 6.1\) | \(-4.0\pm 0.3\) | \(-0.7\pm 0.3\) | \(-83.3\pm 6.1\) | \(21.3\pm 6.1\) | |

EC-OP | \(-50.8\pm 6.2\) | \(23.6\pm 6.2\) | \(-3.1\pm 0.3\) | \(0.9\pm 0.3\) | \(-54.0\pm 6.2\) | \(24.5\pm 6.2\) | |

P\(_1\) (\(+{1}/{2}\) year) | MERRA | \(-10.1\pm 0.3\) | \(0.1\pm 0.3\) | \( 7.1\pm 0.2\) | \(42.1\pm 0.2\) | \(-3.0\pm 0.3\) | \(42.2\pm 0.3\) |

MERRA | \(-10.3\pm 0.3\) | \(1.2\pm 0.3\) | \(7.1\pm 0.3\) | \(41.1\pm 0.3\) | \(-3.2\pm 0.4\) | \(42.3\pm 0.4\) | |

CFSR | \( -6.9\pm 0.3\) | \(7.0\pm 0.3\) | \( 5.1\pm 0.2\) | \(35.6\pm 0.2\) | \(-1.9\pm 0.4\) | \(42.6\pm 0.4\) | |

ERA | \(-17.3\pm 0.3\) | \(0.9\pm 0.3\) | \(13.5\pm 0.2\) | \(45.8\pm 0.2\) | \(-3.8\pm 0.3\) | \(46.7\pm 0.3\) | |

EC-OP | \(-17.0\pm 0.3\) | \(2.5\pm 0.3\) | \(11.2\pm 0.2\) | \(41.5\pm 0.2\) | \(-5.8\pm 0.3\) | \(44.1\pm 0.3\) | |

\(\phi _1\) (\(-{1}/{2}\) year) | MERRA | \( -2.1\pm 0.8\) | \( -6.1\pm 0.8\) | \( 0.9\pm 0.2\) | \(-3.0\pm 0.2\) | \( -1.2\pm 0.8\) | \( -9.1\pm 0.8\) |

MERRA | \(-1.8\pm 0.9\) | \(-5.9\pm 0.9\) | \(0.7\pm 0.3\) | \(-3.3\pm 0.3\) | \(-1.1\pm 0.9\) | \(-9.3\pm 0.9\) | |

CFSR | \(-17.3\pm 1.0\) | \( 1.9\pm 1.0\) | \(-2.1\pm 0.2\) | \(-1.4\pm 0.2\) | \(-19.3\pm 1.0\) | \( 0.6\pm 1.0\) | |

ERA | \(-12.2\pm 0.8\) | \( -8.9\pm 0.8\) | \(-2.5\pm 0.2\) | \(-4.9\pm 0.2\) | \(-14.7\pm 0.8\) | \(-13.9\pm 0.8\) | |

EC-OP | \( 1.8\pm 0.8\) | \(-13.0\pm 0.8\) | \( 0.7\pm 0.2\) | \(-5.7\pm 0.2\) | \( 2.5\pm 0.9\) | \(-18.7\pm 0.9\) | |

\(\pi _1\) (\(+{1}/{3}\) year) | MERRA | \(-2.6\pm 0.2\) | \(4.1\pm 0.2\) | \(1.9\pm 0.2\) | \(-1.4\pm 0.2\) | \(-0.7\pm 0.3\) | \(2.6\pm 0.3\) |

MERRA | \(-2.3\pm 0.2\) | \(4.0\pm 0.2\) | \(2.3\pm 0.3\) | \(-1.6\pm 0.3\) | \(0.0\pm 0.3\) | \(2.4\pm 0.3\) | |

CFSR | \(-2.1\pm 0.2\) | \(3.3\pm 0.2\) | \(2.1\pm 0.2\) | \(-0.7\pm 0.2\) | \(-0.1\pm 0.3\) | \(2.6\pm 0.3\) | |

ERA | \(-2.5\pm 0.2\) | \(2.8\pm 0.2\) | \(1.1\pm 0.2\) | \( 0.3\pm 0.2\) | \(-1.4\pm 0.3\) | \(3.0\pm 0.3\) | |

EC-OP | \(-2.2\pm 0.2\) | \(2.7\pm 0.2\) | \(1.1\pm 0.2\) | \(-0.1\pm 0.2\) | \(-1.1\pm 0.3\) | \(2.7\pm 0.3\) |

Numerical results for our nutation analysis are reported in Table 5, using—with some exceptions—an averaging period from 2004 to 2013 that has been specified after consulting station tide determinations in next section. If the somewhat deficient CFSR results are discarded, S\(_1\) estimates from third-generation reanalyses and EC-OP deviate from each other by less than 22 \(\upmu \)as, which slightly betters the agreement noted by Koot and de Viron (2011) and conforms with the threefold VLBI SD in the prograde annual band (Fig. 1). We have also assessed the stability of the S\(_1\) peak in terms of its frequency through a Morlet wavelet analysis of demodulated filter residuals \(\tilde{H}'\). Deviations from the nominal S\(_1\) ridge at 365.26 days (solar days) are well within 5 days for all datasets except for MERRA, which exhibits a transition from 355 days in 1998 to 385 days in 2002 before leveling off exactly at the annual period (not shown). These minor fluctuations contrast with the 30 day range deduced by Dehant et al. (2003) on the basis of NCEP R1 data during 1958–1999. We surmise, however, that the estimate of Dehant et al. (2003) is less reliable due to the inclusion of reanalysis products prior to 1979, i.e., a period that lacks both satellite retrievals and a broad network of in situ pressure observations in the southern hemisphere.

Our nutation results for the minor solar constituents (\(\psi _1\), P\(_1\), \(\phi _1\), \(\pi _1\)) can be compared with estimates tabulated in Bizouard et al. (1998), Yseboodt et al. (2002), Brzeziński et al. (2004), or Koot and de Viron (2011). Here, we only point out that the harmonics fitted to the four atmospheric datasets agree well for P\(_1\) and \(\pi _1\) but differ substantially in the \(\psi _1\) band, which is of interest for studies of the FCN and Earth’s internal properties (Dehant and Defraigne 1997; Koot and de Viron 2011). Temporal variations of this tide can be large (\(>100\) \(\upmu \)as for individual models), and periods of its wavelet ridge vary within 30 days, probably as a reflection of a strong stochastic atmospheric influence. Nonetheless, a comparatively good inter-model agreement is found for the ip component of \(\psi _1\) (\(\sim \)50 \(\upmu \)as).

### 3.3 Validation of Pressure Tides against In Situ Data

The high-quality backbone of our compilation comprises 16 S\(_1\) estimates from Ray (1998a). We excluded nine solar determinations of this author (e.g., three Hawaiian sites, Ascension, or Tahiti) as they were inconsistent with nearby open-ocean estimates from smaller islands and buoys, presumably as a result of the latent and sensible heat flux over larger landmasses. 18 additional S\(_1\) determinations come from Schindelegger and Dobslaw (2016), who tidally analyzed hourly and 3-h synoptic sea level pressure observations from the ISD (Integrated Surface Database, Smith et al. 2011) to extract the much smaller lunar semidiurnal L\(_2\) tide. Again, care was exercised in selecting stations at sufficiently small islands and atolls. The final subset of estimates, also derived by Schindelegger and Dobslaw (2016), comprises 16 buoy locations that are part of the Tropical Moored Buoy System (McPhaden et al. 2010). Further densifications were attempted but led to clusters and subsequent biases in the statistics given below. The median time series length is 6 years, the maximum is 26 years (Bermuda), and short time spans of only 2 years occurred for eight sites, most of them being buoys.

In Fig. 5a (full compilation), ERA features the largest RMS misfits at about 9 Pa, accumulated through a significant overestimation of S\(_1\left( p\right) \) in the Tropics and an underestimation of the tidal amplitude elsewhere (Fig. 5d). These deficiencies can be expected to lead to a less reliable oceanic S\(_1\) tide. For equatorial Pacific stations, the second ECMWF dataset EC-OP also produces an excess in amplitude (by about 10 Pa) that maps into Fig. 5c but has no effect on the reduced network (Fig. 5d). Yet, median and normalized RMS differences are generally better for EC-OP than for the probed reanalyses and highlight the accuracy of the operational solution in a global domain. The CFSR statistics are comparable to those of MERRA and EC-OP, evidently unaffected by the southern hemispheric pressure anomalies (Sect. 3.2) as these regions are not sampled by our ground truth network.

S\(_1\left( p\right) \) differences of the tested analysis models with 50 station tide determinations, expressed as RMS misfits (Pa), median absolute differences (MAD, Pa), and median phase differences \(\Delta \varphi \) in the sense model-minus-station^{a}

RMS | MAD | \(\Delta \varphi \) | |
---|---|---|---|

MERRA | 5.2 | 5.1 | 6.1° |

CFSR | 5.9 | 5.9 | 6.1° |

ERA | 8.3 | 7.2 | −5.2° |

EC-OP | 6.5 | 5.9 | 2.5° |

A brief numerical comparison of the resulting model pressure tide climatologies against the 50-station set of S\(_1\) estimates is given in Table 6. Median absolute differences (MAD) are included as a more robust supplement to the averaged RMS misfits, underlining the reliability of all analysis models other than ERA. The general level of consistency with ground truth data (5–6 Pa) closely resembles values obtained by Ray and Ponte (2003), but note that these authors have additionally applied a small phase shift \(\Delta \varphi \) to their model estimates. Median phase lag differences for MERRA and CFSR in Table 6 generally confirm Ray and Ponte (2003)’s assumption of \(\Delta \varphi = 5^\circ \), and imposing the corresponding correction on the MERRA tide does indeed reduce the RMS misfit with station data to 4.9 Pa. We have, however, abstained from revising the tidal phases to avoid inconsistencies with the diurnal cycle in AAM.

## 4 Numerical Modeling of the Oceanic S\(_1\) Tide

### 4.1 Ocean Model Configuration

*f*is the Coriolis parameter oriented along the local vertical unit vector \(\hat{z}\),

*g*is the nominal gravitational acceleration, \(\nabla \) signifies the spherical del operator,

*H*is the resting water depth, \(\rho \) is the average density of seawater, \(C_D=0.003\) denotes a dimensionless drag coefficient in the standard expression for quadratic bottom friction, and \(C_{IT}\) is a location-dependent scalar (in units of m s\(^{-1}\)) to represent the drag due to tidal conversion. The forcing terms in the gradient operator of Eq. (5) comprise the gravitational equilibrium tide \(\zeta _{EQ}\), a combination of self-attraction/loading and “memory” elevations \(\zeta _{SAL}\) and \(\zeta _{MEM}\) to realize the SAL scheme of Arbic et al. (2004), as well as the atmospheric pressure tide \(P=\mathrm {S}_1\left( p\right) \). \(A_H \nabla \cdot \varvec{\sigma }\) is a comparatively rigorous implementation of the horizontal turbulent eddy viscosity with a second-order tensor \(\varvec{\sigma }\) related to the Reynolds stress tensor; see Einšpigel and Martinec (2015) for details. Here, we keep this term to eschew possible numerical instabilities in our medium-resolution runs (Egbert et al. 2004), with horizontal viscosity \(A_H\) set to the widely cited value of \(10^3\) m\(^2\) s\(^{-1}\). Our forward tidal solutions and OAM results are insensitive to the exact value of \(A_H\), unless it is used as a tuning parameter of inordinately large magnitude (\(\sim \)10\(^5\) m\(^2\) s\(^{-1}\)); cf. Arbic et al. (2004).

Equations (5) and (6) were solved by finite difference time-stepping on a \({1}/{3}^\circ \) C-grid, covering the latitude range from \(78^\circ \)S to \(78^\circ \)N with rigid walls assumed at the top and the bottom of the domain. This setting does not allow for accurate tidal modeling in the Weddell and Ross Sea, or in the Arctic Ocean. Yet, we readily accept such high-latitude limitations given our interest in the equatorial component of Earth’s rotation that has a peak sensitivity at \(45^\circ \) from the equator. The bottom topography was derived from the bedrock version of the fully global \(1' \times 1'\) ETOPO1 database (Amante and Eakins 2009) by choosing average values over each \({1}/{3}^\circ \) model grid cell and setting depths between 10-m and the 0-m land–sea boundary to 10 m. Coastlines in the Antarctic come from a recent data-assimilative ocean model (Taguchi et al. 2014) and are similar to those of Padman et al. (2002), with the cavities under the floating ice shelves considered as part of the ocean domain; cf. also Arbic et al. (2004) or Carrère et al. (2012). Blocking ice shelf areas as dry cells would lead to a noticeable increase of the tidal variability in southern hemisphere waters, also amplifying the oceanic contribution to the prograde annual nutation by roughly 10 \(\upmu \)as in both ip and op components. However, in these simulations, also the RMS misfit of the gravitationally forced constituents (M\(_2\) and O\(_1\); see below) to altimetry-based reference solutions, (FES2012, Carrère et al. 2012) increases, consistent with similar control runs by Wilmes and Green (2014). We thus proceed on the assumption that vertically displaceable ice shelves allow for a more realistic account of the tides, including S\(_1\).

Other aspects of the DEBOT configuration closely follow Ray and Egbert (2004). We prescribe equilibrium tidal forcing (\(\zeta _{EQ}\)) for M\(_2\) and O\(_1\) with amplitudes and solid Earth tide corrections taken from Table 1 of Arbic et al. (2004). The resulting larger-magnitude background variability appears to aid the fidelity with which S\(_1\) can be simulated in various basins and bays, but an extension to more than one diurnal and semidiurnal gravitational constituent is expendable as it alters our S\(_1\) OAM estimates by less than 3 %. Experiments with different equilibration periods (up to 90 days) showed that the spin-up time of the model could be reduced to 12 days with little effect (cf. Arbic et al. 2004), although in very shallow waters (Gulf of Thailand, Java Sea) the convergence of S\(_1\) takes considerably more time than that of any gravitational tide, presumably due to the vagaries of the pressure forcing near landmasses (Fig. 2). With 12 days reserved for equilibration, we integrated the model in each of our runs for 40 days at a time step of 24 s, harmonically analyzing the last 28 days to deduce the tidal constants of S\(_1\) (as well as M\(_2\) and O\(_1\)) in terms of sea level elevation and barotropic volume transports \(\varvec{u}H\).

### 4.2 Effects of Self-Attraction and Loading (SAL)

*a*is the Earth’s radius and \(\psi \) measures the angular separation of \(\left( \phi ,\lambda \right) \) from the load with spherical coordinates \(\left( \phi ',\lambda '\right) \). For our \({1}/{3}^\circ \) model, values of \(\mathcal {G}\left( \psi \right) \) were interpolated from the SAL kernel function tabulated in Stepanov and Hughes (2004). Explicit usage of Eq. (7) in the momentum equations is computationally unfeasible (Egbert et al. 2004), so alternative implementation schemes are required. To first order, the full convolution with \(\mathcal {G}\) is approximated by a simple scalar multiplication \(\zeta _{SAL} \approx \beta \zeta \) (Accad and Pekeris 1978), with \(\beta \) usually taken to be in the range of about 0.08 to 0.12. This widely used approximation is inappropriate for all locations in the ocean (Ray 1998b) and accurate tidal modeling necessitates a more rigorous handling of the effect. In tide models forced by a suite of individual constituents, the unparameterized formalism of Eq. (7) can be applied in a comparatively simple manner via iteration, that is, repeated model runs where each simulation employs a better approximated SAL term to gradually achieve convergence between the tidal elevations and \(\zeta _{SAL}\). We applied the iteration method of Arbic et al. (2004), initialized by the scalar SAL estimate using a nominal value of \(\beta =0.12\) that is an appropriate choice for diurnal tides; cf. Parke (1982) and Fig. 11 of Einšpigel and Martinec (2015). Once this initial run is completed and harmonically analyzed, the tidal components (sine and cosine terms) of M\(_2\), O\(_1\), and S\(_1\) are inserted into Eq. (7) in an intermediate offline computation to derive a first solution of \(\zeta _{SAL}\) for each tidal constituent. The following simulation then time steps the sum of all partial SAL tides as well as an additional memory term (Arbic et al. 2004)

Convergence of the iterative SAL scheme in terms of global angular momentum mass integrals of the S\(_1\) ocean tide^{a}

Scalar | 1\(\mathrm {st}\) iteration | 2\(\mathrm {nd}\) iteration | 3\(\mathrm {rd}\) iteration | Ray/Egbert | |
---|---|---|---|---|---|

| 1.83 (164\(^\circ \)) | 1.88 (168\(^\circ \)) | 1.86 (167\(^\circ \)) | 1.87 (167\(^\circ \)) | 0.82 (158\(^\circ \)) |

| 2.90 (287\(^\circ \)) | 3.10 (283\(^\circ \)) | 3.10 (283\(^\circ \)) | 3.10 (283\(^\circ \)) | 2.90 (306\(^\circ \)) |

| 1.72 (188\(^\circ \)) | 2.00 (181\(^\circ \)) | 2.03 (183\(^\circ \)) | 2.05 (183\(^\circ \)) | 2.42 (219\(^\circ \)) |

The correction term in Eq. (8) guarantees rapid convergence of the SAL scheme, as exemplified by diminishing RMS discrepancies of the gravitational constituents against FES2012 tides in successive simulations. Specifically, with the choice of \(\beta \) optimized for the diurnal band, our forward solutions of O\(_1\) remain effectively unchanged after the first iteration, whereas sufficient accuracy for semidiurnal tides is reached after three iterations. More to the point, rapid equilibration of tidal dynamics is also observed for the radiational S\(_1\) tide. Table 7 presents successively updated OAM mass values of S\(_1\) as obtained from a three-times iterative DEBOT run with the pressure forcing S\(_1\left( p\right) \) taken from Ray and Egbert (2004) and IT drag (cf. next section) switched off. For equatorial components in particular, the scalar approximation appears to provide reasonably accurate initial OAM estimates, deviating by no more than 5\(^\circ \) in phase and less than 10 % in amplitude from the (arguably) self-consistent third iteration. Yet, the scalar SAL relation is inadequate for both the axial OAM component and the comparison of simulated S\(_1\) surface elevations to coastal tide gauges (Sect. 4.4). Accordingly, results from all of our forward runs presented below have been inferred after completing the second model iteration.

### 4.3 Internal Tide (IT) Drag Scheme

Consistent with previous studies of forward-modeled barotropic tides (Jayne and St Laurent 2001; Arbic et al. 2004; Egbert et al. 2004), surface elevations and tidal energies are poorly represented in DEBOT unless allowance is made for the substantial amount of drag generated by internal tides over major bathymetric features. With this proper dissipation mechanism omitted, area-weighted RMS differences \(\overline{\Delta \zeta }\) ^{1} to the FES2012 reference tide \({\tilde{\zeta }}_R\) (complex sinusoid) are as large as 14.2 cm and 3.0 cm for M\(_2\) and O\(_1\), respectively. These values translate to a mere 72 and 79 % of sea surface height variance explained. Moreover, S\(_1\) charts deduced from IT-free simulations display a number of apparent regional artifacts, such as persistently high amplitudes of the tide in the northern Atlantic (\(\sim \)1 cm) or the South China Sea (\(\sim \)2 cm) that have no correspondence in both the altimetric and hydrodynamic S\(_1\) solutions of Ray and Egbert (2004). Parts of the OAM discrepancy of our initial control run (Table 7) to Ray and Egbert (2004)’s benchmark values can be understood in this light.

*N*follow from the prescription of a horizontally uniform abyssal stratification. Values of

*N*at the ocean bottom (\(N_b\)) as well as vertical averages (\(\overline{N}\)) over the entire water column are calculated from Green and Nycander (2013)

*H*is the resting water depth (in m). Equation (9) is similar in form to the drag coefficient of Jayne and St Laurent (2001) and likewise ignores the influence of critical turning latitudes (where \(\omega = f\), the Coriolis parameter) on the internal wave propagation characteristics. Yet, through scaling by \(\omega \), the scheme is still frequency-dependent and thus applicable to only one specified constituent or, less strictly, to a particular tidal species. Considering our emphasis on the diurnal band, we fixed \(\omega \) to \(\Omega \), though that choice was found to improve the elevation accuracy of semidiurnal fringes (M\(_2\)) as well.

Global angular momentum integrals of the S\(_1\) ocean tide deduced from numerical modeling with varying pressure forcing climatologies^{a}

Ray/Egbert | Control | MERRA | CFSR | ERA | EC-OP | |
---|---|---|---|---|---|---|

| ||||||

| 0.82 (158\(^\circ \)) | 1.01 (163\(^\circ \)) | 0.71 (199\(^\circ \)) | 1.48 (149\(^\circ \)) | 1.62 (161\(^\circ \)) | 1.23 (163\(^\circ \)) |

| 2.90 (306\(^\circ \)) | 2.83 (298\(^\circ \)) | 3.23 (321\(^\circ \)) | 3.50 (319\(^\circ \)) | 2.04 (295\(^\circ \)) | 2.88 (304\(^\circ \)) |

| 2.42 (219\(^\circ \)) | 2.29 (207\(^\circ \)) | 3.00 (239\(^\circ \)) | 3.44 (248\(^\circ \)) | 2.02 (234\(^\circ \)) | 2.23 (218\(^\circ \)) |

| ||||||

| 1.72 (14\(^\circ \)) | 1.60 (3\(^\circ \)) | 1.84 (12\(^\circ \)) | 1.46 (6\(^\circ \)) | 1.26 (312\(^\circ \)) | 1.54 (342\(^\circ \)) |

| 1.62 (226\(^\circ \)) | 1.53 (218\(^\circ \)) | 1.69 (222\(^\circ \)) | 1.18 (209\(^\circ \)) | 1.58 (176\(^\circ \)) | 1.76 (206\(^\circ \)) |

| 2.57 (271\(^\circ \)) | 2.65 (279\(^\circ \)) | 1.66 (279\(^\circ \)) | 2.22 (262\(^\circ \)) | 3.43 (288\(^\circ \)) | 2.71 (291\(^\circ \)) |

S\(_1\) amplitude and phase charts in our updated control runs with IT drag included are nearly indistinguishable from Fig. 3 of Ray and Egbert (2004), with previously noted regional anomalies (South China Sea, North Atlantic) eliminated (not shown). Table 8 underlines this sound agreement on the level of OAM values; cf. also the marked improvement with respect to our original, drag-free solution in Table 7. For both mass and motion components, phases differ by less than 15\(^\circ \) throughout and deviations in amplitude are within \(0.2\cdot 10^{23}\) kg m\(^2\) s\(^{-1}\). We calculated the corresponding contributions to the prograde annual nutation by aid of a standard protocol noted below, obtaining \(21.9+i46.4\) \(\upmu \)as as a credible reproduction of the S\(_1\) excitation value \(20.7+i54.8\) \(\upmu \)as implied by the OAM terms of Ray and Egbert (2004); see Table 10. A moderate underestimation of the op component in DEBOT (\(\sim \)8 \(\upmu \)as) likely relates to differences in bathymetry or the treatment of ice shelves. Note also that the IT drag has no correspondence in the shallow water dynamics formulated by Ray and Egbert (2004), although a pertinent parameterization of tidal conversion, incorporated to the very same numerical model by Egbert et al. (2004), might have gone unmentioned.

Whether our prescription of topographically generated drag at the S\(_1\) frequency is physically justified or not is a potentially interesting issue but not of immediate importance for the topic in hand. One of the vexing problems related to this question is that our forward model operates in the time domain, while internal waves are preferably studied in the frequency domain. Here, we have adopted a diagnostic approach, inferring the need for additional mid-ocean dissipation by comparing our initial results for S\(_1\) and other diurnal tides to established reference charts. On a side note, also the discrepancies to coastal tide gauge estimates of S\(_1\) (next section) are markedly lower when internal wave drag is parameterized.

### 4.4 Hydrodynamic Solutions and Validation with Tide Gauge Data

All computed S\(_1\) realizations agree in terms of the global character of the tide, but basin-wide features can vary substantially in response to different pressure forcing data. Specifically, the secondary peaks of S\(_1\left( p\right) \) around 60\(^\circ \)S in the CFSR climatology (Fig. 2) induce sea level signals in the Southern Ocean that exceed the corresponding tidal variability from MERRA and EC-OP by about 1–5 mm. Amplitudes in the North Atlantic are also comparatively high in the CFSR solution, whereas EC-OP displays the largest S\(_1\) tide in the Tropical Pacific, consistent with the overestimation of equatorial pressure gradients as exposed by Fig. 5c.

Empirical knowledge of S\(_1\) to validate our forward simulations comes from globally distributed coastal tide gauges. Such a point-wise verification may imply little with regard to Earth rotation, yet it is instructive to determine whether individual models systematically perform better than others. Harmonic estimates of S\(_1\) at some 200 places are available in the online datasets of Ponchaut et al. (2001)^{2}, who tidally analyzed multi-year time series of hourly sea level records assembled both by BODC (British Oceanographic Data Centre) and UHSLC (University of Hawaii Sea Level Center). We extracted a subset of 51 estimates from Ponchaut’s compilation, excluding sites where the tide is effectively zero (Hawaii, Japan, Maldives, Central Atlantic) or all model predictions are equivocally different from the observations, e.g., due to unresolved coastal geometries (Prudhoe Bay, Bluff Harbour). Moreover, in order to avoid biases toward densely sampled regions, only a few locations were retained in the equatorial Pacific and further thinning was applied to higher-amplitude S\(_1\) estimates in close proximity to each other (Arabian Sea, Gulf of Alaska). Our final 56-station set also includes five tide gauges from Ray and Egbert (2004) (Karachi, Benoa, Broome, Bermuda, Gibraltar) and is presented in Fig. 9. Valuable additions to this network, e.g., in the seas of Southeast Asia or along the coast of Brazil were pinpointed in the PSMSL (Permanent Service for Mean Sea Level) holdings, yet we refrained from a thorough tidal analysis of these hourly data in the frame of the present work.

_{1}Ocean Tide” at the locations of our 56 gauges and changed the phase reference from the tide-generating potential to the simple radiational S\(_1\) argument; see Ray and Egbert (2004) for details. Tidal components after subtraction of the gravitational signal (usually 1–3 mm) are displayed as phasors in Fig. 9 and cover an amplitude range from 56 mm at Darwin (10-year mean estimate) down to 1.3 mm at St. Helena (4-year mean). Confidence intervals for all small-magnitude S\(_1\) determinations from only a few years of data appear to be sufficiently tight in the analysis of Ponchaut et al. (2001) to warrant the inclusion of these stations in our network. The median time series length over all 56 tide gauges is 8 years.

Simulated S\(_1\) signals at each gauge location were taken from the nearest pelagic point in our \({1}/{3}^\circ \) model and are illustrated for MERRA, CFSR, and EC-OP in Fig. 9. In general, all hydrodynamic solutions agree reasonably well with the observations, even though disparities on the order of a few mm must be accepted at most sites. The unusually large tide at Darwin (56.0 mm after reduction of the gravitational signal) has been addressed by Ray and Egbert (2004) and appears to be a very local modulation of S\(_1\) in a shallow (5-m) inlet that is approximated by a coarse gridpoint of 10 m depth in our bathymetry. MERRA and CFSR amplitudes at Darwin are 35 mm and 37 mm and thus somewhat closer to the observation than the model estimate of Ray and Egbert (2004). In a broader context, the collection of tide gauges across the Atlantic testify to the shortcomings of the CFSR solution, evident, e.g., from the amplitude excess at Puerto Deseado, Port Stanley, and Esperanza. Moreover, most of the EC-OP estimates in the equatorial Pacific are too high, implying that this model must be treated with caution for studies of axial changes in Earth’s rotation.

All stations | Amplitude \(<8\) mm | |||
---|---|---|---|---|

RMS | MAD | RMS | MAD | |

MERRA | 2.6 (0.34) | 2.1 | 1.6 (0.38) | 1.3 |

CFSR | 2.8 (0.40) | 2.2 | 1.9 (0.49) | 1.9 |

ERA | 2.7 (0.35) | 1.9 | 1.9 (0.43) | 1.5 |

EC-OP | 2.8 (0.37) | 2.2 | 1.9 (0.45) | 1.6 |

### 4.5 Contribution of the Oceanic S\(_1\) Tide to Nutation

*x*and

*y*mass terms, i.e., the two single most important components with regard to nutation, are in the order of \(\sim \)1.0 \(10^{23}\) kg m\(^2\) s\(^{-1}\) (160\(^\circ \) phase lag) and \(\sim \)3.0 \(10^{23}\) kg m\(^2\) s\(^{-1}\) (0\(^\circ \) phase lag), respectively. The tabulated harmonics were translated to nutation values in essentially the same manner as AAM through multiplication of mass and motion terms with the proper transfer function coefficients \(\tilde{T}^{p,w}\left( \sigma \right) \); cf. Eqs. (1) and (2). As this scheme is initialized by a demodulation of angular momentum series in the time domain, we first discretized

*x*and

*y*OAM components over a 3-year window by a cosine function using the Doodson argument for the radiational S\(_1\) tide (see Appendix A of Ray and Egbert 2004)

*T*is Universal Time and \(\varphi _H\) are respective phase lag values as given in Table 8. Demodulated equatorial OAM series were then cleansed from non-seasonal signals (that is, the S\(_1\) contribution to prograde polar motion), fitted to the periodic forcing model (Eq. 4), and expressed as nutation harmonics through Eq. (1) with phases referred to the fundamental arguments of gravitational diurnal tides.

Periodic oceanic contributions to the prograde annual nutation (\(\upmu \)as) as inferred from the model-specific OAM values of Table 8 ^{a}

Mass | Motion | Total | ||||
---|---|---|---|---|---|---|

ip | op | ip | op | ip | op | |

Ray/Egbert | 26.2 | 54.7 | \(-5.5\) | 0.1 | 20.7 | 54.8 |

Control | 27.1 | 45.4 | \(-5.2\) | 1.0 | 21.9 | 46.4 |

MERRA | 19.7 | 59.8 | \(-5.8\) | 0.4 | 13.9 | 60.2 |

MERRA | 27.9 | 65.7 | \(-6.4\) | 0.3 | 21.5 | 66.0 |

CFSR | 6.1 | 83.9 | \(-4.1\) | 0.8 | 2.0 | 84.7 |

ERA | −0.1 | 34.1 | \(-2.5\) | 4.3 | −2.6 | 38.4 |

EC-OP | 18.0 | 51.5 | \(-5.0\) | 2.8 | 13.0 | 54.3 |

Table 10 summarizes the various estimates of the oceanic S\(_1\) effect in nutation. Formal errors have been omitted as they are effectively zero given our usage of perfect sinusoids for the angular momentum time series. Overall, the nutation results from all model runs are reasonably consistent, ranging from 0 to 20 \(\upmu \)as in the ip terms and roughly 40 to 60 \(\upmu \)as for the op component, with 90 % of the signal coming from the mass component. MERRA (2004–2013 average) and EC-OP produce a particularly close match within 6 \(\upmu \)as, and the moderate increase in magnitude for the reduced MERRA time span (2004–2010) is in fact expected on grounds of the time-variable amplitudes of S\(_1\left( p\right) \) (Fig. 5). For CFSR, the large-scale enhancement of tidal heights in the Indian Ocean and the North Atlantic (Fig. 7) combine to yield an excessive op estimate of 84.7 \(\upmu \)as. Nonetheless, the spread of nutation values is significantly smaller than that of previous inter-model comparisons, conducted, e.g., by Brzeziński (2008) based on IB-corrected OAM values from much coarser (>1°) barotropic and baroclinic models. We have also mapped the fine-resolution S\(_1\) tide of FES2012 to the prograde annual band, finding a nutation estimate of \(-2.1+i49.1\) \(\upmu \)as that roughly matches our ERA harmonic. This agreement likely relates to similarities in the barometric forcing data, as the hydrodynamic core of FES2012 includes pressure loading from the 3-h ECMWF delayed cutoff stream (Carrère et al. 2012). Prograde annual nutation estimates for FES2012 as well as the model of Ray and Egbert (2004) are also tabulated in Schindelegger et al. (2015), albeit with an internal conversion error at the order of 10 \(\upmu \)as which has been corrected in the frame of the present study.

## 5 Comparison with Geodetic Observations

At an amplitude of \(\sim \)25.6 mas, the prograde annual nutation is among the principal signal components in Earth’s celestial motion and driven almost exclusively by the action of the solar gravitational torque on the equatorial bulge. In the MHB theory, the term is modulated to a minor degree by anelasticity (\(-10-i4\) \(\upmu \)as), electromagnetic torques (\(-14+i6\) \(\upmu \)as for both core mantle and inner core boundaries), geodesic nutation (\(-30+i0\) \(\upmu \)as), and the angular momentum exchange of the solid Earth with the gravitational ocean tide, S\(_1^g\) (\(-21+i22\) \(\upmu \)as); see also Table 2 of Brzeziński et al. (2004). With these contributions accounted for, theory and observation of the prograde annual nutation produce a mismatch of \(-10.4+i108.2\) \(\upmu \)as that has been attributed by MHB to the thermal atmospheric S\(_1\) tide and, implicitly, to the radiational S\(_1\) tide in the ocean.

Realizations of the very same residual have been also derived by Koot et al. (2010) in the frame of a time domain Bayesian inversion of nutation observations including non-linearities and additional terms in the functional model. Koot et al. (2010) used 10 years of additional VLBI data compared to MHB but employed identical corrections for geodesic nutation and the gravitational ocean tide. It is thus not surprising that the empirical S\(_1\) estimate of these authors is numerically very similar to the MHB residual; from a joint inversion of three nutation series from different analysis centers Koot et al. (2010) deduced a harmonic of \(0+i107\) \(\upmu \)as. Corresponding SD in both ip and op components are 4 \(\upmu \)as but probably underestimated and arguably better represented by the single-solution error of 7 \(\upmu \)as; cf. Table 1 of Herring et al. (2002).

_{1}Ocean Tide” (Table 12), applying essentially the same time domain discretization as in Sect. 4.5 but with phases referred to the present-day argument of the gravitational S\(_1\) tide, that is, \(T+295.66^\circ +90^\circ \) (Ray and Egbert 2004). Multiplication of adjusted mass and motion term coefficients (Eq. 4) with the respective transfer ratios (Eq. 1) yielded a harmonic of \(-15.2+i16.8\) \(\upmu \)as. This value is about 5 \(\upmu \)as smaller than the intrinsic MHB estimate in both ip and op components, and a similar decrement is assumed for the analysis of Koot et al. (2010), who also utilized OAM data of Chao et al. (1996). The corresponding correction was imposed on the prograde annual nutation residuals of both studies, resulting in the empirical S\(_1\) terms given in Table 11.

Estimates of the prograde annual nutation (\(\upmu \)as) as driven by the global radiational S\(_1\) tide in the coupled atmosphere–ocean system^{a}

Additional regard must be paid to the distortion of observed nutations through Sun-synchronous thermal deformations of some or all VLBI telescopes (Herring et al. 1991). This effect is now rigorously accounted for in VLBI analyses by means of a conventional procedure using on-site values of temperature, but the matter of discussion is whether a proper deformation correction was employed in the computation of nutation series that underlie the studies of MHB as well as Koot et al. (2010). Here, we draw on different evidences to argue that the effect was sufficiently well modeled, e.g., by early reduction schemes similar to Sovers et al. (1998) (Sect. G, *ibid.*).

Single-session runs with our in-house VLBI software (Böhm et al. 2012) showed that a spurious prograde annual nutation variability of about 20 \(\upmu \)as in both ip and op components is incurred by analyses that explicitly omit corrections for solar heating of VLBI antennas. Hence, a persistent bias of \(\sim \)30 \(\upmu \)as should be evident in the comparison of nutation series from present-day VLBI analyses with the MHB model, assuming that for the latter diurnal deformation signals were neglected. This comparison is actually realized in the form of the IERS CPO data given w.r.t. the MHB series, of which a windowed Fourier analysis in the prograde annual band has already been presented in Fig. 1. The absence of any systematic distortion at the order of 30 \(\upmu \)as is readily apparent, in particular in the post-2000 period that features little uncertainty in the CPO estimates. Moreover, Table 3 of Koot et al. (2010) itself implies a proper modeling of solar heating in previous VLBI solutions. Among the three nutation series inverted by these authors, a thermal deformation correction is unambiguously identified for the IAA (Institute of Applied Astronomy, Moscow) data; see the corresponding documentation available at ftp://ivsopar.obspm.fr/vlbi/ivsproducts/eops/ (accessed 14 October 2015). The treatment of the heating effect is undisclosed in the description of the other two series but, reassuringly, the associated prograde annual nutation residuals are not systematically offset from the IAA solution. Note also that the MHB estimate blends in well with Koot et al. (2010)’s results for various analysis centers. Artificial nutation signals related to antenna structure changes can be therefore deemed insignificant and the collected VLBI-based empirical S\(_1\) terms should be accurate enough to serve as reference values for our geophysical model estimates.

Table 11 also places our final results in the context of previous geophysical modeling efforts by Brzeziński et al. (2004) and Brzeziński (2011). In these studies, a balance with VLBI observations has been mostly impeded by deficiencies in the ip component of the modeled S\(_1\) excitation. To some extent, the choice of meteorological data (NCEP R1, ERA) is critical in either investigation, and further errors relate to insufficiencies of the utilized ocean models regarding the simulation of the S\(_1\) tide. Specifically, the model deployed by Brzeziński (2011) has been optimized for a range of timescales and full baroclinic variability, for which coarse horizontal resolutions (1.875\(^\circ \)) greatly reduce computational costs. We tested the impact of a 1\(^\circ \) discretization in DEBOT, obtaining somewhat anomalous S\(_1\) charts and increasingly negative ip components of the ocean-driven prograde annual nutation (\(-15\) \(\upmu \)as decrement). Brzeziński et al. (2004) analyzed the output of a barotropic model with a comparably coarse domain representation (1.125\(^\circ \) spacing) but also with SAL dynamics neglected. This omission alters nutation amplitudes by roughly 30 \(\upmu \)as, and perturbations of similar size occur if dissipative processes are imperfectly accounted for. Such shortcomings have been redressed in the present work, by drawing on modern insights into the forward modeling of global ocean tides.

## 6 Concluding Discussion

S\(_1\) tidal excitations of nutation in the order of 3.5 mm at the Earth’s surface (\(\sim \)120 \(\upmu \)as) have constituted an anomaly to non-rigid nutation theories for decades. We have put forth an explanation of geodetic observations of the effect based on reanalysis data from MERRA and operational ECMWF analysis fields, complemented by numerical hydrodynamic solutions for the radiational S\(_1\) tide in the ocean. Atmospheric contributions averaged over 2004–2013 are \(-21.9+i45.8\) \(\upmu \)as (MERRA) and \(-22.4+i67.5\) \(\upmu \)as (EC-OP) and combine well with the respective oceanic estimates (\(13.9+i60.2\) \(\upmu \)as, \(13.0+i54.3\) \(\upmu \)as) to match the VLBI-observed S\(_1\) terms within 10 \(\upmu \)as. No attempt was made to rigorously quantify the uncertainty of these geophysical model estimates, but we suppose that the errors are comparable to the threefold VLBI SD in the prograde annual band (21 \(\upmu \)as). In particular, the atmospheric mass term is among the least robust components of the global S\(_1\) excitation given its dependence on the small second-order tesseral surface pressure wave. Table 5 documents an inter-model spread of 25 \(\upmu \)as (excepting CFSR) for the pressure-driven nutation, comprising also uncertainties due to inter-annual S\(_1\) variations that have not been completely removed by the chosen 10-year averaging window. In contrast, our forward simulations of the radiational ocean tide should be fairly reliable on condition that the barometric forcing data themselves are accurate. Only weak (\(<10\) \(\upmu \)as) and possibly counterbalancing influences of bathymetry and drag parameterization have been noted in Sect. 4. We also conducted DEBOT runs on C-grids finer than \({1}/{3}^\circ \), obtaining nutation harmonics of only a few \(\upmu \)as deviation with respect to the estimates given in Table 10.

Differences in the diurnal cycle of modern atmospheric assimilation systems have played one of the recurring themes throughout this paper and are not necessarily smaller than the high-frequency disparities among earlier generation reanalyses; recall, e.g., our assessment of the CFSR pressure data. A more coherent representation of air tides is evidently tied to a near-global observing system with continuous sub-daily sampling (Schindelegger and Dobslaw 2016) but also depends on other aspects of the (re)analysis framework. Poli et al. (2013) emphasized the importance of at least hourly radiation time steps—a condition that is met neither by CFSR nor by JRA-55 (Kobayashi et al. 2015), which was also examined in a preliminary stage of our study but led to deficient AAM/OAM phasors. Moreover, the formulation of the assimilation technique can have implications for tides, considering in particular that the variational analysis (3DVar or 4DVar; see Table 1) is usually performed in sequential 12-h windows without accounting for continuity of state variables at the transition epochs. The resulting perturbations occur at integer fractions of a solar day and potentially fold to an artificial S\(_1\)/S\(_2\) variability. Such spurious signals are, however, minimized in the special case of MERRA through its Incremental Analysis Update method (Rienecker et al. 2008), which might ultimately figure into the good performance of MERRA throughout our study. Finally, the accuracy of S\(_1\) in global analysis models is closely linked to the fidelity with which moist convection and latent heat flux can be simulated. Deficiencies in these quantities relate to imperfect physical paramaterizations or uncorrected biases in observations (Meynadier et al. 2010) and are, e.g., well documented for ERA (Dee et al. 2015).

Displaying little long-term variability both in the celestial pole offsets (Fig. 1) and in the atmospheric S\(_1\) excitation, the 2004–2013 period has provided the ideal setting to study the mean harmonic atmosphere–ocean contribution to the prograde annual nutation. A reliable estimation of the temporal evolution of nutation amplitudes is still challenging, though. These signals differ substantially among the probed models and are masked by noise interferences as well as spurious variabilities when the frozen assimilation routines of reanalyses are confronted with new types and volumes of observations. Judging from Figs. 1, 3, and similar analyses in Bizouard et al. (1998), an upper bound of 30 \(\upmu \)as appears to be a plausible estimate for the irregular departures from a simple sinusoidal S\(_1\) term in nutation. These vacillations dictate the likely accuracy of upcoming nutation models but also serve as an incentive for future foundational research, relating climate signals in geodetic observations with the time-variable excitation quantities from geophysical fluid models.

## Footnotes

- 1.
Computed as \(\overline{\Delta \zeta } = \sqrt{\frac{\iint \left| \tilde{\zeta } - \tilde{\zeta }_R\right| ^2 dA}{2\iint dA}}\), equivalent to the time-averaged expression of Arbic et al. (2004), with grid points poleward of 66\(^\circ \) and waters shallower than 1000 m excluded.

- 2.
http://www.bodc.ac.uk/projects/international/woce/tidal_constants/. Accessed 9 October 2015.

## Notes

### Acknowledgments

Open access funding provided by Austrian Science Fund (FWF). We are indebted to R. Ray for supplying various datasets and patiently answering questions about tides and ocean modeling. Comments from A. Brzeziński, H. Dobslaw, and M. Green are acknowledged, and we also thank C. Mayerhofer for coding some of the basic routines used in this work. The analyzed meteorological data were provided by the ECMWF, NASA’s GMAO, and the Research Data Archive (RDA) of NCAR. FES2012 was produced by Noveltis, Legos and CLS Space Oceanography Division and distributed by Aviso, with support from Cnes (http://www.aviso.altimetry.fr/). We greatly appreciate the Austrian Science Fund for financial support within project I1479-N29. D. Salstein is sponsored in part by Grant ATM-0913780 from the US National Science Foundation (NSF), and D. Einšpigel thanks the Charles University in Prague for supporting him under Grant SVV-2015-260218.

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