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The Global S\(_1\) Tide in Earth's Nutation
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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.
KeywordsEarth 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 agenciesa
Model vintage
Resolution (km)
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
Pressure level
MERRA, assimilated state
\(1.25^{\circ }\)
\(1.25^{\circ }\)
CFSR, analysis & 3-h forecast
\(0.5^{\circ }\)
\(2.5^{\circ }\)
ERA-Interim, analysis
\(0.5^{\circ }\)
\(2.0^{\circ }\)
ECMWF, operational analysis
\(1.0^{\circ }\)
\(1.0^{\circ }\)
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.
3 Atmospheric Contributions to Nutation
3.1 Implementation
\({\tilde{\sigma}} \) (cpsd)
\({\tilde{N}}^p \times 10^3\)
\({\tilde{N}}^w \times 10^3\)
Multipliers of the fundamental arguments of nutation terms that apply to atmospheric tidal linesa
Fundamental arguments
\(\Omega \)
(solar days)
\(\psi _1\)
\(\phi _1\)
\(\pi _1\)
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 2004–2013a
S\(_1\) (\(+1\) year)
\(-25.9\pm 0.4\)
\(18.1\pm 0.4\)
\(4.0\pm 0.2\)
\(27.6\pm 0.2\)
\(-21.9\pm 0.5\)
\(45.8\pm 0.5\)
\(-30.8\pm 0.5\)
\(23.0\pm 0.5\)
\(5.0\pm 0.2\)
\(26.6\pm 0.2\)
\(-25.8\pm 0.6\)
\(49.7\pm 0.6\)
\(-32.3\pm 0.6\)
\(57.5\pm 0.6\)
\(-2.0\pm 0.2\)
\(30.0\pm 0.2\)
\(-34.3\pm 0.6\)
\(87.6\pm 0.6\)
\(-31.8\pm 0.5\)
\(31.7\pm 0.5\)
\(-4.4\pm 0.2\)
\(25.8\pm 0.2\)
\(-36.1\pm 0.5\)
\(57.5\pm 0.5\)
\(-21.9\pm 0.5\)
\(43.8\pm 0.5\)
\(-0.5\pm 0.2\)
\(23.6\pm 0.2\)
\(-22.4\pm 0.5\)
\(67.5\pm 0.5\)
\(\psi _1\) (\(-1\) year)
\(-30.1\pm 5.9\)
\(-44.5\pm 5.9\)
\(-2.5\pm 0.3\)
\(-3.0\pm 0.3\)
\(-32.6\pm 5.9\)
\(-47.5\pm 5.9\)
\(-45.1\pm 6.5\)
\(-48.4\pm 6.5\)
\(-2.7\pm 0.4\)
\(-3.6\pm 0.4\)
\(-47.9\pm 6.5\)
\(-52.0\pm 6.5\)
\(-58.2\pm 7.3\)
\(0.4\pm 7.3\)
\(0.6\pm 0.3\)
\(-3.2\pm 0.3\)
\(-57.5\pm 7.3\)
\(-2.9\pm 7.3\)
\(-79.3\pm 6.1\)
\(22.0\pm 6.1\)
\(-4.0\pm 0.3\)
\(-0.7\pm 0.3\)
\(-83.3\pm 6.1\)
\(21.3\pm 6.1\)
\(-50.8\pm 6.2\)
\(23.6\pm 6.2\)
\(-3.1\pm 0.3\)
\(0.9\pm 0.3\)
\(-54.0\pm 6.2\)
\(24.5\pm 6.2\)
P\(_1\) (\(+{1}/{2}\) year)
\(-10.1\pm 0.3\)
\(0.1\pm 0.3\)
\( 7.1\pm 0.2\)
\(42.1\pm 0.2\)
\(-3.0\pm 0.3\)
\(42.2\pm 0.3\)
\(-10.3\pm 0.3\)
\(1.2\pm 0.3\)
\(7.1\pm 0.3\)
\(41.1\pm 0.3\)
\(-3.2\pm 0.4\)
\(42.3\pm 0.4\)
\( -6.9\pm 0.3\)
\(7.0\pm 0.3\)
\( 5.1\pm 0.2\)
\(35.6\pm 0.2\)
\(-1.9\pm 0.4\)
\(42.6\pm 0.4\)
\(-17.3\pm 0.3\)
\(0.9\pm 0.3\)
\(13.5\pm 0.2\)
\(45.8\pm 0.2\)
\(-3.8\pm 0.3\)
\(46.7\pm 0.3\)
\(-17.0\pm 0.3\)
\(2.5\pm 0.3\)
\(11.2\pm 0.2\)
\(41.5\pm 0.2\)
\(-5.8\pm 0.3\)
\(44.1\pm 0.3\)
\(\phi _1\) (\(-{1}/{2}\) year)
\( -2.1\pm 0.8\)
\( -6.1\pm 0.8\)
\( 0.9\pm 0.2\)
\(-3.0\pm 0.2\)
\( -1.2\pm 0.8\)
\( -9.1\pm 0.8\)
\(-1.8\pm 0.9\)
\(-5.9\pm 0.9\)
\(0.7\pm 0.3\)
\(-3.3\pm 0.3\)
\(-1.1\pm 0.9\)
\(-9.3\pm 0.9\)
\(-17.3\pm 1.0\)
\( 1.9\pm 1.0\)
\(-2.1\pm 0.2\)
\(-1.4\pm 0.2\)
\(-19.3\pm 1.0\)
\( 0.6\pm 1.0\)
\(-12.2\pm 0.8\)
\( -8.9\pm 0.8\)
\(-2.5\pm 0.2\)
\(-4.9\pm 0.2\)
\(-14.7\pm 0.8\)
\(-13.9\pm 0.8\)
\( 1.8\pm 0.8\)
\(-13.0\pm 0.8\)
\( 0.7\pm 0.2\)
\(-5.7\pm 0.2\)
\( 2.5\pm 0.9\)
\(-18.7\pm 0.9\)
\(\pi _1\) (\(+{1}/{3}\) year)
\(-2.6\pm 0.2\)
\(4.1\pm 0.2\)
\(1.9\pm 0.2\)
\(-1.4\pm 0.2\)
\(-0.7\pm 0.3\)
\(2.6\pm 0.3\)
\(-2.3\pm 0.2\)
\(4.0\pm 0.2\)
\(2.3\pm 0.3\)
\(-1.6\pm 0.3\)
\(0.0\pm 0.3\)
\(2.4\pm 0.3\)
\(-2.1\pm 0.2\)
\(3.3\pm 0.2\)
\(2.1\pm 0.2\)
\(-0.7\pm 0.2\)
\(-0.1\pm 0.3\)
\(2.6\pm 0.3\)
\(-2.5\pm 0.2\)
\(2.8\pm 0.2\)
\(1.1\pm 0.2\)
\( 0.3\pm 0.2\)
\(-1.4\pm 0.3\)
\(3.0\pm 0.3\)
\(-2.2\pm 0.2\)
\(2.7\pm 0.2\)
\(1.1\pm 0.2\)
\(-0.1\pm 0.2\)
\(-1.1\pm 0.3\)
\(2.7\pm 0.3\)
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-stationa
\(\Delta \varphi \)
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
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)
Convergence of the iterative SAL scheme in terms of global angular momentum mass integrals of the S\(_1\) ocean tidea
1\(\mathrm {st}\) iteration
2\(\mathrm {nd}\) iteration
3\(\mathrm {rd}\) iteration
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.
Global angular momentum integrals of the S\(_1\) ocean tide deduced from numerical modeling with varying pressure forcing climatologiesa
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.
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 \(<8\) mma
2.6 (0.34)
1.6 (0.38)
2.8 (0.40)
1.9 (0.49)
2.7 (0.35)
1.9 (0.43)
2.8 (0.37)
1.9 (0.45)
4.5 Contribution of the Oceanic S\(_1\) Tide to Nutation
Periodic oceanic contributions to the prograde annual nutation (\(\upmu \)as) as inferred from the model-specific OAM values of Table 8a
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).
Estimates of the prograde annual nutation (\(\upmu \)as) as driven by the global radiational S\(_1\) tide in the coupled atmosphere–ocean systema
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.
- 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.
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.
- Accad Y, Pekeris CL (1978) Solution of the tidal equations for the M\(_2\) and S\(_2\) tides in the world oceans from a knowledge of the tidal potential alone. Philos Trans R Soc London 290:235–266CrossRefGoogle Scholar
- Amante C, Eakins BW (2009) ETOPO1 1 arc-minute global relief model: procedures, data sources and analysis. NOAA Technical Memorandum NESDIS NGDC-24, National Geophysical Data Center, NOAA. doi:10.7289/V5C8276M
- Arbic BK, Garner ST, Hallberg RW, Simmons HL (2004) The accuracy of surface elevations in forward global barotropic and baroclinic tide models. Deep Sea Res II 51:3069–3101CrossRefGoogle Scholar
- Bell TH (1975) Topographically generated internal waves in the open ocean. J Geophys Res Oceans 80:320–327. doi:10.1029/JC080i003p00320 CrossRefGoogle Scholar
- Bizouard C, Brzeziński A, Petrov S (1998) Diurnal atmospheric forcing and temporal variations of the nutation amplitudes. J Geod 72:561–577CrossRefGoogle Scholar
- Bizouard C, Lambert S (2002) Lunisolar torque on the atmosphere and Earth's rotation. Planet Space Sci 50:323–333. doi:10.1016/S0032-0633(01)00120-9 CrossRefGoogle Scholar
- Böhm J, Böhm S, Nilsson T, Pany L, Plank L, Spicakova H, Teke K, Schuh H (2012) The new Vienna VLBI Software VieVS. In: Kenyon S, Pacino MC, Marti U (eds) Proceedings of the IAG scientific assembly 2009, International Association of Geodesy Symposia Series, vol 136, pp 1007–1011. doi:10.1007/978-3-642-20338-1_126
- Brzeziński A (1994) Polar motion excitation by variations of the effective angular momentum function, II: Extended model. Manuscr Geod 19:157–171Google Scholar
- Brzeziński A, Ponte R, Ali A (2004) Nontidal oceanic excitation of nutation and diurnal/semidiurnal polar motion revisited. J Geophys Res 109 (B11407). doi:10.1029/2004JB003054
- Brzeziński A (2008) On the influence of diurnal atmospheric tides on Earth rotation. In: Capitaine N (ed) Proceedings of the Journées Systèmes de Référence Spatio-Temporels 2007, pp 180–183Google Scholar
- Brzeziński A (2011) Diurnal excitation of Earth rotation estimated from recent geophysical models. In: Capitaine N (ed) Proceedings of the Journées Systèmes de Référence Spatio-Temporels 2010, pp 131–186Google Scholar
- Carrère L, Lyard F, Cancet M, Guiloot A, Roblou L (2012) FES 2012: a new global tidal model taking advantage of nearly 20 years of altimetry. Paper presented at the Symposium 20 Years of Progress in Radar Altimetry, ESA SP, Venice, Italy, 24–29 SeptGoogle Scholar
- Chao BF, Ray RD, Gipson JM, Egbert GD, Ma C (1996) Diurnal/Semidiurnal polar motion excited by oceanic tidal angular momentum. J Geophys Res 101(B9):20151–20163. doi:10.1029/96JB01649 CrossRefGoogle Scholar
- Chapman S, Lindzen RS (1970) Atmospheric tides. D Reidel, DordrechtGoogle Scholar
- Compo GP, Whitaker JS, Sardeshmukh PD et al (2011) The Twentieth Century Reanalysis Project. Q J R Meteorol Soc 137:1–28. doi:10.1002/qj.776 CrossRefGoogle Scholar
- Dai A, Wang J (1999) Diurnal and semidiurnal tides in global surface pressure fields. J Atmos Sci 56:3874–3891CrossRefGoogle Scholar
- Dee DP, Uppala SM, Simmons AJ et al (2011) The ERA-Interim reanalysis: configuration and performance of the data assimilation system. Q J R Meteorol Soc 137:553–597. doi:10.1002/qj.828 CrossRefGoogle Scholar
- Dee D, Fasullo J, Shea D, Walsh J (2015) The climate data guide: atmospheric reanalysis: overview & comparison tables. https://climatedataguide.ucar.edu/climate-data/atmospheric-reanalysis-overview-comparison-tables. Accessed 13 July 2015
- Dehant V, Defraigne P (1997) New transfer functions for nutations of a nonrigid Earth. J Geophys Res 102(B12):27659–27687. doi:10.1029/97JB02347 CrossRefGoogle Scholar
- Dehant V, Arias F, Bizouard C et al (1999) Considerations concerning the non-rigid earth nutation theory. Celest Mech Dyn Astron 72:245–310CrossRefGoogle Scholar
- Dehant V, Feissel-Vernier M, de Viron O, Ma C, Yseboodt M, Bizouard C (2003) Remaining error sources in the nutation at the submilliarc second level. J Geophys Res 108(B5):2275. doi:10.1029/2002JB001763 Google Scholar
- Dehant V, Mathews PM (2009) Earth rotation variations. In: Herring T (ed) Treatise on Geophysics, vol 3. Geodesy, pp 295–350Google Scholar
- de Viron O, Boy JP, Goosse H (2004) Geodetic effects of the ocean response to atmospheric forcing in an ocean general circulation model. J Geophys Res 109 (B03411). doi:10.1029/2003JB002837
- Dobslaw H, Thomas M (2005) Atmospheric induced oceanic tides from ECMWF forecasts. Geophys Res Lett 32 (L10615). doi:10.1029/2005GL022990
- Egbert GD, Ray RD, Bills BG (2004) Numerical modeling of the global semidiurnal tide in the present day and in the last glacial maximum. J Geophys Res 109:C03003. doi:10.1029/2003JC001973 Google Scholar
- Einšpigel D, Martinec Z (2015) A new derivation of the shallow water equations in geographical coordinates and their application to the global barotropic ocean model (the DEBOT model). Ocean Model 92:84–100. doi:10.1016/j.ocemod.2015.05.006 Google Scholar
- Fedorov EP, Smith ML, Bender PL (1980) Nutation and the Earth's rotation. In: Proceedings of the of IAU symposium no. 78, D Reidel. Dordrecht, NetherlandsGoogle Scholar
- Fukushima T (1991) Geodesic nutation. Astron Astrophys 244:L11–L12Google Scholar
- Green JAM, Nycander J (2013) A comparison of tidal conversion parameterizations for tidal models. J Phys Oceanogr 43:104–119. doi:10.1175/JPO-D-12023.1 CrossRefGoogle Scholar
- Hendershott M (1972) The effects of solid earth deformation on global ocean tides. Geophys J R Astron Soc 29:389–402CrossRefGoogle Scholar
- Herring TA, Buffett BA, Mathews PM, Shapiro II (1991) Forced nutations of the Earth: influence of inner core dynamics: 3. Very long interferometry data analysis. J Geophys Res 96(B5):8259–8273. doi:10.1029/90JB02177 CrossRefGoogle Scholar
- Herring TA, Mathews PM, Buffett BA (2002) Modeling of nutation-precession: very long baseline interferometry results. J Geophys Res 107(B4):2069. doi:10.1029/2001JB000165 Google Scholar
- Jayne SR, St Laurent LC (2001) Parameterizing tidal dissipation over rough topography. Geophys Res Lett 28:811–814. doi:10.1029/2000GL012044 CrossRefGoogle Scholar
- Jeffreys H, Vicente RO (1957) The theory of nutation and the variation of latitude. Mon Not R Astron Soc 117:142–161CrossRefGoogle Scholar
- Kinoshita H (1977) Theory of the rotation of the rigid Earth. Celestial Mech 15:277–326CrossRefGoogle Scholar
- Kobayashi S, Ota J, Harada J et al (2015) The JRA-55 reanalysis: general specifications and basic characteristics. J Meteorol Soc Jpn 93:5–48. doi:10.2151/jmsj.2015-001 CrossRefGoogle Scholar
- Koot L, Dumberry M, Rivoldini A, de Viron O, Dehant V (2010) Constraints on the coupling at the core-mantle and inner core boundaries inferred from nutation observations. Geophys J Int 182:1279–1294. doi:10.1111/j.1365-246X.2010.04711.x CrossRefGoogle Scholar
- Koot L, de Viron O (2011) Atmospheric contributions to nutations and implications for the estimation of deep Earth's properties from nutation observations. Geophys J Int 185:1255–1265CrossRefGoogle Scholar
- Li Y, Smith RB, Grubišić V (2009) Using surface pressure variations to categorize diurnal valley circulations: experiments in Owens Valley. Mon Weather Rev 137:1753–1769. doi:10.1175/2008MWR2495.1 CrossRefGoogle Scholar
- Lieberman RS, Riggin DM, Ortland DA, Nesbitt SW, Vincent RA (2007) Variability of mesospheric diurnal tides and tropospheric diurnal heating during 1997–1998. J Geophys Res 112 (D20110). doi:10.1029/2007JD008578
- Mathews PM, Herring TA, Buffett BA (2002) Modeling of nutation and precession: new nutation series for nonrigid Earth and insights into the Earth's interior. J Geophys Res 107(B4):2068. doi:10.1029/2001JB000390 Google Scholar
- McPhaden MJ, Ando K, Bourlès B et al. (2010) The Global Tropical Moored Buoy Array. In: Hall J, Harrison DE, Stammer D (eds) Proceedings of the Oceanobs'09: Sustained Ocean Observations and Information for Society Conference, vol 2. ESA Publ WPP-306Google Scholar
- Meynadier R, Bock O, Gervois S, Guichard F, Redelsperger JL, Agusti-Panareda A, Beljaars A (2010) West African monsoon water cycle. 2: assessment of numerical weather prediction water budgets. J Geophys Res 115 (D19107). doi:10.1029/2010JD013919
- Molodensky MS (1961) The theory of nutation and diurnal earth tides. Comm Obs Roy Belgique 188:25–56Google Scholar
- Padman L, Fricker HA, Coleman R, Howard S, Erofeeva L (2002) A new tide model for the Antarctic Ice shelves and seas. Ann Glaciol 34:247–254CrossRefGoogle Scholar
- Parke ME (1982) O\(_1\), P\(_1\), N\(_2\) models of the global ocean tide on an elastic earth plus surface potential and spherical harmonic decompositions for M\(_2\), S\(_2\), and K\(_1\). Mar Geod 6:35–81CrossRefGoogle Scholar
- Petit G, Luzum B, (2010) IERS Conventions 2010. IERS technical note no. 36. Verlag des Bundesamtes für Kartographie und Geodäsie, Frankfurt am Main, GermanyGoogle Scholar
- Poli P, Hersbach H, Tan T et al. (2013) The data assimilation system and initial performance evaluation of the ECMWF pilot reanalysis of the 20th-century assimilating surface observations only (ERA-20C). ERA report series no. 14, European Centre for Medium-Range Weather Forecasts. Reading, UKGoogle Scholar
- Ponchaut F, Lyard F, Le Provost C (2001) An analysis of the tidal signal in the WOCE sea level dataset. J Atmos Ocean Technol 18:77–91CrossRefGoogle Scholar
- Ponte RM, Vinogradov SV (2007) Effects of stratification on the large-scale ocean response to barometric pressure. J Phys Oceanogr 37:245–258CrossRefGoogle Scholar
- Ray RD (1998a) Diurnal oscillations in atmospheric pressure at twenty-five small oceanic islands. Geophys Res Lett 25:3851–3854. doi:10.1029/1998GL900039 CrossRefGoogle Scholar
- Ray RD (1998b) Ocean self-attraction and loading in numerical tidal models. Mar Geod 21:181–192CrossRefGoogle Scholar
- Ray RD, Ponte RM (2003) Barometric tides from ECMWF operational analyses. Ann Geophys 21:1897–1910. doi:10.5194/angeo-21-1897-2003 CrossRefGoogle Scholar
- Ray RD, Egbert GD (2004) The global S\(_1\) tide. J Phys Oceanogr 34:1922–1935CrossRefGoogle Scholar
- Rienecker MM, Suarez MJ, Todling R et al. (2008). The GEOS-5 data assimilation system-Documentation of versions 5.0.1, 5.1.0, and 5.2.0. NASA technical report series on global modeling and data assimilation, vol 27. Greenbelt, Maryland, USAGoogle Scholar
- Rienecker MM, Suarez MJ, Gelaro R et al (2011) MERRA: NASA's Modern-Era Retrospective Analysis for Research and Applications. J Clim 24:3624–3648. doi:10.1175/JCLI-D-11-00015.1 CrossRefGoogle Scholar
- Robertson FR, Bosilovich MG, Chen J, Miller TL (2011) The effect of satellite observing system changes on MERRA water and energy fluxes. J Clim 24:5197–5217. doi:10.1175/2011JCLI4227.1 CrossRefGoogle Scholar
- Saha S, Moorthi S, Pan HL et al (2010) The NCEP Climate Forecast System Reanalysis. Bull Am Meteorol Soc 91:1015–1057CrossRefGoogle Scholar
- Sasao T, Okubo S, Saito M (1980) A simple theory on the dynamical effects of a stratified fluid core upon nutational motion of the Earth. In: Fedorov EP, Smith ML, Bender PL (eds) Proceedings of the IAU symposium no. 78, Nutation and the Earth's Rotation, pp 165–183Google Scholar
- Sasao T, Wahr JM (1981) An excitation mechanism for the free 'core nutation'. Geophys J R Astron Soc 61:729–746CrossRefGoogle Scholar
- Schindelegger M, Böhm S, Böhm J, Schuh H (2013) Atmospheric effects on Earth rotation. In: Böhm J, Schuh H (eds) Atmospheric Effects in Space Geodesy. Springer, pp 181–231Google Scholar
- Schindelegger M, Böhm J, Salstein DA (2015) The global S\(_1\) tide and Earth's nutation. In: Malkin Z, Capitaine N (ed) Proceedings of the Journées Systèmes de Référence Spatio-Temporels 2014, pp 145–150Google Scholar
- Schindelegger M, Dobslaw H (2016) A global ground truth view of the lunar air pressure tide L\(_2\). J Geophys Res Atmos 121:95–110. doi:10.1002/2015JD024243
- Smith A, Lott N, Vose R (2011) The Integrated Surface Database: recent developments and partnerships. B Am Meteorol Soc 92:704–708. doi:10.1175/2011BAMS3015.1 CrossRefGoogle Scholar
- Sovers OJ, Fanselow JL, Jacobs CS (1998) Astrometry and geodesy with radio interferometry: experiments, models, results. Rev Mod Phys 70:1393–1454CrossRefGoogle Scholar
- Stepanov VN, Hughes CW (2004) Parameterization of ocean self-attraction and loading in numerical models of the ocean circulation. J Geophys Res 109 (C03037). doi:10.1029/2003JC002034
- Taguchi E, Stammer D, Zahel W (2014) Inferring deep ocean tidal energy dissipation from the global high-resolution data-assimilative HAMTIDE model. J Geophys Res Oceans 119:4573–4592. doi:10.1002/2013JC009766 CrossRefGoogle Scholar
- Vial F, Lott F, Teitelbaum H (1994) A possible signal of the El Niño-Southern Oscillation in time series of the diurnal tide. Geophys Res Lett 21:1603–1606. doi:10.1029/94GL01016 CrossRefGoogle Scholar
- Wahr J (1981) The forced nutations of an elliptical, rotating, elastic and oceanless earth. Geophys J Int 64:705–727CrossRefGoogle Scholar
- Wilmes SB, Green JAM (2014) The evolution of tides and tidal dissipation over the past 21,000 years. J Geophys Res Oceans 119:4083–4100. doi:10.1002/2013JC009605 CrossRefGoogle Scholar
- Woolard EW (1953) Theory of the rotation of the earth around its center of mass. Astron J 58:2CrossRefGoogle Scholar
- Yseboodt M, de Viron O, Chin TM, Dehant V (2002) Atmospheric excitation of the Earth's nutation: comparison of different atmospheric models. J Geophys Res 107(B2):2036. doi:10.1029/2000JB000042 CrossRefGoogle Scholar
- Zaron ED, Egbert GD (2006) Estimating open-ocean barotropic tidal dissipation: the Hawaiian Ridge. J Phys Oceanogr 36:1019–1035CrossRefGoogle Scholar
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Cjd Enterprises Llc
17254 N 63rd Ave Glendale, Arizona 85308(623) 486-2128
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AboutCjd Enterprises Llc is located at the address 17254 N 63rd Ave in Glendale, Arizona 85308. They can be contacted via phone at (623) 486-2128 for pricing, hours and directions.
Cjd Enterprises Llc has an annual sales volume of 501K - 999,999. .
For maps and directions to Cjd Enterprises Llc view the map to the right. For reviews of Cjd Enterprises Llc see below.
Phone: (623) 486-2128
Employee Size: 1 To 4
Exact Employee Count: 3
Exact Sales Volume: 100000
Location Type: Single Location
Sales Volume: 501K - 999,999
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Brijot Imaging Systems eliminates jobs
Lake Mary-based Brijot Imaging Systems Inc. said it has eliminated an unspecified number of jobs as part of a companywide streamlining. The high-tech maker of personal image-scanning security equipment recently made the layoffs in a cost-cutting effort to boost profits and improve efficiency, Brijot spokeswoman Christine Willard said. Privately held Brijot employs about 50 in the design and production of full-body image scanning systems for security screening at airports, military installations and other security-sensitive facilities. Earlier this summer, the company had a change in its sales division leadership, hiring veteran high-tech executive John Marsala as its vice president for sales and marketing. He was formerly with Brijot competitor L3 Safeview Inc.


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Last month I saw some pants with an ankle tie hem. I knew immediately that I wanted to make some for summer if I could find the right fabric. I used to only wear shorts in the summer, but last summer I was post-partum and couldn't fit into very many clothes. I tried wearing dresses and breathable pants. I actually really enjoyed wearing something other than shorts, and this summer I'm working on expanding my summer wardrobe to be more than shorts. It starts with these pants.
I used the Hudson Pants pattern and just altered the hem. I added two ties near the side seam and then created a front and back hem facing. It got tricky while I was putting it together because I needed a slit above the ties. I really don't even know how I got it done in the end, but here's a simple drawing of what you need to do to add ties to your own pants.
This is how the pants look from all angles. They are semi-fitted all the way down. My alterations to the pattern were to add 1″ to the back waist. I deepened the front pockets and sewed them to the front of the pants. I also added a couple of inches to the legs. After sewing them I needed a little more room in the calves so I let both side seams out at the calves. I eliminated the cuff and did the tie and facing instead to finish the edges. I made a size 2 if I remember correctly. It's one of my tried and true patterns so I perfected it at least a year ago.
I used a stretch woven that I found at a nearby shop for $3 a yard. I got two yards so these pants cost me under $10, especially since I already had the pattern, matching thread, elastic, and I can whip this pattern up so quickly. I don't know exactly what the content of this fabric is, but I'm sure there's some polyester since it doesn't press nicely. It washes nicely and has a nice stretch, so I'm guessing it's some sort of suiting. It actually drapes really well for a opaque, stretch woven. The pants are comfortable with the stretch and not too hot even though there's some synthetic in them. I've worn these pants twice already and I love them. I even wore them to a museum with my kids which was half outdoors in the sun, and I didn't get too hot! I love a good inspiration and a quick sew to add to my handmade wardrobe! The only trouble I had was with the slit and the facing. I should've looked up a tutorial before diving in, but in the end it worked out perfectly.


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Buffon only needed 10 more caps to be the most capped player of all time. Including warm up games he might have made it had they qualified. Watching the interview with him after the game in tears apologising to the country is a bit heart wrenching.
Ugh, thats frustrating.
Looking at the list, surprised how many C Ronaldo has. They play so many these days
I honestly thought Ronaldo would call it a day after the Euros, surprised he decided to keep going.
That said, and speaking of all time record for international football, he needs 31 goals to break Ali Dae's record of 109. And if he decides to carry on until the next world cup he'll most likely break it (albeit he'll be 37 at the time).
I can't see him breaking the record even if he goes on. There's a lot of odder in Europe but not that much, plus he's not playing in a side to his standard, regardless of Euros win.
Yeah Daei's record is fairly ridiculous. There's loads of shit teams in Asia hence why the top 20 International scorers list is full of random Asian players that have filled their boots against minnows. I can see Ronaldo 2nd realistically, which is an achievement in itself. And considering his fitness levels he could get the appearance record too.


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As hard as you work to make experiences great, some customers work harder to make them awful. Here are four real-life situations when customers behaved badly – plus effective tactics to handle them.
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You may never know customer loyalty like professional sports teams do, but it's possible to build enough loyalty to make your competitors envious.
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What do you do when a valued customer makes an unreasonable demand? Or your sales manager gives you an unfair evaluation? Or a colleague takes advantage of a situation?
What a happy customer means to the bottom line [Graphic]
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If you see a pattern of customer complaints, don't just think about fixing the problem. Use the issue to improve service and, possibly, sales. Here's how one company did it.


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'The Interview': Which Theaters Are Showing the Film on Christmas Day
Indie theaters around the country will screen Sony's comedy
Best Places To Cut Your Own Christmas Tree In The Los Angeles Area
You could get a fresh-cut Christmas tree delivered to your door, but the experience of cutting down your own sounds way more fun. Here are some places to do just that.
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?The Interview? released on YouTube, in theaters Thursday
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The U.S. Forest Service needs help counting bald eagles in the San Bernardino National Forest. For the 36th season, the annual winter bald eagle counts in and near the San Bernardino and San Jacinto Mountains will be on the following Saturdays: Dec. 13, Jan. 10, Feb. 14 and March 14. Migrating eagles typically begin arriving
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The enchanted forest light show at Pilochry Perthshire Photograph: Alamy
Keith Carter leads a walk along the River Bollin at Styal, passing ancient and modern tourist attractions
One of my fave walking spots. Curbar and Froggatt edge, Derbyshire.
National Trust's favourite UK walks, day one:the best of Devon's beautiful countryside and coastline


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Reference: Author(s) Last name, Initial(s). (Year) 'Title of lecture/presentation' [Medium], Module Code: Module title. Institution. Day Month.
Example: De Burca, M. (2014) 'Geriatric radiography services in Ireland' [Lecture], RDGY30300: Clinical Practice of Radiography. University College Dublin. 11 May.
|© 2015 University College Dublin Library T + 353 (0)1 7167583 | E firstname.lastname@example.com
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Ireland License


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1300 36 33 32
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Norman Dorsen is Stokes Professor of Law, New York University School of Law, the former president of the ACLU and former Chairman of the Lawyers Committee for Human Rights. He is author or editor of nine books.
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As entrepreneurs there's no shortage of advice, strategies and tools on how to become successful, from time managements systems like the Pomodoro Technique, to apps like Trello and Evernote, to books like Brian Tracy's Eat That Frog.
It's not very often that we are encouraged to return to basics and cultivate our inner qualities that make each of us unique. Check out the full article at Addicted2Success.com.
This shift towards mindfulness will make you think deeply and objectively while being aware of what you're doing. Becoming mindful of your thoughts will help increase your resilience to stress. This, in turn, will enhance your decision-making capabilities. Mindfulness is more than just sitting still and breathing deeply, because while being mindful is simple, most of us lack the passion for sustaining it.
Brendon Burchard has his take on it:[imp_cta_emotion]
Before you go...do you think anyone else might enjoy this content ?


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About a year ago, I posted this picture to /r/amiugly without specifying my sex or race to see how people would perceive my ugliness/attractiveness depending on which sex they believed I was.
(*NOTE* Whether I'm ugly or not was not the objective. Rather I aimed to observe how people would come to their own judgement without me telling them my biological sex.)
Putting aside objectivity, those who perceived me as male were clearly less critical of my appearance and considered me "average, above average" and suggested I could benefit from a haircut. In contrast, those who perceived me as female suggested plastic surgery. More than one comment states that if I'm a male I'm average, but if I'm a female I'm fugly. My unedited, unaltered appearance was considered to be average to those who perceived me as male, but ugly and in need of surgery to those who perceived me as female. Many saw me as androgynous, but it reaffirmed my belief that women are judged more harshly for their appearance.
I found extremely telling of how our society feels entitled to point out every feature they deem unattractive on a woman's natural face, but will hesitate to point out the same exact unattractive features if they exist on a man's face.


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ZWCad 2018 - official
posts: 7
Registered: 2017-9-13
Message 1 of 7
when will You release the official version of ZWCAD 2018?
The link in the download center leads to another version.
The current version is VERNUM = "2017.08.22(21234)_x64" - is it official or still beta?
posts: 572
Registered: 2015-6-9
Message 2 of 7
The current version iVERNUM = "2017.08.22(21234)_x64" - is Official. We update the version in the download page now. Thanks for the correction.
posts: 7
Registered: 2017-9-13
Message 3 of 7
In Poland this release is still beta.
posts: 572
Registered: 2015-6-9
Message 4 of 7
Thanks for your question!
Actually we have local partner in Poland <https://www.zwcad.pl/>, you can contact them for more information in the local market. Or you can directly download the Polish version of ZWCAD 2018 Official from this website to test.
Junior Engineer
posts: 47
Registered: 2011-7-14
Message 5 of 7
I want all drawing should be open in single zwcad window. SDI variable is 0.
I have ZWCad 2018 Pro. Version number 2017.08.22(21234)_x64.
Assistant Engineer
posts: 4
Registered: 2014-11-29
Message 6 of 7
posts: 52
Registered: 2016-4-14
Message 7 of 7
Hi, we tried to 08.22 version, and the latest one, but didn't reproduce it. You can try to upgrade to the latest version and check again.
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A couple of my co-workers have raved about meals that they have had there and their accounts peaked my interest. However, the reviews that I have read on TripAdvisor aren't very flattering. Is this place worth a try?
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Soft and substantial knit
Verified Buyer Reviewed by Perryville 4 from Perryville, MD on Monday, March 14, 2011
I bought two of these in the XL. They are beautiful sweaters, but unfortunately seem sized for a dainty/petite xl, not for a lumberjack like me. Should still be able to use as layering pieces.
- Best Uses: Casual Wear, Wear to School, Wear To Work
- Chest Size: Feels too small
- Cons: Poor Fit
- Describe Yourself: Classic Dresser
- Length: Feels true to length
- Pros: Comfortable, Flattering, Great Color, Lightweight, Soft, Stylish, Versatile, Warm
- Sleeve Length: Feels too short
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