#### The Global S \(_1\) Tide in Earth’s Nutation

Surv Geophys
The Global S1 Tide in Earth's Nutation
Michael Schindelegger 0 1 2 3 4
David Einsˇpigel 0 1 2 3 4
David Salstein 0 1 2 3 4
Johannes Bo¨ hm 0 1 2 3 4
0 School of Cosmic Physics, Dublin Institute for Advanced Studies , Dublin , Ireland
1 Department of Geodesy and Geoinformation, TU Wien , Gußhausstraße 27-29, 1040 Vienna , Austria
2 Michael Schindelegger
3 Atmospheric and Environmental Research, Inc. , Lexington MA , USA
4 Department of Geophysics, Charles University in Prague , Prague , Czech Republic
Diurnal S1 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 S1 terms that deviate by more than 30 las (microarcseconds) from the VLBI-observed harmonic of 16:2 þ i113:4 las. 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 S1 tide gauge determinations. Credence is lent to the global S1 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 þ i106:0 las (MERRA) and 9:4 þ i121:8 las (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.
1 Introduction
Nutation
Geophysical excitation
Atmospheric
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
las (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, K1, P1, O1, and Q1. 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 S1 are capable of exciting
small additional, seasonal nutation waves that exceed the statistical uncertainties of
VLBIbased parameters on the order of 10 las (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 P1 tide,
have been thoroughly addressed by MHB, yet the main S1 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 S1 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 S1 harmonic of somewhat more than 100 las and superimposed
the very same signal as a post-fit correction term to the final nutation series.
From a practical point of view, this approach is legitimate and in fact aided by the
pronounced harmonic character of the unexplained nutation variability in the prograde
annual band. Corrections to the MHB model, published as daily celestial pole offsets by the
International Earth Rotation and Reference Systems Service (IERS), are generally below
20 las in the S1 band, also for more recent years not included in the original MHB
analysis; cf. Fig. 1. However, in the spirit of a theory that should be ultimately free of
empirical adjustments (Fedorov et al. 1980) and also unambiguous in its account of the
various effects at the prograde annual frequency, it is still worthwhile to strive for an
independent S1 estimate from geophysical fluid models.
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
S1ðpÞ, the diurnal pressure variations at the sea surface. Hence, both atmospheric and
oceanic oscillations are closely interrelated aspects of the same ‘‘global S1 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 S1 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
(Brzezi n´ski et al. 2004; de Viron
et al. 2004)
—a terminology that shall, however, not be used in the following.
Investigations of the S1 effect in nutation on the basis of dynamically coupled
atmosphere–ocean models have been pursued primarily by A. Brzezin´ ski and collaborators; cf.
Fig. 1 Prograde annual signal
amplitude in the IAU2000
celestial pole offsets (CPO) w.r.t.
the MHB model. Estimates are
3year sliding window fits, and
error bars indicate standard
deviation (SD) in amplitude that
have been propagated rigorously
from the CPO errors. The MHB
VLBI analysis, documented in
Herring et al. (2002), involves
data up to November 1999 (red
line) and features a threefold SD
of 21 las in the prograde annual
band (dashed black line)
60
50
)s40
a
µ
(
de30
u
lit
p
Am20
10
0
Brzezin´ ski et al. (2004), Brzezin´ ski (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 las. 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 S1-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 las) 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 (P1, S1, and w1), 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 75 las for the S1 out-of-phase term, while de Viron
et al. (2004) specify a value of 38 las. Without further insight into the actual (corrected)
S1 nutation predictions of these authors, we will employ Brzezin´ ski (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 S1 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 (w1, P1, p1, and /1) are
tabulated, even though our prime focus is on the S1 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
Carre`re 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 S1
solutions and respective OAM values that are consistent with the utilized atmospheric datasets.
We adopt the recent barotropic time-stepping model of Einsˇpigel and Martinec (2015),
designated as DEBOT (Barotropic Ocean Tide model developed by D. Einsˇpigel), and
implement the necessary modifications for the problem in hand. Specifically, to obtain S1
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 S1 (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 S1 air tide, even though the
temporal variability of this forcing can be large
(Ray 1998a)
. Seasonal modulations of
S1ðpÞ by 1 cpy (cycle per year) correspond to the P1 and K1 constituents, inducing small
radiational ocean tides that are automatically included in altimetric solutions of P1 and K1
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 S1 averages for the coupled atmosphere–ocean system. To
determine a favorable, i.e., inter-annually ‘‘quiet’’ averaging period, we assess the S1
variability both in the integrated atmospheric nutation values and in surface pressure.
Specifically, the analysis of S1ðpÞ 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 S1 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 S1 band.
2 Meteorological Data for Nutation Studies
Solar tides in the atmosphere are not of immediate relevance to operational or retrospective
analyses, yet a largely realistic model account of these oscillations is guaranteed by the use
of insolational forcing physics in combination with in situ and remotely sensed
meteorological data. Reanalyses, created by various weather agencies on the basis of an
unchanging assimilation scheme over decades, are usually credited with a realistic
longterm variability that also modulates the tides and, by implication, nutation amplitudes.
Their products, issued with an invariable spatial and temporal resolution, have thus become
the preferred means to investigate atmospheric effects in nutation. Table 1, taken from
Schindelegger et al. (2015)
, summarizes some basic information of presently available
reanalysis datasets, sorted by a rough generation index (Dee et al. 2015) that is thought to
reflect the varying degree of sophistication in terms of model physics, resolution, and
Nameb
a The generation index (GI) and information about horizontal resolution and assimilation technique are
taken from Dee et al. (2015), while the model vintage (i.e., the fixation date of the agency’s operational
model) has been extracted from the reanalysis-specific reference articles and differs from Dee et al. (2015)
in individual cases. Yet inaccessible models (e.g., MERRA-2) and twentieth-century reanalyses that
assimilate surface observations only
(e.g., Compo et al. 2011)
are not tabulated. We have also omitted
citations of reanalyses that are not examined in the frame of the present work
b Abbreviations NCEP National Centers for Environmental Prediction, ECMWF European Centre for
Medium-Range Weather Forecasts, ERA ECMWF Reanalysis, JRA-25/JRA-55 Japanese 25-year/55-year
Reanalysis, JMA Japan Meteorological Agency, MERRA Modern-Era Retrospective Analysis for Research
and Applications, GMAO Global Modeling and Assimilation Office, CFSR NCEP Climate Forecast System
Reanalysis
c Abbreviations 3DVar/4DVar 3D/4D Variational Assimilation, IAU Incremental Analysis Update
assimilation technique. Refer to the caption of Table 1 for any model abbreviations used in
the following.
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 Brzezin´ ski 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 Brzezin´ ski (2011). Ignoring differences
due to varying analysis periods, the S1 estimates of these studies exhibit a fair agreement,
roughly at 20–30 las, 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)
.
Operational models, designed for weather prediction on a daily basis, may be thought to
be less suitable for long-term nutation studies. Indeed, S1 estimates of
Yseboodt et al.
(2002)
from early operational systems of ECMWF, NCEP, and JMA differ by as much as
70 las, suggesting that the regular changes to the model and assimilation technique within
each agency are capable of introducing artificial tidal variability (Koot and de Viron 2011).
To some extent, though, the results of
Yseboodt et al. (2002)
are affected by time series
limitations, e.g., data gaps of up to 30 % or occasionally short record lengths (3 years). If
provided continuously, operational datasets might, in fact, still be proper vehicles for tidal
studies, as the underlying analysis systems are optimized to represent the atmospheric
variability on short timescales and as they are also readily adaptable to the introduction of
new data types. By contrast, reanalyses assimilate an evolving observation record through
a predefined framework and are thus prone to spurious variabilities (Dee et al. 2015).
Moreover, and in the context of nutation, most model updates involve resolution and
orography changes that possibly cause local discontinuities in continental surface pressure
but are of no immediate consequence for all other components of the global S1 tide (i.e.,
the wind signal at higher altitudes, the pressure tide over the ocean, and, thus, the full
oceanic S1 variability).
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 S2, which is not properly resolved by four-times-daily analysis products. To
prepare for further subsequent work on S2, we also interlaced 3-h forecast data to the CFSR
analysis fields after verifying that their inclusion had no evident effect on the S1 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 S1 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.
A preliminary comparison of two reanalyses with regard to their diurnal cycle is shown
in Fig. 2 in the form of cotidal S1ðpÞ charts for MERRA and CFSR. Mean amplitude and
phase lag values were obtained from a standard least-squares tidal analysis of 10-year
pressure time series at each grid point location. These climatologies echo the well-known
spatial characteristics of the diurnal barometric tide
(see Ray and Ponte 2003, and
references therein)
but also exemplify that its representation in global analysis models can
diverge. CFSR suggests higher S1 amplitudes almost throughout the world, particularly
over landmasses in latitudes lower than 30 and in valleys for which the local diurnal
oscillation is not resolved by MERRA (e.g., Sierra Nevada). Large regional-scale
differences over flatter terrain (e.g., Sahara, Central Africa, India) portend to difficulties in the
240
210180150
120
models to represent the significant diurnal boundary-layer effects driven by sensible and
latent heating from the ground
(Dai and Wang 1999)
. These non-migrating components
can be excluded from the tidal spectrum by performing a Fourier decomposition of S1ðpÞ
by wavenumber s
(Chapman and Lindzen 1970)
and retaining only the main
Sun-synchronous, migrating S11 tide (s ¼ 1). Forced by tropospheric absorption processes, S11
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 S can be observed for the CFSR solution. These discrepancies in pressure are likely to
have a bearing on the simulation of the oceanic S1 tide.
3 Atmospheric Contributions to Nutation
3.1 Implementation
Computation of the vector equatorial AAM mass term H~p ¼ Hxp þ iHyp (complex notation)
involves weighted-area double integrals of surface pressure, whereas for the motion term
H~w ¼ Hxw þ iHyw a full 3D summation of geometrically weighted horizontal winds is
required. We applied the respective standard formulas, given, e.g., in Sect. 2.5 of
Schindelegger et al. (2013)
on the gridded datasets of Table 2, with lower boundaries in the
vertical integration taken from the model-specific topographies. A priori corrections to the
mass terms for an isostatic (inverted barometer, IB) ocean response to air pressure
variations were categorically avoided, as the oceanic S1 tide is a dynamic phenomenon that
will be rigorously estimated in Sect. 4.
H~p;w are the basic excitation quantities that can be related to nutation through a proper
dynamical theory.
Sasao and Wahr (1981)
devised corresponding expressions for the
geophysically driven nutation from the angular momentum balance equations of a coupled
two-layer Earth, comprising mantle and a fluid core which are allowed to deform
elastically under the action of body tides and atmospheric (oceanic) loads at the Earth’s surface.
Brzezin´ ski (1994) reformulated this pilot equation to a practicable broad-band excitation
scheme for both nutation and polar motion. Yet, for reasons of consistency, the comparison
of geophysical model estimates with the S1 post-fit correction terms of MHB and Koot
et al. (2010) is better accomplished through the excitation scheme of Koot and de Viron
(2011). Their formalism conforms to MHB’s nutation theory for an up-to-date Earth model
with inner core dynamics and anelastic properties. Perturbations n~ r
ð Þ ¼ dX þ idY of the
celestial pole offsets in X and Y in response to changes of AAM at some Earth-referred,
retrograde diurnal frequency r (in cycles per sidereal day, cpsd) are modeled as
n~ r
ð Þ ¼
~p r H~0pðrÞ
T ð Þ XðC AÞ
H~0wðrÞ
T~wðrÞ XðC AÞ
where X is the nominal sidereal angular velocity, A and C denote the equatorial and polar
principal moments of inertia of an axisymmetric solid Earth (i.e., mantle and crust), and
T~p;wðrÞ are transfer functions describing Earth’s nutation response to atmospheric forcing
as conveyed by the periodic terms H~0p;wðrÞ, which are defined below. The transfer
functions read
ð1Þ
T~p;w r
ð Þ ¼
X4 N~ip;w
i¼1 r
r~i
comprising resonances at the frequencies r~i (cpsd) of the four rotational normal modes of a
three-layer Earth: the Chandler wobble (CW), the free core nutation (FCN), the free inner
core nutation, and the inner core wobble (Koot and de Viron 2011). The strengths of these
resonances upon mass and motion excitation are characterized by the coefficients N~ip;w,
specified in Table 3 with a truncation at i ¼ 2 that retains CW and FCN and excludes the
inner core modes without loss of accuracy. If viewed from the surface of the rotating Earth,
the FCN occurs as retrograde nearly diurnal oscillation, thus providing significant
enhancement to excitation effects associated with atmosphere–ocean dynamics at S1 and
adjacent tidal lines. Note, however, that nutations are much more efficiently driven by the
mass term than by relative particle motion
(Sasao and Wahr 1981; Brzezin´ ski 1994)
; cf.
also the excess of N~p at the FCN frequency relative to N~w by a factor of 200 (Table 3).
The forcing terms H~0p;wðrÞ in Eq. (1) are complex coefficients of time dependence
/ eirt as seen from the rotating reference frame (Koot and de Viron 2011). Yet, the
amplitudes of these sinusoids are typically estimated in inertial space after translating the
terrestrial AAM time series H~ðtÞ to their celestial counterparts H~0ðtÞ via the demodulation
(Brzezin´ ski 1994)
that is applicable to both pressure and wind effects (respective superscripts have been
omitted for brevity). The exponent represents a sufficiently accurate linear approximation
for the Greenwich sidereal time
(Bizouard et al. 1998)
, employing a phase offset of U0
referred to t0 at 12 h UT1, 1 January 2000 (J2000.0). The conventional expression for the
Earth rotation angle of the IERS
(Petit and Luzum 2010)
is compatible with Eq. (3) and
implies U0 ¼ 0:7790572732640 rad as well as X ¼ ð2prÞ rad per solar day, where r ¼
1:00273781191135448 scales solar to sidereal time intervals; see also Koot and de Viron
(2011).
The demodulation procedure preserves amplitudes but maps the retrograde (r\0)
quasi-diurnal spectral components to low frequencies, with the center frequency X (i.e., the
K1 band) shifted to zero and S1 appearing as prograde annual line in the celestial frame.
Dominant (intra-)seasonal signals in the original AAM series are mapped to high
frequencies in space and are efficiently removed through filtering
(Bizouard et al. 1998)
. To
this end, we applied an idealized rectangle filter with cutoff at 20 cpy (cycles per year) on
a Theoretical, complex-valued frequencies of CW and FCN are given in the Earth-fixed frame and
correspond to a terrestrial period of 396.06 solar days with quality factor Q ¼ 223 for the CW and a celestial
period of 429:05 solar days with a (terrestrial) quality factor of Q ¼ 19736 for the FCN resonance
the frequency transform of H~0ðtÞ and resampled the proper inverse transform in the time
domain at daily intervals. Experiments with more customary time domain filters and a
range of reasonable cutoff frequencies testified to the insensitivity of our nutation results to
details in the filtering strategy.
To convert the low-frequency, quasi-harmonic celestial AAM time variability
associated with S1 and its seasonal modulations to periodic circular components H~0ðrÞ for use in
Eq. (1) we imposed a Fourier decomposition
ð4Þ
on the complex-valued filter output. The non-dimensional frequencies fmjgj5¼1 of the
demodulated tidal constituents fS1; w1; P1; /1; p1g are f1; 1; 2; 2; 3g=366:26 (Koot and
5
de Viron 2011), and respective phase values fujgj¼1 agree with those from the
corresponding lunisolar nutation terms. The uj are readily computed from the fundamental
arguments of nutation theory
(Petit and Luzum 2010)
, using the integer multipliers
specified in Table 4. A standard least-squares fit of H~0ðtÞ onto these basis functions
provides the unknown parameters a~j composed of real (in-phase, ip) and imaginary
(out-ofphase, op) parts that form the complex-valued forcing terms in Eq. (1) at discrete terrestrial
frequencies rj ¼ mj 1 (cpsd). The constant pole offset contribution from the K1 tide,
conveyed by the estimated c~ term, was excluded from further consideration, as were the
minute secular (precession) contributions associated with the time derivatives of nutation
arguments.
The harmonic decomposition in Eq. (4) exactly follows the model of Koot and de Viron
(2011) except for our inclusion of the /1 component at 2 cpy that suggests some
interannual modulation of the thermal S1 tide. While the existence of such a modulation is
debatable
(Bizouard et al. 1998)
, its impact on nutation is at the level of 10 las and thus
comparable to the usually modeled p1 term. Moreover, for numerical reasons, we
performed the least-squares fit on the basis of prescaled AAM time series H~0ðtÞ=ðXðC AÞÞ
(Eq. 1) in units of las, similar to the classical ‘‘celestial effective angular momentum
functions’’ of Brzezin´ ski (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.
a Periods and phases (referred to J2000.0) result from the linear combinations of series expansions as given
by
Petit and Luzum (2010)
for each argument. The table is identical to Table 1 of Koot and de Viron (2011)
except for the /1 term
Period
(solar days)
3.2 Results
Figure 3 displays the superimposed pressure and wind nutation estimates n~ðrÞ in the
prograde annual band, both as mean contributions over the model-specific time spans as
well as yearly values obtained from repetitions of the analysis in Sect. 3.1 with a 3-year
sliding window. Consistent with illustrations in
Yseboodt et al. (2002)
and Koot and de
Viron (2011), little agreement is seen between all S1 curves in terms of their inter-annual
variability, even though the post-2004 estimates are stable within 20 las for each model.
Roughly 75 % of the observed fluctuations are driven by the pressure term and likely relate
to random perturbations of the second-order tesseral harmonic in surface pressure, i.e., the
only component of the S1ð pÞ wave that efficiently excites Earth’s nutational motion.
Amplitudes of this mode do not exceed 10 Pa, so its representation in atmospheric
assimilation systems is prone to noise interferences.
By contrast, a signal with a possibly physical origin is evident for MERRA during
1997–2001, coinciding in time with a peak El Nin˜ o event in 1997/1998 and subsequent
cold La Nin˜ a conditions up to 2001; cf., e.g., the Oceanic Nin˜ o Index tabulated at http://
www.cpc.noaa.gov/products/analysis_monitoring/ensostuff/ensoyears.shtml (accessed 29
September 2015). The warm phase of ENSO (El Nin˜ o–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
80
)
s
aµ 60
(
e
s
a
h
p
f− 40
o
−
t
u
o
20
0
S
1
−80
2009
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.
In terms of multi-year nutation averages, the ECMWF-based solutions agree
particularly well with each other and with Koot and de Viron (2011)’s results for the
firstgeneration NCEP models from 1979 to July 2002. The ERA-40 estimate of these authors is
anomalous in the ip component ( 58:2 las) due to erratic S1 variations up to the
mid1990s
(cf. Fig. 2 of Koot and de Viron 2011)
. Disregarding the impact of the ‘‘ENSO
swerve’’ on the MERRA solution, the only nutation anomaly in the present work is a large
CFSR estimate, with op pressure term values (50 las as from 1998) exceeding the
predictions from other reanalyses by 20–30 las. This overestimation traces back to the
dubious CFSR pressure oscillations of more than 40 Pa in the Southern Ocean (Fig. 2), a
region that is void of conventional in situ observations and sensitive to the details of
radiance data assimilation. Note also that the quality with which atmospheric tides can be
represented in analysis systems is tied to the time step of radiative processes
(Poli et al.
2013)
. Trading off computational costs and a fine spatial resolution, CFSR integrates its
longwave radiation parameterization every 3 h
(Saha et al. 2010)
, significantly coarser
than the hourly time step recommended by
Poli et al. (2013)
and employed within
MERRA, ERA, and EC-OP.
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, S1 estimates from third-generation reanalyses and EC-OP deviate from each
other by less than 22 las, 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 S1 peak in terms of its frequency through a Morlet
wavelet analysis of demodulated filter residuals H~0. Deviations from the nominal S1 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 (w1, P1, /1, p1) can be compared
with estimates tabulated in
Bizouard et al. (1998)
,
Yseboodt et al. (2002)
, Brzezin´ ski 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 P1 and p1 but differ substantially in the w1
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 las 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 w1
( 50 las).
f
o
d
o
i
r
e
p
s
i
s
y
l
a
n
a
s
trem tlao
T ip
d
m in
o
rf W ip
n
o
it e
a r
t u
u s
n s
e
o rP ip
t
s
n
o
i
t
u
b
i
r
t
n
o
c l
e
c d
i
r o
e
h M
p
s
o
m
t
a
c
i
d
o
i
r
ea
P 3
1
5 0
2
e –
lab 004 rem
T 2 T
5 9 7 7 7 7 2 2 1 4 2 2 2 6 4 9 9 0 3 8
p 4 4 8 5 6 4 5 2 2 4 4 4 4 4 1 1
o
0 5 3 3 3 3 0 0 2 1 5 5 1 3 3 1 4 5
op 27 26 3 2 2 4 4 3 4 4
9: 8: 3: 8: 9: 1: 1: 2: 3: 8: 1: 3: 9: 3: 0: 1: 8: 3: 2: 8:
5 0 2 1 1 0 5 8 9 0 0 0 6 7 7 2 1 7 2 1
2 3 3 3 2 3 4 5 7 5 1 1 1 1 1 1
b
b
b
b
A A A A A A A A
R R b P R R b P R R b P R R b P
þ
(
n
o
r
i
V
t
o
o
K
f
o
2: 3: 2: 2: 2: n
0 0 0 0 0 io
t
n
e
4: 6: 7: 3: 1: v
p 1 1 0 0 0 n
o
o c
d
2: 3: 2: 2: 2: n
0 0 0 0 0 a
9: 3: 1: 1: 1: e
l
1 2 2 1 1 b
a
2: 2: 2: 2: 2: f
0 0 0 0 0 o
s
t
n
1: 0: 3: 8: 7: e
4 4 3 2 2 m
u
g
r
a
m
a
d
2: 2: 2: 2: 2: n
0 0 0 0 0 u
f
e
6: 3: 1: 5: 2: th
2 2 2 2 2
rP ip
d
e
r 0
r
e 1
f 0
e 2
r –
e 4
r 0
a 0
b
s 2
A A t
l R R b P n d
e e o
d R R SR A -O n ir
o E E F R C o e
p p
M M M C E E
)
r
a
e
y
3
=
1
þ
(
1
p
n A
I
3.3 Validation of Pressure Tides against In Situ Data
30°N
0°
30°S
(b)
30°N
0°
30°S
60°N
60°S
60°N
60°S
Stations
Buoys
0
60
120
180
240
300
360
semidiurnal L2 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.
Atmospheric model pressure values were evaluated by bilinear interpolation at the
locations of the 50 ground truth stations and tidally analyzed in a moving 3-year window.
RMS statistics and globally averaged amplitude differences from the comparison of these
windowed S1 solutions to the in situ estimates (non-windowed) are shown in Fig. 5, both
for the full global network as well as for a subset of 20 stations excluding latitudes lower
than 18 . This restriction avoids an over-emphasis on the large migrating pressure tide near
the equator and tests smaller signals in mid-latitudes, i.e., regions of increased importance
for nutation. The resulting network (Fig. 4) is, however, very sparse and dominated by 13
stations in the Pacific.
10 (a)
(c)
)
a
P
(h 9
t
u
r
td 8
n
u
rgo 7
o
t
S 6
M
R
5
5
)a 4
P
(
s 3
e
c
en 2
r
e
iff 1
d
de 0
u
it
lp−1
m
A−2
2005
Year
1995
2000
2010
1995
2005
Year
2010
−13995
2000
2005
Year
2010
−13995
2000
2005
Year
2010
10 (b)
)
a
P
(h 9
t
u
r
td 8
n
u
rgo 7
o
t
S 6
M
R
5
5
)a 4
P
(s 3
e
c
en 2
r
iffe 1
d
de 0
u
t
lip−1
m
A−2
(d)
MERRA
CFSR
ERA
EC−OP
2000
In Fig. 5a (full compilation), ERA features the largest RMS misfits at about 9 Pa,
accumulated through a significant overestimation of S1ð pÞ 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 S1 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.
The 1998 El Nin˜ o and its subsequent reversal during 1999–2001/2002 introduce
irregular tidal behavior and larger RMS values in all three reanalyses when validated
against the climatological in situ solution. Following our prescription of minimal
interannual S1 variability, we excluded model data up to 2002 from further consideration. A
sufficiently long averaging period, required to somewhat conform with the 20-year mean
S1 fit of MHB, might thus be realized by the overlapping 7-year window (2004–2010)
common to all models. Disregarding the partially deficient CFSR analysis, we finally
adopted the time span from 2004 to 2013 as the main analysis window for MERRA, ERA,
and EC-OP, while CFSR results were averaged through 2004–2010, and MERRA
excitation data for 2004–2010 were maintained as well, but only as a secondary option. This
duality for MERRA is motivated by the deterioration of RMS values (Fig. 5a) and the
moderate drifts in nutation estimates (Fig. 3a) as from the year 2010.
A brief numerical comparison of the resulting model pressure tide climatologies against
the 50-station set of S1 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 Du to their model
estimates. Median phase lag differences for MERRA and CFSR in Table 6 generally
confirm
Ray and Ponte (2003)
’s assumption of Du ¼ 5 , 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.
a Model tides are 2004–2013 averages except for CFSR (2004–2010)
4 Numerical Modeling of the Oceanic S1 Tide
4.1 Ocean Model Configuration
DEBOT (Einsˇpigel and Martinec 2015) is a recently developed ephemeris-forced,
barotropic time-stepping model conceived to study the effect of the ocean flow on Earth’s
magnetic field. In the present work, we create a spin-off of the model’s hydrodynamic core
for individual partial tides, with a rigorous treatment of SAL and a parameterized drag term
to account for the conversion from barotropic waves to baroclinic internal tides (IT) over
rough bottom topography. The one-layer shallow water momentum and mass conservation
equations define the horizontal velocity vector u and the tidal surface displacement f
ou
ot þ f z^
u ¼
gr f
CDkuku
H þ f
fEQ
fSAL
fMEM
P=gq
CIT u
H þ f þ AH r
r
of
ot ¼
r ½ðH þ fÞu
ð5Þ
ð6Þ
where f is the Coriolis parameter oriented along the local vertical unit vector z^, g is the
nominal gravitational acceleration, r signifies the spherical del operator, H is the resting
water depth, q is the average density of seawater, CD ¼ 0:003 denotes a dimensionless
drag coefficient in the standard expression for quadratic bottom friction, and CIT 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 fEQ, a combination of self-attraction/loading and ‘‘memory’’ elevations fSAL and fMEM
to realize the SAL scheme of
Arbic et al. (2004)
, as well as the atmospheric pressure tide
P ¼ S1ð pÞ. AH r r is a comparatively rigorous implementation of the horizontal turbulent
eddy viscosity with a second-order tensor r related to the Reynolds stress tensor; see
Einsˇpigel 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 AH set to the widely cited value of 103 m2 s 1. Our forward tidal solutions and
OAM results are insensitive to the exact value of AH , unless it is used as a tuning parameter
of inordinately large magnitude ( 105 m2 s 1); cf.
Arbic et al. (2004)
.
Equations (5) and (6) were solved by finite difference time-stepping on a 1=3 C-grid,
covering the latitude range from 78 S to 78 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 from the equator. The bottom topography was derived from the bedrock version of the
fully global 10 10 ETOPO1 database
(Amante and Eakins 2009)
by choosing average
values over each 1=3 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
Carre`re 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 las in both ip and op
components. However, in these simulations, also the RMS misfit of the gravitationally
forced constituents (M2 and O1; see below) to altimetry-based reference solutions,
(FES2012, Carre`re 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 S1.
Other aspects of the DEBOT configuration closely follow
Ray and Egbert (2004)
.
We prescribe equilibrium tidal forcing (fEQ) for M2 and O1 with amplitudes and solid
Earth tide corrections taken from Table 1 of
Arbic et al. (2004)
. The resulting
largermagnitude background variability appears to aid the fidelity with which S1 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 S1 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 S1 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 S1 (as well as M2 and O1) in terms of sea level elevation and
barotropic volume transports uH.
4.2 Effects of Self-Attraction and Loading (SAL)
Gravitational self-attraction and yielding of the solid Earth to the weight of the water
column (Hendershott 1972) are feedback effects to the tidal dynamics and included in
Eq. (5) as an additional equilibrium-like tide fSAL. This term can be related to the
(unknown) local tidal elevation f through convolution with the global SAL Green’s function G
(Ray 1998b)
fSALð/; kÞ ¼ qa2
ZZ
fð/0; k0ÞGðwÞsin/0d/0dk0
ð7Þ
where a is the Earth’s radius and w measures the angular separation of ð/; kÞ from the load
with spherical coordinates ð/0; k0Þ. For our 1=3 model, values of GðwÞ 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 G
is approximated by a simple scalar multiplication fSAL bf
(Accad and Pekeris 1978)
,
with b 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 fSAL. We applied the iteration method of Arbic et al.
(2004), initialized by the scalar SAL estimate using a nominal value of b ¼ 0:12 that is an
appropriate choice for diurnal tides; cf.
Parke (1982)
and Fig. 11 of Einsˇpigel and Martinec
(2015). Once this initial run is completed and harmonically analyzed, the tidal components
(sine and cosine terms) of M2, O1, and S1 are inserted into Eq. (7) in an intermediate offline
a Tidal solutions have been computed from our hydrodynamic model using atmospheric pressure forcing
from
Ray and Egbert (2004)
. Amplitudes are in units of 1023 kg m2 s 1 and cotidal phases are given relative
to Greenwich noon, consistent with the Doodson convention for the S1 phase as given in
Ray and Egbert
(2004)
. For the respective OAM formulas, refer to
Chao et al. (1996)
computation to derive a first solution of fSAL 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)
fMEM ¼ bðf
fPREV Þ
ð8Þ
that measures the departure of the tidal height f in the current (second) run from the
cumulative M2-O1-S1 elevation fPREV in the previous (first) run. Subsequent iterations are
performed in the same manner, drawing on continuously updated maps of fSAL and fMEM .
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 b optimized for
the diurnal band, our forward solutions of O1 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
S1 tide. Table 7 presents successively updated OAM mass values of S1 as obtained from a
three-times iterative DEBOT run with the pressure forcing S1ð pÞ 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 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 S1 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 Df1 to the FES2012 reference tide f~R
1 Computed as Df ¼
sffiRffiRffiffiffijffif~ffiffiffiffif~ffiffiRffijffiffi2ffidffiffiAffi, equivalent to the time-averaged expression of Arbic et al. (2004), with
2RR dA
grid points poleward of 66 and waters shallower than 1000 m excluded.
(complex sinusoid) are as large as 14.2 cm and 3.0 cm for M2 and O1, respectively. These
values translate to a mere 72 and 79 % of sea surface height variance explained. Moreover,
S1 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 ( 1 cm) or the
South China Sea ( 2 cm) that have no correspondence in both the altimetric and
hydrodynamic S1 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.
To increase the fidelity of our model tides and in particular S1, we implemented the
linear tidal conversion formulation of
Zaron and Egbert (2006)
as described and slightly
modified by Green and Nycander (2013). In this parameterization, the local drag
coefficient is explicitly proportional to the slope of the scattering topography
where C ¼ 50 is a non-dimensional constant, x denotes the frequency of the tidal motion,
and theoretical buoyancy frequencies N follow from the prescription of a horizontally
uniform abyssal stratification. Values of N at the ocean bottom (Nb) as well as vertical
averages (N) over the entire water column are calculated from Green and Nycander (2013)
CIT ¼ CHðrHÞ28Npb2Nx
Nb ¼ N0e H=1300
with N0 ¼ 5:24 10 3 s 1, and 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 x ¼ f , the Coriolis parameter) on the internal wave
propagation characteristics. Yet, through scaling by x, 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 x to X, though that
choice was found to improve the elevation accuracy of semidiurnal fringes (M2) as well.
The IT drag formulation of
Zaron and Egbert (2006)
rather relies on scaling arguments
than on a solid theoretical description of the topographically induced energy flux and thus
contains a free parameter (C) to optimize the performance of the scheme. For practical
reasons, we set C ¼ 50
(Green and Nycander 2013)
and applied a secondary independent
multiplier c at the order of Oð1Þ. With S1ð pÞ taken from
Ray and Egbert (2004)
, we could
choose c in such a way that our model emulates the OAM values of this reference study.
Alternatively, given the resemblance of S1 to the global character of K1 and O1, the RMS
misfit of forward-modeled diurnal gravitational constituents to altimetry-constrained
solutions can be optimized. Both criteria do not lead to fully rigorous tuning experiments,
as ocean dynamics vary from one tide to the other and allowance must be made for subtle
differences of our time-stepping model with respect to
Ray and Egbert (2004)
. In two
separate suites of 40-day simulations with forcing specified for either fM2; O1; S1g or the
purely gravitational combination of fM2; O1; K1g, c was varied in steps of 0.5 within a
range of 0.5–4. Tuning by RMS differences of K1 and O1 to the observed tide favored
c ¼ 1:5, whereas the best match with
Ray and Egbert (2004)
in terms of OAM was
achieved by c values in the vicinity of 3, although the eventual prograde annual nutation
results appeared to be only weakly dependent on the exact value of c (within about 10 las).
ð9Þ
ð10Þ
ð11Þ
1.23 (163 )
2.88 (304 )
2.23 (218 )
a Air pressure tides S1ðpÞ for MERRA, ERA, and EC-OP are averages over 2004–2013, while the CFSR run
draws on a 2004–2010 average. For brevity, the additional MERRA solution computed for the reduced 2004–
2010 window is not tabulated but given in terms of nutation in Table 10. Amplitudes are in units of
1023 kg m2 s 1 and cotidal phases are given relative to Greenwich noon
b S1ðpÞ from
Ray and Egbert (2004)
was deployed for the control run to validate our hydrodynamic model
configuration including the tidal conversion scheme
As a trade-off, we adopted c ¼ 2 as a ‘‘best estimate’’ for all of our S1 runs below. RMS
discrepancies to FES2012 produced by this setting are 5.6 cm (M2), 1.7 cm (O1), and
2.3 cm (K1), implying more than 93 % of sea surface height variance explained for each
tide; cf. similar statistics obtained by
Arbic et al. (2004)
with their 1=2 barotropic model.
S1 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 throughout and deviations in amplitude
are within 0:2 1023 kg m2 s 1. We calculated the corresponding contributions to the
prograde annual nutation by aid of a standard protocol noted below, obtaining
21:9 þ i46:4 las as a credible reproduction of the S1 excitation value 20:7 þ i54:8 las
implied by the OAM terms of
Ray and Egbert (2004)
; see Table 10. A moderate
underestimation of the op component in DEBOT ( 8 las) 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 S1 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 S1 and other diurnal
tides to established reference charts. On a side note, also the discrepancies to coastal tide
gauge estimates of S1 (next section) are markedly lower when internal wave drag is
parameterized.
4.4 Hydrodynamic Solutions and Validation with Tide Gauge Data
Tidal elevation charts obtained from numerical modeling are shown in Figs. 6, 7 and 8 for
MERRA, CFSR, and EC-OP, while ERA has been left aside as the solution with probably
the least accurate forcing data; cf. Sect. 3.3. Similarities with published S1 charts
(e.g.,
Dobslaw and Thomas 2005; Ponte and Vinogradov 2007)
are readily apparent and
particularly striking for our EC-OP model as compared to the S1 tide of
Ray and Egbert
(2004)
, who also employed ECMWF operational analysis data. Measured against Fig. 4 of
these authors, DEBOT appears to underestimate the tide in Baffin Bay and the Sea of
Okhotsk, probably due to differences in seafloor topography or the specification of
dissipative processes. Such small-scale deficiencies in high latitudes are, however, of little
relevance for the global OAM integrals.
All computed S1 realizations agree in terms of the global character of the tide, but
basinwide features can vary substantially in response to different pressure forcing data.
Specifically, the secondary peaks of S1ð pÞ around 60 S in the CFSR climatology (Fig. 2)
induce sea level signals in the Southern Ocean that exceed the corresponding tidal
20
15
10
5
0
360
300
240
180
120
60
0
60°N
30°N
0°
30°S
60°S
60°N
30°N
0°
30°S
60°S
60°E
120°E
180°
120°W
60°W
0°
60°E
120°E
180°
120°W
60°W
0°
Fig. 6 Amplitudes (top, in mm) and Greenwich phase lags (bottom, in deg) for the sea level signal due to
forcing by the S1 atmospheric pressure tide from MERRA (2004–2013 average). Cotidal phases are relative
to Greenwich noon
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 S1
tide in the Tropical Pacific, consistent with the overestimation of equatorial pressure
gradients as exposed by Fig. 5c.
Empirical knowledge of S1 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 S1 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
60°E
120°E
180°
120°W
60°W
0°
Fig. 7 Same as Fig. 6 but for S1ðpÞ from CFSR (2004–2010 average)
2 http://www.bodc.ac.uk/projects/international/woce/tidal_constants/. Accessed 9 October 2015.
60°N
30°N
0°
30°S
60°S
60°N
30°N
0°
30°S
60°S
60°E
120°E
180°
120°W
60°W
0°
20
15
10
5
0
360
300
240
180
120
60
0
few locations were retained in the equatorial Pacific and further thinning was applied to
higher-amplitude S1 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.
The collected harmonics are aggregate measures of both the radiational S1 ocean tide
and the much smaller gravitationally driven component, S1g. The latter must be removed
from the in situ data to rigorously compare with our numerical solutions of S1 that are
60°E
120°E
180°
120°W
60°W
0°
Fig. 8 Same as Fig. 6 but for S1ðpÞ from EC-OP (2004–2013 average)
60°N
30°N
0°
30°S
60°S
60°N
30°N
0°
30°S
60°S
60°E
120°E
180°
120°W
60°W
0°
20
15
10
5
0
360
300
240
180
120
60
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MERRA
CFSR
ERA
EC-OP
Amplitude \8 mma
RMSb
a Of 56 stations, 31 feature in situ amplitudes less than 8 mm
b Numbers in parentheses are amplitude-normalized RMS differences
solely forced by atmospheric pressure. To that end, we evaluated the Sg1 chart given in
Appendix ‘‘The Gravitational S1 Ocean Tide’’ at the locations of our 56 gauges and
changed the phase reference from the tide-generating potential to the simple radiational S1
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 S1 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 S1 signals at each gauge location were taken from the nearest pelagic point in
our 1=3 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 S1 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.
We have also attempted to express the varying accuracies of our hydrodynamic
solutions by global statistical measures in Table 9. Median absolute differences as well as RMS
misfits, given both as absolute and amplitude-normalized values, show little variations
among the four models if all 56 tide gauge locations are considered. This result conforms
with Fig. 9 inasmuch as the simulated tide at larger-amplitude sites tends to differ from the
observation in the same way for all models; see, e.g., Karachi, Port Victoria, Yakutat, or all
Australian stations. Somewhat more instructive statistics are obtained if the network is
limited to stations below a certain amplitude threshold. Table 9 specifies results for an
8-mm threshold that preserves 31 tide gauges, most of them being located at mid-ocean
islands. In this variant, MERRA outperforms all other models both in terms of MAD and
RMS values, of which the reduction from 1.9 mm to 1.6 mm is significant at the 0.15 level.
Results for the second MERRA run associated with the 2004–2010 pressure tide average
(not shown) are slightly inferior, comparable with the MAD statistics of the two ECMWF
models. Note also that the ERA tide, for which we have made reservations with regard to
the forcing data (Sect. 3.3), is among the best models in Table 9 owing to a particularly
good match with tide gauge estimates in the North Pacific.
4.5 Contribution of the Oceanic S1 Tide to Nutation
Global OAM integrals derived from our numerical modeling efforts are compiled in
Table 8 and exhibit a considerable scatter in accordance with the large-scale inter-model
differences noted in the previous section. There is, however, a broad consensus that the x
and y mass terms, i.e., the two single most important components with regard to nutation,
are in the order of 1.0 1023 kg m2 s 1 (160 phase lag) and 3.0 1023 kg m2 s 1 (0
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 T~p;wðrÞ; 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 S1 tide
(see Appendix A of Ray and Egbert 2004)
T þ 180
uH
ð12Þ
where T is Universal Time and uH are respective phase lag values as given in Table 8.
Demodulated equatorial OAM series were then cleansed from non-seasonal signals (that is,
the S1 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.
Table 10 summarizes the various estimates of the oceanic S1 effect in nutation. Formal
errors have been omitted as they are effectively zero given our usage of perfect sinusoids
Table 10 Periodic oceanic contributions to the prograde annual nutation (las) as inferred from the
modelspecific OAM values of Table 8a
Total
ip
op
Ray/Egbert
Control
MERRA
MERRAb
CFSRb
ERA
EC-OP
Mass
ip
a Results are split up into the contributions from tidal heights (mass term) and currents (motion term).
Inand out-of-phase components are referred to the fundamental arguments of nutation (Table 4) and the sign
convention is that of Koot and de Viron (2011)
b Forced by the respective pressure tide average from 2004–2010
for the angular momentum time series. Overall, the nutation results from all model runs are
reasonably consistent, ranging from 0 to 20 las in the ip terms and roughly 40 to 60 las
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 las, 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 S1ð pÞ (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 las. Nonetheless, the spread of
nutation values is significantly smaller than that of previous inter-model comparisons,
conducted, e.g., by
Brzezin´ ski (2008
) based on IB-corrected OAM values from much
coarser ([1 ) barotropic and baroclinic models. We have also mapped the fine-resolution
S1 tide of FES2012 to the prograde annual band, finding a nutation estimate of
2:1 þ i49:1 las 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
(Carre`re 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 las which has been corrected in the frame of the present study.
5 Comparison with Geodetic Observations
At an amplitude of 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 las), electromagnetic torques
( 14 þ i6 las for both core mantle and inner core boundaries), geodesic nutation
( 30 þ i0 las), and the angular momentum exchange of the solid Earth with the
gravitational ocean tide, S1g ( 21 þ i22 las); see also Table 2 of Brzezin´ ski et al. (2004). With
these contributions accounted for, theory and observation of the prograde annual nutation
produce a mismatch of 10:4 þ i108:2 las that has been attributed by MHB to the thermal
atmospheric S1 tide and, implicitly, to the radiational S1 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
nonlinearities 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 S1
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 þ i107 las. Corresponding SD in both ip and op components are 4 las but
probably underestimated and arguably better represented by the single-solution error of
7 las; cf. Table 1 of Herring et al. (2002).
Residual VLBI-based nutations obtained after reduction of known effects do not
necessarily provide a clean account of the rotational signal associated with the global S1 tide.
Both unconsidered Sun-synchronous effects as well as inaccuracies in the incorporated
relativistic or geophysical corrections at the prograde annual frequency might perturb
empirical S1 estimates. However, theoretical values of geodesic nutation are known to
great precision (Fukushima 1991), whereas anelastic and electromagnetic coupling
contributions to the S1 band are too small (\20 las) to leave room for significant changes
even if the MHB treatment of these effects is revised. The contribution from the
gravitational ocean tide is somewhat larger (see above) and in fact subject to uncertainties
owing to the manner in which it has been included in the nutation formalism. In detail,
MHB inferred a harmonic of 21 þ i22 las from OAM estimates of K1, P1, O1, and Q1
(Chao et al. 1996)
via scaling relationships that were optimized for the diurnal band on a
broad scale instead of particular tidal lines. We therefore recomputed the effect based on S1g
OAM integrals deduced in Appendix ‘‘The Gravitational S1 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 S1 tide, that is, T þ 295:66 þ 90
(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 þ i16:8 las.
This value is about 5 las 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 S1
terms given in Table 11.
Additional regard must be paid to the distortion of observed nutations through
Sunsynchronous 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.).
MERRA
MERRAb
CFSRb
ERA
EC-OP
Brzezin´ski et al. (2004)
Brzezin´ski (2011)
VLBI (MHB)
VLBI Koot and de Viron (2011)
1r error
In-phase
8:0
6:1
32:3
38:7
9:4
113.1
60:6
16:2
5:8
7
a MERRA, CFSR, ERA, and EC-OP results are superpositions of the harmonics from Tables 5 and 10. For
comparison, earlier estimates from
Brzezin´ski et al. (2004)
and
Brzezin´ski (2011)
are also shown and have
been multiplied by 1 to account for differences in the definition of nutation amplitudes. VLBI values, with
formal errors taken from Herring et al. (2002), have been cleared of the gravitational S1 tide influence by
using the results of Appendix ‘‘The Gravitational S1 Ocean Tide’’; see the text for further details
b Forced by the respective pressure tide average from 2004–2010
Single-session runs with our in-house VLBI software
(Bo¨ hm et al. 2012)
showed that a
spurious prograde annual nutation variability of about 20 las 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 30 las 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 las 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 S1 terms should be
accurate enough to serve as reference values for our geophysical model estimates.
This excitation balance is elaborated in Table 11 as well as in Fig. 10 and represents the
core result of our study. Both MERRA (either solution) and EC-OP estimates agree with
geodetic observations of the prograde annual nutation at the 10 las level, well below the
threefold SD of the VLBI solutions. A discrepancy of only 3 las is found between
MERRA (2004–2010) and the joint inversion residual of Koot et al. (2010), even though
such a close fit might be fortuitous considering the time variability of S1 excitation terms
3σ VLBI (Koot)
3σ VLBI (MHB)
MERRA
CFSR
ERA
EC−OP
0
20
40
60
80 100 120
out−of−phase (µas)
140
160
180
and the uncertainties of the involved numerical models. By and large, the consistency of
MERRA with VLBI data is in keeping with its good performance in comparison with
atmospheric and oceanic ground truth data. Further correlations between model-specific
in situ statistics and the nutation results in Fig. 10 are less obvious, but a closer inspection
of tide gauge estimates in Durban, Port Elizabeth, Esperance, and Mawson (Fig. 9)
revealed that ERA systematically underestimates the ocean tide in the South Indian Ocean,
by about 2 mm compared to the observed sea level signal and to other simulations
(MERRA, EC-OP). We implemented a split-up of the retrograde OAM mass term into
contributions from different basins, showing that the broadscale S1 features in the South
Indian Ocean have indeed a significant bearing on the op component of the ocean-driven
prograde annual nutation. The 30-las deviation of the ERA model from, e.g., MHB’s
nutation estimate is thus attributed in large part to a regional signal loss in terms of tidal
elevation in the southern hemisphere.
Table 11 also places our final results in the context of previous geophysical modeling
efforts by Brzezin´ ski et al. (2004) and Brzezin´ ski (2011). In these studies, a balance with
VLBI observations has been mostly impeded by deficiencies in the ip component of the
modeled S1 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 S1 tide. Specifically, the model
deployed by Brzezin´ ski (2011) has been optimized for a range of timescales and full
baroclinic variability, for which coarse horizontal resolutions (1.875 ) greatly reduce
computational costs. We tested the impact of a 1 discretization in DEBOT, obtaining
somewhat anomalous S1 charts and increasingly negative ip components of the
oceandriven prograde annual nutation ( 15 las decrement). Brzezin´ ski et al. (2004) analyzed
the output of a barotropic model with a comparably coarse domain representation (1.125
spacing) but also with SAL dynamics neglected. This omission alters nutation amplitudes
by roughly 30 las, 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
S1 tidal excitations of nutation in the order of 3.5 mm at the Earth’s surface ( 120 las)
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 S1 tide in the ocean. Atmospheric contributions
averaged over 2004–2013 are 21:9 þ i45:8 las (MERRA) and 22:4 þ i67:5 las (EC-OP)
and combine well with the respective oceanic estimates (13:9 þ i60:2 las,
13:0 þ i54:3 las) to match the VLBI-observed S1 terms within 10 las. 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 las). In particular, the atmospheric mass term is among the least robust
components of the global S1 excitation given its dependence on the small second-order tesseral
surface pressure wave. Table 5 documents an inter-model spread of 25 las (excepting
CFSR) for the pressure-driven nutation, comprising also uncertainties due to inter-annual
S1 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 las) 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 , obtaining nutation harmonics of only a few las 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 S1/S2 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
S1 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 S1 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 las appears to be a plausible
estimate for the irregular departures from a simple sinusoidal S1 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.
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. Brzezin´ski, 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. Einsˇpigel thanks the Charles University in Prague for supporting
him under Grant SVV-2015-260218.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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1
360
300
240
180
120
60
Appendix 1: The Gravitational S1 Ocean Tide
Following Sect. 2c of
Ray and Egbert (2004)
, a modern-day chart of S1g in the global ocean
is readily computed from observations of the gravitational P1 and K1 tides, which are
separated from the S1 band by only 1 cpy. Tidal heights of K1 were extracted from the
FES2012 atlas on a 1=16 mesh, moderately downsampled, and scaled to local Sg1
amplitudes using a factor of 1.98/368.74, that is, the ratio of gravitational potentials at S1
and K1. Greenwich phase lags were calculated as averages from both the K1 and P1 charts
and are illustrated together with the amplitudes of Sg1 in Fig. 11. Associated barotropic
60°E
120°E
180°
120°W
60°W
0°
Fig. 11 Cotidal elevation charts for the gravitational S1g tide in the global ocean as computed from FES2012
solutions of K1 and P1. Amplitudes (in mm) are shown in the upper panel, Greenwich phase lags ( ) in the
lower panel
60°N
30°N
0°
30°S
60°S
60°N
30°N
0°
30°S
60°S
60°E
120°E
180°
120°W
60°W
0°
currents follow from the velocity (u) grids of K1 and P1 in the FES2012 model, based on
the same interpolation procedures as for the local elevation. To serve Sect. 5, we have
mapped heights and currents of S1g to global OAM values as documented in Table 12. Note
that these harmonics might be also derived from a direct application of admittance
relationships to the angular momentum values of the K1 and P1 tides.
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