Pergamon
PIh S0079-6611(98)00010-X
Prog. Oceanog. Vol. 40, pp. 399-421, 1997
© 1998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
0079-6611/98 $32.00
Oceanic tidal angular momentum and Earth's rotation
variations
B.F. CHAO~'and R.D. RAY 2
1Space Geodesy Branch, NASA's Goddard Space Flight Center, Greenbelt, Maryland 20771,
USA
eHSTX, Code 926, NASA's Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
Abstract - Luni-solar tides affect Earth's rotation in a variety of ways. We give an overview
of the physics and focus on the excitation of Earth rotational variations by ocean tides under the
conservation of angular momentum. Various models for diurnal and semidiurnal tidal height and
tidal current fields have been derived, following a legacy of a number of theoretical tide models,
from the Topex/Poseidon (T/P) ocean altimetry data. We review the oceanic tidal angular momenta
(OTAM) predicted by these T/P models for the eight major tides (Q~, O~, P~, K~, N2, M2, $2, K2),
and their excitations on both Earth's rotational speed variation (in terms of length-of-day or UTI )
and polar motion (prograde diurnal/semidiumal components and retrograde semidiurnal
components). These small, high-frequency effects have been unambiguously observed in recent
years by precise Earth rotation measurements via space geodetic techniques. Here we review the
comparison of the very-long-baseline-interferometry (VLBI) data with the T/P OTAM predictions.
The agreement is good with discrepancies typically within 1-2 microseconds for UTI and 10-30
microarcseconds for polar motion. The eight tides collectively explain the majority of subdaily
Earth rotation variance during the intensive VLBI campaign Cont94. This establishes the dominant
role of OTAM in exciting the diurnal/semidiurnal polar motion and paves the way for detailed
studies of short-period non-OTAM excitations, such as atmospheric and oceanic angular momentum
variations, earthquakes, the atmospheric thermal tides, Earth librations, and the response of the
mantle lateral inhomogeneities to tidal forcing. These studies await further improvements in tide
models and Earth rotation measurements. © 1998 Elsevier Science Ltd. All rights reserved
1. KINEMATICS AND MEASUREMENTS OF EARTH'S ROTATION
Earth's spin has long been h u m a n k i n d ' s best timekeeper, at least until recently. The time it
keeps is called Universal Time (UT), or UT1 after an empirical correction to allow for polar
motion (see below). The first evidence indicating the non-uniformity of UT, i.e. that the Earth
actually spins not at a constant rate, emerged when UT was compared against the Ephemeris
Time kept by motions of solar system bodies according to Newton's gravitational law, or against
the time kept by the best mechanical clocks. The advent of the atomic clocks in the 1950s
provided the truly accurate and uniform time-keeping standard known as the atomic time (AT);
and the difference between UT1 and AT became the standard quantity that reflects the Earth's
spin variation as a function of time. The spin variation is often conveniently expressed in terms
of the length-of-day (LOD), which is proportional to the time derivative of UT1, if the time
scale of interest is longer than one day.
The Earth's spin, of course, is only one component, namely the axial component, of the 3399
400
B.F. CHAO AND R.D. RAY
D Earth rotation vector. The other two equatorial components together give the orientation of
the rotation axis, which also varies slightly with time. The variation in the rotation axis orientation relative to the terrestrial reference frame is known as polar motion. Polar motion was
discovered over a century ago, manifesting as a variation in the latitudes of observatories.
Prior to the space age, Earth rotation was monitored by conventional astrometry using optical
instruments (e.g., some "modern" astrolabes, optical and photographic zenith telescopes). Since
the 1970s the advent of several space geodetic techniques has revolutionized the observation
(for a review see, e.g., EUBANKS,1993). These include satellite Doppler tracking, lunar laser
ranging, satellite laser ranging (SLR), very-long-baseline interferometry (VLBI), and in recent
years the Global Positioning System (GPS). Both VLBI and SLR techniques have been the
major workhorses during the last two decades, and now achieve on daily basis accuracy of a
fraction of a milliarcsecond (1 milliarcsecond of Earth orientation corresponds to about 3 cm
of distance projected onto the Earth surface). Today the measurements are made under the
auspices of the International Earth Rotation Service headquartered in Paris.
Figure 1 provides a historical perspective of all "standard" LOD data. Here three time series
are concatenated: historical data before 1962 at half-year intervals (McCARTHYand BABCOCK,
1986), the official BIH Bureau International de l'Heuer) 5-day solutions since 1962, and the
I
I
I
I
I
I
~0
-2
-3
-4
182(
I
1
I
I
I
I
I
I
1840
1860
1880
1900
1920
1940
1960
1980
Year
Fig. 1. The length-of-day variation since 1820. The curve is a concatenation of three (smoothed)
time series (see text Section 1 ). Tidal signals are largely removed or absent in this record.
2000
Oceanic tidal angular momentum and Earth's rotation variations
401
daily "Space95" series (GRoss, 1996) based on modem space geodetic observations since 1976.
The latter two series have sufficiently high temporal resolution to show prominent seasonal
variations, while all series have been subject to smoothing to a certain extent. The tidal signals
to be studied in this paper are virtually absent from Fig. 1, either because they were averaged
out (as in older observations), or removed via theoretical modeling (as in the case of longperiod tides in newer data), or simply not observed or aliased (as even in modem data which
are far from a continuous sampling as far as the short-period tides are concerned).
Figure 2 plots the Space95 series of the pole position during 1980-1995 in the x (along the
Greenwich Meridian) and y (along the 90°E Longitude) coordinates of the rotating Earth frame.
The quasi-circular, prograde motion manifesting a 6-year "beating" is the combination of the
annual wobble and the 14-month Chandler wobble. Also evident is the polar drift, which is the
gradual departure of the polar motion away from the conventional North Pole, which is by
definition the mean pole position during 1900-1905, indicated here as the origin of the coordinates. Again, in Fig. 2 much of the short-period tidal signals are either smoothed out or aliased.
The primary Earth rotation data that will be used in this paper are those obtained by various
VLBI networks during the last 15 years, typically with 24-hour observing sessions separated by
gaps of a few days. Gappy but unsmoothed, these raw data do allow the extraction of the shortperiod tidal signals because these signals are temporally coherent. Independent algorithms have
been applied by various research groups (SOVERS et al., 1993; HERRING and DONG, 1994; GIPSON, 1996), resulting in spectral estimates of the diumal/semidiumal tidal parameters. These
Polar Motion, 1979-1995
i
J
i
v
i
0
-10o
-200
tO
0
.~_
-300
E
>'-400
-500
-600
-300
-200
- 100
I
I
I
I
0
100
200
300
400
x (milliarcsecond)
Fig. 2. The (smoothed) polar motion path for 1980-1995 (see Section 1 for detail). Tidal signals
are largely removed or absent in this record.
402
B.F. CHAOAND R.D. RAY
estimates for the eight major diurnal/semidiurnal tides (Table 2) are given in the upper half of
Tables 3 - 5 to be discussed below. Similar solutions from SLR measurements to the LAGEOS
satellite have also been published (WATKrNS and EANES, 1994; PAVLIS, 1994) but will not be
used here as they have previously by CHAO et al. (1996).
A caveat is in order here. Tidal variations are characterized by precise tidal periodicities. At
any given periodicity the polar motion can be described, in lieu of its x and y components, as
a wobble composed of two opposite circular motions around a "mean" pole position (MuNK
and MACDONALD, 1960): prograde if counterclockwise as viewed from above the North Pole,
and retrograde for the opposite. In this paper we study the tidal influences on UTI and polar
motion, but we exclude the retrograde diurnal wobbles in the polar motion. The reason is the
empirical ambiguity between polar motion and nutation, which is defined as the variation of the
rotation axis orientation relative to the inertial space. The polar motion and the nutations are
simply different descriptions of the same physical phenomenon. They are well-defined observables, but geodetic measurements (which observe the relative rotation) cannot distinguish them.
Traditionally, they are separated only based on the empirical fact that they have distinct time
signatures: seen by an Earth observer, nutations have diurnal periods while polar motion is of
longer time scales. As the measurement improves in temporal resolution, however, the nutations
become totally inseparable in geodetic measurement from the retrograde diurnal wobbles as they
have the same (tidal) periods. Further complications arise from the existence of the free-corenutation resonance within the retrograde diurnal band. A thorough study of this band is beyond
the scope of the present paper.
2. DYNAMICS AND TIDAL EXCITATION OF EARTH'S ROTATIONAL VARIATIONS
The luni-solar tides influence the Earth's rotation in a variety of ways (e.g., MUNK and MACDONALD, 1960; LAMBECK, 1980). The best-known astronomical variation is probably the astronomical precession and nutation of the Earth's rotational axis in space. The precession is a
motion of the rotation axis which traces out in space a complete cone of radius equal to the
obliquity (present value at 23.5 °) every 25,800 years. It owes its origin to the luni-solar tidal
torque exerted on Earth's permanent equatorial bulge, in a classical condition giving rise to
precessional motion similar to a spinning top - hence the familiar statement that "the Earth
spins like a top". The astronomical precession coupled with external gravitation from other
planets in the solar system slightly perturb the Earth's rotational and orbital parameters in a
quasi-periodic way. Known as the Milankovitch cycles and believed to be the pacemaker of the
ice ages (HAYs et al., 1976), these perturbations typically have periodicities ranging from tens
to hundreds of thousands of years.
Nutations are in a sense shorter-period perturbations in the precession. They are the result of
periodic variations in the orbital elements, and hence in the torque. For example, the largest
nutation term has a period of 18.6 years due to the precession of the Moon's orbit. Other
relatively prominent nutation terms have periods that are half of the main tidal periods, such as
semi-annual and fortnightly. As stated above, nutation periods map into retrograde diurnal periods in the rotation Earth frame.
Similarly, tidal torque exerted on the equatorial ellipticity makes the Earth "librate" like a
physical pendulum (CHAO et al., 1991). Two types of librations arise: semi-diurnal UT1 variations (spin librations), and prograde diurnal wobbles (polar librations). Sharing the same tidal
periods, the librations are relatively small in amplitude but non-negligible in the highest precision
Earth rotation measurements to date (see below).
Oceanic tidal angular momentum and Earth's rotation variations
403
A n o t h e r important astronomical variation is the tidal braking. T h e rotation, coupled with the
E a r t h ' s non-instantaneous response to the tidal forcing (due to inelasticity, - e.g., fluidity of
the o c e a n and the core, viscosity o f the mantle, etc.), "carries" the tidal b u l g e a w a y f r o m the
E a r t h - M o o n (or - S u n ) line by a small angle. This manifests itself as a tidal phase lag to an
o b s e r v e r on Earth. Rotational kinetic energy is dissipated via friction in the process, and the
E a r t h ' s rotation slows d o w n as a result. A n a l y s e s of historical accounts of astronomical events
such as s o l a r / l u n a r eclipses and lunar occultations, and m o d e m lunar laser-ranging data have
yielded the rate o f the secular lengthening o f L O D o f a p p r o x i m a t e l y 2 ms per century (e.g.,
LAMBECK, 1980; STEPHENSON and MORRISON, 1995).
The a b o v e tidal effects are caused by direct external tidal torques as the driving force (see
T a b l e 1 ). T h e tidal effects under consideration in this paper, on the other hand, are those caused
by the tides in the ocean acting as an internal " e x c i t a t i o n " source via the conservation of angular m o m e n t u m .
U n d e r the conservation o f angular m o m e n t u m , the angular m o m e n t u m carried by any geophysical process that i n v o l v e s mass transport on or within the Earth will be reflected in the
solid E a r t h ' s rotation. W e say that these g e o p h y s i c a l processes " e x c i t e " Earth rotational variations. T w o m e c h a n i s m s are i n v o l v e d in this excitation process (e.g., MUNK and MACDONALD,
Table 1. Dynamic effects of the luni-solar tides on Earth's rotation, and their approximate magnitudes.
Type
Astronomical
(external
torques)
Earth system
Phenomenon
Rigid, A = B = C
-0 -
Rigid, C > A = B
Precession
Nutation
Milankovitch
(modified)
(elastic core)
Geophysical
("mass"+
"motion", torque
free)
Period
Rotation axis*
Rotation
speed**
26 Kyr
long-period
41 Kyr, etc.
47 degree cone (r)
10 arcsec (r)
+ 2 degree
(obliquity)
-0 -0 -0 -
diurnal
semidiumal
0.02 mas (p)
-0 -
-0 2 kLs
Rigid, C > A4:B
(elastic core)
Libration
(modified)
Anelastic, Ocean
Tidal braking
secular
-0 -
20 /zs/yr
Axial symmetric
(elastic solid
Earth, core
decoupling)
Zonal (2.0)
Tesseral (2,1)
Sectoral (2,2)
long-period
diurnal
semidiurnal
-0 several mas (r)
-0 -
300 /xs
-0 -0 -
Ocean
Spin-wobble
coupling,
w/dissipation
long-period
diurnal
semidiurnal
several mas (r/p)
0.2 mas (p)
0.3 mas (r/p)
50 ms
20 ms
20 ms
Atmospheric
Thermal "tides"
daily, halfdaily
0.01-0.1 mas (?)
1-10 txs (?)
* Polar motion (r: retrograde, p: prograde), except otherwise indicated.
** UT1 for diurnal/semidiurnal, LOD otherwise.
404
B.F. CHAOAND R.D. RAY
1960): (i) mass redistribution that changes the Earth's inertia tensor (referred to as the "mass"
term); (ii) motion that evokes angular momentum relative to the solid Earth as a whole (referred
to as the "motion" term). In the case of ocean tides, the mass and motion terms are represented
by the tidal height variations and the tidal currents, respectively. They are collectively called
the oceanic tidal angular momentum (OTAM). In the presence of OTAM the conservation of
the total angular momentum of the solid Earth-ocean system dictates changes in the solid Earth's
rotation, in both UT1 and polar motion.
These excitations are also summarized in Table 1. In an axially symmetric Earth, only the
zonal, long-period tidal forcing can affect UT1 (or equivalently LOD) because of symmetry
properties (mathematically expressed as the orthogonality of spherical harmonics). The solid
Earth is indeed nearly axially symmetric and avails a large quantity of mass for tidal deformation
(typically an order of magnitude larger than the oceans tides in terms of mass transport), hence
long-period tidal signals are very prominent in LOD records; for example, the Mf amplitude is
as large as 350 microseconds (ms). These signals have been well studied by various investigators
over the years since H. Jeffreys's pioneering work in the 1920s. The discrepancies between the
observed and the theoretically predicted amplitudes and phases provide geophysical information
about core-mantle coupling, ocean tide contribution, and mantle anelasticity (e.g., WAHR and
BERGEN, 1986; CHAO et al., 1995a). By the same token, only the tesseral, diurnal tidal forcing
can excite polar motion in an axially symmetric Earth. Such retrograde diurnal polar motion,
although prominent, "alias" with nutations in geodetic measurements and will not be discussed
as stated above.
Secondary tidal effects on Earth rotation arise owing to asymmetries in the Earth's mass
distribution. The largest such asymmetry effect comes from the ocean tidal deformation and
currents, subject to the irregular ocean-continent geography and non-equilibrium behavior. Thus,
although the diurnal and semidiurnal ocean tides are produced respectively by the tesseral and
sectoral tidal potentials, they generate net zonal deformation and currents which in turn excite
UTI variations at diurnal and semidiurnal periods. Similarly, polar motion is excited by tesseral
tidal components generated by other types of ocean tides at all tidal periods. Those at the long
periods are discussed in GROSS et al. (1997, in this issue). Here we will review the recent
findings about diurnal/semidiurnal Earth rotation variations and present their comparisons with
OTAM predictions based on ocean tide models.
From their definition, the three components of the OTAM vector are computed by integration
over the ocean as follows. The axial (or z) component of the tidal height (mass) term is
c3(t)=a4p~off~(~,t)sinZOd~
(la)
ocean
while the corresponding tidal current (motion) term is
ha(t)=a3pffu(n,t)H(S~)sinOdn.
(lb)
ocean
The two equatorial (x and y) components in the complex form of x + iy consist of the tidal
height term given by
Oceanic tidal angular momentum and Earth's rotation variations
c(t)=-a4ptoff~(~,t)sinOcosOe'~dO
405
(2a)
ocean
and the tidal current term by
h(t)=-aapf
f[u(~,t)cosO+iv(O,t)]H(f~)eiXd~.
(2b)
ocean
In the above equations, a and to are Earth's mean radius and mean rotational speed; p = 1035
kg m -3 is the mean density of seawater; f~ is an abbreviation for the colatitude 0 and longitude
)t; df~ = sinOdOd)tis the surface element for the integral over the oceans. The tidal height relative
to the seabed is if; u and v are the eastward and northward speed, respectively, of the tidal
current assumed to be barotropic and hence uniform over the water column with depth H. These
tidal variables are provided by tide models described in the next section. In the end, of course,
it is the temporal variation of OTAM that concerns us.
Note that the integrand in ( l a ) may be broken down into two terms: one proportional to the
degree-2 Legendre function P2, the other integrates to become the total water mass change by
the tides. The latter is identically zero because the tides conserve mass. We therefore replace
(la) with an integral of P2. Further discussion of this as to how it relates to possible nonconservation of mass in tide models is in RAY et al. (1997).
While Eq. (2a,b) is expressed in terms of the x and y components, as stated earlier the two
components can be converted into the equivalent prograde and retrograde wobbles for a given
periodicity such as the tides. Our phase convention for prograde and retrograde motions follows
MUNK and MACDONALD (1960, Eq. (6.7.5); for details see CHAO et al., 1996).
A proper linear combination of c3 and h3 that takes into consideration various geophysical
effects yields the axial OTAM's excitation of Earth's spin variation, according to angular
momentum conservation. This excitation, when integrated w.r.t, time, gives the corresponding
excitation of UT1 variation used in the study below. Similarly, a linear combination of c and
h yields the equatorial OTAM's excitation of the polar motion. The geophysical effects that
must be considered include the elastic loading and yielding effects, the non-participation of the
fluid core in the excitation process, and for the polar motion the resonance effects of the Earth's
free Chandler wobble (at 14 month period) and free core nutation (at a retrograde diurnal period,
see, e.g., SASAO and WAHR, 1981). The conversion of OTAM integrals into UT1 and polar
motion variations has been considerably clarified by GROSS (1993), who corrects a number of
oversights and inconsistencies in previous papers.
Note that using this "momentum" approach, one need not address the actual mechanism coupling the ocean and mantle and thereby effecting the angular momentum transfer. As YODER et
al. (1981) discussed, several torques are involved: friction on the sea floor, fluid pressure on
topographic gradients such as the continental slopes, ocean loading, and gravitational attraction.
SEILER (1991 ) shows that bottom friction is negligible and that the primary coupling mechanism
is the pressure torque.
406
B.F. CHAt AND R.D. RAY
3. TIDE MODELS
In this section we review various attempts to determine diurnal and semidiurnal OTAM from
ocean tide models, beginning with the earliest work and extending to the most recent work we
are aware of. The first serious work, by YODER et al. (1981), was published well before subdaily
rotation measurements were possible. In fact, Yoder et al. stressed that such measurements "may
prove elusive," but within a decade they were attained with VLBI (DONG and HERRING, 1990).
The ocean-tide models available to Yoder et al. - early models from the 1970s by Hendershott,
Zahel, and Parke - were not very accurate and they disagreed among themselves by large
amounts. Nonetheless Yoder et al. were able to deduce likely tidal variations in UT1 of order
20 to 70 microseconds and in polar motion of order 200 microarcseconds (/xas), which is essentially correct. These authors, along with WAHR (1979) previously, argued that Earth's core will
be decoupled from the mantle at these high tidal frequencies and that the effective inertia tensor
will be that of the mantle alone.
In computing Earth-rotation effects, YODER et al. (1981) considered only the mass (tidal
height) terms, but shortly thereafter BROSCHE (1982) pointed out that for M2 the motion (tidal
current) terms dominate. This point was expanded upon by BAADER et al. (1983). Subsequent
predictions based on various numerical ocean models include BROSCRE et al. (1989), SELLER
(1991), DICKMAN (1993), and SEILER and WONSCH (1995). These works were based on purely
numerical (or theoretical) ocean models, independent of tidal observations. Dickman's approach
used low-degree spherical harmonic expansions, and, owing to the difficulty of realistically
modeling dissipation in this way, the deduced Earth-rotation predictions were inaccurate. Much
better agreement with observations is obtained with Seiler's model, although even there the
discrepancy in M2 UT1 variations is about a factor of 2 (see, for example, Table 2 of HERRING
and DONG, 1994; also RAY et al., 1994a; CHAt et al., 1996).
The tidal community now has well over three decades of experience in numerical modeling
of global tides. The experience overall has proved to be sobering, and has served primarily to
underscore the extreme difficulty of the task. Indeed, when the basic mechanisms of energy
dissipation are still being debated (e.g., MUNK, 1997, in this issue), it is not surprising that purely
numerical global ocean tide models are found wanting. Models that are tied to observations in
some manner are normally more realistic and accurate. SCHWIDERSKI(1980), who assimilated
harmonic tidal constants from over 2000 coastal gauges by a method that he called "hydrodynamic interpolation," produced a model which was more accurate than any other at that time and
was widely adopted by the geophysical community until it was recently superseded by models
based on satellite altimetry.
Figure 3 compares M2 cotidal charts of the central Atlantic Ocean for the SEILER (1991) and
SCHWIDERSKI (1980) models along with a recent one based on Topex/Poseidon (T/P) satellite
altimetry. Extensive comparisons to in situ and other data have shown that the T/P models are
the most accurate models now available (SHUM et al., 1997). As the figure shows, Seiler's M2
amplitudes are about a factor of 2 too large in the mid-Atlantic off Brazil, while the phases
show an amphidrome in the north that should be much farther west. At the scale of these charts,
Schwiderski's model shows good agreement with the satellite solution. In fact, however, more
detailed comparisons (e.g. SCHRAMA and RAY, 1994, Fig. 1) show that this area is one of the
(open-ocean) regions where Schwiderski's model is in greatest error, with M2 errors approaching
15 cm in some locations. But at this scale, these errors are difficult to detect, and the overall
structure appears qualitatively correct.
The more accurate representations of the ocean have, understandably, shown better agreement
Oceanic tidal angular momentum and Earth's rotation variations
407
Table 2. Basic properties of the eight major diurnal and semidiurnal tides studied in this paper
Tide(type) Period(hr)
Diurnal
Qi (L.
Elliptic)
O1 (L.
Princ. )
Pt (S.
Princ.)
KI (S/L
Declin. )
Semidiurnal
N: (L.
Elliptic )
M 2 (L.
Princ. )
$2 (S.
Princ. )
K~ (S/L
Declin. )
Alias
Period*
(day)
Potential***
(cm)
Doodson argument**
r
s
h
p
26,868
9.13
1
- 2
0
1
- 90 °
5.02
25.819
13.66
1
- 1
0
0
-90 °
26.22
24.066
182.62
1
1
- 2
0
- 90 °
12.20
23.935
m
1
1
0
0
+ 90 °
36.87
12.658
9.13
2
- 1
0
1
0°
12.10
12.421
13.66
2
0
0
0
0°
63.19
12.000
182,62
2
2
- 2
0
0°
29.40
11.967
ac
2
2
0
0
0°
7.99
* as observed in inertial space.
** coefficients listed for the following angles:
--mean lunar Greenwich time
s--mean longitude of Moon
h--mean longitude of Sun
p--mean longitude of lunar perigee
Note: If t is mean solar time, then r = t - s + h.
*** from CARTWRmHTand TAYLER ( 1971 ).
with Earth-rotation observations. RAY et al. (1994a) adopted Schwiderski's height models for
deriving tidal UT1 estimates, and after inferring corresponding currents, they obtained U T I
h a r m o n i c constants that agreed with V L B I analyses to about the 2/xs level. Recent results (CHAO
et al., 1995b, 1996) have relied on T/P altimetry. Although h y d r o d y n a m i c theory must still be
invoked to determine tidal currents, the direct altimetric constraints on tidal heights provide a
distinct advantage w h e n attempting to unravel the ocean t i d e ' s influence on Earth rotation.
For the remainder of this paper we use only tide models based on analyses of T/P altimeter
data. Specifically, we discuss three models which, following CHAO et al. (1996), are referred
to as Models A, B, and C. Model A is an updated version (9405) of that described by SCHRAMA
and RAy (1994); Model B is an updated version (941230) of that described by RAY et al.
(1994b); Model C is T P X O . 2 by EGBERT el al. (1994). Models A and B are both the results
of empirical analyses of the T/P data, the former using a b i n n i n g method, the latter using expansions in special p r e c o m p u t e d oceanic n o r m a l modes. For both models, tidal currents are derived
from the height fields by solving the Laplace m o m e n t u m equations, a u g m e n t e d to account for
ocean self-attraction and loading (CARTWRIGHTe t al., 1992). Model C is an assimilation m e t h o d
that incorporates T/P crossover data into a finite-difference h y d r o d y n a m i c model, using a global
inversion that m i n i m i z e s the misfit to both data and d y n a m i c a l equations according to a priori
408
B.F. CHAOAND R.D. RAY
30
0
30
I
6O
I
I
I
30
I
I
60"
40 °
20 °
60 °
40 °
20 °
Fig. 3. M2 charts of the Mid-Atlantic Ocean from (left) SEILER(1991), (middle) SCHWIDERSKI
(1980), (right) the Topex/Poseidon based solution of RAY et al. (1994b, here Model B). Dashed
lines denote amplitude contours in cm; the contour interval is 25 cm on the left, 20 cm for the
others. Solid lines denote cophase contours; the contour interval is 1 lunar hour on the left, 30°
for the others.
covariances. In the polar regions not overflown by T/P, all three models are supplemented by
hydrodynamic models.
To give an idea of the complicated nature of tidal currents, we show in Fig. 4 charts of the
barotropic volume transports for the M2 and Kj tides of Model C (E6BERT et al., 1994). Volume
transports (current velocities multiplied by local ocean depth) are the appropriate quantity for
insertion into the OTAM integrals ( l b ) and (2b). Figure 4(a,b) give the M2 transports in-phase
and in quadrature with the equilibrium argument at Greenwich; Fig. 4(c,d) give similar diagrams
for K1. (In other words, Fig. 4(a) gives the M2 transports when the mean moon crosses the
Greenwich or 180 ° Meridians; Fig. 4(b) gives the M2 transports 3 lunar hours later. Figure 4(c)
gives the KI transports when the vernal equinox passes the 90 ° W Meridian; Fig. 4(d) gives
the K~ transports 6 sidereal hours later.) Corresponding diagrams of the tidal heights are given
in color plates in EGBERT et al. ( 1 9 9 4 ) , which, of course, are similar to other tidal charts readily
available in many publications.
Although the transport diagrams of Fig. 4 are relatively complicated, simple considerations
of continuity allow one to infer relationships to height. Consider, for example, Fig. 4(a): the
great flux away from the mid-Indian Ocean suggests that high tide there occurred a short time
before, while flood tides are occurring in the Bay of Bengal, the Timor Sea, and elsewhere, while
ebb tides are occurring in the Gulf of Panama, the Gulf of Alaska, and elsewhere. Examination of
cotidal height charts indeed shows that the tidal phase in the mid-Indian to be about 260 °
(indicating high tide about 3 hours before Fig. 4(a)); similarly the height phases indicate high
tides in the Bay of Bengal and Timor Sea a few hours later, and high tides in the Gulfs of
Panama and Alaska a few hours before.
M2 transports are fairly robust throughout the global ocean, excepting perhaps in the South
Pacific. In keeping with the well-known weakness of diurnal tides in the Atlantic Ocean, K~
shows exceeding small vectors there. Kj is largest in the North Pacific and around Antarctica,
where all diurnal tides take on a strong Kelvin-wave character. An important point is that both
diurnal and semidiurnal tides display relatively weak transports in shallow seas even though
current velocities there may be two or more orders of magnitude greater than in the deep ocean;
Oceanic tidal angular momentum and Earth's rotation variations
409
z
k~
t~
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z~
e©
©
©
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e.,
¢.)
C
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.=.
"-5
..=
[..,
410
B.F. CHAO AND R.D. R/~',
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v
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o
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[-..
O
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O
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e,,
,fi
[-..
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Oceanic tidal angular momentum and Earth's rotation variations
411
e~
e-
~g
5,.-:.
0
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b-
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~2
3
.=_
E
g=
,4
d~
412
B.F. CHAO AND R.D. RAY
0
0
0
[..-,0
O
0
2
0
E
,4
Oceanic tidal angular momentum and Earth's rotation variations
413
this is of great benefit when evaluating OTAM integrals, since these shallow-sea velocities are
not well mapped with altimetry.
4. RESULTSAND COMPARISON
As discussed in Section 2, one needs the tidal height field as well as the tidal current field
to compute OTAM. Figure 5 illustrates a break-down of the tidal height and current contributions
for each individual ocean basin according to Model C. In addition, a small contribution from
the Arctic Ocean, which is not available from T/P models, is added according to Ray et al.
(1997) based on KOWALIKand PROSHUTINSKY(1994). The sum of these phasor vectors is, of
course, the total OTAM quoted in Tables 3-5 - for M2 UT1, K1 UT1, M2 retrograde polar
motion, and K~ prograde polar motion. Note that the current terms in general contribute more
than the height term, a fact first realized by BRoscn~ (1982) for the short-period OTAM,
especially the semidiumals. Also note the large cancellations among the basin contributions in
a nearly random-walk fashion. For example, the entire Pacific Ocean has a relatively minor
contribution to the M2 UT1, with height and current terms each acting to cancel the other. For
Kj UT1 the Atlantic is a minor contributor, as may be expected from Fig. 4, and the Indian
Ocean currents and North Pacific currents and height are the dominant influences. The Indian
Ocean and the entire Pacific Ocean are the major contributor to K1 prograde polar motion, while
all basins appear important in M2 retrograde polar motion. Seiler's large errors in the Atlantic
(Fig. 3) and the Atlantic's dominance in M2 UT1 may help explain why the Brosche et al.
estimates for M2 UT1 are about a factor of 2 too large, while their M2 polar motion, although still
somewhat too large, agrees more closely with observations (see Table 7 of CHAO et al., 1996).
Our final T/P OTAM results are given in the lower halves of Tables 3-5: Table 3 for UT1,
Table 4 for prograde polar motion, and Table 5 for retrograde polar motion (excepting the
observations for the diurnals). In comparison, the tables also list the estimates from different
authors based on harmonic analysis of long-term VLBI measurements. The comparison shows
generally good agreement, with discrepancies typically within 1-2/.ts for UT1 and 10-30/.tas
for polar motion for the largest tides, and relatively poorer for smaller tides. GIPSOlq (1996)
provided a quantitative comparison of the goodness of the agreement among various models
and estimates. Overall, the best agreement is found between VLBI estimates by Gipson and the
OTAM predictions by T/P Model C.
Next, let us compare the T/P tide model predictions of OTAM with the VLBI intensive
observations of Earth rotation made during special campaigns. Five intensive campaigns have
so far been conducted: ERDE (October 15-31, 1989), Search92 (July 31-August 9, 1992),
Cont94 (January 12-26, 1994), Cont95 (August 23-28, 1995), and Cont96 (24 selected days
during September 2-November 15, 1996), during which Earth rotation data at sampling intervals
as short as 1 hour were obtained. We will focus on Cont94 as a best example since it has the
lowest formal errors except perhaps for Cont96, whose data have yet to be fully analyzed.
During the two weeks of Cont94 campaign the NASA R and D network of VLBI reached
internal precision of several/.ts in UT1 and roughly 100/.tas in polar motion. Figure 6 shows
those data in comparisonwith the OTAM time series predicted by the sum of eight tides according to T/P Models B and C. The agreement in UT1 as well as in the x and y components of
polar motion is quite good. During Cont94, the eight tides collectively explain as much as 90%
of the sub-dally UT1 variance, and nearly 60% of the variance in sub-dally polar motion (CsAo
et al., 1996).
414
B.F. CHAO AND R.D. RA',
(a)
I
f
I
I
I
UTI( M2 )
TropicalAtla~ic
so~
/x
/
[~ -10
~
"SouthemOcean
"\\,
SouthPacific~
~ - - - - 7 ~ - ~ TropicalPacific
~
NorthPacific
~~
o -15
Arctic
-20 -
i
- w .... =~ heights
) currents
Indian
-25 i
~
- 5
i
-10
r
-5
i
0
5
in-phase (microsec)
(b)
I
I
I
I
I
. . . . . . . . . ~ heights
I0
UTI(
Arctic
K1 )
) currents
5 "
5
.g
o 0
f . . . .
Tropi
r
-5
Pacific
outhAtlantic
///
--...'"~a South
]
"acific /
indian
"4(. . . . -q~TropicalPacific
-10
-10
i
-5
i
0
i
5
in-phase (microsec)
ll0
1~5
Fig. 5. The tide height and tide current contributions from each individual ocean basin, for (a)
M 2 UT1; (b) K~ U T I ; (c) M2 retrograde polar motion, (d) KI prograde polar motion, according
to Topex/Poseidon tide Model C (EOBERT et al., 1994), and RaY et al. (1997) for the Arctic
Ocean. For each basin two vectors, one arising from tidal height and one from tidal current, are
plotted and named at their junction. For purposes of this diagram, the following regions are
defines: Southern Ocean (90°S - 50°S, 180°W-180°E); South Atlantic (50°S - 23°S, 70°W 20°E); Tropical Atlantic (23°S - 23°N, 75°W - 20°E); North Atlantic (23°N - 70°N, 100°W
- 40°E); Indian Ocean (50°S - 30°N, 20°E - 120°E); South Pacific (50°S - 23°S, 120°E 70°W); Tropical Pacific (23°S - 23 °N, 120°E - 75°W); North Pacific (23°N - 70°N, 120°E 120°W).
Oceanic
tidal a n g u l a r
(C)
L
momentum
and Earth's
)
rotation
i
variations
!
J
-50
. ou tlantic
T,o.io
-100
g
-200
-250
........... heights
) currents
-150
(cJ)
-100
I
2OO
~Arctic
-50
in-phase(microarcsec)
I
PM+(K1)
/'
I
I
lnlianOcean
/
150
5o
~Arctlc
loo
"~Tropical
" Pacific
/
50/
,,.SouthAtlantic
"~ropicalAtlantic
Ocetahney Atlant,c~
I
SouthPacific
J
.........) heights
) currents
0 . . . . . i:i . . . . . . . . . . . . . . . . . . . . . .
i
0
t
i
50
100
in-phase(microarcsec)
F i g . 5. C o n t i n u e d .
i
150
415
416
B.F. CHAO AND R.D. RAY
Table 3. The UT1 amplitudes in microseconds (/xs) and Greenwich phase lags in degrees (see text for phase
convention) for the eight diurnal and semidiurnal tides in Table 2. The observations are those derived from longterm VLBI measurements by various authors: SOVERSet al. (1993), HERRING and DONG (1994), GIPSON (1996).
The Topex/Poseidon predictions are computed according to T/P ocean tide models; Model A = SCHRAMAand
RAY (1994), B = RAY et al. (1994b), and C = EGBERT et al. (1994)
Qi
Diurnal tides
Oi
PI
K1
N2
Semidiurnal tides
M2
$2
K2
Observations:
Sovers et al.
Herring &
Dong
Gipson
6.6, 37 °
5.3, 36 °
21.4, 39 °
23.6, 47 °
7.2, 27 °
7.1, 34 °
15.5, 13°
18.9, 20 °
3.0, 221 ° 18.2, 235 ° 5.2, 266 ° 2.8, 251 °
3.2, 240 ° 17.9, 233 ° 8.6, 269 ° 3.8, 282 °
5.6, 31 °
22.2, 37 °
5.8, 25 °
18.6, 29 °
3.7, 239 ° 18.6, 236 ° 8.0, 264 °
20.5, 29 °
23.2, 39 °
20.1, 37 °
8.3, 39 °
5.9, 29 °
22.3, 25 °
24.2, 38 °
19.7, 26 °
19.4, 244 ° 7.7, 262 °
3.8, 250 ° 17.6, 251 ° 7.7, 261 ° 2.1,260 °
4.1,248 ° 17.7, 246 ° 7.6, 267 ° 2.0, 259 °
2.9, 283 °
T/P Predictions:
Model A
Model B
Model C
4.8, 32 °
5.6, 26 °
Table 4. Same as Table 3, but for the prograde polar motion with amplitude in microarcseconds (/xas).
Diurnal tides
Semidiurnal tides
M2
$2
Q1
Ol
Pj
Ki
N2
K2
49, 54 °
132, 54 °
69, 92 °
134, 51 °
23, 125 °
22, 57 °
21, 73 °
32, 160 °
35, 72 °
199, 63 °
60, 54 °
152, 61 °
17, 135 °
58, 91 °
12, 99 °
39, 173 °
33, 81 °
148, 74 °
51, 60 °
166, 63 °
16, 108 °
62, 110 °
14, 89 °
15, 104°
22, 82 °
27, 77 °
135, 76 °
114, 67 °
142, 70 °
49, 57 °
56, 63 °
160, 65 °
150, 57 °
171, 63 °
15, 127 °
17, 135 °
72, 111 °
74, 120°
75, 116 °
23, 82 °
28, 86 °
29, 86 °
7, 90 °
7, 96 °
Observations:
Sovers et
al.
Herring &
Dong
Gipson
Predictions:
Model A
Model B
Model C
T w o d i s c r e p a n c i e s b e t w e e n m o d e l a n d o b s e r v a t i o n in Fig. 6 are w o r t h n o t i n g e v e n t h o u g h
t h e y are at p r e s e n t u n r e s o l v e d . O n e is the c l e a r d i s a g r e e m e n t o n J a n u a r y 18. W e are i n c l i n e d
to attribute this to V L B I errors r e l a t e d to the n e t w o r k d r o p o u t later that day and the n e x t day,
but the o b s e r v e d UT1 signal looks " c l e a n " ( a l t h o u g h the y w o b b l e c o m p o n e n t a p p e a r s noisier),
w h i c h is p u z z l i n g . A s e c o n d n o t i c e a b l e d i s c r e p a n c y is that the tide m o d e l s a l m o s t a l w a y s underp r e d i c t the m e a s u r e m e n t peaks, in b o t h UT1 and p o l a r m o t i o n . I n c l u s i o n o f libration e f f e c t s
r e d u c e s but d o e s not e l i m i n a t e the d i s c r e p a n c y ; n e i t h e r d o e s the i n c l u s i o n o f an additional 16
m i n o r tides o b t a i n e d b y i n f e r e n c e f r o m the tidal a d m i t t a n c e s ( M u N K a n d CARTWRIGHT, 1966).
In fact a spectral analysis o f the residuals s h o w s m o s t significant p o w e r n e a r the M2 and $1
p e r i o d s . M o s t surprisingly, the u n d e r p r e d i c t i o n o f p e a k s also o c c u r s w h e n the p r e d i c t i o n c u r v e
is b a s e d on the V L B I h a r m o n i c t e r m s o f T a b l e s 3 - 5 ( i n c l u d i n g an $1 t e r m - p r e s u m a b l y due
to the a t m o s p h e r e - not t a b u l a t e d h e r e ) , w h i c h a u t o m a t i c a l l y i n c l u d e libration. F u r t h e r m o r e , the
s a m e p h e n o m e n o n o c c u r s w i t h o t h e r V L B I i n t e n s i v e m e a s u r e m e n t c a m p a i g n s (GIPSON, personal c o m m u n i c a t i o n ) .
Oceanic tidal angular m o m e n t u m and Earth's rotation variations
417
Table 5. Same as Table 4, but for the retrograde polar motion. The observed diurnal values are absent for the
reasons given in the text
Diurnal tides
Ol
PI
QI
K1
N2
Semidiurnal tides
M2
$2
K2
Observations:
Sovers et
al.
Herring &
Dong
Gipson
-
-
-
-
37, 267 °
265, 273 °
174, 303 °
62, 286 °
-
-
-
48, 282 °
265, 272 °
120, 304 °
31, 328 °
-
-
-
46, 269 °
257, 272 °
128, 303 °
18, 346 °
46, 267 °
45, 269 °
245, 271 ° 121, 298 °
262, 274 ° 140, 303 °
263, 271 ° 131, 298 °
40, 301 °
34, 301 °
Predictions
Model A
Model B
Model C
167, 305 °
9510, 129 °
200, 314 ° 838, 138 ° 9700, 135 °
132, 297 ° 797, 136 ° 9670, 133 °
68, 321 °
45, 307 °
VLBI Intensive Campaign - Cont94
NASA R&D Network
50 ¸
-50
500
o
x -500
"~ 500
0
>~ -500
]3
]4
15
16
17
18
19
20
.lanunry 1994
21
22
23
24
25
Fig, 6. Comparison o f Earth rotation variations during Cont94 Campaign. The hourly observations
with standard errors are those made by VLB! (the N A S A R &D network); the solid lines are
predictions according to T/P tide Models B (RAY et al., 1994b) and C (EGBERT et al., 1994)
consisting of eight tides (see text). The top panel shows UT1 variation in units of p~s; the lower
two panels for the x and y components of the polar motion (in units of p~as) without the retrograde
diurnal components. (Reproduced from Cr~Ao et al., 1996, courtesy of the American Geophysical
Union.)
418
B.F. CHAOAND R.D. RAy
5. FUTURE IMPROVEMENTS AND STUDIES
Our main results are summarized in Tables 3-5 and Fig. 6. They on one hand clearly establish
the dominant role of OTAM in exciting the diurnal/semidiurnal Earth rotation variations, and
on the other hand pave the way for detailed studies of non-OTAM variations. The excitation
sources for the latter include subdaily atmospheric and oceanic angular momentum variations,
earthquakes, the daily and half-daily atmospheric thermal tides, Earth librations, and the response
of the internal inhomogeneities of the solid Earth to diurnal/semidiumal tidal forcing.
Let us now focus on the Earth-rotation signals at tidal periods of the type presented in Tables
3-5. The solution of these signals takes advantage of the long time span of the observations,
for both VLBI and T/P. The benefits come from both higher frequency resolution and higher
signal-to-noise ratio as observing time increases. Imagine that, after a sufficiently long time of
data collection, one had nearly perfect estimates for Tables 3-5 (presumably with many more
tidal components). Then removing the OTAM estimates from the Earth rotation estimates will
reveal all the smaller non-OTAM signals at the tidal periods. At all tidal periods, there exist
small Earth-rotation excitations due to the mantle's laterally inhomogeneous deformation in
response to the tidal forcing (just as do the asymmetric ocean tides as stated earlier). This effect
has not been studied except for the case of long-period LOD variations (mentioned above by
passing); and such observations can shed light on low harmonic degrees of the mantle inhomogeneity.
Similarly, elucidation of the semidiurnal spin librations and the prograde diurnal polar
librations will be of geophysical importance as they bear sensitively on the equatorial ellipticity
of the core (CHAo et al., 1991; HERRING and DONG, 1994). So far only very preliminary results
have been obtained (CHAOet al., 1996) yet the results are very encouraging. Atmospheric tides
carry angular momentum at Sj and $2 periods; their revelation will provide information for the
modeling of atmospheric thermal responses. Of course, at these periods one must beware of
instrumental errors such as daily heating and cooling of VLBI radio telescopes.
On the other hand, comparisons of the type of Fig. 6 can reveal non-periodic high-frequency
signals once the periodic signals can he confidently removed. Information about the broad-band
atmospheric and oceanic angular momenta can provide global constraints to general circulation
models (e.g., HIDE et al., 1997). On time scales of interest here ranging from hours to days
which have not been explored before, such study can also yield new insight into momentum
coupling mechanisms in the atmosphere-ocean-solid Earth system. Similarly, detection of episodic earthquake signals, which geophysicists have been anticipating for decades, would be
valuable in understanding the source mechanism for large earthquakes (e.g., CHAO and
GROSS, 1987).
Thus, a host of further geophysical investigations rely on both continued Earth rotation
measurement and much improved OTAM estimation. Earth rotation measurement at higher precision and temporal resolution than the current routine is indeed under planning. A new VLBI
era under the "CORE" (Continuous Observation of the Rotation of the Earth) operation will be
launched once the necessary hardware development is completed (MA et al., 1996). Precision
and temporal resolution comparable to those of the intensive campaign Cont94 will be expected
at all times. In addition, SLR will continue to yield high-quality data, and GPS observation will
continue its improvement in precision and temporal resolution.
The outlook for improvements to the OTAM model estimates, and to the tide models in
general, is quite good. Deep-ocean heights are already, of course, well constrained by T/P observations; further data will add small incremental improvements in certain areas and will help
Oceanic tidal angular momentum and Earth's rotation variations
419
constrain some of the smaller tides. The most significant improvements should occur in models
of deep-ocean tidal currents, the dominant term in most of the O T A M integrals. Diagnostic
estimates of currents, in the manner of Models A and B, are prone to inaccuracies especially
near islands and coastlines, near sharp gradients (or noise) in height fields, and near inertial
latitudes (CARTWRIGHX et al., 1992). Incorporation of continuity conditions will improve these
estimates, as will continued experience with assimilation methods. As discussed by EGBERI
(1997, in this issue), there are two aspects to the latter: (1) improvements to computational
methods will allow more rigorous solutions with more realistic and accurate covariance information; (2) better understanding of the physical mechanisms being modeled, especially the
dissipation terms, will clearly improve velocity estimates. As Egbert notes, there is reason to
expect marked improvements in the next few years.
Improvements in the shallow and marginal seas, and especially in the complex tidal regime
surrounding Indonesia, will be more incremental. In shallow regions, the tidal current velocities
are of less concern because the volume transports are rather small (Fig. 4, and Table 4 of CrtAo
et al., 1996), but tidal heights are spatially complex and difficult to map empirically, yet so
large that they may conceivably alias into the degree-2 spherical harmonics of O T A M (Eqs
( l a ) , (2a)). A number of research programs are in place to improve shallow-water tidal fields,
involving both hydrodynamic modeling and assimilation and altimetry from other (current and
future) satellites. Progress is to be expected, but it will be slow. Fortunately, as noted, the
dominant O T A M contributor - deep ocean currents - holds the most promise for rapid improvements in the near future.
6. ACKNOWLEDGEMENTS
We thank Gary Egbert for providing the Topex/Poseidon tide "Model C". We also thank Chopo
Ma and John Gipson for the VLBI data plotted in Fig, 6, and Richard Gross for the "Space95"
data plotted in Figs 1 and 2. This work is supported by the Geophysics Program of NASA.
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