The complete tidal response of an elliptical, rotating, elastic Earth is found to contain small displacements which do not fit into the conventional Love number framework. Corresponding observable tidal quantities (gravity, tilt, strain, Eulerian potential, etc.) are modified by the addition of small latitude dependent terms

Wahr, J. M.

English

NASA Technical Reports Server

Body Tides on an Elliptical Rotating Eartn
John M. Wahr
CIRES, University of Colorado
Boulder, Colorado 80309
N79 21508
Abstract. The complete tidal response of an ellip-
tical, rotating, elastic Earth is found to contain
small displacements which do not fit into the conven-
tional Love number framework. Corresponding observable
tidal quantities (gravity, tilt, strain, Eulerian
potential, etc.) are modified by the addition of small
(<1%) latitude dependent terms.
the relations between an expanded Love number set and
these observables can be disturbingly complex. Conse-
quently, it is often more useful to present results for.
the observational quantities, directly.
As an example, consider the tidal gravity signal as
measured by a standard gravimeter. For a second degree
semi-diurnal tide, it has the form
Introduction
Traditional body tide calculations assume a spheri-'
cally symmetric non-rotating Earth, with surface motion
at a given frequency described by a small set of 'Love
numbers', principally h, k and i. Simple linear com-
binations of these numbers are found to compare with
experiment.
The spherically symmetric case has been extended by
others (Jeffreys and Vicente, 1957a, 1957b; Molodensky,
1961; Shen and Mansinha, 1976; Sasao et al. 1978) to
include effects of non-sphericity on the diurnal tides.
Particular attention has been focused on the dynamical
effects of a rotating fluid core with an accompanying
elliptical core-mantle boundary. The mantle has been
repeatedly modelled as a spherically symmetric, non-
rotating elastic shell, and the fluid core assumed to
be spherically stratified. In each of these expanded
treatments, the computed tidal response consists of the
conventional body tide, represented at the free surface
by the Love numbers h, k and H, together with a toroid-
al component describing the Earth's nutational behav-
ior. In this way the diurnal Love numbers are modified
for the most important effects of rotation and ellip-
ticity. The nutations, traditionally computed for a
rigid Earth (Woolard, 1953; Kihoshita, 1977) are also
improved by including spherically stratified elastic
behavior throughout the Earth.
Qualitative Effects of Rotation and Ellipticity
The work described below extends these results by
also including the effects of rotation in the inner
core and mantle, together with elliptically stratified,
realistic material properties throughout the Earth.
Numerical investigations suggest a computed surface
solution accuracy of at least three parts in a thousand,
the order of the Earth's ellipticity. All important
tidal bands are considered.
It is found that for the complete rotating, ellip-
tical problem, surface motion is too complex to be
represented by the familiar Love numbers: h, k and Si.
Even the inclusion of nutational motion into the
diurnal solution and length of day changes into the
long period solution is not sufficient. Other small
(<1%) unmodelled motions remain. These new terms,
which will not be detailed here, can be handled by
defining more Love numbers. The utility of this pro-
cedure, however, is not always maximal. It is, of
course, physical observables which are the ultimate
computational goal. For an elliptical, rotating mantle
Proc. of the 9th CHOP Conference, An Inlemaliiinal Symposium nn the Appliraliun.\ of
Ce orff.tv to Geoilvnamics. October 2-5.1978. Dept. of Geodetic Science Kept. No. 280. The
Ohio State Univ., Columbus, Ohio 43210.
Gravity = - ±- g Vn=2(w) [<52(u) Y =^2.(6,<J>)
o
64(cu) iwt (1)
where w is the perturbing frequency, Y a surface
spherical harmonic, rQ the Earth's mean radius
(6371km), g the computed gravity acceleration at the
Earth's equator assuming a spherical density distrib-
ution (g = 979.8259 cm/sec2), Vn(o>) the appropriate
term in the luni-solar potential as observed at the
Earth's equator, and 6- (u) and g. (to) are gravimetric
factors which reflect She dynamical behavior of the
Earth at the frequency, u>. (For a spherical Earth
64 = 0).
It is convenient to think of eq. (1) as representing
a latitude dependent gravimetric factor:
(2)
Similar latitude dependent phenomenon are present in
all tidal bands and for all observable quantities
(i.e.), surface gravity, tilt, strain, latitude and
longitude variations, tidal perturbations in inertial
space gravity, inertial space station displacements,
etc. They are found at about the one per cent level
or less. For example, computations of the M2 semi-
diurnal gravimetric factors shown in eq. (1) give,
for Earth model PEM-C (Dziewonski, et al., 1975):
spherical case: 6- = 1.159 6, = 0i 4
rotating, elliptical case:6 = 1.160 $ = -.005
Acknowledgements. This work is supported under NASA
grant NSG 7319.
References
Dziewonski, A. M., A. L. Hales, and E. R. Lapwood,
Parametrically simple Earth models consistent with
geophysical data, Physics of the Earth and Planetary
Interiors. JLO, 12-48, 1975.
Jeffreys, H. and R. 0. Vicente, The theory of nutation
and the variation of latitude, Mon. Not. R. Astr.
Soc. , 112.' 142-161, 1957a.
Jeffreys, H. and R. 0. Vicente, The theory of nutation
and the variation of latitude: the Roche Model Core,
Mon. Not. R. Astr. Soc., 117, 162-173, 1957b.
331
https://ntrs.nasa.gov/search.jsp?R=19790013337 2020-03-21T23:58:35+00:00Z
Kinoshita, H., Theory of the rotation of the rigid
Earth, Celestial Mech. , 15, 277-326, 1977.
Molodensky, M. S., The theory of nutation and diurnal
Earth tides, Comm. Obs. R. Belgique. 288, 25-56,
1961.
Sasao, T. , S. Okubu and M. Saito, A simple theory on
dynamical effects of stratified fluid core upon
nutational motion of the earth, to be published in
the Proceedings of IAU Symposium No. 78 "Nutation
and the Earth's Rotation". Kiev, May 1977), in press.
1978.
Shen, P-Y. and L. Mansinha, Oscillation, nutation and
wobble of an elliptical rotating Earth with liquid
outer core, Geophys. J. R. Astr. Soc., 46, 467-496,
1976.
Woolard, E. W., Theory of the rotation of the earth
around its center of mass. Astronomical Papers for
the American Ephemeris and Nautical Almanac (Gov't.
Printing Office, Washington, DC, 1953), Vol. XV.Pt.l.
1953.
332
N79 21509
Observation of the Nearly Diurnal Resonance ot the Eartn using-
a Laser Strainmeter
Judah Levine *
Time and Frequency Division, National Bureau of Standards
Boulder, Colorado 80302
Abstract. We have used two years of strain
tide data to study the response of the Earth to
the diurnal and semidiurnal tidal excitations.
Our results show that there is significant struc-
ture in the response of the earth to tidal exci-
tations near one cycle/sidereal day. This struc-
ture agrees with the resonance behavior predicted
from the calculations of the forced elastic-
gravitational response of an elliptical, rotating
earth with a liquid outer core. The data can
also be used to test for possible preferred
frames and spatial anisotropies. We find that
upper bounds on the parameterized post-Newtonian
(PPN) parameters which characterize these effects
are a2 s 0,007 and £w < 0.005.
Introduction
We have analyzed approximately two years of
strain-tide data obtained using the 30-meter
laser strainmeter we-have previously described
(Levine and Hall, 1972).
Data Acquisition
The data were obtained using a 30-meter long
laser strainmeter located in the Poorman Mine,
an unworked gold mine located approximately 8 km
west of Boulder, Colorado at latitude 40.03°N,
and longitude 254.67°E.
The heart of the strainmeter is an evacuated
30-meter Fabry-Perot interferometer located along
the length of the tunnel. The axis of the inter-
ferometer is 7° west of North.
The interferometer is illuminated bya3.39-um
helium-neon laser. A servo loop piezoelectri-
cally tunes the-laser to keep its wavelength
coincident with one of the transmission maxima
of the long interferometer. The frequency of
the laser, f, is therefore related to the length
of the interferometer, L, b'y
f = nc
1
 2L
where n is an integer and c is the velocity of
light. Thus
13 AL
M
f
AL
L
A second 3.39-um laser is stabilized using
saturated absorption in methane. The beat fre-
quency between the two lasers is extracted for
further processing. Then
Afbeat AL
L
Proc. of the 9th GEOP Conference, An Internuliottul Symposium on the Applications of
Geotlere in Cemlraai»h:i. October 2-5.1978. Dept. of Geodetic Science Rept. No. 280.The
Ohio State Univ!. Columbus. Ohio 43210.
Af, = 8.85 * 10beat
so that a measurement of the fluctuations in the
beat frequency provides a direct measurement of
the fractional change in the length of the long
path. The relationship between beat frequency
and strain has no adjustable constants or cali-
bration factors.
The beat frequency is digitally recorded 10
times/hour along with other information including
local barometric pressure, etc. The digital data
are bandpass filtered using a symmetric convolu-
tion filter and then decimated to one sample
every two hours for comparison with theory.
Data Analysis
We have used all of the components published
by Cartwright and Edden (1973) in our analysis.
For each component with frequency f^ we construct
a time series of the form
a. cos (2irf,
tc i )Y
m(e,<p)T(n,m,8,Cf>,eo)
n s
where a^ is the amplitude given by Cartwright and
Edden, ^  is the phase of the component, Y™ is
the spherical harmonic computed at the station
co-latitude 6 and East Longitude (f. The quantity
T is a function converting potential to strain
along the axis of the strainmeter 6S (Levine and
Harrison, 1976). The quantity ak is -90° if (n+m)
is odd and is zero otherwise (Cartwright and
Tayler, 1971).
The theoretical series were further modified
by a function to correct for local topography,
local crustal inhomogeneities, cavity effects and
ocean loads (Levine and Harrison, 1976). We as-
sumed initially that these effects do not vary
rapidly with frequency, so that all of the diur-
nal components have the same correction as 01 ,
and that all of the semidiurnal components have
the same correction as M2-
The terms are then grouped by frequency. Each
frequency group contains all of the terms (re-
gardless of parentage) which differ by less than
one cycle/year from each other.
This process produces 48 time series. We fit
these series to our data by the least-squares
method using an adjustable amplitude and an ad-
justable phase for each cycle/year group.
The results of this process are shown in Fig. 1
for the diurnal amplitude, Fig. 2 for the diurnal
phase, and Fig. 3 for the semidiurnal amplitude
and phase. The error bars represent one standard
deviation and are obtained from estimates of the
Fellow, Joint Institute for Laboratory Astro-
physics of the National Bureau of Standards and
the University of Colorado.
333
LU
O
1-
z
e>
2
z
0
I—
o
z
l»
cc
LU
U_
Z
ce
1-
"f t
1.0
0.9
0.8
0.7
0.6
0.5
0 4
*l
j M
M, p1 J'
S
Oi
- i
0|
 x
K,
—
1 1 1 1 1 1
0.80 0.85 0.90 0.95 1.0 1.05 1.10
FREQUENCY (CYCLES/DAY)
Fig. 1. Normalized transfer function for the
diurnal tides. For clarity only the major com-
ponents are plotted. " The error bars are one
standard deviation.
~ 10
o
CO
X
°- -10
CC
!i! LU\l
cc z
I- O
. <
b
0.9
2'
uj
CO
N2 8
M2
?:: WRZ
MZ
I
X2L2T2
1.8 1.9 2.0
FREQUENCY (CYCLES/DAY)
2.1
Fig. 3. Normalized transfer function -amplitude
(lower curve) and phase (upper curve) for the
semidiurnal tides. For clarity only the major
components are platted. The error bars-are one
standard deviation.
l^ /
LU
0
^ 10CO
X
0.
o
H-
CJ
Z -10
LL.
cc
LU
U.
CO .
z
cc
-
- fQI
1 1
f .
T T PlT
o, * S
I Kl
M,.
T
: K
j| .
*,.*,
I I 1
0.80 0.85 0.90 0.95 1.00 1.05 • 1.10
FREQUENCY (CYCLES/DAY)
u 1.0
13
1-
Z 0.9
O
2
Z 0.8
O
0
Z 0.7
D
u_
cc
^ 0.6
CO
£o-5
K,,
I /
t• /M\
£ ' *
/
l^
J|
*l
T. /
_ T
1
/ Qi
/
—
!-
1 1 1 1 1 1
0.80 0.85 0.90 0.95 1.00 1.05 1.10
FREQUENCY (CYCLES/DAY)
Fig. 2. Normalized transfer function phase for
the diurnal tides. For clarity only the major
components are plotted. The error bars are one
standard deviation.
Fig. 4. Normalized transfer function amplitude
for the diurnal tides when-the frequency-depen-
dent Love numbers are inserted into the fitting
function. The dashed line is obtained by fitting
a straight line to the transfer function ampli-
tudes weighted by their respective uncertainties.
334
noise.
The amplitude of the diurnal transfer function
shows a statistically significant structure.
There appears to be a significant decrease in
the transfer function for the lower frequency
diurnal components. We attribute this to a
slow change in the contribution of the ocean
load to our observed data.
More significantly there is a dip in the
transfer function near one cycle/day (we have
not plotted the Sj amplitude at one cycle/day
since it is heavily contaminated by thermoelastic
processes). This dip is consistent with the ef-
fects of the nearly diurnal resonance associated
with the liquid core. The consistency can be
shown more clearly by inserting the resonance
directly into the fitting function. This may be
done using the frequency dependent Love numbers
published by Molodensky (1961) and Shen and
Mansinha (1976). These Love numbers produce
changes in the transfer function at all of the
diurnal frequencies but, except near one cycle/
day, the new transfer function lies within one
standard deviation of the old one. The modified
transfer function is shown in Fig. 4. As can be
seen from the figure the resonance models account
for the dip in the transfer function near one
cycle/day. Unfortunately, the data cannot be
used to infer the fine structure near the reso-
nance. Furthermore, the insertions of realistic
estimates for the energy loss in the nearly diur-
nal band may affect the shapes predicted by all
of the models somewhat. It is unlikely we will
be able to test any of these effects using our
data. The strongest test for the resonance ob-
tainable from our data comes from the components
P! and KI which are really on the tail of the
resonance function. Our measurements at <j>i and
\l>l (where the resonance has the largest effect)
have error bars whose size probably does not per-
mit them to be used in a quantitative comparison
between theory and experiment or in attempts to
differentiate among the various calculations of
the effect of the resonance.
These results may be compared with the analy-
sis of Warburton and Goodkind (1977) using data
from a cryogenic gravimeter, with the gravimeter
data analysis of Abours and Lecolazet (1978) and
with the analyses of Lecolazet and Melchior (1977).
Their results are generally in agreement with
ours, confirming the general shape of the reso-
nance. None of the analyses is able to make a
quantitative comparison with theory because of
the relatively poor signal-to-noise ratio in the
measurements of the crucial components <j>i and ^\.
The semidiurnal transfer function amplitude
shows far less structure, and the agreement be-
tween experiment and theory is quite good. There
is no evidence of anomaly at $2 (two cycles/day)
confirming that the anomaly at Sj is almost cer-
tainly of thermoelastic origin.
We may place upper limits on the various PPN
parameters by calculating the magnitudes of the
anomalous tidal components in terms of the PPN
parameters 02 and Cw. In this way we conclude
a.2 ^  0.007 and C < 0.005.
Conclusions
These results confirm the general correctness
of the various published earth models. From the
point of view of the current discussion they are
also significant in that they illustrate the
sort of measurements that can be performed with
laser strainmeters. These measurements show
that we can make meaningful measurements of
diurnal strain changes at the 10~10 level, and
that almost continuous operation for several
years is possible.
From the geodetic point of view, the limitation
on the utility of laser strainmeters arises from
their sensitivity to local effects, especially to
spurious motions near the piers.
Spurious motions of the piers at the level of
millimeters will even play a significant role in
geodetic measurements made over much longer base-
lines. Motions of this magnitude represent frac-
tional changes of parts in 10 even over 50 km
baselines, so that such effects will make sig-
nificant contributions to the error budget of
any geodetic instrument now in operation or under
construction.
It is important to compare the data obtained
with laser strainmeters with measurements obtained
using electromagnetic distance measuring equipment
operating over parallel baselines. If this com-
parison shows that the two provide a consistent
picture of the regional strain field, then laser
strainmeters may prove useful in measuring re-
gional strain at fixed observatories. In this
service they can provide significantly higher
sensitivity than any other technique.
References
Abours, S. and R. Lecolazet, New results about the
dynamical effects of the liquid outer core as
observed in Strasbourg, Proc. 8th Int. Conf.
on Earth Tides, 1978 (in press).
Berger, J. and J. Levine, The spectrum of the
earth noise from 10~8 to 10+^ Hz, J. Geophys.
Res., 79, 1210-14, 1974.
Cartwright, D.E. and A.C. Edden, Corrected tables
of tidal harmonics, Geophys. J. Roy. Astr.
Soc.. J33, 253-2640, 1973.
Cartwright, D.E. and R.J. Tayler, New computa-
tions of the tide-generating potential,
Geophys. J. Roy. Astr. Soc., 23, 45-74, 1971.
Lecolazet, R. and P. Melchior, Experimental de-
termination of the dynamical effects of the
liquid core of the earth, Ann. Geophys., 33,
11-22, 1977.
Levine, J. and J.L. Hall, Design and operation
of a methane absorption stabilized laser
strainmeter, J. Geophys. Res., 77, 2595-2609,
1972.
Levine, J. and J.C. Harrison, Earth tide strain
measurements in the Poorman Mine near Boulder,
Colorado, J. Geophys. Res., 81, 2543-55, 1976.
Molodensky, M.S., The theory of nutations and
diurnal earth tides, Comm. Obs. R. Belgique,
188, 25-56, 1961.
335
Shen, Po-Yu and L. Mansinha, Oscillation, nuta-
tion and wobble of an elliptical rotating
earth with liquid outer core, Geophys. J. Roy.
Astr. Soc.. 46, 467-496, 1976.
Warburton, R.J. and J.M. Goodkind, Detailed
gravity tide spectrum between 1 and 4 cycles
per day, Geophys. J. Roy. Astr. Soc., 52,
117-130, 1977.
336


Body tides on an elliptical rotating earth

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server