ELSEVIER Physics of the Earth and Planetary Interiors 93 (1996) 223-238
PHYSICS OFTHE EARTH
ANDPLANETARY INTERIORS
The response of the Earth to tidal body forces described by second- and third-degree spherical harmonics as derived
from a 12 year series of measurements with the superconducting gravimeter GWR/T3 in Brussels
P. Melchior *, B. Ducarme, O. Francis Observatoire Royal de Belgique, Avenue Circulaire 3, B-1180 Brussels, Belgium
Received 17 February 1995; revision accepted 12 May 1995
Abstract
A 12 year tidal gravity registration performed in Brussels with a GWR superconducting gravimeter allows us to separate six tidal components derived from the third-degree lunar potential from those derived from the classical second-degree luni-solar potential (39 components) with which they differ in frequency by the lunar perigee period (8.847 year). Four ter diurnal waves are also separated.
Such a separation appears necessary as the results show a non-negligible residue on the third-degree frequency wave M 1 (lunar day period) owing to an amplification of the corresponding oceanic component in the North Atlantic ocean.
The other third-degree components also differ slightly from the theoretical model amplitudes. A correct determination of the free core nutation resonance acting on the ~1 diurnal solar wave is discussed in
relation to atmospheric pressure gravitational effects and climatological disturbances. The zonal fortnightly tide Mf is also deduced with precision but its comparison with the theoretical model raises
a problem.
1. Introduct ion
It is well known (Melchior, 1983) that the purely elastic deformation properties for a Spher- ical, Non Rotating, Elastic, Isotropic (SNREI) Earth depend upon the degree of spherical func- tion. The variations of the intensity of gravity are proportional to
82 = 1 + h 2 - 3 / 2 k 2 for the second-degree potential
* Corresponding author.
and to
8 a = 1 + 2 / 3 h 3 - 4 / 3 k 3
for the third-degree potential
while the numbers (h 3, k 3) are different from the numbers (h 2, k2), higher degrees being more sensitive to the physical properties of the outer parts of the Earth. The most recent W a h r - D e - hant (1995) model of the Earth's interior gives for the main components at Brussels' latitude
224 P. Melchior et aL / Physics o f the Ear th and Planetary Interiors 93 (1996) 223 -238
Table 1 Third-degree main tidal waves in the diurnal and semidiurnal frequency bands
Name Astronomical Doodson Frequency Tamura Beating argument argument (deg. b - 1) amplitude
coefficient
3 M K l a T -- 2S 135.555 13.39401908 2112 M3(3r) - K2(2r + 2s) Q1 "r - 2s + p 135.655 13.39866089 72136 L K I r - p 155.455 14.48741031 10653 L2(2T + s - p ) - Kl(z + s) M1 a r 155.555 14.49205212 6608 N O 1 r + p 155.655 14.49669393 29631 N2(2r - s + p ) - O1(~" - s) J1 7" + 2s - p 175.455 15.58544335 29630 3 M 0 1 a z + 2s 175.555 15.59008516 2412 M3(3~') - O2(2z - 2s) 3MJ2 b 2"r -- 2S + p 235.655 27.89071301 1562 M3(3z) - J l ( r + 2s - p ) 2N 2 2z - 2s + 2p 235.755 27.89535482 23009 3 M K 2 b 2"c - s 245.555 28.43508772 5691 M3(3~') - Kl ( r + s) N 2 2r - s + p 245.655 28.43972953 173881 L 2 2~ + s - p 265.455 29.52847895 25670 3 M 0 2 b 2"r + s 265.555 29.53312076 5250 M3(3"r) - 0 1 ( r - s)
a p1 terms. b p2 terms.
z, mean lunar time (hour angle of the Moon + 180°). s, mean tropical longitude of the Moon. p, mean tropical longitude of the lunar perigee. Tamura amplitude coefficient to be multiplied by 10 -6. We have christened the new waves according to the oceanographers' convention by assembling the Darwin symbols of the beating constituents and subtracting their indices. For third-degree waves we have to put a 3 in front of the symbol to avoid confusion with analogue combinations of second-degree waves.
Theoretical developments of the luni-solar tidal potential have recently been extended from 1000 to 8000 harmonic constituents (Tamura, 1987; Xi Qin Wen, 1987; Roosbeek, 1994, 1995; Hartmann and Wenzel, 1994; Roosbeek and Dehant, 1994a,b) whereas in the analysis of experimental data the selectivity is severely limited by the dura- tion in time of the available data. One is there- fore constrained to combine together those con- stituents that have very close frequencies and cannot be separated from a limited length of data. Such 'groups' are identified by the name of their main constituent wave.
For each group one then calculates, through a least-squares analysis based upon the a priori knowledge of the frequencies, a global amplitude and its phase difference with respect to the tidal force (vector A (A, a)).
One month's data allow us to separate eight groups of waves differing in frequency by the monthly lunar argument (mean longitude of the Moon, s).
One year's data allow us to separate more groups of waves differing in frequency by the
annual solar argument (mean longitude of the Sun, h) i.e. some 37 groups.
With 12 years' data as obtained at Brussels since 1982 with the superconducting gravimeter GWR/T3 we can easily separate groups differing in frequency by the lunar perigee argument p (period 8.847 year), which is up to 50 groups, considering also that with this instrument to achieve a 1% precision in the amplitude determi- nations we must limit the separations to waves having an amplitude of at least 200 ngal that is 2 nm s -2 a
a In view of existing practice in certain fields or countries, the Comit~ International des Poids et Mesures (CIPM) (1978) considered that it was acceptable for those units (listed in Table 10) to continue to be used with SI units, until the CIPM considers their use no longer necessary: the gal is a special unit employed in geodesy and geophysics to express the accel- eration resulting from gravity; 1 gal ffi 10 -2 m s -2. (Bureau International des Poids et Mesures - - Le Syst~me Interna- tional d'Unit~s (SI) - - 6e Edition Pavilion de Breteuil, S~vres, France, 1991 (in French and English): Units in use temporar- ily.)
P. Melchior et al. ~Physics o f the Earth and Planetary Interiors 93 (1996) 223-238 225
Table 2 Number of terms of each degree in the Tamura potential
Wave Potential degree Total
2 3 4
3 M K 1 2 3 0 5 Ql 12 4 1 17 L K 1 15 3 0 18 M 1 2 3 0 5 N O t 4 2 1 7 J1 8 2 1 11 3MO 1 3 3 0 6 3MJ 2 2 3 2 7 2N 2 7 3 0 10 3 M K 2 5 3 0 8 N 2 4 2 1 7 L 2 8 1 0 9 3MO 2 2 4 0 6 M 3 o I1 2 13
In any case, owing to the vectorial composition of different periodic terms with different phases contained in one group, the global amplitude periodically changes with time (Schwahn et al., 1991). The separation of the third-degree terms from the second-order terms is compulsory be- cause the contributions of oceanic tides (attrac- tion and loading) as well as the direct elastic deformation of the 'solid' Ear th are essentially different for the different degrees of harmonics in the potential development.
Dittfeld (1991) succeeded, for the first time, in separating three of the third-degree terms, the diurnal M 1 and the two semidiurnal 3 M K 2 and
3 M O 2 components from their neighbouring sec- ond-degree terms (Tables 1, 2 and 3) in the 17 year long tidal gravity series from Potsdam (1974-1990), obtained with the Askania GS 15 gravimeter no. 222. The mean square errors on the 8 factors found from this series (1.083-1.093) are between 0.006 and 0.010. The error bars are much larger on the other third-degree compo- nents making their determination non-significant.
Another solution, but less efficient, is to weigh, in the adopted potential, the waves deriving from the third and fourth degrees with respect to the second-degree terms, respectively, with the ratios 8 3 / 8 2 and 84/82: as these degrees are not reso- nant it is sufficient to apply a single coefficient in each tidal band (Table 4(a)). This has been done for about the last 20 years in all methods in current use (Venedikov, Hycon, Eterna) for the analysis of series of measurements shorter than 8.8 years.
Moreover, if one aims to analyse the response of the solid Ear th to the third-degree potential by comparison with the Molodensky model I, one has to apply to the 8 factors the corrections for inertia which are different for the different de- grees: for M 1 - - 0.00096, for 3 M O 2 - - 0.00397 and for M a - - 0.00861.
We have chosen here to compare our results with the most recent W a h r - D e h a n t model (1995) and, thus, not to apply such inertial corrections.
The Superconducting gravimeter G W R / T 3 was installed in 1982 at the Royal Observatory of Belgium situated in a suburb (Uccle) of Brussels.
Table 3 Group M I in the Tamura potential
Order Astronomical Doodson Frequency Tamura argument argument (deg h- 1) amplitude
coefficient
3 7 - N 155.545 14.48984571 977 2 ~" - p' 155.554 14.49205016 14 3 M 1 ¢ 155.555 14.49205212 6608 2 ~" + p' 155.556 14.49205408 37 3 ~" + N 155.565 14.49425853 855
¢, mean lunar time (hour angle of the Moon + 180°). - N , mean tropical longitude of the ascending lunar node (period 18.613 years). p', mean tropical longitude of the perihelion (period 20940 years). Tamura amplitude coefficient to be multiplied by 10 -6.
226 P. Melchior et al. / Physics o f the Earth and Planetary Interiors 93 (1996) 223-238
Table 4 Theoretical amplitude factors (a) Third- and fourth-degree potential terms in P~ and p2 groups
Station 0200 Bruxelles-UCCLE Vertical component Belgium 50 47 55N 04 21 2 9 E H 1 0 1 M P 4 M D 9 0 K M 9 8 1 1 1 7 3 0 1 Eocene sediments of the Belgu-French Basin composed of Lutetian sand, lying on Early Cambrian (Devillian) rocks of the Brabant Massif. Those Rocks of the Brabant Massif are of sedimentary origin and are characterized by a very monotonous almost purely elastic rock sequence. Observatoire Royal de Belgique Superconduting gravimeter Analogue registration Digital registration Atmospheric pressure correction TDB-UTC continuously adjusted Calibration Installation Maintenance
Dept.1 GWR T3 Until 87 07 20 Means on 60 s since 87 07 20 - 0.26277 P ( T ) -0.07785 P ( T - 1) -0.01669 P ( T - 2)
According to DB 92/P. Melchior R. Warburton, M. van Ruymbeke, B. Ducarme B. Ducarme, M. van Ruymbeke, R. Laurent, F. Renders
Least-squares analysis/Venedikov filters on 48 h/programming B. Ducarme Potential Tamura Complete development Computing center International Center for Earth Tides/FAGS/Brussels Data processing by L. Vandercoilden on 95/04/21 Computer APOLLO DN10000 Normalization factor to DB 92:0.99073 Residues calculated with respect to Dehant-Wahr Model 1995/Unit microgal Inertial correction not applied T 3 82 6 3/861014 861116/911114 911120/93 228 93 3 8 /94 915 Time Interval 4489.0 days 106752 readings 4 blocks Efficiency 0.99 Quality factors: Q1 68.6, Q2 486.7. 0 1 / K 1 1.0116, 1-O1/1-K l 1.0945, M 2 / O 1 1.0265. a Third-order potential terms.
P. Melchior et al. ~Physics of the Earth and Planetary Interiors 93 (1996) 223-238 227
Table 5 Brussels fundamental station
Wave group Estimated ampl.
Argument N Wave R.M.S.
Ampi. factor
R~M.S. Phase diff.
R.M.S. Residue
Ampl. Phase
l#9 . - l lA. 49 SGQ~ 0.2660.002 123.-126. 23 2Q 1 0.8950.003 127.-135. 31 S i g ~ 1.0730.003 135.-135. 5 a 3MK1 0.2600.002 135.-136. 17 Q1 6.7040.003 137.-13A. 18 RO 1 1.2730.003 142.-145. 32 O1 35.0980.002 146.-149. 26 Tau I 0.4580.003 152.-155. 18 LK1 0.9940.002 155.-155. 5 a M1 0.8100.003 155.-155. 7 NO~ 2.7700.003 156.-158. 18 Chi 1 0.5260.002 160.-162. 7 Pi I 0.9550.003 163.-163. 10 PI 16.3180.003 164.-164. 4 S 1 0.4900.004 165.-165. 18 K 1 48.7960.003 166.-166. 4 Psi t 0.4130.003 167.-169. 13 Phi I 0.7170.003 171.-173 15 TET 1 0.5280.002 174.-175. 11 Jl 2.7770.003 175.-175. 6 a 3M01 0.2970.002 175.-183. 23 SOl 0.4580.002 184.-187. 23 OO1 1.5200.002 190.-1C3. 67 Nul 0.2940.002
2/7.-225. 36 3N2 0.0960.002 226.-234. 25 Eps 2 0.2540.002 235.-235. 7 a 3MJ2 0.1670.002 235.-236. 10 2N2 0.8720.002 237.-244. 31 Mu 2 1.0840.002 245.-245. 8 ~ 3MK 2 0.6140.002 245.-245. 7 NT_ 6.7120.002 246.-249. 25 Nu 2 1.2670.002 252.-259. 56 M 2 35.5330.002 261.-263. 7 Lam 2 0.2540.002 264.-265. 9 LT. 0.9970.002 265.-265. 6 a 3MOz 0.5610.002 265.-268. 17 KNO 2 0.2530.001 270.-273. 9 T 2 0.9780.002 273.-273. 7 S 2 16.7060.002 274.-274. 5 R 2 0.1380.001 274.-278. 23 K 2 4.5480.001 281.-287. 36 Eta 2 0.2530.001 291.-293. 12 2S2 0.0440.001 294.-2C3. 41 2K2 0.0660.001
315.-348. 36 a MN 3 0.1090.001 353.-357. 13 a M3 0.3960.001 362.-367. 14 a ML 3 0.0220.001 373.-3A5. 19 a MK 3 0.0510.001 Standard deviation D 0.560
228 P. Melchior et aL / Physics of the Earth and Planetary Interiors 93 (1996) 223-238
A short description of the site and conditions of measurements is given in Table 5.
The data have been prepared using a remove restore technique, i.e. a synthetic tide based on the preliminary observed tidal factors is sub- tracted from the original data and the residuals are edited.
For the series from June 1982 to June 1987 when analogue recording was used, this proce- dure is applied to hourly data. From June 1987, digital data have been recorded with a 1 min rate and residuals are computed directly on the origi- nal data. To eliminate the perturbations pro- duced by railway activity we use a despiking pro- cedure followed by interpolation using spline function and decimation to hourly values. No jump correction is allowed. We correct the hourly values for atmospheric pressure effects. The hourly residues are then hand-corrected for obvi- ous anomalies resulting from earthquakes or monthly helium fillings. Up to a few days missing data are interpolated taking into account the general drift behaviour. Three remaining inter- ruptions in 12 years are listed in the analysis results (Table 5). Finally, we restore the tidal model prior to final analysis.
As the Venedikov analysis method (1966) elim- inates a second-order polynomial drift on each 48 h interval the long-term drift behaviour does not interfere with the analysis results.
The original Venedikov filters were used to separate the diurnal, semidiurnal and ter-diurnal bands. The least squares programme has been continuously updated since 1966 to introduce, for example, recent potential developments or the difference of barycentric dynamical time-univer- sal time (TDB-UT) which is adjusted by taking into account the leap seconds tabulated by the International Earth Rotation Service (IERS).
The results presented here concern: (1) the resonance owing to the Earth's liquid
core nutation acting very close to the frequency of the $1 solar elliptic diurnal wave whose ampli- tude is 413 ngal at Brussels;
(2) eight principal waves deriving from the third-degree potential (Table 1) - - 3 M K 1 (260 ngal), M 1 (810 ngal) and 3 M O 1 (297 ngal) in the diurnal frequency band (one cycle per day); 3 M J 2
(167 ngal), 3 M K 2 (614 ngal) and 3 M O 2 (561 ngal) in the semidiurnal frequency band (two cycles per day); M N 3 (109 ngal) and M 3 (396 ngal) in the ter-diurnal frequency band (three cycles per day which makes such waves easier to determine);
(3) the fortnightly zonal tide M f (5876 ngal, period 13.6608 day).
The amplitudes here given are the observed amplitudes at the latitude of Brussels (50.7986°N).
The contribution of atmospheric pressure to gravity variations by attraction and loading effects has tentatively been estimated using different ad- mittance factors (from - 0.15 to - 0.35) and sub- tracted from the original data (hourly readings). The most efficient reduction of the variance of postfit residuals has been obtained with a three- term admittance formula
( - 0.2628 _+ 0.0063) P (T)
- (0.0778 + 0.0081)P(T - 1)
-(0.0167 _+ 0.0082)P(T- 2) /zgal mhar -1
T being the time in hours, obtained by the so-called MISO procedure o~ De Meyer (1984).
It reduces the error bars on the observed am- plitudes and phases by a factor 2.75 for the diur- nal waves and 1.25 for the semidiurnal and ter-di- urnal waves.
Of course, tentative improvements of the nu- merical coefficients have been made on the basis of the global 12 years of data now available. The new coefficients (-0.2610 + 0.0058; -0.0722 + 0.0075; -0.0126 _+ 0.0058) do not differ signifi- cantly from the previous ones but, strangely enough, their application deteriorates the solu- tion by increasing the mean square errors. We therefore kept the initial expression.
However, we doubt whether this can really be done for the purely solar waves $1, ~/1, $2 for which the admittance appears to be much lower, eventually close to zero in narrow frequency bands corresponding to these three waves.
Moreover, purely instrumental reactions to pressure and temperature variations are not ex- cluded with this instrument which suffers from a slight helium leak between the gravimeter body and the surrounding vacuum space, increasing the thermal exchanges with the helium bath.
P. Melchior et al. / Physics of the Earth and Planetary Interiors 93 (1996) 223-238 229
The calibration of the instrument has been adjusted on the new system of the International Centre for Earth Tides (ICET) Data Bank DB92 (Melchior, 1994) with a normalisation factor of, 0.99073.
2. The role of the Earth's core
The response of the Earth to the second-de- gree tesseral tidal potential (Dehant and Ducarme, 1987) is strongly frequency dependent because of the existence of a resonance induced by the free nutational modes in the frequency band. The most important one is the free core nutation (FCN) related to a differential rotation of the fluid core with respect to the mantle and with an eigenfrequency in the middle of the diur- nal tidal band. This mode, if excited, is a retro- grade toroidal motion of the entire outer core (like a rigid rotation), which induces important
deformations of the core-mantle boundary. In addition, it induces an important resonance in the body tides as well as the associated astronom- ical forced nutations.
The experimental detection of the nearly diur- nal free wobble or free core retrograde nutation is problematic and many attempts to discover it directly from astronomical observations were not conclusive. As a matter of fact, one can expect that, being excited by different non-securely iden- tified mechanisms its phase and amplitude must be eminently variable.
Therefore, only indirect evidence of it can be obtained from observed resonance in the diurnal Earth tides and their associated forced nutations which are now measured with high precision by the very long base interferometry (VLBI) method.
The analysis of these measurements provides an important tool to investigate the structure and dynamics of the Earth's liquid core and inner core.
Table 6 Main long-period, diurnal and semidiurnal gravity tides at the Brussels station. Superconducting gravimeter GWR T3
Type Wave A 8 a 8 * a * M * WD
Long period L Mf 5876 1.1449 ± 0.0045 0.006 + 0.223* 1.1552
A, observed amplitude in nanogals. 8, tidal gravimetric factor. a: phase (minus represents a lag). 8 *, a * : corrected for oceanic tides attraction and loading (Schwiderski maps). Type of wave: L, lunar wave; S, solar wave; e, elliptic wave; d, declinational. MO: Molodensky I model + inertial effect. WD: Wahr -Dehan t model 1995 at Brussels latitude.
230 P. Melchior et aL / Physics of the Earth and Planetary Interiors 93 (1996) 223-238
The results given by the superconducting gravimeter for those diurnal waves exhibiting a differential behaviour are given in Table 6. The main waves Q1, O1, P1 and K 1 as well as the semidiurnal waves N 2, M 2, S 2, K 2 can be cor- rected for the perturbations owing to the oceanic tides (attraction and loading) on the basis of the Schwiderski world cotidal-corange maps. This is unfortunately impossible for the other smaller waves ~s 1, thl, J~ because of the lack of corre- sponding oceanic maps.
Nevertheless, in Table 6 we give a comparison with the most recent theoretical model (Wahr- Dehant, 1995) hereafter referred to as the Wahr -Dehan t model. The subtraction of the ef- fect of atmospheric pressure, underground water effects and radiation tides at the purely solar frequencies of ~O~ and S 1 where they have the major influence is problematic. Without such cor- rections we have obtained
8(~01) = 1.2699 + 0.0215
Ot(~l) = -- 1.44 ° __ 0.97 °
an amplitude factor which fits VLBI and other superconducting gravimeters results (Defraigne et al., 1994) but an anomalous negative phase. To try to explain this anomaly research is underway to study a possible annual modulation of the meteorological S 1 component directly affecting the ~ wave.
3. Third-degree tidal potential terms
Tidal components derived from the third-de- gree luni-solar tidal potential are of great interest because of their relations with third-degree elas- tic Love numbers and with the characteristics of oceanic attraction and loading which considerably differ from those derived from the second-degree tides (Cartwright, 1975).
However, their amplitudes are much smaller than those of second-degree terms as they are proportional to the fourth power of the ratio 1/60.3 (Earth mean r ad ius /mean distance of the Moon) while the second-degree terms are pro- portional to the third power (Tables 1 and 2).
It was only 30 years ago (Ozawa, 1964;
Barsenkov, 1967; Melchior and Venedikov, 1968; Melchior, 1970; Korba and Korba, 1974) that tentative efforts were made, with some success, to extract the ter-diurnal M 3 wave from experimen- tal data, taking advantage of the fact that this component offers three cycles per day (period 8.279 h).
The 1200 term Tamura tidal potential develop- ment used in this investigation is truncated to 10 -5 of the amplitude of the main semidiurnal wave M 2 which corresponds to 0.8 ngal sensitivity for gravity variations. It is obviously impossible to create individual groups of waves which all derive from only one and the same degree of the tidal potential as shown by Table 2 for the new groups separated in the Brussels' tidal gravity data. The typical example of the important M 1 group of only five waves is shown in Table 3: even if the second-degree side components of M 1 (which could be separated only after 20 940 years!) have extremely small theoretical amplitudes, it is com- pulsory to weight them in proportion to the ratio ~3/82 with a priori modelized values as given in Table 4(b)
82 = 1.15870, 83 = 1.06733
for the Molodensky model I
82 = 1.15242, 83 = 1.06813
for the Wahr -Dehan t model
In groups derived from P~ containing fourth degree components the weighting will be taken proportionally to 83/84 with (Table 4(b))
84 = 1.03698 for the Molodensky model I
~4 = 1.03800 for the Wahr -Dehan t model
3.1. D i u r n a l w a v e s
In the diurnal band, this truncation gives a group of 30 spectral lines usually called 'group M 1' or 'group N O 1' which extends from 152.646 ( r - 3 h + p - N ' + p s ) to 155.755 (z + 2p) Dood- son arguments b.
b For a definition of Doodson arguments (D.A.) see Mel- chior (1983, Section 1.8, p. 33).
P. Melchior et al. / Physics of the Earth and Planetary Interiors 93 (1996) 223-238 231
Eight of these 30 waves derive from the third- degree potential (Tables 1 and 2) - - the principal one being M t - - whereas one wave derives from the fourth degree potential.
The declination of the Moon varies with a tropical month period (27.321582 day) from -28*35' to +28*35'. The mean value being 0", there is no diurnal tide to be christened Mj (Doodson argument 155.555, lunar time fre- quency ¢ = 14.49205210" h -1) with a period of exactly 1 lunar day (24.8412 h) in the develop- ment of the second-degree part of the lunar po- tential (p1 __ sin2~bsin28cosH).
However, a diurnal wave with this exact argu- ment 155.555 (lunar day period) results from the third-degree potential and is called M 1 here.
To separate M 1 with respect to its companions N O 1 and L K 1 one has to subdivide the 30 waves of the group in such a way that they appear in three separate groups.
(1) Group L K 1 - - 18 waves from 152.646 to 155.535 D.A.; frequency interval 0.11635 ° h -1.
(2) Group M 1 - - three third-degree waves and two very small second-degree waves from 155.545 to 155.565 D.A. as given in the Table 3; fre- quency interval 0.00441 ° h-1.
(3) Group N O 1 - - seven waves from 155.645 to 155.755 D.A.; frequency interval 0.00685* h -1.
The frequency difference between the M 1
third-degree wave and the N O t and L K 1
second-degree waves is small (0.00464181* h -1) as it corresponds to the reciprocal of the period of revolution of the lunar mean perigee, i.e. 8.847 year (40.69 ° yea r - 1).
Consequently series of nine years long are needed to separate these three groups of waves properly.
Two other groups of third-degree tidal terms can also be separated in the diurnal band.
(1) A group close to the large Q1 wave which contains five terms from the Tamura potential, around a main one 135.555 D.A. of frequency (7 - 2s), appearing as a beating between M 3 and K2. We call it 3 M K 1 in the tables.
(2) A group close to the J1 wave, 175.455 D.A., its frequency being ~- + 2s we call it 3 M O 1.
We made two different groupings within the many small waves present in this large frequency band and obtained practically the same results (8 = 1.088 + 0.008 and 8 = 1.086 + 0.008).
These two terms 3 M K ~ and 3 M O 1 are declina- tional waves symmetrically located with respect to
M 1 • The results of the analysis of the Brussels
superconducting gravimeter are summarized in Table 7.
Table 7 Tidal gravity waves deriving from the luni-solar potential of third degree
A, ampli tude in nanogals. 8, observed ampli tude factor. a , observed phase in degree (minus corresponds to a lag). WD, W a h r - D e h a n t model 1995. B in nanogals.
232 P. Melchior et al. / Physics of the Earth and Planetary Interiors 93 (1996) 223-238
3.2. S e m i d i u r n a l w a v e s
In the semidiurnal frequency band the same truncation to 10 -5 allowed two important third- degree declinational waves to appear which are close to the large-amplitude second-degree ellip- tic waves N 2 and L 2.
We call these third-degree waves, respectively, 3 M K 2 (argument 2 ~ - - s ) and 3 M O 2 (argument 2~" + s) because they appear as beatings between M 3 and K 1 or O1 (Table 1).
We also identified two elliptic waves, respec- tively, close to 2N2, called 3 M J 2 and close to Kz, called 3 M Q 2 but only 3 M J 2 has a sufficiently large amplitude to be separated.
As in the case of the diurnal wave M1, the third-degree components again differ from N2, L2, 2 N 2 or K 2 by a lunar perigee period so that a series of 9 years long is again needed to perform the separation.
Dittfeld (1991) and Schwahn et al. (1991) have shown that successive series of 480 days' length exhibit ' temporal variations' which result from the non-separation of third-degree diurnal com- ponents and spoil the determination of the main w a v e s N O 1 , N 2 and L 2 s o that, as explained in Section 1, special adjustments of the amplitude of the third-degree terms in the programmes of analysis have been made as long as one cannot perform a 9 year separation.
The results of the analysis of the Brussels superconducting gravimeter are summarized in Table 7.
3.3. Ter -d iurna l w a v e s
With the 12 year series of the Brussels super- conducting gravimeter we obtain
wave M N 3 , frequency 3~-- ( s - p ) :
~3 = 1.055 + 0.009, ot = - 0 . 1 6 ° + 0.49 °,
observed amplitude 109 ngal
wave M 3, frequency 3~':
83 = 1.053 + 0.002, a = 0.62 ° _ 0.14 °,
observed amplitude 396 ngal
in good agreement with the mean obtained by ICET Data Bank 94 from 45 stations in Europe
t~(M3) = 1.057 + 0.008, a = - 0 . 2 7 ° + 0.24 °
However these results, which cannot be cor- rected for oceanic contributions because of the lack of ocean tide models, contradict not only the theoretical models but also a world mean based upon the results of 178 stations collected in the ICET Data Bank, most of them being situated in equatorial and tropical areas where the ter-diur- nal waves have their maximum amplitude, being proportional to the cosine cube of the latitude
t~(M3) = 1.071 + 0.004, a = 0.04 ° + 0.12 °
which fits the W a h r - D e h a n t theoretical model.
3.4. O c e a n - E a r t h t ide in t e rac t ions
Cartwright (1975), Cartwright et al. (1986) pointed out that a M 1 tide 'is shown to have unusually large amplitudes over a sea stretching from Shetland to the Azores, including the North Sea'.
Cartwright explains this observation by the ex- istence in the third-degree tesseral part of the lunar potential P3 ~ of a line at exactly the M 1 frequency as reported here in Section 3.1, at- tributing the magnification of the P~ line relative to the nearby p1 lines (essentially N O 1 =- 155.655 and L K 1 - 155.455 - - see Table 1) as observed at 11 oceanic tidal gauges in the UK, France and Germany to spatial matching with normal modes of oscillation, rather than a resonance in fre- quency.
Platzman (1991) briefly reporting the same ob- servations also expected ' M 1 to dominate by a factor about 6'.
We can confirm this with an analysis of 36 years of oceanic tides at Ostend harbour (Bel- gium) as given in Table 8: M 1 has an amplitude of 9 mm, just like at Dunkerque, France (Cartwright, 1975); N O 1 is only 0.7 mm. If we restrict the analysis to the last 26 years ' data at Ostend, which are of bet ter quality, the M 1 am- plitude is still 9 mm while the N O 1 amplitude is 2 mm (6 mm at Dunkerque). This means a magnifi- cation of M 1 by a factor of about 6 as stated by Platzman.
P. Melchior et al. / Physics of the Earth and Planetary Interiors 93 (1996) 223-238 233
Table 8 Third-degree waves in the oceanic tides at Ostend harbour (latitude 51 ° 14' 24"N): magnification of wave M 1
Group T a m u r a Amplitude Local phase coefficient (mm) (degrees)
36 years' observations 1945-1980. a Given for comparison.
When we look at the results of our supercon- ducting gravimeter, given in Table 7, we observe some interesting features. The phases of third-de- gree waves are different from those of their clos- est companions in the diurnal band and very different in the semidiurnal band, which reflects the differences in the perturbing role of the oceanic tides.
These phase advances are not due to the com- puting procedure as an analysis test applied to a theoretical gravity tide series computed by H.-G. Wenzel (personal communication, 1993) did not reveal any anomalies in the phases.
In the specific case of the M 1 tide, we observe, in agreement with Cartwright, that the residue obtained by subtracting the solid Earth contribu- tion is
B = 14.3 + 3.2 ngal
/3 = 37.8 ° _+ 12.6 °
With respect to a total amplitude of 810 ngal it represents 2% while the N O 1 and L K ~ residues represent only 0.3% and 0.6% of the respective amplitudes.
As shown in Table 7 the ratio between the amplitude factors of M 3 and M 1 third-degree waves also exhibits an amplification of 2% at the M 1 frequency
5( M 1 ) / 5 ( M 3 ) -- 1.0204
We observe in Table 7 that, while the W a h r - Dehant theoretical model proposes an increase of the 8 amplitude factor from the diurnal to the semidiurnal and to the ter-diurnal waves, the measurements show exactly the opposite at the Brussels station. This most probably is due to interactions from the Atlantic Ocean response to the third-degree potential.
4. The fortnightly tide Mf
The argument of this zonal tidal component is twice the mean longitude of the Moon (2s), cor- responding to a frequency of 1.098033 ° h-1 and a period of 13.6608 days. Its amplitude, which is about the same as the tesseral diurnal solar wave P1 in the potential development, is multiplied by the zonal geodetic factor (3sin2~b - 1) which gives a theoretical amplitude of 5.133/~gal at the lati- tude of Brussels for a rigid Earth.
With the remarkable exception of a determi- nation made at the South Pole (where the zonal tides are indeed maximum - - 12.8/~gal for M r )
with the LaCoste Romberg tidal gravimeter no. 4 equipped with electrostatic feedback, in the past the poor stability of classical spring gravimeters did not allow the safe determination of the elastic response of the Earth at periods longer than a few days.
Besides a chandlerian component owing to polar motion (430 days period, 5/~gal amplitude) the superconducting gravimeter installed at Brus- sels has an annual component of 5 /zgal ampli- tude and a small drift not larger than 4 /~gal year -1. The drift is well correlated with the changes of level of underground waters (De- Icourt-Honorez, 1995). These stable components do not perturb the shorter 13.66 day period for which we obtain at Brussels
Amplitude 5.876 + 0.023 /~gal
= 1.1449 + 0.0045 a = 0.006 ° + 0.223 ° = 0.0002 + 0.0085 day
234 P. Melchior et al. / Physics of the Earth and Planetary Interiors 93 (1996) 223-238
The recent Wahr -Dehan t model gives, however, 8 = 1.1552.
The difference should correspond to the at- traction-loading effect of the oceanic tides but the calculation based upon the Mf corange- cotidal Schwiderski map leaves us rather scepti- cal.
We suspect a deficiency in the long-period tides of the Schwiderski maps.
Note that the ice shelf tides in the Weddell and Ross seas are not taken into account as they are not included in the Schwiderski maps. Their amplitudes are known to be large in the diurnal and semidiurnal frequency bands (Melchior, 1995) but unknown for Mr.
It has already been pointed out that the solu- tions for this tidal species do not conserve the water mass (Francis and Mazzega, 1990). The ratio of the non-conserved tidal mass to the tidal amplitude is 0.08, 0.26, 0.54 for Mr, Mm and Ssa, respectively.
As a matter of fact the calculated effect at Brussels amounts to 0.164 /xgal with a phase of 5.07 ° with mass correction and 0.127/~gal with a phase of 8.09 ° without mass correction. If we use these values 8(Mf) becomes as low as 1.113 - - which is of course not acceptable - - with a phase of -0 .14 ° .
To check these results we have calculated the effect of a hypothetical equilibrium &If tide at Brussels and obtained
without mass correction
L = 0 . 1 3 4 /xgal, A = 0 °
with uniform mass correction
L = 0 . 1 9 2 /zgal, A = 0 °
not very different from the Schwiderski map re- sults. This eliminates any possibility of a phase inversion in the case of the zonal waves.
Three years of measurements with the super- conducting gravimeter installed at Bad Homburg, Germany, at a latitude of 50.2285°N, which are not very different from the Brussels' results, have given, according to Richter (1985)
8 = 1 . 1 4 8 + 0 . 0 2 0 , a = 0 . 2 ° + 1 . 0 °
5. Conclusions
The complete listing obtained by analysis of the 12 years' observations with the superconduct- ing gravimeter G W R / T 3 at Brussels is repro- duced in Table 5. It shows results for 48 tidal components: 24 groups of diurnal waves, 20 groups of semidiurnal waves and four groups of ter-diurnal waves of which we discussed only those groups exhibiting specific peculiarities.
From the results obtained for the small I1/1 diurnal wave as well as for six small ter-diurnal waves it appears that a long series of registration with a GWR superconducting gravimeter allows us to determine the amplitudes of tidal waves with an internal precision of 2 or 3 ngal.
It is interesting to examine how the parame- ters corresponding to the 'main' term of a group change when this group is subdivided into the two or three groups given in Table 1.
In the diurnal family we have found that the N O 1 lunar wave is only slightly affected by the separation of its 30 components into the three groups of Table 1: after separation t~(NO 1) changes from its global value 1.1566 + 0.0010 to 1.1572 _+ 0.0011, its phase from 0.221 ° + 0.047 ° to 0.149 ° + 0.053 °.
The Q1 and J1 lunar waves parameters do not change significantly after separation from 3M0 r
In the semidiurnal family, however, the lunar L 2 wave which has long been considered as a most unstable wave is considerably modified as may be seen from the following results
with 32 components
(frequency band width 0.1665 ° )
A = 992 ngal, 8 = 1.1689 _+ 0.0018
ol = 2.24 ° + 0.09 °
when separated
nine components: L 2
(frequency band width 0.0411 ° )
A = 997 ngal, B = 1.1755 + 0.0020
a = 2.88 ° + 0.10 °
P. Melchior et al. / Physics of the Earth and Planetary Interiors 93 (1996) 223-238 235
six components: 3 M O 2
( third-degree potential)
(frequency band width 0.0066 ° )
A = 561 ngal, 8 = 1.0724 + 0.0030
a = 0.25 ° _+ 0.16 °
17 components: K N O 2
(frequency band width 0.1161 °)
A -- 253 ngal, 8 = 1.1919 _+ 0.0060
a = 1.83 ° _+ 0.29 °
This results from the fact that when we sepa- rate the third-degree potential terms 3 M O 2 from the main second-degree potential term L2, it implies also the separation of a group called
KNO2, situated on the other side, in frequency, of 3 M O 2.
This new group has a rather broad frequency band (0.116 ° corresponds to 2 min 52 s) and contains 13 solar waves and four lunar waves. It is therefore not surprising that it has a 8 factor of 1.1919 + 0.0060 close to the nearby solar waves T 2 and S 2 factors, respectively, 1.1976 5:0.0020 and 1.1960 _+ 0.0001.
When dissociated from the third-degree terms, the L 2 parameters tend towards those of the M 2 wave (8 = 1.18359 + 0.00006, a = 2.741 ° :t: 0.003 ° (Table 5) whose L 2 is its smallest elliptic wave.
As N2, the main elliptic wave of M 2 has --1.1678 + 0.0003 it is curious to observe that
after dissociation, t~(L 2) is just the mean of 8 (M 2) and ~(N2).
IJ.
1.25 - -
1.20
1.15
1.10
O .
o
! ~ ~ ~ . r .o~
1[-
v - D "l- f l .
F F
Z
E ¢~
E t model
1 . o 5 ' I ' I ' I ' I ' I
12 13 14 15 16 17
Angular Velocity (degree/hour) Fig. 1. Diurnal wave tidal ampli tude factor 8 as a function of the frequency: • terms derived from the second-degree luni-solar potential; • Q~, 0 I, P1, K1 wave ampli tude factors corrected for the oceanic tides' attraction and loading; * terms derived from the third-degree lunar potential.
236 P. Melchior et al. / Physics of the Earth and Planetary Interiors 93 (1996) 223-238
The M 3 main ter-diurnal wave parameters do not change when we eliminate 33 of the 46 com- ponents of the group.
The diurnal amplitude factors distribution in frequency is illustrated by Fig. 1.
Comparing the observed values (filled circles) with the corresponding values corrected for oceanic effects (filled triangles) for the Q1, O1, P1 and K 1 waves, shows that the differences are small so that one can expect that such a correc- tion is negligible for the smaller components with perhaps the exception of the ~01 factor.
The three third-degree waves 3 M K 1, M 1,
3 M O 1, are of course clearly dissociated from the second-degree waves.
The semidiurnal amplitude factor's frequency dependence is represented in Fig. 2 which shows a larger dispersion.
Important corrections are expected, owing to tidal oceanic attraction and loading, but only the N2, M 2, S 2 and K 2 waves can be corrected. After this correction, an increase of the 8 factors with angular velocity seems to occur (filled triangles in Fig. 2).
As in the case of the diurnal frequency band, the three third-degree waves 3 M J 2 , 3 M K 2 and 3 M O 2 are clearly dissociated from the second-de- gree waves.
However, the separation of 3 M O 2 does not resolve the anomalous response of the A2 wave.
Obviously, the precision reached with super- conducting gravimeters means that it is necessary and urgent to construct accurate corange-cotidal oceanic maps, not only for the main constituents but for the minor ones amongst which ~Ol, ~bl, A2 and L 2 waves appear especially important. The
1.25 - -
LI.
E3
1.20
1.15
1.10
1.05
'-'1
a .
tu ~ :1: I --~ k -
o,I
z ~_~ ,,,
~ - - A
WD model
~ ~ m o a o l
¢'4 o
' I ' t ' I ' I ' I 27 28 29 30 31 32
Angular Velocity (degree/hour) Fig. 2. Semidiurnal wave tidal ampli tude factor $ as a function of the frequency: • terms derived from the second-degree luni-solar potential; • N 2, M 2, S 2, K 2 wave amplitude factors corrected for the oceanic tides' a t t r a c t i o n a n d l o a d i n g ; * t e r m s derived from the third-degree lunar potential.
P. Melchior et al. / Physics of the Earth and Planetary Interiors 93 (1996) 223-238 237
results ob ta ined with superconduc t ing gravime- ters will be useful to val idate such maps.
Acknowledgements
Dur ing the long per iod of con t inuous registra- t ions the daily m a i n t e n a n c e of the equ ipmen t and the regular data storage were per formed by R. Lau ren t u n d e r the supervision of M. Van Ruym- beke.
A great n u m b e r of numer ica l analyses were made by L. Vandercoi lden . We are indeb ted to member s of the Geodynamics section of the Royal Observatory for the care taken in ma in ta in ing the Brussels s tat ion in good operat ion.
Helpful assistance has b e e n provided by R. Warbur ton , especially after the accidental failure of October 1986 when a relevi tat ion had to be made.
Frui t ful discussions were held with V. D e h a n t and P. Defra igne on theoret ical aspects of the
work. Refe ree ing this paper , Wal te r Z i i rn made
helpful comments and gave good advice. We are pleased to thank him for his very careful and thorough analysis. We also thank a second ref-
eree for judicious remarks.
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