Solid earth tide and Arctic oceanic loading tide at Longyearbyen (Spitsbergen)

Physics of the Earth and Planetary Interiors, 13 (1976) 197 —211 © Elsevier Scientific Publishing Company, Amsterdam Printed in The Netherlands

197

SOLID EARTH TIDE AND ARCTIC OCEANIC LOADING TIDE AT LONGYEARBYEN (SPITSBERGEN) M. MOENS Institut d’Astronomie et de Géophysique Georges I,emaftre, Université de Louvain, Lou vain-la-Neuve (Belgium) 1

(Received May 3, 1976; revised and accepted June 14, 1976)

Moens, M., 1976. Solid earth tide and Arctic oceanic loading tide at Longyearbyen (Spitsbergen). Phys. Earth Planet. Inter., 13: 197—211. The three components of the indirect oceanic effect are calculated for the M 2, K1 and O~waves and compared with the observed earth tide. The vertical component of the near Arctic oceanic load explains fairly well the large 45°observed phase lag of M2. The results for the horizontal components are satisfactory; the discrepancies between different tiltmeters are not due to the oceanic perturbations (unless some local or cavity effects are supposed).

1. Introduction The existence of an earth tide station at a very high latitude should theoretically allow a precise determination of the N—S and E—W components of the tesseral tidal waves (K1, 01, P1, ...) because their amplitudes are greatest at the poles. Access, problems of installation and maiutenance are far harder and more difficult to solve at such a location than elsewhere. Prescriptions for horizontal pendulums especially (a deep and stable underground gallery with low noise level) strongly limit the choice of sites. From 1968 to 1970, a well-found station [three Askania (ASK) gravirneters and eight VerbaandertMelchior (VM) horizontal pendulums] was active at Longyearbyen (78°l2’N, 15°36’E,depth under the free surface: 350 m, distance to the sea: about 5 km) as part of the “Astro-geo Project Spitsbergen” (Melchior et aL, 1970). Polar regions are also characterized by very small theoretical amplitudes of the other earth tide constituents: the diurnal vertical and the semi-diurnal 1

. . . Address: Institut d Astronomie et de Geophysique Georges Lemaitre, Université de Louvain, 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium.

horizontal and vertical components are zero at the poles. The oceanic semi-diurnal tide on the other hand, has an appreciable amplitude (nearly 50 cm for M2) along the Spitsbergen coast. TherefoTe, it is clear that the indirect effect (deformations of the earth’s crust under periodical loading of the moving oceanic waters) will considerably modify the observed tidal parameters.

2. The Arctic oceanic tides Within the last few years some numerical solutions of Laplace’s hydrodynamical equation of the World Ocean have been available (Hendershott, 1973) but their discrepancies prevent a satisfactory global view of the phenomenon. We are thus always obliged to describe the oceanic tide in an empirical, and in some manner, subjective way on the basis of the existing observed tidal data. However, the observations of tides in the Arctic Ocean have been episodic, of short duration and often inaccurate. Only one theoretical investigation (Goldsbrough, 1915) gives indications of the peculiarities to be expected in a polar ocean: because of the very small shape of the basin the tide is not generated

198 Care taken in the installation, instrumental corrections

85°N

geophysical and precise calibration phenomenaensure present that in the thesetidal-time constantsspeconly reflect the limit of precision of results the instruments) Any(in observed tidal constituent from the

2cm (34O~)

K

1 ________ Scm (210~)

3cm

K1

trum. superposition the elastic earth of by thethe direct luni-solar earth tide forces) (deformation and of the of oceanic perturbation (having the same frequency, because derived from the same potential):

10~(i91°)

65’N 30w

~5cm7~

A0 cos(w1t

+ Øo)

=~

JAt cos(w~t)+A~ cos(w~t+ (1)

Fig. 1. Schematized tide for O~and K1 waves.

where: in it and cannot co-oscillate with other oceans and so is necessarily derived from the North Atlantic Ocean. We tried (Moens, 1969) to collect all available data disseminated in the literature and to draw a cotidal chart (lines of same amplitude and same phase of the tide) for the principal constituent M2 (Fig. 2, pp. 199— 200). This indicated that the tide propagates as a progressive wave from the North Atlantic Ocean, pefletrating the Arctic Ocean partly between Greenland and Iceland and partly between Iceland and Norway. This wave is gradually damped and deformed by the

A0

=

At

=

A1

=

~o

=

undersea topography of the European and Siberian shelves. Due to the accessibility of the west coast, the oceanic tide in Spitsbergen is fairly well determined (except for the southeast region where uncer-

Theoretical works (Alsop and Kuo, 1964) assign values of Ll6 to ~ and 0.68 toy with a precision better than 3% depending on the choice of the elastic earth’s

tainty about the locality of an amphidrome between Hopen Island and Bjornoya remains). The diurnal constituents Km and 0~have an amplitude of a few centimetres only. An attempt to draw precise cotidal charts for these waves is thus worthless. In order to estimate the associated indirect effect we have adopted an elementary distribution of these tides schematized in Fig. 1.

1~)

= = =

model [Gutenberg-Bullen (G-B) type], in excellent agreement with experimental values (Meichior, 1975). Taking into account the dynamical effect of a liquid core (Molodensky, 1961) one obtains comparable values with a resonance effect in the diurnal spectrum and thus a slight variation as a function of the pulsatance. A precise determination of the indirect effect (A1, ~) should provide values of y and b deduced from ob-

servations by means of eq. 1: cos~0~AicosØi)/At

3. The indirect oceanic effect The signal recorded by tiltmeters and gravimeters is the response to all forces acting on the measuring instrument; the harmonic analysis of the records gives the harmonic constants (amplitude and phase) for the

main constituents (M2, S2, N2, ..., K1, 01, P1,

=

o.~

amplitude of the observed wave amplitude of the theoretical wave (rigid earth) amplitude of the oceanic perturbation phase lag of the observed wave phase lag of the oceanic perturbation pulsatance of the wave under consideration universal time (UT) non-dimensional factors containing the global response of the earth (y for horizontal components, cS for vertical component)

...).

(2)

At present the observed amplitude and phase (A0, m~~) are sufficiently precise, but the evaluation of A~,

~ causes problems. The presence of moving oceanic waters produces

three different effects:

,s~_______

I

\~

U.

\ ~

/%~V

// ~

__

~

I

/

//

~

\~N ,~X Y V~ ~ \ (V~Y ~~Tt7

~\

>c/

I 7//~~NV ~ I

~

~N.

‘

\\

\~ /K/l4rT /~ ~ ~

\

\

\

/

\

N N

/

/

/

N~/S~,4Q ~

‘°iA~

~

/~-

~ /

~ ~-~‘

3~ 2~

~-~J

/

\

/

~ 40

--

~ç1\~J~j~ \~

~

_x \~ \.

\~~j~,fl) ~

t;—~---~

s’~JL-~-~\ ‘~$~ \

// ~/ v-°--~~--.~/ / /

f4~

//

‘0

203

(A) An elastic deformation of the crust under the periodic loading. (B) A gravitational (Newtonian) attraction of the displaced water masses on the station. (C) A modification of the earth’s gravity field by the deformation (mass redistribution). The first effect (A) is often evaluated with a Boussinesq model (elastic and homogeneous infinite half-plane subject ~o a point-load). Caputo (1962), Longman (1963) and particularly Farrell (1972) have adopted spherical models of G-B type more suitable for the gravimetric tide (for which the curvature of the earth is important). Moreover, Farrell’s method allows direct and precise convolution of the ocean tide by means of Green’s functions. The Newtonian attraction (B) is easily calculated by numerical integration over a sphere (or by circular sectors division in a plane model). An explicit computation of the mass-redistribution effect (C) has never been precisely done for plane models. Nishimura (1950) and many later workers evaluated it as proportional to (A) for the tilt corrections. One advantage of Longman’s and Farrell’s work is to give an easy method to take into account for spherical models. Applications of the procedures described here require the digitization of the cotidal charts in a form suitable for computation. Therefore we have divided the Arctic and North Atlantic oceans into 23 concentric zones surrounding the Longyearbyen station. Successive radii were: 5, 10, 15, 25, 50, 75, 100, 125, 250, 375, 1,875, 2,000 km; each zone was azimuthally every ten degrees. This 36 = 828divided spherical trapezia covering thegives: ocean23upX to 2,000 km from the station. A transparent template adapted to the map projection was used to assign three values to each trapezium: (1) The percentage of sea in proportion to land. (2) The mean amplitude of the ocean tide. (3) The mean phase lag of the ocean tide with respect to the maximum of the potential at the zero meridian. ...,

4. Gravimetric tide

4.1. Boussinesq ~smethod Boussinesq’s solution (infinite homogeneous halfplane with elastic Lamé’s parameters X, p) express the vertical displacement w caused by a unit point-mass load with respect to the distance r of the station as:

x + ~+ /1) 1r (3) 4~PO~ This displacement in the unperturbed earth’s gravity field causes a perturbation of the gravity at the station: + 2~ P(r) = a 4irp(X + ~i) r (4) ~i~r)

=

~

where g is the gravity acceleration and a the earth’s radius, P(r) is the Green’s function of the Boussinesq problem. On this flat model, a spherical trapezium with a constant tidal amplitude His approximated by a circular sector of radii r 1 , r2 and side azimuths a~ The gravimetric perturbation associated with such a trapezium is obtained by integrating P(r) on its surface; this takes account of all the water masses: r2

z~,g=

02

ff r1

pgHP(r)rdrdcx

(5)

c51

where p is the specific mass of seawater. Thus: ~ X + 2ji a 4irp(X + p) (~2 ~1)(r2 r1)pH (6) The sum of the values obtained for the 828 trapezia gives the effect of the crust displacement (A). Numerical results in Table I, are obtained Lamé’s param2 and pfor 1.22 - 1012 dyn/ eters ~ =The 1.5choice 1012 of dyn/cm cm2 these numerical values is very important because their combination influences zXg by the multiplying factor (X + 2ji)/ [p(X + p)]. As in the crust and the upper mantle the factor is —‘14p/l 1: it can be observed that the influence of a variation in p is more than seven times that of a variation in X. =

—

--

~‘.

4.2. Plane stratified medium Kuo (1969) had considered the problem of a planestratified medium and solved it numerically with “realistic” elastic parameters adapted from spherical *

To evaluate the precision, the deformation of the crust (A) was computed according to three different methods:

~

This corresponds to the elastic properties at a depth of 500 km in the G-B models. It is generally assumed that the layers situated at a depth equal to half the load-station distance contribute the most to the deformation.

204 TABLE I Indirect gravimetric tide at Longyearbyen: comparison of different models of crustal deformation Elastic model

M

2

Boussinesq: homogeneous half-plane 2 p = 1.22 1012 dyn/cm2 X = 1.50 1012 dyn/cm Kuo: stratified flat model transposed from G-B models Farrell: spherical earth of G-B type a = 6.371 . 10~m

—

Vertical displacement only (A)

K1

01

amplitude (pGal)

phase lag (°)

amplitude (pGal)

phase lag (°)

amplitude (pGal)

phase lag

3.415

143.27

1.256

—46.86

0.422

119.84

2.687

144.77

0.969

—46.59

0.352

118.33

2.323

144.54

0.839

—46.51

0.311

117.97

(0)

Amplitudes expressed in microgals and phase lags given with respect to the maximum of the potential at the local meridian.

models. He provided us with a program giving directly the gravity perturbation on such a model. Results are given in Table I. 4.3. Spherical G-B type models Farrell (1972) has tabulated the deformation Green’s function with respect to the angular distance 0 between station and load, for three spherical G-B type models, In spherical coordinates the gravity perturbation due to the displacement of the station under the load of a spherical trapezium is expressed by:

=

—~ff 2

02

02

a2sin ON(0)pHdOdct

(7)

o~ 0~

where:

=

value of the Green’s function at the distance 0 azimuthal angle

=

limits of the spherical trapezium

=

a (a 1, a2, 01, 02)

-

.

H being constant in a trapezium of 10 opening one 0 b tams. ‘

4 ~gapH

cal model. On the contrary, the choice of the model does not influence the phases of the indirect effect. The vertical component of the Newtonian attraction (B) of a unitary point-mass load on a spherical model

N(0)

=

This expression is numerically integrated, with the values of N(0) interpolated from Farrell’s tables. Table I reproduces the results of this application. It shows that the Boussinesq model is inadequate for the computation of the vertical component of the indirect effect (even for the near zones) because of the necessarily arbitrary choice of a mean value for X and p. With values of Lamé’s parameters corresponding to a depth of 680 km the results would have been more acceptable. The discrepancy between the methods of Kuo and Farrell is less than 15% on the amplitudes and probably reflects the difference between a flat and a spheri-

f

~2

sin ON(O)dO

(8)

G —

(9)

2

4a srn(0/2)

G being the gravitational constant, 0 having the same meaning as in eqs. 7 and 8 (the angular distance be.

tween the over integrated load

and the whole the station). ocean surface. This formula A small must correcbe

tion taking account of the height of the station is to be applied to eq. 9. The redistribution of masses (C) is evaluated by in-

205

tegration of the appropriate Green’s functions (also computed by Farrell) in the same form as eq. 8. Tables lI—IV show tile indirect gravimetric effect for the three waves M2, K1, 01, and the respective polar diagrams showing the calculated values and the observed values with error boxes. These results only

of the World Ocean on the basis of Zahel’s (1970) M2 cotidal chart gives an amplitude of 0.5 pGal, and on the basis of Bogdanov and Magarik’s (1969) 0~chart 0.05 pGal; this is far less than the regional effect. We must also recall that for the diurnal constituents the results are only a rough approximation due to the lack

reflect the effect of the nearest zones (up to a distance of 2,000 km). An evaluation of the influence

of information concerning the oceanic tide. Nevertheless, the calculated contribution seems satisfactory.

TABLE II Indirect gravimetric effect M2

______

ASK. 85AJ direct + near indirect effect

[A~Kt16

ASRT~~ N N

N N N

PHASE 1

N~

2

______

MICROGALS

“N I

Nrheor. Tide

3

xl.16

Predicted values

Predicted amplitude (pGal) (1) Elastic deformation and mass redistribution (A, C) (2) Direct Newtonian attraction (B) (3) Total indirect effect (1 + 2) (4) Direct effect (theoretical tide X 1.16) (5) Predicted effect (3 + 4)

Predicted local phase (0)

1.7599 0.5202

142.63 141.96 142.46 0 36.98

2.3501

3.7649 2.3803

Observed and corrected values Instrument

ASK 85A ASK 116 ASK 206

Observed I

0.6298 0.5914 0.5531

Observed amplitude

Observed phase lag

(pGal)

(0)

2.0450 1.9199

46.43 47.45

±3.41

1.0081

0.87

±0.0403

±3.90

0.9739

—0.32

±0.0858

1.7867

39.21

±8.90

1.0098

—5.32

±0.0374

See Appendix for comment statement.

Corrected I

Corrected phase lag (0)

206 TABLE Ill Indirect gravimetric effect K

1

~ASEIiI15IiII2?lth~rItidex1.,6~:

85~

CROGALS rect+neor I irect effect

Predicted values

(1) Elastic deformation and mass redistribution (A, C) (2) Direct Newtonian attraction (B) (3) Total indirect effect (1 + 2)

(4) Direct effect (theoretical tide x 1.16) + 4)

(5) Predicted effect (3

Predicted

Predicted

amplitude

(pGal)

local phase (°)

0.5984 0.2230 0.8214

—46.47 —46.88 —46.58

23.0793 23.65 14

0 —1.45

Observed and corrected values Instrument

ASK 85A ASK 116 ASK 206

Observed I

1.1623 ±0.0156 1.1216 ±0.0139 1.1831 ±0.0317

Observed

Observed

amplitude (pGal)

phase lag

phase lag

(o)

(o)

23.3755 23.5286 20.2289

1.45 ±0.77 —3.65 ±0.71 —2.00 ±1.57

Corrected I

1.1352 1.0920 1.1494

Corrected

2.98 —2.19 —0.32

See Appendix for comment statement.

5. Tilt tide

Newtonian attraction is:

The computation of the horizontal indirect effect is

quite similar gravimetric one, with the restriction that we have to to the project the calculated contributions azimuthally on the two reference axes: N—S and E—W. Using again Farrell’s procedure, the deformation effect for one spherical trapezium is: ~~NS 2~ ~2 ‘cos ~

~ 1~(~a2pHf ~&FW)

01

sin ON’(O)dO) X

1

1~tNS — —G cos 0/2 X tsin (cosa~) a 2sin2O/2 (11) l~tEwJ 4a Green’s—formalism is also used for the effect of mass

redistribution (C) by introducing corresponding values of N’(0) in eq. 10. Before assessing the results of Tables V—VII, it should be kept in mind that in the horizontal components the indirect effect is very sensitive to local con-

~sin az

i

ditions (elasticity of the upper layers of the crust) as (10)

with a~the load azimuth measured from the station. This is to be integrated in the same way as eq. 8. As in eq. 9 the Green’s function for the direct

pointed out by Beaumont and Lambert (1972) and essentially depends on the very near oceanic tide (80%

of the total effect of the zones up to 2,000 km is concentrated in the first hundred kilometres). The shape of the coast by itself thus greatly influences the results.

207 TABLE IV Indirect gravimetric effect O~ ASK. 206

PHASE

direct + indirect

3

1b

~.

~:ct

1~

!Th~TI

1.16

____

Predicted values

(1) (2) (3) (4)

Elastic deformation and mass redistribution (A, C) Direct Newtonian attraction (B) Total indirect effect (1 + 2) Direct effect (theoretical tide X 1.16) (5) Predicted effect (3 + 4)

Predicted amplitude

Predicted local phase

(pGal)

(0)

0.1918 0.0640 0.2557 12.0137 11.8903

118.73 121.41 119.40 0 1.07

Observed and corrected values

Instrument

Observed I

ASK 85A

1.2173

ASK116 ASK 206

1.1345±0.0184 1.1443 ±0.0501

See Appendix for

±0.0207

Observed amplitude

Observed phase lag

(pGal)

(o)

12.5971

0.21

11.7550 11.8792

Corrected I

Corrected phase lag (0)

±0.98 0.59±0.93 3.91 ±2.51

1.2295 1.1466 1.1551

—0.80 —0.49 2.81

comment statement.

The interpretation of the tilt tide is critical. We

indirect effect while a continental one would, on the

the reader must remember that pendulums VM 99 and VM 100 are, in fact, situated in a second station at the opposite side of the Longyearbyen Valley (-“2 km large, —‘500 m deep, open to the sea in approximately

contrary, increase it. The phase remains the same. In the M2 E—W component, with the same phase, an increase of 30% of amplitude would fit better with pendulums VM 76 and 99, and an increase of 15% with VM 13. Geological maps attribute to Spitsbergen a continental-shelf structure: ourofmodel underestimates the amplitude by perhaps an amount 2%; this cannot

the north direction). In all cases the correction to the M2 wave improves the observed values; this is spectacular for ~ which is divided by nearly 5. The phases of the N—S7EW component K1homogeare diminbecome of more ished while the values of neous. The phase of the E—W component and the val-

explain why the observed values are bigger than the

ues of ~N5 on the contrary are poorly corrected.

have used an elastic G-B model; a modified model with an oceanic crust would diminish the amplitude of the

predicted one. When considering the ob~ervedvalues,

The corrections for 01 are very small.

208 TABLE V Indirect tiltmnetric effect M

2 VM.76 VM.99

VM 101

/

direct + near indirect effect

_______

[vM.102J

/

/

VM.42

I

/

PHASE N.S.

1

Theor Tide *0.63 ,, ‘ i~-~

2

direct + near I indirect effec~j

/

VM.77

\

VM.13

3

/

/

PHASE ,1 Theor Tide x 68 E.W.

1

2

3

10~

Predicted values

Predicted

Predicted

Predicted

N—S

N—S

E-W

Predicted E-W

amplitude

local phase

amplitude

local phase

(10~”)

(0)

(10~”)

(0)

3.9895

119.15

10.5249

0.9889 4.5650

59.15 108.33

4.4108 14.9213

(1) Elastic deformation and mass redistribution

(A,C) (2) Direct Newtonian attraction (B) (3) Total indirect effect (1 + 2) (4)

Direct effect (theoretical tide

X

0.68)

2.2440

(5) Predicted effect (3 + 4)

0

4.4081

2.2916 16.1305

79.43

—29.81 —24.29 —28.18 (+90 = 61.82)

—90

—35.37 (+90 = 54.63)

Observed and corrected values

Compo-

Instrument

Observed

amplitude

Observed phase lag

(10~”)

(0)

1.0948 ±0.0154 1.4261 ±0.0173 3.0313 ±0.0198 1.2675 ±0.0315

3.6128 4.7061 10.0033 4.1828

59.08 ±0.81 91.86 ±0.70 121.73 ±0.38 64.61 ±1.42

1.0654 0.4046 1.7158

—20.55 16.10 132.50

0.9928

—9.75

5.4678 ±0.0188 5.1713 ±0.0275 6.1764 ±0.0313 6.2702 ±0.0537

18.4265 17.4273

56.85 60.02

± ±0.31

0.20

1.1242

36.90

20.8145

56.24

±0.29

21.1306

57.90

±0.49

0.7587 1.8213 1.8774

49.46 42.57 48.62

‘y

nent

N—S N—S N—S N—S

VM VM VM VM

42 102 100 77

E—W E—W E—W E—W

VM VM VM VM

13 101 99 76

*

*

Observed

Corrected y

Correctec phase lag (0)

See Appendix for comment statement. * VM 99 and VM 100 are situated at 2 km from the other pendulums, but on the other side of the valley.

209 TABLE VI Indirect tiltmetric effect K

1 ,~ direct

+

near I

indirect effeç~j

~

~

99

~

.

Predicted values Predicted N—S amplitude

Predicted N—S local phase (0)

(10~”) (1) Elastic deformation and mass redistribution (A, C) (2) Direct Newtonian attraction (B) (3) Total indirect effect (1 + 2)

0.2894 0.2610 0.5503

(4) Direct effect (theoretical tide x 0.68) (5) Predicted effect (3 + 4)

6.7 157 7.0314

—33.87 —31.46 —33.11 (—90 = —123.11) 90

Predicted E—W amplitude

Predicted E—W local phase

(10~”)

(0)

1.1398 0.3861 1.5259

134.64 134.31 134.56

7.1767

93.75

0

6.2021

10.10

(—90 = 3.75) Observed and corrected values Component

Instrument

Observed

‘~‘

N—S N—S

VM 42 VM 102

0.7920 0.7363

± 0.0108 ± 0.0130

N—S

VM 100

0.7775

± 0.0112

E—W E—W E—W

VM 13 VM 101 VM 99

0.6238 ±0.0095 0.65 15 ±0.0120 0.5816 ±0.0140

See Appendix

Observed amplitude (10~”)

Observed phase lag

7.8218 7.2717 7.6786

—9.66 —8.25 —16.43

± 0.78 ± 1.01

6.5836 6.8759

—14.08 —16.93

6.1382

—18.80

Corrected

(0)

Corrected phase lag (o)

0.8158 0.7614 0.7953

—6.07 —4.44 —12.58

± 0.87 ±

0.7510

—19.83

1.05

0.7816

—21.99

± 1.37

0.7138

—24.01

± 0.83

for comment statement.

As exhibited in Tables V—VII the observed N—S components are always worse than the E—W. Except for “valley or gallery effect”, there is no clear reason for such discrepancies, the theoretical amplitudes being of the same magnitude in both components. It must be pointed out that the same phenomenon can be observed in the European stations where the N—S components

generally show large discrepancies, while the E—W cornponents look quite coherent (Melchior, 1975).

6. Conclusions At Longyearbyen station, the gravimetric indirect effect explains fairly well the very important 45°phase

210 TABLE VII Indirect tiltmetric effect O~ VM.99

________

Theor. Tide]

PHASE

1

NS

~

~

_____

~~

1VM

PHASE

~

~o~”

~

13

~iioi

E.W

~

~

-3~

6

Predicted values Predicted N—S amplitude

Predicted N—S local phase (°)

(10~’’)

Predicted E—W amplitude

Predicted E—W local phase

(10~”)

(0)

(1) Elastic deformation and mass

redistribution (A, C) (2) Direct Newtonian attraction (B) (3) Total indirect effect (1 + 2)

0.0682 0.0553 0.1234

—85.62 —91.12 —88.08

0.4860 0.2098 0.6957

(4) Direct effect (theoretical tide X 0.68) (5) Predicted effect (3 + 4)

4.94 16 5.0649

(+90 = 1.92) —90 —89.95 (+90 = 0.05)

5.2795 5.5692

112.00 109.42 111.22 (—180 = —68.78)

180 173.31 (--180

=

—6.69)

Observed and corrected values Component

Instrument

N—S N—S N—S

VM 42 VM 102 VM 100

E—W E—W E—W

VM 13 VM 101 VM 99

Observed -y

—

0.7306 0.6814 0.6415

± 0.0144 ± 0.0173 ± 0.0150

0.8205 ±0.0127 0.8016 ±0.0160 0.7991

± 0.0187

Observed amplitude

Observed phase lag

(10~”)

(0)

5.3093 4.9517 4.6618 6.3704 6.2236 6.2042

—3.01 —2.62 —1.18

Corrected y

Corrected phase lag (0)

± 1.13 ± 1.45 ± 1.34

3.48 ±0.89 0.63 ±1.14 6.24 ±1.34

0.7137

—3.13

0.6645 0.6245

—2.83 —1.26

0.7978

9.62

0.7746

6.85

0.7808

12.61

See Appendix for comment statement.

lag of the observed M2 wave. When comparing the corrected ~c = 1.01 with its theoretical value 1.16, one

has to realize that the difference in fact amounts to 0.4 pGal only (corresponding to a vertical displacement of -=1.5 mm). The schematized repartition of the K1

and O~oceanic waves seems to realize a good approximation of the gravimetric indirect effect that is compatible with the observed values. Thermal perturbations on K1 and the small amplitude of 0~do not allow more

precise information to be obtained by comparing observation and computation. Lacolazet and Wittlinger (1974) tried to explain the discrepancies of observed phase and amplitude of hon. zontal components by “cavity effects” of the global earth tide; Harrison (1976) calculated this effect for some particular cases. But unless one supposes a “cavity effect” to be associated with superficial deformation of

the crust under oceanic loading (more sensitive for M2

211 than for O~and Kl of small amplitude), it will not be possible to explain why the N—S diurnal component is more coherent than the semi-diurnal M 2.

Acknowledgements We wish to express our thanks to all those who went in the Arctic to install the Longyearbyen station (M. Bonatz, P. Melchior, B. Ducarme, J. Blanckenburgh, A.

Vandewinckel), especially Prof. M. Bonatz who spent many winter months for the maintenance of the station in very hard conditions, and to the direction of the mine (Store Norske Kulkompani). Prof. J.T. Kuo made available to us a copy of his program for indirect-effect . correction available and we were very happy to utilize it. It is pleasure to acknowledge Prof. P. Melchior for suggesting the present investigation and for critically reading the manuscript.

Appendix Comment statement for Tables Il—VU The vector diagrams represent: (1) The theoretical tide in the X-axis direction with magnitude proportional to the amplitude (2) The observed tide for each instrument: magnitude is proportional to the amplitude; phase lag (relative to the theoretical tide) is positive counter-clockwise; boxes represent the mean-squares errors deduced from harmonic analysis (3) The calculated indirect effect as a vector (broken line) added to the theoretical tide in order to obtain the predicted tide (direct + near indirect effect) In the tables: (1) Observed values are taken from Bonatz et al. (1971) (2) Observed t is the ratio between the observed amplitude and the theoretical one (3) Corrected I and corrected phase lag are obtained after substraction of the calculated indirect effect from the observed tide. Note that in Tables Il—V the local phases refer to the maximum of the potential at the local meridian while observed and corrected phase lags refers to the maximum of the theoretical tidal components (derivatives of the potential)

References under inclined surface loads. J. Geophys. Res., 74 (12):

Alsop, L.E. and Kuo, J.T., 1964. The characteristic numbers of semi-diurnal Earth tidal components for various Earth models. Ann. Géophys., 20 (Fasc. 3): 286—300. Beaumont, C. and Lambert, A., 1972. Crustal structure from surface load tilts, using a finite element model. Geophys. J.R. Astron. Soc., 29: 203—226. Bogdanov, K.T. and Magarik, V.A., 1969. A numerical solution of the problem of the tidal wave propagation in the World Ocean. Izv. Atmos. Oceanic Phys., 5 (12): 1309—1317. Bonatz, M., Melchior, P. and Ducarme, B., 1971. Station: Longyearbyen. Bull. Obs., Marées Terr., Ohs. R. Belg., IV, Fasc. 1. Caputo, M., 1962. Tables for the deformation of an Earth model by surface mass distributions. J. Geophys. Res., 67 (4): 1611—1616. Farrell, W.E., 1972. Deformation of the Earth by surface loads. Rev. Geophys. Space Phys., 10 (3): 761—797. Goldsbrough, G.R., 1915. The dynamical theory of the tides in a zonal basin. Proc. London Math. Soc., 14: 207. Harrison, J.C., 1976. Cavity and topographic effects in tilt

and strain measurement. J. Geophys. Res., 81(2): 319— 328. Hendershott, MC., 1973. Ocean Tides. Trans. Am. Geophys. Union, 54 (2): 76—86. Kuo, J.T., 1969. Static response of a multilayered medium

3195—3207. Lecolazet, R. and Wittlinger, G., 1974. Sur l’influence perturbatrice de Ia deformation des cavitCs d’observation sur les marécs clinométnques. C.R. Acad. Sci. Paris, Sér. B, 278: 663—666. Longman, l.M., 1963. A Green’s function for determining the deformation of the Earth under surface mass loads. J. Geophys. Res., 68 (2): 485- 496. Melchior, P., 1975. General Report on Earth Tides (1971— 1974). XVI Gen. Assem. I.U.G.G., Grenoble, 1975. Meichior, P., Bonatz, M. and Blanckenburgh, J., 1970. Astrogeo Project Spitsbergen 1968—1970. Ciel Terre, 86 (2): 85—99 (or Comm. Obs. R. BeIg., Sér. B, 5Cr. GCophys., 52: 98). Moens, M., 1969. Dynamique des mar~esocéaniques, Application l’Océan Arctique. Thesis, University of Louvain, Louvain (unpublished). Molodensky, M.S., 1961. The theory of nutation and diurnal Earth tides. Comm. Obs. R. Beig., Sér. Géophys., 188: 58.

a

Nishimura, E., 1950. On Earth tides. Trans. Am. Geophys. Union, 31(3): 357—376. Zahel, W., 1970. Die Reproduktion gezeitenbedingtcr Bewegungsvorg~ingeim Weltozean mittels des hydrodynamisch-nunierischen Verfahrens. Mitt. Inst. Meereskd. Univ. Hamburg, No. XVII.