Transcript ***** 1 - Copernicus.org

ABSTRACT
For calculating the ocean loading component, g LOAD , the tidal
The dissipation, anisotropy, and rotation constraints of the total
oceanic gravimetric effect are analysed.
heights were expanded in the spherical functions. This component is
The dependence of the results on the selected P- and S-velocity
associated with the deflection of the mantle, which causes correspond-
model (i.e., on the structure of the crust and upper mantle of the
ing variation in the potential. According to (Pertsev, 1971), applying
Earth) is considered.
the Kummer's transformation, we obtain the following expression for
For calculating the effect of oceanic loading, we apply the method of
the component:
Legendre polynomial expansion of tidal heights. The CSR3 model data

g LOAD  8 G 0   N

are expanded up to the 720th order. The results yielded by this method closely agree with those calculated from the Green's functions by
the LOAD87 program of the ETERNA software. Remarkable advantage
of our program over other approaches is that it provides high-speed
n 720

n 0
n
 H nm
m 0

n 720

n 0
n

( n   N ) H nm 
m 0

(1)
Here,  0 is the density of seawater, G is gravitational constant,
H
m
n
 Yn are the Laplace's Ys of the oceanic tidal height expansion in
m
processing and does not require introducing the near-field formalism.
n 1
kn
2
is the loading delta-factors of
2n  1
hn 
Application of the pre-computed expansions reduces the time of calcu-
the Legendre polynomials,  n 
lations by two orders of magnitude, compared to LOAD87. This is par-
order n. N is the maximum order of expansion. It is slightly above the
ticularly important when analyzing the geographical distributions of the
real n = 720; however, this does not affect the accuracy of calcula-
loading effect predicted by different models.
tions.
Taking dissipation into account improves the total gravimetric effect
calculated for the M2 wave near the coast of Europe by 0.1-0.2 mcGal
in amplitude and by a few hundredths of degree in phase. Transition
from the PREM model to the IASP91 model which is better suitable for
Europe changes the model predictions by 0.1-0.4 mcGal in amplitude
and by 0.1 to 5-7 degrees in phase.
The direct Newtonian attraction of the water masses was calculated
by the formula
g ATTR
1
 2 G 0 
Yn
n 2n  1
(2)
The files containing the load Love numbers and expansions of cotidal maps are included in our software package ATLANTIDA 3.0 intended for calculating the ocean loading effect (Fig. 1).
Thus, allowance for dissipation together with the use of the refined
data on the crustal and upper-mantle structure of the Earth may contribute, at places, over 0.5 mcGal to the amplitude
and a
few de-
grees to the phase of the total oceanic gravimetric effect. In this relation, particular attention should be paid to the regions about the
Land's End cape (Cape Cornwall) and Cape Saint Mathieu.
CONCLUSIONS
Dissipation contributes 0.1-0.2 gal to the amplitude and, typically, a
few hundredths of a degree to the phase of the total oceanic gravimetric effect near the coast of Europe.
Transition from PREM to the IASP91 model which is more suitable for
CALCULATION PROCEDURE
Europe adds another 0.1-0.4 gal to the amplitude and from 0.1 to 5-7
First, the velocities of longitudinal and transverse seismic waves
degrees to the phase.
for the PREM and IASP91 models of the Earth's structure were recalcu-
Thus, the overall contribution of dissipation and the use of the re-
lated from a reference period of 1 s to the periods of 12 and 24 hours.
fined data on the structure of the Earth's crust and upper mantle may
For doing this, we applied the logarithmic creep function introduced by
attain at places up to 0.5 gal in amplitude and a few degrees in
Lomnitz, i.e. the quality factor Q was assumed to be constant. This
amplitude and a few degrees in phase. In this relation, the vicinity of
approach is sufficiently accurate in the range of periods from 1 s to 1
the Land's End headland on Cornwall Peninsula and Saint Mathieu
day (Zharkov and Molodensky, 1977). Using the velocities determined
promontory in Brittany deserve special attention.
in this way, we calculated the Lame parameters as the functions of the
Comparing and analyzing the Love numbers, the Green's functions,
depth from the surface to the center of the Earth with a step of 100 m.
and the gravimetric effect calculated in our work and in (Pagiatakis,
The upper oceanic layer in the PREM model was replaced by the crust.
1990), we come to the preliminarily conclusion that the effects of rota-
The IASP91 model covers an interval to a depth of 871 km, below
tion and anisotropy, which are taken into account by Pagiatakis, only
which the values predicted by the PREM model were used.
insignificantly change the loading oceanic effect, contributing at most a
Then, based on the obtained Lame parameters and the depth distribution
of
density,
we
evaluated
the
load
The amplitudes and phases of the ocean loading effect in the gravimetric data measured close to the coast of Europe are
shown in Figs. 2 and 3. The amplitude exhibits an overall trend of a moderate increase in its gradient from south northwards.
For example, as the coastline is approached from inland near the Moroccan coast off the mouth of Gibraltar, the amplitude
increases, on average, from 4.5 to 7.0 gal over a distance of 100 km. However, the amplitude near the western coasts of
Portugal and France increases over the same distance from 6 to 10 gal. Farther north, at Cape Saint Mathieu at the
extremity of Brittany and at the Land's End headland of the Cornwall Peninsula in Northern England, the effect changes from
7 to 12.5 gal as the coast is approached. The southwesterly increase in Western Ireland is from 6 to 11 gal, and near the
northeastern coast of Great Britain the effect in the coastal zone attains 5 gal.
Some specificity in the behavior of amplitude against the ambient background is also observed in the English Channel
slightly west of Calais. It is interesting that in the nearby Solent Straight, the tidal flow rates are very high (up to 10 km per
hour). As far as the phase in concerned, here, the nodular features in Fig. 3 near Le Havre and in the eastern part of the Irish
Sea and north of Scotland, close to the Orkney and Shetland Islands are remarkable.
Love
numbers
k’n and h’n up to the order n = 10000. The method of calculation is described in detail by Vinogradova and Spiridonov (2012a; 2012b).
Dissipation causes the variation in the amplitude
immediately close to the coastline, which does not typically
exceed 0.1 gal. This is clearly seen in Fig. 6, where the
difference in amplitudes between the reference periods of 12
h and 1 s is shown for the PREM model. Slightly higher
values (up to 0.2 gal) are only observed close to the Saint
Mathieu Cape and Land's End Cape projecting into the
ocean. Here again, the specific features of the variations in
the Irish Sea and English Channel are remarkable. The
phase discrepancies normally lie within hundredths of a
degree and attain a few degrees only in the mentioned
particular zones.
The differences in the structure of the crust and upper
mantle cause somewhat higher discrepancies in the
amplitudes and phases of the ocean gravimetric effect. The
differences between the amplitudes calculated by the IASP91
and PREM models are presented in Fig. 7. The discrepancies
between the models attain 0.1 gal near the Moroccan coast
and increase to 0.3 gal at the western coasts of Portugal
and France. The maximum phase differences here are at
most 0.1. However, near the tip of Cape Cornwall and close
to the Irish seaboard, the region of Le Havre and Calais, the
amplitudes differ by 0.35--0.4 gal and the phases, by 5-7.
The total result of the above-discussed effects associated
with the differences in the structure of the Earth's crust and
the upper mantle and dissipation is shown in Fig. 8.
We have also studied the total effect caused by rotation and
anisotropy from the data in (Pagiatakis, 1990). This effect
turned out to be very small.
few hundredths of a microgal to it.
TABLE 1
REFERENCES
Vinogradova, O.Yu. and Spiridonov, E.A., Comparative Analysis of Oceanic Corrections to
Gravity Calculated from the PREM and IASP91 Models, Izv. Phys. Solid Earth, 2012, vol. 48,
no. 2, pp. 162-170.
Vinogradova, O.Yu., Oceanic Tidal Loads near European Coast from the Green's Functions,
Izv. Phys. Solid Earth (in press).
Pertsev, B.P., Implications of the Near-Zone Oceanic Tidal Effects for the Observations of
the Earth's Tides, Izv. Akad. Nauk SSSR, Fiz. Zemli, 1976, no. 1, pp. 13-22.
Zharkov, V.N. and Molodenskii, S.M., On the Corrections of the Love Numbers for Dynamical Shear Modulus, Izv. Akad. Nauk SSSR, Fiz. Zemli, 1977, no. 5, pp. 13-22.
Pagiatakis, Spiros D., The response of a realistic Earth to ocean tide loading, Geophys. J.
Int., 103, 541-560, 1990.
N
NAME
LONG
LAT
ALT
113
NEWLYN
-5.55
50.10
13
114
K1
P1
Q1
K2
N2
S2
PROG
AMPL PHASE AMPL PHASE AMPL PHASE AMPL PHASE AMPL PHASE AMPL PHASE AMPL PHASE AMPL PHASE
ATLANTIDA 12.947 60.881 0.658 194.96 0.967 82.817 0.322 91.931 0.246 254.95 1.062 16.196 2.496 81.228 4.123 20.902
LOAD07
12.871 61.065 0.657 195.01 0.963 82.796 0.322 91.888 0.246 254.89 1.056 16.411 2.490 81.356 4.104 21.094
REDRUTH
-5.23
50.23
113
0.076 -0.184 6E-04 -0.047 0.004 0.021 3E-04 0.043 3E-04 0.062 0.006 -0.215 0.006 -0.128 0.019 -0.192
ATTLANTIDA 12.683 57.698 0.645 193.28 0.950 81.516 0.317 90.646 0.241 253.72 1.044 11.835 2.432 77.98 4.043 16.768
LOAD07
12.642 57.69 0.644 193.24 0.946 81.484 0.317 90.605 0.241 253.72 1.040 11.823 2.431 77.933 4.036 16.736
DIF
329 MONTIERNEU
-0.95
45.88
10
0.041
ATTLANTIDA 8.207
LOAD07
8.265
0.008
8E-04
0.044
0.004
0.032
4E-04
0.041
5E-04 5.E-04 0.004
0.012 8E-04 0.047
0.007
0.032
83.943 0.468 207.44 0.628 93.437 0.210 102.74 0.176 263.29 0.718 51.641 1.752 103.62 2.759 53.552
83.986 0.472 207.51 0.628 93.368 0.211 102.75 0.177 263.34 0.724 51.662 1.770 103.66 2.786 53.581
DIF
423
Fig. 1. The ATLANTIDA 3.0 interface
O1
DIF
404
The program calculates: the effect of the direct Newtonian attraction
of water masses, the ocean load effect, and their sum. It is possible
to make calculations for either a point or for nods of a grid.
The kinds of the program from LOAD87 of ETERNA package are:
1) Calculations for different models of Earth’s structure;
2) Absence of necessity of the local area.
M2
The differences in amplitudes of the total oceanic effect calculated by the ATLANTIDA3.0 and LOAD87 programs for the M2
wave are presented in Fig. 4. Just as the amplitudes themselves and their gradients, these differences increase from the
south towards the north; however, they nowhere exceed 0.2 gal in absolute value and vanish 50--100 km offshore. At the
same time, along the coastline up to 45°N, these differences are not larger than 0.05-0.074 gal. The differences increase up
to 0.1 gal in the approach to the Cornwall Peninsula where they attain 0.1--0.2 gal.
Similar values are also observed in local areas near Le Havre and on the western coast of Great Britain. It is worth noting
that overall, the amplitude differences (in percentage) do not exceed 0.5--1% over the major part of the coastline of Western
Europe. This value is half the relative error of tidal height determination in the CSR3.0 oceanic model.
The maximal phase differences are also revealed near the Cornwall Peninsula, on the French coast of the English Channel,
and on the western coast of Great Britain. Instead of the phase differences themselves, Fig. 5 shows the sine of the phase
difference times the amplitude. Overall, contribution of the phase differences to the studied signal along the coast of Europe
does not typically exceed 0.01 gal except for the narrow zones mentioned above, i.e., the phase mismatch is, generally,
substantially smaller than the amplitude difference. This is also true for other tidal waves.
SANTADER
LA CORUNA
-3.81
-8.42
43.47
43.37
25
-0.058 -0.043 -0.004 -0.067 7E-05 0.069 -0.001 -0.010 -9E-04 -0.049 -0.006 -0.021 -0.018 -0.039 -0.027 -0.029
ATTLANTIDA 9.037 94.456 0.526 216.84 0.710 99.679 0.234 108.96 0.195 270.37 0.785 63.513 1.941 114.27 3.022 65.188
LOAD07
9.104 94.471 0.531 216.89 0.711 99.652 0.235 108.99 0.196 270.39 0.791 63.495 1.960 114.28 3.050 65.173
0
DIF
-0.067 -0.015 -0.005 -0.054 -0.001 0.027 -6E-04 -0.032 -0.001 -0.016 -0.006 0.018 -0.019 -0.014 -0.028 0.015
ATTLANTIDA 10.751 107.62 0.654 223.4 0.931 105.11 0.302 114.36 0.232 277.71 0.927 78.172 2.320 127.76 3.576 79.569
LOAD07
10.813 107.64 0.659 223.42 0.932 105.11 0.304 114.40 0.233 277.74 0.932 78.16 2.336 127.79 3.601 79.562
DIF
427
SANTIAGO/L
-8.54
42.88
250
-0.062 -0.018 -0.005 -0.018 -0.002 -0.002 -0.002 -0.036 -0.001 -0.027 -0.005 0.012 -0.016 -0.027 -0.025 0.007
ATTLANTIDA 10.011 109.94 0.619 224.82 0.887 106.08 0.288 115.32 0.220 279.25 0.863 81.403 2.171 129.96 3.333 82.644
LOAD07
9.832 109.85 0.608 224.73 0.874 105.82 0.284 115.06 0.216 279.24 0.848 81.340 2.134 129.88 3.273 82.578
DIF
434
OVIEDO F. C
-5.85
43.35
246
0.179
ATTLANTIDA 8.270
LOAD07
8.230
DIF
480
PORTO
-8.67
41.08
0
0.040
0.095
0.011
0.087
0.013
0.257
0.004
0.257
0.004
0.010
0.015
0.063 0.037 0.083
0.060
0.066
99.726 0.502 218.99 0.722 100.65 0.237 109.75 0.187 273.87 0.709 69.670 1.782 119.74 2.741 71.203
99.718 0.499 218.94 0.721 100.57 0.237 109.65 0.186 273.94 0.705 69.649 1.773 119.74 2.726 71.183
0.008
0.003
0.055
0.001
ATTLANTIDA 9.7047 117.28 0.612 229.99 0.876
LOAD07
9.679 117.29 0.612 229.98 0.871
0.084 4.E-04
0.096
7.E-04 -0.066 0.004
0.021 0.009 -0.003 0.015
0.020
111.3
0.282 120.65 0.211 284.49 0.850 90.935 2.118 136.74 3.267 91.840
111.2
0.282 120.61 0.211 284.54 0.848 90.942 2.117 136.76 3.262 91.851
DIF
481
COIMBRA
-8.43
40.20
50
0.0257 -0.012 2E-04 0.008 0.005 0.103 3E-04 0.045 3E-04 -0.049 0.002 -0.007 9E-04 -0.018 0.005 -0.011
ATTLANTIDA 8.816 119.39 0.561 231.71 0.808 112.57 0.261 121.93 0.194 286.43 0.788 94.086 1.929 138.48 2.987 94.857
LOAD07
8.857 119.45 0.564 231.73 0.809 112.56 0.262 121.98 0.195 286.51 0.783 94.118 1.941 138.54 3.005 94.896
DIF
482
LISBONNE
-9.19
38.71
0
-0.041 -0.057 -0.003 -0.021 -8.E-04 0.013 -0.001 -0.048 -7.E-04 -0.080 0.005 -0.032 -0.012 -0.062 -0.018 -0.039
ATTLANTIDA 9.874
LOAD07
9.842
DIF
0.032
127.18 0.629 236.41 0.894 118.30 0.288 127.83 0.212 291.53 0.893 103.08 2.161 145.51 3.408 103.74
127.18 0.628 236.40 0.888 118.19 0.287 127.78 0.211 291.58 0.890 103.09 2.159 145.53 3.400 103.75
-0.002 9.E-04 0.007
0.006
0.113
8.E-04
0.054
5.E-04 -0.047 0.003 -0.006 0.002 -0.016 0.008 -0.011
In
conclusion,
we
summarize in Table 1 the
amplitudes and phases of the
total
oceanic
gravimetric
effect calculated by the
ATLANTIDA 3.0 and LOAD87
programs at 10 ICET points.
Only those points where the
amplitude of the effect in the
M2 wave exceeds 7 gal are
included in the table. The
calculations do not take into
account dissipation for the
PREM and CSR3.0 models.