The ordinary differential equations system of the sixth order, describing the strain state of the elastic, compressible, gravitating shell with geopotential, density and Lame parameters depending on the latitude was discharged and integrated. Thus, the elliptical shell of the earth is represented in the form of spherical layers. Density and elastic parameters vary with latitude along the surface of the sphere as the average radii of the crossing this surface ellipsoids on which these values are constant. Dissipation is accounted for under the logarithmic creep function. In the process of integrating of the obtained equations it avoided use of approximate methods.

As a result, the curves of amplitude delta factors latitude dependence for tidal waves of second order were obtained. The characteristic growth of tidal waves amplitude delta factors from the equator to the pole on the order of 0.12%, regardless of the frequency of the wave, was established.

After additional accounting of the effects associated with the forces of inertia, as well as dynamic resonance factors, obtained according to the works of other authors, the dependences were compared with SG observations data. This comparison showed а good agreement between the experimental data with the results of the calculations in this paper.

Average values of the calculated amplitude delta factors were strictly between those for DDW /H and DDW /NH models [Dehant V. et al., 1999].

The biaxial elliptical shell of the Earth is modeled by a set of spherical layers. The elastic Lame constants and density in each layer are assumed to depend on the distance r to the center of mass, the colatitude θ, and the ellipticity ε:

(1)

Here, index 0 denotes the mean values of the parameters for a given spherical layer. Formula (1) implies that density and elastic parameters vary with latitude along the surface of the sphere as the mean radii of the ellipsoids that cross this surface and on which the density and elastic parameters are constant.

0 2( , , ) ( , , ) 1 p

The loading potential is a homogeneous polynomial of order n: (2)

(3)

0

r nr

Denote the coordinates of the displacement vector by ur, u, u and the volumetric expansion by δ. Vi is the additional potential associated with the displacement of the mass due to the deformation.

u

With these designations, the equation describing the stress state of the elastic shell has the form::

(4)

Here, Tr is the stress tensor.

0;igrad u gradV V gradV grad Tr

The solution of Eqs. (4) is sought in the form:

(5)

with the additional condition

(6)

Eqs. (5) and (6) contain three auxiliary functions of the depth and latitude: function H characterizing the radial displacement, function T describing the tangential displacement, and function R determining the variation in the potential due to deformation.

The first term in the expression for u is necessary for providing the volumetric expansion to be independent of the derivatives of tidal potential with respect to the coordinates, and formula (6) defines the condition of unique separation of the normal and tangential stresses.

r nu H

r

1n n

T Tu

r r r r

1

sin nu T

r r

20

T T H

r

By adding the Poisson equation to Eq. (4) and eliminating the latitude derivatives of the sought functions H, T and R from the obtained expressions, we obtain the following system of ordinary differential equations of the sixth order:

(7)

(8)

(9)

(10)

(11)

(12)Where

The auxiliary unknowns N, M, L contained in Eqs. (7)–(12) after the corresponding normalization are

where

Potential V has the following form:

where

The boundary conditions on the sphere of radius are

where is the mean density of the Earth and is the gravitational acceleration on the Earth's surface.

Another three conditions are defined on the core–mantle boundary:

Here, ρi is the density of the core on its boundary with the mantle and G is the gravitational constant.

For the diurnal waves, we use the resonance curve (24) of [Dehant V. et al., 1999].

2

1 1 12 1 ;

2 2 1 2 2

nH n H n n T N

r n r

2

12 1 2 ;

nT n H n T M

r r

2

14 2 1 1 ;

nR G n H R n L

r r

2 2 22 2

3 22 14 4 2

1 1 1

p rpn nL G p r H n G T n R p L

n n n r

2 22

42 3 2 1 2 12 1 2 1 6

2 2 2

nM r r V H n n n n p T

n n

2

22 2 1 ;

2

r nR p M r n N

n r

22 2 2

4 4 8 1 2 4 1 12 2 1 /

2 2 2 1N V H V n n T

r r r n r r r n

22 4

11 1 14 ,

2 1 2 1 2

n nnL M n r p N

r n n r n r

00 0 0 2

122 ;

n nN H H T

r r

20

2;M r T H T

r

204 ;L r R G H

2

22

V Vctg

20

2 20

.2

T V Vctg

r

2 2222

1 2 33

2 31 14 ,

5 6x

a

r

r ppV G I qdq I r I

r r

22

1cos ,

3p 2

1

0

;xr

I q dq 52

0

;xr d q

I dqdq

3 .

x

a

r

dI dr

dr

0;N 0;M 20

1(2 1) ,

nL n a g R

a

m 0g V a

2( ) 4 0;iL r n r R G r H 0;M 0.iN R V H

The characteristic increase in the amplitude delta-factors of tidal waves from the equator to the pole by about 0.0014 for PREM and 0.0021 for IASP91 irrespective of the wavelength is revealed.

WAVES EQUATOR POLE MEAN DDW/H DDW/NH DDW_MEAN DIF_MEAN

Q1 1.15288 1.15427 1.15357 1.15280 1.15425 1.15353 0.00005O1 1.15288 1.15426 1.15357 1.15279 1.15424 1.15352 0.00006P1 1.14783 1.14920 1.14851 1.14777 1.14915 1.14846 0.00005K1 1.13324 1.13459 1.13392 1.13283 1.13489 1.13386 0.00006

OO1 1.15489 1.15628 1.15558 1.15478 1.15627 1.15553 0.00006M2 1.16037 1.16176 1.16106 1.16030 1.16172 1.16101 0.00005

Table 1. Amplitude delta-factors in comparison with DDW (PREM)

The obtained model average values of the amplitude delta-factors of tidal waves for PREM model lie strictly halfway between the corresponding average values of DDW/H and DDW/NH models (Table 1). For the IASP91 model, these values practically coincide with DDW/NH (Table 2).

WAVES EQUATOR POLE MEAN DDW/NH DIF_MEANQ1 1.15325 1.15536 1.15431 1.15425 0.00006O1 1.15324 1.15536 1.15430 1.15424 0.00006P1 1.14816 1.15026 1.14921 1.14915 0.00006K1 1.13390 1.13599 1.13494 1.13489 0.00005

OO1 1.15527 1.15739 1.15633 1.15627 0.00006M2 1.16071 1.16284 1.16177 1.16172 0.00005

Тable 2. Amplitude delta-factors in comparison with DDW/NH (IASP91)

The comparison of the calculated delta-factors with the data of the experiment (Tables 3 and 4) demonstrates their close coincidence at least within the accuracy of the presently applied oceanic models.

M2 O1 K1STATION LATITUDE EXP THEOR DIF% EXP THEOR DIF% EXP THEOR DIF%Membach 50.609 1.16142 1.16120 1.9E-02 1.15359 1.15370 -9.9E-03 1.13568 1.13405 1.4E-01Strasbourg 48.622 1.16232 1.16115 1.0E-01 1.15366 1.15366 2.9E-04 1.13547 1.13400 1.3E-01Bad Homb. 50.229 1.16287 1.16119 1.4E-01 1.15450 1.15369 7.0E-02 1.13600 1.13404 1.7E-01Moxa 50.645 1.16159 1.16120 3.4E-02 1.15315 1.15371 -4.8E-02 1.13507 1.13405 9.0E-02Medicina 44.522 1.16237 1.16105 1.1E-01 1.15334 1.15365 -2.7E-02 1.13480 1.13391 7.9E-02Wettzel 49.146 1.16116 1.16116 -3.1E-05 1.15237 1.15356 -1.0E-01 1.13424 1.13401 2.0E-02Vienna 48.248 1.16092 1.16114 -1.9E-02 1.15224 1.15367 -1.2E-01 1.13404 1.13399 4.4E-03

Table 3. The comparison of the calculated delta-factors with the data of the experiment (PREM)

M2 O1 K1STATION LATITUDE EXP THEOR DIF% EXP THEOR DIF% EXP THEOR DIF%

Membach 50.609 1.16142 1.16198 -4.9E-02 1.15359 1.15450 -7.9E-02 1.13568 1.13515 4.7E-02Strasbourg 48.622 1.16232 1.16191 3.6E-02 1.15366 1.15443 -6.7E-02 1.13547 1.13507 3.5E-02Bad Homb. 50.229 1.16287 1.16197 7.8E-02 1.15450 1.15449 9.7E-04 1.13600 1.13513 7.6E-02Moxa 50.645 1.16159 1.16198 -3.4E-02 1.15315 1.15451 -1.2E-01 1.13507 1.13515 -6.8E-03Medicina 44.522 1.16237 1.16176 5.3E-02 1.15334 1.15428 -8.1E-02 1.13480 1.13493 -1.1E-02Wettzel 49.146 1.16116 1.16193 -6.6E-02 1.15237 1.15445 -1.8E-01 1.13424 1.13509 -7.5E-02Vienna 48.248 1.16092 1.16189 -8.4E-02 1.15224 1.15442 -1.9E-01 1.13404 1.13506 -9.0E-02

Тable 4. The comparison of the calculated delta-factors with the data of the experiment (IASP91)