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Tidal variations of the inner mantle potential coefficientsJ. M. Ferrandiz a; Yu. V. Barkin b; J. Getino ca Alicante University, Spainb Sternberg Astronomical Institute, Moscow, Russiac Valladolid University, Spain

Online Publication Date: 01 January 2001To cite this Article: Ferrandiz, J. M., Barkin, Yu. V. and Getino, J. (2001) 'Tidal

variations of the inner mantle potential coefficients', Astronomical & Astrophysical Transactions, 19:6, 845 - 858To link to this article: DOI: 10.1080/10556790108244096URL: http://dx.doi.org/10.1080/10556790108244096

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TIDAL VARIATIONS OF THE INNER MANTLE POTENTIAL COEFFICIENTS

J. M. FERRANDIZ,’ Yu. V. BARKIN,2 and J. GETIN03

Alicante University, Spain Sternberg Astronomical Institute, Moscow, 11 9899, Russia

Valladolid University, Spain

(Received June 25, 1999)

Tidal variations of the inner gravitational field of the Earth’s mantle coefficients of arbitrary order due to elastic deformations of the mantle caused by lunar and solar attraction have been found.

KEY WORDS Inner mantle potential, tidal variations, elastic deformations

1 INTRODUCTION

The Earth’s mantle is a non-spherical, non-homogeneous cover with a quasi-concen- tric distribution of densities. Let &, be the mean radius of the Earth, and &, the radius of the bigger sphere which we can put in the mantle cavity (we assume that the centre of this sphere coincides with the Earth’s centre of mass).

We will consider the mantle as a deformable elastic body subject to the attrac- tion of external celestial bodies (the Moon and the Sun). The lunar and solar tidal deformations of the Earth will be described by the classical model (Takeuchi, 1950) which was studied in detail in the papers of Ferrandiz and Getino (1991-1994) for the construction of the rotation theory of the deformable Earth.

Let us introduce in to consideration the main Cartesian reference system C x y z with the origin at the Earth’s centre of mass and with axes directed along its principal axes of inertia in the undeformed state. Let t o and T be radius vectors of an arbitrary point (or a elementary volume dm) of the mantle in the absence of deformations and in the deformable state. As usual, we assume that particles of the deformable solid mantle deviate slightly from their positions which they occupy in the absence of deformation. The small displacement vector, u(r, t ) , is defined in the following way

845

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846 J. M. FERRANDIZ, Yu. V. BARKIN, AND J. GETINO

where (2, y, z) are the positional coordinates of the particle of the deformable body, and (20, yo, ZO) are those that the same particle would have in the absence of defor- mations, (u, v , w ) being the components of the displacement vector.

The components of the displacement vector during the deformation of the mantle under Newtonian attraction of the external bodies (the Moon, the Sun) are defined as (Takeuchi, 1950)

where W,I, is the harmonic of the n' order of the tidal potential

00

n'=l n'=l (3)

The functions Fnr(r) and Gnl(r) are defined by a set of ordinary differential equations that depend on the model of the radial distribution of density (Ferrandiz and Getino, 1993). In (3), G is the gravitational constant, r is the distance between the origin C and the point within the Earth where the potential is evaluated, m*, r* are the mass of and the distance to the perturbing body (the Moon, the Sun), and S is the angle between the vectors r and r*. P,,I is Legendre's functions.

The main term in the full development of the tidal potential (3) is the second harmonic, since the factor r/r* is of the order of 1/60 for the Moon and 1/23000 for the Sun.

Now we introduce spherical coordinates r , 8 , cp for the elementary mass dm and r*, 6*, a* for the perturbing body (here B and 6* are colatitude and latitude and cp, a* are the longitudes). The expression of the W,I function can be presented in following form (Takeuchi, 1950):

n'

(4) Wnl = C qntm'r n' Pntmr(cos8)(A:rml cosm'cp + B;,,I sinm'cp, m'=O

where

Here 60,t is a Kronecker symbol and 6,' = &,I + 1. For the coefficients A:,,,, B:,,, we have the known formula:

defined as functions of time in the form of Fourier series (Kinoshita, 1977; Getino and Ferrandiz, 1991) with respect to the classical arguments of the Moon's orbital theory.

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TIDAL VARIATIONS OF THE J"ER MANTLE 847

In accordance with the Takeuchi solution the spherical components of the dis- placement vector are defined by

We will assume that linear displacements of the particles of the mantle are small and the small variation of density p is described by the formula of Takeuchi

Po& dPo p = po - ur- - dr (9)

where po is the density corresponding to the undeformable state, Ur = (xu + yv + aw)/r being the radial displacement, and A = div u is the volume divergence. Now we consider the gravitational force function U = -V (V is the gravitational potential) of the non-spherical and deformable mantle. The equations of this force function for some external P and for internal P points are defined by (Douboshin, 1975)

where R, 9, A and R, 4, k are spherical coordinates of the points P and P in the coordinate system Cxya (the angles 9 and 4 are the colatitudes). Pik' are associated Legendre functions. Here rn and riz is the mass of the Earth and of its mantle. & and & are the mean equatorial radius of the external and internal surfaces of the mantle.

Equations (10) and (11) converge, consequently, in the domains R > & and R < & (Douboshin, 1975).

The coefficients Cnk, ~ , , k and C n k , Snk in (10),(11) are non-dimensional and are defined by the following volume integrals:

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848 J. M. FERRANDIZ, Yu. V. BARKIN, AND J. GETINO

and

Here T , 6 , cp are spherical coordinates of elementary mass pdu in the deformable state. In (12), (13) we use special sine 6, = 2 for rn = 0 and 6, = 1 for m > 0. Thus parameter is connected with the Kronecker symbol 60, by the formula

6, = 60, + 1

We can note that the expressions (12) and (13) first case the following value of the density

are practically identical, if in the

is implied. We will use this remark in the construction of the variations of the coefficients (13) due to tidal deformations. Here we follow Ferrandiz and Getino (1993), in which the variations of the coefficients (12) for arbitrary n were obtained.

The integrals (12), (13) are spread over the whole volume of the deformed mantle, so it is convenient to transform the domain of integration to the undeformable body. Taking the relations (1) into account, the Jacobian of this transformation, neglecting the quadratic terms, will have the form (Getino and Ferrandiz, 1990) J M 1 + A.

For a rigid non-spherical mantle (with quasi-spherical distribution of density) the integrals (12), (13) provide a constant value. And in case of the deformable body they will be defined as functions of time as the position vector T (1)-(3) will be some function of time due to the tidal gravitational influence from the Moon and Sun on the mantle as an elastic body. In our approach it is sufficient to assume that the particles of the deformable solid deviate slightly from the position they would occupy in the absence of the deformation. We can neglect the products of the deformations terms (theory of linear elasticity).

The full description of the tidal potential, the definition of the components of the displacement vector, and of linear transformations for the volume integral and other questions are given in detail by Getino and Ferrandiz (1994).

2 TIDAL VARIATION OF C'nm

In this section let us undertake the general expression for the tidal variations of the coefficients en, (m # 0) which will be applied later to get the expressions of Snm.

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TIDAL VANATIONS OF THE INNER MANTLE 849

According with (13) the variation of Cnm is given by the integral

where we have introduced the notation

knm = T-n-1 prim (cos e) cos mv. (15)

Here Pnm = Pi"' are associated Legendre functions.

approximation) will be defined by the formula The tidal variation of the coefficient (or integral (13)) in our linear theory (or

din, = 81 + 8 2 , (16)

where

where u, is the radial displacement (7), and the integral is extended to the 'unper- turbed' volume (in our case on some mantle cover with external and internal radii & and and having a concentric distribution of density).

In reality we have

Inm = 111, + 6inm

QO

Po& 6po = -ur- - dPo dr

au av aw A = - + - + - ax ay a2

and for the first and third terms in the second integral (17) we will have

and for the variation in, we have a representation in the form (15)-(18).

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850 J. M. FERRANDIZ, Yu. V. BARKIN, AND J. GETINO

3 ANALYTICAL EXPRESSION FOR 2,

First let us calculate the variation 6Rnm. We have 00

(20) JRnm = C grad R n m * unl , ‘tLnt = (unl, vnl , wn’ nr

or, in coordinate form,

where the components of the displacement vector in accordance with Takeuchi (1950) can be represented as:

1 u,,, = -[,I Wnl,

T

where

and Wnl is a nth harmonic of the tidal potential (3), (4).

derivatives of the function (18):

lnl = n‘Fnl + T2Gnl (23)

Now we apply some algebra to the expression dKnm. First we obtain the next

Now from (21) we find

1 - -mT-n-3 sin2 e

where (4) nr

W,,I = ~n~m~rn ’Pn~mr(cos~) (A~ l , l cosm’p + Bilml sinm’cp), (26) m’=O

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TIDAL VARIATIONS OF THE INNER MANTLE

For the coefficients A:,,, , BZrmr we have the following known formula:

851

(27)

For the derivatives aWnl/ae and aWnl/acp we have:

n'

(29) a Wnf - = m'qntmtrn'Pn~,~(-A~r,r sinm'cp + B:,ml cosm'cp). '9 m'=O

Now let us substitute formulae (26)-(29) into (25) and reduce the expression bi?,,,:

n'=l m'=O

x [A:,,, cos m'cp + BZIml sin m'cp]} apnl ,,

ae [ A:' ,, cos m'cp + B:, ml sin m'cp] d p n m + T-" -~ F,, - ae - ,.-n--3 F,,, - ~ ~ ~ ~ c o s ~ ~ m s i n m c p ~ m ~ q , ~ , ~ r ~ ~ ~ ~ ~ , ~ ~ ~ ~ ~ ~ ~ 1

sin2 8 x [-A:,,, sin m'cp + Bit,,,, cosm'cp]}

Finally we have:

mm' sin rncp sin m'cp aP aPn, PnmPn,ml + Fnl (T- ae cos rncp cos m'cp + sin2 6

-(n + 1 ) l n ~ P n m P n ~ m ~ cosrncpsinm'cp

mm' sin rncp cos m'cp . (30)

11 I aP,, OPnlml Pnm Pn, rnl cos rncp sin m'cp - + Fn, (r- ae sin2

Let us calculate the volume integral

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852 J. M. FERRANDIZ, Yu. V. BARKIN, AND J. GETINO

Integrating with respect to the variable cp can reduce the integral (31). For this we take into account the trigonometric integrals:

cos mcp cos m'cp dq = A6mml6m, 0

sin mcp sin m'cp dcp = nsmml, P 0

P 0 cos mcp sin m'cp dcp = 0,

where 1 form = m' 2 for m = 0 0 form # m' d m = { 1 form > 0

d,,I =

As a result for integral (31) we obtain the more compact representation:

ap,,apnJm m2 PnmPn~m)] sin B dr do. -(n + l)ln~PnmPntm + F,I -- + - 30 30 sm2e (33)

As a result of integration with respect to 9 we have the following integrals (Getino, 1992):

2 ( n + m ) ! P,, Pntm sin B dB = (2n + 1) (n - m)! Jnnl 1 i 0

On the basis of these formulae we find:

In accordance with (23) 1, = nFn + r2Gn

and we obtain -1, + nF, = -r2G,,

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TIDAL VARIATIONS OF THE INNER MANTLE 853

and as a result (using formula (27)) we obtain

It is interesting to note that in the case of an internal mantle potential, the coefficient function 21, does not depend on the function Fn compared to the case of external geopotential coefficients.

Let us consider the expression for 22:

ffo

where (see (15), (22), (26))

Rnm = r-n-lpnm(cose) cOsrncp,

n'

Wnl = Qn:m:Tn'Pn:m:(COS8)(Ai:,, cosm'q + sinrn'cp). (41) m'=O

Substituting (41) into (40) we have

x cos mcp(A~lml cos m'cp + B:,,: sin m'cp) (42)

Taking into account relations (32) we obtain:

and using formula (34) we find

where

or 1, = nFn + r2Gn

(44)

(45)

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854 J. M. FERRANDIZ, Yu. V. BARKIN, AND J. GETINO

Returning to our initial formula (13), (14) for variations of the coefficients en, we have:

- 2(n - m)!@+l (& + z,) sc,, = mrJ,(n + m)!

2(n - m)!&+’ [ Gm* (5) - - ma, (n + m)! 0*”+1

Formula (47) defines the tidal variations of the coefficients en,, of the inner grav- itational potential of the mantle. A similar formula holds for the tidal variations of the other coefficients Snm. The final results can be presented in the form of the following formulae for the variations of the coefficients of the inner gravitational mantle potential:

where n+ 1

47rG

Here we also present similar formulae which define the tidal variations of the coeffi- cients C,,, S,, of the external gravitational potential of the mantle (12) (Getino and Ferrandiz, 1994):

P,,(sin a*) cos ma* , 1 as,, = 2(n - m)! Dnt [($>”+I P,, (sin 6’) sin ma*

Dnt [ ($) 2(n - m)! am(n + m)!

sc,, =

am(n + m)! where

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TIDAL VARIATIONS OF THE INNER MANTLE a55

In = J (nr2"po[(2n+ 1)Fn + r2Gn] - +rzn+'(nFn + rzGn dr. (51) r 1)

From formulae (48), (49) and (SO), (51) we find the simple relations:

where

For n = 2 the numerical value 1 2 , was obtained as a result of integration over the mantle volume on the basis of the Takeuchi model 2 (Getino and Ferrandiz, 1991):

I2 = 1.917290 x 10'' C.g.S. (53)

with the corresponding numerical values of the parameter

{ 3.185508 x lo3' (the Sun) (54) 6.953379 x (the Moon)

Dzt =

We can produce analogous transformations for the variations of the inner potential coefficients (18). As a result, we have:

where

& = 1 [?(5F2(r) + r2G2(r)) - (2 + F) (2Fz(r) + r'Gz(r))] dr. (56) r

Integral (56) as well as integral (51) is spread over the elastic mantle and depends on its internal structure. Directly from formulae (48),(50),(52) we obtain

d j 2 = k26J2, 6C22 = k26C22, 8322 = k26S22,

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856 J. M. FERRANDIZ, Yu. V. BARKIN, AND J. GETINO

SC21 = k26C21, SS21 = k26S21,

where the coefficient of proportionality is defined as (57)

Using the variations of coefficients (SO), (S l ) , which were studied earlier (Ferrai- idiz and Getino, 1993), and the numerical value of the k-parameter (54), (58), we can calculate the values of the variations of the other coefficients (57). Variations of the coefficients (53) were obtained on the basis of known trigonometric rela- tions of the functions of time which appear in formulae (50) for the corresponding perturbing bodies (Kinoshita, 1977; Getino and Ferrandiz, 1991):

i

where 1 1 1 6 Bi = - - ( ~ c o s ~ E - l)Aio' - - 2 sin2.4:') - - 4 sin2

Ci(7) = --sin2&Ai0) 1 + '(1 1 +7cosE)(-l+ ~TCOSE)A( (1) 4 2

7- + - sinE(1+ . r c o s & ) ~ j ~ ) , 4

1 1 2 4 Di(r) = -- sin2 E A ~ ) + r sin&(1+ r cos~)A!') - -(1 + 7 c o s ~ ) ~ A { ~ ) ,

7 = fl. (60)

Numerical values of the coefficients A?) were obtained by Kinoshita (1977). In the construction of these relations, it was supposed that the angle o between the vector of the angular moment of the Earth G and polar axis Cz is small ( so sino = 0, cos o M 1). E is the angle between ecliptic plane and the intermediate plane, which is orthogonal to the vector G ( E = 23.45').

The argument 0, is a linear combination with numerical coefficients of the ar- guments of the Moon's orbital theory:

Oi = mllM + m2ls + m3F + m4D + m5C

F = + g M ,

D + g M -t h M -1s -9s - h s , a = (mlim2,m3,m4,m5)-

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TIDAL VARIATIONS OF THE INNER MANTLE a57

Here L M , g M l h M and Isl gs, hs are the Delaunay variables for the Moon and the Sun. p + Y is the angle of the Earth’s rotation.

Variations of the coefficients (50) were presented in the following form (Ferrandiz and Getino, 1993):

8 6 2 2 = k2 C K22a(Z) COS(2p + 2V - Oi) + k2 C K 2 2 b ( i ) COS(2p + 2V + Oi), i i

8 3 2 2 = -k2 K22a( i ) h ( 2 p 2v - 0;) - k2

8621 = k2

K22b(i) Sin(2p + 2v 4- Oi), i i

Kzio( i ) sin(p + V - Oi) + k2 c K21b(i) sin(p + v + Oi), i i

8521 = k2 C K21a(i) COS(/J + v - Oi) + k2 C Kzlb(a) C O S ( ~ + v + Oi). (62) i i

In Table 1 the main coefficients of the variations (34) are given; they are listed together with their respective arguments Oi (1 unit = These values were obtained for the value k2 = 0.10743 which was found from formulae (23)’ (28), (30) for the well-known Earth model 1066A (Gilbert and Dziewonski, 1975).

By an analogous method we can obtain variations of the coefficients of the third and higher harmonics of the mantle external and internal potentials due to its tidal deformations.

Acknowledgements

Yu. Barkin’s work has been partially supported by the Russian Foundation for Basic Research (project codes 96-05-65015 and 99-05-64899) and the Russian Uni- versities Programm (topic 9-5240). His stay at University of Alicante has been par- tially funded by grant INV99-01-06 of Generalitat Valenciana. J. Ferrandiz and J.

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858 J. M. FERRANDIZ, Yu. V. BARKIN, AND J. GETINO

1 0 0 - 2 0 1 0 0 0 0 0 0 0 2 0 1 0 2 01 0 0 2 01 0 0 0 0 1

-102 2 2 -102 0 2 1 0 2 0 2 0 0 2 2 2 0 0 2 0 2 0 1 0 0 0 0 1 2 - 2 2 0 0 2 - 2 2

0.2742 1.4337 0.2380 0.2154 1.1251

-1.1372 0.0988

-0.0767 0.5198 0.0829 2.7139 0.2005 0.0740 1.2598

-0.0071 -0.0372 -0.0062 0.0375 0.1959

-0.1981 -0.1912 0.1487

-1.0063 -0.1608 -5.2563 -0.0053 -0.1429 -2.4397

-0.0071 -0.0372 -0.0062 -0.0016 -0.0084 0.0084

-0.0004 0.0003

-0.0019 -0.0003 -0.0098 -0.0053 -0.0003 -0.0046

-0.0656 -0.3429 -0.0569 0.1574 0.8226

-0.8317 0.1586

0.8350 0.1334 4.3613

-0.0480 0.1188 2.0244

-0.1232

-0.0656 -0.3429 -0.0569 -0.0231 -0.1202 0.1216

-0.0070 0.0053

-0.0358 -0.0058 -0.1876 - 0.0480 0.0012 0.0216

Getino work has been partially supported by Spanish Projects ESP97-1816-C0402 (CICYT) and PB95-696 (DGES).

References

Douboshin, G. N. (1975) Celestial Mechanics. General Problems and Methods, Nauka, Moscow. Getino, J. and Ferrandia, J. M. (1990) Celest. Mech. 49, 303. Getino, J. and Ferrandiz, J. M. (1991) Celest. Mech. 61, 35. Getino, J. (1992) Celest. Mech. 63, 11. Getino, J. and Ferrandiz, J. M. (1994) In: Kuvzynska, Berlier, Seidelmann and Wytrzyszczak

(eds.), Dynamics and Astrometry of Natural and Artificial Celestial Bodies, p. 341, Astronom- ical Observatory of A. Michiewic University, Poznan.

Gilbert, F. and Dziewonski, A. M. (1975) Phil. %ns. R. SOC. 287A, 187. Ferrandiz, J. M. and Getino, J. (1993) Dynam. Astron. 67, 279. Ferrandiz, J. M., Getino, J. , and Caballero, R. (1993) Adu. Astron. Sci. 84, Part 11, 915. Kinoshita, H. (1977) Celest. Mech. 16, 277. Takeuchi, H. (1950) %ns. Am. Geophys. Union, 31, 651.