Influence of viscosity on the phase of earth tides

Physics of the Earth and Planetary Interiors, 9 (1974) 141 - 146 © North-Holland Publishing Company, Amsterdam - Printed in The Netherlands

INFLUENCE

OF VISCOSITY

ON THEPHASE

OF EARTH TIDES

BERTALAN BODRI

Department of Geophysics, R. E6tvOs University, Budapest (Hungary) Revised version accepted for publication June 25, 1974 The effect of viscosity on the phase of earth tides is computed. Computations are based on the general theory of the effect of viscosity of the mantle considered as a Maxwell body as it has been elaborated by Molodensky (1963). The problem is solved by the method of variation of constants, starting from the solution for the case of an ideally elastic, spherically symmetrical globe. The computations presented here cover the cases of the earth models B 1 and B2 of Bullen and Haddon (1967). Love numbers k,/, h were computed for tides of second and third order, and values of phase delay due to viscosity were computed for tides of second order.

Molodensky (1963) has shown that the task o f investigation o f the connection between the phase of earth tides and viscosity can be solved by means o f the m e t h o d o f variation o f constants, if the solution of the problem is known for the case of a spherically syrmnetrical, ideally elastic globe. Differential equations for the elastic equilibrium state can be reduced to a set of six linear differential equations. According to Molodensky and Kramer (1961) these equations are:

M - r2II(T' + H - 2T/r) = 0 N - (X + 2 # ) H ' - X [2H/r - n(n + I ) T] = O r2 L - r2(R ' - 41rGpH)= 0 3fl + Nr2 + pr2(R + V'H) - 21a [H'r 2 - Hr + (n 2 + n - 1)T] = 0 (1) L' - n(n + 1)(R - 4rrGp T) = 0 IV'

n(n + 1)M+O__ [L - 4 V'rH + n(n + I ) TV'] r4 r2 +-7-t.21aI2H'-2H/r+n(n+r2 1)T] = 0

where H, T, R, L, M and N are the functions sought for; we will represent them in what follows by ~i; r is radius, reduced to earth's radius at unity; p is density; ?, and ~ are Lam6 parameters; n is degree o f

term of harmonic development of tidal potential; V' = - g , where g denotes gravity acceleration; G is the gravitational constant. Equation system (1) can be rewritten in the following form: 6 i=1

(Ais + IJBis)Cb~+ (Fis + I.tHis)Cbi = 0

(2) ( s = 1 , 2 .... , 6 )

The equations o f system (2) are idependent one from another, thus the solutions can be considered as vectors in a 6-dimensional Euclidean space and we know that every solution o f system (2) can be represented as the linear combination o f a basic system o f solutions. Thus: 6

¢bs = ,~1"=CiYis

(s = 1 , 2 ..... 6)

(3)

where Ci denote arbitrary constants. Let us consider the shear modulus and the functions sought for appearing in the differential equations ( 1 ) t o be complex quantities, i.e.:

ff = l~ + jm ~i = ~i +J~i

(4)

where j = x / - ~ . The bulk modulus ~ + 2~/3 should be assumed to be a real number. This means that energy dissipation in compression is much less than in shear.

142

B. BODRI

The imaginary part of the shear modulus is determined by the rheological properties of the earth. If the earth is regarded as a Maxwell body then the connection between shear stress p and shear strain e can be given by the following expression: de dp p 2U ~/- = ~ - +--r

the determination of unknowns D i the following linear system: 6 i=1

I + L7"P ---I~+jm 1 + (rv) -2

(6)

where m = rvla/[1 + (rv) 2] and r = r///a. In the expression above rl is viscosity. Substituting the expression (4) into the system (2) we get a new system of equations which also include viscosity: 6 i=1

(Zis + IdBis) ~Iti+ (Fis + I'tHis) ~i = 6

- m ~ Bisd~ + aisCbi

(7a)

i=1 6 i=1

(his + llBis)dP~ + (Fis + llnis)dPi =

4 1 bx=~[~-N-3~--~r

(7b)

D i=m

i=1

For the case m = 0 the two equation systems above are identical. Supposing that the value of m is small, we can neglect - as a first approximation - the right-hand side of equation system (7b) and so we may obtain the solution for the case of an ideally elastic globe. Obtaining in that manner the functions q5i the expression on the right side of equation system (7a) becomes determined. The solution of the equation system (7a) is sought for in the following form: 6

xI,s= ~= i l

6

Ci~i+ i~l'= Diebi

( s = l ..... 6)

where the first sum represents the general solution of equation system (1) and D i are functions o f r . Substituting the expression (8) into (7a) we get for

(8)

(9)

-

n(n+l) T)]0 2r 2 (10)

The subscript 0 means the ~i-functions contained there satisfy for m = 0 the given boundary conditions. It is obvious that the constants C i and functions D i do not figure in the equation system (9), because ~i itself belongs to the basic system of the solutions of equation (2). For this investigation we have supposed that on the surface of the earthYis represents a unit orthonormal basis and the values ofYis(r ) have been obtained by numerical integration of system (1). After an inverse transformation of equation (9) we get the following expression: ,

( s = l , 2 .... , 6 )

X + 2p/3[ 1 H

b 2 = (M/ti)O b3 = b4 = 0 b 5 = ~r2bl + (n 2 + n - 2)T 0 b6 = -3bl/r

6

m~Bis~+His~i

(s = 1 ..... 6)

where Yis are the coordinates of the basis-vectors of the equation system (2). In the expression (9)fs = - m b s , where:

(5)

where r denotes relaxation time. In the case of harmonic oscillation of frequency v the equation above gives Hooke's law with complex value of the shear modulus ~: -

Di'Yis =fs

s=lA

bs

(i =1 .... 6)

(11)

where A = det [Yis[ and Eis denotes the minor corresponding to the elementYis of matrix with determinant A. Of course, A4~ 0, since - as mentioned above the equations of system ( I ) are independent from each other. The values of all minors figuring in expression (11) are given only by the structure of the earth's mantle, while the functions b s also depend on the boundary conditions. For the calculation of functions D i we have to know the function m(r). The radial variation o f m has been taken into account in computation by Zharkov's (1960) next approximative formula: -

m = m I 10 a(r- 1)

(12)

where ot = some parameter, and m 1 = Oa/rV)r = 1 is the value of m on the surface of the Earth. We can compute D~/m 1 without knowing the fac-

INFLUENCE OF VISCOSITY ON THE PHASE OF EARTH TIDES

143

tual value o f m 1 and carrying o u t the integration we get the values o f D i / m 1. The six integration constants Ci can be obtained f r o m the b o u n d a r y c o n d i t i o n s o f Molodensky and Kramer (1961) relating to expressions (1) if we substitute into t h e m the c o m p l e x values o f

TABLE I The computed values of the second-and third-order Love numbers in the case of earth models B1 and B2

/~ and ffi" The b o u n d a r y conditions are as follows: M=0 N=0

L + (n+l)rR = 0

h

l

k

B1

2 3

0.6341 0.2993

0.0886 0.0152

0.3219 0.0972

Bz

2 3

0.6335 0.2983

0.0882 0.0150

0.3228 0.0970

for r = 1 (13)

M=O

N + p(HV' + R) = 0 L - (~,r 2 + nr)R + 4nGpr2H = 0

for r = r C

where r c is the value o f the reduced radius at the c o r e mantle b o u n d a r y and the p a r a m e t e r 7 is d e t e r m i n e d by Poisson's equation:

,~, + ,),2 + 2n + 2 - r

n

7 +

4nGp' v'

= h + (CH + DH) / 7-= l + (C T + DT) j k = k + (CR + DR) ]

(15) forr=

1

and so the phase shift o f tide caused by viscosity:

CR + D R (tg~P)R-

1 +k

(16) - 0

C,~ + Dz-- ~ (CR +DR)

(14)

(tg~0)s =

We can solve system (13) w i t h o u t giving the factual value o f m l , thus obtaining the values o f Ci/m 1. Then we can c o m p u t e the c o m p l e x values o f the Love numbers:

l +h-~k The values o f Love numbers h, l, k have been comp u t e d according to the algorithm given above for the case o f tides o f second and third degree. In the corn-

TABLEIIA Generalsolutionofthehomogeneousequationsystem (1):Bl, n = 2 r

Ho

Mo

To

Ro

1.0000 0.9948 0.9714 0.9686 0.9372 0.9058 0.8744 0.8456 0.8117 0.7803 0.7489 0.7175 0.6861 0.6547 0.6233 0.5919 0.5770 0.5483

0.6341 0.6341 0.6342 0.6343 0.6401 0.6450 0.6493 0.6527 0.6562 0.6591 0.6619 0.6647 0.6673 0.6698 0.6717 0.6728 0.6728 0.6713

0.0000 0.0023 0.0157 0.0172 0.0343 0.0506 0.0650 0.0755 0.0842 0.0887 0.0898 0.0877 0.0825 0.0741 0.0627 0.0480 0.0398 0.0216

0.0886 0.0902 0.0941 0.0951 0.1049 0.1128 0.1196 0.1246 0.1291 0.1321 0.1341 0.1353 0.1361 0.1367 0.1372 0.1381 0.1387 0.1404

1.3219 1.3062 1.2564 1.2505 1.1850 1.1207 1.0583 1.0033 0.9425 0.8902 0.8420 0.7983 0.7594 0.7260 0.6988 0.6786 0.6718 0.6636

No 0.0000 0.0067 0.0371 0.0406 0.0774 0.1066 0.1251 0.1328 0.1307 0.1177 0.0934 0.0580 0.0124 -0.0412 -0.0976 -0.1473 -0.1638 -0.1684

Lo 1.0343 1.0135 0.8568 0.8386 0.6469 0.4743 0.3195 0.1917 0.0577 -0.0525 -0.1506 -0.2378 -0.3152 -0.3841 -0.4456 -0.5006 -0.5253 -0.5682

144

B. B O D R I

TABLE liB General solution of the homogeneous equation system (1): B2, n = 2 r

H0

M0

TO

R0

1.0000 0.9948 0.9686 0.9551 0.9372 0.9058 0.8744 0.8456 0.8117 0.7803 0.7489 0.7175 0.6961 0.6547 0.6233 0.5919 0.5770 0.5483

0.6335 0.6339 0.6343 0.6369 0.6401 0.6450 0.6493 0.6527 0.6562 0.6591 0.6619 0.6646 0.6671 0.6693 0.6711 0.6718 0.6716 0.6696

0.0000 0.0023 0.0173 0.0245 0.0339 0.0499 0.0643 0.0749 0.0836 0.0880 0.0891 0.0869 0.0815 0.0730 0.0613 0.0462 0.0378 0.0192

0.0882 0.0906 0.0951 0.0996 0.1048 0.1128 0.1196 0.1247 0.1293 0.1323 0.1344 0.1358 0.1366 0.1372 0.1379 0.1389 0.1395 0.1414

1.3228 1.3062 1.2505 1.2225 1.1858 1.1220 1.0597 1.0046 0.9435 0.8910 0.8426 0.7986 0.7594 0.7257 0.6980 0.6762 0.6686 0.6597

putations, earth models B 1 and B 2 o f Bullen and Haddon (1967) were applied. Numerical integration of system (1) was carried out using the Runge-Kutta method. Table I contains the values o f Love numbers for second- and third-order tides for an ideally elastic earth. Comparing our results with those obtained by Farrell (1972) for earth model A of Gutenberg-Bullen we see:that in that case the difference o f Love numbers is about 0.02. It is rather difficult to decide whether the differences above originate from the differences in earth models used by the authors or

NO

L0

0.0000 0.0067 0.0405 0.0566 0.0766 0.1056 0.1234 0.1324 0.! 308 0.1183 0.0949 0.0605 0.0163 -0.0351 -0.0882 -0.1324 -0.1454 -0.1415

1.0316 1.0135 0.8386 0.7535 0.6461 0.4727 0.3177 0.1899 0.0560 -0.0539 -0.1516 -0.2382 -0.3151 -0.3832 -0.4438 -0.4967 -0.5198 -0.5613

whether they are due to the errors o f the applied method. We note that the values o f Love numbers given in Table I are in good agreement with those obtained from observations. On the basis o f a 5,000 day long series o f earth tide observations Melchior (1972) gives the following values for k and h: k = 0.316-+ 0.010 h = 0.637 -+ 0.016

(17)

Tables IIA and liB show the general solution of the homogeneous equation system (1) in the case o f

TABLE IIIA Values of imaginary parts of complex Love numbers and tide phase lag: B1, n = 2 h/m 1

-8 -4 -2 0 2 4 8

l/m 1

372.45 32.184 6.5316 0.2370 1.5706 0.1430 1.4741 0.1036 0.7615 0.0477 0.5739 0.0296 0.4471 0.0198

6 = 1 +h - 3k/2

3,=l+k-h

k/m I

6/m 1

"r/m x

183.18 97.681 -189.27 3.1426 1 . 8 1 7 7 -3.3890 0.6494 0.5965 -0.9212 0.6166 0.5494 -0.8575 0.4006 0.1606 -0.3609 0.3246 0.0870 -0.2493 0.2451 0.0794 -0.2020

tg~oh

tg~o/

tg~R

,tg~°6

m I

l'H 1

ml

ml

587.93 10.310 2.4792 2.3269 1.2020 0.9059 0.7057

363.21 1 3 8 . 5 7 84.847 -275.18 2.6749 2.3773 1.5789 -4.9273 1.6143 0.4913 0.5181 -1.3393 1.1693 0.4664 0.4770 -1.2467 0.5384 0.3030 0.1395 -0.5247 0.3341 0.2456 0.0756 -0.3625 0.2235 0.1854 0.0690 -0.2937

ml

INFLUENCE OF VISCOSITY ON THE PHASE OF EARTH TIDES

145

TABLE IIIB Values of imaginary parts of complex Love numbers and tide phase lag: B2, n = 2

a

him 1

-8 -4 -2 0 2 4 8

l/m I

373.50 6.8859 1.6360 1.4978 0.7789 0.5891 0.4557

~/m I

32.110 183.98 0.2479 3.2349 0.1508 0.6836 0.1066 0.6349 0.0487 0.4149 0.0302 0.3355 0.0199 0.2528

I

4.0-

k/m 1

I

I

I

I

"rim 1

97.536 - 1 8 9 . 5 2 2.0336 -3.6510 0.6106 -0.9524 0.5455 -0.8629 0.1566 -0.3640 0.0859 -0.2536 0.0765 -0.2029

f

I

I

tg~h ml

I

I 75.4 I

.~(-

,g "~ 2 . 0 -

I _ b

l.O-

I -8

i -6

l

-4

I

I

I

I

I

I

-2

0

2

4

6

8

Parameter,

tg~R ml

589.06 364.05 139.08 10.859 2.8106 2.4455 2.5800 1.7098 0.5168 2.3621 1.2086 0.4800 1.2283 0.5522 0.3137 0.9291 0.3424 0.2536 0.7187 0.2256 0.1911

T 6s.9

t,

tg~l ml

e~

Fig. 1. The dependence of the earth tide phase lag on the parameter a for ~oR and ~o~.For the value c~= -8~0 R = 75.4 ° and ~o8 = 66.9 °.

tg~p8 ml

tg¢,r ml

84.860 - 2 7 4 . 9 5 1.7694 -5.2967 0.5313 -1.3817 0.4746 -1.2518 0.1362 -0.5381 0.0747 -0.3679 0.0666 -0.2943

second-order tide as well as models B1 and B 2. Tables IIIA and IIIB contain the values of the imaginary part of the complex Love numbers and the values of tangents of phase lag divided by constant m 1 at different values of parameter a in the case of earth models mentioned above. If, according to Molodensky (1963) we take m 1 = 0.0276 which corresponds to a relaxation time of order of a day or more, it becomes possible to obtain the factual values of phase lag. In Fig. 1 the values of phase lag f o r tpR and ~6 are shown versus parameter a in the case of earth model B 1. For well known reasons it is rather difficult to obtain the correct value of phase lag from earth tide observations. As the 01 wave corresponds practically to the static theory, a phase lag of 0.5 ° or 1° seems to be probable. Finally we note that the presented task of compu~ tation of the effect of viscosity expressed by formula (4) on the phase of earth tides does not differ significantly from the corresponding problem of theory of elasticity. The effect of viscosity on the amplitude of earth tides and also the more precisely computed effect on the phase can be obtained in the following way. Taking the functions qsi from the first approximation, they can be used for the computation of the right-hand side of equation (7b). The new values of functions q~i obtained in this way can be used for the computation of the right-hand side of equation (7a). Solving the equation system (Ta) with its new righthand side we get the more precise values of functions ,I,i. In the calculation of the effect of viscosity on the phase of earth tides it would be interesting to take into account the effect of the viscosity of the core too. The viscosity of the earth's core probably would play a significant role for small relaxation times in the core.

146

References Bullen, K.E. and Haddon, R.A.W., 1967. Earth models based on compressibility theory. Phys. Earth Planet. Inter., 1:1-13 Farrell, W.E., 1972. Deformation of the earth by surface loads. Rev. Geophys. Space Phys., 10: 761-797. Melchior, P., 1972. Physique et Dynamique Plan~taires, 3. Vander, Louvain, 311 pp.

B. BODRI Molodensky, M.S., 1963. Vliyanie vyazkosti na fazu zemnih prilivov. Izv. Akad. Nauk S.S.S.R., Ser. Geophiz., 10: 1469-1481 (in Russian). Molodensky, M.S. and Kramer, M.V., 1961. Zemnie prilivi i nutaciya Zemli. lzd. Akad. Nauk, Moscow, 40 pp. (in Russian). Zharkov, V.N., 1960. Vyazkost nedr Zemli. Tr. Inst. Fiz. Zemli, 1 1 : 3 6 - 6 0 (in Russian).