Damping of free nutation and relaxation time of the earth

ICARUS 6, 292--297 (1967)

Damping of Free Nutation and Relaxation Time of the Earth H. G E R S T E N K O R N .~ool Isernhagen NB, Gri~nberger Steg l, Western Germany

Communicated by Zden~k Kopal Received August 8, 1966 The dissipation of energy in the mantle of the Earth is held to be the main cause of the damping of the free nutation. The mantle is treated as a "Maxwell body." Under these assumptions the period and the time constant of the damping are calculated in analogy to a former calculation of Jeffreys, who restricted himself to the case of an ideal elastic body. Comparison with the observed period and damping gives Love parameter z ~ 0.3 and relaxation time r ~ 108 sec. It turns out, however, that a layer with r values of 10~ sec below a large zone with r ~ 108 sec would give about the same dissipation: therefore one can't find such a layer from the damping data. Nevertheless, a sufficiently large zone in the mantle with r > 108 sec seems to be necessary in order to excite free nutation. On the other hand, zones of many hundred kilometers in size with relaxation times of r ~ 106 sec may hardly exist in the mantle, otherwise the damping should be more effective. A c a l c u l a t i o n m a d e b y Jeffreys (1952) to d e t e r m i n e t h e p a r a m e t e r s of t h e free n u t a t i o n is generalized. Jeffreys was d e a l i n g o n l y w i t h t h e case of ideal e l a s t i c i t y of t h e E a r t h ' s m a t t e r . B e y o n d t h a t , t h e E a r t h ' s m a n t l e is t r e a t e d as a " M a x w e l l b o d y " in this paper. D e f o r m a t i o n s of such a b o d y a r e c o n n e c t e d w i t h a loss of energy. Owing t o it, a n u t a t i o n c r e a t e d b y a s i n g u l a r p e r t u r b a t i o n dies a w a y a f t e r s o m e time. T h e s t r e n g t h of this d a m p ing d e p e n d s on t h e ~alue of t h e so-called " r e l a x a t i o n t i m e " r of t h e M a x ~ e l l b o d y , c o r r e l a t e d w i t h its r i g i d i t y # for processes w i t h d a m p i n g ~ exp (F/) a c c o r d i n g to

(,)

l,

w h e r e F is a c o m p l e x c o n s t a n t ; i' = ---,/ + i~o. C o m p a r e G e r s t e n k o r n (1967), Eq. (2a). I n t h e ease of t h e M a x w e l l b o d y , t h e c o m p l e x q u a n t i t y ¢ replaces t h e r i g i d i t y ~. F u r t h e r , t h e m a t t e r of t h e m a n t l e is cons i d e r e d as h o m o g e n e o u s a n d i n c o m p r e s s i b l e a g a i n s t s m a l l d e f o r m a t i o n s . U n d e r these s u p p o s i t i o n s we c a l c u l a t e first the p e r i o d a n d t h e d a m p i n g t i m e of the free n u t a t i o n ,

d e p e n d i n g on t h e m e c h a n i c a l d a t e s of t h e Earth's body. L e t t h e figure axis of t h e E a r t h be r e p r e s e n t e d b y t h e unit ~ e c t o r le0, a n d let t h e direction of it be t h e z d i r e c t i o n of a C a r t e s i a n s y s t e m , w i t h t h e e q u a t o r i a l x, y plane fixed w i t h i n t h e E a r t h (Fig. l a ) . T h e m o m e n t s of i n e r t i a will be A = B < C with t h e v a l u e of ( C - A ) / C = 1/305. T h e p r o d u c t s of i n e r t i a are defined b y F = fyzdm,

G = fxzdm,

H = fxydm.

T h e y m a y be different f r o m zero o n l y due to t h e n u t a t i o n . A t a n y time, t h e m o v e m e n t m a y be considered as a rigid r o t a t i o n a r o u n d t h e m o m e l ) t a r y axis W0, t h e a n g u l a r v e l o c i t y of it is given b y a v e c t o r W w i t h t h e comp o n e n t s p, q, u. T h e c o m p o n e n t u is n e a r l y t h e s a m e as t h e f r e q u e n c y of r o t a t i o n ¢o~ a f t e r n u t a t i o n has ceased. T h e a n g u l a r m o m e n t u m has the c o m p o n e n t s L, M , N . T h e n t h e following s y s t e m of e q u a t i o n s holds valid : L = Ap--

Hq--Gu

d L / d t = M u -- N q

292

(2)

293

DAMPING OF FREE NUTATION OF THE EARTH

The body of the E a r t h tries to form the figure of equilibrium corresponding to the m o m e n t a r y rotation around the axis W. In the F0 system, a corresponding perturbing potential appears. If the E a r t h would rotate around the axis F0, the potential would be VF = - ~ E 2 r 2 P 2 ( 0 ) ; if the Earth rotates around W, the potentialis V w = ½ao~E2r~P2(O'). The difference of both potentials gives the perturbing potential V, (a, density; r, distance of the considered point A from the E a r t h ' s center; r ~ R, radius of the Earth; 0, ~', angles, see Fig. 1). If we allow for the smallness of the angle ~ = < W , F0, the development of the Legendre polynomial P2 gives FIG. la. The orientation of the x , y, z system in the E a r t h : C, :Earth's center, W, m o m e n t a r y angular

velocity, F0, unit vector of the figure axis joining the geographical poles, A, any considered point at the surface or in the interior of the Earth.

P2(O') .~ P2(~) + c~ sin ~ cos ~[dP2/d(cos ~)],

and we get the perturbing potential V8 V , = i~o~ar~P~l(O) cos

with

A consideredpoint

P2 ~ = sin O[dP~/d(cos ~)].

(4)

Figure l b shows the time dependence of = ¢ -- ~t 4- ~; ¢o denotes the frequency of the nutation. B y putting a = s 0 e x p ( - - # ) and introducing the spherical harmonics Y., = P2 ~ cos ~, Y'2 = P2 ~ sin (I), V, becomes

x-direction

V~ = --½~o~2~0r2 Re [{ Y2 -

Y'2i} exp

(rt)]. (4a)

The abbreviation Re signifies the real part of the complex quantity; i is the imaginary unity. We have put F = --'y 4- iwt; "y and are real and positive. The potential (4a) produces a radial deformation of the surface [compare Gerstenkorn, 1967, Eq. (8a)] (s~). = - ~ 0 R ~

pole of figure a~is

X Re

19~ + 2~gR { Y2 - iY'~} exp (Ft)|-

Fro. lb. The azimuthal angles ~, ¢, and ~t. This is :Fig. la, seen from above, ~ is the frequency of the nutation. W0 circulates around F0. The instantaneous axis of angular momentum is not drawn in.

We put

and similar equations for M and N. If we neglect the products pq, p F , FG, etc., we get

then we have

(5)

p A -- u O + u~F + u q ( C -- A ) = 0 qA -- u F - -

u~G -

up(C--A)

= 0.

(3)

XRe

~{Y2--iY'~}exp(rt)

•

(ba)

294

H. GERSTENKORN

Next, we calculate the p r o d u c t s of inertia: We use the relations x = r sill 0 COS ~o, y ---rsinOsin¢, z = r c o s O and put r ~ R near the surface

A~i -

+ Bu2a0(C - A) ~

dft = sin OdOd4~ G = f a s ~ , R 4 sin 0 cos 0 cos Cdft. (6) Only, because of the radial deformation of the surface due to the deviation of the axis W of rotation, the quantities F a n d G are not zero. Owing to the dissipation of energy, there is a phase angle between the defornlation s~ a n d the potential V,. Therefore F and G also do not possess the same phase angle, a n d t h e y are also not in phase with the potential V~. This implies t h a t the a n g u l a r m o m e n t u m D is no longer situated in the W, le0 plane as in the case of a rigid b o d y or in the case of ideal elasticity. After some calculation, we get for F and G

1 ~oMR~(~Re

G = -

~-~exp

~ c~oMR20 Re

(Ft)

1

exp (rt)

r -

io~:Xo + ~xo(io~. + r )

i~xo

F

--

Bao(C-

A) R e [

G = --Ba0(C-A)

Re

i~exp(Ft)] ~exp(I't)

(65)

Between F a n d G, there is a phase shift of 90 degrees. In the ease of homogeneity, fl = 1 holds valid. F o r small angles ~ << 1, the following supposition seems to be reasonable: p = ~ 0 exp (--34) cos ¢¢t q = ¢OEa0exp (--34) sin ~t.

1 rl

1 +z--xo

(7)

T h a t is, the complex q u a n t i t y v = p + iq is given b y oJEc~0exp (Ft). T h e angle a between W and le0 decreases exponentially with time, while W moves a r o u n d le0. F r o m (3) a n d (7) there follows

exp (Ft).

(s)

A)/A,

and

- o.

1 +

(9)

z + x +Z--Xo

i~

--

7.

(9a) If we neglect x0 ~ 1/300 << z, we get for the frequency ~o of the n u t a t i o n and for the time constant q, of the d a m p i n g

• (6a)

= 4 t a R 3 ~ 3 is the mass of the E a r t h . F u r t h e r , if we use the relation M R 2 , ~ = 2(C - A), valid for a homogeneous earth, we get

2

M t e r (1) and (5), F and 2 are correlated according to F~ = Fz + ( z / r ) . W i t h B = 1, (9) gives

C--A

1

A

l+z

~o = ~o~

M

exp (rt)

With the n o t a t i o n Xo = ( C since u ~ ore, (8) becomes

1

v -

F=

i2

+ / 3 u a 0 ( C -- A) ~

F = f ( r s e R ~ sin O cos 0 sin ~(1~

F =

i u v ( C - - A)

z

7-1 + z "

(10)

T h e preceding formulas hold only for a homogeneous E a r t h . If we allow for the existence of a h e a v y core, we have to multiply (6a) with a factor ~ma.*l-./~...... If we take into a c c o u n t the increase of the density of the mantle towards the center, an additional factor 8 appears in (6a). This factor can only be estimated b y O'surfaee 1 + "0

1

with the p a r a m e t e r of inhomogeneity ,/ 0.57 for the E a r t h . F o r ~, we have to take a suitable mean value. Further, because of the changed relation MR2~

(6b) remains ~(1 + ½7)/(1 [3 value into equation for

= 2((7 -- .4) 1 + ½7

valid, if we use for fl the value -- -~n) ~ 1 + ½7. Inserting this (9), we get the characteristic F

295

DAMPING OF FREE NUTATION OF THE EARTH

( C - A)/C. The kinetic energy of state 1

r~wr

1Tz

1Tz 1

-- --iwEXo-

z

(8-

1).

(11)

r

(rotation with additional nutation) E1 = ½(D, W) is different from the kinetic energy E2 = ½ C ~ 2 of the state 2 (without nutation,

Generally, this equation has two solutions I'~ and F:. But only that one with a negative real part corresponds to a damped nutation. The other gives an exponentially increasing and must be rejected by physical arguments. As long as one of the two conditions 1 z ---<
1 z --->>wl

or

ritz

ritz

is fulfilled, (11) is solved approximately by F =

1

Z

+ i~l

rlTz

C

with

¢ol=wEXo(1--f~l-~).

(12)

This agrees with the formulas, given by Jeffreys (1952), if we put 3

z

FIG. 2a. The unit vectors Fo (figure axis), Do (direction of angular momentum), a n d Wo (rotation) lie within the same plane. This is the case of a rigid body or of a body with ideal elasticity.

the angular momentum being the same for both states). In a first approximation for small x and a, AE = E1 - - E 2 is given by

kJ~sl"~u'~ = ~ 2 1 T z

AE ~-. ½C~s2xa 2.

With a period of nutation of about 440 sidereal days (Allen, 1963) and values of A / ( C - A) ~ 304 and f~ ~ 1.285, a value of z/(1 + z) ~ 0.24 results, i.e., z ~ 0.318. Since ~ ~ 0.81, this agrees nearly exactly with Jeffreys coefficient k - - 0 . 2 9 . The corresponding damping time, with the amplitude a diminished to l/e, results

The free nutation of an ideal elastic body is distinguished from that of a rigid body by the fact that the angle

To = r(1 + z)/z.

(12a)

The order of magnitude of To is about 3-13 periods (cf. e.g., Bondi and Gold, 1955), i.e., To ~ 1 - 5 X 10s sec. This gives a value of r ~ 0.2-1.2 X 108 sec. Obviously, we have to deal with such high r values in wide ranges of the mantle, at least above the 700-kin level. Next, we compare our results with those of the free nutation of a rigid body (Fig. 2a). In this case, the vectors W and Fo form the angle ~, the vector D is situated in the same plane, and the angle ~ between W and D is much smaller than a: ~ ~ ax, x =

~=


(13)

ax(i-~l--~z)

becomes smaller. For given ~ = < W, F0, D comes nearer to W. Calculating the difference of energy, we must take into account also the elastic and gravitational energies of the deformation. B y this, the order of magnitude of this difference is not changed (factor ~ 1.1 for the Earth). In the case of a Maxwell body, however, there is a characteristic distinction in comparison with the two former cases: D leaves the plane formed by W and F0 (Fig. 2b). This is caused b y the existence of a phase shift of 90 ° between F and G. In the body-fixed Fo system, W circulates again around Fo once in a period. The dimensions of the spherical triangle Wo, F0, Do are shrinking exponentially with time exp (-- ~,t).

296

H.

GERSTENKORN

In Fig. 2b, D is drawn far out of the plane W0, F0: ~ >> a. Now, ~ is given b y = ~x(~ -

2 Re ~ +

[~1~)'/~

with

a = ~/(1 + ~). Figure 2b corresponds to the case x]51 > 1. This implies ~o~r < (1 + z)/z and w~r <<1. Then 5 has in this case a large imaginary part being proportional to 1/r; the deformation s, also contains a component increasing with 1/r, the phase of it being 90 ° against the potential V,. Thereby, relatively great

wg

C

FIG. 2b. For a Maxwell body, the angular momentum D is situated out of the W0, le0 plane. additional terms proportional to 151 appear in the expression for AE. We can interpret the shortness of the damping time To according to Eq. (10) in such a way t h a t the great AE ~ [sls is compensated for b y an even greater dissipation proportional to Is]2 as well as to sin ~ with ~ ~ 90 °. Further, we m a y r e m e m b e r t h a t all considerations hold valid only if the condition

axlSI << 1

(14)

is fulfilled. Otherwise we should have taken into account the higher powers of a in the development. The limiting case r - ~ 0 needs a special consideration, since then for a given a it would follow t h a t ,~r ~

O l / 7 " -----) ~ ) .

Actually, after the preceding results we should expect r values of about 10~ sec throughout the mantle. Smaller r values in particular should be excluded because of

their high dissipation of energy and the shortness of their damping times. But the inhomogeneity of the E a r t h gives rise to peculiarities. Unfortunately as yet, we could not overcome the difficulties of a treatment of the inhomogeneous ease with enormously variable r(r) within the mantle. But if we consider an inhomogeneous structure of the mantle, consisting of an upper zone with high rl values of 108 see and an inner zone with considerably smaller r2 values, we m a y expect, indeed, a rather high damping time corresponding to the order of magnitude of r~, since the 90 ° component of the deformation sr ought to be much smaller than in the case of a homogeneous r2 body. To get an estimation, we choose another standpoint with regard to the excitation of the nutation: When the nutation has been excited, and if it is maintained by some statistical process providing the needed energy, we can treat the nutation approximately as a periodic process. This statistical view seems to be allowed, if m a n y periods are included. Then, the complex constant £ in the differential equations of the Maxwell body is given by

1_1

(

1+/~

1)'

differing from (1). Further, the formulas derived in a previous paper (Gerstenkorn, 1967) for the loss of energy per second hold valid for the homogeneous case. Since the important q u a n t i t y 5 now tends towards unity for wr << 1, contrary to the damped case, the 90 ° component of s~ is not any longer of great importance. For two corresponding values r~ and r2, there are no noticeable differences between the correlated s~, neither in amount nor in phase; rl and r~ are defined [eft Gerstenkorn, 1967, Eq. (15b)] by ~o~-n-~.,_ = [ z / O + z ) b

(1~5)

The loss of energy per second is the same for both eases. The difference of energy for both states, with nutation and without nutation, respectively, has the same order of magnitude as in the damped case ½CcoE2xa~ (apart. from a factor of zero order).

DAMPING OF FREE :NUTATION OF THE EARTH

The loss of energy per second in the whole Earth is given by 47ra2~E4a2R 7 57ur1(1 W z) 2 for

/~R ER

=

19rWE4ot2RS#r2°~2 for 3g2

wrl >> 1

(16a)

¢0r2<<1. (16b)

Equations (16a) and (16b) have the same numerical value, if (15) is fulfilled. Equation (16) holds valid exactly only for homogeneity, but since the imaginary part of s~, being essential for the loss of energy, is the same in both cases (a) and (b), we m a y expect values of ER in the same order of magnitude also for an inhomogeneous mantle, consisting of two layers with rl and r2. On the other hand, if there exists a layer of sufficient size with r values of the intermediate range r2. < r < rl, this layer should give values of Ea that are too high, as in the homogeneous case. If we define a time T'0 by AE/2ER, we get from (16a) and the approximation for AE (13) after some calculation

T'o = ~,[(1 + z)Vz].

(17)

Apart from a factor 1 q - z ~ 1, this agrees with the damping time of (12a). Both times correspond to each other, but in our case, statistical excitation keeps constant the amplitude of the nutation, as we have supposed. It is interesting to see that the same high T'o value of ~ 1 0 ~ sec results as well for an E a r t h consisting of homogeneous rl- or r2-material as for the inhomogeneous structure outlined above. Both substances show nearly the same loss of energy per second per cm 3. Therefore we can't distinguish between them from energetic points of view. Nevertheless, we should remark that our supposition concerning the statistical excitation seems to be reasonable only for the high rl value, since the intervals between the real excitations m a y be some 107 sec or more. At least in the upper mantle, there must be a layer sufficient in size to be responsible for the time constant of the process. If we use To = 10s sec corresponding to

297

rl = 2 X 107 sec, we get with w ~ 1.66 X 10-7 sec-7 the correlated value from (15) of r2 = 105 sec. Values of r between rl and r2 bring about a greater loss of energy. For instance, if we choose r = 1.4 X 106 sec, a maximum loss of energy would result, being seven times the former value. Therefore layers in the mantle of about 103 km in size with such r values should give a stronger damping than that derived from the observations. Without that, a very high damping with a decay to exp ( - 1 ) during only three periods belongs to To = l0 s sec. Starting from a weaker damping [13 periods for exp ( - 1)], the discrepancy increases considerably: To = 5 × 10s sec, rl = 10s sec, r2 = 0.19 X 105 sec. The loss of energy is now 30-40 times smaller than the maximum value, corresponding to r -- 1.4 X 106 scc. Therefore, we can exclude with some certainty larger zones in the mantle with such relaxation times. Summarizing, we can say that a zone of many hundred kilometers in depth with r values of ~ 1 0 s sec is needed in the upper mantle to maintain the excited nutation. Zones in the crust with a higher r contribute only a negligible amount to the damping. Below a depth of 700 kin, r values of 104-10 s sec are most compatible with the observed damping. Likewise smaller r values of 102 sec in the lowest part of the mantle would give little contribution to the damping. Although completely safe statements are difficult to make, there seems to exist a limit with regard to the interval of possible r values in the middle and lower mantle: larger zones with r ~ 106 sec which would give a damping that is too strong, apparently do not exist there. REFERENCES ALLIgN, C. W. (1963). "Astrophysical Quantities," 2nd ed., p. 109. Athlone Press, London. BONDI, It., AND GOLD, W. (1955). On the damping of the free nutation of the Earth. Monthly Notices Roy. Astron. Soc. 115, 41. GERSTENKORN, It. (1967). The Earth as a Maxwell body. Icarus 6, 92. JV.FFREYS, It. (1952). "The Earth," 2nd ed. Cambridge Univ. Press, England.