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LOCAL TSUNAMI WARNING AND MITIGATION
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EFFECT OF ASTRONOMICAL FACTORS ON ENERGY DENSITY VARIATIONS IN THE EARTH SOLID MANTLE
Boris W. Levin1, Vladimir P. Pavlov
1 2

P. P. Shirshov Institute of Oceanology RAS, Moscow, Russia 2 Steklov Institute of Mathematics RAS, Moscow, Russia E-mail: levin@rfbr.ru

The idea of possible linkage between tectonic processes in the lithosphere and astronomical factors, such as tidal forcing, irregularity of Earth rotation and Chandler wobble (pole displacement), has been repeatedly discussed and recently analyzed in details in the geophysical literature [Wahr, 1985; Gor'kavy, 1989; Chao, 1995; Avsjuk, 1996; Levin, 1996]. A more universal approach, which allows to estimate the influence of these factors on the state of the stressed lithosphere, is proposed here. We base this approach on the following modern model of the structure of the Earth: the Earth is a spheroid consisting of a solid crust and a viscous liquid filling its spherical interior, with a denser solid core inside. The nature of tectonic motions and driving forces, which move plates and slabs of the Earth crust, is still a subject of considerable discussion. Traditionally, crust motion and the forces that cause it are associated with various processes deep inside the Earth, such as convective currents, gravitational and chemical differentiation and effects of plumes [Khain, 1973; Zonenstein, 1993; Pushcharovsky, 1999]. It was also postulated that external astronomical factors may have a significant effect on the tectonic and seismic processes [Kant, 1756; Darwin, 1879; Mayer, 1893; Khain, 1960; Kropotkin, 1963; Nalivkin, 1963]. Recently, considerable attention was focused on the influence of the Earth core motion on the various geophysical processes [Avsjuk, 1973; Jacobs, 1995; Avsjuk, 1998], on the consequences of the Earth center displacement [Avsjuk, 1999], and on the analysis of time correlation between regional seismicity and irregular rotation of the Earth [Gor'kavy, 1999]. Levin [1999] has analyzed the specific features of meridional distribution of Earth seismicity and has provided well-grounded arguments confirming the correlation between the seismicity and the Earth rotation. According to Avsjuk [1973, 1996], the solid core moves inside the liquid core subject to gravitational forces of Moon and Sun. Displacements of the solid core with an amplitude of the order of 100 m and specific periods that are known from astronomical observations (approximately 14, 365, 412-437 days and 6-7 years) produce displacements of the Earth center of gravity with the same periods and an amplitude of about 4 m. The Earth rotation axis follows the position of the gravity center and moves inside the Earth body. Consequently, the poles move as well, producing the Chandler wobble of the poles [Chandler, 1892]. These tidal forces also influence the tectonics of the lithosphere [Nadai, 1969; Sadovsky et al.,1987]. Model. We propose a model, which combines the effects of irregular Earth rotation, Chandler wobble and tidal forces on the stressed state of the lithosphere. Density variations in the free litospheric energy provide a qualitative example of such effects. These variations can be calculated using the linear theory of elasticity. They are represented by a stress tensor, which can be reconstructed from equilibrium conditions of a lithosphere element subjected to all of the forces.

Translated by O. Y. Yakovenko, edited by A. B. Rabinovich and J. Cherniawsky.

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It is convenient to write the equilibrium conditions in a rotating coordinate system. In this case, the revolution axis passes through the center of gravity of the Earth, and the lithosphere element is at rest. Let r be the radius vector of the lithosphere element (from the origin, or the Earth gravity center), the density of an element, with all quantities defined per unit volume. The net force on a motionless element is zero: (1)
0 = -

-3 - [[r ]] + x + ( M R - r (R - r ) - a ) + K . r

Here the first term on the right side is due to the Earth gravity, the second term is the centrifugal force ( is the angular velocity vector of the Earth rotation), the third term is inertia force due to irregularity of the displacement x(t ) of the Earth center of gravitation, the forth term is due to the tidal forcing from any astronomical body with mass M , separated from Earth by a distance R , ( is the Newtonian gravitational constant, a is a relative acceleration between the Earth and this body),.The Coriolis force is assumed to be zero. The last term is the force produced by the stress in the crust and is the divergence of the stress tensor (where p is the pressure): (2) Ki = ik / r , ik = -ik p + r -2 (ri rk - 3-1 ik r 2 ) s .

All forces, except those caused by the stress, are the gradients of the corresponding potentials (multiplied by the density). These potentials are: gravity Gr , centrifugal



Cent

= 2 -1 ( 2r 2 - (r ) 2 ) , the potential of inertia forces

Iner

= rDD , and the tidal potential x



Tide

= M R - r - ra 2 -1 M R-5 (r 2 R2 - 3(rR ) 2 )

-1

To make the model more realistic, we can approximate the gravity potential as a homogeneous ellipsoid of revolution (3)

Gr = 0 - 2 0 R 2 (1 - ( + cos 2 )(r / R) 2 ) ,

where the parameters 0 , and are chosen from the condition that the sum of gravity and centrifugal potentials includes the surface of the rotation ellipsoid as a level surface (in this case the Earth surface will be described in the same way). Here 0 is mean density of the Earth, R is polar radius, is a latitude of the lithosphere element. Evaluation of the potentials. Let us evaluate the order of magnitudes of the potentials. Taking = 6.7 10 -8 cm 3 g -1s -2 , 0 5 g cm -3 , and R 6.4 10 8 cm , we obtain Gr 8.6 1011 cm 2 s -2 . For the centrifugal potential, if 7.3 10 -5 s -1 , then
Cent

2 -1 2 R 2 1.1 10 9 cm 2 s -2 . We

showed previously [Levin and Pavlov, 2001], that the magnitude of the potential for inertia force is given by Ry 2 10 -1 2.6 10 -3 cm 2 s -2 , where 2 10 -7 s -1 is the angular velocity of the Earth revolving around the Sun, y 103 cm is the amplitude of the Chandler wobble. The Moon tidal potential is expressed in the form: Tide 2 -1 M R 2 R-3 1.8 10 4 cm 2 s -2 . Finally, if we take into the account the irregularity of the Earth rotation (i.e. forced precession of 50'' per year corresponding to the contribution of the vector with length 1.2 10 -12 s -1 , normal to the ecliptic plane, to the angular velocity ) then the centrifugal potential changes by: Prec = (( )r 2 - ( r )(r )) R 2 3.6 101 cm 2 s -2 . All other potentials have small effects in comparison with gravity and centrifugal potentials. These effects can be calculated using the framework of the perturbation theory.
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Levin B. W., Pavlov V. P.

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Contribution to free energy. The expression (2) for the stress tensor is an assumption of the model. Its first term is related to the volume, and the second term to the shear stress. From linear theory of elasticity we can express the density of free energy of a lithosphere element as:

(4)

F = (18 K ) -1

2 ll

+ (4 µ ) -1 (

2 ik

- 3-1 2 ) = (2 K ) -1 p 2 + (4 µ ) -1 (2 / 3) s 2 . ll

We can use the so-called hydrostatic approximation as a zero-th order approximation of the perturbation theory. In this case the stress tensor is reduced to the first volume term. The equilibrium conditions take into account only gravitational and centrifugal potentials but do not include astronomical factors. Applying a first order perturbation, we obtain a small contribution p to the pressure, while the scalar function s is also a small quantity of the first order. Hence, first order contribution to the free energy due to astronomic factors can be written as: (5)

F = K -1 p p .

The density and the pressure p are dynamic properties of the state of stress, and, generally speaking, depend on the vector variables r , , , a , x(t ) , and so on. If these dynamic characteristics can be described by analytical functions, we can conclude that they depend only on scalar combinations of these vector variables. We have shown previously [Levin and Pavlov, 2001] that the solvability condition of the equilibrium system leads us to conclude that this dependence is specific: the dynamic characteristics are the functions of one scalar variable , which is simply the sum of all the potentials we used in the model. Moreover, the pressure is the integral of the density over this variable: (6)
p = - ( )d ,

0

where 0 is the value of the variable on the Earth surface. In this case, expression (5) for the contribution of pressure to free energy density becomes: (7)

F = K -1 p ,

where = Iner + Prec + Tide is a sum of the contributions due to gravity and centrifugal potentials. These account for the astronomic factors to a first-order approximation of the perturbation theory.
Meridian dependence. Generally speaking, all three factors in the right part of expression (7) depend on latitude of the lithosphere element. However, the pressure and the density are calculated in the zeroth-order approximation of the perturbation theory, while their argument accounts only for gravity and centrifugal potentials. In this case the term proportional to cos 2 is small, of an order of 1/300 (Earth flatness), and it is possible to neglect the meridional dependence of pressure and density.

We showed [Levin and Pavlov, 2001] that for a coarse approximation for the Chandler wobble, (8)
y(t ) = y (cos t - 1, sin t , 0) ,

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where /2 is the frequency of a mode with maximum amplitude (corresponding to the period of 425 days), the displacement of the center of gravity x(t ) can be approximately described by a similar expression (we neglect here declination differences between the Sun and the Moon): (9)
x(t ) = y (sin ) -1 (cos t - 1, sin t , cos ), cos = sin cos t ,

where /2 - is a tilt angle of the rotation axis to the ecliptic, is an angle between the rotation axis and direction to the Sun (Figure 1a shows two extreme axis positions), is the angular velocity of Earth rotation around the Sun. In this case, the variation Iner = rDD during x one half of the Chandler period has the form (10)
Iner

=k

Iner

( x) f

Iner

( ), k

Iner

= 10-1 Ry 2 x, x = r / R, f

Iner

( ) = 1.36 cos + 0.6 sin .

This function depends on latitude and has its maximum at about 23o. In case of a forced precession, it is natural to estimate variation of the corresponding potential Prec for half of a day. Two positions of the lithosphere elements separated by half a day are shown in Figure 1b. We then have: (11)
k
Prec

= r 2 cos (cos(
2 2

Prec

( x) = r x ,

- - ) - cos( - + )) = k 2 2 f Prec ( ) = sin 2 sin

Prec

( x) f

Prec

( ),

,

where is an angle between the equatorial plane and the ecliptic. Finally, the variation of the tidal potential Tide for half a day can be expressed as (see Figure 1b): Tide = 2-13 M r 2 R-3 (cos 2 ( - ) - cos 2 ( + )) = kTide ( x) f Tide ( ),
(12)

k

Tide

( x) = 2-13 M R 2 R- 3 x 2 , f

Tide

( ) = sin 2 sin 2

.

In the last two cases the latitude dependence function has its maximum at 45o.
In t egral vari at i on of free en ergy. It i s useful t o eval uat e t ot al vari at i on of free energy of the lithosphere, which is due to astronomical effects. In order to do this, we prescri be t he dependence of densi t y on dept h and i nt egrat e (7) over t he surface and t he radius-vector r down to a certain depth h. Let assume that h = 100 km and approx imate the density increase with depth by a linear function (the calculations show that correction due to non-linearity is negligible in the transition zones [ Levin and Pavlov, 2001] ),

(13)

= R (1 + (1 - r/R )), 4,5 ,

where R i s t he densi t y on t he surface, and R = R(1 + cos 2 ) is the radius of the point on the surface at a gi ven latitude. Parameter , which can be ex pressed using the parameters of the gravity potential, is of the order of 1/300. Then, for z ero-th order in small parameter ,

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(14)

F =

R -h

dr dS

R

F=K R
-1

1- hR



1

-1

dxx p ( x) ( x)k ( x) d cos f ( )
0

1

After simple calculations and for the same periods of time as in the evaluation of the latitude dependence, we get for Chandler wobble FIner 0.65 10 24 erg, for forced precession FPrec 1.2 10 25 erg, and for t i dal effect s FTide 2.8 10 29 erg.
Discussion . S t ri ct l y speaki ng, precessi on and C handl er effect s are not correct l y accounted for in the first approx imation of the perturbation theory. Indeed, the tidal pot ent i al ex pansi on i s based on t he Lapl ace approx i m at i on, whi ch i s a l eadi ng order ex pansion in a parameter R / R 1.7 10 -2 . The precessi on and C handl er pot ent i al s are, respectively, 3 and 7 orders of magnitude smaller than the tidal potential. However, we found it useful to present the corresponding calculations for these two factors as well.

Latitude functions calculated from the model of free-energy density variations (Figure 2a) coinci de quantitativel y with the latitude function of power fl ux es during earthquakes (Figure 2b). They have a minimum at the equator and max i ma at mid latitudes. Further, a comparison of earthquake energy flux es with total free energy variation shows that the total energy flux es during earthquakes are of the order of 10 25 - 10 26 erg . On t he ot her hand, accordi ng t o a general t heory on peri odi c processes i n non-ideal continuum, there are vibrational energy dissipation due to viscosity and also contributions due to local defects. Ex isting estimates of the energy dissipation are very rough; they depend on the non-ideal nature of the model, and are in the range of 10 -2 - 10 -3 . W e observe that tidal variations can be one of the energy sources during earthquakes.

Figure. 1.

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LOCAL TSUNAMI WARNING AND MITIGATION ______________________________________________________________________

Figure. 2.

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Levin B. W., Pavlov V. P.

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The above conclusions are qualitative. The model deals with quasi-stationary lithosphere and only accounts for density variations with depth. It is obvious that geological structural heterogeneity have to influence significantly the effects from the astronomical factors. Let us further note that the dependence of free energy variations on the depth of the ellipsoidal layer, over which the density is integrated, is quadratic. It is interesting that these variations do not depend on model assumptions about the nature of density variations with depth. They depend only on the small depth of the layer relative to the Earth radius; the first terms of the respective ratio provide the main contribution in integral (14). The term, which is linear with depth, vanishes because the multiplier of this term, which is proportional to the pressure at zero depth, is zero. Instead, this depends on the choice of approximation of free energy in the linear elasticity theory. It is justified by the fact that accounting for non-linear terms (e.g. a cubic in free energy stress) is supposed to decrease the speed of sound with increased pressure, in contradiction with the actual observations in the Earth crust..
Acknowledgment. The authors would like to thank Yu. N. Avsyuk, G. A. Alexeev, G. S. Golitsyn, O. I. Zavialov, A. M. Fridman, and L. O. Tchekov for fruitful discussions. REFERENCES
Avsjuk Yu. N., 1973: On inner core motion. DAN USSR, 212, 5, 1103-1105 (in Russian). Avsjuk Yu. N., 1996: Tidal Forces and Natural Processes. Moscow, UIPE RAS, 188 p. (in Russian). Avsjuk Yu. N., and Levin B. W., 1998: Inner core of the Earth: Process of latitude change and its relation to geodynamo problem. In: Conference on Mathem. Geophys., July 12th-13th, 1998, Cambridge, UK. J. Conf. Abstr. 3(1), 56, Cambridge Publications. Avsjuk Yu. N., and Levin B. W., 1999: To the question by M. V. Lomonosov on the Earth center motion. Vestnik RFBR, 2(16), 4 ­11 (in Russian). Chandler C., 1892: On the variation of the latitude. Astron. J., 11, 12, 97 - 107. Chao B. F., and Gross R. S., 1995: Changes in the Earth`s rotational energy induced by earthquakes. Geophys. J. Int., 122, 776 - 783. Darwin G. H., 1897: The Tides and Kindred Phenomena in the Solar System. London. (Russian translation, 1992). Gor`kavy N. N., Minin V. A., Tajdakova T. A., and Fridman A. M., 1989: Are there any astronomical reasons for the strongest earthquake? Astronomical Circular, 1540, 35-36. Gor'kavy N. N., Levitsky L. S., Tajdakova T. A., Trapeznikov Yu. A., and Fridman A. M., 1999: On the dependence of the correlation between regional Earth seismicity and the irregularity of its rotation on the earthquake focal depth. Earth Physics, 10, 52 ­ 66 (in Russian). Jacobs J. A., 1995: The Earth`s inner core and the geodynamo: determining their roles in the Earth`s history. EOS, 76, 25. Kant I., 1902: Geschichte und Naturbeschreibung der merkwurdischen Verfall der Erdbeben welchesam der 1755 einer grossen Teil de erschwittert hat, 1756. Kant`s Gesammelte Schriften herausgegeben Von Kon. Preussischen Akademie d. Wissenschaften , Bd. I, Berlin. Khain V. E., 1960: Role of astro-geological factors in the Earth crust evolution, and formation of its megastructure. Theses, 3d Astro-geological conference on the problems of the Earth theory. Leningrad, 34-35 (in Russian). Khain V. E., 2001: Tectonics of the Continents and Oceans. Moscow, Nauchny mir, 604 p. (in Russian). PETROPAVLOVSK-KAMCHATSKY TSUNAMI WORKSHOP, SEPTEMBER 10-15, 2002 85


LOCAL TSUNAMI WARNING AND MITIGATION ______________________________________________________________________ Kropotkin P. N. and Trapeznikov Yu. A., 1963: Variations in angular velocity of the Earth rotation, pole swinging and drift velocity of a geomagnetic field; their possible connection with geotectonic processes. Izv. Ak. Nauk USSR, Series ­ Geology 11, 32-50, (in Russian). Levin B. W., 1996: Nonlinear oscillating structures in the earthquake and seaquake dynamics. CHAOS, 6 (3), 405 ­413. Levin B. W., and Chirkov E. B., 1999: Features of latitudinal seismicity distribution and the Earth rotation. Volcanology and Seismology, 6, 65 ­ 69 (in Russian). Levin B. W., and Pavlov V. P., 2001: Theoretical and field method of the stress-tensor reconstruction in the Earth with mobile core. Theoretical and Mathematical Physics. 128, 3, 439-445 (in Russian). Mayer I. R., 1893: Die Mechanik der Warme. Stuttgart. Nadai A., 1969: Plasticity and Solid Body Destruction. Moscow, Mir, 1969. v. 2. 863 p. (translation into Russian) Problems of planetary geology. 1963: The articles. Eds: Nalivkin D. V. and Tupitsyn N. V. Moscow, Gosgeoltechizdat, 342 p. (in Russian). Pushcharovsky Yu. M., and Pushcharovsky D. Yu., 1999: Geospheres of the Earth mantle. Geotectonics. 1, 3 ­ 14 (in Russian). Sadovsky M. A., Bolkhovitinov L. G., and Pisarenko V. F., 1987: Deformation of geophysical medium and a seismic process. Moscow, Nauka, 100 p. (in Russian). Wahr J. M., 1985: Deformation induced by polar motion. J. Geophys. Res., 90, 9363 - 9368. Zonenshain L. P., and Kuz'min M. I., 1993: Paleogeodynamics. Moscow, Nauka, 190 p. (in Russian).

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