that than that that that that that

C H A P T E R 10

RHEOLOGICAL PROBLEMS O F THE EARTH'S INTERIOR B. Gutenberg

I. I n t r o d u c t i o n : Structure of the E a r t h ; Elastic Constants 401 I I . I n t r o d u c t i o n t o R h e o l o g i c a l P r o b l e m s in the E a r t h 406 I I I . Processes R e t a r d i n g Elastic M o t i o n in the E a r t h 408

I V . F l o w Processes in the E a r t h ' s Outer L a y e r s : P o s t - G l a c i a l Uplift 413 V . T h e Coefficient of V i s c o s i t y and the Strength in the E a r t h 418 V I . F l o w Processes in the D e e p e r P o r t i o n of the E a r t h ' s M a n t l e ; Results Based

on D e e p - F o c u s E a r t h q u a k e s 422 V I I . S t r a i n - R e b o u n d Characteristics of E a r t h q u a k e Series and A f t e r s h o c k s . . 423

V I I I . R h e o l o g i c a l P h e n o m e n a in G e o l o g i c a l and G e o p h y s i c a l Processes on a

Small Scale 427 I X . V i s c o s i t y in the E a r t h ' s C o r e ; F l o w in the C o r e as Source of Terrestrial

M a g n e t i s m 429 N o m e n c l a t u r e 430

I. Introduction: Structure o f the E a r t h ; Elastic Constants

A b o u t four t o five billion years ago the earth began t o d e v e l o p as an individual b o d y . Until a b o u t 1940 it was widely believed that it condensed from a h o t cloud. H o w e v e r , m o s t specialists n o w prefer the hypothesis that the earth has developed from accretion of cold particles. It is assumed on b o t h hypotheses that heavier material, especially iron and metals associ- ated with it, settled in the central portion of the earth and formed the core there, which was first suspected from the fact that the mean density of the earth—5.52—is considerably greater than that of the materials found in the upper layers. In 1912 the radius of the core—3450 km.—was deter- mined from travel times of w a v e s refracted through it, and later the o b - served times of waves reflected at its surface, either at the outside or at its inside, confirmed the result. A further complication was introduced when a relatively large increase in the v e l o c i t y of longitudinal w a v e s was estab- l i s h e d at a distance of a b o u t 1300 k m . from the earth's center (Fig. 1, insert). T h e question is still unsolved whether the t w o boundaries, at depths of 2900 and at a b o u t 5000 k m . , are a consequence of a change in state, phase, or of material.

Observations of the tides of the earth's b o d y and of the free m o v e m e n t s of the earth's axis (a period of 4 2 0 ± days instead of 305 days calculated for an absolutely rigid earth) leave n o d o u b t that the average rigidity of

401

402 Β . G U T E N B E R G

F I G. 1. R i g i d i t y G and bulk modulus k as function of d e p t h in the earth. Insert:

Cross section t h r o u g h the earth. (Drafted by Mr. John M. Nordquist.)

the earth's core must be appreciably lower than that of the mantle. Since n o transverse w a v e s h a v e been observed passing through the core (Fig. 1 ) , it is generally believed that at least its outer portion is liquid. T h u s far w e have n o indication of w a v e s which have traversed the inner core as trans- verse waves, but this m a y be due t o the difficulty of observing such w a v e s which are expected t o carry only relatively little energy since a large frac- tion of the already small percentage entering the inner core would be re- flected b a c k into the inner core when the w a v e s reach its b o u n d a r y on their w a y out.

C o m b i n a t i o n of the various hypotheses has led t o an ever-increasing number of c o n c l u s i o n s^{1 ,2} concerning the properties of the core. A p p a r e n t l y , the assumption is still preferred that the whole core consists mainly of iron, which is liquid at least in the outer part. Other scientists prefer the c o m b i - nation of a liquid outer core consisting of rock with an inner core of solid

1 B . G u t e n b e r g , e d . , in "Internal C o n s t i t u t i o n of the E a r t h , " p . 183. D o v e r P u b - lications, N e w Y o r k , 1951.

2 H . Jeffreys, " T h e E a r t h , " 3rd e d . , p . 295. M a c m i l l a n , N e w Y o r k , 1952.

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H Y I N T E R I O R 403

iron. Other hypotheses (summaries in refs. 1, 2) suppose that the whole core consists of rock and that the boundary of the inner core is a result of a change in phase of the material forming the whole core, or that the pres- sure which increases from about 1}^ million atmospheres at its outer bound- ary to about 33^2 million in its center has rendered difficult or even pre- vented an exchange of material in the deeper parts of the earth, so that even hydrogen remaining from the (solar) origin of the earth m a y play an appreciable role in the core.

In the mantle, a well-marked change occurs at a depth of between 900 and 1000 k m . A b o v e this depth the wave velocities and elastic constants (Fig. 1) increase more rapidly with depth than below. It is not certain whether or not the rapid increase between depths of about 200 and 1000 km. can be explained without the assumption of a (gradual) change in material or phase. T h e fact that earthquakes rather suddenly terminate at a depth of about 720 km. is probably not related t o this change occurring at least 200 k m . deeper, but m a y be caused b y a decrease in stresses with depth, or b y a decrease in viscosity resulting in flow rapid enough to pre- vent the reaching of the breaking strength .2 a

A layer with relatively low wave velocity (minimum a few per cent smaller than the velocities a b o v e and below) extends from a depth of about 50 k m . roughly 100 k m . downward and is probably a result of greater effect of the increase in temperature than that of the increase in pressure on the elastic constants in the depth range where the temperature is relatively close to the melting point. A t greater depth the effect of the continued rela- tively rapid increase in pressure prevails again over that of the relatively small increase in temperature.2b

A sharp discontinuity marked b y a sudden increase in wave velocities and elastic constants at depths of 10 to 15 k m . below sea level under the deeper parts of the oceans and at about 30 k m . (lowlands) to 50 k m . (moun- tains) in the continents is called " M o h o r o v i è i é discontinuity" after its discoverer. There are several types of evidence3 that starting at a depth of about 10 k m . wave velocities and elastic constants decrease b y small amounts downward in each of the crustal layers. Whereas the elastic con- stants (Table I ) in the layers above the discontinuity differ regionally (e.g., ref. 4) with probably basaltic material (gabbro, olivine gabbro) forming their deeper portion, there is good evidence that at a given depth below the discontinuity the physical constants are everywhere the same within the

2 a B . Gutenberg, Quart. J. Geol. Soc. London 112, 1, (1956).

2 b B . Gutenberg, Bull. Geol. Soc. Amer. 66, 1203 (1955).

3 Β . G u t e n b e r g , Geophysics 20, 283 (1955).

4 L . H . A d a m s , in " I n t e r n a l C o n s t i t u t i o n of the E a r t h " ( G u t e n b e r g , e d . ) , p . 50.

D o v e r P u b l i c a t i o n s , N e w Y o r k , 1951; see also B . G u t e n b e r g , ibid.} p . 364.

404 Β . G U T E N B E R G

T A B L E 1

SELECTED CHARACTERISTIC V A L U E S OF T H E C O E F F I C I E N T G OF R I G I D I T Y AND T H E B U L K M O D U L U S k IN T H E E A R T H ' S C R U S T ; U N I T : 1 011 D Y N E S / C M2.

G k

A l l u v i u m near the surface in continents 0.1 0 . 2 A l l u v i u m at d e p t h of 1 k m . in c o n t i n e n t s 1 2 T e r t i a r y sandstone at 2 k m . in continents 2 3 Granite at d e p t h of 2 k m . in continents 3 5 C o n t i n e n t a l layers at d e p t h of 20=t k m 4 7 5 ± k m . b e l o w oceanic b o t t o m s (excluding ähelf) 5 10 D e p t h of 8 0 ± km. everywhere Q}i 12

limits of observation. There is no agreement about the types of rock which form the mantle below the Mohoroviëic discontinuity. Olivine, dunite, and others are being considered as representative. T h e suggestion b y L e e s4a

that the discontinuity m a y be produced b y a change in phase and not in material has some merits4b but it needs support b y more results of labora- tory research of the type now undertaken b y K e n n e d y4e and b y M a c - D o n a l d4^

It is still undecided whether at present the earth is cooling or getting hotter. It is believed that more heat is developed inside the continental layers of the earth b y radioactive processes than in the ocean bottoms. On the other hand, recent observations indicate that the heat flow through the earth's surface is of the same order of magnitude (about 1% X 1 0- 6

c a l . / c m .2 sec.) regardless of the location of the point of observation in continents or ocean bottoms. However, reliable data are still very scanty.

A n additional difficulty in the investigation of the temperature in the earth's interior results from the fact that the change of thermal conduc- tivity with depth is not known. It has been found that the electric conduc- tivity increases 1000 fold or more in the upper 200 km. of the earth, and that its deeper parts are good electric conductors. If also in these layers an appreciable increase in the thermal conductivity takes place (a good cor- relation of the t w o coefficients under surface conditions is known only for metals), all deeper portions of the earth must have practically the same temperature. T h e highest and lowest scientifically supported values at a depth of 100 k m . are about 1600 and 600° C , respectively. For the tempera- ture in the core, values between about 1500 and many thousand degrees C .

4 a M . Lees, Quart. J. Geol. Soc. London 109, 217 (1953).

4 B B . Gutenberg, Stille-Festschrift, p . 411. Enke, Stuttgart, 1956.

4 0 G . Kennedy, Oral communication.

4 d G. M a c D o n a l d , Discussion during Conference on Theoretical Geophysics, Wash- ington, 1956.

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H ' S I N T E R I O R 405 are being considered possible, with a current preference for values between 2000 and 4000°. T h i s uncertainty as well as the u n k n o w n melting points of rocks at pressures of the order of one million atmospheres render v e r y d o u b t - ful all estimates of the depth where the melting p o i n t of a given material can b e expected. In addition, the a m o u n t of heat which is being d e v e l o p e d at the various depths is still in d o u b t . A p p r e c i a b l y m o r e heat must h a v e been d e v e l o p e d during the earth's earlier history w h e n m o r e short-lived radioactive elements existed. A n o t h e r p o o r l y k n o w n source of heat has been furnished b y the hypothetical sinking of heavier material and rising of corresponding amounts of lighter material (see also Urey, ref. 4e, p . 1 3 ) .

T h e first numerical values⁶ of the elastic constants in the earth as a func- tion of depth h a v e been obtained in 1923 after the v e l o c i t y of earthquake waves throughout the earth had been calculated and the density in the earth as a function of depth had been obtained t o a first approximation.

In a purely elastic material and under conditions which permit the neglect of second-order terms the coefficient G of rigidity (in seismological papers usually indicated b y μ) and the bulk m o d u l u s k m a y b e found from the density ρ and the velocities V of longitudinal and ν of transverse w a v e s from the following equations:

These equations h a v e been used generally t o calculate the elastic constants in the earth ( T a b l e I, Fig. 1 ) . H o w e v e r , o n the basis of experiments with an organic glass⁶ and theoretical considerations,⁷ K u h n and Vielhauer b e - lieve that the observed velocities of elastic w a v e s are smaller b y increasing amounts than those calculated from the elastic constants b y use of equa- tions (1) and (2) w h e n the viscosity coefficient decreases b e l o w a b o u t 1 0⁴ poises. Consequently, if for a material with such l o w v i s c o s i t y the bulk modulus k is calculated from observed w a v e velocities b y use of equation ( 2 ) , the resulting value m a y b e appreciably smaller than the bulk m o d u l u s for v e r y rapid m o t i o n .

T h e a c c u r a c y of the values for G and k resulting from equations (1) and (2) decreases in general with depth as a consequence of the decreasing ac- curacy with which the density ρ is k n o w n in the deeper parts of the earth.

I n addition, for the earth's core the coefficient of rigidity G is n o t k n o w n .

4 e S y m p o s i u m " L e n o y a u terrestre," Ann. gêophys. 11, 49 (1955).

5 Β . G u t e n b e r g , Physik. Ζ. 24, 296 (1923); " D e r Aufbau der E r d e , " p . 78. B o r n - traeger, Berlin, 1925.

6 W . K u h n and S. Vielhauer, Geochim. et Cosmochim. Acta 3 , 169 (1953).

7 W . K u h n and S. Vielhauer, Z. physik. Chem. 202, 124, 161 (1953).

(1) (2)

406 Β . G U T E N B E R G

I n the outer part of the core (Fig. 1, insert) ν and G are at least several powers of 10 smaller than in the mantle. T h e assumption that in the core G = 0 does n o t lead t o discrepancies between any calculated and observed quantities such as tides of the earth's b o d y or m o v e m e n t s of the earth's axis.⁸ In the inner core G is possibly greater than in the outer core and m a y even b e of the same order of magnitude as in the mantle. T h e possible effect of a corresponding small viscosity in the core on the observed w a v e velocities is an additional source of uncertainty in the calculation of k in the core. K u h n and Vielhauer⁶ estimate that in the core of the earth the actual value of k m a y b e 40 % greater than the value calculated from equa- tion ( 2 ) .

I n the calculations of the bulk m o d u l u s k for Fig. 1 it has been assumed that ν and G are negligible in the outer core between depths of 2900 and 5100 k m . ; for the inner core t w o curves are given, one supposing ν = G = 0, and a second on the hypothesis of B u l l e n^{9 , 10} that, under high pressure, k depends on the pressure o n l y and n o t on the material. T h e hypothetical effect of the l o w viscosity in the core on the w a v e v e l o c i t y has n o t been considered in the calculation of k for F i g . 1. H o w e v e r , if Bullen's h y p o t h e - sis is correct, the l o w viscosity w o u l d rather affect the calculation of the density than that of k.

II. Introduction t o Rheological Problems in the Earth

In most early theoretical studies of geophysical phenomena it is assumed that mechanical processes on a large scale can be treated as purely elastic.

Jeffreys¹¹ was one of the first t o consider nonelastic processes. Originally, he represented delayed elastic processes (elastic afterworking, firmovis- cosity, internal friction) following Sir J. Larmor, and time-dependent flow processes ( " e l a s t i c o - v i s o c i t y " ) following M a x w e l l b y introducing the fol- lowing convenient (but frequently v e r y rough) approximation for a shear- ing strain y p r o d u c e d b y a tangential stress S:

'=l-*4

U

where

η = GX and ν = G\r (4)

G = rigidity, η = "coefficient of v i s c o s i t y , " λ = time of relaxation in vis-

8 P . J. M e l c h i o r , " L e s Marées Terrestres," Monograph 4, Observatoire Royal de Belgique, 1954.

9 K . E . Bullen, Monthly Notices Roy. Astron. Soc, Geophys. Suppl. 6, 50 (1950).

1 0 K . E . Bullen, Ann. geofis. (Rome) 6, 1 (1953).

11 H . Jeffreys, Monthly Notices Roy. Astron. Soc. 77, 449 (1917).

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H 's I N T E R I O R 407

cous flow; ν is designated the "coefficient of retarded elastic m o t i o n " or

"coefficient of internal friction," and Xr is the " t i m e of retardation" in elas- tic processes (firmoviscosity, internal friction), that is, the time in which in such processes the strain falls t o 1/e of its value, if the stress is r e m o v e d . I n the present section, η and ν are expressed in poises ( = g . / c m . s e c ) , G in g . / c m . sec.² ( = d y n e s / c m .2 = baryes), λ and Xr in seconds.

T h e first part of equation ( 3 ) , 7 = S/G, corresponds t o H o o k e ' s l a w ; the following term is important only for short-period stresses (dy/dt not t o o small), and the last term o n l y for stresses lasting for intervals of at least a noticeable fraction of the time λ = η/G. Consequently, in m o s t equations for elastic processes at least one of these additional terms is insignificant.

Jeffreys¹² suggests that t o a c c o u n t for viscous flow (elastico-viscosity) in basic equations for elastic processes concerning the earth, the rigidity G should b e replaced b y G/[I + ( Ω / λ ) ] where Ω is the operator of definite integration defined generally b y

F o r harmonic m o t i o n proportional t o e^iqt, (5) can be evaluated, and it fol- lows that G is t o be replaced b y

In equations representing first-order approximations of processes in which the effect of retarded elastic response (firmoviscosity, internal friction) is significant, the p r o d u c t G f(t) of the rigidity G and any time-dependent function/(0 should b e replaced in the fundamental equations b y

T e r m s containing only the b u l k modulus k remain u n c h a n g e d .³ E q u a t i o n (18) is an example of the use of substitution ( 7 ) . E v e n these rough approxi- mations which in several respects lead t o inconsistencies result in v e r y c o m - plicated equations for which frequently n o general solution can b e given.

It is sometimes overlooked that firmoviscosity (retarded elastic response) in solids corresponds t o viscosity in true liquids. In case of a suddenly starting constant stress, b o t h result in a displacement which is always smaller than that given b y H o o k e ' s law, but approaches it asymptotically, whereas viscous flow in solids results in a displacement which starts with the value given b y H o o k e ' s law b u t continues t o increase [compare also equations (10) and (11)]. Applications of substitution (5) or of the last term in equation (3) t o equations for true liquids, or of equations containing

1 2 H . Jeffreys, " T h e E a r t h , " 3rd e d . , p p . 8 and 242. M a c m i l l a n , N e w Y o r k , 1952.

1 3 B . G u t e n b e r g and H . S c h l e c h t w e g , Physik. Z. 31, 745 (1930).

(5)

Giq\/(1 + iqk) (6)

Gf(f) + vdf(t)/dt (7)

408 Β . G U T E N B E R G

ν or \^r t o equations for viscous flow of solids, lead t o erroneous results. T h e following tabulation illustrates results obtained from equation (3) in special cases (t = t i m e ) .

Assumed Condition Viscous Response in Solids Retarded Elastic Response

Short-period m o t i o n it

~ λ,)

Effect usually negligible

Long-period m o t i o n (t ~ λ)

7 = £ + - [Sdt

G η J (9)

• Effect usually negligible Long-period m o t i o n

(<»λ)

G = 0 and ν > 0 (viscous fluid)

S approaches v(dy/dt) (10)

S = v(dy/dt) (11)

It should b e emphasized again that equation (3) and all equations de- rived from it describe at best v e r y crude approximations t o the processes involved since t h e y h a v e t o fulfill the requirement of leading t o expres- sions which can be solved w i t h o u t t o o great difficulties. E q u a t i o n (3) can be applied t o a delaying process (second term on right side) as well as t o a viscous-type process (last t e r m ) , b u t the representation of details is limited b y the form of these terms and m a y n o t correspond t o the details of the process in nature. Frequently, quantities assumed t o be constant in the equations actually change with time, frequency of vibrations i n v o l v e d , and the a m o u n t of stress or strain. F o r further discussion, see H a r d t w i g .¹⁴ Equations (31) and (32) in Section V I I are examples of equations based on observed nonelastic processes.

Numerical values of quantities related t o nonelastic processes at selected depths in the interior of the earth are summarized in T a b l e I I ; a few cor- responding data found in laboratory experiments and from processes at the earth's surface are added. E x c e p t for the u p p e r m o s t layers, even the order of magnitude of m o s t of these quantities is doubtful, and it also should be considered that t h e y depend on the assumptions m a d e in their determi- nation, o n the duration or period of the process, on the history of the ma- terial and other v a r i a b l e s .^{1 4a}

III. Processes R e t a r d i n g Elastic M o t i o n in the Earth

Various phenomena, such as internal friction, creep, elastic afterworking, may b e considered t o b e " d e l a y e d elastic processes" t o a first approxima-

14 E . Hardtwig, Ann. geofis. (Rome) 7, 143 (1954).

1 4a H . M a t t i c e and P . Lieber, Trans. Am. Geophys. Union 35, 613 (1954).

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H ' S I N T E R I O R 409

T A B L E I I

O R D E R OF M A G N I T U D E OF N O N E L A S T I C C O N S T A N T S (A) I N S O M E M A T E R I A L S AT T H E S U R F A C E OF T H E E A R T H , A N D (B) AT

SELECTED D E P T H S I N T H E I N T E R I O R OF T H E E A R T H

log λ

(A) Material log Str.* log η log ν log \^r

(a) (b)

Solenhofen l i m e -

stone 22± 3 ± 9 ± - 2 J i ±

G a b b r o ^? ? ? ?

Glacier ice ^? 13 t o 14 3 ± - 4 ± ? ?

F l o w i n g l a v a ? < 5 ± ? ? ? ?

F l o w i n g m u d ? < 4 ± ? ? ? ?

(B) Depth

10 k m . ( c o n t i n e n t ) . . 9 ± 22 t o 23 11 4 ± 9=b - 2 t o - 3

100 7? 2 2 ± 10 2 ± 9 ± - 2 t o - 3 ?

700 < 7 ? 2 0 ± ? 8? 1? 9 t o 10? - 3 d b ?

800 t o 2900 < 7 ? < 2 0 ? < 7 ? < 0 ? 10? - 3 ?

> 2 9 0 0 < 7 ? 5?? ? ? ? ?

* Str. = strength resisting v i s c o u s flow, in p o i s e s ; η = coefficient of v i s c o s i t y , in p o i s e s ; λ = c o r r e s p o n d i n g time of relaxation, (a) in s e c o n d s , (b) in y e a r s ; ν — coeffi- cient of retarded elastic response for elastic w a v e s w i t h periods of 0.1 t o 20zh s e c o n d s , in p o i s e s ; \^r = c o r r e s p o n d i n g time of retardation, in s e c o n d s .

tion. T h e only t y p e of this group for which numerical data are available on the basis of geophysical observations is usually called "internal friction."

I t has a time of retardation X^r of a small fraction of a second. M o s t large- scale geophysical phenomena, such as m o v e m e n t s of the earth's axis or the tides of the earth's b o d y , p r o c e e d t o o slowly t o b e delayed n o t i c e a b l y b y interna] friction. In these processes, dy/dt is v e r y small in equation (3) in addition t o the small value of X^r. On the other hand, earthquake w a v e s through the interior of the earth usually consist mainly of w a v e s having periods of the order of 1 t o 20 sec. Consequently, dy/dt m a y b e c o m e large enough t o result in a noticeable decrease in the amplitudes of w a v e s travel- ing o v e r distances of the order of magnitude of 1000 k m . J e f f r e y s^{1 5 , 16} finds that, t o a first approximation, the distance Δ * t o which these waves h a v e t o travel before their amplitude a is reduced t o 1/e as a consequence of retarded elastic m o t i o n alone is given b y

2V q2\r

Δ* (12)

1 5 H . Jeffreys, " T h e E a r t h , " 3rd e d . , p . 242. M a c m i l l a n , N e w Y o r k , 1952.

1 6 H . Jeffreys, Monthly Notices Roy. Astron. /Soc, Geophys. Suppl. 1, 412 (1926).

410 Β . G U T E N B E R G

where q = 2π/Τ. X^r is defined b y equation ( 4 ) , V is the w a v e v e l o c i t y and Τ is the period of the waves. T h e decrease in amplitudes of seismic w a v e s b y absorption (after corrections h a v e been m a d e for the expansion or con- traction of the w a v e front) is usually expressed b y

a = e ~ K\ or Κ = 1 / Δ * (13)

where a is the amplitude of the w a v e at the distance Δ , a⁰ its amplitude at Δ = 0, and Κ is designated the ''coefficient of a b s o r p t i o n / '

F o r longitudinal w a v e s through the interior of the earth (excluding the layers v e r y close t o the surface) the average v a l u e¹⁷ of Κ is a b o u t 0.12 per 1000 k m . T h i s corresponds t o Δ * = 8 0 0 0 ± k m . for the distance along which the amplitude decreases b y absorption t o 1/e. Since q is approximately unity, and V is of the order of 10 k m . / s e c , it follows from equation (12) that X^r is of the order of 0.003 sec. C o m b i n i n g this value with an average coefficient of rigidity of 2 Χ 1 0¹² d y n e s / c m .² for the earth's mantle, equa- tion (4) indicates that at depths of between a b o u t 20 and 2900 k m . ν is of the order of 10⁹ t o 1 0¹⁰ poises. F o r the earth's core the absorption coeffici- ent Κ is of the same order of magnitude as for the m a n t l e .¹⁷ H o w e v e r , n o corresponding values can b e given for X^r and ν on a c c o u n t of the u n k n o w n rigidity and viscosity in the core.

T h e coefficient of absorption of seismic surface w a v e s with wavelengths of a b o u t 100 k m . is of the order of 0.1 t o 0.2 per 1000 km., that is, a b o u t the same as that for longitudinal waves through the earth's interior. I t de- creases noticeably with increasing wavelength. F o r a b o u t 1000 k m . long surface w a v e s of the R a y l e i g h w a v e t y p e with periods of 230 sec. E w i n g and Press¹⁸ have found that Κ is 0.018 per 1000 k m . In this c o n n e c t i o n they have used the dimensionless q u a n t i t y 1/Q t o indicate the internal friction. It is defined b y

1/Q = dE/2vE = δ/ττ (14) where dE is the loss of energy per cycle in a vibrating b o d y , Ε the total

energy and δ the logarithmic decrement of free vibrations. F o r the 1 0 0 0 ± k m . long surface waves, E w i n g and Press find 1/Q = 590 X 10~⁵. F o r elas- tic waves spreading spherically in a h o m o g e n o u s m e d i u m the absorption coefficient Κ is related t o 1/Q approximately (ref. 19, p . 8 8 ) b y

Κ = q/2QV (15)

where again q = 2π/Τ, Τ = w a v e period, V = w a v e v e l o c i t y . C o m b i n i n g

17 B . G u t e n b e r g , Bull. Seismol. Soc. Amer. 35, 57 (1945).

18 M . E w i n g and F . Press, Bull. Geol. Soc. Amer. 63, 1248 (1952).

1 9 F . B i r c h , J. F . Schairer, and H . C . Spicer, Geol. Soc. Amer. Spec. Paper 36, 88-92 (1942).

R H E O L O G I C A L P R O B L E M S O P T H E E A R T H ' S I N T E R I O R 411 equations ( 1 2 ) , ( 1 3 ) , and (15) w e find that for elastic w a v e s with a period T⁷, approximately

Κ = 1/gQ = T/2wQ (16) F o r seismic w a v e s with periods of a few seconds, \^r in seconds should be of

the same order of magnitude as the dimensionless q u a n t i t y l/Q. F o r a n u m b e r of r o c k types, l/Q has been determined from longitudinal as well as torsional vibrations of bars in laboratory experiments (ref. 19, p . 9 2 ) ; m o s t values are between 1CT² and 2 X 10~~³, in agreement with the value of 3 Χ 1 0^{- 3} found from longitudinal waves through the earth's interior.

T h e laboratory experiments also indicate that l/Q depends on a variety of factors, a m o n g t h e m the period T. If Τ increases from 0.0001 t o 1 s e c , l/Q for steel seems t o increase b y a factor 5 ± .

S e z a w a²⁰ has introduced t w o friction terms into the equations for purely elastic waves, one for compressional and one for shear m o t i o n . F o r example, the equation for Νι, the normal stress in the direction of the x-axis, is given on the assumption of H o o k e ' s law b y

ΛΓι = LG + 2G ^ (17) dx

where L = k — ( 2 / 3 ) G . L is one of Lamé's constants; it is usually indicated b y λ in geophysical publications, θ is the change in v o l u m e per unit v o l u m e , u is the displacement c o m p o n e n t in the direction of the x-axis. On Sezawa's assumptions for delayed elastic m o t i o n , equation (17) is replaced b y

Nr = LG + 2Gp + L'ff + * p L (18)

dx at dxdt

where V and ν are constants for the delayed elastic m o t i o n ; v, for delayed shearing m o t i o n , is given b y (4) ; U is a similar constant for delayed c o m - pressional m o t i o n . T h e equations for the tangential stresses Τι, T2, Td

in a purely elastic m e d i u m h a v e t o b e treated similarly. F o r example, if ν and w are the displacements in the direction of the y- and 2-axis, respec- tively, w e find for a m e d i u m with retarded elastic m o t i o n

T h e resulting equations for the change θ in v o l u m e per unit v o l u m e , and the shear ω are

k + \G T, , 9 (20)

d θ 3 . L + Zv dv θ

= V t ) τ

dt2 ρ ρ dt

2 0 Κ . Sezawa, Earthquake Research Inst. Tokyo Bull. 3, 43 (1927).

4 1 2 Β . C U T T E N B E I W Ï

§=^ν

α, + ^ (21)

ον ρ ρ dt

where ρ = density. T h e y cannot be solved without introduction of specific w a v e forms. On the basis of his findings for a number of such forms, Sezawa c a m e t o the conclusion that, as a consequence of internal friction, seismic waves are gradually getting longer and flatter in traveling. T h e v e l o c i t y of the beginning of the disturbance remains unchanged.

Gutenberg and S c h l e c h t w e g¹³ h a v e pointed o u t that the effect of fric- tion in compressional m o t i o n is negligible relative t o its effect in shear m o - tion and h a v e modified Sezawa's equations correspondingly. Especially, the constant V is then given b y

L' = - f (22)

G u t e n b e r g²¹ has studied the resulting changes in w a v e form. Since Sezawa's method cannot b e applied t o sinusoidal waves, G u t e n b e r g defines the w a v e period Τ for the various w a v e forms used b y Sezawa, e.g.,

y = Bb-2xe-x2b~2 (23)

as twice the time between successive crossings of the rest line in opposite directions, and finds that for seismic waves the so defined original w a v e period T0 increases with distance Δ t o the value Ί\ given approximately b y

Γ^{Δ 2} = Γ „² + ^ (24)

where ρ is the density and V the w a v e v e l o c i t y . E q u a t i o n (24) permits the calculation of ν and \^r [equation (4)] from the observed increase in the peri- ods of seismic w a v e s with distance. T h e resulting values of ν and λ^Γ depend on the wavelength. If this increases from 100 m . t o 100 k m . , ν increases from a b o u t 10⁷ poises t o a b o u t 5 Χ 10⁹ poises, and \^r increases approxi- mately from 10~⁴ t o 10~³ s e c .^{2 2}"²⁵ T h e s e values are of the same order o f magnitude as those mentioned a b o v e which result from the absorption coefficient Κ and from torsional vibrations of bars in laboratory experi- ments. H o w e v e r , other p h e n o m e n a are i n v o l v e d in the observed increase

2 1 B . Gutenberg, ed., " H a n d b u c h der G e o p h y s i k , " V o l . 4, p . 22,1932; V o l . 2, p . 553, Borntraeger, Berlin 1933.

2 2 B . Gutenberg, e d . , in ''Internal Constitution of the E a r t h , " p . 385. D o v e r Publications, N e w Y o r k , 1951.

2 3 B . Gutenberg, U. S. Coast and Geodetic Survey, Spec. Puhl. N o . 201, 163 (1936).

2 4 B . Gutenberg, Beitr. angew. Geophys. 6, 125 (1936).

2 5 O. Meisser, Veröffentl. Reichsanstalt Erdbebenforsch. Jena, 9, 70 (1929).

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H ' S I N T E R I O R 413 in prevailing w a v e periods with distance or time, such as decrease of o b - sorption with increasing w a v e l e n g t h ,²⁶ scattering,²⁷ and p h e n o m e n a related to g r o u p v e l o c i t y .^{2 8 , 29} Since the actual process is v e r y c o m p l i c a t e d it is usually assumed that o n l y one t y p e of mechanism is responsible for the c o m b i n e d effect of s e v e r a l .1 4 , 1 4 a'^{3 0} M a t h e m a t i c a l l y the various approaches are different t y p e s of approximations t o fit a given functional relationship between the change in period of the waves on one hand and their distance from the source or their travel time on the other.

IV. Flow Processes in the Earth's O u t e r Layers. Postglacial Uplift In practically all "classical" investigations of flow inside the earth it is assumed that the flow is N e w t o n i a n , i.e., that the ratio of the stress t o the shear rate D does n o t depend on D. T h i s is a rather special assumption, b u t frequently it can be used as a first approximation since usually the available observations are scanty, and the processes are t o o c o m p l i c a t e d t o permit the finding of precise quantities. It should b e n o t e d that in m o s t geophysi- cal and geological publications the expression ' ' v i s c o u s ' ' is used t o indicate N e w t o n i a n flow, the expression " p l a s t i c " for m o r e general t y p e s of flow.

The only p h e n o m e n o n which thus far has p r o v i d e d sufficient data for the qualitative investigation of subcrustal flow is the "postglacial uplift."

In portions of Fennoscandia (Fig. 2 ) , as well as in the Great Lakes-Hudson B a y region, the land is rising at a rate which has its m a x i m u m of the order of 1 m . / c e n t u r y at present in the center of the respective formerly glaciated area, decreases t o w a r d zero near the b o u n d a r y of the former ice sheet, and b e c o m e s slightly negative b e y o n d .

I n nearly all of the extensive literature concerning F e n n o s c a n d i a3 1 , 3 2 , 33

it is assumed that the c o n t e m p o r a r y uplift is mainly a consequence of the removal of the ice sheet. A c c o r d i n g t o D a l y^{3 3a} this had a m a x i m u m thick-

2 6 N . R i c k e r , Geophysics 18, 10 (1953).

27 H . Jeffreys, " T h e E a r t h , " 3rd ed., p . 40. M a c m i l l a n , New Y o r k , 1952.

2 8 H . Jeffreys, Monthly Notices Roy. Astron. Soc., Geophys. Suppl. 1, 286 (1925); see also "Operational M e t h o d s in M a t h e m a t i c a l P h y s i c s , " Cambridge Univ. Press, L o n - don and N e w Y o r k , 1927.

2 9 F . Press and M . Ewing, Trans. Am. Geophys. Union 29, 163 (1948); see also M . Ewing and F . Press, Bull. Seismol. Soc. Amer. 42, 315 (1952).

3 0 H . Menzel, Geophys. Prospecting 2, 139 (1954).

3 1 Β . Gutenberg, Bull. Geol. Soc. Amer. 52, 721 (1941).

3 1a D a t a for Oslo, Stavanger, Bergen, and N a r v i k t o 1952 furnished b y H . S. Jel- strup, Norges Geografiske Oppmâling, Geodet. Afdeling, Oslo.

32 H . Renquist, in "Suomi, A General Handbook of the Geography of Finland,"

40. 1952.

3 3 A. Siren, Fenniald, 1 (1951); Fennia 72; M . Sauramo, Ann. Acad. Sei. Fennicae, Ser. A. I I I . 44 (1955)

3 3a R . Daly, "Our Mobile Earth," p . 189. Scribner, New York, 1926.

414 Β . G U T E N B E R G

F I G. 2. Present uplift in Fennoscandia in centimeters per century based on records of tide gages ( G u t e n b e r g³¹ and recent data from N o r w a y^{3 1}* ) and on lake water levels in Finland ( S i r e n^{3 3}) . (Drafted by Mr. John M. Nordquist.)

ness of at least 2000 m . over the northern B a l t i c ; other estimates are as high as 3000 m . T h e ice sheet began t o melt, slowly at the beginning, a b o u t 40,000 years a g o . A b o u t 9000 years a g o , when the ice was reduced t o patches northwest of the Baltic, the land near the former center of glaciation had risen between 250 and 300 m . A n uplift of a b o u t 260 m . has been added there since. Figure 3 indicates that for the past 8000 years the flow was N e w t o n i a n with a fair approximation. Calculations assuming a viscous process lead t o the conclusion that the center of the area is still a b o u t 200 m . b e l o w its equilibrium position. N e g a t i v e gravity anomalies increase t o a b o u t —40 mgal. as the center of the uplift is a p p r o a c h e d .^{3 5}"³⁸ H o w e v e r , effects of tectonic disturbances which are superposed u p o n those p r o d u c e d b y the postglacial mass deficit result in irregularities in the distribution of

3 4 F . Nansen, Avhandl. Norske Videnskaps-Acad. Oslo, K l . 1, N o . 12 (1928).

3 5 W . Heiskanen, Trans. Am. Geophys. Union 32, 524 (1951).

3 6 E . Niskanen, Ann. Acad. Sei. Fennicae, Ser. A 53, N o . 10 (1939).

3 7 R . A . H i r v o n e n , Veröfftl. Finn. Geodät. Inst. N o . 24 (1937).

3 8 F . Model, Mitt, geogr. Ges. Hamburg 49, 92 (1950).

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H ' S I N T E R I O R 415

-6800 -8000 -5000 -4000 -2000 0 2000

F I G . 3. Depression of the center of glaciation in Fennoscandia since 6900 B.C. and rate of uplift, based on data given b y N a n s e n .³⁴ {Drafted by Mr. John M. Nordquist.)

gravity anomalies. I t is therefore n o t possible t o d e c i d e on the basis of the gravity anomalies h o w m u c h of the present mass deficit of a b o u t 300 or even 400 m . indicated b y the g r a v i t y data for the center of uplift³⁹ is a remainder of that mass deficit, caused b y the melting of the ice.

T h e p h e n o m e n a related t o the uplift in northern N o r t h A m e r i c a corres- p o n d t o those observed in Fennoscandia. H o w e v e r , all data are less plenti- ful and d o n o t p r o v i d e results of the same a c c u r a c y as those found for Fennoscandia. T h e m a x i m u m thickness of the N o r t h A m e r i c a n icecap is estimated t o h a v e been betwen 2 and 3 k m . S t a n l e y⁴⁰ points out that along the east coast of H u d s o n B a y " s o u t h of latitude 55°50' N o r t h neither the islands nor the highest hills of the coastal zone are high enough t o h a v e recorded the marine limits, and a b a n d o n e d beaches m a y b e found u p t o their s u m m i t . " T h e center of the uplift is p r o b a b l y in the southern part of H u d s o n B a y , and its m a x i m u m a m o u n t seems t o exceed 300 meters. T h e m o s t reliable findings for the present uplift in northern N o r t h A m e r i c a are based on records of tide gages in the Great Lakes. I n F i g . 4 selected rela- tive changes in sea level at pairs of tide gage stations are reproduced. T h e y are calculated from differences in five-year means of the observed water

3 9 W . Heiskanen, in " H a n d b u c h der G e o p h y s i k " (Gutenberg, e d . ) , V o l . 1, p . 938.

Borntraeger Berlin, 1936.

4 0 G . M . Stanley, Bull. Geol. Soc. Amer. 50, 1936 (1939).

416 Β . G U T E N B E R G

τ 1 1 1 1 10.8

F I G . 4 . R e l a t i v e change in level (in feet) at pairs of gages in the Great L a k e s .^{3 1, 41} Insert: L o c a t i o n of these gages and lines of equal uplift in centimeters per century, after G u t e n b e r g .³¹ {Drafted by Mr. John M. Nordquist.)

level at pairs of gages. F o r each gage the measurements refer t o an indi- vidual arbitrary reference point and for each pair the difference in the first five-year period is taken as zero.

Unfortunately, there is only one tide gage in the H u d s o n B a y with rec- ords covering a time interval long enough for use in calculations of sea-level changes. I t is a t Churchill, M a n i t o b a , and even there data exist for o n l y 24 years. M o r e o v e r , adverse meterological conditions during a large part of the year, as well as ice which piles u p during the winter and does n o t dis- appear until late in M a y or during June, m a k e it advisable t o use tide-gage data for Churchill for the m o n t h s July t o September only. A p p l i c a t i o n of the m e t h o d of least squares t o the yearly sea level averages for the months July t o September 1928 t o 1951 gives an average rate of 1.05 ± 0.18 m . / c e n t u r y for the uplift at Churchill.⁴¹ Six-year averages for the sea level at Churchill during July t o September give the following values relative t o an arbitrary zero p o i n t :

1 9 2 8 / 3 3 1 9 3 4 / 3 9 1 9 4 0 / 4 5 1 9 4 6 / 5 1

+ 5 =fc 4 - 8 =h 5 - 1 1 d r 4 - 1 5 ± 6 c m .

T h e change from year t o year is rather iregular, b u t this is true also for the changes of sea level in Fennoscandia. A part of the irregularities results

4 1 B . Gutenberg, Arch. Meteorol. Geophys. u. Bioklimatol. 7, 2 4 3 ( 1 9 5 4 ) .

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H ' S I N T E R I O R 417

L^ ^ 5 2eW 4 8 · " — — 44·

i D i s k o ^ ^ * / /

.69° N ^ ^ r ^ ^b s;h α v n j

10·. 3

N L

. 100 KM ,

Ρ-72·Ν-3

4 Ν

Ii

D - 3 KM

» ? :

^ Χ

Λ * ^ \ Ρ ^

0 100 200 300 400 1 I I «

500 600 700 800 KM F I G . 5. Profile t h r o u g h Greenland showing r o c k surface after J o s e t⁴⁴ on the basis of seismic e x p l o r a t i o n . (Drafted by Mr. John M. Nordquist.)

from the fact that the sea level (or lake level) at a given coast depends rather strongly on meterological conditions, especially precipitation and winds during a given year. In addition, the process called "postglacial uplift" does n o t only include uplift in the formerly glaciated area b u t also subcrustal flow from the surrounding regions which correspondingly show a sinking. T e m p o r a r y blocking of the subcrustal currents which control the process must be expected t o slow d o w n or stop the uplift in portions of the affected area from time t o time. E a r t h q u a k e s³² p r o v e the existence of o c - casional increase in strain in and a b o u t areas of postglacial uplift.

F r o m all evidence it follows that the uplift in N o r t h A m e r i c a proceeds in time and space similarly t o that in Fennoscandia and that in these t w o areas the physical constants i n v o l v e d in the process are of the same order of magnitude. T h e r e seems t o be n o justification⁴¹ for the belief which has been expressed repeatedly that the present uplift in northern N o r t h A m e r i c a is n o longer related t o a mass deficit as a consequence of the past glaciation, but has mainly a tectonic cause.

As soon as the uplift in Fennoscandia was found t o be a result of the de- crease in the ice load, it was pointed out that regions which at present are heavily loaded b y ice, e.g., Antarctica and Greenland, should show a cor- responding depression of the rock surface. Nansen expected the thickness of the ice in central Greenland t o be of the order of 2 k m . Alfred W e g e n e r

418 Β . G U T E N B E R G

considered the measurement of the ice thickness in Greenland b y seismic m e t h o d s (recording of echoes from explosions) one of the m o s t important tasks for his expeditions t o Greenland, which ended with his death there.

T h e measurements indicated a rapid increase in ice thickness with increas- ing distance from the coast. A b o u t 40 k m . inland at points roughly 200 k m . northeast of D i s c o (Fig. 5, left t o p ) with the ice surface a b o u t 1570 m . a b o v e sea level, an ice thickness of 1200 m . was found. H o w e v e r , the fur- ther result that in the central part of Greenland the r o c k surface is not far from sea l e v e l ,⁴² was considered t o b e d o u b t f u l⁴³ until it was definitely con- firmed b y the French expedition, Missions Paul-Emile V i c t o r⁴⁴ (Fig. 5 ) . T h e fact that the r o c k surface in centra] Greenland forms a basin under the ice load furnishes a strong support for the t h e o r y of post-glacial uplift.

T h i s uplift is p r o b a b l y the viscous process for which observations cover the longest time interval and the widest area.

V . The Coefficient o f Viscosity a n d the S t r e n g t h in the Earth

T h e coefficient of viscosity η in the earth's upper few hundred kilometers has been calculated on various assumptions. Usually it is supposed that η does n o t change with depth in the layers i n v o l v e d . Haskell⁴⁵ has given a formal solution for the m o t i o n of a highly viscous fluid when a symmetrical pressure is applied at the surface. H e starts with the p r o b l e m of the m o t i o n of a semi-infinite, incompressible, viscous fluid (viscosity η, density p ) under the action of a radially symmetrical load of radius r applied at the time t = 0 u p o n the free surface. H e neglects the curvature of the earth as well as inertial terms in the equations of m o t i o n in comparison with those arising from viscosity. T h e equations of m o t i o n (rate V, stress p) in a gravi- tational field g are

77 V²V = grad ρ - (0, 0, pg) and d i v V = 0 (25) with the positive 2-axis directed d o w n w a r d . In cylindrical coordinates (r, ζ, Φ), setting ρ = ρ — pgz, the c o m p o n e n t s of stress in which w e are interested are

V» = - ( P + m) + 2r, ^ and p„ = V

(j£ + ?~)

A t the free surface, prz = 0, and pzz equals the applied stress; at infinity,

4 2 B . B r o c k a m p , E . Sorge, and K . Wölcken, "Wissenschaftliche Ergebnisse der Deutschen Grönlandexpedition Alfred Wegener, 1929 und 1930/31," V o l . 2. Brock- haus, Leipzig, 1933.

4 3 B . Brockamp, Z. Gletscherkunde 23, 277 (1935).

44 A . Joset, " E x p é d . Polaires Françaises P. E . V i c t o r , " 8er. Sei. 15, (repp, prelim.) 43 (1952); A. Joset and J. J. Holtzscherer, Ann. géophys. 10, 351 (1954).

4 5 Ν . W . H A S K E L L , Physics 6, 265 (1935) ; 7, 56 (1936).

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H ' S I N T E R I O R 419 stresses and velocities are zero. If the equation of the actual free surface is ζ — f (r, t), and if the plane ζ = 0 is taken at the undisturbed surface, the values of the quantities in equation (26), except for pgz, m a y be replaced b y their values at ζ = 0. If the applied pressure is —a(r> t), the b o u n d a r y conditions b e c o m e

p ( r , 0, / ) + « t f ( r , 0 - 2n = <r(r, t) (27)

\ OZ / ( r . O . O

(% + ψ) = 0 and g - F . ( r , 0 , 0 (28)

\ dz dr /z=o at

I n t r o d u c t i o n of functions for Vr, Vz, and p leads t o Bessel equations which are solved b y Haskell. H e has applied the resulting formal equations t o the subsidence of a cylindric b o d y . I n this case a(r, t) is zero for all values of r > r0 and all times t < 0, and constant otherwise. H e has then calculated the change in f o r m of the viscous half space under the load of a cylindric b o d y as well as under the load of an infinite long strip with parallel sides.

Haskell also has applied his results t o the viscous recoil of the earth after the disappearance of the Pleistocene ice on Fennoscandia. His results indi- cate that as a consequence of the melting of the ice load appreciable flow processes must affect m o s t of the earth's mantle. A t a depth equal t o the radius of the original (cylindrical) ice load (order of magnitude 500 k m . ) , the vertical displacement should scarcely decrease under Haskell's assump- tions, and at t w i c e this depth it is a b o u t half that at the surface ( m a n y hundred meters).

On the basis of N a n s e n ' s³⁴ results for the uplift in Fennoscandia Haskell finds, on the assumption that the coefficient of viscosity η is constant throughout the mantle, the following values for η

Starting y e a r 5000 B . C . 4000 B . C . 3000 B . C . 2000 B . C .

η 2.6 Χ 1 0²¹ 3 . 2 Χ 1 0²¹ 3 . 0 Χ 1 0²¹ 2 . 9 Χ 1 0²¹ poises A t t e m p t s t o consider the decrease of viscosity with d e p t h⁴⁶ or the effect of the strength in the upper 5 0 ± miles of the earth, as well as use of m o r e recent data concerning the u p l i f t^{3 1 , 47} h a v e resulted in values between 1 0²² and 1 0²³ poises for the coefficient of viscosity at depths roughly between 50 and 200 miles. T h e corresponding value of the time of relaxation [λ, equa- tion ( 4 ) ] is of the ordet of 8000 years.

T h e r e are other m e t h o d s t o get information a b o u t the value of η inside the earth. O n the basis of the well-established and repeatedly confirmed fact that there is n o phase displacement between the tidal forces and the

4 6 F . A . Vening M e i n e s z , Proc. Koninkl. Akad. Wetenschap. Amsterdam. 40, 654 (1937).

4 7 E . Niskanen, Isostatic Inst. Helsinki I. A. G. Puhl. N o . 20 (1948).

420 Β . G U T E N B E R G

observed tides of the earth's b o d y in excess of the errors of observation ( 5 ± m i n . ) , S c h w e y d a r⁴⁸ in 1912 has concluded that there can b e n o layer in the mantle of a thickness of even 100 k m . with a viscosity η of 109 poises (more than that for pitch at normal temperature) nor a layer with a thick- ness of 600 k m . and a viscosity of 1 0¹³ poises ( a b o u t as viscous as i c e ) . D a t a on the Chandler m o v e m e n t of the poles (variation of latitude) h a v e been discussed b y P r e y .⁴⁹ T h e earth's axis rotates t o a first approximation along the surface of a cone with variable vertex angle. T h i s causes devia- tion of the poles from their average position b y a m a x i m u m a m o u n t of 10 m., and a corresponding variation of latitude, w h i c h depends on the earth's viscosity. If this were zero, the earth's figure w o u l d change immediately t o fit the position of the earth's axis. T h e greater the viscosity the smaller would be the damping effect. H o w e v e r , n o accurate result for the earth's viscosity can be found from these observations. On one hand, the change of viscosity with depth is n o t k n o w n ; on the other hand, in the case of the b o d y tides as well as that of the polar m o v e m e n t , the period of the m o t i o n (order of magnitude one d a y and a b o u t 420 days, respectively) is small compared with the time of relaxation ( p r o b a b l y λ ^ 1 0 0 ± years) c o n - nected with the viscous process, so that observations of these phenomena can furnish at best a lower limit for the time of relaxation λ and the coeffi- cient of viscosity η. In addition, the interference of a yearly period (forced vibration) in the variation of latitude (related t o meteorological p h e n o m - ena) with the free period of the g y r o s c o p i c m o t i o n of the earth's axis of a b o u t 420 days results in a periodic (7-year =h) increase and decrease of the amplitudes.⁵⁰ T h i s renders v e r y difficult⁵¹ the finding of the d e c a y in ampli- tudes and the determination of λ, and m a y result in mistaking a decrease in amplitude from interference of these t w o major periods for a d e c a y as a consequence of viscosity. Finally, effects p r o d u c e d b y a redistribution of angular m o m e n t u m on or within the earth h a v e t o b e c o n s i d e r e d .⁵² ( F o r further discussion of these problems, see refs. 52a (which does n o t consider some of the difficulties discussed a b o v e ) , 5 2 b , 52c, and 52d.)

M o s t data for the b o d y tides (for details, see ref. 8) are based on pendu- l u m observations. A l r e a d y in 1919 S c h w e y d a r has p o i n t e d o u t that on a viscous earth the p e n d u l u m m o t i o n should precede the corresponding theo-

4 8 W . Schweydar, Veröffentl. preuss. geodät. Institut [ Ν . F.] 54 (1912).

4 9 A . Prey, Gerlands. Beitr. Geophys. 56, 155 (1942). ·

5 0 W . Schweydar, Sitzber. preuss. Akad. Wiss. Physik. Math. Kl. 20, 357 (1919).

5 1 H . Jeffreys, " T h e E a r t h , " 3rd e d . , p . 244. Macmillan, New Y o r k , 1952.

5 2 T . G o l d , Nature 175, 526 (1955).

5 2a P . R u d n i c k , Trans. Am. Geophys. Union 37, 137 (1956).

b 2b A . M . Walker and A . Y o u n g , Monthly Notices Roy. Astron. Soc. 115, 443 (1955).

5 2c B . Gutenberg, Nature 177, 887 (1956).

5 2d P . J. Melchior, Atti accad. nazi. Lincei, Rend. [8] 19, 137 (1955),

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H ' S I N T E R I O R 421

retical motion calculated for a purely elastic earth b y an amount a which, for the semi-daily tide, is given b y

cot a = i * ü (29) 2gr0pm

where η = rotational v e l o c i t y of the earth, η = viscosity, g = gravity, r⁰ = radius of the earth, and pm = its mean density. F r o m the fact already mentioned that the phase difference a is zero within the errors of observa- tion of a few degrees it follows that the coefficient of viscosity in the outer part of the earth is greater than 1 0¹⁷ poises.

In instances of viscous m o t i o n of the earth as a w h o l e or of relatively large portions of it, it is frequently accurate enough t o estimate the time of relaxation λ for the respective process. Calculations m a y appear t o give a higher a c c u r a c y , b u t usually other processes than that under considera- tion p r o d u c e effects greater than that resulting from errors in the estimated value of λ. H o w e v e r , details related t o the order of magnitude of displace- ments during the flow and of changes with depth can be estimated only o n a theoretical basis.

T h e r e are other processes which require the assumption of viscous flow w i t h o u t permitting at present an estimate of the time of relaxation or of the coefficient of viscosity. A n indication for the existence of viscous flow in the earth's crust is given b y " i s o s t a s y . " T h i s is an expression for the fact that g r a v i t y anomalies at the earth's surface are relatively small except for narrow belts where earthquakes or other evidence indicate a c t i v e tec- tonic processes. Findings of E w i n g⁵³ and his g r o u p for the P u e r t o R i c o D e e p and of R e v e l l e and his collaborators for the T o n g a D e e p^{5 4} b y means of ex- ploration with elastic w a v e s from artificial explosions indicate d o w n w a r p i n g of lighter material t o relatively great depth (order of 3 0 ± k m . ) , which could result in negative g r a v i t y anomalies of the observed order of magnitude—

100 t o m o r e than 200 m g a l . — o v e r such deep trenches.

Unfortunately, various errors are introduced in the reduction of g r a v i t y observed at the earth's surface t o values w h i c h w o u l d exist at sea level;

there is neither agreement on the best m e t h o d of reduction nor on some quantities i n v o l v e d in the calculations, especially the change in density with depth. H o w e v e r , the fact of relatively small resulting gravity anomalies (usually less than 30 ± milligals) in recently undisturbed areas, regardless of whether the measurements are taken in m o u n t a i n areas or over the ocean, requires the assumption of viscous flow at a depth of the order of 100 k m . or even less under stresses as small as 1 0⁷ or 1 0⁶ d y n e s / c m .² T h e resisting strength which must be surpassed t o permit viscous flow at these depths,

5 3 M . E w i n g and F . Press, Handbuch Physik 27, 256 (1956).

5 4 W . R a i t t , R . Fisher, and R . M a s o n , Geol. Soc. Amer. Spec. Paper 62, 237 (1955).

422 Β . G U T E N B E R G

therefore, cannot exceed 1 0⁷ d y n e s / c m .²; it m a y be m u c h smaller. It is not k n o w n at what depth in approaching the surface the strength increases t o the order of 10⁹ d y n e s / c m .² which is found from the fact that up t o 10-km.

high mountains h a v e not shown an observable decrease in height b y viscous flow during history. T h e transition can b e expected b e l o w a depth of a b o u t 10 k m . where the velocities of elastic w a v e s and the elastic constants begin to decrease.⁴ Jeffreys⁵⁵ estimates that the strength for mountain support should exist at least d o w n t o a depth of 20 or 30 k m . On the other h a n d , experiments of Griggs indicate that under stresses continuing for years—

and still m o r e for geological periods—any a m o r p h o u s or p o l y crystalline substance m a y flow even if the stresses are smaller than the strength. W i t h other words, for the material investigated b y Griggs strength is dependent u p o n the time during w h i c h a stress acts. M o r e such experimental or t h e o - retical data are needed for drawing of conclusions concerning the earth's interior,⁵⁶ especially since there are n o data for the crystalline rocks form- ing the outer layers of the earth's crust.

V I . Flow Processes in the D e e p e r Portions o f the Earth's M a n t l e ; Results B a s e d o n D e e p - F o c u s E a r t h q u a k e s

A s w e g o deeper into the earth the investigation of rheological p r o b l e m s must be based on an increasing n u m b e r of assumptions. Effects of deep- seated processes on observations m a d e at the earth's surface are scarce, and the physical conditions at depth are n o t well k n o w n , as has been pointed out in Section I on the structure of the earth. D o u b t a b o u t the possibility of viscous flow across the whole mantle is expressed b y B i r c h .⁵⁷ H e points out that the observed, rather rapid increase in seismic w a v e velocities at depths between a b o u t 200 and 800 or 900 k m . b e l o w the surface m a y indi- cate a zone of transition t o a high-pressure f o r m of the material and possibly also changes in chemical c o m p o s i t i o n . V e r h o o g e n⁵⁸ disagrees with s o m e of these conclusions. If Birch is correct, circulation between 200 and 900 k m .

" m i g h t b e limited b y the requirement of an exchange of latent heat, p o s - sible irreversibility of phase changes and intrinsic differences of density.

It is n o t clear at present h o w serious any of these effects m a y b e , b u t t o assume unimpeded circulation in this region is t o ignore its definitely tran- sitional n a t u r e . "⁵⁹

Viscous flow, especially in c o n v e c t i o n currents, plays an i m p o r t a n t role in m a n y tectonic theories, but, as mentioned, all basic data are v e r y un-

5 5 H . Jeffreys, " T h e E a r t h , " 3rd e d . , p . 192. M a c m i l l a n , N e w Y o r k , 1952.

5 6 D . T . Griggs, Trans. Am. Geophys. Union 32, 505 ff. (1951).

6 7 F . B i r c h , J. Geophys. Research 57, 227 (1952).

5 8 J. V e r h o o g e n , Geophys. Research 58, 337-346 (1953).

5 9 F . Birch, Trans. Am. Geophys. Union 32, 533 (1951).

R H E O L O G I C A L P R O B L E M S O F T H E E A R T H ' S I N T E R I O R 423 certain and i n c o m p l e t e (see, e.g., A d a m s et al., Gutenberg, Meinesz, N a d a i ,⁶³ S c h e i d e g g e r ,⁶⁴ all with m a n y additional references). H e a t d e v e l o p e d in physical-chemical (especially radioactive) processes is usually c o n - sidered t o b e the source of energy.

A few geophysicists believe that viscous flow in the mantle of the earth is impossible since otherwise earthquakes caused b y rupture along fault sur- faces d o w n t o a depth of a b o u t 700 k m . could n o t exist. H o w e v e r , the time of relaxation for viscous flow is of a higher order of magnitude than the time needed in active earthquake regions for accumulation of strain t o the break- ing p o i n t . In m o s t active belts relatively large shocks follow each other at intervals of the order of one hundred or even ten years. Consequently, the strain which, in such belts, accumulates from tectonic processes should n o t be reduced appreciably b y viscous flow in the outer portion of the earth's mantle with a time of relaxation of the order of thousands of years during the relatively short time intervals between successive deep shocks.

T h e fact that n o deep-focus earthquakes have been observed b e l o w a depth of a b o u t 700 k m . since instrumental records m a k e it possible t o fix their depth—that is, since a b o u t 1905—may either result from a rather rapid decrease in the effects of disturbing processes near this depth or from an increase in speed of flow there, great enough t o prevent an accumulation of strain. I n the latter case, λ w o u l d b e of smaller order of magnitude than 10⁸ sec. ( a b o u t 30 years) b e l o w 700 k m . , and, correspondingly, η w o u l d decrease b e l o w an order of magnitude of 1 0²⁰ poises at a depth of a b o u t 700 k m . O n the other hand, the existence of earthquakes in the upper 700 k m . of the mantle indicates that a b o v e a depth of 7 0 0 ± k m . η p r o b a b l y exceeds 1 0²⁰ p o i s e s .^2a

V I I . S t r a i n - R e b o u n d Characteristics o f E a r t h q u a k e Series a n d Aftershocks I n recent years B e n i o f f^{6 5 - 68} has devised a procedure for determining the relative size of elastic strain-rebound increments from earthquake magni- tudes and has applied it t o series of aftershocks of earthquakes, as well as

6 0 Colloquium., Trans. Am. Geophys. Union 32, 499 (1951).

6 1 B . Gutenberg, ed., in "Internal Constitution of the Earth (Gutenberg, e d . ) , " p . 186. D o v e r Publications, New Y o r k , 1951.

6 2 F . A . Vening-Meinesz, Proc. Koninkl. Akad. Wetenschap. Amsterdam B55, 527 (1952).

6 3 A . N a d a i , Trans. Am. Geophys. Union 33, 274 (1952).

6 4 A . E . Scheidegger, Bull. Geol. Soc. Amer. 64, 127 (1953).

65 H . Benioff, Bull. Geol. Soc. Amer. 60, 1837 (1949).

6 6 H . Benioff, Bull. Seismol. Soc. Amer. 41, 31 (1951).

6 7 H . Benioff, Trans. Am. Geophys. Union 32, 508 (1951).

6 8 H . Benioff, in "Internal Constitution of the E a r t h " (Gutenberg, e d . ) , p . 391.

D o v e r Publications, N e w Y o r k , 1951.

424 Β . G U T E N B E R G

t o series of earthquakes in certain regions and at certain depth ranges.

H e assumes that an earthquake is generated b y the elastic rebound of a v o l u m e of rock having an average rigidity G and an average elastic strain y preceding the rupture. T h e energy stored in a unit v o l u m e of the rock is Gy2/2. Benioff assumes that the conversion efficiency of strain energy into seismic w a v e s is unity, and t h a t consequently 7 is the strain rebound.

T h e square r o o t of the energy J is then proportional t o 7 . F o r a sequence of shocks occurring on a given fault, a graph of the accumulated sum of the increments J⁰'⁵ p l o t t e d against time represents the elastic strain-rebound characteristic (times the constant factor C) of the sequence. F o r a first approximation, values for J⁰'⁵ m a y be derived from Richter's instrumental earthquake magnitude M b y means of the tentative e q u a t i o n⁶⁹

log J⁰'⁵ = a + bM. (30)

Originally Gutenberg and R i c h t e r used a = 6, b = 0.9, b u t later they pre- ferred a = 4.5, b = 0.9; r e c e n t l y6 9* ' ^{6 9b} t h e y h a v e revised the equation.

T h e resulting value of log J⁰*⁵ m a y well b e incorrect b y one unit in large shocks, b u t this does n o t seriously affect the following conclusions of Benioff.

M o s t strain-rebound characteristics of aftershock sequences show for certain time intervals either a form as given b y

Si =

or as given b y

£ 2 = 0 — Ne~PVI (32)

where Z, m, Ο, Ν, Ρ are constants of the process. Si is the strain-rebound given b y G r i g g s⁷⁰ for rocks and b y others for other materials under c o m - pressionalelastic or recoverable creep strain. S² is of a f o r m derived empiri- cally b y M i c h e l s o n⁷¹ for shearing creep r e c o v e r y . H o w e v e r , this form has not been verified b y subsequent experimenters. Unpublished creep tests in shear b y C . L o m n i t z under D r . BeniofFs supervision h a v e yielded only the form ( 3 1 ) . C o n s e q u e n t l y the occurrence of form (32) in aftershock se- quences remains unexplained. In the aftershock sequences of the form of equation ( 3 1 ) , activity followed for intervals varying from 100 days mini- m u m t o 600 days m a x i m u m . On the other hand, sequences having the dual form came t o early definite terminations in intervals varying from 1 t o 500 days. F o r example, the aftershock series (Fig. 6) of the K e r n C o u n t y ,

6 9 B . G u t e n b e r g and C . F . R i c h t e r , Bull. Seismol. Soc. Amer. 32, 163 (1942).

6 9a B. G u t e n b e r g and C. F. R i c h t e r , Bull Seismol. Soc Amer. 46, 105 (1956).

6 9b B . G u t e n b e r g and C . F . R i c h t e r , Ann. geofis. (Rome) 9, 1, (1956).

7 0 D . T . G r i g g s , J. Geol. 47, 225 (1939).

7 1 A . A . M i c h e l s o n , J. Geol. 25, 405 (1917); 28, 18 (1920).