The Shrinkage of the Earth’s Crust.
April 1924.
I49
The Shrinkage of the Earth’s Crust t h o u g h Dirnmisluhg
Rotation. By R. Stoneley.
(Received 1924 March 19.)
I . Introduction.
It has been pointed out by Dr. J. W. Evans * that a cause qualita-
tively competent to produce mountain ranges is l o be found in t h e
diminution of centrifugal force consequent on the slowing down of t h e
earth’s axial rotation. If the potential 4p2(z2+yz) of the centrifugal
forces be split u p in the custoiuary manner into t h e two terms
~ ~ ~ ( ~ ; 2 +and
y ~- +
& w~z ( z~9 )- x2 - y2), t h e former will give rise to
purely radial displaceinents i n a spherically symnietrical earth, and
so will produce a change of surface area, whereas the latter term
occasions a spherical harmonic deformation of the second degree,
which can easily be shown to leave the surface area unchanged to
the first order of small quantities. It is the object of this paper to
examine the effect of the radial term.
2. Homogeneous Earth.
It has been found by JeKreyst tliat the surface of the earth
approximates very closely to the hydrostatic form ; this strongly
suggests that, except possibly in the outermost layers, the stress system
is very nearly that of a rotating liquid in relative equilibrium, that is, the
only stress is a hydrostatic pressure. As the a~igularvelocity diminishes,
this compressible, effectively fluid interior will contract, arid the adjustment of tho outer surface will give rise to niountain rauges.
Suppose then that the deiisity p,, of the earth were initially a
function only of t h e distance yo froni t h e centre ; then on the application of forces derived from a potential J ~ J an
~ velement
~
origi~iallya t 7*0
would move to 7‘- ro + u. W e sliall takc the suffix (G) as referring always
to the undisturbed configuration. By considering a shell originally
comprised between 7*o and r,,+&,), the dilatation is found to be
Let R be t h e body force due to gravitation, and
t h a t arising from the rotation potential.
initially and finally are
The hydrostatic conditions
(2)
(3)
* Geogiaphicctl Joum.,58, No. 3, Sept. rgzr, 213.
t
Memoirs R . A . S . , 60(1g1s), 193.
150
Mr. R. Stoneley, The Shrinkage of the
1.
4,
Now the increase of volume of the element is conditioned by the
difference between the pressure in the undisturbed position arid tlie
pressure i n that now occupied, that is p ( 7 * ) - po(y0),which must equal
- - k ( ~ , ) . A, where k(r,) is the incompressibility. Tlie equation (3) niay
be written
in which
since t h e term depending on w is supposed small compared with that
arising from initial stress.
The form of k and R, depends 011 the law of density : for an earth
of constant density rrl and~ncompressibilityX., the equation beconies
The required solritioii of (6) must vanish a t yo = 0,and also when w = 0.
At tlie free surface, A = 0 .
Writing now yO=n,t, u = a j ; where n is the external radius of the
undisturbecl earth, n 2 = 1 6 i ~ y u , ~ n ~ / 3 tlie
k , , equation (6) reduces to
It can readily be verified that a particular solution of this is C t , where
c = - ~ ~ / 8 7 7 y. c ~ ~.
.
(8)
For t h e cornpleinentary functions, tlie solution that is not infinite a t the
centre is
where H is a n arbitrary constant.
conditions W B find
Taking into account the boundary
April
1924.
Earth's Crust through Diminishifig Rotation.
1.51
The solution (9) corresponds with that obtained for an elastic solid
by Love.* To see how the particular integral would arise in h i s
treatment, it is necessary to modify t h e quantity W and write
in place of his equation ( I I ) (Zoc. cit., p. 92). It can then be shown that,
subject t o the above-mentioned boundary conditions, we again obtain
equation (8). It may also be noted that our boundary condition is
derivable from that of a n elastic sphere, by putting p = o in
AA
+ 2p-dU
= 0.
dr
Adopting t h e values k,= 7 x 1 0 dynes/cni.2
~ ~
(see section 4) ;
7'294 x I O - ~ rad./sec. ; a=6*378 x 1o8 cm.; y=6'658 x 10-8 C.G.S.
units ; = 5-6 gms.lcm.3 we find the surface value of a , to be 0.59 kni.,
and thus the effect of stopping the earth's rotation gradually, so that
progressive adjustment could occur to the hydrostatic state, would be to
shorten every great circle by about 3.7 kiii.
W=
3. Il'e!llecf oj* Gr~auitation.
If we neglect the change in gravitational attraction
011
a particle
as i t iuoves t o its final position, and suppose that the only effect of
gravity is to determine tlie liydrostatic pressnre, the problem is greatly
simplified, and heterogeneity of density and inconipressibilit-j can be
taken into account. If t h e change of pressnre in the neighbouri~oodof
a point distant 'i' from the centre is 8p (the distinction between r and T~
has now been dropped) a shell of thickness 61. becomes conipressed by
an amonnt 4 w 2 . 6 r . 8p//c, and if 1) is known as a function of i', the
approximate diminution of volunie may be found by direct integration.
Since for small changes of the radius the increase in lengtli of a great
a ~ t h e increase in volume, an estimate niay be easily
circle is ~ / z times
made of the distentional cffect of rotation.
With the data of the preceding section, the change in volunie is
8a(+,w2a5/45kl,which is equivalent to n change of the perimeter by
3.0 km. The order of magnitude of the result is therefore not affected
by neglect of the change of the gravitational attraction on a particle as
i t moves from its initial to its final position.
4. Heterogeneous Earth.
IViechert's hypothesis t concerning the distribution of density within
t h e earth has m e t with fairly wide acceptance as an approximate description of the state of the earth's interior, and i n our present ignorance of
the law of density variation this hypothesis will be accepted for warit of
a better one. The earth is supposed to consist of an outer shell of density
* Geodynamics, p. 104,equation (58), withp=o.
t Qottiiige?~Nachrichten, 1897, p. 229.
152
M r . R. Stoneley, The Shri~kageof the
1.
4,
3 2 , enclosing a core of density about 8.2 and radius equal to 0'78 of t h e
earth's external radius. If the earth were formerly molten, i t is impossible
t h a t tlie outer sliell sliould now be of constant density in view of the
lrnown fact that its material is compressible; even if the earth were
built up by continuous aggregation of planetesimals we should have to
assume that, as tlie outer layers collected, the density of the matter
arriving progressively increased in such a way tliat the equilibrium
dcnsity of tlie sliell mould be 3.2 throughout.
If tlie density is known, the clastic constants can be fouiid from a
knowledge of the velocities V and U of tlie primary and secondary
earthquake waves. These have been determined by l i n o t t * for points
a t a distance from the earth's centre greater than 0.4 of the radius.
W i t h i n t h a t part of the Wiechert core where Knott's wave-velocities are
linown we find, from the forniulle
X+zp=pV2;
p=pu2
.
*
(1)
t h a t the elastic constants h + 2p and p are very nearly constant. Since
tlie incompressibility is equal to h $p, its value throughout the core
will be taken as 8.38 x 1o12 dynes/cm.2
In the crust, i t is found that when V2 or U2 is plotted against tllc
distance from the centre i n eartli-radii, tlie grapli never departs by
more than z per cent. from a straight line. W e shall, accordingly,
nssiinie the elastic constants i n this region to vary linearly with 6,and in
tliis way we obtain, approximately, k = (9'63 - 8.666) x 101~. I t happens
conveniently, but most probably qiiite accidently, that both primary and
~econdnrywave velocities, after increasing steadily a s the distance froin
t h e surface increases, suddenly become approximately constant in tlie
neiglibourhood of the Wiecliert surface of discontinuity, [= 0.78.
It is easily verified that the additional preswres in core and crust
a r e respectively
- &w2[u(b2- 1.2) ip(u2 - b2)] and - &z[p(an where p = 3.2, U = 8.2 .
. (2)
+
@)I,
O n substituting appropriate values of It and carrying through a piece of
quite stritightforwarcl integration we find that the iniposition of t h e
radially acting forcive 3 w z r causes a stretcliing of every great circle by
3 . 2 km.
It should be nientioned that tlie value of k used for the calculations
of sections z and 3 was a weighted mean of the values found above.
5 . Comparison with other
Causes.
I t has been found by Jeffreys t that shrinkage through cooling would
shorten every great oircle of the earth by about 1 3 3 km. The contraction we have just found is proportional to tlie square of the angular
velocity. Accordingly, if we suppose the earth to slow down from a
* Proc.R.Sf.E.,39,Part2(rg18), 1 5 7 e t s c l .
32 (1916), 575-591.
t I%il. Mag.,
April 1924. Earth’s Crust through Diminishing Rotation.
I53
Iz-hour day to the present 24-liour day,* every great circle will contract
by about 10 kni., provided time has elapsed for plastic flow to reduce the
stress to t h e hydrostatic value. Thus, diminishing rotation is a niucli
less potent cause of the elevation of mountain ranges than is cooling, but
is sufficiently important to be considered as a large correction to the
cooling theory.
One possible source of error should be pointed out. The values of
A + zp arid p clerived,frorn equations ( I ) of section 4 depend essentially
on phenonie~iawliosu period is of the order of 40 seconds. For such,
the rigidity u. is high ; yet i t is assumed that for phenomena whose period
is very long the rigidity is zero. In the foregoing work the modulus
of incompressibility was calculated as h + +p, and i t was assumed that this
quantity, although written in a form involving p, remains constant for
all tinie. This slight difficulty could be met by using as the fundamental
elastic constants the quantities Iz arid p, as is sometimes done. T h e
velocity of propagation of waves of compression then depends on k + j p ,
which mould, with h constant, diminish with the time. W e are certainly
justitied in inquiring if k is actually independent of the time. If it
diminishes, the rotational effect will be greater.
6 . Homogeneous Elastic fiartli.
The analysis used by Love t for a gravitating compressible planet
subjected to n harmonic deforniation may be extended, as in 5 2 , t o the
case of L: disturbing potential fw2r2, and t h e dilatation A i s then given by
A particular integral of this equation is A = - 3O ~ / S X ~ Uand
~ , the
solution, finite a t the centre, of the corresponding equation with the
term on the right equal to zero is A sin n [ / n l where n is now equal to
~d
~d
-(rO2u) = -(12f), we
Since A is equal to
1 6 a y u ~ ~ a ~ /+3 zp).
(h
roz dr,
tzdc
have,
sin n( - nt cos rnt - _l a_
st
f= u / a = A
. (2)
n3f2
8nyal ’
‘
no additive constant being required since the displacement must vanish,
as does (z), a t the centre.
The condition that the bounding surface is free from traction is
AA
+ z pdf
a=
0,and
by substitution in this equation the constant A may
be found ; on reduction we have
.
-
* This would be the retardation if for the last
1000 million years tidal friction
had been as potent as now. Compare H. Jeffreys, Proc. Roy.. Soc., Ser. A.. 100
(I92I), 129.
t. LOC.c i t . , p. 103 ; the
“initial stress” is taken as a state of hydrostatic pressure.
I2
154
1117.
R. Stonetey, T h e XhriPzkage of the
I.
4,
and at t h e surface
where (1 = n2/(1- n cot n).
If w e adopt t h e values of the elastic constants derived froni Knott's
data coinbined with Wiechert's hypothesis, we shall have for the crust
X+zp=(16.414'75t). 1oI2, p=(5'078-+-571.$. ~ o ~ ~ , for
a n thecore
d
A + z p = 1 3 - 4 7 x 1o12, p = 3'82 x 10l2 dynes/cn?.
d weighted niean iriay
be taken for the whole earth, and gives A + z p = 1 1 - 2 3 i( 10'2 and
p = 3.20 x 1o12clynes/cai2. With these values i t is found that the radial
displacement of the surface is about 97 per cent. of what i t was in the
correspoiiding problem for a liquid earth.
The radial and tangential stresses P and Q are given by
and the stress-difference will be t h e absolute value of
2pA
~-
n3$
{ ( 3 - +<2) sin n<- 3nt cos nE}
0 - P, i e. of
.
Now A has been chosen so that P vaiiislies at the surface, and
.
(2
(6)
is
T h e surface value of Q reduces to
and with the assumed values of t h e elastic constants 4 = 2 . 7 5 , so that
Q is positive in the outer layers, corresponding to a tangential tension.
If then t h e earth, in hydrostatic equilibriiim a t solidification, slows
down subsequently, a state of tangential compression will be set up i n
tlie outer layers.
Taking the value of (CJ - P) given by ( 6 ) , i t is easily seen that when
4 is sinall t h e quantity i n the bracket becomes 2n512/15 , which is positive. and tends to zero with t. The stationary values of the stressdifference are given by the roots of the equation
tan nE=
n3t3- 9116
+n'1<2
-
9
L e t this equation be written tan # =f(#),where 9 = nE. Then i t will
b e seen from the graphs of tan # and f(#) that the only root within the
range # = o to 1.4is at the centre (4 = 0 ) . W i t h the adopted numerical
April 1924. Earth's Crust through Diminishing Rotation.
1-55
data n is I ' I o j radians. It can be inferred, moreover, that Q-Y
increases froni t h e centre to the circumference, the increase being hardly
appreciable until the outer layers are reached. As a further test, we
niay compare t h e value of
(Q - P) there.
d
( Q - P) a t t h e surface with the value of
d4
These two quantities would be the same if the rate of
increase were constant; the actual value of the ratio of the two is
1'85, implying t h a t the stress difference increases more rapidly in the
outer layers.
Thus the effect of the imposition of the force of 8~1%
on an elastic
earth would be to set up small tangential tensions in the crust; if these
were large enough to cause fracture, radial cracks would couiuence
at the surface and proceed inwards much i n the same way as in the
initial stages of a solidifying earth." Conversely, if the earth's rotation
diminishes, t h e stress-dilference (Q - P) becomes negative and a stilte
of stress is developed which, if great enough, mould cause the outer
layers to separate off' as concentric shells, 80 that a t the weaker places of
the crust mountain ranges would he formed, with much overthrusting.
The value of the stress-difference corresponding to a retardation
froni a 12-hour day to a 24-hour clay will he ( - 3) times that set u p b y
the imposition of the present angular velocity, and the data already
used give for the surface value 3'2 x 1 0 9 dynes/cm2. The strength of
hard surface rocks is about 1 0 9 dynes/cm2.
I t is seen, then, that the niaximuni stress-differciice is only about
three tinies as great as the strength of surface rocks. If, as is possible,
rocks may flow for a considerably smaller stress-difference provided it
be rnaintained for a very long time, this putative cause of mountainbuilding may have a moderate effect even if the earth be treated as an
elastic solid rather thaii as a fluid for long-period forces. Nevertheless,
the preceding analysis suggests that the additional stresses set u p by
cliniinisliing rotation have not been such as t o cause any considerable
cruiiipling iii the surface of an elastic earth whose outer layers are of a
strength comparable with that of granite.
For the actual earth, these results inay he greatly iuodified when
heterogeneity of density and elasticity are taken into account, and the
crunipliiig of surface rocks will in part depend 011 the maximum stressdin'erence supportable in the astlienospliere. I t appears then that the fact
that an elastic solid can withstand shearing stresses will entail a considerable departure from what has been found for a fluid earth, so much
so that arising froni the cause postulated there may be little or no
niountain-building.
I n conclusion, I wish t o express iny thanks to lh. H. ?Teffreysfor
his helpful criticism.
* Cf. Jettreys, PWL.Bog. Soc., A., 100 ( I ~ z I ) ,143.