CONVECTION CURRENTS IN THE EARTH.

Transcription

1 372 Mr. A. L. Hales, 3, 9 These numbers are probably lower than the truth, as only those Fellows have been counted whose interest is positively indicated by their occupation, which is not always given or otherwise known to the present writer ; naval officers and Fellows of the Geological and Royal Meteorological Societies arc included in the second class in the absence of other information. The Geophysical Supplement is received by 243 Fellows and Associates of the R.A.S., by 63 subscribers and by 276 institutions on the free and exchange lists. The number of members of the Society (Fellows and Associates) is 948 (1935 December). There are still many active geophysicists in the country who are not Fellows of the R.A.S., and many geophysical papers are still published in less specialized journals that would be seen by more persons interested and more easily found if they came to the Supplement. At present two numbers of the Supplement normally appear each year, but the work actually done would probably fill four or five. Meteorology and geology are not considered in these remarks, as they are adequately provided for by two of the co-operating societies. It is hoped by many that a time will come when a geophysical paper will come to the Geophysical Supplement as naturally as an astronomical one comes to the Monthly Notices or a mathematical one to the London Mathematical Society. The possible formation of a Geophysical Society, either as a separate body or as a section of the R.A.S., has been discussed at intervals, but no steps have been taken. The present system is peculiar, but on the whole works very well, and Geophysics has much reason to be grateful to the Royal Astronomical Society for its parental care. H. J. CONVECTION CURRENTS IN THE EARTH. A. L. Hales, B.A., University of the Witwatersrand, Johannesburg. (Communicated by Dr. Harold Jeffreys) (Received 1935 December 12) I. In discussing the Continental Drift hypothesis Prof. A. Holmes * has suggested that there are steady convection currents in the outer shell of the earth, i.e. outside the dense core. The currents are supposed to be maintained by a supply of heat in excess of that which can be carried away by conduction. If this exceeds a certain limit, largely determined by the viscosity, there will be cellular convection currents ; but if it is too great further disturbances will arise. The latter are not considered in this paper. Dr. H. Jeffreys suggested that it was necessary to investigate whether the theory of convection currents in the outer shell is consistent with the fact that the gravity anomalies show that there are deep-seated stress differences of the order of 5 x 107 dynes/cm.2.t * A. Holmes, Mining Magazine, April-June t H. Jeffreys, The Earth, 2nd edition, p. 20r.

2 1936 April Convection Currents in the Earth 373 The stress differences depend on the magnitude of the departure from the undisturbed symmetrical state, and on the viscosity. For any temperature gradient there is an upper limit to the viscosity if convection currents are to exist, and it follows that for stress differences of the order of 5 x 10' dynes/cm.2 there will be a lower limit to the disturbance. The heat transfer due to the convection currents depends on the square of the disturbance. The question is, therefore, whether the supply of heat necessary to maintain convection currents sufficient to give a disturbance greater than the lower limit referred to above is reasonable. Magnitude of the Instability 2. If the supply of heat be such that the gradient required to carry it off is very much greater than the gradient which just produces instability, - - then the heat transfer across a plane will be cpwv - k- per unit area, where az c is the specific heat, k the thermal conductivity, V the absolute temperature, w the velocity in the z direction, and the bar denotes the average over a large area. av In the body of the fluid - can only be slightly greater than the gradient az for instability, and the greater part of the transfer of heat depends on the first term. At the boundaries w + 0, and therefore K- must be equal to az the whole rate of supply of heat. This means that the solution of the equation (2.1) will not fit at the boundaries where - is considerably greater az than the critical temperature gradient. We shall find the magnitude of the disturbance necessary in order that in the middle of the fluid the heat transfer due to the first term should be of the same order as that due to conduction with a temperature gradient 8" A. per cm. If we substitute V=Pz+ V', P being the temperature gradient in the steady state, the equation satisfied by V' is * where K is the thermometric conductivity, a the coefficient of thermal expansion and Po the adiabatic temperature gradient. Substituting 2nx h 2ny V' = Z sin - sin -, h 2-4, wb b2 I I -@=s+rp' * H. Jeffreys, Phil. Mag.,2,833, 1926, and Proc. Camb. Phil. SOC., 26, 170,1930. \ IC av av av

3 374 MY. A. L. Hales, 31 9 where Z is a function of 5 only and h is the depth of the fluid, the equation (2.1) reduces to where {(Dz- I ) + ~ (p2 + I)~}>Z = 0, (2.2) and The equation was reduced to the form (2.2) by A. R. Low,* and solved in the form 2 = az cos p5 + a,(eqcc+ e-qc3 cos $ + as(eqcc - e+3 sin 4 where hence Also or + a, sin p5 + a3(eqcc - e-q.3 cos 4 + a,(eqd+ e-q.3 sin +,!f =P4 +3P2 +3, -pz + I tan zy = 437, P + 3 c =cos y, s =sin y, * = qs5. In deriving the equation (2.1) it is found that W= w( B - Po) = K vz v ; m2b2 2xx 27ry (B - B O P z(dz - I)Z sin - sin -. A P (2.3) where pc and pa are respectively the density midway between the boundaries, and the density in the steady state. If we substitute for p, V and w in the expression for the heat carried 2xx 27ry 2xx 27ry across a plane, the terms in sin - sin -, and sin3 - sin3 - disappear A P A P 2TX. 2xy in the process of averaging. The averaging of terms such as sinz - sinz - x P introduces a factor a. The heat carried across a plane Z=constant is If the origin be taken midway between the boundaries, and the boundary * Proc. Roy. SOC., A, 125, 180, 1929.

4 1936 April Convection Currents in the Earth 375 conditions be those of no slip, we need only consider the even terms in the solution given in (2.3). It is found that * p = I.285, b = , (p2 4- I)*TPb4 = =-= a2 a ' For the plane midway between the boundaries the solution (2.3) is accordingly Z =a2 + 2u4 = 1.122~~ ; DZ=O, D22= ~,+4651a a, = ~~ ; The transfer of heat due to convection is thus Z(DZ) =o, (2.6) Z{(D2 - I)Z} =2.798az2. (2.7) The transfer due to conduction with a temperature gradient 6" A. per cni. is KP~CS. Equating these we find a, = 1.1 x IO%+(/?, taking h = 2900 km. With this value of a, we find that taking K =O.OI c.g.s.u. w =3-2 x IO-~O((B~ - /3)-W cm./sec. (2.10) Stress Di@rences 3. The stresses are expressed in terms of the velocities by the equation The principal stresses are given by the roots of the cubic in h found by expanding the determinant I I pij - haisij I I. Since we are concerned only with the differences between the principal stresses, we may neglect the symmetrical part of pi, in writing down the determinant. * A. R. Low, loc. cit.

5 376 Mr. A. L. Hales, Therefore we take au av aw P Z X = 2PF ax Pull = 2P-, Pz, = 2P-. ay ax 3, 9 In deriving the equation (2.1) it is found that Substituting for V and w, aw FU av =aka2v. ax ax ay 2u av Krr2b2 a--- n-b 2n-x 2ny D (D2-1)Z sin - sin -. h(b - Po) ) h tl Assuming symmetry between the x and y directions, we find u= -~ 2h2 ZT. 2ny h 2n-X (02 - I>z cos - sin --, h 2n-x 2n-y sin - cos --. h h Considering the values of the stress coefficients in the plane midway between the boundaries, we find 2n-X 27ry - /3)W sin - sin -. p,, =p,, = x 10-~~p(19~ h h Pz, =o- 2TX 2Ty p,, = x 10-~~p(P~ - /3)hSk cos - cos -, h h 2TX 2n-y p,, = x 1o- *p(/3~ - P)-*8$ cos - sin --. h h 2TX 2n-y p,, = x ~o-l~p(p~ - fi)-*sh sin - cos -. h h / (3.7) Since (Po- 8) is at most of order IO-~ A. per cm., it follows that the important stresses are p,, and p,,. The cubic equation for the principal stresses is then and the maximum stress difference h(h2 - Pv: - P*2) = 0, The maximum value of this is 1-3 x 10-~~p(fl,, - fl)-64. In order that convection currents should exist, 2- (3.8) i.e. v < x JO~O(B -Po),

6 1936 April Convection Currents in the Earth 377 taking a =2 x IO-~ c.g.s.u. Substituting this value of y in the expression for the stress difference, we find that-the stress difference must be less than 4.1 x 1012 x (Po- P)W, taking p =4. In order that the stress differences should be of order 5 x 10' dynes/cm.8, 6(fi0-P) must be greater than 1-5 x 10-l~ A. per cm. With a critical temperature gradient 10-6 A. per cn. in excess of the adiabatic, the heat carried up by convection would need to be equivalent to conduction by a temperature gradient of 1.5 x 10-8 A. per cm. If the heat carried upwards by convection be greater than this value, the stress differences will be greater than those necessitated by the gravity anomalies. Gravity Anomalies 4. Dr. G. Vening Meinesz * has painted out that in consequence of the variations in temperature due to the convection currents there would be variations of density in the shell, and that therefore positive gravity anomalies are to be expected over regions where the current is moving downwards, and negative anomalies over regions where it is moving upwards. He considers that this explains the consistently positive anomalies found in his observations of gravity over the oceans. The anomalies due to this cause would be of the same sign over distances of h/2, i.e km. The residuals on isostatic reduction have been found to keep their sign over distances of km. only, so that all residuals are not accounted for in this way.t We shall evaluate the anomalies on the assumption that the horizontal temperature distribution is the same at all depths and equal to its value in the plane midway between the boundaries, i.e. 2TX 279) V' =1.12a2 sin - sin -. h P We require the attraction due to the distribution of matter of density p'. Suppose that the potential due to the distribution of mass is +2 inside the matter and dl outside. Then v%$, = 0, (4.4 or y being the constant of gravitation. Substituting and +8 equal to functions 2TX 27ry involving sin -. sin -, we find that the solutions are h P 2wx. 27ty +1 =(Ale-<+ A#+3 sin - sm -, (4.3) h P * Lecture at the University of the Witwatersrand. f H. Jeffreye, The Earth, p. 201.

7 378 Mr. A. L. HaEes, 3, 9 27rx. 2ny +* = (A$-< + A4e& - As) sin - sm -, h L1 where h as before, and A, - 497Ypaaha x I.12U4. 7rb 7rsbs 27rx 2ny (4.4) - A, sin - sin - is the particular integral of equation (4.2). +1 must h c c I be of order - at infinity and therefore As =o. Z The density distribution is symmetrical about the plane 2 PO and therefore All =A,. The boundary conditions are that + and - a+ are continuous at z =h/2, 7rb 82 i.e Therefore or 2 The attraction at the surface a+ 2 = h/2 is - i.e. ad 7rb -- a+ h at' 7rb -e 27rx 2ry --Ale 2 sin -sin -. h h c c The maximum value of the attraction is then With the value of as given in paragraph 2 this reduces to 3.9 x 1058Q0 - /3)+ c.g.s.u. Physical DisMcssion of the Results 5. In paragraph 3 it was shown that in order that the stress differences should be of the order of 5 x 107 dynes/cm.a, a(&, -p) must be greater than 1.5 x 10-l~. Vening Meinesz * has found that the mean anomaly for a number of ocean stations in the Atlantic is cm./sec.' after isostatic reduction. The maximum anomaly, which is the one calculated in paragraph 4, would be considerably greater. Taking the maximum value of the anomaly to be 0.06, we hd that 8(p0 - /3) must be less than 2.3 x Thus it is possible to satisfy the conditions for both the stress differences and the gravity anomalies by a suitable value for S(po - p). With * G. Vening Meinesz, Geog.gownal, 71, ,1928.

8 1936 April Convection Cuwents in the Earth 379 this value of 6(p0-/3) there is considerable latitude in the values of 6 and (Po - P) taken independently. Dr. Jeffreys has pointed out that it is unlikely that the value of 6 should be only a fraction of /3, for this would mean that the deep-seated heat supply would have to be sufficient to maintain the critical temperature gradient and yet not exceed this value by more than a small fraction. The following table gives corresponding values of 6, (Po - /3) and Y :- Bo -B 8 Lower limit from stress differences 8 Upper limit from gravity V From (3.3) x x x xoy IO-~ 1.5 x 10- I 2.3 x I O-~ 8.1 x I O ~ x x x I O ~ x x x IO ~ is approximately equal to 3 x IO-~ A. per cm. numerically. 6 must therefore be appreciable compared with 3 x IO-~ A. per cm. The values given in the last two rows satisfy this condition and also that implied by the persistence of the variation of latitude and the absorption of seismic waves that in the shell the viscosity is greater than 5 x 10 0 c.g.s.u.* Holmes supposed that the currents were maintained by a deep-seated layer of radioactivity. Jeffreyst has pointed out that this is inconsistent with the fact, shown by Holmes for the Finland granites, that there is a tendency for radioactive material to rise to the top. It seems doubtful whether the assumption of a deep-seated layer of radioactivity is necessary. If the conductivity in the core is greatly in excess of that in the shell, then even with the lower temperature gradient the heat brought to the lower surface of the shell by conduction would be more than could be carried away by conduction through the shell. This supply of heat would therefore maintain the convection currents in the shell. The conductivity of the core is probably sufficiently large for this to be possible. Jeffreys has also suggested that since the pattern of the instability is always changing there would be no uniformity of stress over continental areas and for geological periods of time. The time necessary for an appreciable change of pattern will be large compared with the time for a complete cycle which is of the order of IO~O years, i.e. is far greater than a geological period of time. In conclusion my thanks are due to Dr. Jeffreys for suggesting the problem, and for advice and criticism. Summary The idea of convection currents in the outer shell of the earth is investigated, and it is found that currents sufficient to give the stress differences required by the gravity anomalies are possible. * The Earth, p t EmthqrcakRI and Mountains, p. 170, 1935.