Proc. Nat. A cad. Sci. USA Vol. 68, No. 6, pp. 1111-1113, June 1971

The Magnetic Field Induced by the Bodily Tide in the Core of the Earth (dynamo theory/coupling coefficient) C. L. PEKERIS Department of Applied Mathematics, The Weizmann Institute, Rehovot, Israel Corninunicated March 15, 1971

ABSTRACT The motion in the liquid core of the earth The coupling term V X H in (1) generates combination due to the bodily tide can induce a periodic magnetic field having the frequency a of the tide as well as multiple fre- frequencies, including a steady term of zero frequency. quencies, including a steady term. The coupling coefficient The periodic components of H will not be observed at for the steady term between the convectively inducing the surface of the earth because of damping of the field and induced fields is estimated to be of the order of crH2/X, where H denotes the height of the equilibrium tide, and by conduction in passing through the mantle [3]. The only X = 1/4K7rK, K denoting the electrical conductivity of the component of the magnetic field induced by the bodily core. With a = 1.4 X 10-4 sec-', H = 20 cm, and K = 3 X tide that would be observed at the surface is the one of 10-6 emu, the coupling coefficient comes out only of the order of 10-6, as against unity in the case of the dynamo zero frequency. In the homogeneous dynamo theory theory. [3 ], the steady field is visualized to be maintained through the convection by a process of bootstrapping. At present we do not possess a satisfactory theory of In the process envisaged here, the steady magnetic the origin of the earth's magnetic field, if by satisfactory field does not feed on itself, but on the periodic terms. we mean one that matches in simplicity the nature of For an assumed conductivity K of 3 X 10-6 emu, the the field itself: a simple dipole (to within 20%). The velocity required by the dynamo theory [3 ] is estimated dynamo theory, originally proposed by Larmor [1] to be of the order of 0.1-0.01 cm/sec. In the case of and more recently revived by Elsasser [2], Bullard [3] the 1\M2 bodily tide, the displacement is of the order of and others, attributes the field to convective motion in the equilibrium ocean tide of about 20 cm. With a fre- the fluid core of the earth, which acts as a self-exciting quency of u = 1.4 X 10-4 sec', this gives a velocity dynamo. Without going into the controversy as to of the order of 3 X 10-s cm/sec, which is close enough whether a steady dynamo is possible [3 ], or whether one to the lower limit cited above to merit examination. must assume the motion in the core to be turbulent [2 ], it is clear that the case for the dynamo theory rests at THE FIELD EQUATIONS best on the arguments for the existence of the respective Let the magnetic field be represented by [4], convection currents in the core, which, though plausi- ble, still fall short of a convincing proof. In this in- Hr = KE YE (3) vestigation, I examine the possibility of a steady mag- 6 netic field being induced by the motion in the core due Heo= E M 1i ¢,6 + Lo6 ab]' (4) to the bodily tide. The equation governing the magnetic field H is [4] H Th~=Z[-M~'3+=A[Ma ay + Law bY] (5) OH/bt = XV2H + curl (V X H), (1) sin 0 6T where V is the velocity vector due to the tidal motion, where K, L, and M are functions of r and t, and and Ye= Pfl,'3(cos 0) [cos m#O, sin m,]. (6) X = 1/47rK, (2) Because of the condition K denoting the electrical conductivity. The induction takes place primarily in the liquid core, where K is V.H = 0, (7) estimated to be of the order of 3 X 10-6 emu (about we must have one-third of the conductivity of mercury), compared to a much lower value in the mantle of the earth. 3(3 + 1)L, = rKe + 2Kg,, (8) Now V has the period of the tide, say 12 lunar hours the dot denoting differentiation with respect to r, and in the case of the principal M2 tide, and, consequently, the factor p3(3 + 1) signifying n3(n, + 1). Eqs. (3), (4), H will also vary periodically with the tidal frequency. (5), and (8) are the most general representation of a 1111 Downloaded by guest on September 25, 2021 1112 Geophysics: C. L. Pekeris Proc. Nat. Acad. Sci. USA 68 (1971) vector satisfying (7) [5]. The r-component of (1) be- We shall not assume the material in the liquid core of comes the earth to be incompressible, so that the divergence r sin + of the velocity is not taken to be zero, as is done in the OE [(OK#/t) XK#*]YI dynamo theory [2, 3]. - (6/60) [sin O(urHo - uoH)] In contrast to the velocity field, which is of a single frequency only, the magnetic field given in (3), (4), and - (a/a4) (uH, - uH,) (9) (5) will also contain the harmonics of a-:

where co K, = E [Kkccos kaot + Kk' sin kcrt], (18) K =n -{(d2K/jdr2) + (4/r)(dK,6/dr) - k=O [(32 + A - 2)/r2]K,}. (10) and likewise for the functions LB and MO. Substitution If we multiply (9) by Yy, and integrate over the surface of (15) into (11) gives of the sphere, we get, after some partial integrations, rN,,[(blbt)Ky + XK,*] = Em, (VKj6F.iay an equation for the poloidal field a 6 - UaL#FaiFy + (19) rATN[(bK./zt) + XKY*] = UaMi5Eapy), o2 r where f2o 'rdOE$Hr[u,(Yy/bq1) Fapf do f dOaY [sin 0 ( Y#/60) (. Y,/10) + sin Ou0(bYK/b) I - ur[HO,(bYj/b) + sin OHeY(bY/b))]}, (11) + (1/sin 0)(a6)(aY 16)] (20) r2N' with Ease f2do d Ya[(aY/60)(a /60) Ny = f dfs sin 0d0Yl2. (12) (FcYc/xp)i(ano/fe- au (2i1)) In a similar manner, one can derive from the 0 and q F.a-, can be expressed in terms of the Gaunt integral components of (1) an equation for the toroidal field: Ga{RyGt1= J do snOdOYiesarpY (22) rNy(y + 1) ROM7/t) + XM7,*] = and there exists a relation fd2 df { y(y + 1)(ueH,,- uHo) Fa,67 + F6aY = ya(y + 1)Gap-j. (23) - (6/6r) [r(uHe - uOH,) (aYr/a46) Similarly, (13) becomes + r(ulH^ - uH,) sin @(Y7/60) ] }. (13) rN,,y(7 + 1)[(6/at)M7 + XM.,*I = Here Ah I -y(-y + + a 1)VaMpFiay VaLj6Eap-y Mar* = {-Mt - (2/r)M ,, + [hy(ky + 1)/r2]M.}. (14) - (b/br) [r(VaMqFa,,y + UaLlsEapy I in the first instance, only the largest shall consider, + (24) of the components of the bodily tide, namely the lunar VaKXEa0y)]}. semidiurnal term. Let the velocity components in the. Let bodily tide be given by r = bx, 0 % x S 1, (25)

Ur = E Ua Yay U9 = E Va(a/a0) Ya) where b, the radius of the core of the earth, is equal to a a 3.47 X 108 cm. Using the representations (16), (17), and Uo E (V,/sin 0) (a/0) Ya a (15) (18), and equating each Fourier coefficient in (19) to zero, we get with 2N7X(r2KO*) = bxEj[(VCKjelC Ua = [Uac(r) cos at + Ua8(r) sin at], (16)a + Va8Kils)Feay - (UacLeic + Ua8Lji')FaOy Va = [Vac(r) cos at + Va (r) sin at]. (17) + (UacM.i1C + Ua'Mpj8)Eafry]j (26) Here a = 1.401 X 10-4 sec-' denotes the frequency of the M2 tide. If we disregard the effect of tidal loading 2N-,,[b2x2crKY, + X(r2K.Yi*C)] = and of the earth's as is done in the classical rotation, bxEZE3{ (2Va'cKpo'C + Va'K02 + Va8K,023)Feay theory of the bodily tide, then the sums in (15) consist a ft of a single term only, which in the case of the M2 tide is - (2UaCLsOOc + UaCL02j + UaSL328)Fajgy Ya = P22(cos 0) [cos 2X, sin 24]. + (2UccMOoC + UaCM,02C + UaSM028)Ea.jy (27) Downloaded by guest on September 25, 2021 Proc. Nat. Acad. Sci. USA 68 (1971) Magnetic Field Induced by Earth's Core 1113

2Nj [-b22- Kc + X(r2KI*s)] of the order of unity. Since, with an assumed electrical conductivity K for the core of 3 X 106 emu, bxuE>3 (2VasK,60C - VaSKc2C + Va¢K62)Fz a of X = 2.7 X 104 cm2/sec, X/ab2 = 1.6 X 10-9, (32) - (2UUa8L,6' - UaSL,62C + UaCL623)FacIy it follows that the X-terms on the left-hand sides of (27), + (2UasMjfoc- UaSM62C + UaCM,628)Ea,,j}. (28) (28), (30), and (31) can be neglected in comparison The corresponding equations for the toroidal field, with the respective first terms. In the right-hand sides stemming from (24), are of 'he above equations, we can neglect all but the K7o and M.0 terms, since it is clear that + 1)(r2M o*) = 2XANT7y(y K-sk+l~-VKykl/bo = (H/b)K7,yk (33) bxZjaI-7y(y + 1)(VacM6ic + Va8M6I8)Feca7 ar 6 where H denotes the height of the equilibrium tide and is of the order of 20 cm. It follows from (27) that + (VaCL,6lc + Va8L,618)Eapy - (/lx) [X(Vac M81,c + VasMils)Faioy + X(UaCLilc + Ua8Loi8)Eao, bxK1l (V/lo)Ko, (34) + x(VacKoic + VasKi8)E.,Oy]}, (29) and, hence, by (26), 2N-y(-y + 1) [ablx2M,1s + X(r2Mlc*)] = -2N~,)\[X2(d2Kzo/dx2) + 4x(dKeo/dx) bx>E t -Ye)' + 1) (2VaCMeoc + VacMj2c - (92 + -y - 2)Kzo] _ (V2/a)Kyo = uH2Kyo. (35) a ft The coupling coefficient between the convectively in- + VaSMj628)Fsa7 + (2VaCLjpoC + VacLjP2c ducing field and the induced field is thus, at most, of + VaSL923)Ea,6, - (a/ax) [x(2VcMpoc the order of + V CM,,2C + Va8M,2')Fay + x(2UaCcLpoc VH/X uH2/X - 10-6. (36) + U acL,62c + Ua8Lf28)Ea,3, + x(2VcKooc In the case of the dynamo theory, on the other hand, + VaCK,02C + Va8Ke28)E,,],yIII (30) the coupling coefficient is of the order of (Vb/X), so that a velocity of 10-s cm/sec suffices to bring it above 2N7'yY( + 1)[-ob2X2M71c + X(r2M Yi*)] = unity. ZZ{'-,y(- + 1) (2V.SMjC V-3M02C In order for the field to be self-maintaining, the coupling coefficient has to be of the order of unity. As + VaCM.2s)F,,.gy + (2VaSsL6oc- VaSLif2C an illustration, consider the case of the main dipole + VacL,028)EalY - (b/lx) [x(2VS8M,6OC field of the earth, for which y = 1 and Y, = Pi(cos 0). Let (35) be represented by - VaSM#2C + VCM,62S)Fj6Y + x(2UaSiLoc - UASL2C + UaeCL62S)Ea,.Y + x(2VasK~0c x2K + 4xk = -E2X2K) (37) - VaSKjj2C + VaCKfl23)Ea~,]} (31) of which the solution iS JS/2(EX)/x312. At x = 1 we must have (T/K) = -3, and this condition requires that THE COUPLING COEFFICIENT e = T. As seen from (26) and (29), the steady components of This research was supported by the ONR under Contract the magnetic field, Kyo and MY0, are determined en- N00014-66-C-0080 and by the NSF under Grant GA-12386. tirely by the first harmonic components, Ky1 and M71, 1. Larmor, J., Rept. Brit. As8oc., 159 (1919); Electr. Review, 85, while the latter, in turn, depend on the zero-th and 412 (1919). second harmonics, and, in general, K7,kis determined by 2. Elsasser, W. M., Rev. Mod. Phys., 22, 1 (1950). 3. Bullard, E. C., "Geomagnetic Dynamics", in The Nature of K7,,k+1 and K-,,,k. An order-of-magnitude estimate of the Solid Earth, ed. E. C. Robertson (McGraw-Hill, New the terms in this system of coupled equations may now York, 1971). be made. First, let us assume that the scale of variation 4. Bullard, E. C., and H. Gellman, Phil. Trans. Roy. Soc., 247, of the 213 (1954). functions K,(r) is of the order of the radius b of 5. Lamb, H., Hydrodynamics (Cambridge University Press, the core. In that case, the function (r2K*) is, by (10), 1932), p. 594. Downloaded by guest on September 25, 2021