Non-steady Bullard- Gellman Dynamo Model (2)

Transcription

1 JOURNAL OF GEOMAGNETISM AND GEOELECTRICITY VOL.20,N0.4,1968 Non-steady Bullard- Gellman Dynamo Model (2) Tsuneji RIKITAKE and Yukio HAGIWARA Earthquake Research Institute, University of Tokyo (Received July 26,1968) Abstract In contrast to the non-steady states of a Bullard-Gellman dynamo model as studied in the previous paper ignoring the reaction to the fluid motion of the mechanical force of electromagnetic origin, a few examples of non-steady B-G model are studied taking the equation of motion coupled to the induction equation into account. The zonal flow is approximated by rotation of a rigid sphere in actual calculation. Computational difficulties are proved so serious that no results analogous to reversals of the geomagnetic field or the like are obtained. But it seems likely that the model is stable for a small disturbance. 1. Introduction Study on the non-steady Bullard-Gellman (B-G) dynamo model (Rikitake and Hagiwara, 1968), in which a constant velocity field was assumed, will in the following be extended to cases for which the reaction due to the mechanical force of electromagnetic origin exerted to the fluid is taken into account in an approximate way. 2. Equation of motion The writers would here like to set out an equation of motion of the B-G dynamo on the assumption that the fluid motion can be approximated by a rotation of a rigid sphere. It is certainly not possible to make such a simplification if one wants to deal with the detail of fluid motion. But we are here mostly concerned with the pondermotive force arising as a result of the interaction between inducing and induced fields. In such a case a rigid-sphere approximation might be workable. At any rate it appears to the writers that no other practicable way of tackling a non-steady problem of the B-G dynamo would easily be found even if we use a high-speed computer available in Tokyo. Let us assume that the Tl motion is represented by a uniform rotation of which the angular velocity is denoted by Q. If we further assume that a sphere of radius al is rotating in an infinite conducting medium (conductivity: a) in the presence of a uniform magnetic field Ho which is parallel to the rotation axis, the pondermotive force is calculated as r(t)=aai5ho(t)2u(t, where U(t)FSQ+IS2I(p)I52(ka1)K52(ka1)dp

2 416 T.RIKITAKE and Y. HAGIWARA in which L denotes Bromwich's path of integration and Sl=o+, where p denotes time-operator Fl/at. 15/2 and K512 denote modified Bessel functions. (1) is obtained on the assumption that the sphere starts rotating at t=0 from a state of constant angular velocity Q0. The details of deducing (1) and (2) may be found in articles by the writers (Rikitake, 1962,1966; Rikitake and Hagiwara,1966). Denoting the moment of intertia of the sphere by C and the driving couple by G, the equation of motion of the sphere is then given as C--G-P.a U(t) as defined by (2) is transformed to U(t)=Sdo+ISd(t-z)(z)dz where cct1it1exn(2rria2tlli9(2rcia2tdt t that has been used here is not dimensionless. We are now in a position to seek the relation between the T1 motion introduced by equation (9) of the previous paper and the angular velocity. Assuming a uniform rotation, it is clear that the following relation should hold: U=4warts. It is obvious, however, that such a motion does not give rise to a T2 magnetic because the term involving QT on the righthand-side of the second equation of (11) of the previous paper identically vanishes. A T2 field is to be excited only at the slipping surface of the rotating sphere though it diffuses into both the stationary and rotating conductors. Bullard and Gellman (1954) computed the radial distribution of the steady T2 field for a number of prescribed velocities. Since most of the maxima of the T2 field intensity occur at r= , we may take al/a=0.8 or thereabouts so long as we confine ourselves to an approximation with a rotating rigid sphere. A typical radial distribution of QT may then be assumed as field Um=4raa(1-r)M where e is a constant. In reality, the radial distribution of velocity should be ever changing in a non-steady dynamo. But this point is forced to be ignored by assuming (8). As for the uniform axial field, i.e. Ho, we may assume that Ho is represented by the radial field on 6=0 at the core boundary, so that Hi(t)=2S(ta)

3 Non-steady Bullard-Gellman Dynamo Model(2) 417 We hope that the order of magnitude of the electromagnetic couple would be approximately correct in spite of the above simplification. Judging from the variations of S1 with r as have been seen for various epochs of the B-G dynamo models in which the velocities are prescribed (Figs.1,2 and 3 of the previous paper), the assumption does not seem utterly unreasonable. The effect of the core boundary on the distribution of the T2 field and so on F is ignored in the present estimate. Although this seems to result in an over-estimate of F, it would not be worthwhile trying hard to make an exact estimate because the whole argument is based on a very crude model. In this respect, the reaction or pondermotive force exerted to the eddies represented by the S22c motion may as well be disregarded because the moment of inertia, although no such quantity can be defined in an exact sense, for such eddies is much smaller than that considered for the T1 motions. We may, as a typical case, therefore assume the following relation: QS=4rrca2r3(1-r)2S2, which is the same radial distribution as that studied by Bullard and Gellman (1954), 3. Non-dimensional form of equations The equations for magnetic induction as given by (11) of the previous paper hold good although they are not repeated here. Let us put In that case all the quantities involved in the equations of magnetic fields and motion are made dimensionless. The equations can be written as r2=oral2viol-2h+y1qs/g4

4 418 T.RIKITAKE and Y. HAGIWARA Y(t)=Gn5fh0(t)I2u(t) u(t)=w0+wu)c(t)+i1-y(z)c(t-t)dz Ir(r)ilr. hn(t)=g1l1(1.tl atr,t)=giver(i-r)w(t). a0(r.t)=avr(17)zc(t). where H=(4wa)1 p(t)=-=1t-lexp(-nlt)jc9(nvzt)dt Yl,=ll,Ill- It is therefore seen that ho, hl, h2, h3, h4i r, u, w,q and qs can be determined by solving equations from (12) to (21) together with proper initial and boundary conditions. 4. Initial and boundary conditions Let us think of a case in which the motion starts at t=0 with an angular velocity jump from wo to wo+w(0). At that time the magnetic fields are zero except the Sl field which may be assumed as uniform. The initial conditions are then written as w=w+w(u), h=cr2 h=h1=hd=0. (att=0) Ignoring the conductivity of the mantle, the continuity condition of magnetic field components at the core-mantle interface leads to Ollyc)+h l(rt)=o (atr=1 h2=h3=h4=o Outside the conducting core, it is evident that the magnetic potential is written as W(t)=121(1,t)rF1(cosd)

5 Non-steady Bullard-Gellman Dynamo Model (2) Steady state Going back to (1) and (4), it is clear for a steady state that the following relations should hold: IC=oalHOCQ( G=r where subscript c means that quantities to which it is attached are at the steady state. The first relation of (28) can readily be obtained because 15i2(kal) K5/2 (kal) tends to approach 1/5 as k--o. (27) leads to JU 8TCQa513n2 which can be in the dimensionless form rewritten as 1VW1 4+wC=n-r Meanwhile (9) reduces to hoc=2hicr=i Adopting the steady value of the non-dimensional angular velocity as given by (30), qs and qt are determined as functions of r. As wo+wis inversely proportional to h0c2, (12), (13), (14) and (15), of which the lefthand members vanish for a steady state, provide nonlinear simultaneous differential equations for h1, h2c, h3c and h4combining those equations with (31), we may have a set of differential and integral equations. On solving those equations with a proper account of the boundary conditions as given by (26), the steady state of the system should be obtained. In that case, all the magnetic field intensities as well as their distributions with respect to r could be determined. In the B-G model, however, it has not been able to determine the field intensities because no account of driving force has been taken. The writers are not going to specifically tackle the steady state problem, but timedependent behaviours of the system will be studied in the following in the hope of eventually approaching a steady state. 6. Solutions of equations from (12) to (21) Following the previous paper, all the equations are rewritten respect to the radial distance and time. Starting in difference forms with then from the initial values, the equations are solved step by step with a sufficiently small interval of time. The procedure of solving being more or less the same as that for the previous paper, no detailed account is given here. Special caution should be drawn, however, to equation (17) which is a non-linear integral equation. Fortunately, the condition w(0)=0 enables us to determine quantities for t=t;

6 420 T. RIKITAKE and Y. HAGIWARA from those for t=t11. In actual computation, which was made on a HITAC 5020 computer at the University of Tokyo, the writers met difficulties. When the ratio of magnetic energy to kinetic energy is large, successive integrations become difficult to perform as in the cases of coupled disk dynamo (Hide and Roberts, 1961; Allan, 1962) and Herzenberg dynamo (Rikitake and Hagiwara,1966). The writers cannot tackle problems in which the steady magnetic fields are of the order of the earth's ones. The only cases handled by the writers are Case 1(H0=10-4 Gauss) and Case 2(Hog=x/10x10x4Gauss). Corresponding to these values, it is estimated that wo=360(case1) and wo=36(case 2). Figs. 1 and 2 show two examples of Case 1. It is assumed that a uniform S1 field is given to the already rotating field with a dimensionless angular velocity wo at t=0. Changes in the angular velocity, the T2 field at r=0.8 and the S1 field at r=1 are illustrated for two cases. The initial kicks of the S1 field are 0.01 and 0.1 respectively for Figs. 1 and 2.w(0) =0 is assumed for all the examples. Figs. 3 and 4 are two similar curves for Case 2 although a smaller kick in the S1 field is assumed for Fig. 4. The time unit is exactly the same as that adopted in the previous paper, Fig. 1 Changes in the S1 field at r=1, the T2 field at r= 0.8 and the angular velocity (AV) for w0=360 and c=0.01. All the quantities are dimensionless. The time unit is estimated as 1.47 years for the earth's core. Fig. 2 Changes for wo=360 and c=0.1.

7 Non-steady Bullard-Gellman Dynamo Model (2) 421 Fig. 3 Changes for w0=36 and c=0.1. Fig. 4 Changes for w0=36 and c= i.e. the unit equals to 1.47 years for parameters appropriate to the earth's core. Owing to the tremendous amount of computation time, it does not seem practicable to extend the study to other cases. 7. Discussion Looking at Figs. 1,2,3 and 4, we observe that the pondermotive force is so large, except for the initial S1 field as small as 0.001, that the rotation (T1 motion) is decelerated initially. For an initial kick as large as 0.1, the angular velocity changes its sign some time later growing to an extremely large amplitude, so that large Sl and T2 fields seem to be associated at the same time. It would be expected that the motion is decelerated when the magnetic fields grow tremendously but no computer work has been possible for such a case because of computational diflicultly. It is therefore impossible to see whether the S1 magnetic field experiences a reversal of its sign. For a smaller initial kick, time-dependent behaviours of the motion as well as the magnetic fields are more gentle as is seen in Fig. 1. The S1 field is subjected to an initial decay, but, when the T2 field grows to some extent, the dynamo action seems to be at work. The S1 field then exhibits some growth although it again starts decreasing. It may be said that

8 422 T. RIKITAKE and Y. HAGIWARA the apparent period of these oscillations is of the order of 104 years which is not much different from the time-constant of the system as discussed in the previous paper. In Fig. 4, for which the initial kick is as small as 0.001, we see an acceleration of the rotational motion which is decelerated when the magnetic fields grow to a certain extent. An oscillatory motion occurs in this case although the Sl field seems to decrease monotonically. The writers regret that the examples in this paper would not be sufficient for examining time-dependent behaviour of the B-G model. Judging from the computation time needed for the present study, it is concluded that no good results would be reached within a reasonable time of computation. Recent studies of dynamo action (Gibson and Roberts, 1967; Braginskii, 1965; Tough, 1967) cast serious doubts on the possibility of having a steady state by the fluid motion which is taken by a B-G model. It would therefore appear not very sensible to work hard on a B-G model. But the writers should like to study non-steady states of all the existing dynamo models. They also feel that a dynamo model would have an important bearing on the origin of geomagnetism as long as it maintains a quasi-steady state which lasts over a period of time comparable the time of free decay of the magnetic field. Although the present study did not result in an outstanding behaviour of the model concerned, studies on non-steady states of a more realistic dynamo model should not be discouraged. References Allan, D.W., On the behaviour of systems of coupled dynamos, Proc. Cambr. Phil. Soc., 58, ,1962. Braginskii, S.I., Self-excitation of a magnetic field during the motion of a highly conducting fluid, Trans. in Soviet Physics J.E.T.P., 20, ,1965. Bullard, E.C. and H. Gellman, Homogeneous dynamos and terrestrial magnetism, Phil. Trans. Roy. Soc. London, Ser. A 247, , Gibson, R.D. and P.H. Roberts, Some comments on the theory of homogeneous dynamos, Magnetism and the Cosmos (Oliver and Boyd, Edinburgh), , Hide, R. and P.H. Roberts, The origin of the main geomagnetic field, Phyics and chemistry of the Earth. (Pergamon, London), 4, 27-98,1961. Rikitake, T., Dynamics of spherical eddies in the earth's core, J. Geomag. Geoelectr. 14, 66-69, Rikitake, T., Electromagnetism and the Earth's Interior (Elsevier Publ. Co., Amsterdam), Rikitake, T. and Y. Hagiwara, Non-steady state of a Herzenberg dynamo, J. Geomag. Geoelectr. 18, , Rikitake, T. and Y. Hagiwara, Non-steady Bullard-Gellman dynamo model (1), J. Geomag. Geoelectr. 20, 57-65,1968. Tough, J.G., Nearly symmetric dynamos, Geophys. J., 13, , 1967.