Equations governing the system are given in non-dimensional form as

Transcription

1 Research Note J. Geomag. Geoelectr., 37, , 1985 A Perturbed Disk Dynamo Model and Polarity Reversals of the Earth's Magnetic Field Y. HoNKURA and M. SHIMIZU* Department of Applied Physics, Tokyo Institute of Technology, Ookayama, Tokyo, Japan 1. Introduction (Received August 19, 1985) Cox (1968, 1969) assumed that the earth's magnetic field consists of a dipole field, which undergoes a variation having a period of ten thousand years, and a non-dipole field, which varies at random with a shorter time scale, about one thousand years. The non-dipole field is represented by some radial dipoles existing near the surface of the core. Using this simplified model, Cox (1968, 1969) proposed a mechanism of polarity reversals of the dipole field; the dipole field changes its polarity, when the resultant axial component changes its sign. The Cox model implies that the dipole grows in the same direction as an instantaneous axial component, however small it may be. Such a behavior is well represented in a disk dynamo model which is often referred to as the Bullard model (BULLARD, 1955). However, it seems to be too simple to consider the effect of a non-dipole field variation on a dipole field variation only at the time of polarity reversal. In this respect, we point out that the effect of a non-dipole field variation is important even under the circumstance that the dipole field is varying periodically without changing its polarity. Moreover, in the Cox model, no consideration is given to the velocity field which must be closely related to the magnetic field in hydromagnetic dynamo action. In this research note, we examine the aspects mentioned above on the basis of a single disk dynamo model (the Bullard model), which is subject to perturbation from the outside of the system, since it exhibits a periodic variation (BULLARD, 1955; SHIMIZU and HONKURA, 1985). 2. A Perturbed Disk Dynamo Model Equations governing the system are given in non-dimensional form as *Now at Suwa-Seikosha Company, Suwa, Japan. 1147

2 1148 Y. HONKURA and M. SHIMIZU where x and y are non-dimensional expressions for electric current and angular velocity, x and 9 their derivatives with respect to non-dimensional expression represents a perturbation corresponding to the non-dipole magnetic field in the Cox model, while X represents the dipole field. We assign the total magnetic exhibits a random variation (SHIMIZU and HoNKURA, 1985). Figure 1 shows variations of x and y for a Bullard model which is subject It is striking that not only the amplitude of dipole field (x) but also its period changes sporadically. Hence, the effect of perturbation cannot be neglected even during non-reversal stages. This result suggests that the maintenance of a stable periodic variation is hardly possible in a non-linear self-exciting system. of M implies that the system is more vulnerable to a perturbation from the outside of the system, 3. Statistical Nature of Polarity Reversals in a Perturbed Bullard Model Figure 3 shows the frequency distribution of length of polarity intervals for HONKURA, 1985), the statistical nature of polarity reversals in this model agrees well with that for the earth's magnetic field. Figure 4 shows Fourier and Walsh power spectra, respectively, as derived Compared with corresponding ones for the earth's magnetic field, as shown in Fig. 5, statistical properties of polarity reversals in a perturbed Bullard model are again in good agreement with those for the earth's magnetic field. 4. Concluding Remarks A small perturbation added to a Bullard model drastically affected the system, which is stable without the perturbation, and instability prevails with its manifesta-

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7 A Perturbed Disk Dynamo Model 1153 tion as polarity reversals. Generally speaking, therefore, the Cox model is physically plausible. In detail, however, the effect of the non-dipole field on polarity reversals of the dipole field is by no means as simple as presumed in the Cox model, and the non-linear nature of dynamo problem must be taken into account. It is rather surprising to find, in a perturbed Bullard model, the statistical nature very similar to that in more complicated coupled-disk dynamo models such as an inhomogeneous N-disk dynamo model (SHIMIzu and HONKURA, 1985). We neglected the action of the dominant dipole field on the non-dipole field, although it may be operative in actual dynamo process. If this action is included, the system would become similar to inhomogeneous coupled-disk dynamo models, and in this sense, the present treatment of a perturbed Bullard model should be sufficient. REFERENCES BULLARD, E.C., The stability of a homopolar dynamo, Proc. Cambridge Phil. Soc., 51, , Cox, A., Lengths of geomagnetic polarity intervals,. J. Geophys, Res., 73, , Cox, A., Geomagnetic reversals, Science, 163, , SHIMIZU, M. and Y. HONKURA, Statistical nature of polarity reversals of the magnetic field in coupled-disk dynamo models, J. Geomag. Geoelectr., 37, , 1985.

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