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A new approach to subcritical instability and turbulent transitions in a simple dynamo

Published online by Cambridge University Press:  24 October 2008

K. A. Robbins
K. A. Robbins
Affiliation:
The University of Texas, San Antonio, Texas 78285
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Abstract

The onset of turbulence in many systems appears disorganized and unpredictable. However, the present detailed study of a well-known model reveals a series of well-defined transitions from steady motion to highly non-periodic behaviour. The bifurcation structure of this simple model, which describes both a reversing disc dynamo and Benard convection in thin fluid loops, is characteristic of a class of systems with subcritical instabilities. It is found that the bifurcation curve for this instability does not have a stable branch and that non-periodic and linearly stable steady solutions coexist. The boundary between transient and non-periodic behaviour is marked by a non-uniformity in the number of oscillations between reversals. This non-uniformity, not previously observed, is a striking corroboration of the model's relationship to the geodynamo. Other features of reversal in the disc dynamo such as the presence of two time scales in the oscillations are also exhibited by geomagnetic fields. In addition to a geometric description of the transitions, a one-dimensional mapping first constructed by Lorenz for the system beyond marginal stability, is extended to the subcritical regime. This type of mapping, which can be interpreted in terms of invariant surfaces of the system, may be of value in dissecting more general systems with subcritical Hopf bifurcations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 2 , September 1977 , pp. 309 - 325
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Allan, D. W.On the behaviour of systems of coupled dynamos. Proc. Cambridge Philos. Soc. 58 (1962), 671–693.CrossRefGoogle Scholar
(2)Babcock, H. W.The topology of the sun's magnetic field and the 22-year cycle. Astrophys. J. 133 (1961), 572–587.CrossRefGoogle Scholar
(3)Baker, N. H., Moore, D. W. and Spiegel, E. A.Aperiodic behavior of a nonlinear oscillator. Quart. J. Mech. Appl. Math. 24 (1971), 391–422.CrossRefGoogle Scholar
(4)Bullard, E. C.The stability of a homopolar dynamo. Proc. Cambridge Philos. Soc. 51 (1955), 744–760.CrossRefGoogle Scholar
(5)Cook, A. E. and Roberts, P. H.The Rikitake two-disc dynamo system. Proc. Cambridge Philos. Soc. 68 (1970), 547–569.CrossRefGoogle Scholar
(6)Cox, A. J.Lengths of geomagnetic polarity intervals. J. Geophys. Res. 73 (1968), 3247–3260.CrossRefGoogle Scholar
(7)Cox, A. J.The frequency of geomagnetic reversals and the symmetry of the nondipole field. Rev. Geophys. Space Phys. 13 (1975), 35–51.CrossRefGoogle Scholar
(8)Creveling, H. F., De Paz, J. F., Baladi, J. Y. and Schoenhals, R. J.Stability characteristics of a single-phase free convection loop. J. Fluid Mech. 67 (1975), 65–84.CrossRefGoogle Scholar
(9)Halmos, P. R.Lectures on ergodic theory (Chelsea Publishing, 1956).Google Scholar
(10)Keller, J. B.Periodic oscillations in a model of thermal convection. J. Fluid Mech. 26 (1966), 599–606.CrossRefGoogle Scholar
(11)Lin, C. C.The theory of hydrodynamic stability (Cambridge University Press, 1967).Google Scholar
(12)Lorenz, E. N.Deterministic nonperiodic flow. J. Atmospheric Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
(13)Lorenz, E. N.The mechanics of vacillation. J. Atmospheric Sci. 20 (1963), 448–464.2.0.CO;2>CrossRefGoogle Scholar
(14)Malkus, W. V. R.Reversing Bullard's dynamo. EOS, Trans. Am. Geophys. Union 53 (1972), 617.Google Scholar
(15)Malkus, W. V. R. and Veronis, G.Finite amplitude cellular convection. J. Fluid Mech. 4 (1958), 225–260.CrossRefGoogle Scholar
(16)McLaughlin, J. B. Ph.D. Thesis, Harvard University (1974).Google Scholar
(17)Naidu, P. S.Second-order statistical structure of geomagnetic field reversals. J. Geophys. Res. 80 (1975), 803–806.CrossRefGoogle Scholar
(18)Poincaré, H.Les méthodes nouvelles de la mécanique céleste, vol. 3 (Gauthiers-Villars, 1899).Google Scholar
(19)Robbins, K. A. Ph.D. Thesis, Massachusetts Institute of Technology (1975).CrossRefGoogle Scholar
(20)Saltzman, B. J.Finite amplitude free convection as an initial value problem, I. J. Atmospheric Sci. 19 (1962), 329–341.2.0.CO;2>CrossRefGoogle Scholar
(21)Sattinger, D. H.Topics in stability and bifurcation theory (Springer-Verlag, 1973).CrossRefGoogle Scholar
(22)Soward, A. M.A convection-driven dynamo, I. Philos. Trans. Roy. Soc. London, Ser. A 275 (1974), 611–646.Google Scholar
(23)Tracier, J. D., Switzer, P. and Cox, A.A model relating undetected geomagnetic polarity intervals to the observed rate of reversals. J. Geophys. Res. 80 (1975), 4446–4448.CrossRefGoogle Scholar
(24)Von Kerczek, C. and Davis, S. H.Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62 (1974), 753–773.CrossRefGoogle Scholar
(25)Welander, P.On the oscillatory instability of a differentially heated fluid loop. J. Fluid Mech. 29 (1967), 17–30.CrossRefGoogle Scholar