4.5 The Sun as an oscillator

An even more radical simplification of the solar dynamo problem ignores any spatial dependence in the solutions completely, concentrating on the time dependence only. Spatial derivatives appearing in Equations (14View Equation) and (15View Equation) are estimated as ∇ ∼ 1∕L and the resulting terms Uc ∕L and 2 ηT∕L as 1∕ τ where τ is a characteristic time scale. This results in the pair
A˙= αB − A∕τ , (16 )
˙ B = (Ω ∕L )A − B ∕τ , (17 )
which can be combined to yield
D − 1 2 ¨B = ---2--B − -B˙, (18 ) τ τ
where D = αΩ τ2∕L is the dynamo number. For D < 1, Equation (18View Equation) clearly describes a damped linear oscillator. For D > 1, solutions have a non-oscillatory character. The system described by Equation (18View Equation), then, is not only not a true dynamo (missing the spatial dependence) but it does not even display growing oscillatory solutions that would be the closest counterpart of dynamo-like behaviour in such a system. Nevertheless, there are a number of ways to extend the oscillator model to allow for persistent oscillatory solutions, i.e., to turn it into a relaxation oscillator:

(1) The most straightforward approach is to add a forcing term + sin(ω t) 0 to the r.h.s. of Equations (18View Equation). Damping would cause the system to relax to the driving period 2π ∕ω0 if there were no stochastic disturbances to this equilibrium. Hiremath (2006) fitted the parameters of the forced and damped oscillator model to each observed solar cycle individually; then in a later work (Hiremath, 2008Jump To The Next Citation Point) he applied linear regression to the resulting series to provide a forecast (see Section 3.1 above).

(2) Another trick is to account for the π∕2 phase difference between poloidal and toroidal field components in a dynamo wave by introducing a phase factor i into the first term on the r.h.s. of Equation (17View Equation). This can also be given a more formal derivation as equations of this form result from the substitution of solutions of the form A ∝ eikx, B ∝ ei(kx+π∕2) into the 1D dynamo equations. This route, combined with a nonlinearity due to magnetic modulation of differential rotation described by a coupled third equation, was taken by Weiss et al. (1984). Their model displayed chaotic behaviour with intermittent episodes of low activity similar to grand minima.

(3) Wilmot-Smith et al. (2006) showed that another case where dynamo-like behaviour can be found in an equation like (18View Equation) is if the missing effects of finite communication time between parts of a spatially extended system are reintroduced by using a time delay Δt, evaluating the first term on the r.h.s. at time t − Δt to get the value for the l.h.s. at time t.

(4) Yet another possibility is to introduce a nonlinearity into the model by assuming D = D [1 − f(B )] 0 where f(B = 0) = 0 and f ≥ 0 everywhere. (Note that any arbitrary form of α- or Ω-quenching can be cast in the above form by series expansion.) The governing equation then becomes one of a nonlinear oscillator:

D0 − 1 2 D0 − 1 B¨ = ----2--B − -B˙ − ----2--Bf (B ). (19 ) τ τ τ

In the most commonly assumed quenching mechanisms the leading term in f(B ) is quadratic; in this case Equation (19View Equation) describes a Duffing oscillator (Kanamaru, 2008). For large positive dynamo numbers, D0 ≫ 1, then, the large nonlinear term dominates for high values of B, its negative sign imposing oscillatory behaviour; yet the origin is a repeller so the oscillation will never be damped out. The Duffing oscillator was first considered in the solar context by Paluš and Novotná (1999). Under certain conditions on the parameters, it can be reduced to a van der Pol oscillator (Adomian, 1989Mininni et al., 2002Kanamaru, 2007):

¨ 2 ˙ ξ = − ξ + μ(1 − ξ )ξ , (20 )
with μ > 0. From this form it is evident that the problem is equivalent to that of an oscillator with a damping that increases with amplitude; in fact, for small amplitudes the damping is negative, i.e., the oscillation is self-excited.

These simple nonlinear oscillators were among the first physical systems where chaotic behaviour was detected (when a periodic forcing was added). Yet, curiously, they first emerged in the solar context precisely as an alternative to chaotic behaviour. Considering the mapping of the solar cycle in the differential phase space {B, dB∕dt }, Mininni et al. (2000) got the impression that, rather than showing signs of a strange attractor. The SSN series is adequately modelled by a van der Pol oscillator with stochastic fluctuations. This concept was further developed by Lopes and Passos (2009) who fitted the parameters of the oscillator to each individual sunspot cycle. The parameter μ is related to the meridional flow speed and the fit indicates that a slower meridional flow may have been responsible for the Dalton minimum. This was also corroborated in an explicit dynamo model (the Surya code) – however, as we discussed in Section 2.4, this result of flux transport dynamo models is spurious and the actual effect of a slower meridional flow is likely to be opposite to that suggested by the van der Pol oscillator model.

In an alternative approach to the problem, Nagovitsyn (1997) attempted to constrain the properties of the solar oscillator from its amplitude–frequency diagram, suggesting a Duffing oscillator driven at two secular periods. While his empirical reconstruction of the amplitude–frequency plot may be subject to many uncertainties, the basic idea is certainly noteworthy.

In summary: despite its simplicity, the oscillator representation of the solar cycle is a relatively new development in dynamo theory, and its obvious potential for forecasting purposes has barely been exploited.

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