HELICITY FLUCTUATIONS IN MEAN-FIELD THEORY - AN EXPLANATION FOR THE VARIABILITY OF THE SOLAR-CYCLE

We consider the effect of fluctuations deltaalpha(t) in the mean helic ity alpha0 (both assumed independent of position) on a plane dynamo wa ve. The time scale tau(c) of the fluctuations is much shorter than the diffusion time 1/beta0k2 (beta0k2tau(c) much less than 1; beta0 = tur bulent diffusion coefficient; k = wave number). We distinguish weak an d strong random forcing, according to whether (deltaalpha(r.m.s.)/alph a0) square-root beta0k2tau(c) small or large with respect to 1, and we present a detailed analysis of the weak forcing case. Simple equation s are derived for the phases and the logarithmic amplitudes of the pol oidal and toroidal mean field, in which the forcing terms appear as ad ditive noise. Phase difference and amplitude ratio of the poloidal and toroidal (mean) field are subject to small fluctuations of constant r .m.s. magnitude. Simple expressions are derived for the r.m.s. phase s hift, amplitude drift and quality factor of the toroidal (mean) field. These depend on the fluctuations only through the quantity D = 1/4(de ltaalpha(r.m.s.)/alpha0)2beta0k2tau(c) which plays the role of a diffu sion coefficient. The results are: (1). Phase shift DELTA and logarith mic amplitude LAMBDA each perform a random walk; (2). In the alpha2-li mit these random walks are uncorrelated; the phase is very stable but the amplitude is completely irregular; (3). In the alphaomega-limit th ere exists a correlation: LAMBDA + DELTA congruent-to 0, which persist s for many dynamo periods. The quality factor is then given by Q = 1/D . The model is then applied to the solar dynamo. The predicted correla tion LAMBDA + DELTA congruent-to 0 implies that weaker (stronger) cycl es last longer (shorter) than average, which is a well-known observed feature of the solar cycle. We define LAMBDA and DELTA using the epoch s of solar maxima and the sunspot numbers, and show that LAMBDA + DELT A congruent-to 0 is obeyed rather well. This indicates that fluctuatio ns in the mean helicity are an important mechanism causing the observe d phase and amplitude variations of the solar cycle. Simulations show many features also seen in the solar cycle, such as quasi-periodicity, intermittency and long periods of low activity. Further inferences ar e: (1). The sunspot numbers appear to be proportional to the strength of the toroidal field; (2). The quality factor Q is about 10, which ma kes the solar dynamo a border-line case between weak and strong forcin g; (3). The solar data indicate that it is necessary to allow for nonl inear effects; (4). The mean helicity fluctuations deltaalpha(t) are c aused by very large spatial fluctuations in the local helicity. This c ould explain the discrepancy between theoretical estimates for alpha0 and values derived from mean field models.