Sunspot numbers exhibit large short-timescale (daily-monthly) variation in addition to longer-timescale variation due to solar cycles. A formal statistical framework is presented for estimating and forecasting randomness in sunspot numbers on top of deterministic (including chaotic) models for solar cycles. The Fokker-Planck approach formulated assumes a specified long-term or secular variation in sunspot number over an underlying solar cycle via a driver function. The model then describes the observed randomness in sunspot number on top of this driver function. We consider a simple harmonic choice for the driver function, but the approach is general and can easily be extended to include other drivers which account for underlying physical processes and/or empirical features of the sunspot numbers. The framework is consistent during both solar maximum and minimum, and requires no parameter restrictions to ensure non-negative sunspot numbers. Model parameters are estimated using statistically optimal techniques. The model agrees both qualitatively and quantitatively with monthly sunspot data even with the simplistic representation of the periodic solar cycle. This framework should be particularly useful for solar cycle forecasters and is complementary to existing modeling techniques. An analytic approximation for the Fokker-Planck equation is presented, which is analogous to the Euler approximation, which allows for efficient maximum likelihood estimation of large data sets and/or when using difficult to evaluate driver functions.
The telegraph equation and its generalizations have been repeatedly considered in the models of diffusive cosmic-ray transport. Yet the telegraph model has well-known limitations, and analytical arguments suggest that a hyperdiffusion model should serve as a more accurate alternative to the telegraph model, especially on the timescale of a few scattering times. We present a detailed sideby-side comparison of an evolving particle density profile, predicted by the telegraph and hyperdiffusion models in the context of a simple but physically meaningful initial-value problem, compare the predictions with the solution based on the Fokker-Planck equation, and discuss the applicability of the telegraph and hyperdiffusion approximations to the description of strongly anisotropic particle distributions. Published by AIP Publishing. [http://dx
Recently Pop ({\it Solar Phys.} {\bf 276}, 351, 2012) identified a Laplace
(or double exponential) distribution in the number of days with a given
absolute value in the change over a day, in sunspot number, for days on which
the sunspot number does change. We show this phenomenological rule has a
physical origin attributable to sunspot formation, evolution, and decay, rather
than being due to the changes in sunspot number caused by groups rotating onto
and off the visible disc. We also demonstrate a simple method to simulate daily
sunspot numbers over a solar cycle using the \cite{2012SoPh..276..351P} result,
together with a model for the cycle variation in the mean sunspot number. The
procedure is applied to three recent solar cycles. We check that the simulated
sunspot numbers reproduce the observed distribution of daily changes over those
cycles.Comment: 18 pages, 8 figures. Accepted for publication in Sol. Phys.
(29/09/2012
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