The solar cycle as a forced and damped harmonic oscillator: long-term variations of the amplitudes, frequencies and phases

We model solar activity cycle as a forced and damped harmonic oscillator consisting of two parts, sinusoidal and transient. The amplitudes, frequencies, phases and decay factors of such a harmonic oscillator are determined by fitting the equation of the sinusoidal and transient parts to the sunspot data for the years 1755–1996 (cycles 1–22) with the results: (i) there is a long-term decreasing trend in the phase, while the amplitude and the frequency (or period of ∼11 yr) of the sinusoidal part remain constant for all the solar cycles; (ii) the amplitude of the transient part is phase locked with the phase of the sinusoidal part; (iii) for all the cycles, the period and decay factor (that is much less than 1) of the transient part remain approximately constant; (iv) except in cycles 6 and 15, the phases of the transient part are approximately constant with a magnitude of ∼π/2 radians and; (v) for cycles 6 and 15, the simultaneous change in magnitude of phase difference (∼2π radians) between the transient and sinusoidal parts and of very low sunspot activity may be due to the Maunder minimum type of oscillations. The constancy of the amplitudes and the frequencies of the sinusoidal part and a very small decay factor from the transient part suggests that the solar activity cycle mainly consists of a persistent oscillatory part that might be compatible with long-period (∼22 yr) Alfven oscillations.