Nonlinear Rossby wave-wave and wave-mean flow theory for long term Solar cycle modulations

The Schwabe cycle of solar activity exhibits modulations and frequency fluctuations on slow time scales of centuries and millennia. Plausible physical explanations for the cause of these long-term variations of the solar cycle are still elusive, with possible theories including stochasticity of alpha effect and fluctuations of the differential rotation. It has been suggested recently in the literature that there exists a possible relation between the spatio-temporal structure of Solar cycle and the nonlinear dynamics of magnetohydrodynamic Rossby waves at the solar tachocline, including both wave-wave and wave-mean flow interactions. Here we extend the nonlinear theory of MHD Rossby waves presented in a previous article to take into account long term modulation effects due to a recently discovered mechanism that allows significant energy transfers throughout different wave triads: the precession resonance mechanism. We have found a large number of Rossby-Haurwitz wave triads whose frequency mismatches are compatible with the solar cycle frequency. Consequently, by analyzing the reduced dynamics of two triads coupled by a single mode (five-wave system), we have demonstrated that in the amplitude regime in which precession resonance occurs, the energy transfer throughout the system yields significant long-term modulations on the main $\sim 11$yr period associated with intra-triad energy exchanges. We further show that such modulations display an inverse relationship between the characteristic wave amplitude and the period of intra-triad energy exchanges, which is consistent with the Waldmeier's law for the solar cycle. In the presence of a constant forcing and dissipation, the five-wave system in the precession resonance regime exhibits irregular amplitude fluctuations with some periods resembling the Grand Minimum states.