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The Schwabe cycle of solar activity exhibits modulations and frequency fluctuations on slow timescales 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 the alpha effect and fluctuations of the differential rotation. It has been suggested recently in the literature that there exists a possible relation between the spatiotemporal structure of the solar cycle and the nonlinear dynamics of magnetohydrodynamic (MHD) 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–Hauwirtz wave triads whose frequency mismatches are compatible with the solar cycle frequency. Consequently, by analyzing the reduced dynamics of two triads coupled with 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 ∼11 yr period associated with intratriad energy exchanges. We further show that such modulations display an inverse relationship between the characteristic wave amplitude and the period of intratriad energy exchanges, which is consistent with the Waldmeier 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. [ABSTRACT FROM AUTHOR]

We study the dynamics of a spherical rigid body that rocks and rolls on a plane under the effect of gravity. The distribution of mass is non-uniform and the centre of mass does not coincide with the geometric centre. The symmetric case, with moments of inertia I1 = I2 < I3, is integrable and the motion is completely regular. Three known conservation laws are the total energy E, Jellett's quantity QJ and Routh's quantity QR. When the inertial symmetry I1 = I2 is broken, even slightly, the character of the solutions is profoundly changed and new types of motion become possible. We derive the equations governing the general motion and present analytical and numerical evidence of the recession, or reversal of precession, that has been observed in physical experiments. We present an analysis of recession in terms of critical lines dividing the (QR, QJ) plane into four dynamically disjoint zones. We prove that recession implies the lack of conservation of Jellett's and Routh's quantities, by identifying individual reversals as crossings of the orbit (QR(t), QJ(t)) through the critical lines. Consequently, a method is found to produce a large number of initial conditions so that the system will exhibit recession. [ABSTRACT FROM AUTHOR]

New generalized equations of motion for the Weber-Clebsch potentials that describe both the Navier-Stokes and magnetohydrodynamics (MHD) dynamics are derived. These depend on a new parameter, which has dimensions of time for Navier-Stokes and inverse velocity for MHD. Direct numerical simulations (DNSs) are performed. For Navier-Stokes, the generalized formalism captures the intense reconnection of vortices of the Boratav, Pelz and Zabusky (BPZ) flow, in agreement with the previous study by Ohkitani and Constantin. For MHD, the new formalism is used to detect magnetic reconnection in several flows: the three-dimensional (3D) Arnold, Beltrami and Childress (ABC) flow and the (2D and 3D) Orszag-Tang (OT) vortex. It is concluded that periods of intense activity in the magnetic enstrophy are correlated with periods of increasingly frequent resettings. Finally, the positive correlation between the sharpness of the increase in resetting frequency and the spatial localization of the reconnection region is discussed. [ABSTRACT FROM AUTHOR]