A weak turbulence theory is derived for magnetohydrodynamics under rapid rotation and in the presence of a large-scale magnetic field. The angular velocity $Ω_0$ is assumed to be uniform and parallel to the constant Alfvén speed ${\bf b_0}$. Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length $d \equiv b_0/Ω_0$. In the large-scale limit ($kd \to 0$; $k$ being the wave number), the left- and right-handed waves tend respectively to the inertial and magnetostrophic waves whereas in the small-scale limit ($kd \to + \infty$) pure Alfvén waves are recovered. By using a complex helicity decomposition, the asymptotic weak turbulence equations are derived which describe the long-time behavior of weakly dispersive interacting waves {\it via} three-wave interaction processes. It is shown that the nonlinear dynamics is mainly anisotropic with a stronger transfer perpendicular ($\perp$) than parallel ($\parallel$) to the rotating axis. The general theory may converge to pure weak inertial/magnetostrophic or Alfvén wave turbulence when the large or small-scales limits are taken respectively. Inertial wave turbulence is asymptotically dominated by the kinetic energy/helicity whereas the magnetostrophic wave turbulence is dominated by the magnetic energy/helicity. For both regimes a family of exact solutions are found for the spectra which do not correspond necessarily to a maximal helicity state. It is shown that the hybrid helicity exhibits a cascade whose direction may vary according to the scale $k_f$ at which the helicity flux is injected with an inverse cascade if $k_fd < 1$ and a direct cascade otherwise. The theory is relevant for the magnetostrophic dynamo whose main applications are the Earth and giant planets for which a small ($\sim 10^{-6}$) Rossby number is expected.
Precession-driven dynamos in a full sphere and the role of large scale cyclonic vortices
岁差已被提议作为行星发电机的替代动力源。以前的流体动力学模拟表明，岁差可以在行星液核中产生非常复杂的流动[Y。Lin，P. Marti，and J. Noir，“Shear-driven parametric instability in a precessing sphere，”Physics of Fluids 27，046601（2015）].在本研究中，我们数值研究了旋进球的磁流体动力学。我们展示了不同的流动制度，从层流到湍流的进动驱动发电机。特别是，我们强调的磁场产生的大尺度气旋性涡旋，这是以前没有探索。在这种情况下，发电机可以维持在相对较低的埃克曼数和磁普朗特数，这为行星应用铺平了道路。
On the weak synchronization of a nonlinear solar dynamo oscillator by coherent planetary timing
Académie des Sciences, Institut de France（法兰西科学院，法兰西学院）；Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences（俄罗斯科学院地震预测理论与数学地球物理研究所）；Accademia Nazionale delle Scienze detta dei XL（XL国家科学学院）；DeepField Sensing；Muséum National d’Histoire Naturelle, CNRS, UMR 7196, INSERM U1154（法国国家自然历史博物馆，CNRS，UMR 7196，INSERM U1154）；Sorbonne Université, CNRS, METIS, UMR 7619（索邦大学，CNRS，METIS，UMR 7619）
Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿罗森多夫亥姆霍兹中心）；Abastumani Astrophysical Observatory（阿巴斯图马尼天体物理观测台）；Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE UMR 7342（艾克斯马赛大学，法国国家科学研究中心，马赛中央理工学院，IRPHE UMR 7342）；Department of Applied Mathematics, School of Mathematics, University of Leeds（利兹大学数学学院应用数学系）
