The roughly 11-year Schwabe cycle of solar magnetic activity is modulated on timescales of decades to millennia, including Gleissberg-cycle envelope variations and Grand Minima such as the Maunder Minimum. Raphaldini, Teruya, Raupp and Bustamante (2019) proposed that these long-term modulations arise from nonlinear interactions among magnetohydrodynamic (MHD) Rossby waves in the solar tachocline: two wave triads coupled by a shared mode exchange energy through modulational-type instability and the precession-resonance mechanism, producing ~100-year envelope modulations of an ~11/22-year cycle, Maunder-like suppressed epochs under constant forcing, an amplitude-period relation consistent with Waldmeier's law (the observational rule that stronger cycles are shorter), and an anti-phase relation between zonal-flow strength and Schwabe amplitude consistent with observations of Zhang, Mursula and Usoskin (2015). The result was demonstrated only in a five-wave truncation of a barotropic quasi-geostrophic MHD model with a prescribed background toroidal field. The open problem is whether this mechanism survives scale-up to larger triad clusters, is robust to model parameter choices and neglected physics, can be embedded in a tachocline or full MHD dynamo model that quantitatively reproduces sunspot-number series and butterfly diagrams (the standard plot of sunspot-emergence latitude versus time), and is statistically and observationally validated. Existing work provides partial progress (low-order phase-synchronization models of Maunder-like minima; verification of precession resonance in the full hydrodynamic barotropic vorticity equation for Earth's atmosphere), but the mechanism-validation question in the solar context remains open.

Background

The Schwabe cycle is the approximately 11-year periodic variation of solar magnetic activity seen in sunspot counts; Hale's magnetic polarity cycle doubles it to ~22 years. Superposed on it are slower envelope modulations — the decades-scale Gleissberg cycle, the ~200-year Suess-de Vries cycle, and irregular epochs of strongly suppressed activity called Grand Minima, of which the Maunder Minimum (1645-1715) is the canonical example. The physical origin of these long-term modulations remains unsettled; proposed mechanisms include stochastic fluctuations of the alpha effect (the regenerative poloidal-field term in mean-field dynamo theory) and fluctuations of the differential rotation.

The solar tachocline is the shear layer at the base of the convection zone separating almost uniform rotation in the radiative interior from the differentially rotating envelope. Rossby waves (planetary waves, r-modes) are vorticity-restoring waves intrinsic to any rotating fluid with a latitudinal potential-vorticity gradient; on the sphere their linear eigenmodes are the Rossby-Haurwitz modes, indexed by spherical-harmonic degree and order. In the magnetized tachocline these waves split into fast and slow magnetic branches governed by the Alfvén speed VA=B0/μ0ρV_A=B_0/\sqrt{\mu_0\rho} set by the background toroidal field, and are then called MHD (magnetohydrodynamic) Rossby or magneto-Rossby waves.

Weakly nonlinear waves resonantly interacting in triads — triplets of modes whose wavenumbers and eigenfrequencies satisfy matching conditions — exchange energy on slow timescales through the classical triad equations of wave-turbulence theory. Raphaldini and Raupp (2015) developed the nonlinear triad interaction theory of MHD Rossby waves in a barotropic (vertically averaged) quasi-geostrophic (motions in near-geostrophic balance, where the pressure-gradient force is nearly balanced by the Coriolis force) MHD model of the tachocline and connected intra-triad energy-exchange periods to the solar cycle. Bustamante, Quinn and Lucas (2014) discovered precession resonance: even when a triad is far from exact resonance, highly efficient inter-triad energy transfer occurs when the frequency mismatch of one triad resonates with the nonlinear phase-precession (amplitude-oscillation) frequency of an adjacent triad, a mechanism for transferring energy across widely separated scales in turbulent wave systems. Modulational instability is the related instability of a pump wave pair against a long-wavelength (here zero-zonal-wavenumber, zero-frequency) perturbation that exchanges energy with the mean zonal flow.

Raphaldini, Teruya, Raupp and Bustamante (2019) combined these ingredients. Searching (over spherical harmonics truncated at degree/order 30, Alfvén speeds up to about 1000 m/s) they found hundreds of Rossby-Haurwitz triads whose eigenfrequency mismatches match harmonics of the solar-cycle frequency and that contain a zero-frequency zonal mode. Integrating a representative five-wave system — two triads, [(0,2),(1,10),(1,9)][(0,2),(1,10),(1,9)] and [(1,9),(1,12),(2,10)][(1,9),(1,12),(2,10)], sharing mode (1,9)(1,9) — they showed that precession-resonance-regime energy transfer modulates the ~11/22-year intra-triad exchange at ~100-200-year scales; that the modulation amplitude is inversely related to the exchange period, consistent with Waldmeier's law (stronger cycles last shorter); that under constant divergence forcing applied only to the shared mode and with dissipation, the system exhibits irregular amplitude fluctuations with decades-long suppressed epochs resembling Grand Minima; and that the zonal-flow (zero-mode) amplitude oscillates approximately in anti-phase with the Schwabe-related triad, consistent with the observational negative correlation between solar differential-rotation strength and cycle amplitude found by Zhang, Mursula and Usoskin (2015). All of this was obtained within a barotropic quasi-geostrophic reduction with a prescribed background field Bˉ(θ)=B0cos⁡θ\bar{B}(\theta)=B_0\cos\theta, neglected nonlinear divergence terms, and no coupling to a dynamo.

Problem Statement

Establish, or refute, precession-resonance-mediated nonlinear interaction of MHD Rossby waves as the physical mechanism of the long-term modulation of the solar cycle, going beyond the five-wave demonstration of Raphaldini, Teruya, Raupp and Bustamante (2019). The paper's Conclusions explicitly point to the extension of the analysis to larger clusters of nonlinearly interacting triads and the possibility of emergence of self-organized synchronized states. The documented limitations of the five-wave demonstration frame the remaining open issues: the prescribed background toroidal field profile Bˉ(θ)=B0cos⁡θ\bar{B}(\theta)=B_0\cos\theta, the triad search truncated at spherical-harmonic degree/order 30, the neglected nonlinear divergence-related and divergent magnetic-field terms, the divergence forcing applied only to the shared mode, the barotropic quasi-geostrophic reduction with no coupling to a dynamo, and the high dependence of suppressed-activity durations on the forcing magnitude established without ensemble statistics. It therefore remains open:

  • (i) Scale-up and robustness. Whether the long-term modulation produced by modulational instability and precession resonance among tachocline MHD Rossby-wave triads persists, changes qualitatively, or gives rise to self-organized synchronized states when larger clusters of nonlinearly interacting triads are integrated, and whether it is robust to the modeling choices listed above (each of the six limitations enumerated above is to be tested; a single choice that destroys the modulation counts as a refutation).
  • (ii) Dynamo embedding. Whether the dynamics can be embedded in a tachocline or full magnetohydrodynamic dynamo model producing surface magnetic indicators (sunspot-number time series and butterfly diagrams, the latitude-versus-time pattern of sunspot emergence) quantitatively consistent with solar observations.
  • (iii) Observational validation. Whether the predicted anti-phase relation between zonal-flow amplitude and Schwabe-cycle amplitude holds against extended helioseismic (helioseismology infers the Sun's internal rotation and flow fields from its global oscillations) and sunspot datasets.
  • (iv) Ensemble statistics. How likely and long-lasting Grand-Minimum-like epochs produced by the mechanism are across ensembles of parameter choices and initial phases and amplitudes.

These are the independently refutable tests of a single mechanism claim, and the claim is conjunctive: establishing it requires a result on each of (i)-(iv), whereas a decisive negative result on any one of them — most directly on the scale-up (i), on which (ii)-(iv) depend — refutes the mechanism as the explanation of the long-term modulation. Resolving only one item is partial progress, with the remaining items left open; and 'quantitatively consistent with solar observations' in (ii) is to be read as agreement in the periodicity and amplitude structure of a standard sunspot-number time series (the ~11/22-year cycle and its Gleissberg-scale envelope) and of the latitude-versus-time butterfly pattern, beyond mere qualitative resemblance.

References

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