Bistable attractors in a model of convection-driven spherical dynamos

The range of existence and the properties of two essentially different chaotic attractors found in a model of nonlinear convection-driven dynamos in rotating spherical shells are investigated. A hysteretic transition between these attractors is established as a function of the rotation parameter \ttau. The width of the basins of attraction is also estimated.

💡 Research Summary

The paper investigates a minimal self‑consistent model of convection‑driven dynamos in rotating spherical shells, focusing on the existence of two distinct chaotic attractors that coexist at identical control‑parameter values. The governing equations are the Boussinesq Navier‑Stokes, heat, and induction equations, nondimensionalized by the shell thickness, viscous diffusion time, and appropriate scales for temperature and magnetic field. The four dimensionless parameters are the Rayleigh number R, the Coriolis number τ (proportional to the rotation rate), the Prandtl number P, and the magnetic Prandtl number Pm.

A pseudo‑spectral method employing spherical harmonics up to degree 96 and 41 Chebyshev collocation points in radius is used to solve the equations with stress‑free velocity boundaries and electrically insulating magnetic boundaries. The magnetic field is decomposed into poloidal and toroidal scalars (h,g) and the velocity into (v,w). Energy densities for each component (poloidal, toroidal, axisymmetric, non‑axisymmetric) are monitored, together with kinetic, magnetic, cross‑helicity, and magnetic helicity densities.

Two families of chaotic dynamos are identified:

1. Mean‑Dipole (MD) regime – characterized by a dominant axisymmetric poloidal magnetic energy Mₚ, a very small ratio eMₚ/Mₚ (≲ 0.1), and a largely steady large‑scale dipole field.

2. Fluctuating‑Dipole (FD) regime – where the non‑axisymmetric poloidal component eMₚ is comparable to Mₚ (eMₚ/Mₚ ≈ 0.5–0.7), indicating strong multipole contributions, and the magnetic field exhibits quasi‑periodic polarity reversals.

Both regimes are found at the same values of R≈4 R_c (four times the critical Rayleigh number), P=0.75, Pm=1.5, while τ is varied. A hysteresis loop is observed in the interval τ≈3.9 × 10³ – 1.25 × 10⁴. Starting from an MD solution and decreasing τ leads to a sudden jump to the FD state at τ_FD≈1.25 × 10⁴. Conversely, beginning with an FD solution and increasing τ causes a reverse jump to MD at τ_MD≈3.9 × 10⁴. The transitions are discontinuous and persist over many magnetic diffusion times, confirming true bistability rather than long transients.

The authors quantify the width of the basins of attraction by initializing many random states at fixed τ and recording which attractor is reached. The basin sizes are comparable near the middle of the hysteresis interval and become strongly biased toward one attractor when τ is far from the coexistence range.

Temporal analysis shows that MD dynamos have nearly stationary energy spectra, while FD dynamos display oscillatory behaviour: magnetic flux of opposite polarity is generated near the inner boundary, propagates toward the poles, and replaces the old polarity in a quasi‑periodic cycle. During this cycle kinetic helicity remains roughly constant, whereas magnetic helicity and cross‑helicity oscillate in phase with the dipole reversal.

Spatially, MD solutions exhibit a strong, large‑scale dipole with two toroidal flux tubes near the poles and opposite‑polarity tubes near the equator. FD solutions contain smaller‑scale structures, evident multipole patterns, and fragmented helicity distributions.

When the magnetic field is artificially removed, both MD‑ and FD‑derived initial conditions relax to the same non‑magnetic convective state, which displays relaxation oscillations—a known quasi‑periodic behaviour of pure convection at these parameters. This demonstrates that the bistability is purely magnetic in origin.

Overall, the study establishes that the Coriolis number τ is a decisive control parameter for multistability and hysteresis in rotating convective dynamos. The findings have direct relevance for planetary, stellar, and possibly exoplanetary interiors, where variations in rotation rate could trigger abrupt changes between distinct magnetic regimes. The work highlights how nonlinear feedback between flow and magnetic field can generate multiple chaotic attractors, a phenomenon that may underlie observed magnetic polarity reversals and mode switches in natural dynamos.