Constraining the outer boundary condition for the Babcock-Leighton dynamo models

TL;DR

This work derives a zero radial diffusion outer boundary condition for Babcock-Leighton dynamos from the axisymmetric MHD induction equation and shows it aligns surface field evolution with surface flux transport models. It compares this BC with a vertical-field plus strong pumping BC and with a pumping-free variant, finding that suppressing radial diffusion yields surface evolution consistent with SFT and supports a solar-cycle-like dynamo with a non-purely radial surface field. Numerical experiments on a 2D distributed-shear BL dynamo demonstrate that the zero-diffusion BC reproduces key surface and interior dynamo features, including realistic polarity reversals and torsional-oscillation behavior, while improving dynamo efficiency by limiting diffusion. The results offer a physically motivated boundary condition that bridges BL dynamos and SFT, with potential for extension to 3D and broader solar/stellar-cycle applications.

Abstract

The evolution of the Sun's large-scale surface magnetic field is well captured by surface flux transport models, which can therefore provide a natural constraint on the outer boundary condition (BC) of Babcock-Leighton (BL) dynamo models. For the first time, we propose a zero radial diffusion BC for BL dynamo models, enabling their surface field evolution to align consistently with surface flux transport simulations. We derive a zero radial diffusion BC from the Magnetohydrodynamic induction equation and evaluate its effects in comparison with two alternatives: (i) a radial outer BC, and (ii) a radial outer BC combined with strong near-surface radial pumping. The comparison is carried out both for the evolution of a single bipolar magnetic region and within a full BL dynamo model. The zero radial diffusion outer BC effectively suppresses radial diffusion across the surface, ensuring consistency between the evolution of the bipolar magnetic region in the BL dynamo and the surface flux transport model. With this outer BC, the full BL dynamo model successfully reproduces the fundamental properties of the solar cycle. In addition, the model naturally produces a surface magnetic field that is not purely radial, in closer agreement with solar observations. The physically motivated zero radial diffusion boundary condition paves the way for deeper insight into the solar and stellar cycles.

Figures (4)

• Figure 1: Temporal evolution of an initial BMR on the solar surface obtained from dynamo simulations under the three outer BC settings: (a) zero radial diffusion outer BC (Case 1); (b) radial outer BC along with strong near-surface radial pumping (Case 2); (c) radial boundary without near-surface radial pumping (Case 3).
• Figure 2: Comparison of the latitudinal distribution of the surface radial field $B_r$ from dynamo simulations with three different outer BC settings. Shown are snapshots at $t$ = 0.5 yr (black), $t$ = 1.5 yr (blue), $t$ = 6 yr (red), and $t$ = 10 yr (green). Solid, dashed, and dot–dashed curves correspond to Cases 1, 2, and 3 of the outer BC, respectively.
• Figure 3: Dynamo solution with the zero radial diffusion outer boundary condition. (a) Temporal evolution of subsurface flux density of the toroidal field; (b) Temporal evolution of the surface radial fields. Cycle minimum ($t_{min}$) and cycle maximum ($t_{max}$) are indicated by the vertical dashed and dotted lines, respectively.
• Figure 4: Snapshots of the poloidal field (first row) and toroidal field (second row) over a dynamo cycle for the distributed-shear BL dynamo with the zero radial diffusion outer boundary condition. Solid (dashed) contours in the top panels correspond to clockwise (counterclockwise) poloidal field lines.