Letter to the Editor

Ultra-fast simulations of the solar dipole and open flux

Space Physics and Astronomy Research Unit, University of Oulu, POB 8000, FI-90014 Oulu, Finland

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Received: 24 February 2026
Accepted: 12 April 2026

Abstract

Context. The solar dipole captures important information characterizing the large-scale solar magnetic field. The evolution of the solar magnetic field including the solar dipole can be simulated with a surface flux transport (SFT) model, but these simulations are more extensive than is otherwise necessary to produce the evolution of the dipole alone.

Aims. We present a dipole flux transport (DFT) matrix method that combines the classic SFT model with dipole vector representation of the solar magnetic field, which enables significantly faster simulations of the solar dipole.

Methods. By simulating the evolution of basis vectors of a synoptic map, we were able to construct propagator matrices that produce the time evolution of the solar magnetic field by means of matrix multiplication. The computational speedup was achieved by compressing the propagator matrices to very small fraction (< 10⁻⁴) of their original size with a recent vector sum method.

Results. Depending on the time resolution, the DFT performs 100−1000 times faster than a 4-year SFT simulation of a single active region, while producing equivalent results. For multiple source regions, daily propagation matrices are sufficient to produce results that agree within 1% with the SFT simulation of solar cycle 24, while performing 80 times faster. If the evolution of individual active regions is needed, the DFT is equipped to perform 50000 times faster than the SFT model.

Conclusions. Overall, DFT makes solar dipole simulations extremely fast, making it possible to run thousands of simulations in a few minutes with a basic laptop setup. As the magnitude of the dipole vector closely matches the open solar flux (OSF) from the potential field source surface model, the DFT can be used to study the development of OSF in various scenarios in an extremely efficient way.