Algebraic quantification of an active region's contribution to the solar cycle

The solar dipole moment at cycle minimum is considered to be the most successful precursor for the amplitude of the subsequent cycle. Numerical simulations of the surface flux transport (SFT) model are widely used to effectively predict the dipole moment at cycle minimum. Recently an algebraic method has been proposed to quickly predict the contribution of an active region (AR) to the axial dipole moment at cycle minimum instead of SFT simulations. However, the method assumes a bipolar magnetic region (BMR) configuration of ARs. Actually most ARs are asymmetric in configuration of opposite polarities, or have more complex configurations. Such ARs evolve significantly differently from that of BMR approximations. We propose a generalized algebraic method to describe the axial dipole contribution of an AR with an arbitrary configuration, and evaluate its effectiveness compared to the BMR-based method. We employ mathematical deductions to obtain the generalized method. We compare the results of the generalized method with SFT simulations of observed ARs, artificially created BMRs, and ARs with more complex configurations. We also compare the results with that from the BMR-based method. The generalized method is equivalent to the SFT model, and precisely predicts the ARs' contributions to the dipole moment. The method has a much higher computational efficiency than SFT simulations. Although the BMR-based method has similar computational efficiency as the generalized method, it is only accurate for symmetric bipolar ARs. The BMR-based method systematically overestimates the dipole contributions of asymmetric bipolar ARs, and randomly miscalculate the contributions of more complex ARs. The generalized method provides a quick and precise quantification of an AR's contribution to the solar cycle evolution, which paves the way for the application into the physics-based solar cycle prediction.