The Spectral Difference with Divergence Cleaning Method for 3D Simulations of Solar Magnetic Cycles

The spectral difference method is a versatile high-order method that has shown its high-order accuracy and robustness for predicting compressible viscous flows on structured grids of multi-domains and unstructured grids [1-3]. The spectral difference method can also be massively parallelized to solve non-linear MHD problems [4,5]. In this study, the Spectral Difference with Divergence Cleaning (SDDC) method is further developed to solve several three-dimensional kinematic dynamo problems. The first kinematic dynamo problem is solved in a cubic periodic box, where the SDDC method is verified successfully to capture the “Cigar” structures published in the literature [6]. The second kinematic dynamo problem adopts a spherical shell geometry [7] and perfectly conducting and perfectly radial boundary conditions [8,9] for solving three-dimensional induction equations. Finally, the SDDC method is employed to model Sunspot cycles. References: [1] Kopriva, D. A., “Multidomain Spectral Solution of Compressible Viscous Flows,” Journal of Computational Physics, Vol. 115, No. 1, 1994, pp. 184–199. [2] Sun, Y., Wang, Z. J., and Liu, Y., “High-Order Multidomain Spectral Difference Method for the Navier-Stokes Equations on Unstructured Hexahedral Grids,” Communications in Computational Physics, Vol. 2, 2007, pp. 310–333. [3] Liang, C., Jameson, A., and Wang, Z. J., “Spectral difference method for compressible flow on unstructured grids with mixed elements,” Journal of Computational Physics, Vol. 228, 2009, pp. 2847–2858. [4] K. Chen and C. Liang, An arbitrarily high-order Spectral Difference method with Divergence Cleaning (SDDC) for compressible magnetohydrodynamic simulations on unstructured grids, The Astrophysical Journal, published online, June 2022. https://doi.org/10.3847/1538-4357/ac6e61 [5] Russell Hankey, Kuangxu Chen, and Chunlei Liang (2023) "Extending the Spectral Difference Method with Divergence Cleaning (SDDC) to the Hall MHD Equations," Northeast Journal of Complex Systems (NEJCS): Vol. 5 : No. 1 , Article 1. https://doi.org/10.22191/nejcs/vol5/iss1/1 [6] Shishir Biswas, Rajaraman Ganesh; Effect of flow shear on the onset of dynamos. Phys. Plasmas 1 November 2023; 30 (11): 112902. https://doi.org/10.1063/5.0170796 [7] Kuangxu Chen, Chunlei Liang, Minping Wan; Arbitrarily high-order accurate simulations of compressible rotationally constrained convection using a transfinite mapping on cubed-sphere grids. Physics of Fluids 1 August 2023; 35 (8): 086120. https://doi.org/10.1063/5.0158146 [8] Mark Miesch and Kinfe Teweldebirhan; A three-dimensional Babcock–Leighton solar dynamo model: Initial results with axisymmetric flows; Advances in Space Research 58.8 2016; 1571-1588. https://doi.org/10.1016/j.asr.2016.02.018 [9] A. R. Yeates, A. Muñoz-Jaramillo, Kinematic active region formation in a three-dimensional solar dynamo model, Monthly Notices of the Royal Astronomical Society, Volume 436, Issue 4, 21 December 2013, Pages 3366–3379, https://doi.org/10.1093/mnras/stt181