Methodology for estimating the magnetic Prandtl number and application to solar surface small-scale dynamo simulations

Context. A crucial step in the numerical investigation of small-scale dynamos in the solar atmosphere consists of an accurate determination of the magnetic Prandtl number, Pr-m, stemming from radiative magneto-hydrodynamic (MHD) simulations. Aims. The aims are to provide a reliable methodology for estimating the effective Reynolds and magnetic Reynolds numbers, Re and Re-m, and their ratio Pr-m & x2004;=& x2004;Re-m/Re (the magnetic Prandlt number), that characterise MHD simulations and to categorise small-scale dynamo simulations in terms of these dimensionless parameters. Methods. The methodology proposed for computing Re and Re-m is based on the method of projection on proper elements and it relies on a post-processing step carried out using higher order accurate numerical operators than the ones in the simulation code. A number of radiative MHD simulations with different effective viscosities and plasma resistivities were carried out with the (COBOLD)-B-5 code, and the resulting growth rate of the magnetic energy and saturated magnetic field strengths were characterised in terms of Re and Re-m. Results. Overall, the proposed methodology provides a solid estimate of the dissipation coefficients affecting the momentum and induction equations of MHD simulation codes, and consequently also a reliable evaluation of the magnetic Prandtl number characterising the numerical results. Additionally, it is found that small-scale dynamos are active and can amplify a small seed magnetic field up to significant values in (COBOLD)-B-5 simulations with a grid spacing smaller than h & x2004;=& x2004;12 & x2006;km, even at Pr-m & x2004;similar or equal to & x2004;0.65. However, it is also evident that it is difficult to categorise dynamo simulations in terms of Pr-m alone, because it is not only important to estimate the amplitude of the dissipation coefficients, but also at which scales energy dissipation takes place.

Automated summary

This study proposes a method to accurately estimate important parameters in MHD simulations and categorizes small-scale dynamo simulations. The results show that under certain conditions, a small seed magnetic field can rapidly amplify, but it is difficult to categorize dynamo simulations based solely on the magnetic Prandtl number.