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Context. A crucial step in the numerical investigation of small-scale dynamosin the solar atmosphere consists of an accurate determination of the magneticPrandtl number, Prm, stemming from radiative magneto-hydrodynamic (MHD)simulations. Aims. The aims are to provide a reliable methodology forestimating the effective Reynolds and magnetic Reynolds numbers, Re and Rem,and their ratio Prm=Rem/Re (the magnetic Prandlt number), that characterise MHDsimulations and to categorise small-scale dynamo simulations in terms of thesedimensionless parameters. Methods. The methodology proposed for computing Reand Rem is based on the method of projection on proper elements and it relieson a post-processing step carried out using higher order accurate numericaloperators than the ones in the simulation code. A number of radiative MHDsimulations with different effective viscosities and plasma resistivities werecarried out with the CO5BOLD code, and the resulting growth rate of themagnetic energy and saturated magnetic field strengths were characterised interms of Re and Rem. Results. Overall, the proposed methodology provides asolid estimate of the dissipation coefficients affecting the momentum andinduction equations of MHD simulation codes, and consequently also a reliableevaluation of the magnetic Prandtl number characterising the numerical results.Additionally, it is found that small-scale dynamos are active and can amplify asmall seed magnetic field up to significant values in CO5BOLD simulations witha grid spacing smaller than h=12 km, even at Prm=0.65. However, it is alsoevident that it is difficult to categorise dynamo simulations in terms of Prmalone, because it is not only important to estimate the amplitude of thedissipation coefficients, but also at which scales energy dissipation takesplace.

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