Magnetic field distribution in the quiet Sun: a simplified model approach

Context. The quiet Sun presents magnetized plasma whose field strengths vary from zero to about 2 kG. The probability density function of the magnetic field strength B efficaciously describes the statistical properties of the quiet Sun magnetic field. Aims. We simulate the dynamics and the evolution of quiet Sun magnetic elements to produce a probability density function of the field strengths associated with such elements. Methods. The dynamics of the magnetic field are simulated by means of a numerical model in which magnetic elements are driven passively by an advection field characterized by spatio-temporal correlations that mimick the granulation and mesogranulation scales observed on the solar surface. The field strength can increase due to an amplification process that occurs where the magnetic elements converge. Starting from a δ-like probability density function centered on B = 30 G, we obtain magnetic field strengths of up to 2 kG (in absolute value). To derive the statistical properties of the magnetic elements, several simulation runs are performed. Results. Our model is able to produce kG magnetic fields in a time interval of the order of the granulation timescale. The mean unsigned flux density and the mean magnetic energy density of the synthetic quiet Sun reach values of 〈B〉 ≃ 100 G and 〈B2〉1/2 ≃ 350 G respectively in the stationary regime. The derived probability density function of the magnetic field strength decreases rapidly from B = 30 G to B ∼ 100 G and has a secondary maximum at B = 2 kG. From this result, it follows that magnetic fields ≥ 700 G dominate the unsigned flux density and magnetic energy density, although the probability density function of the field strength reaches a maximum at B ∼ 10 G. Conclusions. A photospheric advection field with spatio-temporal correlations, driving the magnetic elements, and reduced magnetic amplification rules are able to create a realistic probability density function of the quiet Sun magnetic field. It has been found that they naturally produce an excess of magnetic fields around 2 kG if an upper limit is imposed on the field strength.