11:00 AM - 1:00 PM
[PEM16-P05] MHD simulations of the magnetorotational instability with a low-dissipation Riemann solver
Keywords:MHD simulation, Magnetorotational instability, Magnetic Prandtl number, Approximate Riemann solver, All-speed scheme
An MHD simulation is a powerful tool for studying the macroscopic dynamics of laboratory, space, and astrophysical plasmas. To tackle a situation including supersonic flows, many modern MHD simulation codes adopt shock-capturing schemes, which is built based on the solution to the Riemann problem in one-dimensional hyperbolic conservation laws. In particular, the Roe-type (Brio and Wu 1988; Balsara 1998) and Harten-Lax-van Leer discontinuities (HLLD; Miyoshi and Kusano 2005) approximate Riemann solves are extensively implemented by virtue of their robustness and accuracy. In practical multidimensional MHD simulations, however, these schemes may suffer from numerical difficulties, which include a numerical shock instability for high Mach number flows and a degradation of the solution accuracy for low Mach number flows. These difficulties severely restrict a range of Mach numbers available for simulations. The preservation of the solenoidal condition of the magnetic field is also a problem for the MHD simulation with the shock-capturing scheme. We proposed new techniques to overcome these difficulties (Minoshima et al. 2019, 2020, 2021). Especially, Minoshima et al. (2021) proposed a low-dissipation HLLD (LHLLD) approximate Riemann solver to avoid the numerical shock instability and improve the accuracy of low Mach number flows. The scheme is quite easy for current HLLD scheme users to implement.
To demonstrate the performance of the LHLLD scheme in practical problems, we revisit the gas pressure dependence of the saturation level of the magnetorotational instability (MRI) presented by Sano et al. (2004). They conducted the nonlinear simulation of the MRI with a large set of simulation parameters, and they showed weak positive dependence of the saturation level on the gas pressure. However, subsequent study by Minoshima et al. (2015) have argued that the gas pressure dependence could be affected by the numerical magnetic Prandtl number, depending on the design of the numerical code. We re-examine the gas pressure dependence with the LHLLD and HLLD schemes. We find that the solution with the LHLLD scheme is independent of the pressure (consistent with the incompressible limit), whereas the solution with the HLLD scheme shows positive dependence (similar to Sano et al.). This stems from the fact that the HLLD scheme is designed to give the numerical viscosity with a scale of the fast magnetosonic speed and the numerical resistivity with a scale of the Alfven speed, and hence the numerical magnetic Prandtl number increases with the gas pressure and eventually it affects the solution. The LHLLD scheme corrects the numerical viscosity with a scale of the Alfven speed (for high beta cases) so that the numerical magnetic Prandtl number remains constant with varying gas pressure. Therefore, the LHLLD scheme is suitable to survey a wide parameter range. Using this scheme, we attempt to derive a scaling law of the saturation level of the MRI, particularly focusing on the magnetic Prandtl number dependence, which could be of importance for systems having extreme magnetic Prandtl numbers (~0.01-100) such as accretion disks around compact objects (e.g., Balbus et al. 2008).
To demonstrate the performance of the LHLLD scheme in practical problems, we revisit the gas pressure dependence of the saturation level of the magnetorotational instability (MRI) presented by Sano et al. (2004). They conducted the nonlinear simulation of the MRI with a large set of simulation parameters, and they showed weak positive dependence of the saturation level on the gas pressure. However, subsequent study by Minoshima et al. (2015) have argued that the gas pressure dependence could be affected by the numerical magnetic Prandtl number, depending on the design of the numerical code. We re-examine the gas pressure dependence with the LHLLD and HLLD schemes. We find that the solution with the LHLLD scheme is independent of the pressure (consistent with the incompressible limit), whereas the solution with the HLLD scheme shows positive dependence (similar to Sano et al.). This stems from the fact that the HLLD scheme is designed to give the numerical viscosity with a scale of the fast magnetosonic speed and the numerical resistivity with a scale of the Alfven speed, and hence the numerical magnetic Prandtl number increases with the gas pressure and eventually it affects the solution. The LHLLD scheme corrects the numerical viscosity with a scale of the Alfven speed (for high beta cases) so that the numerical magnetic Prandtl number remains constant with varying gas pressure. Therefore, the LHLLD scheme is suitable to survey a wide parameter range. Using this scheme, we attempt to derive a scaling law of the saturation level of the MRI, particularly focusing on the magnetic Prandtl number dependence, which could be of importance for systems having extreme magnetic Prandtl numbers (~0.01-100) such as accretion disks around compact objects (e.g., Balbus et al. 2008).