Finite Dissipation in Anisotropic Magnetohydrodynamic Turbulence

Abstract

In presence of an externally supported, mean magnetic field a turbulent, conducting medium, such as plasma, becomes anisotropic. This mean magnetic field, which is separate from the fluctuating, turbulent part of the magnetic field, has considerable effects on the dynamics of the system. In this paper, we examine the dissipation rates for decaying incompressible magnetohydrodynamic (MHD) turbulence with increasing Reynolds number, and in the presence of a mean magnetic field of varying strength. Proceeding numerically, we find that as the Reynolds number increases, the dissipation rate asymptotes to a finite value for each magnetic field strength, confirming the Kármán-Howarth hypothesis as applied to MHD. The asymptotic value of the dimensionless dissipation rate is initially suppressed from the zero-mean-field value by the mean magnetic field but then approaches a constant value for higher values of the mean field strength. Additionally, for comparison, we perform a set of two-dimensional (2DMHD) and a set of reduced MHD (RMHD) simulations. We find that the RMHD results lie very close to the values corresponding to the high mean-field limit of the threedimensional runs while the 2DMHD results admit distinct values far from both the zero mean field cases and the high mean field limit of the three-dimensional cases. These findings provide firm underpinnings for numerous applications in space and astrophysics wherein von Kármán decay of turbulence is assumed.

Enhanced Energy Transfer Rate in Solar Wind Turbulence Observed near the Sun from Parker Solar Probe

“…These values are derived from the dimensionless dissipation rate, C , in MHD turbulence: α ± = 2C /9 √ 3 (Usmanov ). The precise values of α ± depend on several parameters, e.g., the mean-magnetic-field strength, cross helicity, magnetic helicity, but usually remain close to the generic values used here (Matthaeus et al 2004;Linkmann et al 2015Linkmann et al , 2017Bandyopadhyay et al 2018a).…”

On Exact Laws in Incompressible Hall Magnetohydrodynamic Turbulence

“…Based on these remarks and on the behavior of the three laws we conclude that, as expected, b 0 does not contribute explicitly to the incompressible energy cascade rate and, in the purpose of computing ε Hall , that it should be removed from the simulation of the spacecraft data beforehand in order to minimize the possible numerical errors that it can generate. Note that this property does not mean that b 0 has no influence on the nonlinear dynamics (Galtier et al 2000;Wan et al 2012;Oughton et al 2013): it is actually expected that the energy cascade rate ε Hall decreases with increasing b 0 , as shown recently with DNSs (Bandyopadhyay et al 2018). It is worth mentioning that the situation is very different in compressible law (e.g., Banerjee & Galtier 2013) where the b 0 dependence is explicit and cannot a priori be ruled out (Hadid et al 2017).…”

Contextual Predictions forParker Solar Probe. II. Turbulence Properties and Taylor Hypothesis

“…Other notations are: V A = B(4πρ) −1/2 is the mean Alfvén velocity, σ D = v 2 − b 2 /Z 2 is the normalized energy difference that we continue treating as a constant parameter (= −1/3) derived from observations, α and β are the Kármán-Taylor constants (see Matthaeus et al 1996;Smith et al 2001;Breech et al 2008), and Matthaeus et al 2004). The last term on the right-hand side of Equation (1) is the von Kármán turbulence heating rate (de Kármán & Howarth 1938) adapted for MHD (Hossain et al 1995;Wan et al 2012;Bandyopadhyay et al 2018) and plasma (Wu et al 2013). The fluctuation energy loss due to von Kármán decay is balanced in a quasi-steady state by internal energy supply in the pressure equations, with Q T = αf + (σ c )Z 3 /(2λ).…”