Residual energy in weakly compressible turbulence with a mean guide field

TL;DR

This study analyzes residual energy $E_r=E_{kin}-E_{mag}$ in weakly compressible MHD turbulence with a strong guide field, using DNS with the PENCIL code across sub-Alfvénic, quasi-static forcing and three beta values. By comparing magnetic versus kinetic driving, it shows magnetically driven turbulence yields an approximately zero $E_r$ in the inertial range with a $E(k)\propto k^{-3/2}$ cascade (dynamic alignment), while kinetically driven turbulence produces a positive $E_r$ and a shallower $E(k)\propto k^{-1}$ cascade, with the residual-energy slope $\alpha$ depending on $\beta$ as $\alpha \in [-2,-5/3], [-5/3,-3/2],$ or $\approx -1$ for $\beta=4.0,1.0,0.3$ respectively. The work links incompressible and highly compressible regimes by attributing the observed scalings to the driving mechanism: Alfvénic, locally imbalanced cascades under magnetic forcing versus reflection-driven cascades under kinetic forcing, modulated by density inhomogeneities and compressible effects. The findings have implications for interpreting Solar wind turbulence, suggesting where and how positive residual energy may arise in interplanetary space.

Abstract

The energy distribution is a fundamental property of magnetohydrodynamic (MHD) turbulence. In strongly magnetized turbulence energy imbalances can arise, quantified by the so-called residual energy: $E_r~=~(E_{kin}~ - ~E_{mag})$; $E_{kin}$ and $E_{mag}$ stand for the volume-averaged kinetic and magnetic energy, respectively. Numerical simulations of incompressible turbulence yield $E_r < 0$, which is consistent with Solar wind observations, while in highly compressible turbulence simulations $E_r > $ 0. Differences arise in the cascade of $E_r$ between the two regimes. We explore the properties of $E_r$ in weakly compressible MHD turbulence in the presence of an initially strong (guide) magnetic field. We study the influence of different driving mechanisms and field strengths on the cascade of $E_r$. We run a suite of direct numerical simulations with the PENCIL code. All simulations are maintained through forcing in a quasi-static regime with sonic Mach numbers close to 0.1. We solely change the Alfvén Mach number, or equivalently the plasma beta ($β$) of the simulations. We drive turbulence by either injecting velocity or magnetic fluctuations at large scales and study the power spectra of kinetic, magnetic, density, and $E_r$. Magnetically-driven simulations show locally imbalanced Alfvénic fluctuations and a $\propto k^{-3/2}$ cascade, consistent with the dynamic alignment theory. Kinetically-driven simulations give rise to a $\propto k^{-1}$ scaling, consistent with interactions between Alfvén waves scattered by density inhomogeneities -- a hallmark of reflection-driven turbulence. Residual energy is positive with a spectral slope ($α$) depending on $β$ as: for $β= 4.0$, $-2 \lesssim α\lesssim -5/3$, for $β= 1.0$, $-5/3 \lesssim α\lesssim -3/2$, and for $β= 0.3$, $α\approx -1$.

Figures (10)

• Figure 1: Time evolution of kinetic (solid) and magnetic (dashed) energy of kinetically- (black) and magnetically- (blue) driven turbulence with $\beta \approx 1$ and $\mathcal{M}_{S}$$\approx 0.1$. Energy is in dimensionless units and time is normalized with the eddy turnover time. Driving affects the saturation level of the magnetic energy, yielding $\mathcal{R_A}$$\lesssim 1$ for magnetic, and $\mathcal{R_A}$$\approx 2.5$ for kinetic driving in the quasi-static regime, $t/t_{eddy}~\epsilon~[5, 12]$.
• Figure 2: Alfvén ratio as a function of plasma beta of magnetically- (blue) and kinetically- (black) driven turbulence. Horizontal line corresponds to perfect balance between kinetic and magnetic energies. The impact of forcing in the obtained volume-averaged energetics is evident: kinetic driving yields $\mathcal{R_A}$$> 1$ and magnetic driving $\mathcal{R_A}$$\lesssim 1$,
• Figure 3: Fluctuating-to-order magnetic field ratio as a function of Alfvén Mach number. Kinetically- and magnetically-driven simulations are shown as black and blue dots respectively. Black dashed line corresponds to linear scaling $\delta u \sim \delta B$, while cyan to $\delta u \sim \sqrt{\delta B B_0}$. The numerical data strongly favor the linear scaling for both types of driving.
• Figure 4: Kinetic (black) and magnetic (blue) compensated power spectra. Results of mangetically-driven turbulence are shown in the bottom row, while of kinetically-driven in the top. From let to right the initial magnetic field strength increases, or equivalently $\mathcal{M}_{A}$ (and $\beta$) decreases. Dashed and dashed-dotted black lines correspond to the kinetic power spectrum of the solenoidal and compressible modes, obtained from Helmholtz decomposition. The two power law scalings (-1 and -3/2) are shown for comparison. Incompressible modes carry the majority of kinetic energy in all simulations. In kinetically-driven turbulence the turbulence cascade is shallower than magnetically-driven simulations. There is a systematic excess in kinetic energy which leads to a positive residual energy. In kinetically-driven simulations, the scaling of compressible modes is -3/2, which is different than the -1 scaling of incompressible modes.
• Figure 5: Slow mode pressure balance (Eq. \ref{['eq:slow_mode']}) as a function of plasma beta. Black and blue points correspond to kinetic and magnetic driving respectively. The slow mode relation accurately describes the numerical results, especially of simulations with $\beta \geq 1$.