Residual energy in weakly compressible turbulence with a mean guide field

Technical Summary: Residual Energy in Weakly Compressible Turbulence with a Mean Guide Field

Problem Statement
Magnetohydrodynamic (MHD) turbulence is characterized by the distribution of energy across spatial scales, a property heavily influenced by the sonic ($M_S$) and Alfvénic ($M_A$) Mach numbers, as well as the driving mechanisms. A key metric in this distribution is the residual energy, $E_r = E_{kin} - E_{mag}$, which quantifies the imbalance between kinetic and magnetic energy. Previous studies have established distinct behaviors for different regimes: incompressible, strongly magnetized turbulence typically yields $E_r < 0$ (magnetic excess), consistent with solar wind observations, while highly compressible, supersonic turbulence often exhibits $E_r > 0$ (kinetic excess). However, the properties of residual energy in the intermediate, weakly compressible regime ($M_S \sim 0.1$) with a strong mean guide field remain less understood, particularly regarding how different forcing mechanisms (kinetic vs. magnetic) influence the cascade and spectral scaling of $E_r$.

Methodology
The authors conducted a suite of direct numerical simulations (DNS) using the PENCIL code to solve the isothermal, hyper-visco-resistive MHD equations. The simulations were performed in a periodic cubic domain with a resolution of $256^3$ (and $512^3$ for convergence tests).

• Regime: The turbulence was maintained in a quasi-static regime with a sonic Mach number $M_S \approx 0.1$ to isolate weak compressibility effects.
• Parameters: Three initial magnetic field strengths ($B_0 = 0.5, 1.0, 2.0$) were tested, corresponding to plasma beta values of $\beta = 4.0, 1.0,$ and $0.3$. The Alfvén Mach number ($M_A$) varied accordingly.
• Forcing: Two distinct driving mechanisms were employed:
  1. Kinetic Driving ($K$): Velocity fluctuations injected at large scales ($k_f \sim 1.5$).
  2. Magnetic Driving ($M$): Magnetic field fluctuations injected at large scales.
• Analysis: The study analyzed volume-averaged energetics, power spectra of kinetic, magnetic, density, and residual energy, and the correlation lengths of solenoidal and compressible modes. The authors also examined the normalized cross-helicity ($\sigma_c$) and residual energy ($\sigma_r$) distributions.

Key Results
The simulations revealed that the forcing mechanism is the dominant factor determining the energy partition and spectral scaling, even when global Mach numbers are comparable.

1. Magnetically Driven Turbulence:

   • Energy Balance: These simulations exhibit a balanced cascade with $E_r \approx 0$ throughout the inertial range. The Alfvén ratio ($R_A = E_{kin}/E_{mag}$) is approximately $0.7-0.8$($E_r < 0$).
   • Scaling: Both kinetic and magnetic power spectra follow an Iroshnikov-Kraichnan (IK) scaling of $E(k) \propto k^{-3/2}$.
   • Dynamics: The turbulence is dominated by Alfvénic fluctuations. While the global cascade is balanced, local patches exhibit strong imbalances in cross-helicity, consistent with the theory of dynamic alignment. Compressible modes passively follow the Alfvénic cascade, satisfying the slow-mode pressure balance relation.

2. Kinetically Driven Turbulence:

   • Energy Balance: These simulations show a systematic excess of kinetic energy over magnetic energy ($E_r > 0$) across all scales in the inertial range, with $R_A \approx 2.5 - 3.4$.
   • Scaling: The power spectra for both kinetic and magnetic energy are significantly shallower, scaling as $E(k) \propto k^{-1}$.
   • Residual Energy Scaling: The cascade of residual energy ($E_r \propto k^\alpha$) depends on $\beta$:
     • $\beta = 4.0$: $-2 \lesssim \alpha \lesssim -5/3$
     • $\beta = 1.0$: $-5/3 \lesssim \alpha \lesssim -3/2$
     • $\beta = 0.3$: $\alpha \approx -1$
   • Dynamics: The $k^{-1}$ scaling suggests weakly interacting modes that preserve high coherence. The normalized cross-helicity is weak and fragmented, indicating a lack of strong dynamic alignment. Density fluctuations are directly injected and advected, creating a slowly varying inhomogeneous background that sustains coupling between Elsasser fields.

3. Fluctuation Scaling:
   Contrary to some incompressible studies suggesting $\delta u \sim \sqrt{B_0}\delta B$ for kinetic driving, this work finds a linear Alfvénic scaling ($\delta u \sim \delta B$) for both driving types in the weakly compressible regime.

Significance and Conclusions
The paper concludes that the energy partition in weakly compressible turbulence is strongly dictated by the forcing mechanism, not just by the global Mach numbers.

• Mechanism Dependence: Magnetic driving leads to a balanced, Alfvénic cascade ($k^{-3/2}$) with negligible residual energy, consistent with dynamic alignment theory. Kinetic driving generates a shallow cascade ($k^{-1}$) with positive residual energy, driven by the persistence of large-scale density inhomogeneities that couple Elsasser fields.
• Solar Wind Context: The results suggest that the positive residual energy and $k^{-1}$ scaling observed in specific solar wind regions (e.g., slow wind streams or sheath regions) may be attributed to kinetic-like driving or density inhomogeneities, whereas the standard $k^{-3/2}$ scaling and magnetic excess ($E_r < 0$) are characteristic of magnetically driven, Alfvénic turbulence.
• Theoretical Implication: The study bridges the gap between incompressible and highly compressible regimes, demonstrating that even weak compressibility ($M_S \sim 0.1$) allows density inhomogeneities to fundamentally alter the turbulence cascade when driven kinetically, sustaining correlations that would otherwise decorrelate in purely magnetic driving scenarios.