Rugged magneto-hydrodynamic invariants in weakly collisional plasma turbulence: Two-dimensional hybrid simulation results

Abstract

Aims. We investigated plasma turbulence in the context of solar wind. We concentrated on properties of ideal second-order magneto-hydrodynamic (MHD) and Hall MHD invariants. Methods. We studied the results of a two-dimensional hybrid simulation of decaying plasma turbulence with an initial large cross helicity and a negligible magnetic helicity. We investigated the evolution of the combined energy and the cross, kinetic, mixed, and magnetic helicities. For the combined energy and the cross, kinetic, and mixed helicities, we analysed the corresponding Kármán-Howarth-Monin (KHM) equation in the hybrid (kinetic proton and fluid electron) approximation. Results. The KHM analysis shows that the combined energy decays at large scales. At intermediate scales, this energy cascades (from large to small scales) via the MHD non-linearity and this cascade partly continues via Hall coupling to sub-ion scales. The cascading combined energy is transferred (dissipated) to the internal energy at small scales via the resistive dissipation and the pressure-strain effect. The Hall term couples the cross helicity with the kinetic one, suggesting that the coupled invariant, referred to here as the mixed helicity, is a relevant turbulence quantity. However, when analysed using the KHM equations, the kinetic and mixed helicities exhibit very dissimilar behaviours to that of the combined energy. On the other hand, the cross helicity, in analogy to the energy, decays at large scales, cascades from large to small scales via the MHD+Hall non-linearity, and is dissipated at small scales via the resistive dissipation and the cross-helicity equivalent of the pressure-strain effect. In contrast to the combined energy, the Hall term is important for the cross helicity over a wide range of scales. The magnetic helicity is scantily generated through the resistive term and does not exhibit any cascade.

Figures (13)

• Figure 1: Evolution of different quantities as a function of time: (a) Relative changes in the kinetic energy $\Delta E_{\text{kin}}$ (dashed line), magnetic energy $\Delta E_{\text{mag}}$ (solid line), internal energy $\Delta E_{\text{int}}$ (dash-dotted line), and total energy $\Delta E_{\text{tot}}$ (dotted line), (b) resistive dissipation rate $Q_\eta$ (dashed line) and pressure-strain effective dissipation rate $\psi$ (solid line), (c) relative changes in the kinetic helicity $\Delta H_k$ (dash-dotted line), mixed helicity $\Delta H_x$ (solid line), and cross helicity $\Delta H_c$ (dashed line), (d) resistive cross-helicity dissipation rate $\epsilon_{\eta Hc}$ (dotted line), pressure-strain effective cross-helicity dissipation rate $\psi_{Hc}$ (dashed line), pressure-strain effective mixed-helicity dissipation rate $\psi_{Hx}$ (solid line), and pressure-strain effective kinetic-helicity dissipation rate $\psi_{Hk}$ (dash-dotted line), (e) relative change in the magnetic helicity $\Delta H_m$, and (f) resistive magnetic-helicity dissipation rate $\epsilon_{Hm}$.
• Figure 2: Omnidirectional spectral properties of different quantities at $t\Omega_i = 700$: (a) Power spectral densities of (solid) the magnetic field $\boldsymbol{B}$, $P_B$ (solid), and compensated proton velocity field $\boldsymbol{w}$, $P_w$ (dashed), as a function of $k$ normalised to $d_i$. (b) Cross helicity co-spectrum $P_{Hc}$ (solid) and (absolute value of) kinetic helicity co-spectrum $P_{Hk}$ (dashed) as a function of $k$. The dotted lines show a spectrum $\propto k^{-5/3}$ for comparison.
• Figure 3: Isotropised energy KHM analysis at $t\Omega_i=700$: Validity test $O$ (black line) as a function of $l$ along with the different contributing terms: decay rate $-\partial_t S$ (blue), MHD cascade rate $K_\text{MHD}$ (green), Hall cascade rate $K_\text{Hall}$ (orange), resistive dissipation rate $-D$ (red), and pressure-strain rate $-\varPsi$ (magenta). All the quantities are normalised to the effective total dissipation rate $Q$.
• Figure 4: Evolution of isotropised energy KHM results. Shown are the different KHM terms as a function of time $t$ and $l$: (a) decay rate $\partial_t S$, (b) MHD cascade rate $K_\text{MHD}$, (c) Hall cascade rate $K_\text{Hall}$, (d) resistive dissipation rate $D$, and (e) pressure-strain rate $\varPsi$. All the quantities are normalised to the effective total dissipation rate $Q$ (averaged over $600 \le t\Omega_i \le 700$).
• Figure 5: Isotropised mixed-helicity KHM analysis at $t\Omega_i=700$: Validity test $O_{Hx}$ (black line) as a function of $l$ along with the different contributing terms: decay rate $-\partial_t {S\!}_{Hx}$ (blue), MHD term $K_{Hx}$ (green), resistive dissipation rate $-D_{Hc}$ (red), and pressure-strain rate $-\varPsi_{Hx}$ (magenta). All the quantities are normalised to the effective total (cross-helicity) dissipation rate $\epsilon_{Hc}$.