Issue 
A&A
Volume 690, October 2024



Article Number  A174  
Number of page(s)  12  
Section  The Sun and the Heliosphere  
DOI  https://doi.org/10.1051/00046361/202450313  
Published online  08 October 2024 
Rugged magnetohydrodynamic invariants in weakly collisional plasma turbulence: Twodimensional hybrid simulation results
^{1}
Astronomical Institute of the Czech Academy of Sciences, Prague, Czech Republic
^{2}
Institute of Atmospheric Physics of the Czech Academy of Sciences, Prague, Czech Republic
^{3}
Department of Electromagnetism and Electronics, University of Murcia, Murcia, Spain
Received:
10
April
2024
Accepted:
14
August
2024
Aims. We investigated plasma turbulence in the context of solar wind. We concentrated on properties of ideal secondorder magnetohydrodynamic (MHD) and Hall MHD invariants.
Methods. We studied the results of a twodimensional 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 (kinetic plus magnetic) energy and the cross, kinetic, and mixed helicities, we analysed the corresponding KármánHowarthMonin (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 nonlinearity and this cascade partly continues via Hall coupling to subion scales. The cascading combined energy is transferred (dissipated) to the internal energy at small scales via the resistive dissipation and the pressurestrain 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 nonlinearity, and is dissipated at small scales via the resistive dissipation and the crosshelicity equivalent of the pressurestrain effect. In contrast to the combined energy, the Hall term is important for the cross helicity over a wide range of scales (even well above ion scales). In contrast, the magnetic helicity is scantily generated through the resistive term and does not exhibit any cascade.
Key words: magnetohydrodynamics (MHD) / turbulence / solar wind
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
Fluids in a turbulent regime transfer the energy of fluctuations across scales due to their nonlinear interactions, in a process known as energy cascade. Nonlinearities not only act upon the energy of fluctuations, but also on other turbulence properties as well. This enables the formation of multiple cascades in the system, with the development of one affecting that of another (Alexakis & Biferale 2018). Properties of incompressible magnetohydrodynamic (MHD) fluid are strongly determined by three secondorder (rugged) invariants: combined (kinetic plus magnetic) energy, and cross and magnetic helicities. While the magnetic helicity remains an ideal invariant even in the compressible Hall MHD, the presence of the Hall term couples the cross and kinetic helicities; a combination of the two, the mixed helicity (also called X helicity; see Pouquet & Yokoi 2022) is the ideal invariant appropriate for the incompressible Hall MHD. These ideal invariants are particularly relevant in turbulence. Cascade and dissipation of the combined energy is one of the central problems of (Hall) MHD turbulence and those processes are influenced by the different helicities, which in turn may exhibit analogous cascade and dissipation phenomena to the energy ones. All these processes constrain and change the dynamics of turbulent systems. For instance, cross helicity may reduce the energy cascade (Dobrowolny et al. 1980) and plays an important role in the generation of magnetic fields through turbulence dynamo processes (Yokoi 1999); it may also affect the repartition of heating between ions and electrons (Squire et al. 2023).
The solar wind exhibits important fluctuations of the magnetic field, plasma bulk velocity, and other quantities over a wide range of scales (Kiyani et al. 2015; Alexandrova et al. 2021). These fluctuations often have powerlawlike spectral properties and are nonlinearly coupled; the solar wind system is an example of a turbulent system in weakly collisional plasmas (Bruno & Carbone 2013). While slow solar wind streams have a weak average cross helicity (however see e.g. D’Amicis & Bruno 2015; Shi et al. 2021), fast solar wind streams have typically relatively strong average cross helicities. These highcross helicity streams are also called Alfvénic streams as they exhibit important correlations or anticorrelations between the magnetic field and the plasma bulk velocities, which are characteristic of Alfvén waves (Belcher & Davis 1971; Grappin et al. 1991). Plasma turbulence in highcross helicity streams and the behaviour of the combined energy and cross helicity are often studied in the incompressible MHD approximation and the energy and helicity is replaced by pseudoenergies in the terms of the Elsässer variables (combinations of the magnetic field in the Alfén units and the plasma bulk velocity; see Tu et al. 1989). In situ observations in the solar wind, as well as numerical simulations, indicate that the two pseudoenergies cascade in parallel, which means that the combined energy and cross helicity cascade in parallel. However, at variance with predictions of incompressible MHD (Dobrowolny et al. 1980), the combined energy in the solar wind decreases more slowly with the radial distance than the cross helicity, and consequently the relative cross helicity decreases. It is therefore important to determine the influence of compressive and nonideal (Hall and kinetic) effects.
Compressive effects break the invariance of the combined energy and the mixed helicity, but the solar wind exhibits only weak density fluctuations (Shi et al. 2021), meaning that compressibility effects are likely not dominant in solar wind turbulence; the compressible MHD turbulence simulations of MontagudCamps et al. (2022) behave qualitatively similar to incompressible simulations, again in contrast with observations. For weakly collisional solarwind plasmas, it is also necessary to investigate the consequences of formation of nonMaxwellian particle distribution functions, which requires the tensor description of the particle pressure (Del Sarto et al. 2016). One of these consequences is the pressurestrain coupling, which appears to be very important, as it may work as an effective dissipation rate for the combined energy (Yang et al. 2017; Matthaeus et al. 2020). In this paper, we address the question how the pressurestrain effect influences the different helicities in order to complete our understanding of the rugged invariants in weakly collisional plasma turbulence. We analysed results of the twodimensional pseudospectral hybrid simulation of decaying, highcross helicity plasma turbulence. We used the KármánHowarthMonin (KHM) equation (de Kármán & Howarth 1938; Politano & Pouquet 1998a,b) in the hybrid approximation for the energy (Hellinger et al. 2024) and we derived the equivalent equations for the helicities. The paper is organised as follows: In Section 2 we describe the pseudospectral hybrid code and in Section 3 we summarise the governing equations and their conservation properties. Section 4 presents the overall simulation results and in Section 5 we analyse the KHM properties of the energy (subsect. 5.1) and the cross, kinetic, and mixed helicities (subsect. 5.2). Finally, in Section 6 we summarise and discuss the simulation results. In the Appendix we complement our KHM results with spatial and spectral filtering techniques.
2. Numerical code
For the numerical simulation, we used the 2D pseudospectral version of the hybrid code based on the model of Matthews (1994). In this code, the electrons are considered as a massless chargeneutralising fluid, with a constant temperature, whereas ions are treated as particles. Ions have positions and velocities separated by half time steps and are advanced by the Boris scheme, which requires the fields to be known at a half time step ahead of the particle velocities. This is obtained by advancing the current density to this time (with only one computational pass through the particle data at each time step). The same grid is used for all the fields and their spatial derivatives, needed for the time advance (cf. Valentini et al. 2007), are calculated with the fast Fourier transform (Frigo & Johnson 2005). The particle contribution to the current density at the relevant nodes is evaluated with bilinear weighting followed by smoothing over three points. No smoothing is performed on the electromagnetic fields. A small resistivity is used in the Ohm’s law to avoid an accumulation of energy at small scales. The magnetic field is advanced in time with a modified midpoint method, which allows time substepping to advance the field.
The units and parameters of the simulation are as follows: units of space and time are the ion inertial length d_{i} = c/ω_{pi} and the (inverse of the) ion gyrofrequency Ω_{i}, respectively, where ω_{pi} = (n_{p}e^{2}/m_{p}ϵ_{0})^{1/2} is the proton plasma frequency. In these expressions, n_{p} and B_{0} are the density of the plasma protons and magnitude of the initial magnetic field, respectively, while e and m_{p} are the proton electric charge and mass, respectively; and, finally c, ϵ_{0}, and μ_{0} are the speed of light and the dielectric and magnetic permeabilities of vacuum, respectively. The spatial resolution is Δx = Δy = d_{i}/8. There are 16 384 particles per cell for protons; β_{i} = β_{e} = 0.5. The simulation box is in the xy plane and is assumed to be periodic in both dimensions. The fields and moments are defined on a 2D grid with dimensions 2048 × 2048. The time step for the particle advance is dt = 0.01Ω_{i}^{−1} while the magnetic field B is advanced with a smaller time step, dt_{B} = dt/20. (The vector potential A is initialised from the magnetic field assuming the Coulomb gauge and is then evolved in time in the code.) The background magnetic field B_{0} is perpendicular to the simulation plane. Following Franci et al. (2015), we initialised the system with an isotropic 2D spectrum of modes with random phases, linear Alfvén polarisation (δB ⊥ B_{0}) over large scales k ≤ 0.1d_{i}^{−1} with a flat 1D spectrum. The system initially has the (rms) amplitude of magnetic field fluctuations δB/B_{0} = 0.3, and the relative cross helicity σ_{c} = 0.6. The magnetic and kinetic helicities are initially almost zero.
3. Governing equations
We investigated a system governed by the following equations for the plasma density ρ, the plasma mean velocity u, and the magnetic field B:
$$\begin{array}{cc}\hfill \frac{\partial \rho}{\partial t}+(\mathit{u}\xb7\mathbf{\nabla})\rho & =\rho \mathbf{\nabla}\xb7\mathit{u},\hfill \end{array}$$(1)
$$\begin{array}{cc}\hfill \frac{\partial \mathit{u}}{\partial t}+(\mathit{u}\xb7\mathbf{\nabla})\mathit{u}& =\frac{\mathit{J}\times \mathit{B}}{\rho}\frac{\mathbf{\nabla}\xb7\mathbf{P}}{\rho}\hfill \end{array}$$(2)
$$\begin{array}{cc}\hfill \frac{\partial \mathit{B}}{\partial t}& =\mathbf{\nabla}\times [(\mathit{u}\mathit{j})\times \mathit{B}]+\eta {\mathrm{\nabla}}^{2}\mathit{B}.\hfill \end{array}$$(3)
Here P is the plasma pressure tensor, η is the electric resistivity, J is the electric current density, and j is the electric current density in velocity units, j = J/ρ_{c} = u − u_{e} (ρ_{c} and u_{e} being the charge density and the electron velocity, respectively). We assume SI units except for the magnetic permeability μ_{0}, which is set to one (SI results can be obtained by the rescaling B → Bμ_{0}^{−1/2}). Equation (3) (except the resistive term) can be derived taking moments of the Vlasov equation for protons and electrons, and assuming massless electrons. We added a resistive dissipation, as used in the hybrid code (cf. Hellinger et al. 2024).
We investigated properties of the different energies and helicities in electron–proton plasma in the hybrid approximation. For the sum of the (proton) kinetic (E_{kin} = ⟨ρu^{2}⟩/2) and the magnetic (E_{mag} = ⟨B^{2}⟩/2) energies averaged over a closed volume (denoted by angle brackets ⟨ • ⟩) we obtained the following budget equation:
$$\begin{array}{c}\hfill {\partial}_{t}({E}_{\phantom{\rule{0.333333em}{0ex}}}\text{kin}+{E}_{\phantom{\rule{0.333333em}{0ex}}}\text{mag})=Q,\end{array}$$(4)
where Q = Q_{η} + ψ denotes the total (effective) dissipation rate consisting of the pressurestrain rate ψ = −⟨P : ∇u⟩, and the resistive dissipation rate Q_{η} = η⟨J^{2}⟩. Following Kida & Orszag (1990), we defined a densityweighted velocity field w = ρ^{1/2}u and we represented the kinetic energy as a secondorder positively definite quantity, E_{kin} = ⟨w^{2}⟩/2.
For the averaged cross helicity (H_{c} = ⟨u ⋅ B⟩), we obtained the following equation:
$$\begin{array}{c}\hfill {\partial}_{t}{H}_{c}=\langle \mathit{\omega}\xb7(\mathit{j}\times \mathit{B})\rangle {\psi}_{\mathit{Hc}}{\u03f5}_{\eta Hc},\end{array}$$(5)
where ω = ∇ × u is the vorticity field, ψ_{Hc} = −⟨P : ∇(B/ρ)⟩ is the cross helicity equivalent of the pressurestrain rate (generalisation of the pressure–dilation coupling; see MontagudCamps et al. 2022), and ϵ_{ηHc} = η⟨ω ⋅ J⟩ is the cross helicity resistive dissipation rate. The first term on the righthand side of Eq. (5) couples the cross helicity to the kinetic helicity.
For the (rescaled) kinetic helicity (H_{k} = m⟨u ⋅ ω⟩/(2e)), we derived the dynamic equation
$$\begin{array}{c}\hfill {\partial}_{t}{H}_{k}=\langle \mathit{\omega}\xb7(\mathit{j}\times \mathit{B})\rangle {\psi}_{\mathit{Hk}},\end{array}$$(6)
where ψ_{Hk} = −m⟨P : ∇(ω/ρ)⟩/e is the kinetic helicity equivalent of the pressurestrain rate. In the above formulas, we chose to represent the kinetic helicity with the renormalisation factor m/e, that is, the proton masstocharge ratio, in order to have the cross and kinetic helicities in the same units. The mixed helicity, the sum of the cross and (renormalised) kinetic helicities, H_{x} = H_{c} + H_{k}, is then an ideal invariant of the hybrid system, and behaves as
$$\begin{array}{c}\hfill {\partial}_{t}{H}_{x}={\psi}_{\mathit{Hc}}{\u03f5}_{\eta Hc}{\psi}_{\mathit{Hk}}.\end{array}$$(7)
The magnetic helicity is a separate ideal invariant of Hall MHD as well as of the hybrid system. In this paper, we investigate properties of a simulated system with periodic boundary conditions and a background uniform magnetic field B_{0} = ⟨B⟩. In this case, the vector potential A_{0} generating B_{0} = ∇ × A_{0} is not periodic and affects the magnetic helicity conservation (Matthaeus & Goldstein 1982); we studied properties of the modified magnetic helicity H_{m} = ⟨A_{1} ⋅ (B_{1} + 2B_{0})⟩, where B_{1} and A_{1} are the fluctuating components of the magnetic and vector potential fields, B_{1} = B − B_{0} and A_{1} = A − A_{0}, respectively. For the averaged modified magnetic helicity, we obtained the following conservation properties:
$$\begin{array}{c}\hfill {\partial}_{t}{H}_{m}={\u03f5}_{\mathit{Hm}},\end{array}$$(8)
where ϵ_{Hm} = 2η⟨B ⋅ J⟩ is the resistive dissipation rate.
4. Simulation results
Figure 1 shows the evolution of different quantities as a function of time. In the simulation, the proton kinetic energy E_{kin} and the magnetic energy E_{mag} oscillate with opposite phases, suggesting energy exchanges, but overall these two quantities decrease. On the other hand, the proton internal energy, E_{int} = 3⟨p⟩/2, increases (here p is the scalar pressure, p = tr(P), tr being the trace). This heating (energisation) is weak initially (tΩ_{i} ≲ 200) and gets progressively stronger as turbulence develops. The total energy E_{tot} = E_{kin} + E_{int} + E_{mag} slowly decreases owing to resistive dissipation because electrons are assumed to be massless and isothermal. This behaviour is seen in Fig. 1a, which displays the relative changes in the kinetic (ΔE_{kin}), magnetic (ΔE_{mag}), internal (ΔE_{int}), and total (ΔE_{tot}) energies. The proton energisation is driven by the pressurestrain coupling, which plays the role of an effective dissipation channel. Figure 1b displays the pressurestrain effective dissipation rate ψ and the resistive dissipation rate Q_{η} (dashed line). The pressurestrain rate ψ exhibits a large initial surge owing to a relaxation of the initial conditions and then slowly increases and saturates; ψ oscillates in time largely owing to compressive effects. The resistive dissipation rate Q_{η} slowly increases, exhibits a local maximum at around tΩ_{i} ≃ 240 (due to the onset of reconnection). The resistive dissipation rate reaches the maximum at tΩ_{i} ≃ 550 and slowly decreases thereafter.
Fig. 1. Evolution of different quantities as a function of time: (a) Relative changes in the kinetic energy ΔE_{kin} (dashed line), magnetic energy ΔE_{mag} (solid line), internal energy ΔE_{int} (dashdotted line), and total energy ΔE_{tot} (dotted line), (b) resistive dissipation rate Q_{η} (dashed line) and pressurestrain effective dissipation rate ψ (solid line), (c) relative changes in the kinetic helicity ΔH_{k} (dashdotted line), mixed helicity ΔH_{x} (solid line), and cross helicity ΔH_{c} (dashed line), (d) resistive crosshelicity dissipation rate ϵ_{ηHc} (dotted line), pressurestrain effective crosshelicity dissipation rate ψ_{Hc} (dashed line), pressurestrain effective mixedhelicity dissipation rate ψ_{Hx} (solid line), and pressurestrain effective kinetichelicity dissipation rate ψ_{Hk} (dashdotted line), (e) relative change in the magnetic helicity ΔH_{m}, and (f) resistive magnetichelicity dissipation rate ϵ_{Hm}. 
The cross helicity H_{c} decreases with time, initially slowly, and later (tΩ_{i} ≳ 200) the decrease is faster and approximately linear in time; the kinetic helicity H_{k} very slightly increases at the beginning (from its nearly zero initial value) and remains roughly constant later on (tΩ_{i} ≳ 200); consequently, the mixed helicity H_{x} = H_{c} + H_{k} closely follows the evolution of the cross helicity H_{c}. This is shown in Fig. 1c, which displays the evolution of the relative changes in the cross (ΔH_{c}), mixed (ΔH_{x}), and kinetic (ΔH_{k}) helicities. We expected this evolution of the different helicities to be driven by the respective (effective) dissipation rates. Figure 1d shows the resistive crosshelicity dissipation rate ϵ_{ηHc}, and the equivalents of the pressurestrain effective dissipation: the crosshelicity rate ψ_{Hc}, the kinetichelicity rate ψ_{Hk}, and the mixedhelicity rate ψ_{Hx}. The resistive crosshelicity dissipation rate is negligible but the crosshelicity pressurestrain rate ψ_{Hc} is strong and reaches a roughly constant value at tΩ_{i} ≳ 240, which is compatible with the rate of decrease in the cross helicity. We interpret ψ_{Hc} as an effective dissipation rate of the cross helicity. On the other hand, the kinetichelicity ψ_{Hk} and the mixedhelicity ψ_{Hx} rates become negative for later times (tΩ_{i} ≳ 300), which is at variance with the time evolution of H_{k} and H_{x}; interpretations of the terms ψ_{Hk} and ψ_{Hx} are unclear. The pressurestrain terms ψ_{Hc} and ψ_{Hk} are dominated by incompressive fluctuations, and their compressive components ⟨p(B ⋅ ∇)ρ^{−1}⟩ and ⟨p(ω ⋅ ∇)ρ^{−1}⟩ are negligible due to small density variations (here p is the scalar pressure, p = tr(P), tr being the trace). The combined energy and the cross helicity decrease in time but the latter process is slower; consequently the relative cross helicity increases with time.
Finally, the (modified) magnetic helicity increases with time in a quadratic manner as displayed in Fig. 1e, which shows the relative change in this quantity (ΔH_{m}). This evolution is driven by the resistive magnetichelicity dissipation rate ϵ_{Hm}, which becomes negative and decreases approximately linearly for tΩ_{i} ≳ 200 (see Fig. 1f). In summary, Figure 1 indicates that, for tΩ_{i} ≳ 550, the different dissipation rates are relatively constant, a property which is expected in a fully developed turbulence system. Once this state is attained in a free decaying simulation, temporal variations of the fluctuating quantities can be considered negligible (we test this expectation using the KHM equations in Section 5).
During the evolution, the kinetic and magnetic energies spread over a wide range of scales and similar evolution is observed for the cross helicity. Figure 2 shows their omnidirectional spectral properties at tΩ_{i} = 700: Figure 2a displays power spectral densities of B and w fields, P_{B} and P_{w}, respectively. These power spectral densities exhibit a powerlawlike behaviour at large scales (with a slope somewhat less steep than −5/3) and they steepen at small scales, P_{w} at around kd_{i} ≃ 1 and P_{B} at around kd_{i} ≃ 3. The kinetic spectral density P_{w} becomes dominated by the noise for about kd_{i} ≳ 6, whereas the magnetic spectral density P_{B} seems to be affected by the noise for kd_{i} ≳ 20.
Figure 2b shows cospectra of the cross helicity P_{Hc} and the (absolute value of the) kinetic helicity P_{Hk}. P_{Hc} exhibits a powerlaw like behaviour at large scales with a slope similar to that of the energy power spectral densities and steepens somewhere between kd_{i} ≃ 1 and kd_{i} ≃ 3, which are the spectral break points of P_{w} and P_{B}. The kinetichelicity cospectrum P_{Hk} oscillates between positive and negative values and constitutes a small fraction of P_{Hc} at large scales and at small scales P_{Hk} becomes comparable to P_{Hc}; for kd_{i} ≳ 6, both quantities are dominated by the noise. As the magnetic helicity remains small during the whole simulation, we did not investigate its spectral or spatial decomposition properties.
Fig. 2. Omnidirectional spectral properties of different quantities at tΩ_{i} = 700: (a) Power spectral densities of (solid) the magnetic field B, P_{B} (solid), and compensated proton velocity field w, P_{w} (dashed), as a function of k normalised to d_{i}. (b) Cross helicity cospectrum P_{Hc} (solid) and (absolute value of) kinetic helicity cospectrum P_{Hk} (dashed) as a function of k. The dotted lines show a spectrum ∝k^{−5/3} for comparison. 
5. The KármánHowarthMonin equations
To understand crossscale transfers (cascades), exchanges, and dissipations of energies and helicities, we analysed the hybrid simulation results using the corresponding KHM equations. We started with the combined energy.
5.1. Energy
We characterised the spatial scale decomposition of the kinetic and magnetic (and their sum) energies using the secondorder structure functions:
$$\begin{array}{c}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{w}=\frac{1}{4}\langle {\delta \mathit{w}}^{2}\rangle ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{S\phantom{\rule{0.166667em}{0ex}}}_{B}=\frac{1}{4}\langle {\delta \mathit{B}}^{2}\rangle ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.333333em}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}S={S\phantom{\rule{0.166667em}{0ex}}}_{w}+{S\phantom{\rule{0.166667em}{0ex}}}_{B},\end{array}$$
where the deltas denote increments of the corresponding quantities; for example, δw = w(x + l)−w(x), with x being the position and l the separation (lag) vector. For the secondorder structure function, S, we obtain (Hellinger et al. 2024)
$$\begin{array}{c}\hfill {\partial}_{t}S={K}_{\phantom{\rule{0.333333em}{0ex}}}\text{MHD}+{K}_{\phantom{\rule{0.333333em}{0ex}}}\text{Hall}\Psi D,\end{array}$$(9)
where K_{MHD} and K_{Hall} are the MHD and Hall nonlinear crossscale transfer (cascade) rates, respectively, $\Psi $ represents the pressurestrain effect, and D accounts for the effects of resistive dissipation. We expressed the cascade rates K_{MHD} and K_{Hall} (as well as $\Psi $ and D) in the following form:
$$\begin{array}{cc}\hfill {K}_{\phantom{\rule{0.333333em}{0ex}}}\text{MHD}& =\frac{1}{2}\langle \delta \mathit{w}\xb7\delta [(\mathit{u}\xb7\mathbf{\nabla})\mathit{w}+\frac{1}{2}\mathit{w}(\mathbf{\nabla}\xb7\mathit{u})]\rangle \hfill \\ \hfill & \frac{1}{2}\langle \delta \mathit{w}\xb7\delta \left(\frac{\mathit{J}\times \mathit{B}}{\sqrt{\rho}}\right)\rangle +\frac{1}{2}\langle \delta \mathit{J}\xb7\delta (\mathit{u}\times \mathit{B})\rangle \hfill \end{array}$$(10)
$$\begin{array}{cc}\hfill {K}_{\phantom{\rule{0.333333em}{0ex}}}\text{Hall}& =\frac{1}{2}\langle \delta \mathit{J}\xb7\delta (\mathit{j}\times \mathit{B})\rangle \hfill \end{array}$$(11)
$$\begin{array}{cc}\hfill \Psi & =\frac{1}{2}\langle \delta \mathit{w}\xb7\delta \left(\frac{\mathbf{\nabla}\xb7\mathbf{P}}{\sqrt{\rho}}\right)\rangle \hfill \end{array}$$(12)
$$\begin{array}{cc}\hfill D& =\frac{1}{2}\eta \langle {\delta \mathit{J}}^{2}\rangle ={Q}_{\eta}\frac{1}{2}\eta {\mathrm{\nabla}}^{2}{S\phantom{\rule{0.166667em}{0ex}}}_{B}.\hfill \end{array}$$(13)
The KHM equation constitutes a crossscale energy conservation equation; we define the validity test O as
$$\begin{array}{c}\hfill O={\partial}_{t}S+{K}_{\phantom{\rule{0.333333em}{0ex}}}\text{MHD}+{K}_{\phantom{\rule{0.333333em}{0ex}}}\text{Hall}\Psi D,\end{array}$$(14)
which measures the error of the code owing to numerical issues. Figure 3 displays the isotropised energy KHM analysis of the simulation results at the end of the simulation, tΩ_{i} = 700. Figure 3 shows the validity test O and the different contributing terms, −∂_{t}S, K_{MHD}, K_{Hall}, $\Psi $, and −D (normalised to the total effective dissipation rate Q) as a function of the scale separation l = l (normalised to d_{i}). The KHM Equation (9) is relatively well satisfied (O/Q ≲ 10 %); this error is connected with the noise level owing to the particleincell scheme. The energy decay ∂_{t}S is strong at large scales l ≳ 10d_{i}, but becomes less important and is negligible somewhat below l = d_{i}. The MHD cascade rate is around zero at large scales and becomes important at intermediate scales (6d_{i} ≳ l ≳ 1d_{i}), where it reaches a maximum value of about the dissipation rate Q. At small scales, part of the cascade continues via the Hall term and K_{Hall} attains a maximum value of about 20% of Q; at these scales, the resistive dissipation and the pressurestrain interaction are also active.
Fig. 3. Isotropised energy KHM analysis at tΩ_{i} = 700: Validity test O (black line) as a function of l along with the different contributing terms: decay rate −∂_{t}S (blue), MHD cascade rate K_{MHD} (green), Hall cascade rate K_{Hall} (orange), resistive dissipation rate −D (red), and pressurestrain rate $\Psi $ (magenta). All the quantities are normalised to the effective total dissipation rate Q. 
Figure 3 shows the properties of welldeveloped decaying turbulence (see Fig. 3 of Hellinger et al. 2024). The combined energy decays at large scales, cascades at intermediate scales, and dissipates at small scales. As the KHM equation remains relatively well satisfied throughout the entire simulation, including the turbulence onset, we can analyse the temporal evolution of each KHM term at all times. We can determine which term or process dominates (at a given scale) during the onset and once the cascade is fully developed. Figure 4 shows the evolution of the isotropised KHM results, the different contributing terms as a function of time, and the separation scale l normalised to the total effective dissipation rate Q (averaged over the time interval 600 ≤ tΩ_{i} ≤ 700). Figure 4 demonstrates that the features of welldeveloped turbulence seen in Fig. 3 only appear at later times. During an initial phase (0 ≤ tΩ_{i} ≤ 200), the system is governed by ∂_{t}S ≃ K_{MHD}, and increasingly small scales are generated through the (MHD) nonlinear coupling as ∂_{t}S is positive at intermediate scales. At later times (for about tΩ_{i} ≲ 550), ∂_{t}S becomes negative, monotonically decreasing with l as expected. As the energy transfer towards smaller scales continues, the resistive and the effective pressurestrain dissipation effects gradually appear (their behaviour corresponds to the evolution of the resistive Q_{η} and pressurestrain rates ψ in Fig. 1b). The Hall term gradually sets in around 180 ≲ tΩ_{i} ≲ 230.
Fig. 4. Evolution of isotropised energy KHM results. Shown are the different KHM terms as a function of time t and l: (a) decay rate ∂_{t}S, (b) MHD cascade rate K_{MHD}, (c) Hall cascade rate K_{Hall}, (d) resistive dissipation rate D, and (e) pressurestrain rate $\Psi $. All the quantities are normalised to the effective total dissipation rate Q (averaged over 600 ≤ tΩ_{i} ≤ 700). 
5.2. Cross and kinetic helicities
Next we continued with the cross, kinetic, and mixed helicities. First, we characterised the cross helicity using a secondorder structure function (cf. Banerjee & Galtier 2016, 2017) as
$$\begin{array}{c}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hc}}=\frac{1}{2}\langle \delta \mathit{u}\xb7\delta \mathit{B}\rangle .\end{array}$$(15)
For S_{Hc} we get the following dynamic KHM equation:
$$\begin{array}{c}\hfill {\partial}_{t}{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hc}}={K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hc}+{K}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}Hc}{\Psi}_{\mathit{Hc}}{D}_{\mathit{Hc}},\end{array}$$(16)
where
$$\begin{array}{cc}\hfill {K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hc}& =\frac{1}{2}\langle \delta \mathit{B}\xb7\delta [(\mathit{u}\xb7\mathbf{\nabla})\mathit{u}]\rangle +\frac{1}{2}\langle \delta \mathit{B}\xb7\delta \left(\frac{\mathit{J}\times \mathit{B}}{\rho}\right)\rangle \hfill \\ \hfill & +\frac{1}{2}\langle \delta \mathit{\omega}\xb7\delta (\mathit{u}\times \mathit{B})\rangle \hfill \end{array}$$(17)
$$\begin{array}{cc}\hfill {K}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}Hc}& =\frac{1}{2}\langle \delta \mathit{\omega}\xb7\delta (\mathit{j}\times \mathit{B})\rangle \hfill \end{array}$$(18)
$$\begin{array}{cc}\hfill {D}_{\mathit{Hc}}& =\frac{\eta}{2}\langle \delta \mathit{\omega}\xb7\delta \mathit{J}\rangle \hfill \end{array}$$(19)
$$\begin{array}{cc}\hfill {\Psi}_{\mathit{Hc}}& =\frac{1}{2}\langle \delta \mathit{B}\xb7\delta \left(\frac{\mathbf{\nabla}\xb7\mathbf{P}}{\rho}\right)\rangle .\hfill \end{array}$$(20)
Here, K_{MHDHc} and K_{HallHc} represent the MHD and Hall nonlinear coupling terms, respectively, D_{Hc} accounts for the effects of resistive dissipation, whereas $\Psi $ characterises the crosshelicity equivalent of the pressurestrain effect.
Similarly, we define a secondorder structure function for the kinetic helicity:
$$\begin{array}{c}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hk}}=\frac{m}{4e}\langle \delta \mathit{u}\xb7\delta \mathit{\omega}\rangle .\end{array}$$(21)
For S_{Hk}, the KHM equation reads
$$\begin{array}{c}\hfill {\partial}_{t}{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hk}}={K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hk}+{K}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}Hk}{\Psi}_{\mathit{Hk}},\end{array}$$(22)
where
$$\begin{array}{cc}\hfill {K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hk}& =\frac{m}{2e}\langle \delta \mathit{\omega}\xb7\delta [(\mathit{u}\xb7\mathbf{\nabla})\mathit{u}]\rangle ,\hfill \end{array}$$(23)
$$\begin{array}{cc}\hfill {K}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}Hk}& ={K}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}Hc},\hfill \end{array}$$(24)
$$\begin{array}{cc}\hfill {\Psi}_{\mathit{Hk}}& =\frac{m}{2e}\langle \delta \mathit{\omega}\xb7\delta \left(\frac{\mathbf{\nabla}\xb7\mathbf{P}}{\rho}\right)\rangle .\hfill \end{array}$$(25)
Here, analogously to the cross helicity, K_{MHDHk} and K_{HallHk} represent the MHD and Hall nonlinear coupling terms, respectively, and ${\Psi}_{\mathit{Hk}}$ characterises the kinetichelicity equivalent of the pressurestrain effect.
Finally, for the mixed helicity, we define a secondorder structure function as a sum of the two previous ones,
$$\begin{array}{c}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hx}}={S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hc}}+{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hk}},\end{array}$$(26)
and we obtain the corresponding KHM equation (again as a sum of the two previous ones):
$$\begin{array}{c}\hfill {\partial}_{t}{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hx}}={K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hx}{\Psi}_{\mathit{Hx}}{D}_{\mathit{Hc}},\end{array}$$(27)
where
$$\begin{array}{cc}\hfill {K}_{\mathit{Hx}}& ={K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hc}+{K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hk}\hfill \end{array}$$(28)
$$\begin{array}{cc}\hfill {\Psi}_{\mathit{Hx}}& ={\Psi}_{\mathit{Hc}}+{\Psi}_{\mathit{Hk}}.\hfill \end{array}$$(29)
In the mixedhelicity KHM equation, K_{Hx} represents the MHD nonlinear coupling term (the Hall contribution cancels out), D_{Hc} remains the crosshelicity resistive term, and ${\Psi}_{\mathit{Hx}}$ characterises the mixedhelicity equivalent of the pressurestrain effect.
Starting with the mixed helicity, we define the validity test O_{Hx},
$$\begin{array}{c}\hfill {O}_{\mathit{Hx}}={\partial}_{t}{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hx}}+{K}_{\mathit{Hx}}{\Psi}_{\mathit{Hx}}{D}_{\mathit{Hc}},\end{array}$$(30)
as in the energy case. Figure 5 displays the isotropised mixedhelicity KHM analysis of the simulation results at the end of the simulation, tΩ_{i} = 700, showing the validity test O_{Hx} and the different contributing terms (−∂_{t}S_{Hx}, K_{Hx}, ${\Psi}_{\mathit{Hx}}$, −D_{Hc}) as a function of the scale separation l=. These quantities are normalised to the total effective dissipation rate ϵ_{Hc} = ϵ_{ηHc} + ψ_{Hc}, a sum of the crosshelicity resistive and pressurestrain rates. As for the energy, the mixedhelicity KHM Eq. (27) is relatively well satisfied (O_{Hx}/ϵ_{Hc} ≲ 5 %).
Fig. 5. Isotropised mixedhelicity KHM analysis at tΩ_{i} = 700: Validity test O_{Hx} (black line) as a function of l along with the different contributing terms: decay rate −∂_{t}S_{Hx} (blue), MHD term K_{Hx} (green), resistive dissipation rate −D_{Hc} (red), and pressurestrain rate ${\Psi}_{\mathit{Hx}}$ (magenta). All the quantities are normalised to the effective total (crosshelicity) dissipation rate ϵ_{Hc}. 
Figure 5 exhibits a very different behaviour from that seen in Fig. 3 for the energy. The decay term ∂_{t}S_{Hx} is important (and negative) at large scales. The dissipation term D_{Hc} is positive but small, and the resistive dissipation of cross and mixed helicities is weak. While ∂_{t}S_{Hx} and D_{Hc} behave similarly to the energy counterparts, K_{Hx} and ${\Psi}_{\mathit{Hx}}$ are of opposite sign. The mixedhelicity pressurestrain term ${\Psi}_{\mathit{Hx}}$ has a negative sign, which suggests that the pressurestrain coupling generates the mixed helicity at small scales. The nonlinear term K_{Hx} also has a negative sign, which may indicate an inverse cascade of the mixed helicity. These two suggestions are at variance with the overall decrease in the mixed helicity; consequently, we cannot interpret K_{Hx} as the cascade rate and ${\Psi}_{\mathit{Hx}}$ as the effective dissipation rate of the mixed helicity.
Fig. 6. Isotropised kinetichelicity KHM analysis at tΩ_{i} = 700: Validity test O_{Hk} (black line) as a function of l along with the different contributing terms: the decay rate −∂_{t}S_{Hk} (blue), the MHD cascade rate K_{MHDHk} (green), the Hall cascade rate K_{HallHk} (orange), and the pressurestrain rate ${\Psi}_{\mathit{Hk}}$ (magenta). All the quantities are normalised to the effective total (crosshelicity) dissipation rate ϵ_{Hc}. 
To better understand the behaviour of the mixed helicity, we looked at its two contributions, H_{c} and H_{k}, separately. We started with the kinetic helicity KHM Equation (22), defining the corresponding validity test O_{Hk},
$$\begin{array}{c}\hfill {O}_{\mathit{Hk}}={\partial}_{t}{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hk}}+{K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hk}+{K}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}Hk}{\Psi}_{\mathit{Hk}}.\end{array}$$(31)
Figure 6 displays the isotropised kinetichelicity KHM analysis of the simulation results at the end of the simulation, tΩ_{i} = 700; it shows the validity test O_{Hk} and the different contributing terms, −∂_{t}S_{Hk}, K_{MHDHx}, K_{HallHx}, and ${\Psi}_{\mathit{Hk}}$ (normalised to the total effective dissipation rate ϵ_{Hc}) as a function of the scale separation l. The kinetichelicity KHM equation is well satisfied (O_{Hk}/ϵ_{Hc} ≲ 10 %), but the behaviours of the different terms are very different from the energy KHM results in Fig. 3. The decay term ∂_{t}S_{Hk} and the MHD coupling term K_{MHDHx} are negligible; this is in agreement with the evolution of the kinetic helicity, which is roughly constant. In contrast, the Hall coupling term K_{HallHx} and the pressurestrain term ${\Psi}_{\mathit{Hk}}$ are strong and mostly compensate each other,
$$\begin{array}{c}\hfill {K}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}Hk}\simeq {\Psi}_{\mathit{Hk}}.\end{array}$$(32)
Therefore, the Hall term couples to the kinetichelicity pressurestrain term.
Fig. 7. Isotropised crosshelicity KHM analysis at tΩ_{i} = 700: Validity test O_{Hc} (black line) as a function of l along with the different contributing terms: the decay rate −∂_{t}S_{Hc} (blue), the MHD term K_{MHDHc} (green), the Hall term K_{HallHc} (orange), the resistive dissipation rate −D_{Hc} (red), and the pressurestrain rate ${\Psi}_{\mathit{Hc}}$ (magenta). All the quantities are normalised to the effective total (crosshelicity) dissipation rate ϵ_{Hc}. 
We continued with the crosshelicity KHM Equation (16): we define the corresponding validity test O_{Hc},
$$\begin{array}{c}\hfill {O}_{\mathit{Hc}}={\partial}_{t}{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hc}}+{K}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}Hc}+{K}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}Hc}{\Psi}_{\mathit{Hk}}{D}_{\mathit{Hc}}.\end{array}$$(33)
Figure 7 displays the isotropised crosshelicity KHM analysis of the simulation results at the end of the simulation, tΩ_{i} = 700; it shows the validity test O_{Hc} and the different contributing terms, −∂_{t}S_{Hc}, K_{MHDHc}, K_{HallHc}, ${\Psi}_{\mathit{Hc}}$, and −D_{Hc}, as a function of the scale separation l. The crosshelicity KHM equation is relatively well satisfied (O_{Hc}/ϵ_{Hc} ≲ 15 %). The results of Fig. 7 become somewhat similar to those of the energy KHM results of Fig. 3: the decay term ∂_{t}S_{Hc} is negative and dominates at large scales; and the resistive and the crosshelicity pressurestrain rates behave as dissipation channels and are important at small scales. However, the MHD and Hall coupling terms remain strange. Nevertheless, when we combine them into one term, K_{Hc} = K_{MHDHc} + K_{HallHc}, the resulting quantity exhibits an analogous behaviour to the energy cascade rate (see Fig. 3).
Fig. 8. Rearranged isotropised crosshelicity KHM analysis of Fig. 7: Validity test O_{Hc} (black line), decay rate −∂_{t}S_{Hc} (blue), MHD+Hall term K_{Hc} (green), resistive dissipation rate −D_{Hc} (red), and pressurestrain rate ${\Psi}_{\mathit{Hc}}$ (magenta) as a function of l. All the quantities are normalised to the effective total (crosshelicity) dissipation rate ϵ_{Hc}. 
Figure 8 repeats the results of Fig. 7 but combines the MHD and Hall contributions into one term, K_{Hc}. This indicates that K_{Hc} describes the total (MHD+Hall) cascade rate of the cross helicity. Figure 8 indicates that the cross helicity decays at large scales, cascades at intermediate scales, and dissipates at small scales via the resistive and effective pressurestrain dissipation effects, at least at the end of simulation. We used the crosshelicity KHM equation to look at the development of the system. Figure 9 shows the evolution of the isotropised KHM results, the different contributing terms as a function of time, and the separation scale l normalised to the total effective dissipation rate ϵ_{Hc} (averaged over the time interval 600 ≤ tΩ_{i} ≤ 700). Figure 9 reveals a very similar behaviour to the energy properties in Fig. 4. During the initial phase (0 ≤ tΩ_{i} ≤ 200), the system is governed by ∂_{t}S_{Hc} ≃ K_{Hc}, and increasingly small scales are generated through the (MHD) nonlinear coupling as ∂_{t}S_{Hc} is positive at intermediate scales. At later times (for about tΩ_{i} ≲ 550), ∂_{t}S_{Hc} becomes mostly negative and negligible at small scales. As the energy transfer towards smaller scales continues, the resistive dissipation and more importantly the pressurestrain effective dissipation gradually appear (corresponding to the evolution of the resistive ϵ_{ηHc} and pressurestrain rates ψ_{Hc} in Fig. 1d).
Fig. 9. Evolution of isotropised crosshelicity KHM results. Shown are the different KHM terms as a function of time t and l: (a) decay rate ∂_{t}S_{Hc}, (b) MHD+Hall cascade rate K_{Hc}, (c) resistive dissipation rate D_{Hc}, and (d) pressurestrain rate ${\Psi}_{\mathit{Hc}}$. All the quantities are normalised to the effective total dissipation rate ϵ_{Hc} (averaged over 600 ≤ Ω_{i} ≤ 700). 
6. Discussion
In this paper we present an investigation of 2D decaying plasma turbulence in a weakly collisional plasma with an outofplane background magnetic field using a pseudospectral hybrid code. The plasma system is initialised with substantial fluctuating kinetic and magnetic energies as well as cross helicity at relatively large scales; these initial fluctuations exhibit no overall magnetic or kinetic helicities. We analyse the simulation results using the KHM equation for the combined (kinetic plus magnetic) energy, and cross, kinetic, and mixed helicities. These different KHM equations, which characterise the crossscale conservation of the given quantity, are well satisfied. These code properties are partly attributable to the pseudospectral scheme, which improves the crossscale conservation properties compared to a finitedifference scheme. The KHM results show that the combined energy, which is initially distributed at large scales, transfers to smaller scales owing to the nonlinear MHD term (and later the Hall one). After a transient period, a fully developed turbulent system arises, where the energy decays at large scales, cascades at intermediate scales, and is dissipated at small scales via the resistive term and the pressurestrain term; this system plays the role of an effective dissipation mechanism. These results agree with previous hybrid simulation results (Hellinger et al. 2022, 2024).
The properties of the cross, kinetic, and mixed helicities are more complicated. In the hybrid approximation, we expected the mixed helicity (combination of the cross and kinetic helicities) to be the relevant quantity, which decays, cascades, and dissipates. However, the simulation results show that the cross helicity is the relevant quantity in weakly collisional plasmas. The corresponding KHM equation indicates that the cross helicity behaves analogously to the combined energy: it decays at large scales, cascades via the MHD and Hall nonlinear coupling at intermediate scales, and dissipates at small scales via the resistive term and, more importantly, via the cross helicity equivalent of the pressurestrain term, which again plays a role of an effective dissipation mechanism.
The combined energy and the cross helicity decrease over time, but the latter process is slower, and consequently the relative cross helicity increases. This is in agreement with the theoretical incompressible MHD expectations and numerical simulations (e.g. Dobrowolny et al. 1980; MontagudCamps et al. 2022), but is at variance with observations (Bavassano et al. 1998). This effect may be related to largescale gradients in the solar wind due to its expansion (Grappin et al. 2022) and/or velocity shears (Roberts et al. 1992); we note that, assuming homogeneity, these effects can be included in the KHM equation (Wan et al. 2009; Stawarz et al. 2011; Hellinger et al. 2013).
The magnetic helicity is a separate ideal invariant in the hybrid system (similarly to the Hall MHD case; see Pouquet & Yokoi 2022). In the simulation, the magnetic helicity is initially negligible and is scantily generated via the resistive term (as the magnetic helicity is not a positively definite quantity, this term may lead to its production). We did not observe any coupling between the magnetic and cross helicities, in agreement with theoretical expectations. We also derived and analysed the KHM equation for the magnetic helicity; these results indicate that the magnetic helicity does not exhibit any cascade (not shown here). We investigated a system where the mixed and magnetic helicities are not coupled, but in a more general situation it is necessary to investigate the generalised (canonical) helicity, that is, a combination of the mixed and magnetic helicities (cf. Pouquet & Yokoi 2022). In such a situation, there may be coupling between the cross and magnetic helicities, which could lead to a reduction of the energy cascade, a phenomenon called helicity barrier (Meyrand et al. 2021). Squire et al. (2023) interpreted some hybrid simulation results as being a consequence of the helicity barrier. Our results, however, show that the helicity barrier is not relevant in hybrid (or Hall MHD) systems.
In this work, we used the KHM equations to analyse numerical simulation results for the properties of the energy and the cross helicity. Based on the similarities between the KHM results for the two quantities, we conclude that, even though the cross helicity is not an ideal invariant, it is this quantity that decays, cascades, and dissipates in parallel with the combined energy. As our results could simply be due to some peculiar properties of the KHM approach, we investigated analogous spectraltransfer and coarsegraining approaches (see Appendices A and B). We obtained equivalent results for the energy and the cross helicity, allowing us to conclude that our results are robust. However, we investigated only one 2D simulation with a particular set of parameters; more simulations are needed to study the roles of the various plasma and turbulence parameters. In particular, it would be interesting to check how different values of magnetic helicity (and its cascade) influence properties of plasma turbulence (cf. Stribling et al. 1995). Furthermore, in the simulation, the initial kinetic helicity was negligible and remained negligible during the whole simulation. It is not clear what would happen if the simulation were to start with a strong kinetic helicity. This work also needs to be extended to three dimensions to account for the anisotropy induced by the background magnetic field (Verdini et al. 2015; MontagudCamps et al. 2022; Hellinger et al. 2024). It is also necessary to study a fully kinetic regime; the electron pressure strain effect may act at relatively large scales for the energy (Yang et al. 2022; Manzini et al. 2024) and may influence the evolution of the cross helicity.
In conclusion, this paper we show that the dynamics of cross helicity in weakly collisional plasmas is strongly affected over a wide range of scales (even well above ion scales) by the Hall and pressurestrain physics. Observational tests of these results are challenging; the relevant quantities contain the vorticity field, which is impossible to measure with just one spacecraft. The incompressible MHD KHM equation is often used to estimate energy and crosshelicity cascade rates from in situ spacecraft measurements in the solar wind (SorrisoValvo et al. 2007; MacBride et al. 2008; Stawarz et al. 2009; Marino & SorrisoValvo 2023). Our results, along with those of previous studies, show that at large scales this approximation is sufficient for the energy; the Hall correction appears at the ion scale (Bandyopadhyay et al. 2020) and the pressurestrain effect for the energy also becomes important at around this scale (Yang et al. 2022; Manzini et al. 2024). On the other hand, our results indicate that the MHD estimates of the crosshelicity cascade rate are inadequate in the solar wind; in our simulation, the crosshelicity MHD cascade rate is weaker than the Hall cascade rate and they even have the opposite sign.
Acknowledgments
This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic through the eINFRA CZ (ID:90254). The authors thank T. Tullio for a continuous enriching support and acknowledge useful discussions with A. Verdini, S. Landi, L. Matteini, Emanuele Papini, and L. Franci.
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Appendix A: Energy
A.1. Spectral transfer approach
Another possibility how to quantify the turbulence properties is the spectral approach (cf. Mininni et al. 2007; Grete et al. 2017). We characterised the combined energy by a lowpass filtered quantity (Hellinger et al. 2021, 2024)
$$\begin{array}{cc}\hfill {E}_{k}& =\frac{1}{2}\sum _{{\mathit{k}}^{\prime}\le k}(\widehat{\mathit{w}}{}^{2}+{\widehat{\mathit{B}}}^{2}),\hfill \end{array}$$(A.1)
where the wide hat denotes the Fourier transform. For the spectrallydecomposed quantity E_{k} we obtained the following dynamic equation
$$\begin{array}{c}\hfill {\partial}_{t}{E}_{k}+{S\phantom{\rule{0.166667em}{0ex}}}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}k}+{S\phantom{\rule{0.166667em}{0ex}}}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}k}={\mathrm{\Psi}}_{k}{D}_{k},\end{array}$$(A.2)
where S_{MHDk} and S_{Hallk} represent the MHD and Hall energy transfer rates, respectively; Ψ_{k} describes the pressurestrain effect; and D_{k} is the resistive dissipation rate for modes with wavevector magnitudes smaller than or equal to k. They can be expressed as
$$\begin{array}{cc}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}k}& =\mathfrak{R}\sum _{{\mathit{k}}^{\prime}\le k}[{\widehat{\mathit{w}}}^{\ast}\xb7\widehat{(\mathit{u}\xb7\mathbf{\nabla})\mathit{w}}+\frac{1}{2}{\widehat{\mathit{w}}}^{\ast}\xb7\widehat{\mathit{w}(\mathbf{\nabla}\xb7\mathit{u})}\hfill \\ \hfill & {\widehat{\mathit{w}}}^{\ast}\xb7\widehat{{\rho}^{1/2}\mathit{J}\times \mathit{B}}{\widehat{\mathit{B}}}^{\ast}\xb7\widehat{\mathbf{\nabla}\times (\mathit{u}\times \mathit{B})}],\hfill \end{array}$$(A.3)
$$\begin{array}{cc}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}k}& =\mathfrak{R}\sum _{{\mathit{k}}^{\prime}\le k}{\widehat{\mathit{B}}}^{\ast}\xb7\widehat{\mathbf{\nabla}\times (\mathit{j}\times \mathit{B})},\hfill \end{array}$$(A.4)
$$\begin{array}{cc}\hfill {\mathrm{\Psi}}_{k}& =\mathfrak{R}\sum _{{\mathit{k}}^{\prime}\le k}{\widehat{\mathit{w}}}^{\ast}\xb7\widehat{{\rho}^{1/2}\mathbf{\nabla}\xb7\mathbf{P}},\hfill \end{array}$$(A.5)
$$\begin{array}{cc}\hfill {D}_{k}& =\eta \sum _{{\mathit{k}}^{\prime}\le k}{\mathit{k}}^{\prime}{}^{2}{\widehat{\mathit{B}}}^{2}.\hfill \end{array}$$(A.6)
In these expression the asterisk denotes the complex conjugate, and ℜ denotes the real part. For fully developed turbulence S_{MHDk} and S_{Hallk} are the MHD and Hall cascade rates, respectively.
Similarly to the KHM case, we defined the validity test of the spectral transfer (ST) equation, Eq. (A.2), as
$$\begin{array}{c}\hfill {O}_{k}={\partial}_{t}{E}_{k}+{S\phantom{\rule{0.166667em}{0ex}}}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}k}+{S\phantom{\rule{0.166667em}{0ex}}}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}k}+{\mathrm{\Psi}}_{k}+{D}_{k}.\end{array}$$(A.7)
Fig. A.1. Spectral transfer (ST) analysis of the combined energy, the validity test of the isotropic ST O_{k}, Eq. (A.7), as a function of k at tΩ_{i} = 700 (black line) along with the different contributing terms: decay rate ∂_{t}E_{k} (blue), MHD cascade rate S_{MHDk} (green), Hall cascade rate S_{Hallk} (orange), resistive dissipation rate D_{k} (red), and pressurestrain rate Ψ_{k} (magenta). All the quantities are given in units of the total (effective) dissipation rate Q. 
Figure A.1 shows the ST validity test O_{k} and the contributing terms as functions of k at tΩ_{i} = 700 The ST equation (A.2) is relatively well satisfied (O_{k}/Q ≲ 14 %). Figure A.1 quantifies the turbulence properties, the energy decays at large scales, cascades at intermediate scales, and at small scales the cascade partly continues via the Hall term and partly is dissipated by the resistive and pressurestrain effects. The ST results in Fig. A.1 are quantitatively equivalent to the corresponding KHM results (see Fig. 3) through the inverse proportionality $kl=\sqrt{2}$. Using this relationship we got
$$\begin{array}{c}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}k}\simeq {K}_{\phantom{\rule{0.333333em}{0ex}}}\text{MHD},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{S\phantom{\rule{0.166667em}{0ex}}}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}k}\simeq {K}_{\phantom{\rule{0.333333em}{0ex}}}\text{Hall}.\end{array}$$(A.8)
The other terms exhibit a complementarity properties; the ST approach is by construction (spectral) lowpass filter whereas the KHM approach exhibits rather highpass (or spatial lowpass) filter behaviour (see Hellinger et al. 2021, 2024).
A.2. Coarsegraining approach
An alternative way how to analyse the turbulence properties is the spatial filtering (Eyink & Aluie 2009; Aluie 2011; Manzini et al. 2022). In this approach the scale decomposition of the combined energy is done using a spatial filter (coarse graining)
$$\begin{array}{c}\hfill {E}_{l}=\frac{1}{2}\langle \overline{\rho}\stackrel{\sim}{\mathit{u}}{}^{2}+{\overline{\mathit{B}}}^{2}\rangle \end{array}$$(A.9)
where the line denotes a spatial filtering
$$\begin{array}{c}\hfill {\overline{\mathit{a}}}_{l}(\mathit{x})=\int {\mathrm{d}}^{2}x{G}_{l}(\mathit{r})\mathit{a}(\mathit{x}+\mathit{r})\end{array}$$(A.10)
and the tilde denotes the corresponding densityweighted or Favre filtering (Favre 1969)
$$\begin{array}{c}\hfill {\stackrel{\sim}{\mathit{a}}}_{l}=\frac{{\overline{\rho \mathit{u}}}_{l}}{{\overline{\rho}}_{l}}.\end{array}$$(A.11)
For the filtered energy we obtained this dynamic equation
$$\begin{array}{c}\hfill {\partial}_{t}{E}_{l}+{\mathrm{\Pi}}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}l}+{\mathrm{\Pi}}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}l}={\mathrm{\Phi}}_{l}{\mathcal{D}}_{l}\end{array}$$(A.12)
where Π_{MHDl} and Π_{Halll} represent the MHD and Hall energy transfer rates, respectively; Φ_{l} and 𝒟_{l} describe the pressurestrain and resistive dissipation effects, respectively. These terms can be expressed as
$$\begin{array}{cc}\hfill {\mathrm{\Pi}}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}l}& =\langle \mathbf{\nabla}\stackrel{\sim}{\mathit{u}}:\overline{\rho}\stackrel{\sim}{\mathit{\tau}}\rangle \langle \stackrel{\sim}{\mathit{u}}\xb7\overline{\mathit{J}\times \mathit{B}}\rangle \langle \overline{\mathit{J}}\xb7\overline{(\mathit{u}\times \mathit{B})}\rangle \hfill \end{array}$$(A.13)
$$\begin{array}{cc}\hfill {\mathrm{\Pi}}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}l}& =\langle \overline{\mathit{J}}\xb7\overline{(\mathit{j}\times \mathit{B})}\rangle \hfill \end{array}$$(A.14)
$$\begin{array}{cc}\hfill {\mathrm{\Phi}}_{l}& =\langle \mathbf{\nabla}\stackrel{\sim}{\mathit{u}}:\overline{\mathbf{P}}\rangle \hfill \end{array}$$(A.15)
$$\begin{array}{cc}\hfill {\mathcal{D}}_{l}& =\eta \langle \overline{\mathit{J}}{}^{2}\rangle \hfill \end{array}$$(A.16)
where $\stackrel{\sim}{\mathit{\tau}}=\stackrel{\sim}{\mathit{u}}\stackrel{\sim}{\mathit{u}}\stackrel{\sim}{\mathit{u}\mathit{u}}$. As before we defined the validity test of the coarsegraining (CG) equation, Eq. (A.12), as
$$\begin{array}{c}\hfill {\mathcal{O}}_{l}={\partial}_{t}{E}_{l}+{\mathrm{\Pi}}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}l}+{\mathrm{\Pi}}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}l}+{\mathrm{\Phi}}_{l}+{\mathcal{D}}_{l}\end{array}$$(A.17)
Fig. A.2. Coarsegraining (CG) analysis of the combined energy: Validity test of the isotropic CG 𝒪_{l}, Eq. (A.17), as a function of k at tΩ_{i} = 700(black line) along with the different contributing terms: decay rate ∂_{t}E_{l} (blue), MHD cascade rate Π_{MHDl} (green), Hall cascade rate Π_{Halll} (orange), resistive dissipation rate 𝒟_{l} (red), and pressurestrain rate Φ_{l} (magenta). All the quantities are given in units of the total (effective) dissipation rate Q. 
Figure A.2 shows results of the coarsegraining for the spatial lowpass filter using the normalized boxcar window function. The CG validity test is relatively well satisfied (𝒪_{l}/Q ≲ 10 %); the respective errors of the KHM, ST, and CG approaches are comparable. Figure A.2 repeats the turbulence picture we saw in the KHM and ST approaches, the energy decays at large scales, cascades at intermediate scales, and at small scales the cascade partly continues via the Hall term and partly is dissipated by the resistive and pressurestrain effects. For the spatial lowpass filter and the same separation and filter scale l we obtained quantitatively equivalent MHD and Hall cascade rates compared to the KHM equation (and, consequently, compared to the ST equation as we saw in the previous subsection):
$$\begin{array}{c}\hfill {\mathrm{\Pi}}_{\phantom{\rule{0.333333em}{0ex}}\text{MHD}l}\simeq {K}_{\phantom{\rule{0.333333em}{0ex}}}\text{MHD},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathrm{\Pi}}_{\phantom{\rule{0.333333em}{0ex}}\text{Hall}l}\simeq {K}_{\phantom{\rule{0.333333em}{0ex}}}\text{Hall}.\end{array}$$(A.18)
The other respective terms in the CG and KHM approaches exhibit a complementarity behaviour, which is similar to that of the ST approach in subsection A.1. We note that it is possible to use the spectral filtering in a way analogical to the CG approach (Arró et al. 2022). It is possible to use spectral lowpass filters, which lead to an inverse dependence of the different quantities on l (i.e. l → −l).
Appendix B: Cross helicity
B.1. Spectral transfer
For the spectral scale decomposition of the cross helicity we chosen
$$\begin{array}{cc}\hfill {H}_{\mathit{ck}}& =\mathfrak{R}\sum _{{\mathit{k}}^{\prime}\le k}({\widehat{\mathit{u}}}^{\ast}\xb7\widehat{{\mathit{B}}_{1}})\hfill \end{array}$$(B.1)
In analogy with the KHM equation (where the increment representation losses the information about the background magnetic field B_{0}) we used the fluctuating magnetic field B_{1} = B − B_{0}. For H_{ck} we obtained the following dynamic equation
$$\begin{array}{c}\hfill {\partial}_{t}{H}_{\mathit{ck}}+{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hck}}={\mathrm{\Psi}}_{\mathit{Hck}}{D}_{\mathit{Hck}},\end{array}$$(B.2)
where S_{Hck} represents the (joint MHD and Hall) cross helicity transfer rate, and Ψ_{Hck} and D_{Hck} describe the pressurestrain and resistive dissipation effects, respectively. These terms can be expressed as
$$\begin{array}{cc}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hck}}& =\mathfrak{R}\sum _{{\mathit{k}}^{\prime}\le k}[{\widehat{{\mathit{B}}_{1}}}^{\ast}\xb7\widehat{(\mathit{u}\xb7\mathbf{\nabla})\mathit{u}}{\widehat{{\mathit{B}}_{1}}}^{\ast}\xb7\widehat{\frac{\mathit{J}\times \mathit{B}}{\rho}}\hfill \\ \hfill & {\widehat{\mathit{\omega}}}^{\ast}\xb7\widehat{\mathit{u}\times \mathit{B}}+{\widehat{\mathit{\omega}}}^{\ast}\xb7\widehat{\mathit{j}\times \mathit{B}}]\hfill \end{array}$$(B.3)
$$\begin{array}{cc}\hfill {\mathrm{\Psi}}_{\mathit{Hck}}& =\mathfrak{R}\sum _{{\mathit{k}}^{\prime}\le k}{\widehat{{\mathit{B}}_{1}}}^{\ast}\xb7\widehat{\frac{\mathbf{\nabla}\xb7\mathbf{P}}{\rho}}\hfill \end{array}$$(B.4)
$$\begin{array}{cc}\hfill {D}_{\mathit{Hck}}& =\eta \mathfrak{R}\sum _{{\mathit{k}}^{\prime}\le k}{{\mathit{k}}^{\prime}}^{2}{\widehat{\mathit{u}}}^{\ast}\xb7\widehat{\mathit{B}}\hfill \end{array}$$(B.5)
Here again we defined the validity test of the ST crosshelicity equation (B.2) as
$$\begin{array}{c}\hfill {O}_{\mathit{Hck}}={\partial}_{t}{H}_{\mathit{ck}}+{S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hck}}+{\mathrm{\Psi}}_{\mathit{Hck}}+{D}_{\mathit{Hck}}.\end{array}$$(B.6)
Fig. B.1. ST analysis of the cross helicity: Validity test of the isotropic ST O_{Hck}, Eq. (B.6), as a function of k at tΩ_{i} = 700 (black line) along with the different contributing terms: decay rate ∂_{t}H_{ck} (blue), cascade rate S_{Hck} (green), resistive dissipation rate D_{Hck} (red), and pressurestrain rate Ψ_{kHc} (magenta). All the quantities are given in units of the total (effective) crosshelicity dissipation rate ϵ_{Hc}. 
Figure B.1 shows that the cross helicity ST equation (B.2) is relatively well satisfied (O_{Hck}/ϵ_{Hc} ≲ 15%). In the Fig. B.1 we recovered the KHM results, the cross helicity decays at large scales, cascades at intermediate scales, and at small scales it is dissipated by the resistive and pressurestrain effects, with the latter process being the dominant one. Similarly to the case of the energy ST and KHM equations the cross helicity ST and KHM approaches give equivalent cascade rates
$$\begin{array}{c}\hfill {S\phantom{\rule{0.166667em}{0ex}}}_{\mathit{Hck}}\simeq {K}_{\phantom{\rule{0.333333em}{0ex}}}\text{Hc}\end{array}$$(B.7)
for $kl=\sqrt{2}$. The other terms have again complementary behaviour.
B.2. Coarse graining
For the scaledecomposition of the cross helicity we choose the following filtered quantity
$$\begin{array}{c}\hfill {H}_{\mathit{cl}}=\langle \overline{{\mathit{B}}_{1}}\xb7\overline{\mathit{u}}\rangle .\end{array}$$(B.8)
For the coarsegrained cross helicity H_{cl} we obtained this dynamic equation
$$\begin{array}{c}\hfill {\partial}_{t}{H}_{\mathit{cl}}+{\mathrm{\Pi}}_{\mathit{Hcl}}={\mathrm{\Phi}}_{\mathit{Hcl}}{\mathcal{D}}_{\mathit{Hcl}}.\end{array}$$(B.9)
where Π_{Hcl} represents the cross helicity transfer rate, and Φ_{Hcl} and 𝒟_{Hcl} describe the pressurestrain and resistive dissipation effects, respectively. These terms can be expressed as
$$\begin{array}{cc}\hfill {\mathrm{\Pi}}_{\mathit{Hcl}}& =\langle \overline{{\mathit{B}}_{1}}\xb7\overline{(\mathit{u}\xb7\mathbf{\nabla})\mathit{u}}\rangle \langle \overline{{\mathit{B}}_{1}}\xb7\overline{\frac{\mathit{J}\times \mathit{B}}{\rho}}\rangle \langle \overline{\mathit{\omega}}\xb7\overline{\mathit{u}\times \mathit{B}}\rangle \hfill \\ \hfill & +\langle \overline{\mathit{\omega}}\xb7\overline{\mathit{j}\times \mathit{B}}\rangle \hfill \end{array}$$(B.10)
$$\begin{array}{cc}\hfill {\mathrm{\Phi}}_{\mathit{Hcl}}& =\langle \overline{{\mathit{B}}_{1}}\xb7\overline{\frac{\mathbf{\nabla}\xb7\mathbf{P}}{\rho}}\rangle \hfill \end{array}$$(B.11)
$$\begin{array}{cc}\hfill {\mathcal{D}}_{\mathit{Hcl}}& =\eta \langle \overline{\mathit{\omega}}\xb7\overline{\mathit{J}}\rangle \hfill \end{array}$$(B.12)
As before, we defined the validity test of the CG crosshelicity equation (B.9) as
$$\begin{array}{c}\hfill {\mathcal{O}}_{\mathit{Hcl}}={\partial}_{t}{H}_{\mathit{cl}}+{\mathrm{\Pi}}_{\mathit{Hcl}}+{\mathrm{\Phi}}_{\mathit{Hcl}}+{\mathcal{D}}_{\mathit{Hcl}}\end{array}$$(B.13)
Figure B.2 shows that the cross helicity CG equation is relatively well satisfied (𝒪_{Hcl}/ϵ_{Hc} ≲ 16%). Figure B.2 again repeats the same information seen in the corresponding KHM and ST analyses: the cross helicity decays at large scales, cascades at intermediate scales, and at small scales it is dissipated by the resistive and pressurestrain effects. The KHM and CG approaches give quantitatively similar results, for the cascade rates
$$\begin{array}{c}\hfill {\mathrm{\Pi}}_{\mathit{Hcl}}\simeq {K}_{\mathit{Hc}}.\end{array}$$(B.14)
The other respective terms in the CG and KHM approaches exhibit a complementarity behaviour, which is similar to that of the ST approach in subsection B.1.
Fig. B.2. CG analysis of the cross helicity: Validity test of the isotropic CG 𝒪_{Hcl}, Eq. (B.13), as a function of k at tΩ_{i} = 700 (black line) along with the different contributing terms: decay rate ∂_{t}H_{cl} (blue), MHD cascade rate Π_{Hcl} (green), resistive dissipation rate 𝒟_{Hcl} (red), and pressurestrain rate Φ_{Hcl} (magenta). All the quantities are given in units of the total (effective) crosshelicity dissipation rate ϵ_{Hc}. 
All Figures
Fig. 1. Evolution of different quantities as a function of time: (a) Relative changes in the kinetic energy ΔE_{kin} (dashed line), magnetic energy ΔE_{mag} (solid line), internal energy ΔE_{int} (dashdotted line), and total energy ΔE_{tot} (dotted line), (b) resistive dissipation rate Q_{η} (dashed line) and pressurestrain effective dissipation rate ψ (solid line), (c) relative changes in the kinetic helicity ΔH_{k} (dashdotted line), mixed helicity ΔH_{x} (solid line), and cross helicity ΔH_{c} (dashed line), (d) resistive crosshelicity dissipation rate ϵ_{ηHc} (dotted line), pressurestrain effective crosshelicity dissipation rate ψ_{Hc} (dashed line), pressurestrain effective mixedhelicity dissipation rate ψ_{Hx} (solid line), and pressurestrain effective kinetichelicity dissipation rate ψ_{Hk} (dashdotted line), (e) relative change in the magnetic helicity ΔH_{m}, and (f) resistive magnetichelicity dissipation rate ϵ_{Hm}. 

In the text 
Fig. 2. Omnidirectional spectral properties of different quantities at tΩ_{i} = 700: (a) Power spectral densities of (solid) the magnetic field B, P_{B} (solid), and compensated proton velocity field w, P_{w} (dashed), as a function of k normalised to d_{i}. (b) Cross helicity cospectrum P_{Hc} (solid) and (absolute value of) kinetic helicity cospectrum P_{Hk} (dashed) as a function of k. The dotted lines show a spectrum ∝k^{−5/3} for comparison. 

In the text 
Fig. 3. Isotropised energy KHM analysis at tΩ_{i} = 700: Validity test O (black line) as a function of l along with the different contributing terms: decay rate −∂_{t}S (blue), MHD cascade rate K_{MHD} (green), Hall cascade rate K_{Hall} (orange), resistive dissipation rate −D (red), and pressurestrain rate $\Psi $ (magenta). All the quantities are normalised to the effective total dissipation rate Q. 

In the text 
Fig. 4. Evolution of isotropised energy KHM results. Shown are the different KHM terms as a function of time t and l: (a) decay rate ∂_{t}S, (b) MHD cascade rate K_{MHD}, (c) Hall cascade rate K_{Hall}, (d) resistive dissipation rate D, and (e) pressurestrain rate $\Psi $. All the quantities are normalised to the effective total dissipation rate Q (averaged over 600 ≤ tΩ_{i} ≤ 700). 

In the text 
Fig. 5. Isotropised mixedhelicity KHM analysis at tΩ_{i} = 700: Validity test O_{Hx} (black line) as a function of l along with the different contributing terms: decay rate −∂_{t}S_{Hx} (blue), MHD term K_{Hx} (green), resistive dissipation rate −D_{Hc} (red), and pressurestrain rate ${\Psi}_{\mathit{Hx}}$ (magenta). All the quantities are normalised to the effective total (crosshelicity) dissipation rate ϵ_{Hc}. 

In the text 
Fig. 6. Isotropised kinetichelicity KHM analysis at tΩ_{i} = 700: Validity test O_{Hk} (black line) as a function of l along with the different contributing terms: the decay rate −∂_{t}S_{Hk} (blue), the MHD cascade rate K_{MHDHk} (green), the Hall cascade rate K_{HallHk} (orange), and the pressurestrain rate ${\Psi}_{\mathit{Hk}}$ (magenta). All the quantities are normalised to the effective total (crosshelicity) dissipation rate ϵ_{Hc}. 

In the text 
Fig. 7. Isotropised crosshelicity KHM analysis at tΩ_{i} = 700: Validity test O_{Hc} (black line) as a function of l along with the different contributing terms: the decay rate −∂_{t}S_{Hc} (blue), the MHD term K_{MHDHc} (green), the Hall term K_{HallHc} (orange), the resistive dissipation rate −D_{Hc} (red), and the pressurestrain rate ${\Psi}_{\mathit{Hc}}$ (magenta). All the quantities are normalised to the effective total (crosshelicity) dissipation rate ϵ_{Hc}. 

In the text 
Fig. 8. Rearranged isotropised crosshelicity KHM analysis of Fig. 7: Validity test O_{Hc} (black line), decay rate −∂_{t}S_{Hc} (blue), MHD+Hall term K_{Hc} (green), resistive dissipation rate −D_{Hc} (red), and pressurestrain rate ${\Psi}_{\mathit{Hc}}$ (magenta) as a function of l. All the quantities are normalised to the effective total (crosshelicity) dissipation rate ϵ_{Hc}. 

In the text 
Fig. 9. Evolution of isotropised crosshelicity KHM results. Shown are the different KHM terms as a function of time t and l: (a) decay rate ∂_{t}S_{Hc}, (b) MHD+Hall cascade rate K_{Hc}, (c) resistive dissipation rate D_{Hc}, and (d) pressurestrain rate ${\Psi}_{\mathit{Hc}}$. All the quantities are normalised to the effective total dissipation rate ϵ_{Hc} (averaged over 600 ≤ Ω_{i} ≤ 700). 

In the text 
Fig. A.1. Spectral transfer (ST) analysis of the combined energy, the validity test of the isotropic ST O_{k}, Eq. (A.7), as a function of k at tΩ_{i} = 700 (black line) along with the different contributing terms: decay rate ∂_{t}E_{k} (blue), MHD cascade rate S_{MHDk} (green), Hall cascade rate S_{Hallk} (orange), resistive dissipation rate D_{k} (red), and pressurestrain rate Ψ_{k} (magenta). All the quantities are given in units of the total (effective) dissipation rate Q. 

In the text 
Fig. A.2. Coarsegraining (CG) analysis of the combined energy: Validity test of the isotropic CG 𝒪_{l}, Eq. (A.17), as a function of k at tΩ_{i} = 700(black line) along with the different contributing terms: decay rate ∂_{t}E_{l} (blue), MHD cascade rate Π_{MHDl} (green), Hall cascade rate Π_{Halll} (orange), resistive dissipation rate 𝒟_{l} (red), and pressurestrain rate Φ_{l} (magenta). All the quantities are given in units of the total (effective) dissipation rate Q. 

In the text 
Fig. B.1. ST analysis of the cross helicity: Validity test of the isotropic ST O_{Hck}, Eq. (B.6), as a function of k at tΩ_{i} = 700 (black line) along with the different contributing terms: decay rate ∂_{t}H_{ck} (blue), cascade rate S_{Hck} (green), resistive dissipation rate D_{Hck} (red), and pressurestrain rate Ψ_{kHc} (magenta). All the quantities are given in units of the total (effective) crosshelicity dissipation rate ϵ_{Hc}. 

In the text 
Fig. B.2. CG analysis of the cross helicity: Validity test of the isotropic CG 𝒪_{Hcl}, Eq. (B.13), as a function of k at tΩ_{i} = 700 (black line) along with the different contributing terms: decay rate ∂_{t}H_{cl} (blue), MHD cascade rate Π_{Hcl} (green), resistive dissipation rate 𝒟_{Hcl} (red), and pressurestrain rate Φ_{Hcl} (magenta). All the quantities are given in units of the total (effective) crosshelicity dissipation rate ϵ_{Hc}. 

In the text 
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