⇦Chinese Physics Letters, 2017, Vol. 34, No. 1, Article code 015201 Helical Mode Absolute Statistical Equilibrium of Ideal Three-Dimensional Hall Magnetohydrodynamics * Zhen-Wei Xia(夏振伟)1**, Chun-Hua Li(李春华)2, Dan-Dan Zou(邹丹旦)3, Wei-Hong Yang(杨维纮)1 Affiliations 1Department of Modern Physics, University of Science and Technology of China, Hefei 230026 2Department of Information Engineering, Hefei University of Technology, Hefei 230009 3School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013 Received 11 September 2016 *Supported by the National Natural Science Foundation of China under Grant Nos 11375190 and 11547137.
**Corresponding author. Email: zhenwx@mail.ustc.edu.cn Citation Text: Xia Z W, Li C H, Zou D D and Yang W H 2017 Chin. Phys. Lett. 34 015201 Abstract Using the Fourier helical decomposition, we obtain the absolute statistical equilibrium spectra of left- and right-handed helical modes in the incompressible ideal Hall magnetohydrodynamics (MHD). It is shown that the left-handed helical modes play a major role on the spectral transfer properties of turbulence when the generalized helicity and magnetic helicity are both positive. In contrast, the right-handed helical modes will play a major role when both are negative. Furthermore, we also find that if the generalized helicity and magnetic helicity have opposite signs, the tendency of equilibrium spectra to condense at the large or small wave numbers will be presented in different helical sectors. This indicates that the generalized helicity dominates the forward cascade and the magnetic helicity dominates the inverse cascade properties of the Hall MHD turbulence. DOI:10.1088/0256-307X/34/1/015201 PACS:52.35.Ra, 52.30.Ex, 47.27.Gs © 2017 Chinese Physics Society Article Text For understanding the spectral transfer properties of turbulence,[1-3] classical Gibbs ensemble methods are commonly used to investigate the nonlinear dynamics of related ideal systems.[4-15] In the case of hydrodynamics turbulence, it was named as the absolute statistical equilibrium theory.[6] In this theory, the basic dynamic variables are the Fourier coefficients of turbulent fields within a periodic box. Assuming that they are Gaussian distribution, the equilibrium spectra of global quadratic invariants in the ideal systems can be obtained through Gibbs ensemble average. Furthermore, as shown by Kraichnan et al.,[5,16] the ideal equilibrium spectra have a tendency to peak in different regions of wave number space when more than one quadratic invariant is present. The tendency corresponds to the cascade properties when the dissipative turbulence is considered. Applications of this theory to two-dimensional (2D) and three-dimensional (3D) magnetohydrodynamics (MHD) have successfully predicted the inverse cascades of mean square vector potential[8] and magnetic helicity,[7] which provide a basis for understanding the dynamo problem[17-19] in astrophysics. In recent years, the small-scale plasma physics (not covered by the MHD model) have been reported to be important for solar wind turbulence,[20] magnetic reconnection[21] and dynamo action.[22,23] To investigate these physical phenomena at scales comparable with the ion skin depth, the Hall MHD model[24,25] is usually used for study. For example, the first direct numerical simulations of turbulent Hall dynamos were presented in 2003.[26] The results showed that the Hall current can strongly enhance or suppress the generation of large-scale magnetic energy depending on the relative values of the length scales of the system. In 2008, Servidio et al.[13] investigated the absolute statistical equilibrium of incompressible, ideal Hall MHD. It was shown that the magnetic helicity, as that in MHD, cascades toward large scales. In addition, a weak enhancement of generalized helicity at small scales was also found. Fourier helical decomposition is a conventional technique to investigate 3D hydrodynamics and magnetohydrodynamics turbulence when incompressibility constraints are imposed. It is first proposed by Constantin et al.,[27] and exploited in great detail by Waleffe.[28] At present, great progress has been made in the spectral dynamics of 3D fluid flow[29] and the weak turbulence of electron (Hall) MHD[30,31] with the Fourier helical decomposition. In particular, Biferal et al.[32] recently performed direct numerical simulations on the decimated Navier–Stokes equations. The results showed that an inverse energy cascade may occur in the subsystem of 3D fluid flow when one chiral sector is retained. In 2014, Zhu et al.[33] applied the Fourier helical decomposition to the absolute statistical equilibrium of 3D Navier–Stokes equations and obtained the pure helical spectra. They found that the pure helical spectra allow a negative temperature state to support the argument of inverse energy cascade by Birferal. In this Letter, we extend the work on the 3D Hall MHD absolute statistical equilibrium through the Fourier helical decomposition and give more insights about the spectral transfer properties of the Hall MHD turbulence. The dimensionless equations of the incompressible, ideal Hall MHD are given as follows: $$\begin{align} &\frac{\partial {\boldsymbol u}}{\partial t}=-\nabla p^\ast +{\boldsymbol u}\times ({\nabla \times {\boldsymbol u}})+( {\nabla \times {\boldsymbol b}})\times {\boldsymbol b},~~ \tag {1} \end{align} $$ $$\begin{align} &\frac{\partial {\boldsymbol b}}{\partial t}=\nabla \times ({{\boldsymbol u}\times {\boldsymbol b}})-\varepsilon \nabla \times [{({\nabla \times {\boldsymbol b}})\times {\boldsymbol b}}],~~ \tag {2} \end{align} $$ $$\begin{align} &\nabla \cdot {\boldsymbol u}=0,~~ \tag {3} \end{align} $$ $$\begin{align} &\nabla \cdot {\boldsymbol b}=0,~~ \tag {4} \end{align} $$ where ${\boldsymbol u}$ and ${\boldsymbol b}$ represent the normalized velocity and magnetic field, respectively. The normalized parameters are the Alfvén speed $V_{\rm A}=B_{\rm 0}/\sqrt {4\pi \rho}$ ($\rho$ is the uniform mass density) and the characteristic magnetic field $B_0$. Coulomb gauge is used for the magnetic vector potential. Here $p^\ast$ is the normalized total pressure and can be obtained by taking the divergence of Eq. (1). The parameter $\varepsilon$ is the ratio between the ion skin depth $\delta _{\rm i}$ ($\delta _{\rm i}=c/\omega _{\rm pi}$, $\omega _{\rm pi}$ is the ion plasma frequency, and $c$ is the speed of light in vacuum) and the characteristic length scale $L_0$. The characteristic time is the Alfvén time $L_0 /V_{\rm A}$. By considering a 3D periodic box with sides of dimensionless length $2\pi$, the velocity and magnetic field can be naturally expanded in Fourier series. Furthermore, due to the incompressibility constraints of Eqs. (3) and (4), the vector Fourier coefficients of the velocity and magnetic field are required to be orthogonal to the wave vector ${\boldsymbol k}$ (i.e., ${\boldsymbol k}\cdot {\boldsymbol u}_{\boldsymbol k}=0$ and ${\boldsymbol k}\cdot {\boldsymbol b}_{\boldsymbol k}=0$) and can be projected on the Fourier helical basis, $$\begin{align} {\boldsymbol u}({\boldsymbol r},t)=\,&\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {{\boldsymbol u}_{\boldsymbol k} e^{i{\boldsymbol k}\cdot {\boldsymbol r}}}=\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\sum\limits_{s_{\boldsymbol k}} {u_{\boldsymbol k}^{s_{\boldsymbol k}} {\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k}} e^{i{\boldsymbol k}\cdot {\boldsymbol r}}}},~~ \tag {5} \end{align} $$ $$\begin{align} {\boldsymbol b}({\boldsymbol r},t)=\,&\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {{\boldsymbol b}_{\boldsymbol k} e^{i{\boldsymbol k}\cdot {\boldsymbol r}}}=\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\sum\limits_{s_{\boldsymbol k}} {b_{\boldsymbol k}^{s_{\boldsymbol k}} {\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k}} e^{i{\boldsymbol k}\cdot {\boldsymbol r}}}},~~ \tag {6} \end{align} $$ where ${\boldsymbol N}$ is a collection of 3D wave vectors ${\boldsymbol k}$ with the integer components such that $k_{\min} \leq|{\boldsymbol k}|\leq k_{\max}$ ($|{\boldsymbol k}|=\sqrt {m^2+n^2{\rm +}l^2}$, with $m,n,l=0,\pm 1,\pm 2,\ldots$). We denote $N$ as the number of wave vectors that belong to ${\boldsymbol N}$. In the box, $N=({2N_{\rm box} +1})^3-1$. Here ${\boldsymbol u}_{\boldsymbol k}$ and ${\boldsymbol b}_{\boldsymbol k}$ are the abbreviations for the vector Fourier coefficients ${\boldsymbol u}({\boldsymbol k},t)$ and ${\boldsymbol b}({\boldsymbol k},t)$, respectively, and $u_{\boldsymbol k}^{s_{\boldsymbol k}}$ and $b_{\boldsymbol k}^{s_{\boldsymbol k}}$ are the projections of ${\boldsymbol u}_{\boldsymbol k}$, ${\boldsymbol b}_{\boldsymbol k}$ on the helical basis ${\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k}}$ and satisfy the reality conditions $u_{-{\boldsymbol k}}^{s_{\boldsymbol k}}=u_{\boldsymbol k}^{s_{\boldsymbol k} ^\ast}$ and $b_{-{\boldsymbol k}}^{s_{\boldsymbol k}}=b_{\boldsymbol k}^{s_{\boldsymbol k} ^\ast}$, respectively. The Fourier helical basis is defined as $$\begin{align} {\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k}}=\hat {{\boldsymbol \phi}}_{\boldsymbol k} \times \hat {{\boldsymbol k}}+is_{\boldsymbol k} \hat {{\boldsymbol \phi}}_{\boldsymbol k},~~ \tag {7} \end{align} $$ where $i^2=-1$, $\hat {{\boldsymbol k}}={\boldsymbol k}/k$ and $s_{\boldsymbol k}=\pm 1$ (where + represents the left-handed mode and $-$ represents the right-handed one), and $\hat {\boldsymbol\phi}_{\boldsymbol k}$ is the unit vector orthogonal to the wave vector ${\boldsymbol k}$. To ensure the reality of the velocity and magnetic field, one can choose $\hat {{\boldsymbol \phi}}_{\boldsymbol k}={\boldsymbol z}\times \hat {{\boldsymbol k}}/|{{\boldsymbol z}\times \hat {{\boldsymbol k}}}|$ with ${\boldsymbol z}$ being an arbitrary vector. Therefore, the Fourier helical basis has the properties $i{\boldsymbol k}\times {\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k}}=s_{\boldsymbol k} k{\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k}}$, ${\boldsymbol h}_{-{\boldsymbol k}}^{s_{\boldsymbol k}}={\boldsymbol h}_{\boldsymbol k}^{-s_{\boldsymbol k}}={\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k} ^\ast}$ and ${\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k} ^\ast} \cdot {\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k} ^\prime}=2\delta _{s_{\boldsymbol k},s_{\boldsymbol k} ^\prime}$. Then, replacing Eqs. (5) and (6) for ${\boldsymbol u}$ and ${\boldsymbol b}$ in Eqs. (1) and (2) and projecting on ${\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k}}$ ($s_{\boldsymbol k}=\pm 1$) leads to the following dynamical system for the helical modes' evolution $$\begin{alignat}{1} \frac{\partial u_{\boldsymbol k}^{s_{\boldsymbol k}}}{\partial t}=\,&\frac{1}{2}\sum\limits_{{\boldsymbol k}+{\boldsymbol p}+{\boldsymbol q}=0} \sum\limits_{s_{\boldsymbol p},s_{\boldsymbol q}} g(s_{\boldsymbol p} p\\ &-s_{\boldsymbol q} q) ({u_{\boldsymbol p}^{s_{\boldsymbol p}} u_{\boldsymbol q}^{s_{\boldsymbol q}} -b_{\boldsymbol p}^{s_{\boldsymbol p}} b_{\boldsymbol q}^{s_{\boldsymbol q}}})^\ast,~~ \tag {8} \end{alignat} $$ $$\begin{alignat}{1} \frac{\partial b_{\boldsymbol k}^{s_{\boldsymbol k}}}{\partial t}=\,&\frac{1}{2}\sum\limits_{{\boldsymbol k}+{\boldsymbol p}+{\boldsymbol q}=0} \sum\limits_{s_{\boldsymbol p},s_{\boldsymbol q}} g[(-s_{\boldsymbol k} k)(u_{\boldsymbol p}^{s_{\boldsymbol p}} b_{\boldsymbol q}^{s_{\boldsymbol q}}\\ &-u_{\boldsymbol q}^{s_{\boldsymbol q}} b_{\boldsymbol p}^{s_{\boldsymbol p}})^\ast+\varepsilon s_{\boldsymbol k} k({s_{\boldsymbol p} p-s_{\boldsymbol q} q})(b_{\boldsymbol p}^{s_{\boldsymbol p}} b_{\boldsymbol q}^{s_{\boldsymbol q}})^\ast],~~ \tag {9} \end{alignat} $$ where $g$ is a function of ${\boldsymbol k}$, ${\boldsymbol p}$, ${\boldsymbol q}$, $s_{\boldsymbol k}$, $s_{\boldsymbol p}$ and $s_{\boldsymbol q}$ defined in Ref. [28] by $$\begin{align} g=-\frac{{\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k} ^\ast} \cdot [{\boldsymbol h}_{\boldsymbol p}^{s_{\boldsymbol p}^\ast} \times {\boldsymbol h}_{\boldsymbol q}^{s_{\boldsymbol q}^\ast}]}{{\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k} ^\ast} \cdot {\boldsymbol h}_{\boldsymbol k}^{s_{\boldsymbol k}}}.~~ \tag {10} \end{align} $$ It is known that the ideal Hall MHD model conserves three global quadratic invariants, when the boundary conditions are periodic. They are the total energy $E$, the generalized helicity $H_{\rm g}$ and the magnetic helicity $H_{\rm m}$. The expressions for the three quadratic invariants can be written in terms of these helical modes, $$\begin{alignat}{1} E=\,&\frac{1}{2V}\int(|{\boldsymbol u}|^2+|{\boldsymbol b}|^2)dV\\ =\,&\frac{1}{2}\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\sum\limits_{s_{\boldsymbol k}} {({|{u_{\boldsymbol k}^{s_{\boldsymbol k}}}|^2+| {b_{\boldsymbol k}^{s_{\boldsymbol k}}}|^2})}}=E^{\rm +}{\rm +}E^-,~~ \tag {11} \end{alignat} $$ $$\begin{alignat}{1} H_{\rm g}=\,&\frac{1}{2V}\int {[{{\boldsymbol u}\cdot {\boldsymbol b}+\frac{\varepsilon}{2}({{\boldsymbol u}\cdot {\boldsymbol \omega}})}]}dV \\ =\,&\frac{1}{4}\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} \sum\limits_{s_{\boldsymbol k}} [u_{\boldsymbol k}^{s_{\boldsymbol k}} b_{\boldsymbol k}^{s_{\boldsymbol k} ^\ast} +u_{\boldsymbol k}^{s_{\boldsymbol k} ^\ast} b_{\boldsymbol k}^{s_{\boldsymbol k}} +s_{\boldsymbol k} \varepsilon k|u_{\boldsymbol k}^{s_{\boldsymbol k}}|^2] \\ =\,&H_{\rm g}^+ +H_{\rm g}^-,~~ \tag {12} \end{alignat} $$ $$\begin{alignat}{1} H_{\rm m}=\,&\frac{1}{2V}\int {({{\boldsymbol a}\cdot {\boldsymbol b}})} dV \\ =\,&\frac{1}{2}\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} \sum\limits_{s_{\boldsymbol k}} {\frac{|{b_{\boldsymbol k}^{s_{\boldsymbol k}}}|^2}{s_{\boldsymbol k} k}}=H_{\rm m}^+ +H_{\rm m}^-,~~ \tag {13} \end{alignat} $$ where the integrals are evaluated over a box of volume $V=({2\pi})^3$, ${\boldsymbol \omega}=\nabla \times {\boldsymbol u}$ is the vorticity, the magnetic vector potential ${\boldsymbol a}$ is defined by ${\boldsymbol b}=\nabla \times {\boldsymbol a}$ and $\nabla \cdot {\boldsymbol a}=0$. It can be proved that the above invariants survive under truncation in the wave number space from Eqs. (8) and (9). Considering a generalized phase space with coordinates that are the real and imaginary parts of the helical modes $u_{\boldsymbol k}^{s_{\boldsymbol k}}$ and $b_{\boldsymbol k}^{s_{\boldsymbol k}}$, the Fourier helical representation of Eqs. (1)-(4) satisfy Liouville's theorem.[34] This allows one to use the classical Gibbs equilibrium ensembles to investigate statistical properties of the Hall MHD. The generalized probability distribution in phase space can be written as $$\begin{align} P=\frac{1}{Z}e^{-({\beta E+\gamma H_{\rm g} +\theta H_{\rm m}})},~~ \tag {14} \end{align} $$ where $\beta$, $\gamma$ and $\theta$ are generalized inverse temperatures corresponding to $E$, $H_{\rm g}$ and $H_{\rm m}$, respectively. The partition function $Z$ is given by $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!&Z=\int_{-\infty}^{\infty}e^{-(\beta E+\gamma H_{\rm g}+\theta H_{\rm m})}d\xi=\prod_{{\boldsymbol k}\in {\boldsymbol N}}\prod_{s_{\boldsymbol k}}Z_{\boldsymbol k}^{s_{\boldsymbol k}}\\ \!\!\!\!\!\!\!\!\!=\,&\!\!\prod\limits_{{\boldsymbol k}\in {\boldsymbol N}}\prod\limits_{s_{\boldsymbol k}}\!\frac{(2\pi)^24k}{2s_{\boldsymbol k}\varepsilon \beta\gamma k^2\!+\!(4\beta^2\!+\!2\varepsilon\gamma\theta \!-\!\gamma^2)k\!+\!4s_{\boldsymbol k}\beta\theta},~~ \tag {15} \end{alignat} $$ with $d\xi$ being the differential phase volume element $$\begin{align} d\xi=\prod\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\prod\limits_{s_{\boldsymbol k}} {du_{{\boldsymbol k}{\rm R}}^{s_{\boldsymbol k}} du_{{\boldsymbol k}{\rm I}}^{s_{\boldsymbol k}} db_{{\boldsymbol k}{\rm R}}^{s_{\boldsymbol k}} db_{{\boldsymbol k}{\rm I}}^{s_{\boldsymbol k}}}},~~ \tag {16} \end{align} $$ where $u_{{\boldsymbol k}{\rm R}}^{s_{\boldsymbol k}}$ and $u_{{\boldsymbol k}{\rm I}}^{s_{\boldsymbol k}}$ are the real and imaginary parts of $u_{\boldsymbol k}^{s_{\boldsymbol k}}$, and $b_{{\boldsymbol k}{\rm R}}^{s_{\boldsymbol k}}$ and $b_{{\boldsymbol k}{\rm I}}^{s_{\boldsymbol k}}$ are the real and imaginary parts of $b_{\boldsymbol k}^{s_{\boldsymbol k}}$, respectively. The requirement that $Z$ is finite and real implies the following relations for the inverse temperatures $$\begin{alignat}{1} \!\!\!\!\!&2\beta +s_{\boldsymbol k} \varepsilon \gamma k>0,~~ \tag {17} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!&2s_{\boldsymbol k} \varepsilon \beta \gamma k^2+({4\beta ^2+2\varepsilon \gamma \theta -\gamma ^2})k+4s_{\boldsymbol k} \beta \theta >0.~~ \tag {18} \end{alignat} $$ It should be noted that the inequalities (17) and (18) must be satisfied simultaneously for $s_{\boldsymbol k}=+1$ and $s_{\boldsymbol k}=-1$. By the differentiation of the partition function $Z$ $$\begin{align} \langle {E^{s_{\boldsymbol k}}({\boldsymbol k})}\rangle=\,&-\frac{\partial\ln Z_{\boldsymbol k}^{s_{\boldsymbol k}}}{\partial\beta},~~ \tag {19} \end{align} $$ $$\begin{align} \langle {H_{\rm g}^{s_{\boldsymbol k}} ({\boldsymbol k})} \rangle=\,&-\frac{\partial \ln Z_{\boldsymbol k}^{s_{\boldsymbol k}}}{\partial \gamma},~~ \tag {20} \end{align} $$ $$\begin{align} \langle {H_{\rm m}^{s_{\boldsymbol k}} ({\boldsymbol k})} \rangle=\,&-\frac{\partial \ln Z_{\boldsymbol k}^{s_{\boldsymbol k}}}{\partial \theta},~~ \tag {21} \end{align} $$ the helical modes equilibrium spectra of three quadratic invariants are obtained, $$\begin{align} \langle {E^{s_{\boldsymbol k}}({\boldsymbol k})} \rangle=\,&\frac{8\beta k+4s_{\boldsymbol k} \theta +2s_{\boldsymbol k} \varepsilon \gamma k^2}{D^{s_{\boldsymbol k}}(k)},~~ \tag {22} \end{align} $$ $$\begin{align} \langle {H_{\rm g}^{s_{\boldsymbol k}} ({\boldsymbol k})} \rangle=\,&\frac{2s_{\boldsymbol k} \varepsilon \beta k^2+2\varepsilon \theta k-2\gamma k}{D^{s_{\boldsymbol k}}(k)},~~ \tag {23} \end{align} $$ $$\begin{align} \langle {H_{\rm m}^{s_{\boldsymbol k}} ({\boldsymbol k})} \rangle=\,&\frac{2\varepsilon \gamma k+4s_{\boldsymbol k} \beta}{D^{s_{\boldsymbol k}}(k)},~~ \tag {24} \end{align} $$ $$\begin{align} D^{s_{\boldsymbol k}}(k)=\,&2s_{\boldsymbol k} \varepsilon \beta \gamma k^2+(4\beta ^2+2\varepsilon \gamma \theta -\gamma ^2)k+4s_{\boldsymbol k} \beta \theta.~~ \tag {25} \end{align} $$ We should point out that the generalized inverse temperatures cannot change independently. They strongly depend on the global quadratic invariants and can be numerically obtained by solving the system of nonlinear polynomial equations $$\begin{alignat}{1} E=\,&\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\langle {E({\boldsymbol k})} \rangle}=\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\sum\limits_{s_{\boldsymbol k}} {\langle {E^{s_{\boldsymbol k}}({\boldsymbol k})} \rangle}},~~ \tag {26} \end{alignat} $$ $$\begin{alignat}{1} H_{\rm g}=\,&\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\langle {H_{\rm g} ({\boldsymbol k})} \rangle}=\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\sum\limits_{s_{\boldsymbol k}} {\langle {H_{\rm g}^{s_{\boldsymbol k}} ({\boldsymbol k})} \rangle}},~~ \tag {27} \end{alignat} $$ $$\begin{alignat}{1} H_{\rm m}=\,&\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\langle {H_{\rm m} ({\boldsymbol k})} \rangle}=\sum\limits_{{\boldsymbol k}\in {\boldsymbol N}} {\sum\limits_{s_{\boldsymbol k}} {\langle {H_{\rm m}^{s_{\boldsymbol k}} ({\boldsymbol k})} \rangle}}.~~ \tag {28} \end{alignat} $$ In the following we will examine some properties of these equilibrium spectra. In Fig. 1, the equilibrium spectra of left- and right-handed helical modes of three quadratic invariants are given for a particular choice of parameters (see caption). We can see that the left-handed helical sector manifests the simultaneous condensation at the large and small wave numbers. However, the right-handed helical sector does not have any condensation in the region of wave number space. We know that the condensation of equilibrium spectra corresponds to the cascade properties of turbulence when driving and dissipation are reinstated. Therefore, we can infer that the left-handed helical modes play a major role on the inverse and forward cascade properties of the Hall MHD turbulence when the generalized helicity and magnetic helicity both are positive.
cpl-34-1-015201-fig1.png
Fig. 1. The equilibrium spectra of left-handed and right-handed helical modes (upper and lower panels, respectively) for the given parameters $E=1$, $H_{\rm g}=0.73$, $H_{\rm m}=0.72$, $N_{\rm box}=32$, $k_{\min}=1$, $k_{\max}=32\sqrt 3$. The vertical bar indicates the ion skin depth wave vector $k_{\rm h}=1/\varepsilon=3$.
cpl-34-1-015201-fig2.png
Fig. 2. The equilibrium spectra of left-handed (upper panel) and right-handed (middle panel for energy and magnetic helicity, lower panel for generalized helicity) helical modes for given values of parameters, namely $E=1$, $H_{\rm g}=0.73$, $H_{\rm m}=-0.72$, $N_{\rm box}=32$, $k_{\min}=1$, $k_{\max}=32\sqrt 3$, and the vertical bar indicates again $k_{\rm h}$.
At variance with energy, the generalized helicity and magnetic helicity are not positively defined and can be positive or negative. Figure 2 shows the equilibrium spectra of the helical modes when the generalized helicity is positive and the magnetic helicity is negative. From the comparison between Figs. 2 and 1, we find that the condensation at the small wave numbers disappears from the left-handed helical sector, while it appears in the right-handed helical sector. Thus we can conclude that the magnetic helicity dominates the inverse cascade properties of Hall MHD. By the way, it should be pointed out that we only draw the single-log figure for the right-handed helical spectrum of generalized helicity because the spectrum has different signs in the region of wave number space ($\langle {H_{\rm g}^-({k=1})}\rangle >0$ and $\langle {H_{\rm g}^- ({k>1})} \rangle < 0$). Furthermore, in Fig. 3, the generalized helicity is negative and the magnetic helicity is positive. The condensation at the large wave numbers appears in the right-handed helical sector compared with Fig. 1. It indicates that the generalized helicity dominates the forward cascade properties of the Hall MHD. From the above results, we can also expect the simultaneous condensations at the large and small wave numbers in the right-handed sector when the generalized helicity and the magnetic helicity both are negative. This implies that the right-handed helical modes play a major role in the cascade properties of the Hall MHD turbulence.
cpl-34-1-015201-fig3.png
Fig. 3. The equilibrium spectra of the left-handed and right-hand helical modes (upper and lower panels, respectively) for the given parameters $E=1$, $H_{\rm g}=-0.73$, $H_{\rm m}=0.72$, $N_{\rm box}=32$, $k_{\min}=1$, $k_{\max}=32\sqrt 3$. The vertical bar indicates again $k_{\rm h}$.
In addition, it is of interest to compare the cascade properties of the Hall MHD turbulence with that of drift-wave turbulence.[9] In the drift-wave turbulence, the electrostatic potential spectrum can be obtained using the Gibbs ensemble method. Based on the electrostatic potential spectrum, the dual cascades (energy towards large scales and the potential vorticity towards small scales) are predicted. However, the cascade process may be different from that in the Hall MHD turbulence. Because the potential vorticity spectrum and energy spectrum have a definite relation $U_{k}=k^2W_{k}$, there is no such relation between the magnetic helicity spectrum and the generalized helicity spectrum. In the above discussion, we assume that the fluctuating fields are isotropic and only the omni-directional spectrum is considered. However, if a mean magnetic field is present, the Hall MHD fluctuations will develop in anisotropies. For such a situation, the spectral cascades take place preferentially in the direction perpendicular to the mean magnetic field, rather than along the parallel direction.[31,35] In addition, the magnetic helicity in periodic geometry will not be conserved anymore due to the presentation of a mean magnetic field.[36] In summary, the equilibrium spectra of left- and right-handed helical modes of the incompressible ideal Hall MHD have been presented for given parameters. It is shown that the tendency of equilibrium spectra to condense in the region of wave number space is affected by the signs of the generalized helicity and magnetic helicity. Further analysis also reveals that the generalized helicity dominates the forward cascade and the magnetic helicity dominates the inverse cascade properties of the Hall MHD turbulence. These results provide us with valuable information on the nonlinear dynamics of the Hall MHD turbulence. Helical modes are basic in the incompressible Hall MHD and have been used to model the solar wind turbulence. To obtain more details on the nonlinear interaction of helical modes, we explore the numerical analysis for the Hall MHD equations in future work. References Statistical theory of magnetohydrodynamic turbulence: recent resultsInertial Ranges in Two-Dimensional TurbulenceHelical turbulence and absolute equilibriumPossibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulenceHigh-beta turbulence in two-dimensional magnetohydrodynamicsPseudo-three-dimensional turbulence in magnetized nonuniform plasmaPossible inverse cascade behavior for drift-wave turbulenceStatistical properties of ideal three-dimensional magnetohydrodynamicsSpectral properties of decaying turbulence in electron magnetohydrodynamicsStatistical properties of ideal three-dimensional Hall magnetohydrodynamics: The spectral structure of the equilibrium ensembleGlobal invariants in ideal magnetohydrodynamic turbulenceStatistical properties of three-dimensional two-fluid plasma modelIs there a statistical mechanics of turbulence?Strong MHD helical turbulence and the nonlinear dynamo effectModeling of short scale turbulence in the solar windHall Magnetic Reconnection RateDynamo Action in Hall MagnetohydrodynamicsEnergy transfer in Hall-MHD turbulence: cascades, backscatter, and dynamo actionHall Magnetohydrodynamics and Its Applications to Laboratory and Cosmic PlasmaDynamo Action in Magnetohydrodynamics and Hall‐MagnetohydrodynamicsThe Beltrami spectrum for incompressible fluid flowsThe nature of triad interactions in homogeneous turbulenceThe joint cascade of energy and helicity in three-dimensional turbulenceAnisotropic weak whistler wave turbulence in electron magnetohydrodynamicsWave turbulence in incompressible Hall magnetohydrodynamicsInverse Energy Cascade in Three-Dimensional Isotropic TurbulencePurely helical absolute equilibria and chirality of (magneto)fluid turbulenceTwo-dimensional turbulenceAnisotropy in Hall MHD turbulence due to a mean magnetic fieldNonlinear decay of magnetic helicity in magnetohydrodynamic turbulence with a mean magnetic field
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