2.1. Governing equations

The basic fluid equations governing the electron dynamics in an incompressible (dissipationless) plasma are

(2.1)\begin{gather} \partial_t \boldsymbol{u}_e + (\boldsymbol{u}_e \boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u}_e ={-}\frac{1}{\rho_e} \boldsymbol{\nabla} P_e - \frac{q_e}{m_e} ( \boldsymbol{u}_e \times \boldsymbol{B} + \boldsymbol{E} ), \end{gather}

(2.2)\begin{gather}\partial_t \boldsymbol{B} ={-} \boldsymbol{\nabla} \times \boldsymbol{E}, \end{gather}

(2.3)\begin{gather}\boldsymbol{\nabla} \times \boldsymbol{B} = \mu_0 \boldsymbol{J}, \end{gather}

(2.4)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}_e = 0, \end{gather}

(2.5)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{B} = 0 , \end{gather}

where $\boldsymbol {u}_e(\boldsymbol {x},t)$ is the electron velocity, $\rho _e(\boldsymbol {x},t)= m_e n_0$ is the constant electron mass density, with $m_e$ the electron mass and $n_0$ the density, $P_e(\boldsymbol {x},t)$ is the electron pressure, $q_e>0$ is the modulus of the electron charge, $\boldsymbol {B}(\boldsymbol {x},t)$ is the magnetic field, $\boldsymbol {E}(\boldsymbol {x},t)$ is the electric field, $\mu_0$ is the vacuum permeability, $\boldsymbol {J}(\boldsymbol {x},t) = n_0 q_e (\boldsymbol {u}_i - \boldsymbol {u}_e )$ is the electric current and $\boldsymbol {u}_i(\boldsymbol {x},t)$ is the ion velocity (assumed to be zero). Normalizing the magnetic field to the (electron) Alfvén velocity and then taking the rotational of (2.1) combined with the Maxwell–Faraday law (2.2), one obtains

(2.6)\begin{equation} \partial_t ( d_e^{2} \nabla^{2} -1) \boldsymbol{b} + ( \boldsymbol{u}_e \boldsymbol{\cdot} \boldsymbol{\nabla} ) (d_e^{2} \nabla^{2} -1)\boldsymbol{b} = ( d_e^{2} \nabla^{2} - 1 ) \boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}_e, \end{equation}

where $d_e = \sqrt {m_e/(n_0 q_e^{2} \mu _0)}$ is the electron inertial length. Now, we introduce a relatively strong and uniform (normalized) magnetic field $\boldsymbol {b}_0 = b_0 \hat {\boldsymbol {e}}_\|$ that defines the parallel direction. In the limit of IEMHD, the spatial variations of $\boldsymbol {b}$ are done on a characteristic length $L \ll d_e$ and mainly in the plane perpendicular to $\hat {\boldsymbol {e}}_\|$. Thus, at the leading order, we have

(2.7)\begin{equation} ( \partial_t + {\boldsymbol{u}_e}_\perp \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp )d_e^{2} \nabla^{2}_\perp \boldsymbol{b} = d_e^{2} (\nabla^{2}_\perp \boldsymbol{b}_\perp \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp) \boldsymbol{u}_e - (\boldsymbol{b}_0 \boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u}_e, \end{equation}

and also $\boldsymbol {J} = - n_0 q_e \boldsymbol {u}_e$, which can be written $d_e \boldsymbol {j} = - \boldsymbol {u}_e$ with the normalized electric current $\boldsymbol {j} \equiv \boldsymbol {\nabla } \times \boldsymbol {b}$. The magnetic field having a zero divergence, we define $\boldsymbol {b} \equiv \boldsymbol {b}_0 - \boldsymbol {\nabla } \times ( g \boldsymbol {e}_x + \psi \boldsymbol {e}_z )$, where $\hat {\boldsymbol {e}}_z$ is a unit vector (hereafter, we will assume $\hat {\boldsymbol {e}}_z = \hat {\boldsymbol {e}}_\|$ which is valid at leading order for a relatively strong uniform magnetic field $\boldsymbol {b}_0$), $\psi (\boldsymbol {x},t)$ a streamfunction and $g(\boldsymbol {x},t)$ a function satisfying the relation $\partial _y g \equiv b_\|$. We obtain the relation

(2.8)\begin{equation} \nabla^{2}_\perp \boldsymbol{b} = ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp ) (\nabla^{2}_\perp \psi) + \nabla^{2}_\perp b_\| \hat{\boldsymbol{e}}_\|, \end{equation}

where, hereafter, the $z$-derivative is assumed to be negligible compared with the perpendicular derivative. Replacing $\boldsymbol {b}$ by its expression, the electron velocity can be expressed as a function of the magnetic field components

(2.9)\begin{equation} \boldsymbol{u}_e = d_e ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp b_\| - \nabla_\perp^{2} \psi \hat{\boldsymbol{e}}_\| ). \end{equation}

Projecting equation (2.7) in the perpendicular plane to $\hat {\boldsymbol {e}}_\|$, we find

(2.10)\begin{align} & ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp ) [\partial_t ( d_e^{2} \nabla_\perp^{2} \psi ) ] + d_e^{2} ( {\boldsymbol{u}_e}_\perp \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp ) [ ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp ) \nabla^{2}_\perp \psi ] \nonumber\\ & \quad = d_e^{3} ( \nabla^{2}_\perp \boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp ) ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp b_\| ) - d_e b_0 \partial_\| ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp ) b_\| . \end{align}

The non-trivial relation

(2.11)\begin{align} & ( {\boldsymbol{u}_e}_\perp \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp ) [ ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp ) \nabla^{2}_\perp \psi ] \nonumber\\ & \quad =(\hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp ) [ ({\boldsymbol{u}_e}_\perp \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp ) \nabla^{2}_\perp \psi ] + d_e ( \nabla^{2}_\perp \boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp ) ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp b_\| ), \end{align}

allows us to simplify the previous equation and, by expressing $\boldsymbol {u}_e$ as a function of $\psi$, we obtain, after some algebraic manipulations,

(2.12)\begin{equation} \partial_t ( \nabla_\perp^{2} \psi ) + d_e [ ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp b_\| ) \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp ] \nabla_\perp^{2} \psi ={-} \varOmega_e \partial_\| b_\|, \end{equation}

with $\varOmega _e \equiv b_0 / d_e$ the cyclotron frequency of electrons (note that, here, $\varOmega _e$ is constant due to the assumption of incompressibility).

Now, a projection of (2.7) in the $\hat {\boldsymbol {e}}_\|$ direction gives directly

(2.13)\begin{equation} \partial_t ( d_e^{2} \nabla_\perp^{2} b_\| ) + d_e^{2} ( {\boldsymbol{u}_e}_\perp \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp ) \nabla^{2}_\perp b_\| ={-} d_e^{3} ( \nabla^{2}_\perp \boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp )\nabla_\perp^{2} \psi + d_e b_0 \partial_\| ( \nabla_\perp^{2} \psi ). \end{equation}

It is straightforward to show that the first term of the right-hand side is exactly zero. Then, by expressing $\boldsymbol {b}$ and $\boldsymbol {u}_e$ as functions of $\psi$ and $b_\|$, we obtain

(2.14)\begin{equation} \partial_t ( \nabla_\perp^{2} b_\| ) + d_e [ ( \hat{\boldsymbol{e}}_\| \times \boldsymbol{\nabla}_\perp b_\| ) \boldsymbol{\cdot} \boldsymbol{\nabla}_\perp ] \nabla_\perp^{2} b_\| = \varOmega_e \partial_\| ( \nabla_\perp^{2} \psi ) . \end{equation}

Equations (2.12) and (2.14) describe the dynamics of electrons at inertial scales. They have been derived in a more general framework and using kinetic arguments by Chen & Boldyrev (Reference Chen and Boldyrev2017) and Passot et al. (Reference Passot, Sulem and Tassi2017). Here, we have used the incompressibility condition to propose a (less accurate but more) fast derivation of a system that a priori describes only IWW. However, it is interesting to note that at inertial electron scales: (i) IKAW and IWW can have the same dispersion relation and the only difference is that the transition to the inertial regime occurs at $k_\perp ^{2} d_e^{2} \simeq 1$ for IWW rather than $k_\perp ^{2} d_e^{2} \simeq 1 + 2/ \beta _i$ for IKAW; (ii) the nonlinear equations governing the dynamics of IKAW and IWW are mathematically similar (up to a change of variable from $b_z$ to $\rho _e$ Chen & Boldyrev Reference Chen and Boldyrev2017; Passot et al. Reference Passot, Sulem and Tassi2017), which means that the physics of wave turbulence developed in this paper applies to both waves. A similar situation is found at scales larger than $d_e$: in the presence of a strong $\boldsymbol {B_0}$, the equations describing the nonlinear dynamics of kinetic Alfvén waves and whistler waves have exactly the same mathematical form, which means that the physics of wave turbulence is similar for both problems (Galtier & Meyrand Reference Galtier and Meyrand2015).

2.2. Three-dimensional quadratic invariants

In the absence of forcing and dissipation, the system (2.12)–(2.14) has two quadratic invariants. The first invariant is the energy which is written at the leading order

(2.15)\begin{equation} E = d_e^{2} \langle \boldsymbol{j}^{2} \rangle = E_\perp{+} E_\| = d_e^{2} \langle (\nabla_\perp b_\| )^{2} + (\nabla_\perp^{2} \psi )^{2} \rangle, \end{equation}

where $\langle \rangle$ is a spatial average or, equivalently by ergodicity, an ensemble average. Here, $E$ can also be interpreted as the kinetic energy of electrons. As shown in Appendix A, both $E_\perp$ and $E_\|$ are separately conserved at the nonlinear level, however, energy is exchanged between the two at the linear level, thanks to the presence of waves. This definition of energy is valid for both IWW and for IKAW in the limit of small $\beta _i$.

The second quadratic invariant is the momentum that can be written at the leading order

(2.16)\begin{equation} H = d_e^{2} \langle (\nabla_\perp^{2} \psi) (\nabla_\perp^{2} b_\|) \rangle. \end{equation}

Here, $H$ can be interpreted as the kinetic helicity of electrons. Unlike energy, the momentum is not positive defined. As we will see later, the wave kinetic equations conserve these two invariants on the resonant manifold.

2.3. Dispersion relation

In the linear regime, the Fourier transform of (2.12) and (2.14) gives

(2.17)\begin{gather} \partial_t \psi_k = {\rm i} \varOmega_e k_\| k_\perp^{{-}2} b_k, \end{gather}

(2.18)\begin{gather}\partial_t b_k = {\rm i} \varOmega_e k_\| \psi_k, \end{gather}

where the Fourier transform used is

(2.19)\begin{equation} \psi(\boldsymbol{k},t) \equiv \psi_k = \int_{\mathbb{R}^{3}} \psi(\boldsymbol{x},t) \, \mathrm{e}^{-{\rm i} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{x}} \,\mathrm{d} \boldsymbol{x}. \end{equation}

Hereafter, we use the notation $b_k \equiv {b_\|}_k$. If the wavevector $\boldsymbol {k}$ is decomposed as $\boldsymbol {k} = k_\perp \hat {\boldsymbol {e}}_\perp + k_\| \hat {\boldsymbol {e}}_\|$, then the linear dispersion relation reads

(2.20)\begin{equation} \left( \frac{\omega_k}{\varOmega_e} \right)^{2} = \left( \frac{k_\|}{k_\perp} \right)^{2} . \end{equation}

One can find the following solutions to the linear IEMHD equations in Fourier space:

(2.21)\begin{gather} \psi_k(k_\perp,t) = f ( k_\perp ) \cos(\omega_k t ) + i g ( k_\perp ) k_\perp^{{-}1} \sin (\omega_k t), \end{gather}

(2.22)\begin{gather}b_k(k_\perp,t) = g ( k_\perp ) \cos(\omega_k t ) + i f ( k_\perp ) k_\perp \sin (\omega_k t), \end{gather}

with $f$ and $g$ two arbitrary functions.

5.1. Definition of the energy density tensor

We now move on to a statistical description. We use the ensemble average $\langle \rangle$ and define the following spectral correlators (cumulants) for homogeneous turbulence (we assume $\langle a_k^{s_k}\rangle =0$):

(5.1)\begin{equation} \langle a_{\boldsymbol{k}}^{s_k} a_{\boldsymbol{k'}}^{s_k'} \rangle = e_{\boldsymbol{k}'}^{s_k'} \delta_{kk'}\delta^{s_k}_{s_k'}, \end{equation}

with ${\rm e}^{s_k'}( \boldsymbol {k}') = e_{\boldsymbol {k}'}^{s_k'}$. We observe the presence of the delta function $\delta ^{s_k}_{s_k'}$ meaning that two-point correlations of opposite polarities have no long-time influence in the wave turbulence regime. The other delta function is the consequence of the statistical homogeneity assumption. The objective of the wave turbulence theory is to derive a self-consistent equation for the time evolution of this spectral correlator; this is the kinetic equation. In this development, we have to face the classical closure problem: a hierarchy of statistical equations of increasingly higher order emerges. In contrast to strong turbulence, in the weak wave turbulence regime we can use the time scale separation to achieve a natural closure of the system (Benney & Saffman Reference Benney and Saffman1966; Newell et al. Reference Newell, Nazarenko and Biven2001). After a lengthy (but classical) algebra, we obtain the time evolution equation of the energy density tensor (we leave the details of the derivation to Appendix B)

(5.2)\begin{align} \partial_t e_{\boldsymbol{k}}^{s_k} & = \frac{{\rm \pi} \epsilon^{2}}{16} \sum_{s_p s_q} \int_{\mathbb{R}^{6}} s_k \omega_k \vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} ( s_k \omega_k e_{\boldsymbol{p}}^{s_p} e_{\boldsymbol{q}}^{s_q} + s_p \omega_p e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{q}}^{s_q} + s_q \omega_q e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{p}}^{s_p} ) \nonumber\\ & \quad \times \delta(\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q}, \end{align}

where

(5.3)\begin{equation} \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \equiv \frac{L_{\boldsymbol{kpq}}^{s_k s_p s_q}}{s_k \omega_k} , \end{equation}

and $\varOmega _{kpq} \equiv s_k \omega _k + s_p \omega _p + s_q \omega _q$ and $\delta _{kpq} = \delta (\boldsymbol {k} + \boldsymbol {p} + \boldsymbol {q})$. This equation is the main result of the wave turbulence formalism. It describes the statistical properties of IWW or IKAW turbulence at the leading order, i.e. for three-wave interactions.

5.2. Detailed conservation of quadratic invariants

In § 2.2 we introduced the three-dimensional invariants of IEMHD. The first test that the wave turbulence equations must pass is the detailed conservation – i.e. for each triad ($\boldsymbol {k}, \boldsymbol {p}, \boldsymbol {q}$) – of these invariants. Starting from the definitions (2.15) and (2.16), we define the energy and momentum spectra

(5.4)\begin{gather} E(\boldsymbol{k}) \equiv \sum_{s_k={\pm}} e_{\boldsymbol{k}}^{s_k} = e_{\boldsymbol{k}}^{+} + e_{\boldsymbol{k}}^{-}, \end{gather}

(5.5)\begin{gather}H(\boldsymbol{k}) \equiv \sum_{s_k={\pm}} s_k k_\perp e_{\boldsymbol{k}}^{s_k} = k_\perp ( e_{\boldsymbol{k}}^{+} - e_{\boldsymbol{k}}^{-} ). \end{gather}

Before checking the energy conservation, it is interesting to note that, when one of the polarized energy density tensors $e_{\boldsymbol {k}}^{\pm }$ is zero, the other invariant is extremal and verifies the relation $H(\boldsymbol {k}) = \pm k_\perp E(\boldsymbol {k})$, which is in agreement with the realizability condition (Schwarz inequality) $\vert H(\boldsymbol {k}) \vert \le k_\perp E(\boldsymbol {k})$. From (5.2), we obtain the equation for the (total) energy

(5.6)\begin{align} \partial_t E(t) & \equiv \partial_t \int_{\mathbb{R}^{3}} \sum_{s_k} e_{\boldsymbol{k}}^{s_k} \, \mathrm{d}\boldsymbol{k} \nonumber\\ & = \frac{{\rm \pi} \epsilon^{2}}{16} \sum_{s_k s_p s_q} \int_{\mathbb{R}^{9}} s_k \omega_k\vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} ( s_k \omega_k e_{\boldsymbol{p}}^{s_p} e_{\boldsymbol{q}}^{s_q} + s_p \omega_p e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{q}}^{s_q} + s_q \omega_q e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{p}}^{s_p} ) \nonumber\\ & \quad \times \delta(\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{k} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q}. \end{align}

Without forcing and dissipation, energy must be conserved and this conservation is done at the level of triadic interactions (detailed energy conservation). The demonstration is straightforward. By applying a cyclic permutation of wavevectors and polarizations, we find

(5.7)\begin{align} \partial_t E(t) & = \frac{{\rm \pi} \epsilon^{2}}{48} \sum_{s_k s_p s_q} \int_{\mathbb{R}^{9}} \varOmega_{kpq} s_k \omega_k \vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} ( s_k \omega_k e_{\boldsymbol{p}}^{s_p} e_{\boldsymbol{q}}^{s_q} + s_p \omega_p e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{q}}^{s_q} + s_q \omega_q e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{p}}^{s_p} ) \nonumber\\ & \quad \times \delta(\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{k} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q} = 0 , \end{align}

which proves the conservation of (kinetic) energy on the resonant manifold for each triadic interaction.

For the second invariant $H(t)$, one has

(5.8)\begin{align} \partial_t H(t) & \equiv \partial_t \int_{\mathbb{R}^{3}} \sum_{s_k} s_k k_\perp e_{\boldsymbol{k}}^{s_k} \, \mathrm{d}\boldsymbol{k} \nonumber\\ & = \frac{{\rm \pi} \epsilon^{2} \varOmega_e}{16} \sum_{s_k s_p s_q} \int_{\mathbb{R}^{9}} k_\| \vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} ( s_k \omega_k e_{\boldsymbol{p}}^{s_p} e_{\boldsymbol{q}}^{s_q} + s_p \omega_p e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{q}}^{s_q} + s_q \omega_q e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{p}}^{s_p} ) \nonumber\\ & \quad \times \delta(\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{k} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q}. \end{align}

The same manipulations as before lead immediately to

(5.9)\begin{align} \partial_t H(t) & = \frac{{\rm \pi} \epsilon^{2} \varOmega_e}{48} \sum_{s_k s_p s_q} \int_{\mathbb{R}^{9}} \vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} ( s_k \omega_k e_{\boldsymbol{p}}^{s_p} e_{\boldsymbol{q}}^{s_q} + s_p \omega_p e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{q}}^{s_q} + s_q \omega_q e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{p}}^{s_p} ) \nonumber\\ & \quad \times ( k_\| + p_\| + q_\| ) \delta (\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{k} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q} = 0. \end{align}

This proves the conservation of momentum (kinetic helicity) on the resonant manifold for each triadic interaction.

5.3. Helical turbulence

From the wave turbulence equation (5.2), we can deduce several general properties. First, we observe that there is no coupling between the waves associated with the $\boldsymbol {p}$ and $\boldsymbol {q}$ wavevectors when these wavevectors are collinear. Second, the nonlinear coupling disappears whenever the wavenumbers $p_\perp$ and $q_\perp$ are equal if their associated polarities $s_p$ and $s_q$ are also equal. These properties are also observed in EMHD (for scales larger than $d_e$) and more generally for other helical waves (Kraichnan Reference Kraichnan1973; Waleffe Reference Waleffe1992; Turner Reference Turner2000; Galtier Reference Galtier2003; Galtier & Bhattacharjee Reference Galtier and Bhattacharjee2003). Note that they can already be deduced directly from the fundamental equation (3.5). Third, the wave modes ($k_\| > 0$) are decoupled from the slow mode ($k_\|=0$) which is not described by these wave kinetic equations. This situation is thus different from wave turbulence in incompressible MHD where the slow mode has a profound influence on the nonlinear dynamics.

6.1. Wave kinetic equations for the invariants

The objective of this section is to derive, in the stationary case, the exact power law solutions of the kinetic equations for the two invariants, energy and momentum. To do so, it is necessary to simplify the equations, written for $E(\boldsymbol {k})$ and $H(\boldsymbol {k})$, using the axisymmetric assumption. First of all, we have

(6.1)\begin{align} \partial_t \begin{pmatrix} E(\boldsymbol{k}) \\ H(\boldsymbol{k}) \end{pmatrix} & = \frac{{\rm \pi} \epsilon^{2}}{16} \sum_{s_k s_p s_q} \int_{\mathbb{R}^{6}} s_k \omega_k\vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} ( s_k \omega_k e_{\boldsymbol{p}}^{s_p} e_{\boldsymbol{q}}^{s_q} + s_p \omega_p e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{q}}^{s_q} + s_q \omega_q e_{\boldsymbol{k}}^{s_k} e_{\boldsymbol{p}}^{s_p} ) \nonumber\\ & \quad \times \begin{pmatrix} 1 \\ s_k k_\perp \end{pmatrix}\delta(\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q}. \end{align}

We now develop the energy density tensors inside the integral in terms of energy and momentum spectra. We note that only terms containing the products of two $E(\boldsymbol {k})$ or two $H(\boldsymbol {k})$ will survive for energy, whereas only the products of $E(\boldsymbol {k}) H(\boldsymbol {k})$ will survive for helicity. After some algebra, we find for the energy

(6.2)\begin{align} \partial_t E(\boldsymbol{k}) & = \frac{{\rm \pi} \epsilon^{2}}{64} \sum_{s_k s_p s_q} \int_{\mathbb{R}^{6}} s_k \omega_k \vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} \nonumber\\ & \quad \times \left[ s_k \omega_k E (\boldsymbol{p}) E (\boldsymbol{q}) + s_p \omega_p E (\boldsymbol{k}) E (\boldsymbol{q}) + s_q \omega_q E (\boldsymbol{k}) E (\boldsymbol{p}) \vphantom{\frac{H(\boldsymbol{p}) H(\boldsymbol{q})}{p_\perp q_\perp}}\right. \nonumber\\ & \quad \left. +\, s_k s_p s_q \left( \omega_k\frac{H(\boldsymbol{p}) H(\boldsymbol{q})}{p_\perp q_\perp} + \omega_p \frac{H(\boldsymbol{k}) H(\boldsymbol{q})}{k_\perp q_\perp} + \omega_q\frac{H(\boldsymbol{k}) H(\boldsymbol{p})}{k_\perp p_\perp} \right) \right] \nonumber\\ & \quad \times \delta (\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q}, \end{align}

and for the momentum

(6.3)\begin{align} \partial_t H(\boldsymbol{k})& = \frac{{\rm \pi} \epsilon^{2}}{64} \sum_{s_k s_p s_q} \int_{\mathbb{R}^{6}} \omega_k k_\perp \vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} \left( s_k \omega_k \left[ E(\boldsymbol{p}) \frac{H(\boldsymbol{q})}{s_q q_\perp} + E(\boldsymbol{q}) \frac{H(\boldsymbol{p})}{s_p p_\perp} \right] \right.\nonumber\\ & \quad \left. + \,s_p \omega_p \left[ E(\boldsymbol{k}) \frac{H(\boldsymbol{q})}{s_q q_\perp} + E(\boldsymbol{q}) \frac{H(\boldsymbol{k})}{s_k k_\perp} \right] \right. \nonumber\\ & \quad \left. + \,s_q \omega_q \left[E(\boldsymbol{k}) \frac{H(\boldsymbol{p})}{s_p p_\perp} + E(\boldsymbol{p}) \frac{H(\boldsymbol{k})}{s_k k_\perp} \right] \right) \delta(\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q}. \end{align}

If we exchange in the integrand the dummy variables, $\boldsymbol {p}$ and $\boldsymbol {q}$, as well as $s_p$ and $s_q$, we can simplify further the previous expressions to obtain

(6.4)\begin{equation} \partial_t \left( \begin{array}{c}E(\boldsymbol{k}) \\ H(\boldsymbol{k}) \end{array}\right) = \frac{{\rm \pi} \epsilon^{2}}{32} \sum_{s_k s_p s_q} \int_{\mathbb{R}^{6}} \vert \tilde L_{\boldsymbol{kpq}}^{s_k s_p s_q} \vert^{2} s_k \omega_k s_p \omega_p \left( \begin{array}{c} X_E \\ X_H \end{array}\right) \delta(\varOmega_{kpq} ) \delta_{kpq} \, \mathrm{d}\boldsymbol{p} \, \mathrm{d}\boldsymbol{q}, \end{equation}

with

(6.5)\begin{equation} \left( \begin{array}{c} X_E \\ X_H \end{array}\right) = \left( \begin{array}{c} \displaystyle E (\boldsymbol{q})[ E (\boldsymbol{k}) - E (\boldsymbol{p}) ] + \dfrac{H(\boldsymbol{q})}{s_q q_\perp} \left( \dfrac{H(\boldsymbol{k})}{s_k k_\perp} - \dfrac{H(\boldsymbol{p})}{s_p p_\perp} \right) \\ \displaystyle s_k k_\perp \left[ E(\boldsymbol{q}) \left( \dfrac{H(\boldsymbol{k})}{s_k k_\perp} - \dfrac{H(\boldsymbol{p})}{s_p p_\perp} \right) + \dfrac{H(\boldsymbol{q})}{s_q q_\perp} [ E(\boldsymbol{k}) - E(\boldsymbol{p})] \right] \end{array}\right). \end{equation}

6.2. The axisymmetric wave turbulence equations

To simplify the problem, we will consider an axial symmetry with respect to the external magnetic field and introduce the two-dimensional anisotropic spectra

(6.6)\begin{gather} E_k = E ( k_\perp, k_\| ) = 2{\rm \pi} k_\perp E ( \boldsymbol{k}_\perp, k_\| ), \end{gather}

(6.7)\begin{gather}H_k = H ( k_\perp, k_\| ) = 2{\rm \pi} k_\perp H ( \boldsymbol{k}_\perp, k_\| ), \end{gather}

which result from an integration over the angles in the plane perpendicular to the mean magnetic field (see figure 1). In polar coordinates $\mathrm {d}\boldsymbol {p} \, \mathrm {d}\boldsymbol {q} = p_\perp \, \mathrm {d} \alpha _q \, \mathrm {d} p_\perp \, \mathrm {d} p_\| \, \mathrm {d} q_\|$ and, thanks to the Al-Kashi formula: $q_\perp ^{2} = k_\perp ^{2} + p_\perp ^{2} - 2 k_\perp p_\perp \cos \alpha _q$, we find at fixed $k_\perp$ and $p_\perp$, $q_\perp \, \mathrm {d} q_\perp = k_\perp p_\perp \sin \alpha _q \, \mathrm {d} \alpha _q$. Using expression (5.3), we then obtain the kinetic equations

(6.8)\begin{align} \partial_t \begin{pmatrix} E_k \\ H_k \end{pmatrix} & = \frac{\epsilon^{2} \varOmega_e^{2}}{2^{12}} \sum_{s_k s_p s_q} \int_{\varDelta_\perp} \frac{s_k s_p k_\| p_\|}{k_\perp^{2} p_\perp^{2} q_\perp^{2}} \left(\frac{s_q q_\perp{-} s_p p_\perp}{ s_k \omega_k}\right)^{2} ( s_k k_\perp{+} s_p p_\perp{+} s_q q_\perp )^{2} \sin \alpha_q \nonumber\\ & \quad \times \begin{pmatrix} \tilde{X}_E \\ \tilde{X}_H \end{pmatrix} \delta (\varOmega_{kpq} ) \delta_{k_\|p_\|q_\|} \, \mathrm{d} p_\perp \, \mathrm{d} q_\perp \, \mathrm{d} p_\| \, \mathrm{d} q_\|, \end{align}

where $\varDelta _\perp$ the integration domain verifies the resonance condition $\boldsymbol {k}_\perp + \boldsymbol {p}_\perp + \boldsymbol {q}_\perp = \boldsymbol {0}$ and

(6.9)\begin{equation} \begin{pmatrix} \tilde{X}_E \\ \tilde{X}_H \end{pmatrix} = \begin{pmatrix} \displaystyle E_q ( p_\perp E_k - k_\perp E_p ) +\frac{H_q}{s_q q_\perp} \left( \frac{p_\perp}{s_k k_\perp}H_k - \frac{k_\perp}{s_p p_\perp}H_p \right) \\ \displaystyle s_k k_\perp \left[ E_q \left( \frac{p_\perp}{s_k k_\perp}H_k - \frac{k_\perp}{s_p p_\perp}H_p \right) +\frac{H_q}{s_q q_\perp} ( p_\perp E_k - k_\perp E_p ) \right] \end{pmatrix} , \end{equation}

with $\alpha _q$ the angle between $\boldsymbol {k}_\perp$ and $\boldsymbol {p}_\perp$ in the triangle defined by the triadic interaction $( \boldsymbol {k}_\perp, \boldsymbol {p}_\perp, \boldsymbol {q}_\perp )$ (see figure 1). Equation (6.8) will be used to derive exact solutions also called Kolmogorov–Zakharov spectra.

Figure 1. Triadic relation $\boldsymbol {k}_\perp + \boldsymbol {p}_\perp + \boldsymbol {q}_\perp = \boldsymbol {0}$.

6.3. Kolmogorov–Zakharov spectra

Equation (6.8) has sufficient symmetry to apply the bi-homogeneous conformal Kuznetsov –Zakharov transformation (Zakharov et al. Reference Zakharov, L'Vov and Falkovich1992). This transformation has been applied to several problems involving anisotropy (Kuznetsov Reference Kuznetsov2001; Galtier Reference Galtier2003, Reference Galtier2006). It is a generalization of the Zakharov transformation applied for isotropic turbulence (in the context of strong two-dimensional hydrodynamic turbulence, see also Kraichnan Reference Kraichnan1967). With such an operation, we are able to find the exact stationary solutions of the kinetic equations in power law form. The bihomogeneity of the integrals in the wavenumbers $k_\perp$ and $k_\|$ allows us to use the transformations (see figure 2)

(6.10)\begin{gather} p_\perp \to k_\perp^{2}/ p_\perp, \end{gather}

(6.11)\begin{gather}q_\perp \to k_\perp q_\perp{/} p_\perp, \end{gather}

(6.12)\begin{gather}p_\| \to k_\|^{2}/ p_\|, \end{gather}

(6.13)\begin{gather}q_\| \to k_\| q_\| / p_\|. \end{gather}

Figure 2. Illustration of the Kuznetsov–Zakharov transformation. It consists of swapping regions I and III with regions II and IV, respectively. We specify that the grey band is defined up to infinity and corresponds to the domain $\varDelta _\perp$. The same manipulation is done on the parallel wavenumbers.

We apply this transformation first on the energy equation (6.8) which means that we are looking for constant energy flux solutions. We seek stationary solutions in the power law form,

(6.14a,b)\begin{equation} E (k_\perp, k_\| ) = C_E k_\perp^{{-}x} \vert k_\| \vert ^{{-}y} \quad {\rm and} \quad H ( k_\perp, k_\| ) = C_H k_\perp^{-\tilde{x}} \vert k_\| \vert^{-\tilde{y}} , \end{equation}

where $C_E$ and $C_H$ are two constants with $C_E \ge 0$. (We consider only positive parallel wavenumber since it is symmetric in $k_\|$.) The new form of the integral, resulting from the summation of the integrand in its primary form and after the Kuznetsov–Zakharov transformation, can be written as

(6.15)\begin{align} \partial_t E_k & = \frac{\epsilon^{2} \varOmega_e^{2}}{2^{13}} \sum_{s_k s_p s_q} \int_{\varDelta_\perp} \frac{s_k s_p k_\| p_\|}{k_\perp^{2} p_\perp q_\perp^{2}} \left(\frac{s_q q_\perp{-} s_p p_\perp}{ s_k \omega_k}\right)^{2} ( s_k k_\perp{+} s_p p_\perp{+} s_q q_\perp )^{2} \sin \alpha_q \nonumber\\ & \quad \times ( C_E^{2} \varXi_E + s_k s_q C_H^{2} \varXi_H) \delta (\varOmega_{kpq} ) \delta_{k_\|p_\|q_\|} \, \mathrm{d} p_\perp \, \mathrm{d} q_\perp \, \mathrm{d} p_\| \, \mathrm{d} q_\|, \end{align}

with the pure energy contribution

(6.16)\begin{equation} \varXi_E = k_\perp^{{-}x} {\vert k_\| \vert}^{{-}y} q_\perp^{{-}x} {\vert q_\| \vert}^{{-}y} \left[ 1 - \left( \frac{p_\perp}{k_\perp}\right)^{{-}1-x} \left\vert \frac{p_\|}{k_\|}\right\vert^{{-}y} \right]\left[ 1 - \left( \frac{p_\perp}{k_\perp}\right)^{{-}5+2x} \left\vert \frac{p_\|}{k_\|}\right\vert^{{-}1+2y} \right], \end{equation}

and the pure helicity contribution

(6.17)\begin{align} \varXi_H = k_\perp^{{-}1-\tilde{x}} {\vert k_\| \vert}^{-\tilde{y}} q_\perp^{{-}1-\tilde{x}} {\vert q_\| \vert}^{-\tilde{y}} \left[ 1 - s_k s_p \left( \frac{p_\perp}{k_\perp}\right)^{-\tilde{x}-2} \left\vert \frac{p_\|}{k_\|}\right\vert^{-\tilde{y}}\right] \left[ 1 - \left( \frac{p_\perp}{k_\perp}\right)^{2\tilde{x}-3} \left\vert \frac{p_\|}{k_\|}\right\vert^{2\tilde{y}-1} \right]. \end{align}

We can distinguish two different types of solutions. First, there are the thermodynamic equilibrium solutions, which correspond to the equipartition state for which the energy flux is zero. The power laws which verify this condition are

(6.18)\begin{gather} E (k_\perp, k_\| ) = C_E k_\perp, \end{gather}

(6.19)\begin{gather}H (k_\perp, k_\| ) = C_H k_\perp^{2}. \end{gather}

These results can be easily verified by a direct substitution in the original kinetic equations. In general, this stationary state cannot be reached in the presence of helicity because the value $s_k s_p = -1$ prevents the cancellation of the integral. There is, however, a particular case where the solutions exist: it is the state of maximal helicity for which either $e_{\boldsymbol {k}}^{+}=0$ or $e_{\boldsymbol {k}}^{-}=0$. Then, we have the relation $H_k=\pm k_\perp E_k$. But this state is not viable as we can see on (5.2): for example, if $e_{\boldsymbol {k}}^{-}=0$ at time $t=0$, it will not remain zero at time $t>0$. This means that this solution is only possible if there is an external mechanism that forces the system to remain in the maximal helicity state.

The most interesting solutions are those for which the energy flux is constant, non-zero and finite. These exact solutions are called Kolmogorov-Zakharov (KZ) spectra and correspond to the values which make the integral cancel in a non-trivial way and independently of the polarizations. These spectra are

(6.20)\begin{gather} E (k_\perp, k_\| ) = C_E k_\perp^{{-}5/2} { \vert k_\| \vert^{{-}1/2}}, \end{gather}

(6.21)\begin{gather}H (k_\perp, k_\| ) = C_H k_\perp^{{-}3/2} { \vert k_\| \vert^{{-}1/2}}. \end{gather}

There are not constrained by the polarization and can therefore be reached by the system even in the presence of helicity.

For the helicity equation, using the same manipulations as before, we obtain

(6.22)\begin{align} \partial_t H_k & = \frac{\epsilon^{2} \varOmega_e^{2}}{{2^{13}}} C_E C_H \sum_{s_k s_p s_q} \int_{\varDelta_\perp} \frac{s_k s_p k_\| p_\| }{k_\perp^{2} p_\perp q_\perp^{2}} \left(\frac{s_q q_\perp{-} s_p p_\perp}{ s_k \omega_k}\right)^{2} ( s_k k_\perp{+} s_p p_\perp{+} s_q q_\perp )^{2} \sin \alpha_q \nonumber\\ & \quad \times k_\perp^{{-}x-\tilde{x}} \vert k_\| \vert^{{-}y-\tilde{y}} \left[1 - s_k s_p \left(\frac{p_\perp}{k_\perp}\right)^{x+\tilde{x}-4} \left\vert \frac{p_\|}{k_\|}\right\vert^{y+\tilde{y}-1} \right] \nonumber\\ & \quad \times \left[\left(\frac{q_\perp}{k_\perp}\right)^{{-}x} \left\vert \frac{q_\|}{k_\|}\right\vert^{{-}y} \left(1 - s_k s_p \left(\frac{p_\perp}{k_\perp}\right)^{-\tilde{x}-2} \left\vert \frac{p_\|}{k_\|}\right\vert^{-\tilde{y}} \right) \right. \nonumber\\ & \quad \left. + s_k s_q \left(\frac{q_\perp}{k_\perp}\right)^{-\tilde{x}-1} \left\vert \frac{q_\|}{k_\|}\right\vert^{-\tilde{y}} \left( 1 - \left(\frac{p_\perp}{k_\perp}\right)^{{-}x-1} \left\vert \frac{p_\|}{k_\|}\right\vert^{{-}y} \right)\right] \delta (\varOmega_{kpq} ) \delta_{k_\|p_\|q_\|} \, \mathrm{d} p_\perp \, \mathrm{d} q_\perp \, \mathrm{d} p_\| \, \mathrm{d} q_\|. \end{align}

The zero helicity flux solutions satisfy

(6.23)\begin{gather} E (k_\perp, k_\| ) = C_E k_\perp, \end{gather}

(6.24)\begin{gather}H (k_\perp, k_\| ) = C_H k_\perp^{2}, \end{gather}

which correspond to the thermodynamic spectra found for energy (this can be seen directly from (6.9)). For the KZ spectra, we have a family of solutions that meet the following criteria:

(6.25)\begin{gather} x + \tilde{x} = 4, \end{gather}

(6.26)\begin{gather}y + \tilde{y} = 1. \end{gather}

The situation is worse than for energy because none of the constant helicity flux solutions (thermodynamic or KZ) can be reached in general because of the presence of the product $s_ks_p$ which, let us recall, prevents the cancellation of the term in the right-hand side of expression (6.22). Only the maximal helicity state allows the existence of these stationary spectra but, as said above, it is not a naturally viable state. (Note that this property found in weak wave turbulence may not be true in strong turbulence.)

In conclusion, the most relevant solutions are the KZ spectra at constant energy flux. In § 8 we will further investigate the corresponding exact solution for $H=0$ in order to find the direction of the energy cascade and the expression of the Kolmogorov constant. In space plasma physics, we often compare theoretical predictions with the magnetic spectrum $E_k^{B}$ which is well measured by spacecraft (with the Taylor hypothesis, the frequency is used as a proxy for the wavenumber). In our case, a simple dimensional analysis based on the definition of energy (2.15), leads to the relation $E_k \sim k_\perp ^{2} E_k^{B}$. Consequently, we obtain $E_k^{B} \sim k_\perp ^{-9/2}$, which is steeper than the predictions made at scales larger than $d_e$.

6.4. Locality condition

We have seen that the most interesting exact solutions of the kinetic equations are the KZ spectra at constant energy flux. However, these solutions are only fully relevant if they satisfy the locality condition. Mathematically, this condition means that the integral must be convergent. If it is not the case, it means physically that the inertial range is not independent of the largest or smallest scales, where forcing and dissipation are expected. The calculation of the locality condition is highly non-trivial in this anisotropic case. It requires a careful treatment that we leave to Appendix C. Note that the study of locality is still a subject of investigation (Dematteis, Polzin & Lvov Reference Dematteis, Polzin and Lvov2022). In the absence of helicity, we find the following conditions:

(6.27)\begin{gather} 3 < x + 2 y < 4, \end{gather}

(6.28)\begin{gather}2 < x + y < 4. \end{gather}

We obtain a classical result for wave turbulence in the sense that the power law indices of the KZ spectra fall exactly in the middle of the convergence domain (see figure 3).

Figure 3. Domain of convergence of the energy integral. The black dot at the centre of the domain corresponds to the KZ energy spectrum.

8.1. Direct energy cascade

In this section, we will study the sign of the energy flux from the kinetic equations (6.15) and prove that the cascade in the perpendicular direction is direct. In cylindrical coordinates (see figure 4), we have (Zakharov et al. Reference Zakharov, L'Vov and Falkovich1992)

(8.1)\begin{equation} \frac{\partial E(\boldsymbol{k})}{\partial t}={-} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\varPi} ={-}\frac{1}{k_\perp}\frac{\partial( k_\perp \varPi_\perp(\boldsymbol{k}) ) }{\partial k_\perp} - \frac{\partial \varPi_\|(\boldsymbol{k}) }{\partial k_\|}, \end{equation}

where $\boldsymbol {\varPi }$ is the energy flux vector, $\varPi _\perp$ and $\varPi _\|$ its perpendicular and parallel components (axisymmetric turbulence is assumed), respectively. Introducing the axisymmetric spectra $E_k \equiv 2 {\rm \pi}k_\perp E(\boldsymbol {k})$, $\varPi _\perp \equiv 2 {\rm \pi}k_\perp \varPi _\perp (\boldsymbol {k})$ and $\varPi _\| \equiv 2 {\rm \pi}k_\perp \varPi _\|(\boldsymbol {k})$, we obtain

(8.2)\begin{equation} \partial_t E_k ={-} \frac{\partial \varPi_\perp}{\partial k_\perp} - \frac{\partial \varPi_\|}{\partial k_\|} . \end{equation}

Figure 4. Schematic representation of an axisymmetric flux in Fourier space. Each cylindrical shell corresponds to a specific value of $k_\perp$. In theory, they form a continuum but here their discrete nature serves as an illustration.

We now introduce the adimensional variables $\tilde {p}_\perp \equiv p_\perp / k_\perp$, $\tilde {q}_\perp \equiv q_\perp / k_\perp$, $\tilde {p}_\| \equiv p_\|/ k_\|$ and $\tilde {q}_\| \equiv q_\| / k_\|$. We seek power law solutions of the form (6.14a,b) and then obtain

(8.3)\begin{equation} \partial_t E_k = \frac{\epsilon^{2}}{{2^{13}} \varOmega_e} [ k_\perp^{4-2x} \vert k_\| \vert^{{-}2y} C_E^{2} I_E(x,y) + k_\perp^{2-2\tilde{x}} \vert k_\| \vert^{{-}2\tilde{y}} C_H^{2} I_H(\tilde{x},\tilde{y})], \end{equation}

where

(8.4)\begin{align} I_E(x,y) & = \sum_{s_k s_p s_q} \int_{\varDelta_\perp} s_k s_p \frac{\tilde{p}_\|}{\tilde{p}_\perp \tilde{q}_\perp^{2}} ( s_q \tilde{q}_\perp{-} s_p \tilde{p}_\perp )^{2} ( s_k + s_p \tilde{p}_\perp{+} s_q \tilde{q}_\perp )^{2} \sin \alpha_q \tilde{q}_\perp^{{-}x} \vert \tilde{q}_\| \vert^{{-}y} \nonumber\\ & \quad \times (1-\tilde{p}_\perp^{{-}1-x} \vert \tilde{p}_\| \vert^{{-}y})(1 - \tilde{p}_\perp^{{-}5+2x} \vert \tilde{p}_\| \vert^{{-}1+2y}) \nonumber\\ & \quad \times \delta \left( s_k + s_p \frac{\tilde{p}_\|}{\tilde{p}_\perp} + s_q \frac{\tilde{q}_\|}{\tilde{q}_\perp} \right) \delta ( 1 + \tilde{p}_\| + \tilde{q}_\| ) \, \mathrm{d} \tilde{p}_\perp \, \mathrm{d} \tilde{q}_\perp \, \mathrm{d} \tilde{p}_\| \, \mathrm{d} \tilde{q}_\|, \end{align}

and

(8.5)\begin{align} I_H(\tilde{x},\tilde{y}) & = \sum_{s_k s_p s_q} \int_{\varDelta_\perp} s_p s_q \frac{\tilde{p}_\|}{\tilde{p}_\perp \tilde{q}_\perp^{3}} ( s_q \tilde{q}_\perp{-} s_p \tilde{p}_\perp )^{2} ( s_k + s_p \tilde{p}_\perp{+} s_q \tilde{q}_\perp )^{2} \sin \alpha_q \tilde{q}_\perp^{-\tilde{x}} \vert \tilde{q}_\| \vert^{-\tilde{y}} \nonumber\\ & \quad \times ( 1 - s_k s_p \tilde{p}_\perp^{-\tilde{x}-2} \vert \tilde{p}_\| \vert^{-\tilde{y}}) ( 1 - \tilde{p}_\perp^{2\tilde{x}-3} \vert \tilde{p}_\| \vert^{2\tilde{y}-1} ) \nonumber\\ & \quad \times \delta \left( s_k + s_p \frac{\tilde{p}_\|}{\tilde{p}_\perp} + s_q \frac{\tilde{q}_\|}{\tilde{q}_\perp} \right) \delta ( 1 + \tilde{p}_\| + \tilde{q}_\| ) \, \mathrm{d} \tilde{p}_\perp \, \mathrm{d} \tilde{q}_\perp \, \mathrm{d} \tilde{p}_\| \, \mathrm{d} \tilde{q}_\|. \end{align}

Taking the limits, corresponding to the KZ spectra, $( x, y, \tilde {x}, \tilde {y} ) \rightarrow ( 5/2, 1/2, 3/2, 1/2 )$, thanks to l'H$\hat{{\rm o}}$pital's rule, we can write

(8.6)\begin{equation} \left( \begin{array}{c} \varPi_\perp^{\mathrm{KZ}} \\ \varPi_\|^{\mathrm{KZ}} \end{array}\right) = \frac{\epsilon^{2}}{ {2^{13}} \varOmega_e} \left( \begin{array}{c} k_\|^{{-}1} \\ k_\perp^{{-}1} \end{array}\right) \left[ C_E^{2} \left( \begin{array}{c} I_\perp \\ I_\| \end{array}\right)+ C_H^{2} \left( \begin{array}{c} J_\perp \\ J_\| \end{array}\right) \right], \end{equation}

where

(8.7)\begin{align} \left(\! \begin{array}{c} I_\perp \\ I_\| \end{array}\!\right) & \equiv \sum_{s_k s_p s_q} \int_{\varDelta_\perp} s_k s_p \frac{\tilde{p}_\|}{\tilde{p}_\perp \tilde{q}_\perp^{9/2} \vert \tilde{q}_\| \vert^{1/2}} ( s_q \tilde{q}_\perp{-} s_p \tilde{p}_\perp )^{2} ( s_k + s_p \tilde{p}_\perp{+} s_q \tilde{q}_\perp )^{2} \sin \alpha_q \log \left\vert \left(\! \begin{array}{c} \tilde{p}_\perp \\ \tilde{p}_\| \end{array}\!\right) \right\vert \nonumber\\ & \quad \times (1- \tilde{p}_\perp^{{-}7/2} \vert \tilde{p}_\| \vert^{{-}1/2}) \delta \left( s_k + s_p \frac{\tilde{p}_\|}{\tilde{p}_\perp} + s_q \frac{\tilde{q}_\|}{\tilde{q}_\perp} \right) \delta ( 1 + \tilde{p}_\| + \tilde{q}_\| ) \, \mathrm{d} \tilde{p}_\perp \, \mathrm{d} \tilde{q}_\perp \, \mathrm{d} \tilde{p}_\| \, \mathrm{d} \tilde{q}_\|, \end{align}

and

(8.8)\begin{align} \left(\! \begin{array}{c} J_\perp \\ J_\| \end{array}\!\right) & \equiv \sum_{s_k s_p s_q} \int_{\varDelta_\perp} s_p s_q \frac{\tilde{p}_\|}{\tilde{p}_\perp \tilde{q}_\perp^{9/2} \vert \tilde{q}_\| \vert^{1/2}} ( s_q \tilde{q}_\perp{-} s_p \tilde{p}_\perp )^{2} ( s_k + s_p \tilde{p}_\perp{+} s_q \tilde{q}_\perp )^{2} \sin \alpha_q \log \left\vert \left(\! \begin{array}{c} \tilde{p}_\perp \\ \tilde{p}_\| \end{array}\!\right) \right\vert \nonumber\\ & \quad \times (s_k - s_p \tilde{p}_\perp^{{-}7/2} \vert \tilde{p}_\| \vert^{{-}1/2}) \delta \left( s_k + s_p \frac{\tilde{p}_\|}{\tilde{p}_\perp} + s_q \frac{\tilde{q}_\|}{\tilde{q}_\perp} \right) \delta ( 1 + \tilde{p}_\| + \tilde{q}_\| ) \, \mathrm{d} \tilde{p}_\perp \, \mathrm{d} \tilde{q}_\perp \, \mathrm{d} \tilde{p}_\| \, \mathrm{d} \tilde{q}_\|. \end{align}

Therefore, the ratio of the two fluxes is

(8.9)\begin{equation} \frac{\varPi_\|^{\mathrm{KZ}}}{\varPi_\perp^{\mathrm{KZ}}} = \frac{k_\|}{k_\perp} \frac{C_E^{2} I_\| + C_H^{2} J_\|}{C_E^{2} I_\perp{+} C_H^{2} J_\perp}. \end{equation}

Since it is proportional to $k_\| / k_\perp$, we expect $\varPi _\|^{\mathrm {KZ}} \ll \varPi _\perp ^{\mathrm{KZ}}$, which is in agreement with the analysis based on the resonance condition to find the direction of the cascade. In the absence of helicity, the ratio (8.9) only depends on $k_\| / k_\perp \ll 1$ and $I_\| / I_\perp$; we numerically find $I_\| / I_\perp \simeq 0.73$, then the previous expectation is fulfilled.

We can also find the sign of the energy flux and thus prove the direction of the cascade. Since the perpendicular flux is dominant, we will neglect the parallel flux and only look for the sign of $I_\perp$. A numerical evaluation reveals a positive value, which means that $\varPi _\perp > 0$ and that the energy cascade is direct in the transverse direction.

In figure 5, we show the sign of the integrands of $I_\perp$ and $I_\|$ obtained from a numerical evaluation of expressions (8.7). We see that for $I_\perp$ the integrand is always positive, while for $I_\|$ the integrand can be either positive or negative depending on the perpendicular wavenumbers (for large perpendicular wavenumbers it is always positive) but overall, after integration, the positive sign dominates in the sense that the integral $I_\| >0$. Therefore, the parallel cascade is also direct but it is composed of different contributions, with (a minority of) triadic interactions contributing to an inverse transfer.

Figure 5. (a) Integrand of $I_\perp$ as a function of $\tilde {p}_\perp$ and $\tilde {q}_\perp$; a positive value is always observed. (b) Integrand of $I_\|$ which changes sign as a function of (small) $\tilde {p}_\perp$ and $\tilde {q}_\perp$.

8.2. Kolmogorov constant

If we neglect the parallel flux and helicity, we can also obtain the expression of the Kolmogorov constant $C_K$ for which we can numerically get an estimate. To do so, we take advantage of the Dirac distributions to integrate the parallel wavenumbers. Then, since $I_\perp$ is only defined on the region $\varDelta _\perp$, we introduce the change of variable $\tilde {q}_\perp \equiv \xi - \tilde {p}_\perp$ where $\xi \in [1,+\infty [$ and $\tilde {p}_\perp \in [({\xi -1})/{2},\ ({\xi +1})/{2} ]$ that confines the integration to this domain. One finds at a given $k_\parallel$,

(8.10)\begin{equation} E_k = \sqrt{\varPi_\perp \varOmega_e} C_K k_\perp^{{-}5/2} \quad {\rm with} C_K = 64 \sqrt{\frac{2}{I_\perp}} \simeq 8.474 . \end{equation}

The numerical convergence of $C_K$ to this value in shown in figure 6.

Figure 6. Convergence of $C_K$ as a function of $\xi$.