2.3.1 Limiting forms of the invariants

The first term on the right-hand side of (2.32) corresponds to the magnetic energy, the second one to the parallel electron kinetic energy, while the third one includes both internal energy and a term which, at MHD scales, is dominated by the perpendicular ion kinetic energy. The latter becomes subdominant at small scales due to ion FLR effects, as a result of the $\unicode[STIX]{x1D6EC}$ operator in the second term of the right-hand side of (2.36). It then reduces to the perpendicular electron kinetic energy modified by electron FLR. These latter contributions are here retained only through the $\unicode[STIX]{x1D6FF}^{2}$ term in (2.5) which is relevant when $\unicode[STIX]{x1D6FD}_{i}$ is of order unity or larger. The complete form of the energy in the sub- $d_{e}$ range is given by (5.13) of Passot et al. (Reference Passot, Sulem and Tassi2017).

It is also of interest to discuss the forms taken by the generalized cross-helicity when considered in the large- or small-scale limit.

1. (i) In the MHD range, $N_{e}=\unicode[STIX]{x1D6E5}_{\bot }\unicode[STIX]{x1D711}$ which also corresponds to the ion vorticity (with a single component $\unicode[STIX]{x1D714}_{iz}$ along the $z$ direction). The invariant ${\mathcal{C}}$ thus reads ${\mathcal{C}}=-\int \unicode[STIX]{x1D714}_{iz}A_{\Vert }\,\text{d}^{3}x$ which, after integration by parts, also rewrites ${\mathcal{C}}=-\int \boldsymbol{u}_{\bot i}\boldsymbol{\cdot }\boldsymbol{B}_{\bot }\,\text{d}^{3}x$ , i.e. the opposite of the usual cross-helicity. Alternatively, in terms of magnetic helicity, one has for small $\unicode[STIX]{x1D70F}$ , ${\mathcal{C}}=(2/\unicode[STIX]{x1D6FD}_{e}+1)\int B_{z}A_{\Vert }\,\text{d}^{3}x$ .

2. (ii) At sub-ions scales (leaving out the electron inertia and FLR contributions), $B_{z}$ and $N_{e}$ are proportional to $\unicode[STIX]{x1D711}$ , and thus the generalized cross-helicity is proportional to the magnetic helicity which arises in imbalanced EMHD studied by Kim & Cho (Reference Kim and Cho2015). More precisely, from $N_{e}=-M_{2}M_{1}^{-1}B_{z}=-(L_{3}L_{2}^{-1}L_{1}+L_{4})B_{z}$ , one has

   (2.40) $$\begin{eqnarray}{\mathcal{C}}=\int (L_{3}L_{2}^{-1}L_{1}+L_{4})B_{z}(L_{e}A_{\Vert })\,\text{d}^{3}x.\end{eqnarray}$$
   At these scales, where one can take $\unicode[STIX]{x1D6E4}_{0}=0$ and $\unicode[STIX]{x1D6E4}_{1}=0$ ,
   (2.41) $$\begin{eqnarray}\displaystyle {\mathcal{C}} & = & \displaystyle \int \left(1+\frac{1}{1+1/\unicode[STIX]{x1D70F}}\left(\frac{2}{\unicode[STIX]{x1D6FD}_{i}}-\frac{2\unicode[STIX]{x1D6FF}^{2}}{\unicode[STIX]{x1D6FD}_{e}}\unicode[STIX]{x1D6E5}_{\bot }\right)\right)B_{z}(L_{e}A_{\Vert })\,\text{d}^{3}x\end{eqnarray}$$
   (2.42) $$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle \int (L_{e}B_{z})(L_{e}A_{\Vert })\,\text{d}^{3}x+\frac{2}{\unicode[STIX]{x1D6FD}_{e}+\unicode[STIX]{x1D6FD}_{i}}\int B_{z}(L_{e}A_{\Vert })\,\text{d}^{3}x.\end{eqnarray}$$
   In (2.42), $\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6E5}_{\bot }\ll 1$ has been assumed. It is interesting to note that in the incompressible limit where the beta parameters tend to infinity, ${\mathcal{C}}=\int (L_{e}B_{z})(L_{e}A_{\Vert })\,\text{d}^{3}x$ , thus recovering, in the quasi-transverse limit, the generalized magnetic helicities of EMHD (Biskamp et al. Reference Biskamp, Schwarz, Zeiler, Celani and Drake1999), or of extended MHD when the ion velocity is taken to zero ((35) of Abdelhamid, Lingam & Mahajan (Reference Abdelhamid, Lingam and Mahajan2016)).

   The fact that the quantity ${\mathcal{C}}$ provides a generalization of the cross-helicity at sub-ion scales is easily seen when considering (2.39). This expression clearly indicates the way the operator $V_{ph}$ renormalizes the usual cross-helicity, a quantity measuring the degree of imbalance between forward- and backward-propagating waves.

7.1.1 Zero generalized cross-helicity flux

When $\unicode[STIX]{x1D702}=0$ , we can prescribe vanishing spectra at infinity, and solve as

(7.12) $$\begin{eqnarray}\unicode[STIX]{x1D70C}^{2}(k_{\bot })=\frac{2\unicode[STIX]{x1D700}}{C}\int _{k_{\bot }}^{\infty }\frac{1}{k_{\bot }^{\prime 7}v_{ph}(k_{\bot }^{\prime })}\,\text{d}k_{\bot }^{\prime },\end{eqnarray}$$

with $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{0}$ given by $\tanh (\unicode[STIX]{x1D719}_{0})=v_{ph}(k_{0})E_{c}(k_{0})/E(k_{0})$ . The spectra thus also behave like power laws with, for spectra $E^{\pm }(k_{\bot })\sim k_{\bot }^{-m^{\pm }}$ , the relations $m^{+}=m^{-}=2+m_{v}/2$ and $m_{c}=2+3m_{v}/2$ . The ratio $E_{\bot }^{+}(k)/E_{\bot }^{-}(k)=\text{e}^{2\unicode[STIX]{x1D719}_{0}}=(1+v_{ph}(k_{0})E_{C}(k_{0})/E(k_{0}))/(1-v_{ph}(k_{0})E_{C}(k_{0})/E(k_{0}))$ is independent of the wavenumber.

7.1.2 Finite generalized cross-helicity flux

Let us first address the case of pure MHD (i.e. with a constant $v_{ph}$ equal to $s$ ), for which solutions can be continued to infinity, making (7.12) valid. The energy flux $\unicode[STIX]{x1D700}$ is necessarily non-zero and we find $\unicode[STIX]{x1D70C}^{2}(k_{\bot })=\unicode[STIX]{x1D700}/(3Cs)k_{\bot }^{-6}$ . One then obtains, when integrating from a wavenumber $k_{0}$ to $k_{\bot }$ ,

(7.13) $$\begin{eqnarray}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{0}-\int _{k_{0}}^{k_{\bot }}\frac{\unicode[STIX]{x1D702}}{Ck_{\bot }^{\prime 7}\unicode[STIX]{x1D70C}^{2}(k_{\bot }^{\prime })}\,\text{d}k_{\bot }^{\prime }=\unicode[STIX]{x1D719}_{0}-\frac{3\unicode[STIX]{x1D702}s}{\unicode[STIX]{x1D700}}\ln \frac{k_{\bot }}{k_{0}},\end{eqnarray}$$

and thus

(7.14) $$\begin{eqnarray}E^{\pm }(k_{\bot })=\frac{1}{2}\left(\frac{\unicode[STIX]{x1D700}}{3Cs}\right)^{1/2}\text{e}^{\pm \unicode[STIX]{x1D719}_{0}}k_{\bot }^{-2}\left(\frac{k_{\bot }}{k_{0}}\right)^{\mp 3\unicode[STIX]{x1D702}s/\unicode[STIX]{x1D700}}.\end{eqnarray}$$

Writing $\unicode[STIX]{x1D700}^{\pm }=(\unicode[STIX]{x1D700}\pm \unicode[STIX]{x1D702}s)/2$ , one gets

(7.15) $$\begin{eqnarray}\displaystyle & \displaystyle m^{+}=\frac{5\unicode[STIX]{x1D700}^{+}-\unicode[STIX]{x1D700}^{-}}{\unicode[STIX]{x1D700}^{+}+\unicode[STIX]{x1D700}^{-}} & \displaystyle\end{eqnarray}$$

(7.16) $$\begin{eqnarray}\displaystyle & \displaystyle m^{-}=\frac{5\unicode[STIX]{x1D700}^{-}-\unicode[STIX]{x1D700}^{+}}{\unicode[STIX]{x1D700}^{+}+\unicode[STIX]{x1D700}^{-}}, & \displaystyle\end{eqnarray}$$

which satisfy the entanglement condition $m^{+}+m^{-}=4$ . In this case, the nonlinear eigenvalue problem does not have a unique solution. The spectra can a priori intersect at a wavenumber which can only be determined by additional physical effects. For example, in the presence of viscosity, the pinning condition that both spectra become equal at the dissipation scale (taken to be the same for the two waves), and remain so in the whole dissipation range, selects a unique solution, as discussed in Lithwick & Goldreich (Reference Lithwick and Goldreich2003) and Chandran (Reference Chandran2008). Note also that there is a constraint on the fluxes for the spectral exponents to remain in a range ensuring locality of the interactions (Galtier & Meyrand Reference Galtier and Meyrand2015).

Let us now turn to the more delicate situation where both helicity flux and dispersion are present. At small scales, where $v_{ph}(k_{\bot })\sim k_{\bot }$ , assuming again that $\unicode[STIX]{x1D70C}$ vanishes at infinity, $\unicode[STIX]{x1D70C}^{2}(k_{\bot })\sim \unicode[STIX]{x1D700}k_{\bot }^{-7}$ , and thus $\unicode[STIX]{x1D719}(k_{\bot })\sim a+bk_{\bot }$ where $a$ and $b$ are constants, the latter being negative and proportional to $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}$ . It results that $E^{\pm }(k_{\bot })\sim k_{\bot }^{-5/2}\text{e}^{\pm bk_{\bot }}$ . These spectra are not physically relevant in the entire spectral range, as one of the solutions diverges as $k_{\bot }\rightarrow +\infty$ . In fact, the phase velocity should saturate at $k_{\bot }d_{e}\geqslant 1$ due to electron inertia, so that the exponential behaviour of the spectra is only a transient that can be acceptable if $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}$ and the saturated value of $v_{ph}$ are sufficiently small. The above behaviour nevertheless indicates an exponential decrease of the imbalance at scales where the system starts becoming dispersive. To support this point we calculate from (7.3) the local slopes of the energy spectra defined as $m^{\pm }(k_{\bot })=-d\ln (E^{\pm }(k_{\bot }))/d\ln (k_{\bot })$ and find

(7.17) $$\begin{eqnarray}m^{\pm }(k_{\bot })=\frac{\unicode[STIX]{x1D700}\pm \unicode[STIX]{x1D702}v_{ph}}{4C^{\prime }k_{\bot }^{6}v_{ph}}\frac{\widetilde{k}_{\Vert }^{\mp }}{u^{+}(k_{\bot })u^{-}(k_{\bot })}-1.\end{eqnarray}$$

We thus have

(7.18) $$\begin{eqnarray}m^{+}(k_{\bot })-m^{-}(k_{\bot })=\frac{2\unicode[STIX]{x1D702}v_{ph}k_{f}}{4C^{\prime }k_{\bot }^{6}v_{ph}u^{+}(k_{\bot })u^{-}(k_{\bot })},\end{eqnarray}$$

which is positive (zero when $\unicode[STIX]{x1D702}=0$ and growing with $k_{\bot }$ otherwise), showing that the spectrum of the dominant wave is steeper than that of the smaller-amplitude one and, as a consequence, that the two spectra will cross each other.

7.1.3 Numerical solutions

In order to illustrate the previous discussion, we numerically integrated (7.3). Since the original nonlinear eigenvalue problem does not have solutions in the general case, we resorted to prescribing both the transfer rates $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ and the values of $u^{\pm }$ at a finite wavenumber $k_{d}$ . Inspection of (7.10)–(7.11) shows that the value of $\unicode[STIX]{x1D70C}$ at $k_{d}$ should be chosen large enough to prevent the occurrence of singularities. In this case, for $k_{\bot }>k_{d}$ , $\unicode[STIX]{x1D719}$ becomes constant, leading to absolute equilibrium spectra. The obtained solution is similar to a warm cascade (Connaughton & Nazarenko Reference Connaughton and Nazarenko2004). Nevertheless, while the range of absolute equilibrium can be suppressed by an accurate prescription of the energy flux in the Kolmogorov cascade (see appendix B), in the present situation the ultraviolet divergence cannot be prevented. From a physical point of view, $k_{d}$ can be thought of as mimicking the wavenumber where dissipation starts acting, the model being unable to describe the regime of arbitrarily small dissipation. At this dissipation scale, pinning of the two spectra is often reported in numerical simulation of viscous-diffusive MHD (Perez et al. Reference Perez, Mason, Boldyrev and Cattaneo2012).

It turns out that both in the weak and strong turbulence regimes, there exists a critical value $u_{c}=r(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ of $u^{\pm }(k_{d})$ (with a coefficient $r$ of order unity that we empirically estimate by a dichotomy process) such that the small-scale spectrum displays a thermodynamic behaviour or a singularity depending on whether $u^{\pm }(k_{d})$ is chosen above or below $u_{c}$ .

In all the simulations discussed below, we prescribe $\unicode[STIX]{x1D6FD}_{e}=2$ ,Footnote ³ $\unicode[STIX]{x1D70F}=1$ and zero electron inertia. It turns out that using Bessel functions in the definition of $v_{ph}$ leads to a strong increase in the computer time. We thus replaced the function $\unicode[STIX]{x1D6E4}_{0}(x)$ by its usual Padé form $1/(1+x)$ and use for $\unicode[STIX]{x1D6E4}_{1}(x)$ the function $(x/2)(1+0.8x^{2})^{-1}$ . Both functions provide the correct asymptotic behaviour at $x=0$ and decay for large $x$ . The coefficient $0.8$ was adjusted to provide a good global fit at the level of the function $v_{ph}$ , with a maximal error less than $3\,\%$ , localized near $x=1$ .

We show, in figure 1, $E(k_{\bot })$ (red solid line) and $|E_{C}(k_{\bot })|$ (green solid line) (left panel), as well as $E^{+}(k_{\bot })$ (red) and $E^{-}(k_{\bot })$ (green) (middle panel) for a case of weak generalized cross-helicity transfer rate ( $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}=0.008$ ), but relatively large $k_{d}=120$ . Additional parameters are $\widetilde{k}_{\Vert }^{\pm }=1$ , $C^{\prime }=1$ , $\unicode[STIX]{x1D700}=1$ and $u^{+}(k_{d})=u^{-}(k_{d})=(1/2)(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ . The value $1/2$ of the coefficient slightly exceeds the critical value $r$ . The right panel displays the local slopes $m^{+}(k_{\bot })$ and $m^{-}(k_{\bot })$ , together with their sum, showing in particular that, beyond the transition range, where the slopes reach 2.5, the sub-ion range is dominated by an exponential zone. The sum of the slopes is $4$ in the MHD range and $5$ in the sub-ion one. Close to $k_{d}$ , the cross-helicity spectrum becomes negative. Note that changing the sign of $\unicode[STIX]{x1D702}$ leads to exchange $E^{+}(k_{\bot })$ and $E^{-}(k_{\bot })$ and thus the sign of $E_{C}(k_{\bot })$ .

Figure 1. (a) Spectra $E(k_{\bot })$ (red) and $|E_{C}(k_{\bot })|$ (green); (b) spectra $E^{+}(k_{\bot })$ (red) and $E^{-}(k_{\bot })$ (green); (c) slopes $m^{+}(k_{\bot })$ (red), $m^{-}(k_{\bot })$ (green) and $m^{+}+m^{-}$ (blue). The simulation was performed with $\unicode[STIX]{x1D6FD}_{e}=2$ , $\unicode[STIX]{x1D70F}=1$ in the weak turbulence regime in the case of a weak transfer rate of generalized cross-helicity $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}=0.008$ , but a relatively large value of $k_{d}=120$ . Other parameters are $\widetilde{k}_{\Vert }^{\pm }=1$ , $C^{\prime }=1$ , $\unicode[STIX]{x1D700}=1$ , $u^{+}(k_{d})=u^{-}(k_{d})=(1/2)(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ .

As mentioned in Chandran (Reference Chandran2008) for imbalanced MHD, as a consequence of pinning, the larger the dissipation wavenumber, the smaller $\unicode[STIX]{x1D700}^{+}/\unicode[STIX]{x1D700}^{-}$ should be for a given value of $E^{+}/E^{-}$ at the outer scale. The situation is here similar, pinning being replaced by the boundary condition at $k_{d}$ . This point is illustrated in figure 2 which displays the same graphs as figure 1, but for $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}=0.05$ and $k_{d}=15$ .

Figure 2. Same as figure 1, but for $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}=0.05$ and $k_{d}=15$ .

7.2.1 MHD regime

When concentrating on scales large enough for dispersion to be negligible, it is possible to look for cascades associated with power-law transverse spectra $E^{\pm }(k_{\bot })\sim k_{\bot }^{-m^{\pm }}$ and constant fluxes $\unicode[STIX]{x1D700}^{\pm }$ . Several regimes can be distinguished, depending on the amplitude of the fluctuations.

1. (i) When both the parallel- and anti-parallel-propagating Alfvén modes have a strong amplitude, it is legitimate to take $k_{f}=0$ . One has

   (7.19) $$\begin{eqnarray}-\frac{\unicode[STIX]{x1D700}^{(r)}}{4C^{\prime }s}=k_{\bot }^{9/2}\sqrt{E^{(-r)}}\left(\frac{E^{+}}{E^{-}}\right)^{-\unicode[STIX]{x1D708}(r+1)/4}\frac{\text{d}}{\text{d}k_{\bot }}\left(\frac{E^{(r)}}{k_{\bot }}\right),\end{eqnarray}$$
   which, by simple power counting, implies for the spectral indices
   (7.20) $$\begin{eqnarray}\displaystyle & \displaystyle \left(1-\frac{\unicode[STIX]{x1D708}}{2}\right)m^{+}+\frac{1+\unicode[STIX]{x1D708}}{2}m^{-}=\frac{5}{2} & \displaystyle\end{eqnarray}$$
   (7.21) $$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{1}{2}}m^{+}+m^{-}={\textstyle \frac{5}{2}}, & \displaystyle\end{eqnarray}$$
   leading to $m^{\pm }=5/3$ when $\unicode[STIX]{x1D708}\neq 1$ (Lithwick et al. Reference Lithwick, Goldreich and Sridhar2007) and to the sole entanglement relation $m^{+}+2m^{-}=5$ , as in Chandran (Reference Chandran2008), for $\unicode[STIX]{x1D708}=1$ .

   1. (1) For $\unicode[STIX]{x1D708}\neq 1$ , the nonlinear eigenvalue problem (fixing $E^{\pm }(k_{0})=E_{0}^{\pm }$ at a finite transverse wavenumber $k_{0}$ together with $E^{\pm }=0$ at infinity) has a well-defined solution given by

      (7.22) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}^{+}=\frac{32C^{\prime }s}{3}k_{0}^{5/2}(E_{0}^{+})^{1-\unicode[STIX]{x1D708}/2}(E_{0}^{-})^{(1+\unicode[STIX]{x1D708})/2} & \displaystyle\end{eqnarray}$$
      (7.23) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}^{-}=\frac{32C^{\prime }}{3}k_{0}^{5/2}E_{0}^{-}\sqrt{E_{0}^{+}}, & \displaystyle\end{eqnarray}$$
      which leads to
      (7.24) $$\begin{eqnarray}r_{\unicode[STIX]{x1D700}}\equiv \frac{\unicode[STIX]{x1D700}^{+}}{\unicode[STIX]{x1D700}^{-}}=\left(\frac{E_{0}^{+}}{E_{0}^{-}}\right)^{(1-\unicode[STIX]{x1D708})/2}.\end{eqnarray}$$
      The relation $E_{0}^{+}/E_{0}^{-}=r_{\unicode[STIX]{x1D700}}^{2}$ predicted by the model of Lithwick et al. (Reference Lithwick, Goldreich and Sridhar2007) is recovered for $\unicode[STIX]{x1D708}=0$ . Direct numerical MHD simulations by Beresnyak & Lazarian (Reference Beresnyak and Lazarian2009) where the Elsasser fields are randomly driven, seem to indicate that a similar relation holds but with a slightly larger power of the flux-rate ratio, that can easily be fitted with an appropriate choice of $\unicode[STIX]{x1D708}$ . From figure 13 of Beresnyak & Lazarian (Reference Beresnyak and Lazarian2009), one can estimate $\unicode[STIX]{x1D708}\approx 0.2$ , comparable to the value $\unicode[STIX]{x1D708}=1/4$ corresponding to the model of Beresnyak & Lazarian (Reference Beresnyak and Lazarian2008). Nevertheless, the numerical simulations display spectral indices that differ from $-5/3$ , an effect that can be related to finite Reynolds number or hyperviscosity effects.

   2. (2) For $\unicode[STIX]{x1D708}=1$ , the problem is in contrast under-determined and we find, as in Chandran (Reference Chandran2008), that $\unicode[STIX]{x1D700}^{+}$ and $\unicode[STIX]{x1D700}^{-}$ should be equal for infinitely extended power-law spectra to exist. Other values of $r_{\unicode[STIX]{x1D700}}$ are possible if different boundary conditions are prescribed, although the model cannot accommodate for a ratio $r_{\unicode[STIX]{x1D700}}$ larger than $2$ . Indeed, equations (7.19) imply $r_{\unicode[STIX]{x1D700}}=(m^{+}+1)/(m^{-}+1)$ which, supplemented by the entanglement relation, gives, for $r_{\unicode[STIX]{x1D700}}=2$ , $m^{+}=3$ and $m^{-}=1$ , which are the limiting values of the spectral exponents ensuring the convergence of $\int E^{-}(k_{\bot })\,\text{d}k_{\bot }$ . For given fluxes such that $r_{\unicode[STIX]{x1D700}}<2$ , the spectral exponents are uniquely defined as

      (7.25) $$\begin{eqnarray}\displaystyle & \displaystyle m^{+}=(7r_{\unicode[STIX]{x1D700}}-2)/(r_{\unicode[STIX]{x1D700}}+2) & \displaystyle\end{eqnarray}$$
      (7.26) $$\begin{eqnarray}\displaystyle & \displaystyle m^{-}=(6-r_{\unicode[STIX]{x1D700}})/(r_{\unicode[STIX]{x1D700}}+2). & \displaystyle\end{eqnarray}$$

      To specify the amplitudes and thus obtain a complete solution of the problem, pinning at the dissipation scale is requested, as in the weak turbulence case. Equations (7.22)–(7.23) determines $\sqrt{E_{0}^{+}}E_{0}^{-}$ and the equality of the spectra at the wavenumber $k_{d}$ provides the extra condition to fix the amplitudes. Differently, if the amplitudes are given, the extra pinning condition permits the determination of the fluxes, and thus of the spectral slopes.

2. (ii) For a strongly imbalanced regime where the amplitude of one type of waves is large and that of the other sufficiently small for $\widetilde{k}_{\Vert }^{-}$ to be dominated by $k_{f}$ , the spectral exponents obey the conditions $m^{+}+m^{-}=4$ and $m^{+}+2m^{-}=5$ , which implies $m^{+}=3$ and $m^{-}=1$ . Interestingly, the constraint on the fluxes mentioned for the weak turbulence regime is no longer necessary.

The behaviour of the spectra, together with the local slopes, in the case where they are taken equal at the wavenumber $k_{d}$ with a ratio $E^{+}/E^{-}=1000$ prescribed at $k_{\bot }=10^{-6}$ is displayed in figure 3 for $\unicode[STIX]{x1D708}=0$ , $\unicode[STIX]{x1D708}=0.8$ and $\unicode[STIX]{x1D708}=1$ . For any value of $\unicode[STIX]{x1D708}\neq 1$ , both spectral exponents become equal to $5/3$ at scales that are larger and larger when $\unicode[STIX]{x1D708}$ approaches 1. When $\unicode[STIX]{x1D708}=1$ , they stay different at all the scales. In all the cases, the entanglement relation $m^{+}+2m^{-}=5$ is satisfied, except for $k_{\bot }$ very close to $k_{d}$ . The case $\unicode[STIX]{x1D708}=0.25$ is almost undistinguishable from the one with $\unicode[STIX]{x1D708}=0$ .

Figure 3. $E^{+}$ (red) and $E^{-}$ (green) spectra (a–c) together with the local slopes (bottom) $m^{+}$ (red), $m^{-}$ (green) and $m^{+}+2m^{-}$ (blue) in the strong turbulence regime, for $\unicode[STIX]{x1D708}=0$ (a,d), $\unicode[STIX]{x1D708}=0.8$ (b,e) and $\unicode[STIX]{x1D708}=1$ (c,f). The other parameters are $\unicode[STIX]{x1D700}=1$ , $k_{f}=0$ , $k_{d}=100$ with $u^{+}(k_{d})=u^{-}(k_{d})=50(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ . The values $\unicode[STIX]{x1D702}=0.93869$ for $\unicode[STIX]{x1D708}=0$ , $\unicode[STIX]{x1D702}=0.33477$ for $\unicode[STIX]{x1D708}=0.8$ and $\unicode[STIX]{x1D702}=0.07115$ for $\unicode[STIX]{x1D708}=1$ are chosen such that $E^{+}(10^{-6})/E^{-}(10^{-6})\approx 1000$ in all the cases.

7.2.2 Dispersive regime with $\unicode[STIX]{x1D702}=0$

In the case $\unicode[STIX]{x1D702}=0$ , we have

(7.27) $$\begin{eqnarray}m^{+}(k_{\bot })-m^{-}(k_{\bot })=\frac{\unicode[STIX]{x1D700}\sqrt{u^{-}(k_{\bot })}}{4C^{\prime }k_{\bot }^{4}v_{ph}u^{+}(k_{\bot })u^{+}(k_{\bot })}\left(\left(\frac{u^{+}(k_{\bot })}{u^{-}(k_{\bot })}\right)^{(\unicode[STIX]{x1D708}-1)/2}-1\right).\end{eqnarray}$$

When $\unicode[STIX]{x1D708}=1$ , we find that $m^{+}(k_{\bot })=m^{-}(k_{\bot })$ (equal to $5/3$ or $7/3$ in the MHD or sub-ion ranges respectively), whatever the ratio $u^{+}(k_{\bot })/u^{-}(k_{\bot })$ . In contrast, when $\unicode[STIX]{x1D708}<1$ , if we assume $u^{+}(k_{\bot })>u^{-}(k_{\bot })$ , we find that $m^{+}(k_{\bot })<m^{-}(k_{\bot })$ , indicating that the spectrum of the more energetic wave is shallower than that of the lower-amplitude one. Under this hypothesis, the two spectra diverge from each other as $k_{\bot }$ increases, which is unphysical. The physically relevant solution is thus $u^{+}(k_{\bot })=u^{-}(k_{\bot })$ for all $k_{\bot }$ corresponding to $m^{+}=m^{-}$ (equal to $5/3$ or $7/3$ depending on the wavenumber range). We can then conclude that in the case $\unicode[STIX]{x1D708}=1$ , an imbalanced regime can be obtained with a zero helicity flux, as in the weak turbulence regime (see discussion at the end of § 7.1) or, in the presence of a finite $k_{d}$ , with a value of $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}$ that is smaller as $k_{d}$ is increased. Such a regime is impossible for $\unicode[STIX]{x1D708}<1$ .

7.2.3 The dispersive regime with $\unicode[STIX]{x1D702}\neq 0$

The typical behaviour of (7.3) at small scales is easily obtained when assuming $\unicode[STIX]{x1D702}v_{ph}\gg \unicode[STIX]{x1D700}$ , a situation where one can reasonably assume that $k_{f}$ is negligible. The equations then reduce to

(7.28) $$\begin{eqnarray}\displaystyle & \displaystyle \sqrt{u^{+}(k_{\bot })}\frac{\text{d}}{\text{d}k_{\bot }}u^{-}(k_{\bot })=\frac{\unicode[STIX]{x1D702}}{4C^{\prime }k_{\bot }^{5}} & \displaystyle\end{eqnarray}$$

(7.29) $$\begin{eqnarray}\displaystyle & \displaystyle \sqrt{u^{-}(k_{\bot })}\frac{\text{d}}{\text{d}k_{\bot }}u^{+}(k_{\bot })=-\frac{\unicode[STIX]{x1D702}}{4C^{\prime }k_{\bot }^{5}}\left(\frac{u^{+}(k_{\bot })}{u^{-}(k_{\bot })}\right)^{\unicode[STIX]{x1D708}/2}, & \displaystyle\end{eqnarray}$$

from which it follows that

(7.30) $$\begin{eqnarray}(E^{+}(k_{\bot }))^{(1-\unicode[STIX]{x1D708})/2}=\unicode[STIX]{x1D706}k_{\bot }^{(1-\unicode[STIX]{x1D708})/2}-(E^{-}(k_{\bot }))^{(1-\unicode[STIX]{x1D708})/2},\end{eqnarray}$$

where $\unicode[STIX]{x1D706}$ is a positive constant.

Figure 4. $E^{+}$ (red) and $E^{-}$ (green) spectra in the strong turbulence regime. (a) For $\unicode[STIX]{x1D708}=1$ , $\unicode[STIX]{x1D700}=1$ , $\unicode[STIX]{x1D702}=0.01$ , $k_{d}=200$ , $k_{f}=0$ with $u^{+}(k_{d})=u^{-}(k_{d})=0.23552\;(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ ; (b) for $\unicode[STIX]{x1D708}=0$ , $\unicode[STIX]{x1D700}=0.1$ , $k_{f}=0.5$ , $k_{d}=60$ with $u^{+}(k_{d})=u^{-}(k_{d})=(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ , $\unicode[STIX]{x1D702}=0.053752\unicode[STIX]{x1D700}$ (singular point $E^{-}=0$ at $k_{\ast }\approx 23$ ); (c) case $k_{\ast }>k_{d}$ for $\unicode[STIX]{x1D708}=0$ , $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D702}=0.015$ (giving $k_{\ast }\approx 82$ )  $k_{f}=0.5$ , $k_{d}=60$ when choosing the critical value $u^{+}(k_{d})=u^{-}(k_{d})=u_{c}\equiv 0.405176\;(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ for which $E^{+}$ decays but $E^{-}$ tends to the absolute equilibrium.

Figure 5. A case with $\unicode[STIX]{x1D708}=1$ , $\unicode[STIX]{x1D700}=1$ , $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}=0.005$ , $k_{d}=500$ , $u^{+}(k_{d})=u^{-}(k_{d})=10^{3}(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ , $k_{f}=0$ . In (a,b) are shown the $E^{\pm }$ spectra (a,c), and $\widetilde{k}_{\Vert }^{\pm }$ (b,d), while in (c,d) are displayed the local slopes $m^{\pm }(k_{\bot })$ together with $m^{+}+2m^{-}$ (in blue) (a,c) and the nonlinearity parameters $\unicode[STIX]{x1D712}^{\pm }=(k_{\bot }^{3}E^{\pm }(k_{\bot }))^{1/2}/\widetilde{k}_{\Vert }^{\pm }$ (b,d). In all these graphs, red (green) colour refers to $+$ ( $-$ ) waves.

Figure 6. A case with $\unicode[STIX]{x1D708}=1$ , $\unicode[STIX]{x1D700}=1$ , $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}=0.045$ , $k_{d}=200$ , $u^{+}(k_{d})=u^{-}(k_{d})=2\times 10^{3}(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ , $k_{f}=5\times 10^{-3}$ where the $-$ component remains in the weak turbulence regime, while the $+$ one undergoes a transition to strong turbulence. In the top panels are shown the $E^{\pm }$ spectra (a,c), and $\widetilde{k}_{\Vert }^{\pm }$ (b,d), while in the bottom panels are displayed the local slopes $m^{\pm }(k_{\bot })$ together with $m^{+}+2m^{-}$ (in blue) (a,c) and the nonlinearity parameters $\unicode[STIX]{x1D712}^{\pm }=(k_{\bot }^{3}E^{\pm }(k_{\bot }))^{1/2}/\widetilde{k}_{\Vert }^{\pm }$ (b,d). In all these graphs, red (green) colour refers to $+$ ( $-$ ) waves.

For $\unicode[STIX]{x1D708}=1$ , we find

(7.31) $$\begin{eqnarray}E^{+}(k_{\bot })E^{-}(k_{\bot })=\unicode[STIX]{x1D706}k^{2},\end{eqnarray}$$

compatible with small-scale absolute equilibria and showing that both spectra cannot tend to zero at infinity. A nearly singular behaviour is depicted in figure 4(a) for $\unicode[STIX]{x1D700}=1$ , $\unicode[STIX]{x1D702}=0.01$ , $k_{d}=200$ , $k_{f}=0$ , when choosing $u^{+}(k_{d})=u^{-}(k_{d})=r(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ with $r=0.23552$ . For a slightly smaller value of $r$ , $E^{-}$ (respectively $E^{+}$ ) tends to infinity (respectively zero) at a finite wavenumber, while for larger values of $r$ , they both tend to absolute equilibria.

For $\unicode[STIX]{x1D708}\neq 1$ , equation (7.30) implies that, for $\unicode[STIX]{x1D702}\neq 0$ , it is impossible to prescribe that both spectra vanish at infinity (in fact one of them will diverge), as was already mentioned at the beginning of this section. It also indicates that $E^{-}(k_{\bot })$ vanishes at a wavenumber $k_{\ast }$ such that $\unicode[STIX]{x1D702}v_{ph}(k_{\ast })\approx \unicode[STIX]{x1D700}$ , close to which the above approximation is no longer valid. This regime is exemplified in figure 4(b) which displays the $E^{\pm }$ spectra for $\unicode[STIX]{x1D708}=0$ , $\unicode[STIX]{x1D700}=0.1$ , $k_{f}=0.5$ , starting the integration at $k_{d}=60$ with $u^{+}(k_{d})=u^{-}(k_{d})=(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ and choosing (by a dichotomy process) $\unicode[STIX]{x1D702}=0.053752\unicode[STIX]{x1D700}$ such that the singular wavenumber $k_{\ast }$ is slightly smaller than $23$ , satisfying $\unicode[STIX]{x1D702}v_{ph}(k_{\ast })=\unicode[STIX]{x1D700}$ . This result indicates that it is more physically appropriate to choose a value of $\unicode[STIX]{x1D702}$ such that $k_{\ast }>k_{d}$ .

Figure 7. (a) Total energy spectra obtained for $\unicode[STIX]{x1D702}=0$ (red), $\unicode[STIX]{x1D702}=2\times 10^{-3}$ (green), $\unicode[STIX]{x1D702}=5\times 10^{-3}$ (blue), $\unicode[STIX]{x1D702}=8\times 10^{-3}$ (black) and $\unicode[STIX]{x1D702}=9\times 10^{-3}$ (violet), with $\unicode[STIX]{x1D700}=1$ , $k_{d}=200$ , $k_{f}=0$ , $u^{+}(k_{d})=u^{-}(k_{d})=0.6(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ ; (b) normalized cross-helicity spectrum $E_{C}(k_{\bot })/E(k_{\bot })$ for $\unicode[STIX]{x1D702}=2\times 10^{-3}$ and $\unicode[STIX]{x1D702}=9\times 10^{-3}$ , the other parameters being the same as in (b).

The case $k_{\ast }>k_{d}$ is illustrated in figure 4(c) in a situation where $\unicode[STIX]{x1D708}=0$ , $\unicode[STIX]{x1D700}=0.1$ , $\unicode[STIX]{x1D702}=0.015$ (giving $k_{\ast }\approx 82$ ) $k_{f}=0.5$ , $k_{d}=60$ with $u^{+}(k_{d})=u^{-}(k_{d})$ . We find (again by dichotomy) that there exists a value $u_{c}=0.405176(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ of $u^{+}$ , such that if $u^{+}(k_{d})>u_{c}$ , absolute equilibrium spectra establish at small scale and if $u^{+}(k_{d})<u_{c}$ , a singularity is present. When choosing $u^{+}(k_{d})=u_{c}$ , one can ensure that the $E^{+}$ spectrum decays. The $E^{-}$ spectrum nevertheless tends to the absolute equilibrium solution. The smaller $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}$ is, the larger is the wavenumber $k_{\ast }$ and the smaller is the imbalance at large scale. Note that retaining electron inertia in the function $v_{ph}$ does not change qualitatively the above conclusion, and in particular the fact that one of the spectra displays an absolute equilibrium range at small scales. Figure 4 also shows that the ratio $E^{+}/E^{-}$ is small in the MHD range and tends to increase at dispersive scales before decreasing to unity at the pinning scale. As announced, this observation suggests that in the presence of dispersion, the case $\unicode[STIX]{x1D708}=1$ provides a better model. Indeed, figure 5(a,c) shows in this case (for $\unicode[STIX]{x1D700}=1$ , $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}=0.005$ , with $k_{d}=500$ , $u^{+}(k_{d})=u^{-}(k_{d})=10^{3}(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ and $k_{f}=0$ ) that, in the MHD range, the $E^{\pm }$ spectra display a significant imbalance (in spite of a ratio $\unicode[STIX]{x1D700}^{+}/\unicode[STIX]{x1D700}^{-}=1.01$ ) with similar slopes close to $-5/3$ (c). Note the large values of the coefficient entering the expression of $u^{\pm }(k_{d})$ . This choice is necessary, when $k_{d}$ is taken large, to ensure moderate values of $E^{+}(k_{\bot })/E^{-}(k_{\bot })$ in the MHD range.

In the sub-ion range, the low-amplitude wave displays a clear transition towards a $k_{\bot }^{-7/3}$ energy spectrum, while the spectrum of the more energetic one is not a power law, the local slope increasing up to the pinning wavenumber. This phenomenon is also visible on the parallel wavenumber (identical for both waves) displayed in (b), which scales like $k_{\bot }^{2/3}$ at large scales and undergoes a transition with local slopes smaller than $1/3$ . As expected, the nonlinearity, parameters $\unicode[STIX]{x1D712}^{\pm }=(k_{\bot }^{3}E^{\pm }(k_{\bot }))^{1/2}/\widetilde{k}_{\Vert }^{\pm }$ (d) are such that $\unicode[STIX]{x1D712}^{+}$ is equal to unity and $\unicode[STIX]{x1D712}^{-}\ll 1$ , with a tendency to increase at small scales.

A situation displaying a transition from weak to strong turbulence within the framework of the model with $\unicode[STIX]{x1D708}=1$ is displayed in figure 6, where $\unicode[STIX]{x1D700}=1$ , $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D700}=0.045$ , with $k_{d}=200$ , $u^{+}(k_{d})=u^{-}(k_{d})=2\times 10^{3}(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ and $k_{f}=5\times 10^{-3}$ . In (b,c) are shown the $E^{\pm }$ spectra (left) and $\widetilde{k}_{\Vert }^{\pm }$ (right), while in (c,d) are displayed the local slopes $m^{\pm }=-(d/d\ln k)\ln E^{\pm }(k_{\bot })$ , together with $m^{+}+2m^{-}$ (in blue) (left) and the nonlinearity parameters $\unicode[STIX]{x1D712}^{\pm }$ (right). We clearly see that the ‘ $+$ ’ component starts in a weak regime but rapidly undergoes a transition towards strong turbulence, while the ‘ $-$ ’ component remains in the weak regime. In the graph displaying the parallel wavenumber, we have superimposed a $2/3$ power law showing that the critical balance is reached for $k_{\bot }>0.1$ , as confirmed by the values of the nonlinearity parameters.

Another issue concerns the sensitivity of the energy spectrum to the degree of imbalance. We considered the strong turbulence model with $\unicode[STIX]{x1D708}=1$ , as it does not lead to unphysical divergences. When prescribing $\unicode[STIX]{x1D702}=0$ and varying the ratio $E^{+}/E^{-}$ , the energy spectrum always reduces to power laws with exponents $-5/3$ at MHD scales and $-7/3$ in the sub-ion range. More interesting is the case of a non-zero generalized flux of cross-helicity. Figure 7(a) displays the energy spectra obtained for $\unicode[STIX]{x1D702}=0$ (red), $\unicode[STIX]{x1D702}=2\times 10^{-3}$ (green), $\unicode[STIX]{x1D702}=5\times 10^{-3}$ (blue), $\unicode[STIX]{x1D702}=8\times 10^{-3}$ (black) and $\unicode[STIX]{x1D702}=9\times 10^{-3}$ (violet), keeping all the other parameters fixed ( $\unicode[STIX]{x1D700}=1$ , $k_{d}=200$ , $k_{f}=0$ , $u^{+}(k_{d})=u^{-}(k_{d})=0.6(\unicode[STIX]{x1D700}/k_{d}^{7})^{1/2}$ ). The imbalance, measured by the ratio $E^{+}/E^{-}$ at $k_{0}=10^{-4}$ increases with $\unicode[STIX]{x1D702}$ , taking the respective values $0$ , $9.55$ , $425.56$ , $36\,175$ and $1.91\times 10^{5}$ . As $\unicode[STIX]{x1D702}$ increases, the energy spectrum becomes slightly steeper than $k_{\bot }^{-5/3}$ in the MHD range while, in the sub-ion range, the $k^{-7/3}$ domain that establishes in the balanced regime reduces, being replaced by a decay faster than a power law. Such a steepening of the sub-ion KAW spectrum was also observed in fully kinetic particle-in-cell simulations in the imbalanced regime (Grošelj et al. Reference Grošelj, Mallet, Loureiro and Jenko2018). We also show in figure 7(b) the normalized cross-helicity spectrum $E_{C}(k_{\bot })/E(k_{\bot })$ for $\unicode[STIX]{x1D702}=2\times 10^{-3}$ and $\unicode[STIX]{x1D702}=9\times 10^{-3}$ , the other parameters being the same as in the left panel. This quantity is constant in the MHD range, as observed in the solar wind (Podesta & Borovsky Reference Podesta and Borovsky2010). However, while the decay of $E_{C}/E$ observed at small scales is due, as mentioned by the authors, to numerical noise in the data processing, in the present model, the $k_{\bot }^{-1}$ range obtained at sub-ion scales is a physical effect.

7.2.4 Effect of co-propagating waves interactions

It is remarkable that for strong turbulence with $\unicode[STIX]{x1D708}<1$ , the imbalance measured as the ratio $E^{+}/E^{-}$ increases with $k_{\bot }$ in the dispersion range. In order to check that this behaviour is not a consequence of the absence of the co-propagating waves couplings, following Voitenko & De Keyser (Reference Voitenko and De Keyser2016), we heuristically modify (7.3) to account for self-interactions between wavenumbers which are comparable but not necessarily asymptotically close. Inspection of (2.58) suggests defining two nonlinear characteristic frequencies for the mode $\boldsymbol{k}$ propagating in the $\pm$ direction, $\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{\pm (\uparrow \uparrow )}=V_{\boldsymbol{k}}^{(\uparrow \uparrow )}|a_{\boldsymbol{k}}^{\pm }|$ and $\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{\pm (\uparrow \downarrow )}=V_{\boldsymbol{k}}^{(\uparrow \downarrow )}|a_{\boldsymbol{k}}^{\mp }|$ , associated with the interactions between co-propagating $(\uparrow \uparrow )$ or counter-propagating $(\uparrow \downarrow )$ waves respectively. These frequencies are phenomenologically estimated by noting that $|a_{\boldsymbol{k}}^{\pm }|=k_{\bot }\sqrt{u^{\pm }(k_{\bot })}$ and using for $V_{\boldsymbol{k}}^{(\uparrow \uparrow )}$ or $V_{\boldsymbol{k}}^{(\uparrow \downarrow )}$ a semi-local approximation of the vertex $V_{\boldsymbol{k}\boldsymbol{p}\boldsymbol{q}}^{\unicode[STIX]{x1D70E}_{k}\unicode[STIX]{x1D70E}_{p}\unicode[STIX]{x1D70E}_{q}}$ given by (4.6). This vertex includes a factor $s^{-1}(\unicode[STIX]{x1D70E}_{p}v_{ph}(p_{\bot })-\unicode[STIX]{x1D70E}_{q}v_{ph}(q_{\bot }))$ which vanishes when the interactions between co-propagating waves $(\unicode[STIX]{x1D70E}_{p}=\unicode[STIX]{x1D70E}_{q})$ are strongly local. In contrast, for moderately local interactions, this factor is not zero and, since $v_{ph}/s\approx \sqrt{1+[1-1/(1+s^{2})]k_{\bot }^{2}}$ for $k_{\bot }\ll 1$ and $v_{ph}\sim k_{\bot }$ for $k_{\bot }\gg 1$ , it behaves at large scales like a constant or like $k_{\bot }^{2}$ depending on whether counter- or co-propagating waves are interacting, while at small scales it scales like $k_{\bot }$ in both cases. Furthermore, the factor $s(\unicode[STIX]{x1D70E}_{k}k_{\bot }^{2}/v_{ph}(k_{\bot })+\unicode[STIX]{x1D70E}_{p}p_{\bot }^{2}/v_{ph}(p_{\bot })+\unicode[STIX]{x1D70E}_{q}q_{\bot }^{2}/v_{ph}(q_{\bot }))$ in (4.6) is proportional to $k_{\bot }^{2}/v_{ph}(k_{\bot })$ with an undetermined numerical factor which, in the case of strongly local interactions, is equal to 3 or 1 for co- or counter-propagating waves. The global inverse transfer time writes

(7.32) $$\begin{eqnarray}(\unicode[STIX]{x1D70F}_{tr,G}^{\pm })^{-1}=\frac{(\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{\pm (\uparrow \uparrow )})^{2}}{v_{ph}\widetilde{k}_{\Vert }^{\pm }}+\frac{(\unicode[STIX]{x1D6FE}_{\boldsymbol{k}}^{\pm (\uparrow \downarrow )})^{2}}{v_{ph}\widetilde{k}_{\Vert }^{\mp }}.\end{eqnarray}$$

Noting that $(V_{\boldsymbol{k}}^{(\uparrow \downarrow )})^{2}|a_{\boldsymbol{k}}^{\mp }|^{2}/v_{ph}\widetilde{k}_{\Vert }^{\mp }=(\unicode[STIX]{x1D70F}_{tr}^{\pm })^{-1}=(k_{\bot }^{3}v_{ph}\bar{E}^{\mp })/\widetilde{k}_{\Vert }^{\mp }$ , and defining $\unicode[STIX]{x1D6FC}(k_{\bot })=V_{\boldsymbol{k}}^{(\uparrow \uparrow )}/V_{\boldsymbol{k}}^{(\uparrow \downarrow )}$ , which is an increasing function of $k_{\bot }$ that is essentially zero in the MHD range and saturates to a finite value in the dispersive range, the new inverse transfer time rewrites

(7.33) $$\begin{eqnarray}(\unicode[STIX]{x1D70F}_{tr,G}^{\pm })^{-1}\approx \frac{(k_{\bot }^{4}v_{ph}u^{\mp })}{\widetilde{k}_{\Vert }^{\mp }}\left(1+\unicode[STIX]{x1D6FC}^{2}(k_{\bot })\frac{u^{\pm }(k_{\bot })}{u^{\mp }(k_{\bot })}\frac{\widetilde{k}_{\Vert }^{\mp }}{\widetilde{k}_{\Vert }^{\pm }}\right).\end{eqnarray}$$

Equation (7.3) is thus replaced by

(7.34) $$\begin{eqnarray}\frac{\text{d}}{\text{d}k_{\bot }}u^{\pm }(k_{\bot })=-\frac{\unicode[STIX]{x1D700}\pm \unicode[STIX]{x1D702}v_{ph}}{4C^{\prime }k_{\bot }^{7}v_{ph}}\frac{1}{\displaystyle \left(\frac{u^{\mp }(k_{\bot })}{\widetilde{k}_{\Vert }^{\mp }}+\unicode[STIX]{x1D6FC}^{2}(k_{\bot })\frac{u^{\pm }(k_{\bot })}{\widetilde{k}_{\Vert }^{\pm }}\right)}.\end{eqnarray}$$

The function $\unicode[STIX]{x1D6FC}$ is taken as $\unicode[STIX]{x1D6FC}(k)=3((1+ck^{2})^{1/2}-1)/((1+ck^{2})^{1/2}+1)$ in Voitenko & De Keyser (Reference Voitenko and De Keyser2016), where $c$ is a slowly varying function of $k_{\bot }$ that we can here assume to be constant.Footnote ⁴ Within (7.34), the energy spectra $E^{\pm }$ still separate as $k_{\bot }$ increases in the dispersive range, thus not qualitatively changing the predictions of the original model. This feature also holds in the case $\unicode[STIX]{x1D708}=1$ which, in the absence of counter-propagating waves, was displaying a satisfactory behaviour. On the other hand, if one makes the (unjustified) assumption that $\unicode[STIX]{x1D700}\pm \unicode[STIX]{x1D702}v_{ph}$ is independent of $k_{\bot }$ , the results of Voitenko & De Keyser (Reference Voitenko and De Keyser2016) are qualitatively recovered by integrating the resulting system with $\unicode[STIX]{x1D708}=0$ .