2.1 Helical decomposition

The decomposition of the Fourier transform of a solenoidal vector field in circularly polarised waves as proposed by Constantin & Majda (Reference Constantin and Majda1988) has been used in several investigations of hydrodynamic turbulence (Waleffe Reference Waleffe1992; Biferale et al. Reference Biferale, Musacchio and Toschi2012; Biferale & Titi Reference Biferale and Titi2013) in order to establish the properties of energy transfer depending on the kinetic helicity. For conciseness we only review the fundamental properties of the helical decomposition and refer to the relevant literature for details and derivations.

The action of the curl operator on a square integrable real vector field $\boldsymbol{v}(\boldsymbol{x})$ can be viewed in spectral space as the action of a linear operator on the Fourier transform $\hat{\boldsymbol{v}}(\boldsymbol{k})$ of $\boldsymbol{v}(\boldsymbol{x})$ ,

(2.6) $$\begin{eqnarray}\displaystyle I_{k}:\mathbb{C}^{3} & \longrightarrow & \displaystyle \mathbb{C}^{3},\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle \hat{\boldsymbol{v}}(\boldsymbol{k}) & \longrightarrow & \displaystyle \text{i}\boldsymbol{k}\times \hat{\boldsymbol{v}}(\boldsymbol{k}).\end{eqnarray}$$

As such the linear operator $I_{k}(\cdot )=\text{i}\boldsymbol{k}\times (\cdot )$ has a set of linearly independent eigenvectors defining a basis of $\mathbb{C}^{3}$ , thus $\hat{\boldsymbol{v}}(\boldsymbol{k})$ can be expanded in this basis. That is, it can be expressed as a linear combination of eigenvectors $\text{i}\boldsymbol{k}$ , $\boldsymbol{h}_{+}(\boldsymbol{k})$ and $\boldsymbol{h}_{-}(\boldsymbol{k})$ of the curl operator $I_{k}$ , where

(2.8) $$\begin{eqnarray}\displaystyle \text{i}\boldsymbol{k}\times \boldsymbol{h}_{s_{k}} & \hspace{-2.0pt}=\hspace{-2.0pt} & \displaystyle s_{k}k\boldsymbol{h}_{s_{k}},\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle -\text{i}\boldsymbol{k}\times \boldsymbol{h}_{s_{k}}^{\ast } & \hspace{-2.0pt}=\hspace{-2.0pt} & \displaystyle s_{k}k\boldsymbol{h}_{s_{k}}^{\ast },\end{eqnarray}$$

$s_{k}=\pm 1$ and $s_{k}k=\pm k$ are the non-zero eigenvalues of the curl operator in spectral space and $\ast$ denotes the complex conjugate. (Note that the curl operator can have eigenvectors with non-zero eigenvalues, as it involves the cross product of two complex vectors. This is not necessarily orthogonal to the plane spanned by the two complex vectors, instead it is orthogonal to the plane spanned by the complex conjugates of the two vectors.) The complex eigenvectors are orthogonal to each other and are fully helical. They are normalised to unit vectors for the remainder of this paper.

Since $\hat{\boldsymbol{u}}(\boldsymbol{k})$ and $\hat{\boldsymbol{b}}(\boldsymbol{k})$ are solenoidal, they can be expressed in terms of $\boldsymbol{h}_{-}$ , $\boldsymbol{h}_{+}$ only

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{u}}(\boldsymbol{k},t)=u_{-}(\boldsymbol{k},t)\boldsymbol{h}_{-}(\boldsymbol{k})+u_{+}(\boldsymbol{k},t)\boldsymbol{h}_{+}(\boldsymbol{k})=\mathop{\sum }_{s_{k}}u_{s_{k}}(\boldsymbol{k},t)\boldsymbol{h}_{s_{k}}(\boldsymbol{k}), & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{b}}(\boldsymbol{k},t)=b_{-}(\boldsymbol{k},t)\boldsymbol{h}_{-}(\boldsymbol{k})+b_{+}(\boldsymbol{k},t)\boldsymbol{h}_{+}(\boldsymbol{k})=\mathop{\sum }_{s_{k}}b_{s_{k}}(\boldsymbol{k},t)\boldsymbol{h}_{s_{k}}(\boldsymbol{k}), & \displaystyle\end{eqnarray}$$

where $u_{s_{k}}$ and $b_{s_{k}}$ are complex coefficients. These coefficients can be calculated by taking the inner product of the basis vectors with the appropriate fields

(2.12) $$\begin{eqnarray}u_{s_{k}}(\boldsymbol{k},t)=\frac{\boldsymbol{h}_{s_{k}}^{\ast }(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{u}(\boldsymbol{k},t)}{\boldsymbol{h}_{s_{k}}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{h}_{s_{k}}^{\ast }(\boldsymbol{k})},\end{eqnarray}$$

and

(2.13) $$\begin{eqnarray}b_{s_{k}}(\boldsymbol{k},t)=\frac{\boldsymbol{h}_{s_{k}}^{\ast }(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{b}(\boldsymbol{k},t)}{\boldsymbol{h}_{s_{k}}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{h}_{s_{k}}^{\ast }(\boldsymbol{k})}.\end{eqnarray}$$

In order to keep the derivations concise we suppress the dependence of the coefficients on time and wavevector from now on in the notation.

The helical decomposition of a solenoidal vector field was first applied to incompressible MHD flows by Lessinnes et al. (Reference Lessinnes, Plunian and Carati2009), who derived a dynamical system in Fourier space describing helical triadic interactions in MHD. This system was subsequently used to construct a helical shell model of MHD turbulence. In the following section we briefly review the derivation carried out by Lessinnes et al. (Reference Lessinnes, Plunian and Carati2009).

4.1 The steady solutions

The system (3.3)–(3.4) of six coupled ODEs has several steady solutions that one can linearise about. To simplify the notation, a (formal) solution of the system (3.3)–(3.4) consisting of helical $\hat{\boldsymbol{u}}$ - and $\hat{\boldsymbol{b}}$ -field modes interacting in a given triad $\boldsymbol{k},\boldsymbol{p},\boldsymbol{q}$ is written as:

(4.1) $$\begin{eqnarray}(u_{s_{k}},u_{s_{p}},u_{s_{q}};b_{s_{k}},b_{s_{p}},b_{s_{q}}).\end{eqnarray}$$

In order to find the steady solutions of the system (3.3)–(3.4), we assume (without loss of generality) that the middle components $b_{s_{p}}=B_{s_{p}}$ and $u_{s_{p}}=U_{s_{p}}$ are constant in time. Then (3.3) and (3.4) require the other four components to vanish by the following argument. A steady solution requires $\partial _{t}u_{s_{k}}=0$ , and the only way that this can happen non-trivially is if both products $u_{s_{p}}^{\ast }u_{s_{q}}^{\ast }$ and $b_{s_{p}}^{\ast }b_{s_{q}}^{\ast }$ vanish (this requires assuming that no cancellations occur. However, the occurrence of cancellations would require the system to be in a specific state, which is unlikely to happen frequently in a chaotic system). Since $u_{s_{p}}=U_{s_{p}}$ is constant in time, $u_{s_{q}}=0$ and similarly $b_{s_{q}}=0$ . This leaves us with

(4.2) $$\begin{eqnarray}(u_{s_{k}},U_{s_{p}},0;b_{s_{k}},B_{s_{p}},0).\end{eqnarray}$$

Applying the same argument to $\partial _{t}u_{s_{q}}$ , it follows that $u_{s_{k}}$ and $b_{s_{k}}$ must also vanish. Therefore a steady solution of the system (3.3)–(3.4) has the form

(4.3) $$\begin{eqnarray}(0,U_{s_{p}},0;0,B_{s_{p}},0).\end{eqnarray}$$

It can now be checked for consistency that $\partial _{t}b_{s}=0$ for $k$ , $p$ and $q$ also. Therefore the solution is steady for the magnetic field and for the velocity field alike. Asides from the just explained example, steady solutions of the form $(U_{s_{k}},0,0;B_{s_{k}},0,0)$ and $(0,0,U_{s_{q}};0,0,B_{s_{q}})$ are obtained in the same way.

Thus the steady solutions of (3.3)–(3.4) are of the same form as for the hydrodynamic case (Waleffe Reference Waleffe1992), where at least two of the three interacting modes vanish. However, there are two special cases: one where the magnetic field component $B_{s}$ also vanishes, while $U_{s}\neq 0$ and the other, where the velocity field component $U_{s}$ vanishes, while $B_{s}\neq 0$ . The former case may perhaps be connected to a dynamo process. At this point we note that for the kinematic dynamo, where the back reaction of the magnetic field on the velocity field can be neglected, the (linear) stability of the velocity field coefficients $u_{s}$ is only determined by hydrodynamic interactions. We will come back to this point in § 8.

4.2 Linear stability analysis

In order to assess whether a given steady solution is linearly stable in our particular setting, we assume without loss of generality that the coefficients $u_{s_{p}}$ and $b_{s_{p}}$ corresponding to wavevector $\boldsymbol{p}$ are non-zero and constant in time, that is, we study the linear stability of the solution $(0,U_{s_{p}},0;0,B_{s_{p}},0)$ with respect to infinitesimal perturbations of the four modes that had been set to zero. As the first-order equations involve the coupling of all three modes of a given triad, little information can be obtained from them at first sight. Therefore we pass to second-order time derivatives of the evolution equations for the perturbations $u_{s_{k}}$ , $b_{s_{k}}$ , $u_{s_{q}}$ and $b_{s_{q}}$ . Taking time derivatives on both sides of (3.3)–(3.4) and subsequently substituting any occurrence of a first-order time derivative on the right-hand side by the appropriate evolution equation, we obtain

(4.4) $$\begin{eqnarray}\displaystyle \partial _{t}^{2}u_{s_{k}} & = & \displaystyle |g_{kpq\hspace{2.22198pt}}|^{2}(s_{p}p-s_{q}q)\left[\left((s_{k}k-s_{p}p)|U_{s_{p}}|^{2}+s_{q}q\hspace{2.22198pt}|B_{s_{p}}|^{2}\right)u_{s_{k}}\right]\nonumber\\ \displaystyle & & \displaystyle -\,|g_{kpq\hspace{2.22198pt}}|^{2}(s_{p}p-s_{q}q)[((s_{k}k-s_{p}p)U_{s_{p}}^{\ast }B_{s_{p}}+s_{q}q\hspace{2.22198pt}U_{s_{p}}B_{s_{p}}^{\ast })b_{s_{k}}],\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle \partial _{t}^{2}b_{s_{k}} & = & \displaystyle |g_{kpq\hspace{2.22198pt}}|^{2}s_{k}k[(s_{q}q\hspace{2.22198pt}U_{s_{p}}^{\ast }B_{s_{p}}+(s_{k}k-s_{p}p)U_{s_{p}}B_{s_{p}}^{\ast })u_{s_{k}}]\nonumber\\ \displaystyle & & \displaystyle -\,|g_{kpq\hspace{2.22198pt}}|^{2}s_{k}k\left[\left(s_{q}q\hspace{2.22198pt}|U_{s_{p}}|^{2}+(s_{k}k-s_{p}p)|B_{s_{p}}|^{2}\right)b_{s_{k}}\right],\end{eqnarray}$$

where terms of second order in small quantities (such as e.g. $u_{s}^{2}$ ) have been neglected. Note that these equations do not depend on modes at wavenumber $q$ . The evolution equations of the helical coefficients $u_{s_{q}}$ and $b_{s_{q}}$ can be obtained similarly and show no dependence on $k$ , therefore we restrict our attention to the evolution of $u_{s_{k}}$ and $b_{s_{k}}$ .

The system (4.4) and (4.5) can be written as a matrix ODE

(4.6) $$\begin{eqnarray}\ddot{\boldsymbol{x}}=\left(\begin{array}{@{}cc@{}}{\it\alpha} & {\it\beta}\\ {\it\gamma} & {\it\delta}\end{array}\right)\boldsymbol{ x},\end{eqnarray}$$

where $\boldsymbol{x}\equiv (u_{s_{k}},b_{s_{k}})$ and the matrix elements are

(4.7) $$\begin{eqnarray}\displaystyle {\it\alpha} & = & \displaystyle |g_{kpq\hspace{2.22198pt}}|^{2}(s_{p}p-s_{q}q)\big[(s_{k}k-s_{p}p)|U_{s_{p}}|^{2}+s_{q}q\hspace{2.22198pt}|B_{s_{p}}|^{2}\big],\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle {\it\beta} & = & \displaystyle -|g_{kpq\hspace{2.22198pt}}|^{2}(s_{p}p-s_{q}q)\big[(s_{k}k-s_{p}p)U_{s_{p}}^{\ast }B_{s_{p}}+s_{q}q\hspace{2.22198pt}U_{s_{p}}B_{s_{p}}^{\ast }\big],\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle {\it\gamma} & = & \displaystyle |g_{kpq\hspace{2.22198pt}}|^{2}s_{k}k\hspace{2.22198pt}\big[s_{q}q\hspace{2.22198pt}U_{s_{p}}^{\ast }B_{s_{p}}+(s_{k}k-s_{p}p)U_{s_{p}}B_{s_{p}}^{\ast }\big],\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle {\it\delta} & = & \displaystyle -|g_{kpq\hspace{2.22198pt}}|^{2}s_{k}k\hspace{2.22198pt}\big[s_{q}q\hspace{2.22198pt}|U_{s_{p}}|^{2}+(s_{k}k-s_{p}p)|B_{s_{p}}|^{2}\big].\end{eqnarray}$$

The linear stability of this system can be determined from the eigenvalues ${\it\lambda}_{1}$ and ${\it\lambda}_{2}$ of the matrix in (4.6). These eigenvalues depend not only on the helicities of the interacting modes and on the magnitudes of $U_{s_{p}}$ and $B_{s_{p}}$ relative to each other, but also on the alignment between the magnetic and velocity field modes at wavevector $\boldsymbol{p}$ , that is, on the cross-helicity. For a given steady solution to be unstable the perturbations have to be exponentially growing, and so at least for one of the eigenvalues, $\sqrt{{\it\lambda}_{i}}$ (for $i=1,2$ ) must have a positive real part. We will now assess under which conditions this is possible.

The eigenvalues ${\it\lambda}_{i}$ ( $i=1,2$ ) are given by

(4.11) $$\begin{eqnarray}{\it\lambda}_{1,2}=\frac{{\it\alpha}+{\it\delta}}{2}\pm \sqrt{\frac{({\it\alpha}+{\it\delta})^{2}}{4}-{\it\alpha}{\it\delta}+{\it\beta}{\it\gamma}}.\end{eqnarray}$$

For convenience we define

(4.10a,b ) $$\begin{eqnarray}x\equiv \frac{{\it\alpha}+{\it\delta}}{2}\quad \text{and}\quad Q\equiv {\it\alpha}{\it\delta}-{\it\beta}{\it\gamma},\end{eqnarray}$$

such that

(4.13) $$\begin{eqnarray}\displaystyle x & = & \displaystyle -\frac{|g_{kpq}|^{2}}{2}|U_{s_{p}}|^{2}[s_{k}ks_{q}q+(s_{k}k-s_{p}p)(s_{q}q-s_{p}p)]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{|g_{kpq}|^{2}}{2}|B_{s_{p}}|^{2}[s_{k}k(s_{k}k-s_{p}p)+s_{q}q(s_{q}q-s_{p}p)],\end{eqnarray}$$

and

(4.14) $$\begin{eqnarray}Q=|g_{kpq}|^{4}s_{k}ks_{q}q(s_{k}k-s_{p}p)(s_{q}q-s_{p}p)((|U_{s_{p}}|^{2}+|B_{s_{p}}|^{2})^{2}-4H_{c}(p)^{2}),\end{eqnarray}$$

hence the cross-helicity $H_{c}(p)$ enters the dynamics through the parameter $Q$ . The derivation of (4.14) can be found in appendix A. Note that the term $|U_{s_{p}}|^{4}+|B_{s_{p}}|^{4}+2|U_{s_{p}}|^{2}|B_{s_{p}}|^{2}-4H_{c}(p)^{2}$ is always positive, regardless of the value of $H_{c}$ since $|H_{c}(p)|\leqslant |U_{s_{p}}||B_{s_{p}}|$ , thus the sign of $Q$ is determined by the helicities of the interacting modes and the wavenumber ordering.

The eigenvalues ${\it\lambda}_{i}$ can now be written more concisely as

(4.15) $$\begin{eqnarray}{\it\lambda}_{1,2}=x\pm \sqrt{x^{2}-Q},\end{eqnarray}$$

therefore the possibility of finding exponential solutions of the system (4.6) depends on the values of $x$ and $Q$ . Apart from the trivial case, where $x=0$ and $Q=0$ , there is only one case for which no linear instability occurs: this is if $x<0$ and $|x|>|\sqrt{x^{2}-Q}|$ , since then $\sqrt{{\it\lambda}_{1}}$ and $\sqrt{{\it\lambda}_{2}}$ are imaginary numbers allowing only oscillatory solutions of the matrix ODE (4.6). All other cases lead to exponentially growing, as well as exponentially decaying, solutions.

Cases in which $x>0$ and $Q<0$ result in the largest eigenvalues and thus in the fastest growing exponential solution. These cases are therefore of special interest, as within the framework of the instability assumption they may lead to the largest energy transfer and thus can yield information about which combination of parameters facilitates most of the energy and helicity transfers. We will consider related points in further detail in §§ 7 and 8.

As can be seen from the structure of the terms $x$ and $Q$ , the relative magnitudes and the ordering of the wavenumbers in a given triad will influence the stability of steady solutions. In view of the continuous interest in non-locality of interactions in MHD turbulence (Brandenburg Reference Brandenburg2001; Alexakis et al. Reference Alexakis, Mininni and Pouquet2005; Debliquy et al. Reference Debliquy, Verma and Carati2005; Cho Reference Cho2010; Müller et al. Reference Müller, Malapaka and Busse2012), we point towards specific results for local and non-local interactions where appropriate. Following Waleffe (Reference Waleffe1992), for wavenumbers ordered $k<p<q$ , the non-local limit is defined as $k\ll p\simeq q$ , while local interactions are characterised by $k\simeq p\simeq q$ .

5.1 The case $s_{k}=s_{q}\neq s_{p}$

Since the expressions in square brackets of (4.13) become

(5.2a,b ) $$\begin{eqnarray}kq+(k+p)(q+p)>0\quad \text{and}\quad k(k+p)+q(q+p)>0,\end{eqnarray}$$

one obtains $x=({\it\alpha}+{\it\delta})/2<0$ . For an unstable solution $|x|<|\sqrt{x^{2}-Q}|$ , however, we obtain $Q>0$ since

(5.3) $$\begin{eqnarray}Q\sim s_{k}s_{q}kq(s_{k}k-s_{p}p)(s_{q}q-s_{p}p),\end{eqnarray}$$

which is positive for $s_{k}=s_{q}\neq s_{p}$ . Furthermore we obtain $Q<x^{2}$ (see appendix B) and thus $|x|>|\sqrt{x^{2}-Q}|$ , which results in negative eigenvalues of the matrix in (4.6). Therefore we do not obtain unstable solutions for the case $s_{k}=s_{q}\neq s_{p}$ , and this is independent of the ordering of the wavenumbers $k,p$ and $q$ . Note that this implies that exponentially growing solutions of (4.6) are impossible if the perturbations $u_{s_{k}}$ , $u_{s_{q}}$ , $b_{s_{k}}$ and $b_{s_{q}}$ have helicities opposite to the helicities of the modes $U_{s_{p}}$ and $B_{s_{p}}$ constituting the steady solution.

For the remaining helicity combinations, which do result in unstable solutions, the ordering of wavenumbers matters. The arguments used to decide whether or not an exponentially growing solution becomes possible are similar to the procedure employed for the case $s_{k}=s_{q}\neq s_{p}$ described above.

5.2 The case $s_{k}\neq s_{p}=s_{q}$

In this case we obtain

(5.4) $$\begin{eqnarray}Q\sim kq(k+p)(q-p),\end{eqnarray}$$

and

(5.5) $$\begin{eqnarray}x=-\frac{|g_{kpq}|^{2}}{2}|U_{s_{p}}|^{2}[-kq-(k+p)(q-p)]-\frac{|g_{kpq}|^{2}}{2}|B_{s_{p}}|^{2}[k(k+p)+q(q-p)].\end{eqnarray}$$

The stability of a steady solution depends on the signs of these terms which in turn depend on wavenumber ordering, cross-helicity and the ratio $|U_{s_{p}}|/|B_{s_{p}}|$ .

(1) For $k<p<q$ we obtain unstable solutions if $|U_{s_{p}}|\gtrapprox |B_{s_{p}}|$ , since then $x>0$ . For $|B_{s_{p}}|>|U_{s_{p}}|$ unstable solutions are still possible, provided $H_{c}(p)$ is small and $|B_{s_{p}}|$ not much larger than $|U_{s_{p}}|$ . Thus in regions of large cross-helicity unstable solutions only occur for weak magnetic fields. The method by which these results are obtained is explained in appendix C.

For non-local interactions ( $k\ll p\simeq q$ ) we obtain $Q=0$ and the sign of $x$ determines whether unstable solutions occur. The term $x$ is now of the form

(5.6) $$\begin{eqnarray}x\simeq \frac{|g_{kpq}|^{2}}{2}kq(|U_{s_{p}}|^{2}-|B_{s_{p}}|^{2}),\end{eqnarray}$$

hence non-local interactions lead to unstable solutions if $|B_{s_{p}}|<|U_{s_{p}}|$ .

(2) For $k<q<p$ , $Q$ will become negative, leading to unstable solutions regardless of the ratio $|U_{s_{p}}|/|B_{s_{p}}|$ and the value of $H_{c}(p)$ .

(3) For $p<k<q$ again we obtain unstable solutions if $|U_{s_{p}}|\gtrapprox |B_{s_{p}}|$ , since then $x>0$ . For $|B_{s_{p}}|>|U_{s_{p}}|$ unstable solutions are still possible, provided $H_{c}(p)$ is small and $|B_{s_{p}}|/|U_{s_{p}}|$ not $\gg 1$ (see appendix C). Non-local interactions ( $p\ll k\simeq q$ ) lead to unstable solutions if $|U_{s_{p}}|>|B_{s_{p}}|$ , because then

(5.7) $$\begin{eqnarray}x\simeq |g_{kpq}|^{2}k^{2}(|U_{s_{p}}|^{2}-|B_{s_{p}}|^{2})>0.\end{eqnarray}$$

In summary, a given steady solution in this case is more likely to be stable if the non-zero mode is at medium or low wavenumbers in regions of high cross-helicity.

5.3 The case $s_{k}=s_{p}=s_{q}$

In this case we obtain

(5.8) $$\begin{eqnarray}Q\sim kq(k-p)(q-p),\end{eqnarray}$$

and

(5.9) $$\begin{eqnarray}x=-\frac{|g_{kpq}|^{2}}{2}|U_{s_{p}}|^{2}[kq+(k-p)(q-p)]-\frac{|g_{kpq}|^{2}}{2}|B_{s_{p}}|^{2}[k(k-p)+q(q-p)].\end{eqnarray}$$

(1) For $k<p<q$ we obtain $Q<0$ and thus $x+\sqrt{x^{2}-Q}\geqslant 0$ , leading to exponentially growing solutions independent of $H_{c}(p)$ and the ratio $|U_{s_{p}}|/|B_{s_{p}}|$ . We note that both velocity and magnetic field modes have positive and negative contributions to the sign of $x$ . This is of interest since if $x>0$ the resulting eigenvalue would be larger and thus the solution would grow faster. However, in this case this cannot be determined from the ratio $|U_{s_{p}}|/|B_{s_{p}}|$ and thus there is little information about what contributes to a faster growing exponential and thus to a more unstable solution.

For both local ( $k\simeq p\simeq q$ ) and non-local ( $k\ll p\simeq q$ ) interactions we obtain $Q=0$ and the sign of $x$ determines whether unstable solutions occur. For the non-local case only the magnetic field term is positive, and $x$ has the form

(5.10) $$\begin{eqnarray}x\simeq \frac{|g_{kpq}|^{2}}{2}kq(|B_{s_{p}}|^{2}-|U_{s_{p}}|^{2})\end{eqnarray}$$

leading to unstable solutions if $|B_{s_{p}}|>|U_{s_{p}}|$ , while for local interactions no instability occurs as the only term in $x$ that does not vanish is $-|g_{kpq}|^{2}|U_{s_{p}}|^{2}kq<0$ .

(2) For $k<q<p$ , the possibility of exponentially growing solutions depends on the ratio $|U_{s_{p}}|/|B_{s_{p}}|$ and on the relative magnitudes of the wavenumbers $k$ , $p$ and $q$ , as now $Q>0$ . Since the magnetic field term in $x$ is now positive, instabilities occur for $|U_{s_{p}}|/|B_{s_{p}}|<1$ . If $|U_{s_{p}}|/|B_{s_{p}}|>1$ it depends also on the cross-helicity whether instabilities occur. For maximal $H_{c}(p)$ we obtain $x^{2}-Q>0$ , hence the solutions will be stable. If $H_{c}(p)=0$ and $|U_{s_{p}}|/|B_{s_{p}}|$ is not too small, instabilities will occur, depending also on the shape of the triad (see appendix C for further details). In general, the smaller $|U_{s_{p}}|/|B_{s_{p}}|$ the more unstable is the solution.

(3) For $p<k<q$ we obtain $x<0$ and $Q>0$ , furthermore $x^{2}-Q>0$ independent of $|U_{s_{p}}|/|B_{s_{p}}|$ and $H_{c}(p)$ (see appendix B), thus no unstable solutions occur. Non-local interactions ( $p\ll k\simeq q$ ) do not lead to unstable solutions, since

(5.11) $$\begin{eqnarray}x\simeq -|g_{kpq}|^{2}[k^{2}-kp](|U_{s_{p}}|^{2}+|B_{s_{p}}|^{2})<0.\end{eqnarray}$$

5.4 The case $s_{k}=s_{p}\neq s_{q}$

The terms determining the stability in this case are

(5.12) $$\begin{eqnarray}Q\sim kq(k-p)(q+p),\end{eqnarray}$$

and

(5.13) $$\begin{eqnarray}x=-\frac{|g_{kpq}|^{2}}{2}|U_{s_{p}}|^{2}[-kq-(k-p)(q+p)]-\frac{|g_{kpq}|^{2}}{2}|B_{s_{p}}|^{2}[k(k-p)+q(q+p)].\end{eqnarray}$$

(1) For $k<p<q$ unstable solutions occur independent of the ratio $|U_{s_{p}}|/|B_{s_{p}}|$ , and since both magnetic and velocity field terms have positive and negative contributions to the sign of $x$ , we are in a similar situation to the previous case. However, in the present case $Q\simeq 0$ only for local ( $k\simeq p\simeq q$ ) interactions. It is now the velocity field term $|g_{kpq}|^{2}|U_{s_{p}}|^{2}kq>0$ which ensures that exponentially growing solutions exist for local interactions provided $|U_{s_{p}}|>2|B_{s_{p}}|$ .

(2) For $k<q<p$ the result is the same, since reversing the relative ordering of $p$ and $q$ does not change the sign of $Q$ . That is, exponentially growing solutions occur.

(3) For $p<k<q$ the term $Q$ is positive and the term proportional to $|U_{s_{p}}|^{2}$ is positive while the term proportional to $|B_{s_{p}}|^{2}$ is negative. Thus instabilities occur if $|U_{s_{p}}|/|B_{s_{p}}|\gtrapprox 1$ . For $|U_{s_{p}}|/|B_{s_{p}}|<1$ the occurrence of instabilities depends on the value of $H_{c}(p)$ . If $H_{c}(p)$ is maximal and magnetic and velocity field are fully aligned, then the solutions are stable. For zero cross-helicity and $|B_{s_{p}}|$ being not much larger than $|U_{s_{p}}|$ , solutions are unstable (see appendix C).

This type of helicity combination is another possibility for non-local interactions of the type $p\ll k\simeq q$ leading to exponentially growing solutions if $|U_{s_{p}}|>|B_{s_{p}}|$ , since then

(5.14) $$\begin{eqnarray}x\simeq |g_{kpq}|^{2}k^{2}(|U_{s_{p}}|^{2}-|B_{s_{p}}|^{2})>0.\end{eqnarray}$$

The results of the dependence of the occurrence of unstable solutions on combinations of helicities, wavenumber ordering, relative magnitudes of the $\boldsymbol{u}$ and $\boldsymbol{b}$ modes and cross-helicities at wavenumber $p$ are summarised in tables 1–3.

Table 1.

Summary of possible unstable solutions for the middle wavenumber modes $k<p<q$ .

Table 2.

Summary of possible unstable solutions for the largest wavenumber modes $k<q<p$ .

Table 3.

Summary of possible unstable solutions for the smallest wavenumber modes $p<k<q$ .

7.1 Total energy transfer

In the absence of dissipation the total energy is conserved and the transfer term $T(k,p,q)$ in the spectral evolution equation of the total energy redistributes energy between the Fourier modes and vanishes if integrated over all space. Therefore the flux of total energy through wavenumber $k$ due to a given interaction $(i)$ ,

(7.9) $$\begin{eqnarray}{\it\Pi}^{(i)}(k)=-\int _{0}^{k}\,\text{d}k^{\prime }\int _{k}^{\infty }\int _{k}^{\infty }T^{(i)}(k^{\prime },p,q)\,\text{d}p\,\text{d}q,\end{eqnarray}$$

can be written as the sum of two contributions: the flux of total energy into all modes at wavenumber $k^{\prime }$ due to triads with $p,q<k<k^{\prime }$ minus the flux of total energy into all modes at $k^{\prime }$ due to triads with $k^{\prime }<k<p,q$

(7.10) $$\begin{eqnarray}{\it\Pi}^{(i)}(k)=\frac{1}{2}\int _{k}^{\infty }\,\text{d}k^{\prime }\int _{0}^{k}\int _{0}^{k}T^{(i)}(k^{\prime },p,q)\,\text{d}p\,\text{d}q-\frac{1}{2}\int _{0}^{k}\,\text{d}k^{\prime }\int _{k}^{\infty }\int _{k}^{\infty }T^{(i)}(k^{\prime },p,q)\,\text{d}p\,\text{d}q.\end{eqnarray}$$

We now follow the procedure introduced by Waleffe (Reference Waleffe1992) in order to render the two integrals in (7.10) independent of $k$ . This is achieved using a scaling argument, where the two integrals are treated separately. For conciseness we only outline the procedure briefly for the first integral on the right-hand side of (7.10) and refer to the original work of Waleffe (Reference Waleffe1992) for the full derivation. The aim is to express the transfer function in the first integral on the right-hand side of (7.10) in terms of new variables

(7.11a−c ) $$\begin{eqnarray}v=\frac{q}{p},\quad w=\frac{k^{\prime }}{p},\quad u=\frac{k}{p},\end{eqnarray}$$

in order to remove $k$ from the integration limits. Since $T_{HD}^{(i)}(k^{\prime },p,q)$ may scale differently compared to $T_{LF}^{(i)}(k^{\prime },p,q)$ and $T_{mag}^{(i)}(k^{\prime },p,q)$ , the term $T^{(i)}(k^{\prime },p,q)$ in (7.10) must be replaced by the individual transfer terms. The transfer terms are now expressed individually in terms of the new variables $u$ , $v$ and $w$

(7.12) $$\begin{eqnarray}\displaystyle T_{HD}^{(i)}(k^{\prime },p,q)=p^{-{\it\beta}}T_{HD}^{(i)}(w,1,v)=\left(\frac{k}{u}\right)^{-{\it\beta}}T_{HD}^{(i)}(w,1,v), & & \displaystyle\end{eqnarray}$$

(7.13) $$\begin{eqnarray}\displaystyle T_{LF}^{(i)}(k^{\prime },p,q)=p^{-{\it\beta}^{\prime }}T_{LF}^{(i)}(w,1,v)=\left(\frac{k}{u}\right)^{-{\it\beta}^{\prime }}T_{LF}^{(i)}(w,1,v), & & \displaystyle\end{eqnarray}$$

and

(7.14) $$\begin{eqnarray}T_{mag}^{(i)}(k^{\prime },p,q)=p^{-{\it\beta}^{\prime }}T_{mag}^{(i)}(w,1,v)=\left(\frac{k}{u}\right)^{-{\it\beta}^{\prime }}T_{mag}^{(i)}(w,1,v),\end{eqnarray}$$

where ${\it\beta}$ is related to the exponent of the kinetic energy spectrum provided it has a power-law dependence on $k$ , while the exponent ${\it\beta}^{\prime }$ is related to the exponents of the kinetic and magnetic energy spectra as explained in further detail in appendix E. We point out that it order to write down (7.12)–(7.14) we assumed both the magnetic energy spectrum and the kinetic energy spectrum to display power-law scaling in the inertial range. This assumption is made solely to allow estimates of the direction of the flux. The first term on the right-hand side of (7.10) then becomes

(7.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \frac{1}{2}\int _{k}^{\infty }\,\text{d}k^{\prime }\int _{0}^{k}\int _{0}^{k}T^{(i)}(k^{\prime },p,q)\,\text{d}p\,\text{d}q\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad =k^{3-{\it\beta}}\int _{0}^{1}\,\text{d}v~\int _{1}^{1+v}\,\text{d}w\int _{1}^{w}\,\text{d}u\left(\frac{1}{u}\right)^{4-{\it\beta}}T_{HD}^{(i)}(w,1,v)\nonumber\\ \displaystyle & & \displaystyle \displaystyle \qquad +\,k^{3-{\it\beta}^{\prime }}\int _{0}^{1}\,\text{d}v\int _{1}^{1+v}\,\text{d}w\int _{1}^{w}\,\text{d}u\left(\frac{1}{u}\right)^{4-{\it\beta}^{\prime }}\left[T_{LF}^{(i)}(w,1,v)+T_{mag}^{(i)}(w,1,v)\right].\qquad\end{eqnarray}$$

The second term on the right-hand side of (7.10) can be treated similarly (Waleffe Reference Waleffe1992), and we obtain

(7.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \frac{1}{2}\int _{0}^{k}\,\text{d}k^{\prime }\int _{k}^{\infty }\int _{k}^{\infty }T^{(i)}(k^{\prime },p,q)\,\text{d}p\,\text{d}q\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad =k^{3-{\it\beta}}\int _{0}^{1}\,\text{d}v\int _{1}^{1+v}\,\text{d}w\int _{v}^{1}\,\text{d}u\left(\frac{1}{u}\right)^{4-{\it\beta}}T_{HD}^{(i)}(v,1,w)\nonumber\\ \displaystyle & & \displaystyle \displaystyle \qquad +\,k^{3-{\it\beta}^{\prime }}\int _{0}^{1}\,\text{d}v\int _{1}^{1+v}\,\text{d}w\int _{v}^{1}\,\text{d}u\left(\frac{1}{u}\right)^{4-{\it\beta}^{\prime }}\left[T_{LF}^{(i)}(v,1,w)+T_{mag}^{(i)}(v,1,w)\right].\qquad\end{eqnarray}$$

Combining the two results and integrating over $u$ leads to the following expression for the total energy transfer flux

(7.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-10.00002pt}{\it\Pi}^{(i)}(k)=k^{3-{\it\beta}}\int _{0}^{1}\,\text{d}v\int _{1}^{1+v}\,\text{d}w\left(T_{HD}^{(i)}(w,1,v)\left[\frac{w^{{\it\beta}-3}-1}{{\it\beta}-3}\right]+T_{HD}^{(i)}(v,1,w)\left[\frac{v^{{\it\beta}-3}-1}{{\it\beta}-3}\right]\right)\nonumber\\ \displaystyle & & \displaystyle \hspace{-10.00002pt}\quad +\,k^{3-{\it\beta}^{\prime }}\int _{0}^{1}\,\text{d}v\int _{1}^{1+v}\,\text{d}w\left(T_{LF}^{(i)}(w,1,v)\left[\frac{w^{{\it\beta}^{\prime }-3}-1}{{\it\beta}^{\prime }-3}\right]+T_{LF}^{(i)}(v,1,w)\left[\frac{v^{{\it\beta}^{\prime }-3}-1}{{\it\beta}^{\prime }-3}\right]\right)\nonumber\\ \displaystyle & & \displaystyle \hspace{-10.00002pt}\quad +\,k^{3-{\it\beta}^{\prime }}\int _{0}^{1}\,\text{d}v\int _{1}^{1+v}\,\text{d}w\left(T_{mag}^{(i)}(w,1,v)\left[\frac{w^{{\it\beta}^{\prime }-3}-1}{{\it\beta}^{\prime }-3}\right]+T_{mag}^{(i)}(v,1,w)\left[\frac{v^{{\it\beta}^{\prime }-3}-1}{{\it\beta}^{\prime }-3}\right]\right)\!,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $0\leqslant v\leqslant 1\leqslant w\leqslant 1+v$ due to the triad geometry. This now enables us to study the contribution to the total energy transfer from a given interaction $(i)$ , where the scaling of the magnetic and kinetic energy spectra will influence the transfer through the exponents ${\it\beta}$ and ${\it\beta}^{\prime }$ . In the inertial range of total energy the energy transfer flux through a given wavenumber $k$ does not depend on that wavenumber, which leads to the characteristic values of the scaling exponents ${\it\beta}^{\prime }={\it\beta}=3$ , making the split of the total energy transfer term into its individual components redundant in this wavenumber range. In § 7.3.1 we concentrate on the contributions of the different interactions to transfers in the inertial range of total energy and set ${\it\beta}=3$ , thus taking into account only the region in wavenumber space where this scaling is established. Since the values of ${\it\beta}$ and ${\it\beta}^{\prime }$ may influence the direction of energy transfer, a similar approach may be useful to calculate energy and helicity transfer at the very low wavenumbers. However, this awaits consensus on the low-wavenumber scaling of the magnetic and kinetic energy spectra. Furthermore, the integrals must be cut off at some wavenumber such that a single scaling exponent for the wavenumber range of interest can be studied. As the extent of the inertial range will grow with increasing Reynolds number, contributions from the production and dissipation ranges can safely be neglected, as they will become very small compared to the extent of the inertial range. However, in the low-wavenumber region, this argument is not applicable and further work is necessary in order to establish if very non-local interactions contribute significantly to the transfers of magnetic energy and helicity in the low-wavenumber range or not.

7.2 Magnetic helicity transfer

Using the decomposition into helical modes, the transfer term in the evolution equation of the magnetic helicity can be expressed through the transfer term in the evolution equation of the magnetic energy, that is

(7.18) $$\begin{eqnarray}T_{H}^{(i)}(k,p,q)=\frac{s_{k}}{k}T_{mag}^{(i)}(k,p,q),\end{eqnarray}$$

and only the transfer term which originates from the induction equation is present, since $H_{mag}$ is a purely magnetic quantity and as such only implicitly depends on the evolution of the velocity field.

Since the magnetic helicity is an ideal invariant, the transfer term in the spectral evolution equation of the magnetic helicity vanishes if integrated over all space, therefore, similar to the flux of total energy, the flux of magnetic helicity through wavenumber $k$ due to a given interaction $(i)$ ,

(7.19) $$\begin{eqnarray}{\it\Pi}_{H}^{(i)}(k)=-\int _{0}^{k}\frac{s_{k^{\prime }}}{k^{\prime }}\,\text{d}k^{\prime }\int _{k}^{\infty }\int _{k}^{\infty }T_{mag}^{(i)}(k^{\prime },p,q)\,\text{d}p\,\text{d}q,\end{eqnarray}$$

can be written as the sum of two contributions

(7.20) $$\begin{eqnarray}\displaystyle {\it\Pi}_{H}^{(i)}(k) & = & \displaystyle \frac{1}{2}\int _{k}^{\infty }\frac{s_{k^{\prime }}}{k^{\prime }}\,\text{d}k^{\prime }\int _{0}^{k}\int _{0}^{k}T_{mag}^{(i)}(k^{\prime },p,q)\,\text{d}p\,\text{d}q\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{2}\int _{0}^{k}\frac{s_{k^{\prime }}}{k^{\prime }}\,\text{d}k^{\prime }\int _{k}^{\infty }\int _{k}^{\infty }T_{mag}^{(i)}(k^{\prime },p,q)\,\text{d}p\,\text{d}q.\end{eqnarray}$$

Following the approach explained in § 7.1 the integral becomes independent of $k$ and one obtains the following expression for the flux of magnetic helicity through $k$

(7.21) $$\begin{eqnarray}\displaystyle {\it\Pi}_{H}(k) & = & \displaystyle k^{2-{\it\beta}^{\prime }}\int _{0}^{1}\,\text{d}v\int _{1}^{1+v}\frac{\text{d}w}{w}\nonumber\\ \displaystyle & & \displaystyle \times \left(s_{w}T_{mag}^{(i)}(w,1,v)\left[\frac{w^{{\it\beta}^{\prime }-2}-1}{{\it\beta}^{\prime }-2}\right]+s_{v}T_{mag}^{(i)}(v,1,w)\left[\frac{v^{{\it\beta}^{\prime }-2}-1}{{\it\beta}^{\prime }-2}\right]\right).\quad\end{eqnarray}$$

7.3 Cascades and wavenumber-dependent transfers of total energy and magnetic helicity

From the expressions (7.17) and (7.21) for the fluxes of total energy and magnetic helicity, respectively, it is now possible to determine the sign of the fluxes and hence the direction of energy and magnetic helicity transfers using the results from the stability analysis. If the total energy flux is positive, energy is transferred from smaller to larger wavenumbers and if it is negative, energy is transferred from larger to smaller wavenumbers. As the magnetic helicity is not positive definite, the situation is slightly different. For positive magnetic helicity a positive flux indicates forward transfer just as for the total energy. For negative magnetic helicity a negative flux indicates forward transfer while a positive flux indicates inverse transfer. However, as this situation is symmetric we assume positive helicity throughout the analysis.

In § 6 unstable solutions of (3.3) and (3.4) were interpreted as leading to energy transfer out of the unstable mode into the two modes it interacts with for a given helical mode interaction $(i)$ . If $U_{s_{p}}$ and $B_{s_{p}}$ are the unstable modes, this interpretation leads to

(7.22) $$\begin{eqnarray}\partial _{t}|B_{s_{p}}|^{2}=T_{mag}^{(i)}(p,k,q)<0,\end{eqnarray}$$

and

(7.23) $$\begin{eqnarray}\partial _{t}|U_{s_{p}}|^{2}=T_{HD}^{(i)}(p,k,q)+T_{LF}^{(i)}(p,k,q)<0.\end{eqnarray}$$

The instability assumption therefore attributes signs to the transfer terms, which will determine their respective contributions to the overall energy (and magnetic helicity) transfer. Note that $\partial _{t}|U_{s_{p}}|^{2}$ and $\partial _{t}|B_{s_{p}}|^{2}$ cannot have different signs, as both signs are determined from the existence of exponentially growing solutions of the system (4.6).

We now treat the three helicity combinations which lead to unstable solutions separately assuming without loss of generality that $s_{p}=1$ . Having determined the signs of the transfer terms within our framework, we now use these results to calculate the contributions of the individual transfer terms to the fluxes of total energy and magnetic helicity though a given wavenumber.

7.3.1 Total energy cascades

For the (inertial range) energy cascade the flux is wavenumber independent leading to ${\it\beta}=3$ in (7.17). Hence the integrand in (7.17), which determines the sign of the total energy flux, becomes

(7.24) $$\begin{eqnarray}I_{E}=T^{(i)}(w,1,v)\ln w+T^{(i)}(v,1,w)\ln v,\end{eqnarray}$$

where we remind the reader of the wavenumber ordering $v\leqslant 1\leqslant w$ . That is, the term $T^{(i)}(w,1,v)$ describes energy transfer in and out of the largest wavenumber modes while $T^{(i)}(v,1,w)$ describes energy transfer in and out of the smallest wavenumber modes.

Using the signs of the transfer terms determined for the different helicity combinations depending on wavenumber ordering, we can now deduce which helicity combinations contribute to forward or inverse cascades of total energy.

(1) $s_{v}=s_{1}=s_{w}$

For this case we can deduce from the results of the stability analysis summarised in tables 1–3 that $T^{(i)}(1,v,w)<0$ , as modes corresponding to the middle wavenumber are unstable, while $T^{(i)}(v,1,w)>0$ , as modes corresponding to the smallest wavenumber are stable and hence these modes can only receive energy from the modes at higher wavenumbers. The sign of $T^{(i)}(w,1,v)$ depends on the values of cross-helicity and the ratio of magnetic to kinetic energy. For a magnetically dominated system $T^{(i)}(w,1,v)<0$ and we obtain in this case an inverse cascade of total energy, as

(7.25) $$\begin{eqnarray}I_{E}=T^{(i)}(w,1,v)\ln w+T^{(i)}(v,1,w)\ln v<0.\end{eqnarray}$$

If the kinetic energy is much larger than the magnetic energy, cancellations between the two terms in $I_{E}$ occur. The term $T^{(i)}(w,1,v)\ln w$ is now positive, since the modes at the largest wavenumber can only receive energy, thus contributing to a forward cascade. For intermediate cases the value of the cross-helicity becomes decisive as high cross-helicity quenches the inverse transfer in this case. In summary, we expect inverse cascade contributions from this combination of helicities if the magnetic energy dominates, while for larger kinetic energy high values of cross-helicity quench the inverse transfer contribution to some extent.

(2) $s_{v}\neq s_{1}=s_{w}$

From tables 1–3 the instability assumption imposes $T^{(i)}(w,1,v)<0$ and $T^{(i)}(v,1,w)>0$ as modes corresponding to the largest wavenumbers are unstable, while modes corresponding to the smallest wavenumber are stable. This implies

(7.26) $$\begin{eqnarray}I_{E}=T^{(i)}(w,1,v)\ln w+T^{(i)}(v,1,w)\ln v<0,\end{eqnarray}$$

thus we conclude that this combination of helicities leads to an inverse energy cascade as $I_{E}<0$ , and we note that this case behaves differently to its hydrodynamic analogue, where it led to an inverse cascade of kinetic energy (Waleffe Reference Waleffe1992) for non-local interactions and a direct cascade for local interactions. We also note that this inverse cascade should always be present, as it is not subject to constraints from $H_{c}(p)$ and $|U_{s_{p}}|/|B_{s_{p}}|$ .

(3) $s_{v}=s_{1}\neq s_{w}$

Analogously, we obtain $T^{(i)}(1,v,w)<0$ and $T^{(i)}(w,1,v)>0$ , since the modes corresponding to the middle wavenumber are unstable while modes corresponding to the largest wavenumber are stable. As the stability of the remaining transfer term $T^{(i)}(v,1,w)$ depends on several constraints, no clear assessment is possible. If we assume the lowest wavenumber modes to be unstable, that is $T^{(i)}(v,1,w)<0$ , we obtain a contribution towards a direct cascade. However, if we assume them to be stable, contributions to inverse and direct cascades are possible. We note that the instability leading to forward transfer in this case is damped by high values of $H_{c}(p)$ .

(4) $s_{1}\neq s_{w}=s_{v}$

In this case we obtain $T^{(i)}(1,v,w)>0$ and $T^{(i)}(w,1,v)<0$ , since the modes corresponding to the middle wavenumber are stable while modes corresponding to the largest wavenumber are unstable. Again the sign of the remaining transfer term $T^{(i)}(v,1,w)$ depends on several constraints. If we assume the lowest wavenumber modes to be stable, that is $T^{(i)}(v,1,w)>0$ , we obtain a contribution towards an inverse cascade. However, if we assume them to be unstable, contributions to inverse and direct cascades are possible. We note that the instability leading to inverse transfer in this case is damped by high values of $H_{c}(p)$ .

7.3.2 Magnetic helicity transfer in the inertial range of total energy

For ${\it\beta}^{\prime }=3$ , the integrand $I_{H}$ in (7.21) becomes

(7.27) $$\begin{eqnarray}I_{H}=T_{mag}^{(i)}(w,1,v)s_{w}(w-1)+T_{mag}^{(i)}(v,1,w)s_{v}(v-1).\end{eqnarray}$$

Using the signs of the transfer terms determined for the three helicity combinations, we can now deduce which helicity combinations contribute to a forward or inverse cascade of magnetic helicity. As can be seen in (7.27), there is an explicit dependence of the magnetic helicity flux on the helicities of the interacting modes. In the following we assume $s_{1}=1$ .

(1) $s_{v}=s_{1}=s_{w}$

The integrand $I_{H}$ becomes

(7.28) $$\begin{eqnarray}I_{H}=T_{mag}^{(i)}(w,1,v)(w-1)+T_{mag}^{(i)}(v,1,w)(v-1).\end{eqnarray}$$

As the signs of the magnetic energy transfer term deduced from the stability analysis are the same as for the total energy and $\ln w$ and $w-1$ are both positive while $\ln v$ and $v-1$ are both negative, the result for the helicity transfer reflects the results for the total energy cascade. Thus, for this helicity combination, total energy and magnetic helicity will be transferred in the same direction, which can be both forward and inverse in this case.

(2) $s_{v}\neq s_{1}=s_{w}$

The integrand $I_{H}$ becomes

(7.29) $$\begin{eqnarray}I_{H}=T_{mag}^{(i)}(w,1,v)(w-1)-T_{mag}^{(i)}(v,1,w)(v-1),\end{eqnarray}$$

where the contributions from the smallest wavenumber modes now enter with the opposite sign. Compared to the total energy flux, which was purely inverse in this case, we obtain the possibility of simultaneously a forward helicity flux and an inverse energy flux.

(3) $s_{v}=s_{1}\neq s_{w}$

The integrand $I_{H}$ becomes

(7.30) $$\begin{eqnarray}I_{H}=-T_{mag}^{(i)}(w,1,v)(w-1)+T_{mag}^{(i)}(v,1,w)(v-1),\end{eqnarray}$$

where the contributions from the largest wavenumber modes now enter with the opposite sign. Compared to the total energy cascade, again we find that it is possible to have a transfer of magnetic helicity in the opposite direction to the transfer of total energy.

(4) $s_{1}\neq s_{w}=s_{v}$

The integrand $I_{H}$ becomes

(7.31) $$\begin{eqnarray}I_{H}=-T_{mag}^{(i)}(w,1,v)(w-1)-T_{mag}^{(i)}(v,1,w)(v-1),\end{eqnarray}$$

where the contributions from both transfer terms now enter with the opposite sign. That is, magnetic helicity and total energy are transferred in opposite directions.

In this subsection we determined the direction of the magnetic helicity transfer in the inertial range of total energy for different combinations of helicities and compared the results to those for the total energy cascade. We found that a cascade of total energy is possible in one direction while the transfer of magnetic helicity may proceed in the opposite direction.

7.3.3 Magnetic helicity cascades

In the inertial range of magnetic helicity the flux of magnetic helicity is wavenumber independent resulting in ${\it\beta}^{\prime }=2$ in (7.21). Therefore the integrand $I_{H}$ in (7.21) becomes

(7.32) $$\begin{eqnarray}I_{H}=T_{mag}^{(i)}(w,1,v)s_{w}\ln w+T_{mag}^{(i)}(v,1,w)s_{v}\ln v.\end{eqnarray}$$

For the different helicity combinations this leads to

(1) $s_{v}=s_{1}=s_{w}$

The integrand in this case is of the same form as the integrand $I_{E}$ for the total energy cascade (that is, if ${\it\beta}=3$ in $I_{E}$ )

(7.33) $$\begin{eqnarray}I_{H}=T_{mag}^{(i)}(w,1,v)\ln w+T_{mag}^{(i)}(v,1,w)\ln v,\end{eqnarray}$$

hence the results for the cascades of magnetic helicity are the same as for the cascades of total energy.

(2) $s_{v}\neq s_{1}=s_{w}$

The integrand in this case has a different form compared to the integrand $I_{E}$ for the total energy

(7.34) $$\begin{eqnarray}I_{H}=T_{mag}^{(i)}(w,1,v)\ln w-T_{mag}^{(i)}(v,1,w)\ln v,\end{eqnarray}$$

hence the results for the cascades of magnetic helicity are different from the total energy cascades. In particular, this case may lead to a non-helical reverse energy transfer while the helicity cascade may be forwards, due to the contribution from $T_{mag}^{(i)}(v,1,w)$ now having the opposite sign in $I_{H}$ compared to $I_{E}$ .

(3) $s_{v}=s_{1}\neq s_{w}$

Again, the integrand in this case has a different form compared to the integrand $I_{E}$ for the total energy

(7.35) $$\begin{eqnarray}I_{H}=-T_{mag}^{(i)}(w,1,v)\ln w+T_{mag}^{(i)}(v,1,w)\ln v,\end{eqnarray}$$

hence the results for the cascades of magnetic helicity differ from the total energy cascades. In particular, this case may lead to a forward energy transfer while the helicity cascade may be backwards, due to the contribution from $T_{mag}^{(i)}(w,1,v)$ now having the opposite sign in $I_{H}$ compared to $I_{E}$ .

(4) $s_{1}\neq s_{w}=s_{v}$

Now $I_{H}$ and $I_{E}$ have opposite signs

(7.36) $$\begin{eqnarray}I_{H}=-T_{mag}^{(i)}(w,1,v)\ln w-T_{mag}^{(i)}(v,1,w)\ln v=-I_{E},\end{eqnarray}$$

hence this case leads to helicity transfer and energy transfer in opposite directions.

7.3.4 Magnetic energy transfer in the inertial range of magnetic helicity

For ${\it\beta}^{\prime }=2$ , the contributions to the integrand $I_{E}$ due to magnetic energy transfer are

(7.37) $$\begin{eqnarray}I_{E_{mag}}=-T_{mag}^{(i)}(w,1,v)\left(\frac{1}{w}-1\right)-T_{mag}^{(i)}(v,1,w)\left(\frac{1}{v}-1\right).\end{eqnarray}$$

The signs of $T_{mag}$ and $T$ are the same by (7.22)–(7.23), and $\ln w$ and $w-1$ are both positive while $\ln v$ and $v-1$ are both negative. Hence, the result for the contributions of these terms to the total energy transfer in the inertial range of magnetic helicity is the same as in the inertial range of total energy for all helicity combinations. That is, magnetic energy transfer and conversion in the inertial ranges of total energy and magnetic helicity proceed in the same direction.

This assessment of contributions to forward and inverse transfers and cascades is based on an analysis of the nonlinear terms in the MHD equations only, thus neglecting the symmetry-breaking effect of dissipation creating an energy sink at the small scales. Accounting for this effect, it is plausible that the contributions from transfer terms leading to forward transfer are higher weighted than contributions leading to inverse transfer. This is particularly relevant in interactions where forwards and reverse contributions are present and the overall transfer depends on cancellations between the two terms. It would perhaps be safest to attribute these cases to forwards rather than inverse energy cascades.

Although it is not possible to exactly determine which helical interactions are higher weighted than others, some information can be obtained from the magnitude of the geometric factor $g_{kpq}$ defined in (3.5). The magnitude of $g_{kpq}$ depends on the helicity combinations since it involves the helicity-dependent factor $I=s_{k}k+s_{p}p+s_{q}q$ . Therefore it parametrises the strength of a given helical interaction, and the case of all helicities being of the same sign gives the largest value of $|I|$ , since in this case $|I|=|k+p+q|$ .

For the reverse transfers, that is, for $k<p,q$ , the factor $|I|$ takes the smallest value for the case $s_{k}=s_{p}\neq s_{q}$ , since $|I|=|k+(p-q)|$ . Note that in this case $I$ becomes small for small $k$ even in the non-local limit $k\ll p\simeq q$ , suggesting that the non-helical reverse transfer found in this case is less efficient in increasing spectral power at the very low wavenumbers. The remaining class of helical interactions $s_{k}\neq s_{p}=s_{q}$ leads to $|I|=|k-(p+q)|$ . In this case $|I|$ does not necessarily become small for small $k$ which is due to the contribution of non-local interactions, where $p$ and $q$ are large compared to $k$ . According to the results from the stability analysis, in the non-local limit unstable solutions occur for the case $s_{k}=s_{p}=s_{q}$ only if $|B_{s_{p}}|>|U_{s_{p}}|$ and for the case $s_{k}\neq s_{p}=s_{q}$ if $|U_{s_{p}}|>|B_{s_{p}}|$ .

It is therefore possible to deduce within the framework of the instability assumption that most of the increase in energy at the very largest scales (in a magnetically dominated system) is mainly due to a breaking of mirror symmetry, which had been established before by Frisch et al. (Reference Frisch, Pouquet, Léorat and Mazure1975) using a different approach. That is, it is due to the presence of kinetic and magnetic helicity, since interactions of the type $s_{k}=s_{p}=s_{q}$ , which account for most of the inverse transfer, can only occur in significant numbers for fields consisting of many modes with the same helicity. Recent numerical results in hydrodynamics showed that there is an overall reverse flux of energy only when the system mainly contains helical modes of the same sign. As soon as a small amount of oppositely polarised modes is introduced, the usual direct cascade is recovered (Sahoo, Bonaccorso & Biferale Reference Sahoo, Bonaccorso and Biferale2015).

In summary, in this section we determined the direction of total energy and magnetic helicity transfers in their respective inertial ranges. Not surprisingly, we found that fully helical magnetic fields lead to inverse cascades of magnetic helicity and inverse transfer of magnetic energy, but the analysis also showed that an inverse energy cascade is possible for non-helical magnetic fields, which is a new theoretical result. However, due to the coupling of the momentum and induction equations, within this framework it is not possible to determine the nature of the energy transfers resulting from an instability of a given steady solution, since the same eigenvalue controls the growth of the exponential solution of (4.6) for both the magnetic and the velocity field. Nevertheless, for some special cases the evolution equations (3.3)–(3.4) decouple and more detailed information becomes available. These cases are treated in the following section.

8.1 The kinematic dynamo

For small magnetic fields the Lorentz force is small compared to inertial forces, and can be neglected in the momentum equation. This decouples the momentum equation from the induction equation and defines the kinematic dynamo problem. In our setting, it corresponds to $|U_{s_{p}}|/|B_{s_{p}}|\gg 1$ , and terms proportional to $|B_{s_{p}}|$ can be neglected as they are very small compared to terms proportional to $|U_{s_{p}}|$ .

Alternatively, one could also consider the steady solution $B_{s_{p}}=0$ while $U_{s_{p}}\neq 0$ . This would correspond to a stability analysis of a flow field at a particular length scale subject to small perturbations of the magnetic and velocity fields, where the magnetic field perturbation may be viewed as the magnetic seed field to be amplified by dynamo action. In this setting we observe from (3.3) that the term corresponding to the Lorentz force disappears while in (4.5) terms involving $B_{s_{p}}$ disappear, thus the system simplifies to

(8.1) $$\begin{eqnarray}\displaystyle & \partial _{t}^{2}u_{s_{k}}=|g_{kpq}|^{2}(s_{p}p-s_{q}q)(s_{k}k-s_{p}p)\,|U_{s_{p}}|^{2}u_{s_{k}}, & \displaystyle\end{eqnarray}$$

(8.2) $$\begin{eqnarray}\displaystyle & \partial _{t}^{2}b_{s_{k}}=-|g_{kpq}|^{2}s_{k}k\,s_{q}q\,|U_{s_{p}}|^{2}b_{s_{k}}. & \displaystyle\end{eqnarray}$$

As the only contribution to the evolution of the magnetic field now comes from the velocity field, we associate the remaining terms in (3.4) with dynamo action. From (8.2) we observe that this system has exponential solutions leading to magnetic field growth if $s_{k}\neq s_{q}$ , regardless of wavenumber ordering. So for energy transfer from $U_{s_{p}}$ into $b_{s_{k}}$ (and $b_{s_{q}}$ ) to become possible, the magnetic modes at wavenumbers $k$ and $q$ should be of opposite helicity.

For small $k$ , non-local interactions with $k\ll p\simeq q$ provide most transfer into $b_{s_{k}}$ . This is because the eigenvalue determining the growth of the exponential solution of (8.2) is larger for $q\gg k$ than for $q\simeq k$ , thus the perturbations should grow faster in the former than in the latter case. Hence, according to the instability assumption, $U_{s_{p}}$ loses energy in favour of $b_{s_{k}}$ mainly due to non-local interactions if $b_{s_{k}}$ describes the largest scales of the system.

8.2 Excitation of a flow by the Lorentz force

For the other special solution $(0,0,0;0,B_{s_{p}}\neq 0,0)$ the system (4.4)–(4.5) simplifies to

(8.3) $$\begin{eqnarray}\displaystyle \partial _{t}^{2}u_{s_{k}}=|g_{kpq}|^{2}(s_{p}p-s_{q}q)s_{q}q\,|B_{s_{p}}|^{2}u_{s_{k}}, & & \displaystyle\end{eqnarray}$$

(8.4) $$\begin{eqnarray}\displaystyle \partial _{t}^{2}b_{s_{k}}=-|g_{kpq}|^{2}s_{k}k(s_{k}k-s_{p}p)\,|B_{s_{p}}|^{2}b_{s_{k}}, & & \displaystyle\end{eqnarray}$$

and we note that the inertial term in (3.3) and the ‘dynamo’ term in (3.4) are now absent and the system of coupled ODEs has split into two decoupled ODEs. This case may perhaps be associated with the generation of turbulence caused by the action of the Lorentz force on the fluid (i.e. energy conversion from $B_{s_{p}}$ to $u_{s_{k}}$ or $u_{s_{q}}$ ) and interscale transfer of magnetic energy from $B_{s_{p}}$ into $b_{s_{k}}$ or $b_{s_{q}}$ . Exponentially growing solutions of (8.4) only occur if $s_{p}=s_{k}$ and $k<p$ , leading to a reverse transfer of magnetic energy. Exponentially growing solutions of (8.3) occur for $p>q$ and $s_{p}=s_{q}$ leading to forward and reverse transfers corresponding to $k>p$ and $k<p$ , respectively. Interestingly, energy transfer only becomes possible if the magnetic field is helical and the helicity of the velocity field mode does not affect the analysis.