2.1 Spectral evolution equations of magnetohydrodynamics

The evolution equation of a conducting fluid of fluctuating velocity $u_{i}$ , submitted to the Laplace force (magnetic part of the Lorentz force), reads

(2.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}t}+u_{l}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{l}}=-\frac{\unicode[STIX]{x2202}p^{\ast }}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}u_{i}}{\unicode[STIX]{x2202}x_{l}\unicode[STIX]{x2202}x_{l}}+b_{l}\frac{\unicode[STIX]{x2202}b_{i}}{\unicode[STIX]{x2202}x_{l}}+B_{l}^{0}\frac{\unicode[STIX]{x2202}b_{i}}{\unicode[STIX]{x2202}x_{l}},\end{eqnarray}$$

where $b_{i}$ is the fluctuating magnetic field, $\boldsymbol{B}^{0}$ a possible external mean magnetic field, $\unicode[STIX]{x1D708}$ the kinematic viscosity and $p^{\ast }$ the total pressure. Both $u_{i}$ and $b_{i}$ are solenoidal, i.e. $\text{div}(\boldsymbol{u})=\text{div}(\boldsymbol{b})=0$ . The equation of $b_{i}$ is obtained by combining the Maxwell–Faraday law, the Ohm law (without the Hall term) and the Maxwell–Ampère law, so that

(2.2) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}b_{i}}{\unicode[STIX]{x2202}t}+u_{l}\frac{\unicode[STIX]{x2202}b_{i}}{\unicode[STIX]{x2202}x_{l}}=b_{l}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{l}}+B_{l}^{0}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{l}}+\unicode[STIX]{x1D702}\frac{\unicode[STIX]{x2202}^{2}b_{i}}{\unicode[STIX]{x2202}x_{l}\unicode[STIX]{x2202}x_{l}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D702}$ is the magnetic diffusivity, or resistivity. We choose a unit magnetic Prandtl number, so that $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D702}$ . When dropping $\boldsymbol{B}^{0}$ in the latter equation, one remarks that it is similar to the equation of vorticity $\unicode[STIX]{x1D74E}$ , which is the so-called Batchelor analogy (Batchelor Reference Batchelor1950). Furthermore, it is worth noting that $\boldsymbol{b}$ is a pseudo-vector like $\unicode[STIX]{x1D74E}$ , since it is defined as $\unicode[STIX]{x1D735}\times \boldsymbol{a}$ , where the true vector $\boldsymbol{a}$ is the magnetic potential (some considerations about the modelling of $\boldsymbol{a}$ are proposed in appendix A.1).

The spectral counterpart of (2.1) and (2.2) are given by

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D708}k^{2}\right)\hat{u} _{i}(\boldsymbol{k})=-\text{i}P_{imn}(\boldsymbol{k})\,\widehat{u_{m}u_{n}}(\boldsymbol{k})+\text{i}P_{imn}(\boldsymbol{k})\,\widehat{b_{m}b_{n}}(\boldsymbol{k})+\text{i}B_{l}^{0}k_{l}\hat{b}_{i}(\boldsymbol{k}), & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D702}k^{2}\right)\hat{b}_{i}(\boldsymbol{k})=-\text{i}k_{l}\widehat{b_{i}u_{l}}(\boldsymbol{k})+\text{i}k_{l}\widehat{u_{i}b_{l}}(\boldsymbol{k})+\text{i}B_{l}^{0}k_{l}\hat{u} _{i}(\boldsymbol{k}), & \displaystyle\end{eqnarray}$$

where the Kraichnan operator is given by $2P_{imn}=k_{m}P_{in}+k_{n}P_{im}$ , with the projector $P_{ij}=\unicode[STIX]{x1D6FF}_{ij}-\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6FC}_{j}$ and $\unicode[STIX]{x1D6FC}_{i}=k_{i}/k$ , and $\hat{(\cdot )}$ denotes the Fourier transform. In homogeneous turbulence, the spectral second-order moments of interest, namely the Reynolds tensor $\hat{\unicode[STIX]{x1D619}}_{ij}$ , the magnetic tensor $\hat{\unicode[STIX]{x1D609}}_{ij}$ , and the velocity–magnetic correlation $\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}$ , are defined as

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D619}}_{ij}(\boldsymbol{k},t)\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p})=\langle \hat{u} _{i}^{\ast }(\boldsymbol{p},t)\hat{u} _{j}(\boldsymbol{k},t)\rangle , & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D609}}_{ij}(\boldsymbol{k},t)\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p})=\langle \hat{b}_{i}^{\ast }(\boldsymbol{p},t)\hat{b}_{j}(\boldsymbol{k},t)\rangle , & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(\boldsymbol{k},t)\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p})=\langle \hat{u} _{i}^{\ast }(\boldsymbol{p},t)\hat{b}_{j}(\boldsymbol{k},t)\rangle , & \displaystyle\end{eqnarray}$$

where $(\cdot )^{\ast }$ denotes the complex conjugate, and $\langle \cdot \rangle$ an ensemble average. Their evolution equations read

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+2\unicode[STIX]{x1D708}k^{2}\right)\hat{\unicode[STIX]{x1D619}}_{ij}(\boldsymbol{k})=T_{ij}^{\text{V(hyd)}}(\boldsymbol{k})+T_{ij}^{\text{V(mag)}}(\boldsymbol{k})+\text{i}B_{l}^{0}k_{l}\left(\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(\boldsymbol{k})-\hat{\unicode[STIX]{x1D60F}}_{ji}^{c\ast }(\boldsymbol{k})\right), & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+2\unicode[STIX]{x1D702}k^{2}\right)\hat{\unicode[STIX]{x1D609}}_{ij}(\boldsymbol{k})=T_{ij}^{\text{B(mag)}}(\boldsymbol{k})+\text{i}B_{l}^{0}k_{l}\left(\hat{\unicode[STIX]{x1D60F}}_{ji}^{c\ast }(\boldsymbol{k})-\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(\boldsymbol{k})\right), & \displaystyle\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+(\unicode[STIX]{x1D708}+\unicode[STIX]{x1D702})k^{2}\right)\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(\boldsymbol{k})=T_{ij}^{c}(\boldsymbol{k})+\text{i}B_{l}^{0}k_{l}\left(\hat{\unicode[STIX]{x1D619}}_{ji}(\boldsymbol{k})-\hat{\unicode[STIX]{x1D609}}_{ij}(\boldsymbol{k})\right), & \displaystyle\end{eqnarray}$$

where the time dependence has been omitted for clarity. One can note from (2.10) that in the presence of a mean magnetic field $\boldsymbol{B}^{0}$ , kinetic helicity and magnetic helicity could produce the cross-correlation $\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}$ . The nonlinear kinetic transfer $T_{ij}^{\text{V(hyd)}}$ , which is always present in hydrodynamics turbulence, is the same as in HIT, meaning

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D61B}_{ij}^{\text{V(hyd)}}(\boldsymbol{k},t)=P_{imn}(\boldsymbol{k})\int \unicode[STIX]{x1D61A}_{njm}^{uuu}(\boldsymbol{k},\boldsymbol{p},t)\,\text{d}^{3}\boldsymbol{p}+P_{jmn}(\boldsymbol{k})\int \unicode[STIX]{x1D61A}_{nim}^{uuu\ast }(\boldsymbol{k},\boldsymbol{p},t)\,\text{d}^{3}\boldsymbol{p},\end{eqnarray}$$

where $\unicode[STIX]{x1D61A}_{ijn}^{uuu}$ is the spectral triple velocity correlation

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D61A}_{ijn}^{uuu}(\boldsymbol{k},\boldsymbol{p},t)\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q})=\text{i}\langle \hat{u} _{i}(\boldsymbol{q},t)\hat{u} _{j}(\boldsymbol{k},t)\hat{u} _{n}(\boldsymbol{p},t)\rangle .\end{eqnarray}$$

The nonlinear magnetic transfer $\unicode[STIX]{x1D61B}_{ij}^{\text{V(mag)}}$ represents the retro-action of the magnetic field on the velocity one through the Laplace force, and reads

(2.13) $$\begin{eqnarray}\unicode[STIX]{x1D61B}_{ij}^{\text{V(mag)}}(\boldsymbol{k},t)=-P_{imn}(\boldsymbol{k})\int \unicode[STIX]{x1D61A}_{njm}^{ubb}(\boldsymbol{k},\boldsymbol{p},t)\,\text{d}^{3}\boldsymbol{p}-P_{jmn}(\boldsymbol{k})\int \unicode[STIX]{x1D61A}_{nim}^{ubb\ast }(\boldsymbol{k},\boldsymbol{p},t)\,\text{d}^{3}\boldsymbol{p},\end{eqnarray}$$

where $\unicode[STIX]{x1D61A}_{ijn}^{ubb}$ is the spectral velocity–magnetic–magnetic correlation defined as

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D61A}_{ijn}^{ubb}(\boldsymbol{k},\boldsymbol{p},t)\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q})=\text{i}\langle \hat{u} _{i}(\boldsymbol{q},t)\hat{b}_{j}(\boldsymbol{k},t)\hat{b}_{n}(\boldsymbol{p},t)\rangle .\end{eqnarray}$$

Then, for the magnetic equation, one has

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D61B}_{ij}^{\text{B(mag)}}(\boldsymbol{k},t)=k_{l}\int \left(\unicode[STIX]{x1D61A}_{jli}^{ubb\ast }(\boldsymbol{p},\boldsymbol{k},t)+\unicode[STIX]{x1D61A}_{ilj}^{ubb}(\boldsymbol{p},\boldsymbol{k},t)-\unicode[STIX]{x1D61A}_{lji}^{ubb\ast }(\boldsymbol{p},\boldsymbol{k},t)-\unicode[STIX]{x1D61A}_{lij}^{ubb}(\boldsymbol{p},\boldsymbol{k},t)\right)\text{d}^{3}\boldsymbol{p}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Finally, for the cross-helicity, the nonlinear transfer involves two additional three-point correlations

(2.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D61B}_{ij}^{c}(\boldsymbol{k},t) & = & \displaystyle k_{l}\int \left(\unicode[STIX]{x1D61A}_{lji}^{uub\ast }(\boldsymbol{p},\boldsymbol{k},t)-\unicode[STIX]{x1D61A}_{jli}^{uub\ast }(\boldsymbol{p},\boldsymbol{k},t)\right)\text{d}^{3}\boldsymbol{p}\nonumber\\ \displaystyle & & \displaystyle +\,P_{imn}(\boldsymbol{k})\int \left(\unicode[STIX]{x1D61A}_{njm}^{uub}(\boldsymbol{k},\boldsymbol{p},t)-\unicode[STIX]{x1D61A}_{njm}^{bbb}(\boldsymbol{k},\boldsymbol{p},t)\right)\text{d}^{3}\boldsymbol{p},\end{eqnarray}$$

with

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D61A}_{ijn}^{uub}(\boldsymbol{k},\boldsymbol{p},t)\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q})=\text{i}\langle \hat{u} _{i}(\boldsymbol{q},t)\hat{b}_{j}(\boldsymbol{k},t)\hat{u} _{n}(\boldsymbol{p},t)\rangle , & \displaystyle\end{eqnarray}$$

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D61A}_{ijn}^{bbb}(\boldsymbol{k},\boldsymbol{p},t)\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q})=\text{i}\langle \hat{b}_{i}(\boldsymbol{q},t)\hat{b}_{j}(\boldsymbol{k},t)\hat{b}_{n}(\boldsymbol{p},t)\rangle . & \displaystyle\end{eqnarray}$$

More details about cross-helical nonlinear transfers are given later in § 4 for imbalanced MHD.

2.2 EDQNM procedure and eddy damping

In this part, the EDQNM procedure is briefly presented, and more details can be found in Sagaut & Cambon (Reference Sagaut and Cambon2008) for the non-conducting case. The first step is to express $\unicode[STIX]{x1D61B}_{ijn}^{\text{xxx}}$ , which is the nonlinear transfer in the equation for the triple correlation $\unicode[STIX]{x1D61A}_{ijn}^{\text{xxx}}$ , as function of the previous second-order moments thanks to the quasi-normal approximation, where $x$ stands for either $u$ or $b$ . Then, an eddy-damping term is used to model the departure of the statistics from normal laws. In hydrodynamics turbulence, this yields

(2.19) $$\begin{eqnarray}\unicode[STIX]{x1D61B}_{ijn}^{\text{uuu}}(\boldsymbol{k},\boldsymbol{p})=\unicode[STIX]{x1D61B}_{ijn}^{\text{QN,uuu}}(\boldsymbol{k},\boldsymbol{p})-\left(\unicode[STIX]{x1D707}_{1}(k)+\unicode[STIX]{x1D707}_{1}(p)+\unicode[STIX]{x1D707}_{1}(q)\right)\unicode[STIX]{x1D61A}_{ijn}^{\text{uuu}}(\boldsymbol{k},\boldsymbol{p}),\end{eqnarray}$$

where $\unicode[STIX]{x1D61B}_{ijn}^{\text{QN,uuu}}$ is the quasi-normal expression of $\unicode[STIX]{x1D61B}_{ijn}^{\text{uuu}}$ , and $\unicode[STIX]{x1D707}_{1}$ is the classical eddy-damping term

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{1}(k,t)=A_{1}\sqrt{\int _{0}^{k}x^{2}E_{V}(x,t)\,\text{d}x},\quad A_{1}=0.355, & & \displaystyle\end{eqnarray}$$

with $E_{V}$ the kinetic energy spectrum. This procedure must be repeated in MHD for $\unicode[STIX]{x1D61B}_{ijn}^{\text{QN,bbb}}$ , $\unicode[STIX]{x1D61B}_{ijn}^{\text{QN,uub}}$ and $\unicode[STIX]{x1D61B}_{ijn}^{\text{QN,ubb}}$ . Using the definitions of the three-point correlations and the expressions of the nonlinear transfers (2.11)–(2.18) one gets after some algebra

(2.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D61B}_{ijn}^{\text{QN,ubb}}(\boldsymbol{k},\boldsymbol{p}) & = & \displaystyle 2P_{jpq}(\boldsymbol{k})\left[\hat{\unicode[STIX]{x1D60F}}_{pi}^{c}(\boldsymbol{q})\hat{\unicode[STIX]{x1D60F}}_{qn}^{c}(\boldsymbol{p})-\hat{\unicode[STIX]{x1D609}}_{pi}(\boldsymbol{q})\hat{\unicode[STIX]{x1D609}}_{qn}(\boldsymbol{p})\right]\nonumber\\ \displaystyle & & \displaystyle +\,p_{l}\left[\hat{\unicode[STIX]{x1D619}}_{lj}(\boldsymbol{k})B_{ni}(\boldsymbol{q})+\hat{\unicode[STIX]{x1D60F}}_{li}^{c}(\boldsymbol{q})\hat{\unicode[STIX]{x1D60F}}_{jn}^{c}(-\boldsymbol{k})-\hat{\unicode[STIX]{x1D60F}}_{ni}^{c}(\boldsymbol{q})\hat{\unicode[STIX]{x1D60F}}_{jl}^{c}(-\boldsymbol{k})-\hat{\unicode[STIX]{x1D619}}_{nj}(\boldsymbol{k})B_{li}(\boldsymbol{q})\right]\nonumber\\ \displaystyle & & \displaystyle +\,q_{l}\left[\hat{\unicode[STIX]{x1D619}}_{lj}(\boldsymbol{k})B_{in}(\boldsymbol{p})+\hat{\unicode[STIX]{x1D60F}}_{ln}^{c}(\boldsymbol{p})\hat{\unicode[STIX]{x1D60F}}_{ji}^{c}(-\boldsymbol{k})-\hat{\unicode[STIX]{x1D60F}}_{in}^{c}(\boldsymbol{p})\hat{\unicode[STIX]{x1D60F}}_{jl}^{c}(-\boldsymbol{k})-\hat{\unicode[STIX]{x1D619}}_{ij}(\boldsymbol{k})B_{ln}(\boldsymbol{p})\right],\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D61B}_{ijn}^{\text{QN,uub}}(\boldsymbol{k},\boldsymbol{p}) & = & \displaystyle 2P_{npq}(\boldsymbol{p})\left[\hat{\unicode[STIX]{x1D60F}}_{pj}^{c}(\boldsymbol{k})\hat{\unicode[STIX]{x1D619}}_{qi}(\boldsymbol{q})-\hat{\unicode[STIX]{x1D609}}_{pj}(\boldsymbol{k})\hat{\unicode[STIX]{x1D60F}}_{iq}^{c}(-\boldsymbol{q})\right]\nonumber\\ \displaystyle & & \displaystyle \!+\,2P_{ipq}(\boldsymbol{q})\left[\hat{\unicode[STIX]{x1D60F}}_{pj}^{c}(\boldsymbol{k})\hat{\unicode[STIX]{x1D619}}_{qn}(\boldsymbol{p})-\hat{\unicode[STIX]{x1D609}}_{pj}(\boldsymbol{k})\hat{\unicode[STIX]{x1D60F}}_{nq}^{c}(-\boldsymbol{p})\right]\nonumber\\ \displaystyle & & \displaystyle \!+\,k_{l}\left[\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(-\boldsymbol{q})\hat{\unicode[STIX]{x1D619}}_{ln}(\boldsymbol{p})+\hat{\unicode[STIX]{x1D60F}}_{nj}^{c}(-\boldsymbol{p})\hat{\unicode[STIX]{x1D619}}_{li}(\boldsymbol{q})-\hat{\unicode[STIX]{x1D60F}}_{nl}^{c}(-\boldsymbol{p})\hat{\unicode[STIX]{x1D619}}_{ji}(\boldsymbol{q})-\hat{\unicode[STIX]{x1D60F}}_{il}^{c}(-\boldsymbol{q})\hat{\unicode[STIX]{x1D619}}_{jn}(\boldsymbol{p})\right]\!,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.23) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D61B}_{ijn}^{\text{QN,bbb}}(\boldsymbol{k},\boldsymbol{p}) & = & \displaystyle k_{l}\left[\hat{\unicode[STIX]{x1D60F}}_{li}^{c}(\boldsymbol{q})\hat{\unicode[STIX]{x1D609}}_{jn}(\boldsymbol{p})+\hat{\unicode[STIX]{x1D60F}}_{ln}^{c}(\boldsymbol{p})\hat{\unicode[STIX]{x1D609}}_{ji}(\boldsymbol{q})-\hat{\unicode[STIX]{x1D60F}}_{ji}^{c}(\boldsymbol{q})\hat{\unicode[STIX]{x1D609}}_{ln}(\boldsymbol{p})-\hat{\unicode[STIX]{x1D60F}}_{jn}^{c}(\boldsymbol{p})\hat{\unicode[STIX]{x1D609}}_{li}(\boldsymbol{q})\right]\nonumber\\ \displaystyle & & \displaystyle +\,p_{l}\left[\hat{\unicode[STIX]{x1D60F}}_{li}^{c}(\boldsymbol{q})\hat{\unicode[STIX]{x1D609}}_{nj}(\boldsymbol{k})+\hat{\unicode[STIX]{x1D609}}_{ni}(\boldsymbol{q})\hat{\unicode[STIX]{x1D60F}}_{lj}^{c}(\boldsymbol{k})-\hat{\unicode[STIX]{x1D60F}}_{ni}^{c}(\boldsymbol{q})\hat{\unicode[STIX]{x1D609}}_{lj}(\boldsymbol{k})-\hat{\unicode[STIX]{x1D60F}}_{nj}^{c}(\boldsymbol{k})\hat{\unicode[STIX]{x1D609}}_{li}(\boldsymbol{q})\right]\nonumber\\ \displaystyle & & \displaystyle +\,q_{l}\left[\hat{\unicode[STIX]{x1D60F}}_{ln}^{c}(\boldsymbol{p})\hat{\unicode[STIX]{x1D609}}_{ij}(\boldsymbol{k})+\hat{\unicode[STIX]{x1D609}}_{in}(\boldsymbol{p})\hat{\unicode[STIX]{x1D60F}}_{lj}^{c}(\boldsymbol{k})-\hat{\unicode[STIX]{x1D60F}}_{in}^{c}(\boldsymbol{p})\hat{\unicode[STIX]{x1D609}}_{lj}(\boldsymbol{k})-\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(\boldsymbol{k})\hat{\unicode[STIX]{x1D609}}_{ln}(\boldsymbol{p})\right].\end{eqnarray}$$

These expressions are the most general one can get: only homogeneity was assumed, not isotropy. They could be simplified by using classical $\boldsymbol{p}\leftrightarrow \boldsymbol{q}$ symmetry. Hereafter, isotropic MHD turbulence is considered, so that there is no mean magnetic field $\boldsymbol{B}^{0}$ . Firstly, in this framework with mirror symmetry invariance, also called isotropic balanced MHD, the cross-correlation $\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}$ vanishes so that $\unicode[STIX]{x1D61B}_{imn}^{\text{QN,uub}}=\unicode[STIX]{x1D61B}_{imn}^{\text{QN,bbb}}=0$ , and only part of $\unicode[STIX]{x1D61B}_{imn}^{\text{QN,ubb}}$ remains. The kinetic and magnetic helicities, contained in $\hat{\unicode[STIX]{x1D619}}_{ij}$ and $\hat{\unicode[STIX]{x1D609}}_{ij}$ and defined in the following section for generality purposes, are zero as well. Non-zero cross-helicity is considered in § 4.

Moreover, as intensively discussed in Pouquet et al. (Reference Pouquet, Frisch and Léorat1976), Baerenzung et al. (Reference Baerenzung, Politano, Ponty and Pouquet2008), an additional eddy-damping term $\unicode[STIX]{x1D707}_{A}$ should be added to $\unicode[STIX]{x1D707}_{1}$ to take into account the propagation of Alfvén waves, where

(2.24) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{A}(k,t)=\sqrt{\frac{2}{3}}\,k\,\sqrt{\int _{0}^{k}E_{B}(x,t)\,\text{d}x},\end{eqnarray}$$

where $E_{B}$ is the magnetic energy spectrum. Furthermore $E_{V}$ is replaced in (2.20) by the total energy spectrum $E=E_{V}+E_{B}$ , linked to the inviscid invariant total energy. The constant in $\unicode[STIX]{x1D707}_{A}$ is chosen so that when $k\rightarrow \infty$ , $\unicode[STIX]{x1D707}_{A}$ is the Alfvén time: $\unicode[STIX]{x1D707}_{A}^{-1}=(kb_{0})^{-1}=\unicode[STIX]{x1D70F}_{A}$ , where the magnetic energy in IMHDT reads $3b_{0}^{2}/2$ .

Combining the quasi-normal approximation, the eddy-damping terms and the classical Markovianization step to ensure realizability of the spectra in HIT (Orszag Reference Orszag1970; Lesieur Reference Lesieur2008; Sagaut & Cambon Reference Sagaut and Cambon2008), it is possible to express the triple correlations as function of the second-order moments for MHD, thus permitting to close the evolution equations (2.8)–(2.10)

(2.25) $$\begin{eqnarray}\unicode[STIX]{x1D61A}_{ijn}^{\text{xxx}}(\boldsymbol{k},\boldsymbol{p},t)=\unicode[STIX]{x1D703}_{kpq}\unicode[STIX]{x1D61B}_{ijn}^{\text{QN,xxx}}(\boldsymbol{k},\boldsymbol{p},t),\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{kpq}$ is the characteristic time of the third-order correlations, containing the eddy-damping terms, according to

(2.26a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{kpq}=\frac{1-\text{e}^{-\unicode[STIX]{x1D707}_{kpq}t}}{\unicode[STIX]{x1D707}_{kpq}},\quad \unicode[STIX]{x1D707}_{kpq}=\unicode[STIX]{x1D707}_{k}+\unicode[STIX]{x1D707}_{p}+\unicode[STIX]{x1D707}_{q},\quad \unicode[STIX]{x1D707}_{k}=(\unicode[STIX]{x1D708}+\unicode[STIX]{x1D702})k^{2}+\unicode[STIX]{x1D707}_{1}(k)+\unicode[STIX]{x1D707}_{A}(k). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

One can observe in figure 1(a) that $\unicode[STIX]{x1D707}_{A}>\unicode[STIX]{x1D707}_{1}$ in the entire inertial range, so that ${\unicode[STIX]{x1D707}_{A}}^{-1}$ becomes the smallest time scale of the problem. The inertial scaling of the total energy spectrum being determined by the smallest time scale through nonlinear transfers (Lee et al. Reference Lee, Brachet, Pouquet, Mininni and Rosenberg2010), it follows that the choice of $\unicode[STIX]{x1D707}_{A}$ prescribes the inertial scaling of the total energy spectrum $E$ : within this isotropic EDQNM modelling, this amounts to $E\sim k^{-3/2}$ as revealed in § 3. This is also true in the presence of cross-helicity, see later in § 4. Note that if one does not ensure $\unicode[STIX]{x1D707}_{A}>\unicode[STIX]{x1D707}_{1}$ , realizability is not guaranteed anymore, meaning that either $E_{V}$ or $E_{B}$ can become negative. Consequently in what follows, we do not investigate whether one has $E\sim k^{-3/2}$ or $E\sim k^{-5/3}$ in IMHDT, but rather the asymptotic results one can obtain from a given inertial scaling.

Moreover, it appears in figure 1(a) that for sufficiently large $k$ , one has $\unicode[STIX]{x1D707}_{A}(k)\sim k$ , as expected (Pouquet et al. Reference Pouquet, Frisch and Léorat1976). Indeed, as previously mentioned, when $k\rightarrow \infty$ , one has $\unicode[STIX]{x1D707}_{A}\simeq kb_{0}$ since $\int _{0}^{\infty }E_{B}\,\text{d}k=3b_{0}^{2}/2$ . This further illustrates that the Alfvén time $\unicode[STIX]{x1D70F}_{A}=(kb_{0})^{-1}$ is the characteristic time of the inertial range. Also, in the inertial range, the classical eddy-damping part $\unicode[STIX]{x1D707}_{1}$ evolves as $k^{3/4}$ , which is consistent with previous arguments: from its definition (2.20), dimensional analysis yields $\unicode[STIX]{x1D707}_{1}\sim \sqrt{k^{3}E}=(\unicode[STIX]{x1D716}b_{0}k^{3})^{1/4}$ : this is different from HIT where $\unicode[STIX]{x1D707}_{1}\sim \unicode[STIX]{x1D70F}_{V}^{-1}\sim (k^{2}\unicode[STIX]{x1D716}_{V})^{1/3}$ .

Figure 1. (a) Different contributions in the eddy-damping term $\unicode[STIX]{x1D707}_{k}$ , defined in (2.26), of the characteristic time $\unicode[STIX]{x1D703}_{kpq}$ at $Re_{\unicode[STIX]{x1D706}}=6\times 10^{3}$ , along with the integral and dissipative wavenumbers $k_{L}$ and $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ , defined later on. (b) Total energy spectrum $E(k,t)$ , with initial condition (2.44) and $E_{V}(k,t=0)=0$ at $Re_{\unicode[STIX]{x1D706}}=260$ , compared to spectra obtained in Lee et al. (Reference Lee, Brachet, Pouquet, Mininni and Rosenberg2010) with various initial conditions (I6, A6, C6).

2.3 Spherically averaged equations for isotropic MHD

In isotropic MHD turbulence without a mean magnetic field, the solenoidal spectral tensors $\hat{\unicode[STIX]{x1D619}}_{ij}$ and $\hat{\unicode[STIX]{x1D609}}_{ij}$ are decomposed as

(2.27) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D619}}_{ij}(\boldsymbol{k},t)=P_{ij}(\boldsymbol{k}){\mathcal{E}}^{V}(\boldsymbol{k},t)+\text{i}\unicode[STIX]{x1D716}_{ijn}\unicode[STIX]{x1D6FC}_{n}\frac{{\mathcal{H}}^{V}(\boldsymbol{k},t)}{k}, & \displaystyle\end{eqnarray}$$

(2.28) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D609}}_{ij}(\boldsymbol{k},t)=P_{ij}(\boldsymbol{k}){\mathcal{E}}^{B}(\boldsymbol{k},t)+\text{i}\unicode[STIX]{x1D716}_{ijn}\unicode[STIX]{x1D6FC}_{n}\frac{{\mathcal{H}}^{B}(\boldsymbol{k},t)}{k}, & \displaystyle\end{eqnarray}$$

where ${\mathcal{E}}^{V}$ and ${\mathcal{E}}^{B}$ are the kinetic and magnetic energy densities, and ${\mathcal{H}}^{V}$ and ${\mathcal{H}}^{B}$ are the kinetic and ‘super’ magnetic helicity densities, given by

(2.29a,b ) $$\begin{eqnarray}{\mathcal{E}}^{V}(\boldsymbol{k},t)={\textstyle \frac{1}{2}}\hat{\unicode[STIX]{x1D619}}_{ii}(\boldsymbol{k},t),\quad {\mathcal{H}}^{V}(\boldsymbol{k},t)=-{\textstyle \frac{1}{2}}\text{i}\unicode[STIX]{x1D716}_{ijl}k_{l}\hat{\unicode[STIX]{x1D619}}_{ij}(\boldsymbol{k},t).\end{eqnarray}$$

The definitions are similar for ${\mathcal{E}}^{B}$ and ${\mathcal{H}}^{B}$ . At this point, it is of importance to precise that the magnetic helicity $H_{M}$ , which is an inviscid invariant of the MHD equations, is linked to ${\mathcal{H}}^{B}$ through

(2.30a,b ) $$\begin{eqnarray}H_{M}(t)=\frac{1}{2}\langle \boldsymbol{a}\boldsymbol{\cdot }\boldsymbol{b}\rangle =\int k^{-2}{\mathcal{H}}^{B}(\boldsymbol{k},t)\,\text{d}^{3}\boldsymbol{k},\quad \frac{1}{2}\langle \,\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{b}\rangle =\int {\mathcal{H}}^{B}(\boldsymbol{k},t)\,\text{d}^{3}\boldsymbol{k},\end{eqnarray}$$

with $\boldsymbol{j}$ the normalized current density. In the following developments, IMHDT is considered (with mirror symmetry) so that both ${\mathcal{H}}^{V}$ and ${\mathcal{H}}^{B}$ vanish. Then, the evolution equations of the energy densities ${\mathcal{E}}^{V}$ and ${\mathcal{E}}^{B}$ read

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+2\unicode[STIX]{x1D708}k^{2}\right){\mathcal{E}}^{V}(\boldsymbol{k},t)=T_{{\mathcal{E}}}^{\text{V(hyd)}}(\boldsymbol{k},t)+T_{{\mathcal{E}}}^{\text{V (mag)}}(\boldsymbol{k},t), & \displaystyle\end{eqnarray}$$

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+2\unicode[STIX]{x1D702}k^{2}\right){\mathcal{E}}^{B}(\boldsymbol{k},t)=T_{{\mathcal{E}}}^{\text{B(mag)}}(\boldsymbol{k},t), & \displaystyle\end{eqnarray}$$

where the nonlinear transfers are given by

(2.33) $$\begin{eqnarray}\displaystyle & \displaystyle T_{{\mathcal{E}}}^{\text{V(hyd)}}(\boldsymbol{k},t)=\frac{1}{2}\unicode[STIX]{x1D61B}_{ii}^{\text{V(hyd)}}(\boldsymbol{k},t)=P_{imn}(\boldsymbol{k})\int \Re \left(\unicode[STIX]{x1D61A}_{nim}^{uuu}(\boldsymbol{k},\boldsymbol{p},t)\right)\text{d}^{3}\boldsymbol{p}, & \displaystyle\end{eqnarray}$$

(2.34) $$\begin{eqnarray}\displaystyle & \displaystyle T_{{\mathcal{E}}}^{\text{V(mag)}}(\boldsymbol{k},t)=\frac{1}{2}\unicode[STIX]{x1D61B}_{ii}^{\text{V(mag)}}(\boldsymbol{k},t)=-P_{imn}(\boldsymbol{k})\int \Re \left(\unicode[STIX]{x1D61A}_{nim}^{ubb}(\boldsymbol{k},\boldsymbol{p},t)\right)\text{d}^{3}\boldsymbol{p}, & \displaystyle\end{eqnarray}$$

(2.35) $$\begin{eqnarray}\displaystyle & \displaystyle T_{{\mathcal{E}}}^{\text{B(mag)}}(\boldsymbol{k},t)=\frac{1}{2}\unicode[STIX]{x1D61B}_{ii}^{\text{B(mag)}}(\boldsymbol{k},t)=k_{l}\int \Re \left(\unicode[STIX]{x1D61A}_{ili}^{ubb}(\boldsymbol{p},\boldsymbol{k},t)-\unicode[STIX]{x1D61A}_{lii}^{ubb}(\boldsymbol{p},\boldsymbol{k},t)\right)\text{d}^{3}\boldsymbol{p}, & \displaystyle\end{eqnarray}$$

with $\Re$ denoting the real part. The final equations for isotropic MHD turbulence are obtained by spherically averaging the evolution equations (2.31)–(2.32) of ${\mathcal{E}}^{V}$ and ${\mathcal{E}}^{B}$ : the kinetic energy and magnetic energy spectra are then defined as

(2.36a,b ) $$\begin{eqnarray}E_{V}(k,t)=\int _{S_{k}}{\mathcal{E}}^{V}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k},\quad E_{B}(k,t)=\int _{S_{k}}{\mathcal{E}}^{B}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k},\end{eqnarray}$$

where $S_{k}$ is a sphere of radius $k$ . From the definitions of the kinetic and magnetic spectra and equations (2.31)–(2.32), one straightforwardly obtains

(2.37) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+2\unicode[STIX]{x1D708}k^{2}\right)E_{V}(k,t)=S^{\text{V(hyd)}}(k,t)+S^{\text{V(mag)}}(k,t)=S^{\text{V}}(k,t), & \displaystyle\end{eqnarray}$$

(2.38) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+2\unicode[STIX]{x1D702}k^{2}\right)E_{B}(k,t)=S^{\text{B(mag)}}(k,t). & \displaystyle\end{eqnarray}$$

The spherically averaged nonlinear kinetic hydrodynamic transfer is

(2.39) $$\begin{eqnarray}\displaystyle S^{\text{V(hyd)}}(k,t) & = & \displaystyle \int _{S_{k}}T_{{\mathcal{E}}}^{\text{V(hyd)}}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k}\nonumber\\ \displaystyle & = & \displaystyle 16\unicode[STIX]{x03C0}^{2}\int _{\unicode[STIX]{x1D6E5}_{k}}\unicode[STIX]{x1D703}_{kpq}k^{2}p^{2}q(xy+z^{3}){\mathcal{E}}_{0}^{V^{\prime \prime }}({\mathcal{E}}_{0}^{V^{\prime }}-{\mathcal{E}}_{0}^{V})\,\text{d}p\,\text{d}q,\end{eqnarray}$$

with $\unicode[STIX]{x1D6E5}_{k}$ the domain where $k$ , $p$ and $q$ are the lengths of the sides of the triangle formed by the triad, and $x$ , $y$ and $z$ are the cosines of the angles formed by $\boldsymbol{p}$ and $\boldsymbol{q}$ , $\boldsymbol{q}$ and $\boldsymbol{k}$ and $\boldsymbol{k}$ and $\boldsymbol{p}$ . The spherically averaged nonlinear kinetic MHD transfer reads

(2.40) $$\begin{eqnarray}\displaystyle S^{\text{V(mag)}}(k,t) & = & \displaystyle \int _{S_{k}}T_{{\mathcal{E}}}^{\text{V(mag)}}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k}\nonumber\\ \displaystyle & = & \displaystyle 16\unicode[STIX]{x03C0}^{2}\int _{\unicode[STIX]{x1D6E5}_{k}}\unicode[STIX]{x1D703}_{kpq}k^{2}p^{2}qz(1-y^{2}){\mathcal{E}}_{0}^{B^{\prime \prime }}({\mathcal{E}}_{0}^{B^{\prime }}-{\mathcal{E}}_{0}^{V})\,\text{d}p\,\text{d}q,\end{eqnarray}$$

and the spherically averaged nonlinear magnetic transfer is

(2.41) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle S^{\text{B(mag)}}(k,t)=\int _{S_{k}}T_{{\mathcal{E}}}^{\text{B(mag)}}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k}\nonumber\\ \displaystyle & & \displaystyle \displaystyle \quad =16\unicode[STIX]{x03C0}^{2}\int _{\unicode[STIX]{x1D6E5}_{k}}\unicode[STIX]{x1D703}_{kpq}k^{2}p^{2}q\left((xy+z){\mathcal{E}}_{0}^{V^{\prime \prime }}({\mathcal{E}}_{0}^{B^{\prime }}-{\mathcal{E}}_{0}^{B})+z(1-x^{2}){\mathcal{E}}_{0}^{B^{\prime \prime }}({\mathcal{E}}_{0}^{V^{\prime }}-{\mathcal{E}}_{0}^{B})\right)\text{d}p\,\text{d}q,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where ${\mathcal{E}}_{0}^{V}=E_{V}(k)/(4\unicode[STIX]{x03C0}k^{2})$ , ${\mathcal{E}}_{0}^{V^{\prime }}=E_{V}(p)/(4\unicode[STIX]{x03C0}p^{2})$ , ${\mathcal{E}}_{0}^{V^{\prime \prime }}=E_{V}(q)/(4\unicode[STIX]{x03C0}q^{2})$ and similarly for the magnetic energy spectrum. About the nonlinear transfers: $S^{\text{V(hyd)}}$ is conservative, with zero integral over $k$ , whereas only the sum $S^{\text{V(mag)}}+S^{\text{B(mag)}}$ is conservative. The expressions of the two later transfer terms are in agreement with Kraichnan & Nagarajan (Reference Kraichnan and Nagarajan1967), Zhou, Schilling & Ghosh (Reference Zhou, Schilling and Ghosh2002), and there are typo errors in Pouquet et al. (Reference Pouquet, Frisch and Léorat1976), Müller & Grappin (Reference Müller and Grappin2004), Baerenzung et al. (Reference Baerenzung, Politano, Ponty and Pouquet2008). The typos in the reference paper Pouquet et al. (Reference Pouquet, Frisch and Léorat1976) were corrected in Léorat, Pouquet & Frisch (Reference Léorat, Pouquet and Frisch1981), precisely pages 441–442 therein.

The kinetic energy and its dissipation rate are given by

(2.42a,b ) $$\begin{eqnarray}K_{V}=\frac{1}{2}\langle u_{i}u_{i}\rangle =\int _{0}^{\infty }E_{V}(k)\,\text{d}k,\quad \unicode[STIX]{x1D716}_{V}=\unicode[STIX]{x1D708}\left\langle \frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}\right\rangle =2\unicode[STIX]{x1D708}\int _{0}^{\infty }k^{2}E_{V}(k)\,\text{d}k.\end{eqnarray}$$

The magnetic quantities are obtained by changing $()_{V}\rightarrow ()_{B}$ , $u_{i}\rightarrow b_{i}$ and $\unicode[STIX]{x1D708}\rightarrow \unicode[STIX]{x1D702}$ . Since only the total energy $K=K_{V}+K_{B}$ is an inviscid invariant of MHD, we define also the total energy dissipation rate as $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D716}_{V}+\unicode[STIX]{x1D716}_{B}$ , and the integral scale as

(2.43) $$\begin{eqnarray}L=\frac{3\unicode[STIX]{x03C0}}{4K}\int _{0}^{\infty }\frac{E_{V}(k)+E_{B}(k)}{k}\,\text{d}k.\end{eqnarray}$$

Further information about the evolution equations of the total energy $K$ and total dissipation $\unicode[STIX]{x1D716}$ can be found in appendix B.1.

2.4 Numerical set-up and initial conditions

The time evolution of the kinetic and magnetic spectra $E_{V}(k,t)$ and $E_{B}(k,t)$ is obtained by solving two coupled integro-differential equations using a third-order Runge–Kutta scheme with implicit treatment of viscous terms. The wavenumber space is discretized using a logarithmic mesh $k_{i+1}=rk_{i}$ for $i=1,\ldots ,n$ , where $n$ is the total number of modes and $r=10^{1/f}$ , $f=15$ is the number of points per decade. This mesh spans from $k_{\text{min}}=10^{-4}k_{L}$ to $k_{\text{max}}=10k_{\unicode[STIX]{x1D702}}$ , where $k_{L}=1/L$ is the integral wavenumber, and $k_{\unicode[STIX]{x1D702}}=(\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}^{3})^{1/4}$ is the Kolmogorov wavenumber. It will be shown hereafter that the relevant dissipation wavenumber $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ for MHD is smaller than $k_{\unicode[STIX]{x1D702}}$ . The characteristic time used for normalization is the eddy turnover time $\unicode[STIX]{x1D70F}_{0}=K(0)/\unicode[STIX]{x1D716}(0)$ . The initial Reynolds number is varied between $5\times 10^{3}\leqslant Re_{\unicode[STIX]{x1D706}}(0)\leqslant 5\times 10^{4}$ depending on the cases considered.

The following initial condition is chosen for the magnetic energy spectrum if not mentioned otherwise

(2.44) $$\begin{eqnarray}E_{B}(k,t=0)=C_{B}k^{\unicode[STIX]{x1D70E}_{B}}\exp \left(-\frac{\unicode[STIX]{x1D70E}_{B}}{2}k^{2}\right),\end{eqnarray}$$

with $C_{B}$ so that $\int _{0}^{\infty }E_{B}(k,t=0)\,\text{d}k=1$ . The kinetic energy spectrum is initially either zero, or equal to $E_{B}$ : the following results, and notably the inertial scalings and decay laws, are not modified asymptotically by choosing different initial relative intensities between the kinetic and magnetic fields. In both cases, the infrared slope $\unicode[STIX]{x1D70E}_{B}$ of the magnetic energy spectrum is the large scale initial condition (recall that in hydrodynamics turbulence, the infrared slopes $\unicode[STIX]{x1D70E}_{V}=2$ and $\unicode[STIX]{x1D70E}_{V}=4$ for the kinetic energy spectrum correspond respectively to Saffman and Batchelor turbulence). In agreement with classical arguments given in Lesieur & Ossia (Reference Lesieur and Ossia2000), Lesieur (Reference Lesieur2008), if initially zero, $E_{V}$ has an infrared slope $\unicode[STIX]{x1D70E}_{V}=4$ once created. If $E_{V}=E_{B}$ initially, then $\unicode[STIX]{x1D70E}_{V}=\unicode[STIX]{x1D70E}_{B}$ . A particular attention has to be given to the time step. Indeed in MHD, one needs to take into account the characteristic propagation time of Alfvén waves $\unicode[STIX]{x1D70F}_{A}\sim (kb_{0})^{-1}$ . This is essential and considerably reduces the time step. Consequently, even with EDQNM, the duration of the simulations can be rather long at large Reynolds numbers if one wants to capture several turnover times, but is still much less than with DNS.

3.1 Kinetic energy and magnetic spectra $E_{V}(k,t)$ and $E_{B}(k,t)$

The phenomenological arguments given by Kraichnan (Reference Kraichnan1965) to derive the $k^{-3/2}$ isotropic inertial scaling are first briefly recalled. Unlike HIT where the local time of the cascade is $\unicode[STIX]{x1D70F}_{V}=(k^{3}E_{V})^{-1/2}$ , the characteristic transfer time $\unicode[STIX]{x1D70F}_{\text{tr}}$ of energy in MHD is longer. This MHD transfer time $\unicode[STIX]{x1D70F}_{\text{tr}}$ can be seen as the characteristic time of waves packet distortion, and is linked to the characteristic Alfvén time $\unicode[STIX]{x1D70F}_{A}\sim (kb_{0})^{-1}$ which reflects the duration of interactions between two counter-propagating Alfvén waves. Indeed, in Kraichnan’s phenomenology, the transfer of energy at a scale $l\sim k^{-1}$ results from the cumulative effects of multiple interactions and collisions between counter propagating Alfvén wave packets, which evolve along the magnetic field of scales larger than $l$ . Consequently, the transfer time in MHD turbulence is $\unicode[STIX]{x1D70F}_{\text{tr}}=\unicode[STIX]{x1D70F}_{V}^{2}/\unicode[STIX]{x1D70F}_{A}\sim lb_{0}/u^{2}$ (Pouquet Reference Pouquet1996; Gomez, Politano & Pouquet Reference Gomez, Politano and Pouquet1999; Galtier Reference Galtier2016), unlike HIT where $\unicode[STIX]{x1D70F}_{\text{tr}}=\unicode[STIX]{x1D70F}_{V}\sim l/u$ . From these considerations, one can express the total energy dissipation rate $\unicode[STIX]{x1D716}$ as the ratio of the local square velocity $u^{2}\sim (kE)$ to the transfer time according to

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D716}\sim \frac{kE(k)}{\unicode[STIX]{x1D70F}_{\text{tr}}}\sim \frac{kE(k)\unicode[STIX]{x1D70F}_{A}}{\unicode[STIX]{x1D70F}_{V}^{2}}\sim \frac{kE(k)(kb_{0})^{-1}}{(k^{3}E)^{-1}}=k^{3}b_{0}^{-1}E(k)^{2},\end{eqnarray}$$

so that one has in the inertial range

(3.2) $$\begin{eqnarray}E(k,t)=C_{\text{IK}}\sqrt{\unicode[STIX]{x1D716}b_{0}}\,k^{-3/2}.\end{eqnarray}$$

This scaling is referred to as the Iroshnikov–Kraichnan (IK) one, and this analysis also gives that $E_{V}$ and $E_{B}$ have a $k^{-3/2}$ inertial scaling like $E$ . The constant is found here to be of order $2\leqslant C_{\text{IK}}\leqslant 3$ for $10^{3}\leqslant Re_{\unicode[STIX]{x1D706}}\sim 10^{4}$ , which is higher than the Kolmogorov constant in HIT, and this feature is in agreement with Beresnyak (Reference Beresnyak2011).

Within the isotropic EDQNM framework, the $k^{-3/2}$ scaling is a consequence of the choice of the additional eddy-damping term $\unicode[STIX]{x1D707}_{A}$ defined in (2.24). This is illustrated in figure 2 for $\unicode[STIX]{x1D70E}_{B}=2$ . Note that the $k^{-3/2}$ inertial scaling also holds for $E_{+}$ and $E_{-}$ associated with the Elsässer variables $\boldsymbol{z}^{\pm }=\boldsymbol{u}\pm \boldsymbol{b}$ . The kinetic energy spectrum, initially zero in figure 2, is created within the first turnover time by $S^{\text{V(mag)}}$ , with an infrared slope $\unicode[STIX]{x1D70E}_{V}=4$ , and consequently experiences a backscatter of energy (Lesieur & Ossia Reference Lesieur and Ossia2000). Whereas at large scales the magnetic spectrum scales with $E_{B}\sim k^{2}$ . Two other cases ( $\unicode[STIX]{x1D70E}_{B}=4$ with $E_{V}=0$ ; $\unicode[STIX]{x1D70E}_{B}=2$ with $E_{V}=E_{B}$ ) presented in appendix A.2 show that the $k^{-3/2}$ inertial scaling is independent of initial conditions after a few turnover times. This notably implies that the inertial scaling does not depend on the kinetic to magnetic energy ratio $K_{V}/K_{B}$ , which only slightly varies with initial conditions (see later figure 11(d)).

Figure 2. Time evolution of the kinetic energy and magnetic energy spectra $E_{V}(k,t)$ and $E_{B}(k,t)$ for $\unicode[STIX]{x1D70E}_{B}=2$ , from $t=0$ to $t=10^{3}\unicode[STIX]{x1D70F}_{0}$ , with initially $E_{V}=0$ . (a) Kinetic energy spectrum $E_{V}(k,t)$ . (b) Magnetic energy spectrum $E_{B}(k,t)$ .

This $k^{-3/2}$ scaling for $E_{V}$ and $E_{B}$ is in agreement with numerical simulations of IMHDT (Grappin et al. Reference Grappin, Frisch, Léorat and Pouquet1982; Mininni & Pouquet Reference Mininni and Pouquet2007; Yoshimatsu Reference Yoshimatsu2012). However, still in the isotropic case, some studies report rather a Kolmogorov $k^{-5/3}$ inertial scaling, at odds with the previous IK $k^{-3/2}$ inertial scaling (Müller & Biskamp Reference Müller and Biskamp2000; Müller & Grappin Reference Müller and Grappin2004, Reference Müller and Grappin2005; Alexakis, Mininni & Pouquet Reference Alexakis, Mininni and Pouquet2005). Finally, in some cases both scalings are possible (Haugen, Brandenburg & Dobler Reference Haugen, Brandenburg and Dobler2004; Lee et al. Reference Lee, Brachet, Pouquet, Mininni and Rosenberg2010; Alexakis Reference Alexakis2013), depending on the Reynolds number, the possible forcing terms and initial parameters. The total energy spectrum $E(k,t)$ , obtained with EDQNM and the initial condition (2.44) with $E_{V}(k,t=0)=0$ , is presented in figure 1(b) at moderate Reynolds numbers, and compared to spectra obtained by Lee et al. (Reference Lee, Brachet, Pouquet, Mininni and Rosenberg2010) for different initial conditions. One can see that at $Re_{\unicode[STIX]{x1D706}}=260$ , the difference between the $k^{-5/3}$ and $k^{-3/2}$ inertial scalings is quite subtle. Above all, our spectrum $E$ is consistently contained between the different DNS results, obtained after six turnover times, thus showing some quantitative good agreement between EDQNM and DNS.

3.2 Spectral kinetic and magnetic nonlinear transfers

The emphasis is now put on the nonlinear transfers of kinetic and magnetic energies. It is firstly observed in figure 3(a) that all kinetic and magnetic spherically averaged nonlinear transfer terms bring energy from large to small scales, in agreement with Alexakis et al. (Reference Alexakis, Mininni and Pouquet2005). The fluxes, computed according to $\unicode[STIX]{x1D6F1}(k)=-\int _{0}^{k}S(p)\,\text{d}p$ , reveal several important features:

1. (i) The purely hydrodynamic isotropic kinetic flux $\unicode[STIX]{x1D6F1}^{\text{V(hyd)}}$ is conservative.

2. (ii) The magnetic part $\unicode[STIX]{x1D6F1}^{\text{V(mag)}}$ of the total kinetic flux $\unicode[STIX]{x1D6F1}^{\text{V}}=\unicode[STIX]{x1D6F1}^{\text{V(hyd)}}+\unicode[STIX]{x1D6F1}^{\text{V(mag)}}$ is not conservative, so that $\unicode[STIX]{x1D6F1}^{\text{V}}$ is not.

3. (iii) The total magnetic flux $\unicode[STIX]{x1D6F1}^{\text{(mag)}}=\unicode[STIX]{x1D6F1}^{\text{B(mag)}}+\unicode[STIX]{x1D6F1}^{\text{V(mag)}}$ is conservative, in agreement with pioneering developments in Kraichnan & Nagarajan (Reference Kraichnan and Nagarajan1967).

4. (iv) Finally, consistently with the total energy being an inviscid invariant, the total flux $\unicode[STIX]{x1D6F1}^{\text{V(tot)}}=\unicode[STIX]{x1D6F1}^{\text{V(hyd)}}+\unicode[STIX]{x1D6F1}^{\text{V(mag)}}+\unicode[STIX]{x1D6F1}^{\text{B(mag)}}$ is conservative.

One can remark in figure 3(a) that the dissipative wavenumber displayed is not the Kolmogorov wavenumber $k_{\unicode[STIX]{x1D702}}=(\unicode[STIX]{x1D716}_{V}/\unicode[STIX]{x1D708}^{3})^{1/4}$ , but its equivalent for MHD turbulence, namely $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ (Biskamp & Welter Reference Biskamp and Welter1989; Diamond & Biskamp Reference Diamond and Biskamp1990; Pouquet Reference Pouquet1996). This characteristic wavenumber is obtained by equating $(\unicode[STIX]{x1D708}{k_{\unicode[STIX]{x1D702}}^{\text{MHD}}}^{2})^{-1}$ and $\unicode[STIX]{x1D70F}_{\text{tr}}(k_{\unicode[STIX]{x1D702}}^{\text{MHD}})=\sqrt{b_{0}/\unicode[STIX]{x1D716}k_{\unicode[STIX]{x1D702}}^{\text{MHD}}}$ , so that

(3.3) $$\begin{eqnarray}k_{\unicode[STIX]{x1D702}}^{\text{MHD}}=\left(\frac{\unicode[STIX]{x1D716}}{b_{0}\unicode[STIX]{x1D708}^{2}}\right)^{1/3}.\end{eqnarray}$$

The fact that $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ is more relevant than $k_{\unicode[STIX]{x1D702}}$ in MHD turbulence is illustrated in figure 11 in appendix A.2.

Figure 3. Kinetic and magnetic normalized nonlinear transfers and fluxes, at $Re_{\unicode[STIX]{x1D706}}\simeq 10^{4}$ , for $\unicode[STIX]{x1D70E}_{B}=2$ , along with the integral and dissipative wavenumbers $k_{L}$ , and $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ . (a) Kinetic $S^{\text{V(hyd)}}+S^{\text{V(mag)}}=S^{\text{V}}$ and magnetic $S^{\text{(mag)}}=S^{\text{V(mag)}}+S^{\text{B(mag)}}$ nonlinear transfers. (b) Corresponding kinetic and magnetic fluxes.

3.3 Decay of the total energy $K(t)$

In this part, the decay of the total energy $K=K_{V}+K_{B}$ in isotropic MHD turbulence, which evolves according to $\text{d}K/\text{d}t=-\unicode[STIX]{x1D716}$ is addressed. One can define the algebraic time exponents $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FC}_{L}$ as $K\sim t^{\unicode[STIX]{x1D6FC}}$ and $L\sim t^{\unicode[STIX]{x1D6FC}_{L}}$ , where $L$ is the total integral scale, defined in (2.43). It is assumed for the developments hereafter that the kinetic and magnetic integral scales evolve similarly, which is assessed numerically. From figure 2 and the IK scaling (3.2), it was shown that in the inertial range, the total energy spectrum scales in $E=E_{V}+E_{B}\sim \sqrt{\unicode[STIX]{x1D716}b_{0}}k^{-3/2}$ . Secondly, at large scales, the infrared slope of $E$ is the one of the most energetic (smallest infrared slope) spectrum: given the initial conditions (2.44) chosen here, one has $E\sim k^{\unicode[STIX]{x1D70E}-p}$ , with $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{B}$ , and where $p$ is a backscatter parameter which accounts for strong inverse transfers in hydrodynamics Batchelor turbulence (Lesieur & Ossia Reference Lesieur and Ossia2000; Meldi & Sagaut Reference Meldi and Sagaut2013a ; Briard et al. Reference Briard, Gomez, Sagaut and Memari2015).

The prediction of the decay of the total energy in MHD turbulence was addressed in several papers (Hossain et al. Reference Hossain, Gray, Pontius, Matthaeus and Oughton1995; Pouquet Reference Pouquet1996; Galtier et al. Reference Galtier, Politano and Pouquet1997, Reference Galtier, Zienicke, Politano and Pouquet1999; Kalelkar & Pandit Reference Kalelkar and Pandit2004; Banerjee & Jedamzik Reference Banerjee and Jedamzik2004). Since in Hossain et al. (Reference Hossain, Gray, Pontius, Matthaeus and Oughton1995) the infrared slopes of the spectra are not considered, we rather focus on the other references. Notably, in Galtier et al. (Reference Galtier, Politano and Pouquet1997), the predictions are, for the total energy and the integral scale,

(3.4a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=-\frac{\unicode[STIX]{x1D70E}+1}{\unicode[STIX]{x1D70E}+2},\quad \unicode[STIX]{x1D6FC}_{L}=\frac{1}{\unicode[STIX]{x1D70E}+2}.\end{eqnarray}$$

These predictions were assessed in two-dimensional simulations in Galtier et al. (Reference Galtier, Politano and Pouquet1997) within the first 10 turnover times, and are further analysed in Galtier et al. (Reference Galtier, Zienicke, Politano and Pouquet1999). One of the conclusions of these theoretical decay exponents is that MHD turbulence decays more slowly than non-magnetic isotropic turbulence. This is nevertheless at odds with Banerjee & Jedamzik (Reference Banerjee and Jedamzik2004), where stronger decay rates for the kinetic and magnetic energies are obtained, closer to HIT predictions than (3.4).

However, two-dimensional turbulence might be more representative of strong MHD turbulence, where the cascade of energy occurs preferentially in the direction perpendicular to the mean magnetic field $\boldsymbol{B}^{0}$ , than isotropic MHD turbulence with no mean magnetic field. In this view, we propose new theoretical time exponents for the total energy and the integral scale in IMHDT. The method is similar to what is done in HIT (George Reference George1992), and extended to MHD in Pouquet (Reference Pouquet1996), Galtier et al. (Reference Galtier, Politano and Pouquet1997).

First, for a total energy spectrum scaling in $E\sim k^{\unicode[STIX]{x1D70E}-p}$ at large scales, the total energy evolves as $K\sim L^{-(\unicode[STIX]{x1D70E}-p+1)}\sim t^{-\unicode[STIX]{x1D6FC}_{L}(\unicode[STIX]{x1D70E}-p+1)}$ , with $L\sim t^{\unicode[STIX]{x1D6FC}_{L}}$ . Since $K\sim t^{\unicode[STIX]{x1D6FC}}$ , one gets

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=-(\unicode[STIX]{x1D70E}-p+1)\unicode[STIX]{x1D6FC}_{L}.\end{eqnarray}$$

This relation, also verified in HIT, is given in Galtier et al. (Reference Galtier, Politano and Pouquet1997) without the backscatter parameter $p$ . The second step is to link the decay exponent $\unicode[STIX]{x1D6FC}-1$ of the total dissipation rate $\unicode[STIX]{x1D716}$ to the expression (3.5). In HIT, one uses $\unicode[STIX]{x1D716}_{V}\sim u^{3}/L_{V}$ (George Reference George1992). For MHD turbulence, (3.1) is used instead, with $\unicode[STIX]{x1D70F}_{\text{tr}}=Lb_{0}/u^{2}$ , so that $\unicode[STIX]{x1D716}\sim u^{4}/(Lb_{0})$ . From this point, our development differs from Galtier et al. (Reference Galtier, Politano and Pouquet1997): in the latter reference, $b_{0}\sim B_{0}$ is independent of time and might reflect a background mean magnetic field. This gives $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{L}-1$ so that (3.4) is recovered.

On the contrary, we claim that in isotropic MHD turbulence without mean magnetic field, $b_{0}=\sqrt{2K_{B}/3}$ varies with time because the magnetic energies does. Furthermore, we assume that total, kinetic and magnetic energies, have the same asymptotic decay exponent $\unicode[STIX]{x1D6FC}$ , even if they have different intensities, so that $b_{0}\sim t^{\unicode[STIX]{x1D6FC}/2}$ . This provides $\unicode[STIX]{x1D6FC}_{L}=1+\unicode[STIX]{x1D6FC}/2$ . Combined with (3.5), this eventually yields our theoretical predictions for the decay of the total energy and integral scale in isotropic MHD turbulence

(3.6a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{L}=\frac{2}{\unicode[STIX]{x1D70E}-p+3},\quad \unicode[STIX]{x1D6FC}=-2\frac{\unicode[STIX]{x1D70E}-p+1}{\unicode[STIX]{x1D70E}-p+3},\quad \left\{\begin{array}{@{}l@{}}p(\unicode[STIX]{x1D70E}=4)=1/3,\quad \\ p(\unicode[STIX]{x1D70E}\leqslant 3)=0.\quad \end{array}\right.\end{eqnarray}$$

The striking feature is that these predictions are identical to the predictions for kinetic energy in HIT (Meldi & Sagaut Reference Meldi and Sagaut2013a ; Briard et al. Reference Briard, Gomez, Sagaut and Memari2015), despite the non-Kolmogorov scaling of both $E_{V}(k,t)$ and $E_{B}(k,t)$ . Only the backscatter parameter $p$ is changed for $\unicode[STIX]{x1D70E}=4$ from $p=0.55$ in HIT to $p=1/3$ in IMHDT, which is a rather small difference. These theoretical decay rates span from $-1$ for $\unicode[STIX]{x1D70E}=1$ to $-1.4$ for $\unicode[STIX]{x1D70E}=4$ : these values are closer to the three-dimensional simulations of Banerjee & Jedamzik (Reference Banerjee and Jedamzik2004) than the two-dimensional ones of Galtier et al. (Reference Galtier, Politano and Pouquet1997). Note that our prediction (3.6) is in agreement with the self-similar analysis of Campanelli (Reference Campanelli2016).

These new predictions for $K$ and $L$ are successfully assessed numerically in figure 4 for $\unicode[STIX]{x1D70E}_{B}=2$ and $\unicode[STIX]{x1D70E}_{B}=4$ . Supplementary simulations revealed that both the decay exponents of kinetic energy $\unicode[STIX]{x1D6FC}_{V}$ and magnetic energy $\unicode[STIX]{x1D6FC}_{B}$ are in agreement with (3.6) for the different initial conditions investigated in appendix A.2: in particular, if $\unicode[STIX]{x1D70E}_{B}=\unicode[STIX]{x1D70E}=2$ and $\unicode[STIX]{x1D70E}_{V}=4$ , one has still $\unicode[STIX]{x1D6FC}_{V}=\unicode[STIX]{x1D6FC}_{B}=\unicode[STIX]{x1D6FC}$ since the large scale dynamics is driven by the magnetic field. Also, even though not presented here, we obtained that (i) the decay exponent $\unicode[STIX]{x1D6FC}$ of the total energy follows the usual predictions for kinetic energy in HIT at low Reynolds numbers as well; (ii) in the saturated case where $k_{\text{min}}/k_{L}(0)=1$ , one has $K\sim t^{-2}$ , like kinetic energy in HIT (take $\unicode[STIX]{x1D70E}\rightarrow \infty$ in (3.6)); and (iii) the value of $\unicode[STIX]{x1D6FC}$ , for a given $\unicode[STIX]{x1D70E}$ , is independent of the kinetic to magnetic energy ratio $K_{V}/K_{B}$ .

Figure 4. Time exponents $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FC}_{L}$ of the total energy $K$ and integral scale $L$ for $\unicode[STIX]{x1D70E}=2$ (black) and $\unicode[STIX]{x1D70E}=4$ (grey), with initial conditions (2.44). Lines represent the present simulations, and symbols the theoretical predictions (3.6); ◯ for $\unicode[STIX]{x1D6FC}$ and $\Box$ for $\unicode[STIX]{x1D6FC}_{L}$ . (a) Total energy (——); see appendix A.2 for the decay exponent $\unicode[STIX]{x1D6FC}_{R}$ of residual energy ( $--$ ). (b) Integral scale growth.

Some additional features should be pointed out about (3.6): first, for $\unicode[STIX]{x1D70E}=1$ , one has $K\sim t^{-1}$ , a decay rate discussed in Hossain et al. (Reference Hossain, Gray, Pontius, Matthaeus and Oughton1995), Galtier et al. (Reference Galtier, Politano and Pouquet1997), and which is in agreement with a different approach (Kalelkar & Pandit (Reference Kalelkar and Pandit2004), equation (11) therein, for $q=1$ , $E\sim k^{q}$ ). Furthermore, these theoretical decay exponents are fully consistent with the IK scaling (3.2): indeed, the continuity of the total energy spectrum in $k_{L}=1/L$ gives

(3.7) $$\begin{eqnarray}k_{L}^{\unicode[STIX]{x1D70E}-p+3/2}\sim \sqrt{\unicode[STIX]{x1D716}b_{0}}\sim \sqrt{\unicode[STIX]{x1D716}}(K_{B})^{1/4},\end{eqnarray}$$

from which (3.6) directly follows.

In this section, it has been shown that the total energy decays in isotropic MHD similarly to kinetic energy in HIT, with a slight difference due to inverse nonlinear transfers only when $\unicode[STIX]{x1D70E}=4$ . This prediction, assessed numerically, is an important finding of the present work, and strongly differs from the predictions by Galtier et al. (Reference Galtier, Politano and Pouquet1997), essentially because we insisted on the fact that $b_{0}$ varies with time, which is a reasonable statement in the absence of a mean magnetic field.

4.1 Modelling of the spectral cross-correlation $\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}$

Helical decompositions of the spectral fluctuating velocity and magnetic fields are firstly considered, i.e.

(4.1a,b ) $$\begin{eqnarray}\displaystyle \hat{u} _{i}(\boldsymbol{k})=u_{+}(\boldsymbol{k})N_{i}(\boldsymbol{k})+u_{-}(\boldsymbol{k})N_{i}^{\ast }(\boldsymbol{k}),\quad \hat{b}_{i}(\boldsymbol{k})=b_{+}(\boldsymbol{k})N_{i}(\boldsymbol{k})+b_{-}(\boldsymbol{k})N_{i}^{\ast }(\boldsymbol{k}), & & \displaystyle\end{eqnarray}$$

where $N_{i}$ are the helical modes (Cambon & Jacquin Reference Cambon and Jacquin1989) which verify $k_{i}N_{i}=0$ , $N_{i}N_{i}=0$ and $N_{i}(-\boldsymbol{k})=N_{i}^{\ast }(\boldsymbol{k})$ . Because of Hermitian symmetry for $\hat{u} _{i}$ and $\hat{b}_{i}$ , namely $\hat{u} _{i}^{\ast }(\boldsymbol{k})=\hat{u} _{i}(-\boldsymbol{k})$ and $\hat{b}_{i}^{\ast }(\boldsymbol{k})=\hat{b}_{i}(-\boldsymbol{k})$ , this property is verified as well for the $\pm$ components, $u_{\pm }^{\ast }(\boldsymbol{k})=u_{\pm }(-\boldsymbol{k})$ and $b_{\pm }^{\ast }(\boldsymbol{k})=b_{\pm }(-\boldsymbol{k})$ . Using these decompositions, it directly follows that

(4.2) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(\boldsymbol{k}) & = & \displaystyle {\mathcal{H}}_{c}(\boldsymbol{k})P_{ij}(\boldsymbol{k})+\text{i}\unicode[STIX]{x1D716}_{ijn}\unicode[STIX]{x1D6FC}_{n}F^{\text{EM}}(\boldsymbol{k})\nonumber\\ \displaystyle & & \displaystyle +\,\langle u_{+}^{\ast }(\boldsymbol{k})b_{-}(\boldsymbol{k})\rangle N_{i}^{\ast }(\boldsymbol{k})N_{j}^{\ast }(\boldsymbol{k})+\langle u_{-}^{\ast }(\boldsymbol{k})b_{+}(\boldsymbol{k})\rangle N_{i}(\boldsymbol{k})N_{j}(\boldsymbol{k}),\end{eqnarray}$$

where the time dependence has been omitted for clarity. The cross-helicity density ${\mathcal{H}}_{c}$ is half the trace of the spectral cross-correlation

(4.3) $$\begin{eqnarray}{\mathcal{H}}_{c}(\boldsymbol{k},t)={\textstyle \frac{1}{2}}\hat{\unicode[STIX]{x1D60F}}_{ii}^{c}(\boldsymbol{k},t)=\langle u_{+}^{\ast }b_{+}\rangle +\langle u_{-}^{\ast }b_{-}\rangle ,\end{eqnarray}$$

is real, and verifies ${\mathcal{H}}_{c}^{\ast }(\boldsymbol{k})={\mathcal{H}}_{c}(-\boldsymbol{k})$ . The cross-helicity $K_{c}$ is linked to the cross-helical spectrum $H_{c}$ through

(4.4) $$\begin{eqnarray}K_{c}(t)=\frac{1}{2}\langle \boldsymbol{u}.\boldsymbol{b}\rangle =\int _{0}^{\infty }\int _{S_{k}}{\mathcal{H}}_{c}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k}\,\text{d}k=\int _{0}^{\infty }H_{c}(k,t)\,\text{d}k.\end{eqnarray}$$

A non-zero cross-helicity breaks the mirror symmetry of the flow since $\boldsymbol{b}$ is a pseudo-vector, consistently with the decomposition of Frisch et al. (Reference Frisch, Pouquet, Léorat and Mazure1975). Nevertheless, it is possible to consider cross-helicity without magnetic helicity and kinetic helicity.

The second right-hand side term of (4.2) can be non-zero in non-helical turbulence. This term $F^{\text{EM}}$ is linked to the cross-correlation through

(4.5) $$\begin{eqnarray}F^{\text{EM}}(\boldsymbol{k},t)=\langle u_{+}^{\ast }b_{+}\rangle -\langle u_{-}^{\ast }b_{-}\rangle =-{\textstyle \frac{1}{2}}\text{i}\unicode[STIX]{x1D716}_{ijl}\unicode[STIX]{x1D6FC}_{l}\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(\boldsymbol{k},t).\end{eqnarray}$$

The quantity $F^{\text{EM}}$ verifies $F^{\text{EM *}}(\boldsymbol{k})=F^{\text{EM}}(-\boldsymbol{k})$ . It follows that $F^{\text{EM}}$ is linked to the electromotive force through

(4.6) $$\begin{eqnarray}\langle \boldsymbol{u}\times \boldsymbol{b}\rangle =2\text{i}\int _{0}^{\infty }\int _{S_{k}}\frac{\boldsymbol{k}}{k}F^{\text{EM}}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k}\,\text{d}k=\int _{0}^{\infty }\int _{S_{k}}\unicode[STIX]{x1D716}_{pql}\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6FC}_{l}\hat{\unicode[STIX]{x1D60F}}_{pq}^{c}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k}\,\text{d}k.\end{eqnarray}$$

However, the electromotive force is not considered here in the framework of isotropic MHD turbulence: indeed it would create a mean magnetic field, and thus break the isotropy of the flow, according to

(4.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{B}^{0}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times \langle \boldsymbol{u}\times \boldsymbol{b}\rangle +\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{B}^{0}.\end{eqnarray}$$

The two last right-hand side terms of (4.2) might be non-zero in complex flows such as rotating or sheared MHD turbulence, and may be interpreted as a mixed kinetic-magnetic polarization. But they always vanish in isotropic MHD turbulence with or without helicity, so that they are discarded from this point. Consequently, the decomposition is similar to the one proposed in Frisch et al. (Reference Frisch, Pouquet, Léorat and Mazure1975), and one has

(4.8) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}(\boldsymbol{k})=\hat{\unicode[STIX]{x1D60F}}_{ij}^{c\ast }(-\boldsymbol{k})={\mathcal{H}}_{c}(\boldsymbol{k})P_{ij}(\boldsymbol{k})+\text{i}\unicode[STIX]{x1D716}_{ijn}\unicode[STIX]{x1D6FC}_{n}F^{\text{EM}}(\boldsymbol{k}).\end{eqnarray}$$

In what follows, the spherically averaged equation for the cross-helical spectrum is derived, along with its corresponding spherically averaged nonlinear transfer, and with the additional kinetic and magnetic transfer terms resulting from the presence of $H_{c}(k,t)$ .

4.2 Equation for the cross-helical spectrum $H_{c}(k,t)$

In this part, we derive the evolution equation of the cross-helical spectrum $H_{c}(k,t)$ in a manner analogous to what was done in § 2 for $E_{V}(k,t)$ and $E_{B}(k,t)$ . Starting from (2.10), and using the modelling of the cross-correlation studied previously, which amounts to $\hat{\unicode[STIX]{x1D60F}}_{ij}^{c}={\mathcal{H}}_{c}P_{ij}$ in IMHDT, yields

(4.9) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+(\unicode[STIX]{x1D708}+\unicode[STIX]{x1D702})k^{2}\right){\mathcal{H}}_{c}(\boldsymbol{k},t)=\frac{1}{2}\unicode[STIX]{x1D61B}_{ii}^{c}(\boldsymbol{k},t).\end{eqnarray}$$

Then, the EDQNM procedure is applied, and the quasi-normal expressions (2.22) and (2.23) are used to compute explicitly $\unicode[STIX]{x1D61B}_{ii}^{c}$ from (2.16). After classical algebra, this gives the final equation for the cross-helical spectrum

(4.10) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+(\unicode[STIX]{x1D708}+\unicode[STIX]{x1D702})k^{2}\right)H_{c}(k,t)=S^{c}(k,t),\end{eqnarray}$$

with $S^{c}$ the spherically averaged cross-helical nonlinear transfer

(4.11) $$\begin{eqnarray}\displaystyle S^{c}(k,t) & = & \displaystyle \frac{1}{2}\int _{S_{k}}\unicode[STIX]{x1D61B}_{ii}^{c}(\boldsymbol{k},t)\text{d}^{2}\boldsymbol{k}\nonumber\\ \displaystyle & = & \displaystyle 8\unicode[STIX]{x03C0}^{2}\int _{\unicode[STIX]{x1D6E5}_{k}}\unicode[STIX]{x1D703}_{kpq}k^{2}pq\left[2k(y^{2}-z^{2}){\mathcal{H}}_{c}^{\prime \prime }({\mathcal{E}}_{0}^{V^{\prime }}+{\mathcal{E}}_{0}^{B^{\prime }})+p\left((xy+z^{3})({\mathcal{H}}_{c}^{\prime \prime }{\mathcal{E}}_{0}^{B}-{\mathcal{H}}_{c}{\mathcal{E}}_{0}^{V})\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\,(xy+z)({\mathcal{H}}_{c}^{\prime \prime }{\mathcal{E}}_{0}^{V}-{\mathcal{H}}_{c}{\mathcal{E}}_{0}^{V^{\prime \prime }})+z(1-y^{2})({\mathcal{H}}_{c}^{\prime \prime }{\mathcal{E}}_{0}^{B}-{\mathcal{H}}_{c}{\mathcal{E}}_{0}^{B^{\prime \prime }})\nonumber\\ \displaystyle & & \displaystyle +\,\left.\left.z(1-x^{2})({\mathcal{H}}_{c}^{\prime \prime }{\mathcal{E}}_{0}^{V}-{\mathcal{H}}_{c}{\mathcal{E}}_{0}^{B^{\prime \prime }})\right)\right]\text{d}p\,\text{d}q,\end{eqnarray}$$

with ${\mathcal{H}}_{c}=H_{c}(k)/(4\unicode[STIX]{x03C0}k^{2})$ , ${\mathcal{H}}_{c}^{\prime }=H_{c}(p)/(4\unicode[STIX]{x03C0}p^{2})$ and ${\mathcal{H}}_{c}^{\prime \prime }=H_{c}(q)/(4\unicode[STIX]{x03C0}q^{2})$ . Since the cross-helicity $K_{c}(t)$ is an inviscid invariant of isotropic MHD turbulence, its nonlinear transfer $S^{c}$ is conservative, in agreement with the evolution equation

(4.12a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}K_{c}}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D716}_{c}(t),\quad \unicode[STIX]{x1D716}_{c}(t)=\frac{1}{2}(\unicode[STIX]{x1D708}+\unicode[STIX]{x1D702})\left\langle \frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}\frac{\unicode[STIX]{x2202}b_{i}}{\unicode[STIX]{x2202}x_{j}}\right\rangle ,\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{c}$ is the cross-helicity dissipation rate. Furthermore, according to the quasi-normal expression (2.21), the cross-helical spectrum also impacts the magnetic nonlinear transfers $S^{\text{V(mag)}}$ and $S^{\text{B(mag)}}$ . Here are the additional contributions created by a non-zero cross-helicity in the evolution equations of $E_{V}(k,t)$ and $E_{B}(k,t)$ :

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle S^{\text{V(cross)}}(k,t)=16\unicode[STIX]{x03C0}^{2}\int _{\unicode[STIX]{x1D6E5}_{k}}\unicode[STIX]{x1D703}_{kpq}k^{2}p^{2}q{\mathcal{H}}_{c}^{\prime \prime }\left[z(1-y^{2}){\mathcal{H}}_{c}-(xy+z^{3}){\mathcal{H}}_{c}^{\prime }\right]\text{d}p\,\text{d}q, & \displaystyle\end{eqnarray}$$

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle S^{\text{B(cross)}}(k,t)=16\unicode[STIX]{x03C0}^{2}\int _{\unicode[STIX]{x1D6E5}_{k}}\unicode[STIX]{x1D703}_{kpq}k^{3}pq(1-y^{2}+z^{2}+xyz){\mathcal{H}}_{c}^{\prime \prime }({\mathcal{H}}_{c}-{\mathcal{H}}_{c}^{\prime })\,\text{d}p\,\text{d}q. & \displaystyle\end{eqnarray}$$

Both $S^{\text{V(cross)}}$ and $S^{\text{B(cross)}}$ are not conservative separately, but the sum has zero integral over the whole wavenumber space. From now on, we write $S^{\text{B}}=S^{\text{B(mag)}}+S^{\text{B(cross)}}$ and $S^{\text{V}}=S^{\text{V(hyd)}}+S^{\text{V(mag)}}+S^{\text{V(cross)}}$ . These explicit expressions for the three nonlinear transfers involving the cross-helical spectrum are new. The different contributions of the cross-helical, kinetic and magnetic spectra can be disentangled easily, and this does not prevent us from analysing afterwards $E_{+}$ and $E_{-}$ , linked to the Elsässer variables $\boldsymbol{z}^{\pm }=\boldsymbol{u}\pm \boldsymbol{b}$ . On the contrary, in Grappin et al. (Reference Grappin, Frisch, Léorat and Pouquet1982, Reference Grappin, Pouquet and Léorat1983), the cross-helical transfer terms must be deduced from those associated with $E_{+}$ and $E_{-}$ , with $H_{c}=(E_{+}-E_{-})/4$ .

At each time and each wavenumber $k$ , the cross-helical spectrum must satisfy the realizability condition given in Frisch et al. (Reference Frisch, Pouquet, Léorat and Mazure1975), namely

(4.15) $$\begin{eqnarray}H_{c}(k,t)\leqslant \sqrt{E_{V}(k,t)E_{B}(k,t)}.\end{eqnarray}$$

For the numerical simulations of isotropic imbalanced MHD, we choose $E_{V}(k,t=0)=E_{B}(k,t=0)$ given by (2.44), with an infrared slope $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{B}=\unicode[STIX]{x1D70E}_{V}$ which is either  $\unicode[STIX]{x1D70E}=2$ or $\unicode[STIX]{x1D70E}=4$ . To ensure that (4.15) is always verified, we choose $H_{c}(k,t=0)=\sqrt{E_{V}(k,t=0)E_{B}(k,t=0)}/c_{\text{cond}}$ with $c_{\text{cond}}=10$ : for smaller values of $c_{\text{cond}}$ down to unity, the realizability condition can be violated at large scales after a few turnover times.

It is worth noting that the complete spectral evolution equations of $E_{V}$ , $E_{B}$ and $H_{c}$ are entirely solved numerically unlike in Grappin et al. (Reference Grappin, Frisch, Léorat and Pouquet1982). In the latter reference, reduced equations are obtained with the assumption that equipartition between kinetic and magnetic energy is reached, by imposing a strong mean magnetic field with random orientation to remain statistically isotropic.

In what follows, four different features are addressed in imbalanced IMHDT: the nonlinear transfers, the decay of cross-helicity, the spectral scaling of the main spectra with notably the impact of moderate Reynolds numbers, and finally pressure spectra.

4.3 Spectral nonlinear transfers in imbalanced MHD

In this section, we investigate the nonlinear spectral transfers of kinetic energy, magnetic energy and cross-helicity, and their associated fluxes. It is recalled that both $K=K_{V}+K_{B}$ and $K_{c}$ are inviscid invariants of MHD turbulence. To visualize the imbalance of MHD turbulence due to a non-zero cross-correlation $\langle u_{i}b_{i}\rangle$ , it is convenient to work with the Elsässer spectra

(4.16a,b ) $$\begin{eqnarray}E_{\pm }(k,t)=E_{V}(k,t)+E_{B}(k,t)\pm 2H_{c}(k,t),\quad K_{\pm }=\int _{0}^{\infty }E_{\pm }(k,t)\,\text{d}k,\end{eqnarray}$$

whose integrals over the whole wavenumber space, which we call the Elsässer energies $K_{\pm }$ , are inviscid invariants of MHD turbulence as well (in the balanced case, one has $E_{+}=E_{-}=E=E_{V}+E_{B}$ ).

The different fluxes, normalized by the total dissipation rate $\unicode[STIX]{x1D716}$ , are first presented in figure 5(a). The Elsässer, cross-helical and total fluxes $\unicode[STIX]{x1D6F1}_{+}$ , $\unicode[STIX]{x1D6F1}_{-}$ , $\unicode[STIX]{x1D6F1}^{c}$ , and $\unicode[STIX]{x1D6F1}^{\text{V}}+\unicode[STIX]{x1D6F1}^{\text{B}}$ , linked to the inviscid invariants $K_{+}$ , $K_{-}$ , $K_{c}$ and $K$ , are consistently conservative. The + Elsässer flux in slightly more intense than the $-$ one. Moreover, the Elsässer and total fluxes are much stronger than the cross-helical one, which was multiplied by 10 for readability reasons. One can remark that around the integral wavenumber $k_{L}$ , the cross-helical flux is negative, meaning that part of the cross-helicity is transferred towards large scales. This is not a transient effect, and remains rather subdominant since the main part of the cross-helical flux is positive and constant in the inertial range. This could also be a signature of inviscid invariants not positive–definite: indeed, a similar feature was observed for kinetic helicity in HIT (Briard & Gomez Reference Briard and Gomez2017).

Figure 5. Nonlinear transfers and fluxes, normalized by $\unicode[STIX]{x1D716}$ , at $Re_{\unicode[STIX]{x1D706}}(t=30\unicode[STIX]{x1D70F}_{0})=4\times 10^{4}$ for $\unicode[STIX]{x1D70E}=2$ , along with the integral and dissipative wavenumbers $k_{L}$ and $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ . For better readability, the cross-helical transfer and flux were multiplied by $10$ . (a) Cross-helical flux $\unicode[STIX]{x1D6F1}^{c}$ , Elsässer fluxes $\unicode[STIX]{x1D6F1}_{+}$ and $\unicode[STIX]{x1D6F1}_{-}$ and total energy flux $\unicode[STIX]{x1D6F1}^{\text{V}}+\unicode[STIX]{x1D6F1}^{\text{B}}$ . (b) Kinetic energy transfers (black): $S^{\text{V(hyd)}}$ , $S^{\text{V(mag)}}$ and $S^{\text{V(cross)}}$ ; Magnetic energy transfers (grey): $S^{\text{B(mag)}}$ , and $S^{\text{B(cross)}}$ . (c) Total transfer $S^{\text{V}}+S^{\text{B}}$ (black) and cross-helical transfer $S^{c}$ (grey); arrows indicate the direction of the transfers.

This is better illustrated in figure 5(b), where the five kinetic and magnetic nonlinear spherically averaged transfers are displayed, namely $S^{\text{V(hyd)}}$ , $S^{\text{V(mag)}}$ , $S^{\text{V(cross)}}$ , $S^{\text{B(mag)}}$ and $S^{\text{B(cross)}}$ . These transfer terms can be decomposed into two groups: the ones already present in balanced MHD, $S^{\text{V(hyd)}}$ , $S^{\text{V(mag)}}$ and $S^{\text{B(mag)}}$ , which are direct transfers from large to small scales. Whereas the terms arising from the presence of cross-helicity, $S^{\text{V(cross)}}$ and $S^{\text{B(cross)}}$ , are inverse transfers stronger at large scales, and as intense as the others.

Furthermore, in figure 5(c), the total and cross-helical nonlinear transfers $S^{\text{V}}+S^{\text{B}}$ and $S^{c}$ are presented. The transfer of total energy is much more intense than the one of cross-helicity: total energy is taken from a large range of scales and brought to smaller scales close to $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ . Whereas $S^{c}$ is much less intense and transfers cross-helicity in a more complex way: indeed, most of the cross-helicity is taken from a narrow band of large scales to a wide range of smaller scales. But in addition, large scales cross-helicity is also brought at scales bigger than $k_{L}^{-1}$ , illustrating some strong inverse transfer mechanisms. Moreover, one can observe that around $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ , cross-helicity is also slightly transferred to larger scales. These different directions of transfers are sketched with arrows in figure 5(c).

4.4 Decay of cross-helicity

The impact of a non-zero initial cross-helicity on the decay is illustrated in figure 6(a): it is shown that with a sufficient initial velocity–magnetic correlation, here for $H_{c}(k,t=0)=0.5\sqrt{E_{V}(k,t=0)E_{B}(k,t=0)}$ , the decay of the total energy $K$ is slightly delayed. If the amount of initial cross-helicity is too weak, there is no visible effect on $K$ . Note that this kind of initial delay is analogous to what was observed in HIT with kinetic helicity (André & Lesieur Reference André and Lesieur1977; Briard & Gomez Reference Briard and Gomez2017). Furthermore, during the decay of $K_{V}$ , $K_{B}$ and $K_{c}$ , the normalized correlation $\unicode[STIX]{x1D70C}_{c}=K_{c}/\sqrt{K_{V}K_{B}}$ , linked to the averaged cosine of the angle between $\boldsymbol{u}$ and $\boldsymbol{b}$ , increases linearly time, from the initial value to unity, meaning toward alignment in the same direction of the velocity and magnetic fields.

The asymptotic decay of cross-helicity is finally briefly discussed. As mentioned above, because of the slow increase of the normalized correlation $\unicode[STIX]{x1D70C}_{c}$ , the realizability condition (4.15) is eventually broken for large times. In other words, this means that long-time decay with several hundreds of turnover times, as in figure 4, cannot be easily performed. Note that this is not a limitation for DNS because in practice only a few turnover times are computed. We nevertheless propose some quantitative information about the decay exponent $\unicode[STIX]{x1D6FC}_{c}$ of cross-helicity, by choosing $K_{c}(t=0)=10^{-2}$ , so that a few hundred turnover times occur with the realizability condition satisfied.

Three conclusions can be drawn from figure 6(b): (i) the decay of cross-helicity is very slow compared to the decay of total energy; (ii) the decay exponent $\unicode[STIX]{x1D6FC}_{c}$ of $K_{c}$ seems to be independent of $\unicode[STIX]{x1D70E}$ , since one has $\unicode[STIX]{x1D6FC}_{c}\simeq -0.2$ for both $\unicode[STIX]{x1D70E}=2$ and $\unicode[STIX]{x1D70E}=4$ ; and (iii) the presence of cross-helicity does not modify the decay exponent of the total energy, since the previous theoretical predictions (3.6) are recovered: see inset of figure 6(b) where $\unicode[STIX]{x1D6FC}=-1.2$ for $\unicode[STIX]{x1D70E}=2$ and $\unicode[STIX]{x1D6FC}=-1.4$ for $\unicode[STIX]{x1D70E}=4$ . It follows from these three features that the decay of cross-helicity in isotropic MHD is quite different from the decay of kinetic helicity in isotropic hydrodynamic turbulence (Briard & Gomez Reference Briard and Gomez2017).

Figure 6. Decay of total energy and cross-helicity. (a) Impact of initial cross-helicity $K_{c}(t=0)$ on the early decay of the total energy $K$ compared to the balanced case (in grey) for $\unicode[STIX]{x1D70E}=2$ . (b) Long-time decay of cross-helicity with $K_{c}(t=0)=10^{-2}$ , for $\unicode[STIX]{x1D70E}=2$ (black) and $\unicode[STIX]{x1D70E}=4$ (grey); the inset represents the decay of total energy during the same simulation.

4.5 The inertial scaling of the cross-helical spectrum $H_{c}(k,t)$

In this part, the emphasis is put on the inertial scaling of the cross-helical spectrum $H_{c}(k,t)$ . First, the total energy and cross-helical spectra $E(k,t)$ and $H_{c}(k,t)$ are presented in figure 7(a) for $\unicode[STIX]{x1D70E}=2$ . The inertial scaling of $E$ remains $k^{-3/2}$ , similarly to the balanced case: even though not presented, $E_{V}$ and $E_{B}$ still scale in $k^{-3/2}$ despite the presence of cross-helicity. The effects of the cross-helicity are more visible on the Elsässer spectra which are discussed in the next section.

The cross-helical spectrum itself scales with a slope close to $k^{-5/3}$ , and is negative at small scales, similarly to what was obtained for the kinetic helicity in skew–isotropic turbulence (Briard & Gomez Reference Briard and Gomez2017). One could conclude from these observations, at least within the EDQNM framework, that spectra of quantities not positive–definite, such as $\langle \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D74E}\rangle$ and $\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{b}\rangle$ , exhibit negative values at small scales, if $\langle \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D74E}\rangle$ and $\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{b}\rangle$ are positive initially.

Figure 7. Cross-helical spectra for imbalanced MHD for $\unicode[STIX]{x1D70E}=2$ . (a) Total energy and cross-helical spectra $E(k,t)$ and $H_{c}(k,t)$ at $Re_{\unicode[STIX]{x1D706}}(t=30\unicode[STIX]{x1D70F}_{0})=4\times 10^{4}$ along with the integral and dissipative wavenumbers $k_{L}$ and $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ . (b) Spectral slope of $H_{c}$ , given by $\log H_{c}/\log k$ for different $Re_{\unicode[STIX]{x1D706}}$ : the $-5/3$ and $-2$ slopes are indicated in grey; Inset: focus on the inertial range.

The present $k^{-5/3}$ scaling is different from the $k^{-2}$ one of Grappin et al. (Reference Grappin, Frisch, Léorat and Pouquet1982): there are at least two arguments to explain the difference, in favour of the present $k^{-5/3}$ inertial range. Firstly, in Grappin et al. (Reference Grappin, Frisch, Léorat and Pouquet1982), as already explained, simplified EDQNM equations are solved numerically, with the assumption that there is a strong uniform magnetic field of random orientation that causes equipartition of kinetic and magnetic energies, i.e. $E_{V}=E_{B}$ at all scales. The difference between the present $k^{-5/3}$ and $k^{-2}$ scalings could thus result from this assumption, but this is rather difficult to quantify. The second argument is a strong Reynolds number effect. In Grappin et al. (Reference Grappin, Frisch, Léorat and Pouquet1982), the Reynolds number is lower than in our case, where $Re_{\unicode[STIX]{x1D706}}=4\times 10^{4}$ , corresponding to almost two additional decades in wavenumber space. To better illustrate this feature, the slope of $H_{c}$ for different $Re_{\unicode[STIX]{x1D706}}$ is presented in figure 7(b): there is a clear influence of the Reynolds number on the spectral slope of the cross-helical spectrum. Indeed, for moderate $Re_{\unicode[STIX]{x1D706}}\sim 250$ , the slope of $H_{c}$ is close to $k^{-2}$ , whereas very large Reynolds numbers are needed to obtain a steady result, close to $k^{-5/3}$ here. Nevertheless, the tendency of our results is in agreement with Grappin et al. (Reference Grappin, Frisch, Léorat and Pouquet1982) regarding the spectral slope of $H_{c}$ , in a sense that it is steeper than the one of $E$ .

Finally, one can remark that the $-5/3$ slope of the cross-helical spectrum is similar to the one of the kinetic helicity spectrum in HIT, framework in which kinetic helicity is an inviscid invariant, unlike in MHD. If we go further assuming the $k^{-5/3}$ dependence, dimensional analysis provides (i) $H_{c}(k)\sim \unicode[STIX]{x1D716}_{c}^{a}\unicode[STIX]{x1D716}^{b}k^{-5/3}$ with $a+b=2/3$ , and $a\neq 0$ since spectra of inviscid invariants are expressed in general as function of their dissipation rate, and (ii) no dependence on $b_{0}$ . The latter somehow means that the dynamics of $H_{c}$ becomes much less dependent on the Alfvèn time $\unicode[STIX]{x1D70F}_{A}=(kb_{0})^{-1}$ , which is consistent with the cross-helicity reducing initially the nonlinear transfers, as shown in figure 6(a). Compensating $H_{c}$ with the different possibilities for $a$ and $b$ indicates that a relevant scaling could be

(4.17) $$\begin{eqnarray}H_{c}(k,t)=C_{c}\,(\unicode[STIX]{x1D716}\unicode[STIX]{x1D716}_{c})^{1/3}\,k^{-5/3},\end{eqnarray}$$

which gives $C_{c}\simeq 3$ , similar to the value of $C_{\text{IK}}$ for $E$ . Choosing a ‘linear’ dependency on $\unicode[STIX]{x1D716}_{c}$ ( $a=1$ and $b=-1/3$ ) yields a way too large constant compared to other usual constants ( $C_{c}\geqslant 30$ ): the fact that $H_{c}$ should not depend linearly on $\unicode[STIX]{x1D716}_{c}$ is consistent with some arguments proposed by Grappin et al. (Reference Grappin, Frisch, Léorat and Pouquet1982).

Now that the inertial scaling of $H_{c}$ has been addressed and the impact of moderate Reynolds numbers highlighted, the effects of cross-helicity on the Elsässer spectra and on the short-time dynamics are analysed.

4.6 $E_{+}$ , $E_{-}$ and the short-time dynamics

It has been shown in figure 6(a) that a sufficient amount of initial cross-helicity $K_{c}(t=0)$ could slightly delay the initial decay of total energy $K$ , a feature similar to what is obtained in HIT with initial kinetic helicity (André & Lesieur Reference André and Lesieur1977). Furthermore, it was revealed in the previous section that the presence of cross-helicity was not changing the inertial scaling of the kinetic, magnetic and total spectra $E_{V}$ , $E_{B}$ and $E=E_{V}+E_{B}$ . Therefore, it is proposed in this part to briefly illustrate the impact of cross-helicity on the inertial scalings of the Elsässer spectra $E_{+}$ and $E_{-}$ defined in (4.16), and on the short-time dynamics as well.

One can observe in figure 8(a) that adding a greater amount of initial cross-helicity tends to increase the kinetic to magnetic energy ratio $K_{V}/K_{B}$ , along with the positive Elsässer energy $K_{+}$ , the latter being expected from the definition (4.16). Some justification are proposed hereafter to justify the increase of kinetic energy by adding cross-helicity in the flow. Regarding the asymptotic value of $K_{V}/K_{B}$ for the balanced case, it is worth noting that it is rather independent of the Reynolds number and only slightly depends on the initial conditions (see figure 11(d) in appendix A.2 for more details).

Figure 8. (a) Impact of the initial cross-helicity $K_{c}(t=0)$ on the short-time dynamics, for $\unicode[STIX]{x1D70E}=2$ and $E_{V}(k,t=0)=E_{B}(k,t=0)$ , on the ratios $K_{V}/K_{B}$ (black) and $K_{-}/K_{+}$ (grey). (b) Elsässer spectra $E_{+}$ and $E_{-}$ , for $\unicode[STIX]{x1D70E}=2$ and $Re_{\unicode[STIX]{x1D706}}(t=30\unicode[STIX]{x1D70F}_{0})=2\times 10^{3}$ , with initial condition $H_{c}(k,t=0)=0.1\sqrt{E_{V}(k,t=0)E_{B}(k,t=0)}$ ; The inset gives the slope $\log E_{\pm }/\log k$ in the inertial range.

The Elsässer spectra $E_{+}$ and $E_{-}$ are presented in figure 8(b). Unlike for $E_{V}$ , $E_{B}$ and $E$ , a slight departure is observed for their inertial scaling with respect to the IK prediction in $k^{-3/2}$ : the spectral slope of $E_{+}$ is steeper, between $k^{-1.55}$ and $k^{-5/3}$ , whereas the opposite happens for $E_{-}$ , where the slope is approximately between $k^{-1.2}$ and $k^{-4/3}$ . It is rather difficult to determine the exact scalings of $E_{+}$ and $E_{-}$ since the inertial slopes slightly vary with both $K_{c}(t=0)$ and the Reynolds number. Nevertheless, the finding that the slope of $E_{+}$ becomes steeper than $-3/2$ and the one of $E_{-}$ less steep is a robust feature, already obtained in Grappin et al. (Reference Grappin, Frisch, Léorat and Pouquet1982). In fact, even though equipartition is not assumed here unlike the latter reference, the inertial spectral slopes obtained for $E_{+}\sim k^{-m_{+}}$ and $E_{-}\sim k^{-m_{-}}$ are in agreement with the prediction of Grappin and coworkers, namely $m_{+}+m_{-}\simeq 3$ .

Now, we wish to interpret physically the meaning of the departure of $E_{+}$ and $E_{-}$ from the IK $k^{-3/2}$ scaling. It is clear from figure 8(b) that the Elsässer spectrum $E_{+}$ is the most intense, consistently with the fluxes displayed earlier in figure 5(a). In terms of the Elsässer variables $\boldsymbol{z}^{\pm }=\boldsymbol{u}\pm \boldsymbol{b}$ , it means that adding some positive $\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{b}\rangle$ in the flow enhances the dynamics of $\boldsymbol{z}^{+}$ , thus making $E_{-}$ less intense than $E_{+}$ . Therefore, $\boldsymbol{z}^{+}$ is transported by a subdominant field at large scales, leading to a Kolmogorov-like local transfer mechanism for $E_{+}$ , with the characteristic time $\unicode[STIX]{x1D70F}_{V}=(\unicode[STIX]{x1D716}k^{2})^{-1/3}$ like in HIT. This further justifies that the scaling of $E_{+}$ tends to $k^{-5/3}$ . On the contrary, since the transport of $\boldsymbol{z}^{-}$ is amplified, the ‘Alfvénization’ phenomenon is increased for $E_{-}$ and the nonlinear transfers are slowed down, thus explaining the slope less steep than $-3/2$ , close to $E_{-}\sim k^{-4/3}$ . Incidentally, the fact that the Kolmogorov-type dynamics is dominant (because $E_{+}>E_{-}$ at the most energetic scales) is consistent with the growth of the kinetic to magnetic energy ratio in figure 8(a), and with the phenomenology of Dobrowolny, Mangeney & Veltri (Reference Dobrowolny, Mangeney and Veltri1980) as well, where $E_{+}$ should prevail if positive cross-correlation is initially dominant.

Based on these observations, one could propose theoretical scalings for the Elsässer spectra in imbalanced IMHDT. Proceeding as in § 3.1, one would have $\unicode[STIX]{x1D716}_{+}=kE_{+}(k)/\unicode[STIX]{x1D70F}_{V}(k)$ so that $E_{+}\sim \unicode[STIX]{x1D716}_{+}\unicode[STIX]{x1D716}^{-1/3}k^{-5/3}$ and $\unicode[STIX]{x1D716}_{-}=kE_{-}/\unicode[STIX]{x1D70F}_{\text{tr}}(k)$ with $\unicode[STIX]{x1D70F}_{\text{tr}}(k)=(k^{2}\unicode[STIX]{x1D716})^{-2/3}/(kb_{0})^{-1}$ so that $E_{-}\sim \unicode[STIX]{x1D716}_{-}b_{0}\unicode[STIX]{x1D716}^{-2/3}k^{-4/3}$ .

The conclusion is that cross-helicity affects both the early dynamics of the decaying turbulent flow, and the inertial scalings of the Elsässer spectra as well, with different mechanisms at stake for $E_{+}$ and $E_{-}$ . If one injects instead negative correlation $\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{b}\rangle$ , the previous findings are still verified by changing the $()_{+}$ and $()_{-}$ quantities. The $k^{-5/3}$ and $k^{-4/3}$ scalings for $E_{+}$ and $E_{-}$ respectively deserve more investigation, since two different effects are in competition: a larger Reynolds number which brings the slopes closer to $k^{-3/2}$ , and a greater normalized cross-helicity $\unicode[STIX]{x1D70C}_{c}$ which on the contrary makes the slopes depart from $k^{-3/2}$ .

4.7 Pressure spectrum $E_{P}(k,t)$

Now that the impact of cross-helicity on the kinetic energy, magnetic energy, and Elsässer spectra has been discussed, the emphasis is put here on pressure spectra. The evolution equation of the pressure fluctuations, the so-called Poisson equation, is obtained by taking the divergence of the velocity fluctuations equation (2.1)

(4.18) $$\begin{eqnarray}-\frac{\unicode[STIX]{x2202}^{2}p^{\ast }}{\unicode[STIX]{x2202}x_{j}\unicode[STIX]{x2202}x_{j}}=\frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}-\frac{\unicode[STIX]{x2202}b_{j}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}b_{i}}{\unicode[STIX]{x2202}x_{j}}.\end{eqnarray}$$

The spectral two-point second-order pressure correlation is then defined as

(4.19) $$\begin{eqnarray}{\mathcal{E}}_{P}(\boldsymbol{k},t)\unicode[STIX]{x1D6FF}(\boldsymbol{k}-\boldsymbol{p})=\langle \hat{p}(\boldsymbol{k},t)\hat{p}^{\ast }(\boldsymbol{p},t)\rangle ,\end{eqnarray}$$

where the spectral fluctuating pressure is given by

(4.20) $$\begin{eqnarray}\hat{p}(\boldsymbol{k},t)=\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6FC}_{j}\left(\widehat{b_{i}b_{j}}(\boldsymbol{k},t)-\widehat{u_{i}u_{j}}(\boldsymbol{k},t)\right).\end{eqnarray}$$

The spectral second-order pressure correlation reads, after some algebra,

(4.21) $$\begin{eqnarray}\displaystyle {\mathcal{E}}_{P}(\boldsymbol{k})=2\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6FC}_{j}\unicode[STIX]{x1D6FC}_{p}\unicode[STIX]{x1D6FC}_{q}\int _{\boldsymbol{k}=\boldsymbol{p}+\boldsymbol{q}}\left[\hat{\unicode[STIX]{x1D619}}_{ip}(\boldsymbol{p})\hat{\unicode[STIX]{x1D619}}_{jq}(\boldsymbol{q})+\hat{\unicode[STIX]{x1D609}}_{ip}(\boldsymbol{p})\hat{\unicode[STIX]{x1D609}}_{jq}(\boldsymbol{q})-2\Re \left(\hat{\unicode[STIX]{x1D60F}}_{ip}^{c}(\boldsymbol{p})\hat{\unicode[STIX]{x1D60F}}_{jq}^{c}(\boldsymbol{q})\right)\right]\text{d}^{3}\boldsymbol{p}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The pressure spectrum, obtained by spherically averaging ${\mathcal{E}}_{P}$ , is then

(4.22) $$\begin{eqnarray}\displaystyle E_{P}(k,t) & = & \displaystyle \displaystyle \int _{S_{k}}{\mathcal{E}}_{P}(\boldsymbol{k},t)\,\text{d}^{2}\boldsymbol{k}=E_{P}^{\text{(hyd)}}(k,t)+E_{P}^{\text{(mag)}}(k,t)+E_{P}^{\text{(cross)}}(k,t)\nonumber\\ \displaystyle & = & \displaystyle \displaystyle 16\unicode[STIX]{x03C0}^{2}\int _{\unicode[STIX]{x1D6E5}_{k}}kpq(1-y^{2})(1-z^{2})({\mathcal{E}}_{0}^{V^{\prime }}{\mathcal{E}}_{0}^{V^{\prime \prime }}+{\mathcal{E}}_{0}^{B^{\prime }}{\mathcal{E}}_{0}^{B^{\prime \prime }})\,\text{d}p\,\text{d}q\nonumber\\ \displaystyle & & \displaystyle -\,\displaystyle 32\unicode[STIX]{x03C0}^{2}\int _{\unicode[STIX]{x1D6E5}_{k}}kpq(1-y^{2}-z^{2}+y^{2}z^{2}){\mathcal{H}}_{c}^{\prime }{\mathcal{H}}_{c}^{\prime \prime }\,\text{d}p\,\text{d}q,\end{eqnarray}$$

where $E_{P}^{\text{(hyd)}}$ is the classical pressure spectrum of HIT (Meldi & Sagaut Reference Meldi and Sagaut2013b ; Briard, Iyer & Gomez Reference Briard, Iyer and Gomez2017), $E_{P}^{\text{(mag)}}$ is the additional contribution in MHD resulting from the Laplace force and $E_{P}^{\text{(cross)}}$ corresponds to the impact of cross-helicity. In the following simulations, the pressure spectrum is set to zero initially: it is always found that the infrared scaling is $k^{2}$ , as in HIT (Lesieur et al. Reference Lesieur, Ossia and Metais1999).

In balanced MHD (with $E_{P}^{\text{(cross)}}=0$ ), it follows that the scaling of the pressure spectrum is different from its $k^{-7/3}$ inertial scaling in HIT: indeed, by dimensional analysis, one obtains $E_{P}\sim kE_{B}(k)^{2}\sim kE_{V}(k)^{2}$ , so that

(4.23) $$\begin{eqnarray}E_{P}(k,t)\sim b_{0}\,\unicode[STIX]{x1D716}\,k^{-2}.\end{eqnarray}$$

This $k^{-2}$ inertial scaling is assessed in figure 9(a) for $E_{P}^{\text{(mag)}}$ and $E_{P}^{\text{(iso)}}$ , and thus for $E_{P}$ as well. One can remark that at large scales, $E_{P}^{\text{(mag)}}$ is more intense than $E_{P}^{\text{(hyd)}}$ : this is because of the initial conditions (2.44), where $E_{B}\sim k^{2}$ and $E_{V}\sim k^{4}$ at large scales. Note that the pressure variance, defined as $K_{P}=\int _{0}^{\infty }E_{P}(k)\,\text{d}k$ , decays at a rate $2\unicode[STIX]{x1D6FC}$ , with $K\sim t^{\unicode[STIX]{x1D6FC}}$ , for dimensional reasons since $p^{2}\sim u^{4}$ , similarly to HIT (Meldi & Sagaut Reference Meldi and Sagaut2013b ).

Figure 9. Pressure spectra for balanced and imbalanced isotropic MHD, for $\unicode[STIX]{x1D70E}=2$ at $Re_{\unicode[STIX]{x1D706}}=3\times 10^{3}$ , along with the integral and dissipative wavenumbers $k_{L}$ and $k_{\unicode[STIX]{x1D702}}^{\text{MHD}}$ , with initial condition $E_{B}(k,t=0)=E_{V}(k,t=0)$ . (a) Balanced case: $E_{P}(k,t)$ with its isotropic and magnetic parts $E_{P}^{\text{(hyd)}}(k,t)$ and $E_{P}^{\text{(mag)}}(k,t)$ . (b) Imbalanced case: $E_{P}^{\text{(hyd)}}(k,t)$ , $E_{P}^{\text{(mag)}}(k,t)$ and $E_{P}^{\text{(cross)}}(k,t)$ .

For imbalanced MHD, if an inertial $k^{-5/3}$ scaling is assumed for the cross-helical spectrum, as shown previously in figure 7(a), it follows that $E_{P}^{\text{(cross)}}\sim kH_{c}(k)^{2}\sim k^{-7/3}$ : this scaling is recovered here in figure 9(b), and one can observe that $E_{P}^{\text{(cross)}}$ is mainly negative, which is expected given its expression (4.22). Other possible effects of cross-helicity on the hydrodynamic and MHD parts are rather difficult to quantify, since the scaling of $E_{P}^{\text{(mag)}}$ and $E_{P}^{\text{(iso)}}$ remains like $k^{-2}$ .