2.1. Main equations

We begin by considering the MHD equations in the incompressible non-relativistic limit, given as (Biskamp Reference Biskamp2003)

(2.1)\begin{gather} \frac{\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u} ={-}\boldsymbol{\nabla} p + {\boldsymbol{b}} \boldsymbol{\cdot}\boldsymbol{\nabla} {\boldsymbol{b}} + {\nu}\nabla^2 \boldsymbol{u}, \end{gather}

(2.2)\begin{gather}\frac{\partial {\boldsymbol{b}}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{\nabla}{\boldsymbol{b}}={\boldsymbol{b}} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u} + {\eta} \nabla^2 {\boldsymbol{b}} , \end{gather}

(2.3a,b)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol{v}}=0,\quad \boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol{b}} = 0, \end{gather}

where the modified pressure $p$ is given by $p=P/\rho +b^2/2$, ${\boldsymbol {b}}={\boldsymbol {B}}\sqrt {4{\rm \pi} \rho }$, where $\rho$ is the mass density, $\boldsymbol {u}({\boldsymbol {x}}, t)$ is the flow velocity, $P({\boldsymbol {x}}, t)$ is the thermal pressure, ${\boldsymbol {B}}({\boldsymbol {x}}, t)$ is the magnetic induction field, $\nu$ is the viscosity and $\eta$ is the magnetic diffusivity.

2.2. Scale-by-scale analysis

A considerable amount of work in analysing the transfer of energy among scales has been performed in Fourier space with significant results (Verma Reference Verma2004; Alexakis et al. Reference Alexakis, Mininni and Pouquet2005; Mininni et al. Reference Mininni, Alexakis and Pouquet2005; Alexakis et al. Reference Alexakis, Mininni and Pouquet2007; Domaradzki et al. Reference Domaradzki, Teaca and Carati2009; Teaca et al. Reference Teaca, Verma, Knaepen and Carati2009, Reference Teaca, Carati and Andrzej Domaradzki2011; Verma Reference Verma2019, Reference Verma2022). This kind of approach while useful in understanding and quantifying the scale-by-scale energy budget of turbulence, does not allow one to simply link the related energy transfers with local properties of the flow and thus limits the applicability to subgrid-scale models.

To analyse the local-in-space scale-by-scale dynamics, we use the local filtering approach (Germano Reference Germano1992). We introduce a filter such that

(2.4)\begin{equation} \tilde{{\boldsymbol{a}}}_\ell({\boldsymbol{x}}) = \int {\rm d}r^3\,G_\ell({\boldsymbol{r}}) {\boldsymbol{a}}({\boldsymbol{x}}+{\boldsymbol{r}}) , \end{equation}

where $G({\boldsymbol {r}})$ is a smooth filtering function, spatially localized and such that $\int {\rm d}r^3 \,G({\boldsymbol {r}})=1$ and $\int {\rm d}r^3 \,\vert {\boldsymbol {r}} \vert ^2 G({\boldsymbol {r}}) \approx 1$. The function $G_\ell$ is rescaled with $\ell$ as $G_\ell ({\boldsymbol {r}}) = \ell ^{-3}G({\boldsymbol {r}}/\ell )$. Applying this filter to the MHD equations, one obtains (Aluie Reference Aluie2017)

(2.5)\begin{gather} \partial_{t} \tilde{{\boldsymbol{u}}}_\ell + (\tilde{{\boldsymbol{u}}}_\ell\boldsymbol{\cdot}\boldsymbol{\nabla})\tilde{{\boldsymbol{u}}}_\ell ={-} \boldsymbol{\nabla}\tilde{p}_{\ell} + (\tilde{\boldsymbol{b}}_{\ell}\boldsymbol{\cdot}\boldsymbol{\nabla})\tilde{\boldsymbol{b}}_{\ell} -\boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{\tau}_\ell^{uu}-\boldsymbol{\tau}_\ell^{bb}) +\nu\nabla^{2}\tilde{{\boldsymbol{u}}}_\ell, \end{gather}

(2.6)\begin{gather}\partial_{t} \tilde{{\boldsymbol{b}}}_\ell + (\tilde{{\boldsymbol{u}}}_\ell\boldsymbol{\cdot}\boldsymbol{\nabla})\tilde{{\boldsymbol{b}}}_\ell -(\tilde{{\boldsymbol{b}}}_\ell\boldsymbol{\cdot}\boldsymbol{\nabla})\tilde{{\boldsymbol{u}}}_\ell ={-} \boldsymbol{\nabla}\boldsymbol{\cdot} (\boldsymbol{\tau}^{ub}_\ell-\boldsymbol{\tau}^{bu}_\ell) + \eta\nabla^{2}\tilde{{\boldsymbol{b}}}_\ell, \end{gather}

(2.7)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\tilde{{\boldsymbol{b}}}_\ell=\boldsymbol{\nabla}\boldsymbol{\cdot}\tilde{{\boldsymbol{u}}}_\ell=0. \end{gather}

Here $\boldsymbol {\tau }^{uu}_\ell$, $\boldsymbol {\tau }^{bb}_\ell$, $\boldsymbol {\tau }^{ub}_\ell$ and $\boldsymbol {\tau }^{bu}_\ell$ are the subscale stress tensors which describe the force exerted on scales larger than $\ell$ by fluctuations at scales smaller than $\ell$. They are defined as

(2.8a,b)\begin{equation} \boldsymbol{\tau}^{uu}_\ell = \widetilde{({\boldsymbol{u}}{\boldsymbol{u}})}_\ell -\tilde{\boldsymbol{u}}_{\ell}\tilde{\boldsymbol{u}}_{\ell}, \quad \boldsymbol{\tau}^{bb}_\ell =\widetilde{({\boldsymbol{b}}{\boldsymbol{b}})}_\ell - \tilde{\boldsymbol{b}}_{\ell}\tilde{\boldsymbol{b}}_{\ell}, \end{equation}

which are the subscale Reynolds stress and the subscale Maxwell stress, respectively, and finally

(2.9a,b)\begin{equation} \boldsymbol{\tau}^{ub}_\ell = \widetilde{({\boldsymbol{u}}{\boldsymbol{b}})}_\ell - \tilde{\boldsymbol{u}}_{\ell}\tilde{\boldsymbol{b}}_{\ell}, \quad \boldsymbol{\tau}^{bu}_\ell = \widetilde{({\boldsymbol{b}}{\boldsymbol{u}})}_\ell - \tilde{\boldsymbol{b}}_{\ell}\tilde{\boldsymbol{u}}_{\ell} \end{equation}

are cross-field tensors for which one is the transpose of the other, $\boldsymbol {\tau }^{bu}_\ell =[\boldsymbol {\tau }^{ub}_\ell ]^{\rm T}$.

Using this notation we can write an equation for the large-scale energy densities $\tilde {\mathcal {E}}^u=\frac {1}{2}|\tilde {\boldsymbol {u}}|^2$ and $\tilde {\mathcal {E}}^b=\frac {1}{2}|\tilde {\boldsymbol {b}}|^2$ as

(2.10)\begin{gather} \partial_t \tilde{\mathcal{E}}_\ell^u + \boldsymbol{\nabla}\boldsymbol{\cdot}{\mathcal{J}}^u ={-}\varPi_\ell^{uu} - \varPi_\ell^{bb} + \mathcal{W}_{L} -\mathcal{D}_u , \end{gather}

(2.11)\begin{gather}\partial_t \tilde{\mathcal{E}}^b_\ell + \boldsymbol{\nabla}\boldsymbol{\cdot} {\mathcal{J}}^b ={-}\varPi_\ell^{bu} -\varPi_\ell^{ub} - \mathcal{W}_{L} -\mathcal{D}_b, \end{gather}

where the currents $\mathcal {J}^u,\mathcal {J}^b$ are given by

(2.12)\begin{gather} \mathcal{J}^u_j = \left(\tfrac{1}{2}|\tilde{\boldsymbol{u}}|^2 +\bar{p} \right) \tilde{u}_j + (\tau^{uu}_{\ell,ij} - \tau^{bb}_{\ell,ij})\tilde{u}_i - \nu \tfrac{1}{2}\partial_j|\tilde{\boldsymbol{u}}|^{2} , \end{gather}

(2.13)\begin{gather}\mathcal{J}^b_j = \left(\tfrac{1}{2} |\tilde{\boldsymbol{b}}|^2 \right) \tilde{u}_j + (\tau^{ub}_{\ell,ij} - \tau^{bu}_{\ell,ij})\tilde{b}_i - (\tilde{\boldsymbol{u}}\boldsymbol{\cdot}\tilde{\boldsymbol{b}})\tilde{b}_j -\eta (\tilde{\boldsymbol{b}} \times \boldsymbol{\nabla} \times \tilde{\boldsymbol{b}}) \end{gather}

and express the transport of energy in space. The rate of work done on the flow originating from the Lorentz force is given by

(2.14)\begin{equation} \mathcal{W}_{L} = \tilde{u}_i\tilde{b}_j\partial_j\tilde{b}_i. \end{equation}

This term is responsible for the transfer of kinetic energy to magnetic energy in the large scales. The viscous and Ohmic energy dissipation rates are given by

(2.15a,b)\begin{equation} \mathcal{D}_u = \nu (\partial_{j}\tilde{u}_{i})(\partial_{j}\tilde{u}_{i})\quad\mathrm{and}\quad \mathcal{D}_b = \eta (\boldsymbol{\nabla} \times \tilde{\boldsymbol{b}})\boldsymbol{\cdot} (\boldsymbol{\nabla} \times \tilde{\boldsymbol{b}}) \end{equation}

and express the rate that energy is dissipated by viscous and Ohmic effects, respectively. These terms can be shown to be negligible if the filter scale is large and the viscous/resistive coefficients are small (Aluie Reference Aluie2017). Finally the four fluxes in scale space $\varPi _\ell ^{uu},\varPi _\ell ^{bb},\varPi _\ell ^{ub}$ and $\varPi _\ell ^{bu}$, explicitly given by

(2.16)\begin{gather} \varPi_\ell^{uu} ={-}\tau_{\ell,ij}^{ub}\partial_{i}\tilde{u}_{j} ={-}[ \widetilde{(u_iu_j)}_\ell - \tilde{u}_{\ell,i}\tilde{u}_{\ell,j}]\partial_{i}\tilde{u}_{j}, \end{gather}

(2.17)\begin{gather}\varPi_\ell^{bb} ={+}\tau_{\ell,ij}^{bb}\partial_{i}\tilde{u}_{j} ={+}[ \widetilde{(b_ib_j)}_\ell - \tilde{b}_{\ell,i}\tilde{b}_{\ell,j}]\partial_{i}\tilde{u}_{j}, \end{gather}

(2.18)\begin{gather}\varPi_\ell^{ub} ={-}\tau_{\ell,ij}^{ub}\partial_{i}\tilde{b}_{j} ={-}[ \widetilde{(u_ib_j)}_\ell - \tilde{u}_{\ell,i}\tilde{b}_{\ell,j}]\partial_{i}\tilde{b}_{j}, \end{gather}

(2.19)\begin{gather}\varPi_\ell^{bu} ={+}\tau_{\ell,ij}^{bu}\partial_{i}\tilde{b}_{j} ={+}[ \widetilde{(b_iu_j)}_\ell - \tilde{b}_{\ell,i}\tilde{u}_{\ell,j}]\partial_{i}\tilde{b}_{j}, \end{gather}

express the rate of gain (if $\varPi _\ell <0$) or loss (if $\varPi _\ell >0$) of energy of the large scales to the small filtered-out scales. Their sum gives the total energy flux:

(2.20)\begin{equation} \varPi_\ell({\boldsymbol{x}}) = \varPi_\ell^{uu}({\boldsymbol{x}})+\varPi_\ell^{bb}({\boldsymbol{x}})+\varPi_\ell^{ub}({\boldsymbol{x}})+\varPi_\ell^{bu}({\boldsymbol{x}}). \end{equation}

The fluxes are chosen so that they are invariant under Galilean transformations ${\boldsymbol {u}} \to {\boldsymbol {u}}+{\boldsymbol {U}}_0$ and also under ${\boldsymbol {b}} \to {\boldsymbol {b}}+{\boldsymbol {B}}_0$ for any flow realization $\boldsymbol {u},\boldsymbol {b}$, where $\boldsymbol {U}_{\boldsymbol {0}},\boldsymbol {B}_{\boldsymbol {0}}$ are constant in space vector fields. The importance of Galilean invariance has been noted in the past (Speziale Reference Speziale1985; Aluie & Eyink Reference Aluie and Eyink2009, Reference Aluie and Eyink2010). We note, however, that while the defined fluxes are invariant under ${\boldsymbol {b}} \to {\boldsymbol {b}}+{\boldsymbol {B}}_0$, the MHD equations are not. Therefore, the introduction of a constant ${\boldsymbol {B}}_0$ in the dynamics of the MHD equations will alter the statistical behaviour of the fields $\boldsymbol {u},\boldsymbol {b}$ and as a result of the fluxes as well.

The fluxes in (2.16)–(2.19) comprise the main object of the present work.

2.3. Filters

Although we formulate our filtering procedure in physical space, since we will be working in periodic domains it is useful to define the filters through their Fourier transforms:

(2.21)\begin{equation} \hat{G}_q ({\boldsymbol{k}}) = \int G_\ell({\boldsymbol{x}}) {\rm e}^{{\rm i}\boldsymbol{k}\boldsymbol{\cdot} x}\, {\rm d} {{\boldsymbol{x}}}, \end{equation}

where $q=1/\ell$ is the wavenumber corresponding to the filter length $\ell$, not to be confused with the Fourier wavenumber $k$. For the first filter we consider a Gaussian kernel:

(2.22)\begin{equation} \hat{G}_q({\boldsymbol{k}})=\exp\left[-\frac{k^2}{2q^2}\right]. \end{equation}

For an infinite domain this filter corresponds to the Gaussian filter in real space $G_\ell (r) =\exp (-\frac {1}{2}r^2/\ell ^2) /(2{\rm \pi} \ell ^{2})^{3/2}$. We note that this filtering is not a projection and in general $\widetilde {({\tilde {\boldsymbol {u}}_\ell })_\ell }\ne \tilde {\boldsymbol {u}}_\ell$. The second filter we consider is a sharp spectral filter such that

(2.23)\begin{equation} \tilde{{\boldsymbol{u}}}_\ell({\boldsymbol{x}},t)=\sum_{\vert {\boldsymbol{k}} \vert < q}\hat{{\boldsymbol{u}}}({\boldsymbol{k}},t) {\rm e}^{{\rm i}{\boldsymbol{k}} \boldsymbol{\cdot} {\boldsymbol{x}} }. \end{equation}

This filtering is a projector $\widetilde {({\tilde {\boldsymbol {u}}_\ell })_\ell }= \tilde {\boldsymbol {u}}_\ell$ and is based on a Galerkin truncation for all wavenumbers larger than the given cutoff $q=1/\ell$. With regard to the representation of energy fluxes, this filter has been shown in the past not to be optimal as it leads to wider fluctuations of the local energy flux (Buzzicotti et al. Reference Buzzicotti, Linkmann, Aluie, Biferale, Brasseur and Meneveau2018; Alexakis & Chibbaro Reference Alexakis and Chibbaro2020), something that as we show also holds in MHD.

4.1. Numerical set-up

To investigate the local energy fluxes described before, we use the results from DNS using the pseudo-spectral code ghost (Mininni et al. Reference Mininni, Rosenberg, Reddy and Pouquet2011) evolving the MHD equations (2.1)–(2.3a,b) in a cubic triple periodic domain of side $L=2{\rm \pi}$ so that $|\boldsymbol {k}|=\boldsymbol {1}$ gives the smallest non-zero wavenumber. The forcing is limited only to wavenumbers with $|{\boldsymbol {k}}|\leqslant k_f=2$ in all cases and is random and delta-correlated in time so that the mean energy injection rate $\epsilon = \langle {\boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {f}_{\boldsymbol {u}}}\rangle + \langle {\boldsymbol {b} \boldsymbol {\cdot } \boldsymbol {f}_{\boldsymbol {b}}}\rangle =1$ is independent of the flow state. All runs have the same energy injection rate and unit Prandtl number with $\nu =\mu =0.0001$. The involved Reynolds number based on the input parameters $\epsilon$, $L$ and $\nu$ is given by $Re=\epsilon ^{1/3}k_f^{4/3}/\nu$ that is thus fixed to $Re=3968$. The resolution was fixed for all runs at $N=1024$ grid points in each direction that was sufficient for the flow to be well resolved so that an exponential decrease of the energy spectrum is observed at large $k$. No artificial dissipation was used.

Four different cases were considered. In the first, no magnetic forcing was introduced and the magnetic field was set identically to zero $\boldsymbol {b}=\boldsymbol {0}$ so that the flow reduced to a hydrodynamic run, repeating essentially the results of Alexakis & Chibbaro (Reference Alexakis and Chibbaro2020). This run is used as a reference to identify the differences of MHD from hydrodynamic runs and is referred to in the subsequent figures as ‘hydro’. The remaining three runs were designed to have different levels of magnetic energy as in Alexakis (Reference Alexakis2013). The second run was thus a dynamo run. No magnetic forcing was used but an initial small magnetic field was introduced that was amplified by the turbulent motions until a steady state was reached where the magnetic energy fluctuated around a mean value. We note that the forcing was mirror-symmetric so no helicity is injected in the system. A result of the absence of helicity and the randomness of the flow is that magnetic energy is concentrated in the small scales. This run is referred to in the figures as ‘dynamo’. In the third run a magnetic forcing was also used of equal magnitude to the mechanical flow. The resulting flow at steady state has magnetic energy at equipartition with the kinetic energy at all scales. The helicity and magnetic helicity injection rate for this flow are also set to zero. Results from this run are referred to as ‘moderate’. Finally the fourth flow was also both mechanically and magnetically forced but in this case the magnetic helicity was weak but not zero. This leads to the formation of a slowly growing large-scale helical magnetic field. As a result this flow has magnetic energy that exceeds the kinetic energy of the flow. This run is referred to in the figures as ‘strong’.

It is worth noting that all the statistics we present in the following are obtained through time-averaging over a large amount of samples during the steady state to ensure statistical convergence. Whenever possible, several field realizations have been used also to average over space, because of spatial homogeneity.

4.2. Energy spectra and magnetic field

Figure 1 shows the resulting time-averaged energy spectra for the four runs at the steady state. For the dynamo run in particular, analysis is made at the state such that magnetic energy no longer grows. All runs show energy spectra compatible with a $k^{-5/3}$ power-law exponent. The magnetically forced runs moderate and strong have equipartition magnetic energy across wavenumbers except for $k=1$ in the strong case for which magnetic energy is larger. For the dynamo run the magnetic energy is weaker than kinetic at small wavenumbers but the reverse is true for large wavenumbers. This behaviour is typical for randomly forced non-helical small-scale dynamos (Schekochihin et al. Reference Schekochihin, Cowley, Taylor, Maron and McWilliams2004; Moll et al. Reference Moll, Graham, Pratt, Cameron, Müller and Schüssler2011; Brandenburg, Sokoloff & Subramanian Reference Brandenburg, Sokoloff and Subramanian2012). Dynamo flows with steady non-helical forcing, like Taylor–Green flows for example (Ponty et al. Reference Ponty, Mininni, Laval, Alexakis, Baerenzung, Daviaud, Dubrulle, Pinton, Politano and Pouquet2008), produce spectra closer to those of the moderate run. This figure points out also that a quasi-inertial range is roughly displayed over about a decade between $k=4$ and $k=64$, as standard with the present resolution.

Figure 1. Energy spectra for the four different cases examined.

In figure 2, it is possible to get some qualitative insights into the different cases studied. Notably, the visualization of the magnetic energy permits one to observe the structure of the large scales, through its space distribution. The dynamo case displays very little dispersion of the magnetic field amplitude, and energy is concentrated in flux tubes. The moderate case shows a larger distribution and some regions of intense field. Finally the strong case shows a much more widely distributed energy, with some regions characterized by large values of the magnetic field, and in particular large-scale structures are visible. To give a quantitative picture, we also show in figure 2 the probability distribution of the magnetic amplitude $|\boldsymbol {b}|$ for the three MHD cases examined. In the dynamo case the maximum is at small $|{\boldsymbol {b}}|$, with a peaked distribution. The width of the distribution increases in the other two cases, with a broad distribution for the strong case. In this last case, the tails appear substantial.

Figure 2. Visualizations of the magnetic energy density for the dynamo case (top left), the moderate case (top right) and the strong case (bottom left). The bottom-right panel shows the p.d.f. of $|{\boldsymbol {b}}|$ for the three different MHD flows examined.

6.1. Field gradients

In order to help the construction of subgrid-scale models that are based on the gradients of the resolved fields, we need to reveal what correlation exists between the gradients and the local energy flux. In figure 6 we present the joint p.d.f.s between $\varPi _\ell$ and the modulus squared of the symmetric and antisymmetric stress tensors $\varOmega _{u,\ell }^2$, $\varOmega _{b,\ell }^2$, $S_{u,\ell }^2$ and $S_{b,\ell }^2$. The results are for a given wavenumber in the inertial range $q=32$.

Figure 6. Joint p.d.f.s of $\varPi _\ell$ (for $q=32$) with the modulus of the different strain rates for (a) hydrodynamic case, (b) dynamo case, (c) moderate case and (d) strong case. From top to bottom the examined strain is $\varOmega _{u,\ell }^2$, $S_{u,\ell }^2$, $\varOmega _{b,\ell }^2$ and $S_{b,\ell }^2$. The colour map is logarithmic covering ten orders of magnitude with bright colours indicating high probability. The yellow dashed line indicates the Smagorinsky scaling $\varPi _\ell \propto \varOmega _\ell ^{3/2}$ and $\varPi _\ell \propto S_\ell ^{3/2}$.

The results obtained for the pure hydrodynamic configuration are shown for comparison in figure 6(a). In this case, the energy flux is essentially uncorrelated with the antisymmetric part of the strain $\varOmega _\ell$ with high-probability events for a given $\varOmega _\ell$ concentrated around $\varPi _\ell =0$. On the other hand, a visible correlation is observed with the modulus of the symmetric part of the strain tensor $S_{\ell }$, with high-probability events for a given $S_\ell$ concentrated around a non-zero value of $\varPi _\ell$ that increases with $S_\ell$. The yellow line corresponds to the Smagorinsky scaling of (3.4). The flux $\varPi _\ell$ is concentrated around this line giving support to the Smagorinsky model.

Such correlations are less clear for the MHD results. For the dynamo case in the $\varPi _\ell$–$\varOmega _{u\ell }$ diagram, high-probability events are concentrated around $\varPi _\ell =0$ again. For the $\varPi _\ell$–$S_{u\ell }$ diagram, a much weaker correlation is observed compared with the hydro case. The probability density shows long tails with large $\varPi _\ell$ for small $S_{u\ell }$ indicating that $S_{u\ell }$ is not dominant in driving the cascade. A stronger correlation is observed with $\varOmega _{b,\ell }^2$ and $S_{b,\ell }^2$. High-probability events are aligned with the Smagorinsky scaling (yellow dashed line) but the spread is much higher than in the hydro case. The fact that the correlations with $\varOmega _{b,\ell }^2$ and $S_{b,\ell }^2$ look very similar indicates that neither of the two alone is an optimal proxy to estimate the energy flux to the small scales.

The lack of correlation of $\varPi _\ell$ with the velocity gradients is even more clear for the moderate and strong cases. For these cases no correlation is seen either with $\varOmega _{u,\ell }^2$ or with $S_{u,\ell }^2$. Although there is some correlation with the magnetic field gradients $\varOmega _{b,\ell }^2$ and $S_{b,\ell }^2$ with the maximum of probability (for fixed $\varOmega _{b,\ell }^2$ or $S_{b,\ell }^2$) following the Smagorinsky scaling (yellow dashed line), the spread is very large with highly probable negative flux events that become more frequent for the strong field case. This indicates that although there is some correlation with the magnetic field gradients, modelling the subscale stresses with $\varOmega _{b,\ell }^2$ and $S_{b,\ell }^2$ alone is still missing important ingredients.

6.2. Local magnetic field

We pursue the analysis of the statistical phenomenology of the cascade process looking at correlations between the flux and the magnetic field. Figure 7 shows the joint p.d.f. of $\varPi _\ell$ and $\tilde {\boldsymbol {b}}$ for the different cases examined. The three cases display different distributions, yet with similar characteristics. Generally speaking, the maximum probability of the flux as well as most extreme values of it are obtained in the region around the average value of the magnetic field. Furthermore, the larger the average magnetic field, the wider the distribution. In this concern, it is interesting to note that the isolines are almost flat over a large range, indicating that in such a large region the flux distribution, and therefore the cascade process, is basically independent of the value of the magnetic field. That is particularly true for the strong case.

Figure 7. Joint p.d.f. of the local flux $\varPi$ and the local amplitude of the magnetic field $b$ for the three cases: dynamo (left), moderate (centre) and strong (right).

More insight is gained by looking at the p.d.f. of the energy flux conditioned on the knowledge of the magnetic field amplitude. This corresponds at looking at ‘slices’ of figure 7 for fixed values of $\tilde {\boldsymbol {b}}$ and allows a more clear look at the extreme events. The results are displayed in figure 8. Seen this way some differences can be noted between the three different cases. For the dynamo case there is a monotonic increase of the tails of the distribution with respect to increasing magnetic field. Both negative and positive tails are wider when related to a larger magnetic field amplitude. In this case large values of $\tilde {\boldsymbol {b}}$ indicate more extreme events.

Figure 8. Conditioned p.d.f., i.e. probability of having flux $\varPi _\ell$ given the magnetic field amplitude $|{\boldsymbol {b}}|$. In the dynamo case, large $|{\boldsymbol {b}}|$ enhances large fluctuations.

In the moderate case, the probability distribution seems to be almost insensitive to changes in the magnetic field amplitude. The distribution would point to an energy cascade somewhat independent of the magnetic field amplitude. Some small differences are actually present in the distributions, but they are not much more important than statistical errors.

For the strong case, the negative tails increase monotonically with the amplitude of the magnetic field. The positive tails, on the other hand, are monotonically decreased by increasing the magnetic field. Therefore, in strongly magnetized flows, large $|{\boldsymbol {b}}|$ reduces extreme direct cascade fluctuations, while increasing inverse cascade fluctuations. In this sense, the skewness of the distribution seems to be affected by the amplitude of the magnetic field. It appears that in regions of very large magnetic field the distribution is made symmetric. Possibly this behaviour occurs because locally in regions of large $\tilde {\boldsymbol {b}}$ turbulence transitions to a wave turbulence regime that has weaker forward energy flux or by rendering the flow locally quasi-two-dimensional. This effect, however, would require further investigation to be fully understood, and to assess possible finite-size effects.