3. Results

Figure 1(a) shows a snapshot of the magnetic field modulus in a section perpendicular to $\boldsymbol{b}_{\mathbf{0}}$ for case A. The large-scale coherent structures are mostly the signature of the initial condition, whereas the incoherent small-scale structures are produced by the nonlinear dynamics and the direct energy cascade. It is thought that the presence of such patchy structures is a signature of the weak MHD turbulence regime (we have checked that they are not present in case B). A close-up of the current density modulus is also given (figure 1 b). It reveals a hierarchy of current sheets, which are the dominant dissipative structures. Interestingly, we also see the formation of less intense filaments.

Figure 1.

(a) Snapshot of the magnetic field modulus (on a linear scale) in a section perpendicular to $\boldsymbol{b}_{\mathbf{0}}$ (case A). (b) Close-up of the current density modulus (on a logarithmic scale) corresponding to the marked (square) region in (a). The line integral convolution technique (Cabral & Leedom Reference Cabral and Leedom1993) reveals a hierarchy of current sheets as well as the formation of less intense filaments.

We shall quantify the turbulence statistics by introducing the bi-dimensional axisymmetric Elsässer energy spectra $E^{\pm }(k_{\bot },k_{\Vert })$ , which are linked to the Elsässer energies ${\mathcal{E}}^{\pm }=\langle {\boldsymbol{z}^{\pm }}^{2}\rangle /2$ of the system ( $\langle \,\rangle$ denotes an integration over the physical space) by the double integral ${\mathcal{E}}^{\pm }=\iint E^{\pm }(k_{\bot },k_{\Vert })\text{d}k_{\bot }\text{d}k_{\Vert }$ . Figure 2 shows the results for case A. In order to improve the statistics the spectrum $E^{+}$ is plotted after an integration from $k_{\Vert }=1$ to 50 (red curve). In this way we suppress the contribution of the 2D modes and limit the cumulative contribution of the dissipative parallel scales, which can eventually alter the scaling law at the smallest scales. A spectrum compatible with the WT prediction in $k_{\bot }^{-2}$ is clearly observed (see inset). Hence, this observation and the small value of ${\it\chi}$ (not shown) may be considered as a clear signature of WT. In addition, we plot the spectrum $E^{-}$ for the 2D modes (black curve) in order to show that it behaves very differently, possessing a flat spectrum. The spectrum $E^{-}$ (or $E^{+}$ in the latter case) behaves similarly, as is expected for zero cross-helicity. Note that the $-2$ spectrum appears only after a transition in which the scaling is shallower ( ${\sim}\!-3/2$ ). A possible interpretation of this transition is that, in order to emerge, WT needs a sufficient amount of resonant triadic interactions (otherwise one can fall in the discrete WT regime (Nazarenko Reference Nazarenko2011)), which can be found only at sufficiently small scales.

Figure 2.

Transverse wavenumber Elsässer energy spectra, $\int _{1}^{50}E^{+}(k_{\bot },k_{\Vert })\text{d}k_{\Vert }$ (red) and $E^{-}(k_{\bot },k_{\Vert }=0)$ (black) for case A. The former spectrum is also shown when it is compensated by $k_{\bot }^{2}$ (inset). The wavelet spectrum from the sampled slices (see text) is also shown (blue). The shaded area corresponds to the scales for which the intermittency study is conducted.

The detection of the WT regime can also be made directly from the wavenumber–frequency spectrum. In order to do this, during a time window around $t_{\ast }$ , we follow the magnetic energy fluctuations in Fourier space at a given $k_{\bot }$ ( $=64$ to be in the inertial range) and for $k_{\Vert }$ between 0 and $\pm 64$ . Then, we perform a time-Fourier transform of these signals and build the dispersion relation ${\it\omega}(k_{\Vert })$ . The result is shown in figure 3. As we can see, the signal is mainly concentrated near the linear dispersion relation of the Alfvén wave, confirming that the waves are weakly nonlinear. The weak nonlinear effects manifest themselves in a broadening of the ( ${\it\omega},k_{\Vert }$ )-distribution with respect to the ${\it\omega}=2k_{\Vert }b_{0}$ curve. This result suggests that we are not in the discrete regime of WT (Nazarenko Reference Nazarenko2011). Importantly, we also see the contribution of coherent structures which correspond to a band at constant low frequency. Interestingly the same kind of observation has been made in the context of Bose–Einstein condensates (Nazarenko & Onorato Reference Nazarenko and Onorato2007).

Figure 3.

Wavenumber–frequency spectrum of the magnetic energy fluctuations at $k_{\bot }=64$ . The colour map is normalized to the maximal value of the spectrum at each fixed $k_{\Vert }$ . Note the presence of a factor $1/2$ in the frequency normalization because the energy (square of a field) is used.

We use wavelet coefficients to define the Elsässer field increments, denoted by ${\it\delta}z^{\pm }(\boldsymbol{x},\boldsymbol{\ell }_{\bot })$ , between two points separated by a vector $\boldsymbol{\ell }_{\bot }$ transverse to $\boldsymbol{b}_{\mathbf{0}}$ . For more details on the wavelet method used and the justifications for using it please refer to Kiyani et al. (Reference Kiyani, Chapman, Sahraoui, Hnat, Fauvarque and Khotyaintsev2013). Intermittency can be investigated through the PDFs of these increments for different spatial lags $\ell _{\bot }$ . We do not report an intermittency analysis for vector separations along $\boldsymbol{b}_{\mathbf{0}}$ because the corresponding range of scales is too narrow. The results from this analysis are shown in figure 4. For case A (a), strong intermittency is revealed through the development of more extended and heavier tails at a shorter distance $\ell _{\bot }$ . The result is drastically different when the interaction with the 2D modes is artificially reduced (case B, b) – in this case intermittency is strongly reduced, with PDFs approaching closer to a Gaussian form. Removing the interactions between the 2D modes and other wave modes is equivalent theoretically to removing the three-wave resonant interactions which support the WT dynamics. Therefore, what we see in case B is mainly the result of the non-resonant (including quasi-resonant) triadic interactions, which are asymptotically negligible in the pure regime of WT (the contribution of four-wave resonances can be discarded because it leads theoretically to a steep $-7/3$ spectrum (Sridhar & Goldreich Reference Sridhar and Goldreich1994) that is not observed – see below).

Figure 4.

PDFs of the Elsässer field increments (given by the wavelet coefficients) ${\it\delta}z^{+}$ for case A (a) and B (b), and for four distances $\ell _{\bot }$ ( $L_{0}$ is the size of the numerical box) corresponding to the circles inside the shaded area in figure 2. The PDFs are shifted vertically for clarity.

We further analyse intermittency through the symmetric structure functions:

where ${\it\zeta}(\,p)$ are the scaling exponents that are to be measured in the inertial range (homogeneous axisymmetric turbulence is assumed) and $C_{p}$ are some constants. The unusual definition (3.1) for the structure functions is related to the exact solution of the WT equations, which involves a power-law relation for the product of the Elsässer spectra. This information will be used below to develop a WT intermittency model. The study is conducted on several transverse planes and the fluctuations at different scales $\ell _{\bot }$ are calculated using the undecimated discrete wavelet transform detailed in Kiyani et al. (Reference Kiyani, Chapman, Sahraoui, Hnat, Fauvarque and Khotyaintsev2013); a 12-point wavelet was chosen to overcome the limitations of normal two-point structure functions. Utilizing the periodic boundary conditions of the simulation allows us to construct large contiguously sampled signals in each plane; the ensemble of fluctuations is then constructed from a union of all the fluctuations generated from the signal in each of the planes. This ensures that we have a large sample to form our statistics. Figure 5 shows that case A is characterized by strong intermittency such that ${\it\zeta}(\,p)$ cannot be fitted with a trivial (linear) law, nor by the MHD log-Poisson model previously derived (Müller, Biskamp & Grappin Reference Müller, Biskamp and Grappin2003, equation (3)) when $g=2$ (hereafter MBG-weak) or $g=4$ (hereafter MBG-IK), which correspond respectively to weak (Galtier et al. Reference Galtier, Nazarenko, Newell and Pouquet2000) and strong (Kraichnan Reference Kraichnan1965) turbulence (with ${\it\delta}z\sim \ell _{\bot }^{1/g}$ ). We also report the scaling exponents for case B, which behave differently as a result of the $k_{\Vert }=0$ modes being removed. Interestingly, for case B, the data follow the same curvature as the strong MHD turbulence model (MBG-IK) and, as can be seen from the value of ${\it\zeta}(2)$ , are compatible with a $k_{\bot }^{-3/2}$ spectrum. Note that in this case elongated structures (current sheets) are also present. Also, the origin of the $-2$ scaling found in case A seems not to be linked to any field discontinuities (since then we should probably get a $k_{\bot }^{-2}$ spectrum for case B).

Figure 5.

Scaling exponents ${\it\zeta}(\,p)$ for cases A and B with a superimposed plot of the log-Poisson model (equation (3.4)) for $C_{0}=1.08$ . The models MBG-weak and MBG-IK are also plotted (see details in text) for comparison. The errors quoted correspond to one standard error of the parameter fits.

We want to build a model that fits the exact solution of WT (energy spectrum in $k_{\bot }^{-2}$ ), which in physical space implies ${\it\zeta}(2)=1$ . Even if this power-law relation between the Fourier and physical spaces is not mathematically exact, it is clearly observed in figure 5 and can be considered as an excellent constraint for the intermittency model. Following the original development (She & Leveque Reference She and Leveque1994) we then define:

where $\langle {\it\varepsilon}\rangle$ is the mean dissipation rate of energy and $\langle {\it\varepsilon}^{p/2}\rangle \sim \ell _{\bot }^{{\it\mu}_{p/2}}$ . The latter relation is the so-called refined similarity hypothesis (Kolmogorov Reference Kolmogorov1962). The log-Poisson distribution for the dissipation leads to the general relation (She & Leveque Reference She and Leveque1994; Biskamp Reference Biskamp2003):

where ${\it\Delta}$ and ${\it\beta}$ are linked to the co-dimension $C_{0}$ of the dissipative structures such that $C_{0}={\it\Delta}/(1-{\it\beta})$ . We shall consider the co-dimension as a free parameter that will be estimated directly from the data. The system is closed by defining the value of ${\it\Delta}$ , which is related to the dissipation of the most singular structures, such that $\ell _{\bot }^{-{\it\Delta}}\sim E_{\infty }/{\it\tau}_{\infty }$ , where $E_{\infty }$ is the energy dissipated in these most singular structures and  ${\it\tau}_{\infty }\sim \ell /v_{\ell }$ is the associated time scale. ${\it\Delta}$ may be obtained by considering the following remarks. Weak MHD turbulence behaves very differently from isotropic MHD, because in the former case the regime is driven at leading order by three-wave resonant interactions, with the scattering of two of these waves on a 2D/third mode. (In fact, weak MHD turbulence is not applied in a thin layer around $k_{\Vert }=0$ (Galtier et al. Reference Galtier, Nazarenko, Newell and Pouquet2000) where strong turbulence may be expected.) These 2D modes are also important in characterizing dissipative structures (see figure 1): these structures, which look like vorticity/current sheets, are strongly elongated along the parallel direction and are therefore mainly localized around the $k_{\Vert }=0$ plane in Fourier space. If we assume that the dynamics of the 2D modes are similar to the dynamics of two-dimensional strong turbulence, then it seems appropriate to consider that the time scale entering in the intermittency relation may be determined by (Biskamp Reference Biskamp2003) $v_{\ell }\sim \ell _{\bot }^{1/4}$ , hence the value ${\it\Delta}=3/4$ . Note that the importance of the 2D modes on intermittency has already been emphasized in figure 4, where the PDFs are closer to a Gaussian when the interactions with them are greatly reduced. Taking all this into account, from (3.1) to (3.3) we finally obtain the intermittency model:

The plot for this model is superimposed in figure 5(top, red line) for a fractal co-dimension $C_{0}=1.08$ , which is the value that best fits the data (a nonlinear least-squares regression is used). The model fits the numerical simulation perfectly, thus we may infer that weak MHD turbulence is mainly characterized by vorticity/current sheets. Note that the parameter ${\it\beta}$ in this model measures the degree of intermittency: non-intermittent turbulence corresponds to ${\it\beta}=1$ , whereas the limit ${\it\beta}=0$ represents an extremely intermittent state in which the dissipation is concentrated in one singular structure. According to the value obtained here, ${\it\beta}\simeq 1/4$ (with $C_{0}\simeq 1$ ), we might conclude that weak MHD turbulence appears more intermittent than strong isotropic MHD turbulence, for which ${\it\beta}=1/3$ .