3.1. Behaviour of flux rates

In figure 2(a), we plot the total energy flux $\varPi _{Total} (=\varPi _u + \varPi _b + \varPi _{Hall})$ as a function of wavenumber $k$ , using the expressions provided in (2.9)–(2.11). For the sake of comparison and to delineate the role of the Hall term, we plot $\varPi _{Total}$ with and without $\varPi _{Hall}$ . As it is clearly observed, $\varPi _{Total}$ closely follows the injection flux rate $\varPi _{inj}$ (dashed orange curve in figure 2 a) across a range of intermediate wavenumbers, indicating a Kolmogorov-like universal regime for the total energy cascade. For clarity, we also plot the dissipation flux rate $-\varPi _{D}$ (dotted orange curve in figure 2 a). Evidently, $-\varPi _{D}$ remains negligible compared with $\varPi _{Total}$ up to the hyperviscous Kolmogorov wavenumber $k_d$ , where it eventually takes over, ensuring an approximately dissipation-free inertial cascade regime. To further probe into the behaviour of various components of $\varPi _{Total}$ , we plot $\varPi _u$ , $\varPi _b$ and $\varPi _{Hall}$ as a function of $k$ in figure 2(b). For length scales greater than $d_i$ , i.e. $k d_i\lt 1$ , the magnetic energy transfer rate $\varPi _b$ dominates over $\varPi _u$ and $\varPi _{Hall}$ . Similar to previous studies (Mininni et al. Reference Mininni, Alexakis and Pouquet2007; Gómez et al. Reference Gómez, Mininni and Dmitruk2010; Halder et al. Reference Halder, Banerjee, Chatterjee and Sharma2023), $\varPi _{Hall}$ becomes slightly negative across the said range of scales. However, for $k d_i \gt 1$ , $\varPi _{Hall}$ becomes positive and gradually exceeds $\varPi _b$ and $\varPi _u$ at sufficiently small scales. Interestingly, both $\varPi _u$ (pink curve in figure 2 b) and $\varPi _b + \varPi _{Hall}$ (green curve in figure 2 b) remain approximately constant across the inertial range, suggesting a decoupled cascade of kinetic and magnetic energies. In ordinary MHD, such decoupling is attributed to the suppression of magnetic field-line stretching in the inertial regime (Bian & Aluie Reference Bian and Aluie2019). However, for HMHD, a similar type of reasoning requires further investigation.

The flux rates discussed so far only provide the net transfer rate to a particular mode from all the others. In order to get a more detailed picture of the energy cascade and to investigate its locality, it is appropriate to study the corresponding S2S transfer rates using (2.6).

3.2. Calculation of shell-to-shell transfer rates

Figure 3.

The S2S transfer rates corresponding to MHD nonlinear terms as a function of giver and receiver shells $(m, n)$ . Here, $T^{u, u}_{n, m}$ and $T^{u, b}_{n, m}$ denote the transfer rate of kinetic energy through $\boldsymbol{u}$ -to- $\boldsymbol{u}$ , $\boldsymbol{b}$ -to- $\boldsymbol{u}$ channels respectively, and $T^{b, u}_{n, m}$ and $T^{b, b}_{n, m}$ denote the transfers rates of magnetic energy through $\boldsymbol{u}$ -to- $\boldsymbol{b}$ and $\boldsymbol{b}$ -to- $\boldsymbol{b}$ channels respectively.

First, we examine $T^{u, u}_{n, m}$ and $T^{u, b}_{n, m}$ which denote the S2S transfer rates of kinetic energy through the $\boldsymbol{u}$ -to- $\boldsymbol{u}$ and $\boldsymbol{b}$ -to- $\boldsymbol{u}$ channels, respectively. For a given value of the receiver shell number $n$ , if $T^{x, y}_{n, m}$ is positive for $n\gt m$ or negative for $n\lt m$ , then we have a direct transfer and vice versa. From figure 3, both $T^{u, u}_{n, m}$ and $T^{b, b}_{n, m}$ represent direct transfers that are mainly local in nature i.e. for a receiver shell number $n$ , significant contribution primarily comes from the neighbouring shells. In contrast, for both cases of cross-transfer $T^{u, b}_{n, m}$ and $T^{b, u}_{n, m}$ , a non-negligible non-local transfer is observed involving the scale of energy injection. This agrees with the previous findings of non-local transfers in MHD due to the effect of large-scale energy-containing structures (Alexakis et al. Reference Alexakis, Mininni and Pouquet2005; Carati et al. Reference Carati, Debliquy, Knaepen, Teaca and Verma2006; Cho Reference Cho2010).

Figure 4.

Panels (a) and (c) show the S2S transfer rates for the channel $\mathcal{BB}$ of the Hall term as a function of giver and receiver shell numbers ( $m$ , $n$ ) and (b) and (d) show the same for the channel $\mathcal{JB}$ as a function of giver and receiver shell numbers ( $m$ , $n$ ).

In figures 4(a) and 4(b), we plot the S2S transfer rates corresponding to channels $\mathcal{BB}$ and $\mathcal{JB}$ of the Hall term as a function of giver and receiver shell numbers $(m, n)$ , respectively. As it is apparent, both the channels comprise local and non-local transfers. For $\mathcal{H}^{b, b}_{n, m}$ , close to the diagonal, for a fixed giver shell $m$ , positive transfer is observed for a receiver shell $n\lt m$ (red squares above the diagonal) whereas negative transfer is observed for a receiver shell $n\gt m$ (blue squares below the diagonal). This clearly signifies a local inverse transfer of magnetic energy. On the other hand, far from the diagonal, the nature of the transfer is reversed and therefore a negative transfer is found above the diagonal while a positive transfer is found below the diagonal, indicating a direct non-local transfer of magnetic energy. Interestingly, the strength of the non-local transfers becomes increasingly important at small scales i.e. for large values of $m$ and $n$ (see figure 4 a). The nature of the energy transfer due to $\mathcal{H}^{b, j}_{n, m}$ , on the other hand, is remarkably different (see figure 4 b). For shells with $m,n \lt 12$ , both the local and non-local transfer rates of $\mathcal{H}^{b, j}_{n, m}$ are found to be relatively weaker compared with those of $\mathcal{H}^{b, b}_{n, m}$ whereas the transfer rate increases considerably at smaller scales where it is principally nourished by the local S2S transfers. Furthermore, the colour code of the transfers clearly indicates an inverse transfer of energy for small wavenumbers whereas the transfer becomes direct at sufficiently small scales. Interestingly, in both channels, a considerable transfer associated with $m, n = 3$ is observed which is possibly induced by the interaction between the injection and the Hall wavenumbers.

Table 2.

Receiver shell indices and the corresponding wavenumber ranges.

Table 2. Long description

A table titled Receiver shell indices and the corresponding wavenumber ranges. The table has two columns and seven rows. The first column is labeled Shell no. and the second column is labeled Wavenumber range. The table presents the following data: Row 1: Shell no. 5, Wavenumber range 9.19 to 10.56. Row 2: Shell no. 9, Wavenumber range 16.00 to 18.38. Row 3: Shell no. 10, Wavenumber range 18.38 to 21.11. Row 4: Shell no. 11, Wavenumber range 21.11 to 24.25. Row 5: Shell no. 13, Wavenumber range 27.86 to 32.00. Row 6: Shell no. 15, Wavenumber range 36.76 to 42.22. Row 7: Shell no. 19, Wavenumber range 64.00 to 73.52.

To further compare the strength of local and non-local transfer rates for the two channels of the Hall term, we plot $\mathcal{H}^{b, b}_{n, m}$ and $\mathcal{H}^{b, j}_{n, m}$ as functions of $m$ for $n=5, 9, 11, 13, 15$ and $19$ (see figures 4 c and 4 d). The corresponding wavenumber ranges are summarised in table 2. Such a diagram is useful to recognise the principal contributors for a fixed receiver shell $n$ . For every single $n$ , the peaks of the transfer are situated at neighbouring shells, indicating a predominant local transfer for $\mathcal{H}^{b, b}_{n, m}$ . In addition, the amplitudes of the local transfers increase with receiver shell number $n$ running from $5$ to $15$ . A similar trend is also observed for the negative transfers at large $m$ . These two features offer an alternative way to delineate the increasing strength of both the local and non-local transfers at small scales. For $n = 19$ , however, the amplitude for the local transfer is found to decrease along with a substantial non-local contribution from small values of $m$ . For $\mathcal{H}^{b, j}_{n, m}$ , the situation is notably different. For $n\lt 15$ , we observe a comparatively weaker inverse transfer made up of comparable contributions both from local and non-local giver shells. For $n = 15$ and beyond, the direction of the energy transfer flips and local contributions become quite large with respect to the non-local contributions. Interestingly, similar to the $\mathcal{H}^{b, b}_{n, m}$ , here also a considerable (both local and non-local) transfer is observed around $m, n = 3$ .

In order to exclusively capture the contributions from the local shells, we calculate the total magnetic energy flux rate up to second nearest neighbour shell (see figures 5 a and 5 c). As an example, for receiver shell $n$ , the total flux rate is computed for giver shells $m = n-1$ and $m = n-2$ . Note that there is no standard threshold value to distinguish between the local and the non-local transfers. For the present study, a rough idea about the extent of the local transfers can be obtained from the S2S transfers in MHD (see figure 3). As it is apparent, for a given receiver shell $n$ , the transfers die down beyond $m = n\pm 2$ . In our work, this is taken as a benchmark for designating local and non-local transfers. In fact, from figures 4(c) and 4(d), one can see that, even for the Hall term, such a choice is justified as the nearby contributions to a shell $n$ mostly come from $m = n\pm 2$ before the first fall of the transfer rates. As expected, for lower wavenumbers, the local contribution for the channel $\mathcal{BB}$ is stronger compared with the channel $\mathcal{JB}$ . Around shell number 15, the local contribution of the latter sharply increases and becomes positive, signifying direct S2S transfer. Evidently, the local contributions for the channel $\mathcal{JB}$ are primarily responsible for the observed enhancement of magnetic energy transfer towards smaller scales discussed previously. In contrast, the non-local contributions for both the channels $\mathcal{BB}$ and $\mathcal{JB}$ are inverse and of comparable magnitudes for shells below $ n = 15$ . Beyond that, both become direct, with $\mathcal{H}^{b, b}_{n, m}$ being slightly greater than $\mathcal{H}^{b, j}_{n, m}$ (see figures 5 b and 5 d).

Figure 5.

Panels (a) and (b) show the local and non-local magnetic energy transfer rates due to $\mathcal{H}^{b, b}_{n, m}$ (plotted in blue colour) as a function of $n$ , respectively, and (c) and (d) show similar plots for $\mathcal{H}^{b, j}_{n, m}$ (in red colour). For a given $n$ , the local transfer rates are computed only for $m=n-1$ and $n-2$ .

Figure 5. Long description

The image contains four scatter plots labeled (a), (b), (c), and (d). Each plot shows the magnetic energy transfer rates as a function of n. Panel A: The scatter plot shows the local magnetic energy transfer rate due to a variable plotted in blue color. The x-axis is labeled n, and the y-axis ranges from -0.005 to 0.005. The data points show a trend that starts near zero, fluctuates slightly, and then increases. Panel B: The scatter plot shows the non-local magnetic energy transfer rate due to the same variable plotted in blue color. The x-axis is labeled n, and the y-axis ranges from -0.005 to 0.005. The data points show a trend that starts near zero, fluctuates slightly, and then increases. Panel C: The scatter plot shows the local magnetic energy transfer rate due to another variable plotted in red color. The x-axis is labeled n, and the y-axis ranges from -0.005 to 0.005. The data points show a trend that starts near zero, fluctuates slightly, and then increases. Panel D: The scatter plot shows the non-local magnetic energy transfer rate due to the same variable plotted in red color. The x-axis is labeled n, and the y-axis ranges from -0.005 to 0.005. The data points show a trend that starts near zero, fluctuates slightly, and then increases.

Figure 6.

Value of $|\mathcal{H}^{b, j}_{n, m}| - |\mathcal{H}^{b, b}_{n, m}|$ as a function of giver–receiver shell numbers $(m, n)$ .

Finally, to summarise the relative strength of the two channels of the Hall term, in figure 6, we plot $|\mathcal{H}^{b, j}_{n, m}| - |\mathcal{H}^{b, b}_{n, m}|$ as a function of $(m, n)$ . For a given $(m, n)$ , positive and negative values of the transfer rate denote the dominance of $|\mathcal{H}^{b, j}_{n, m}|$ and $|\mathcal{H}^{b, b}_{n, m}|$ , respectively. As it is apparent, local transfers at smaller scales (high values of $m$ and $n$ ) are dominated by $\mathcal{H}^{b, j}_{n, m}$ whereas large-scale (low values of $m$ and $n$ ) local transfers are primarily governed by $\mathcal{H}^{b, b}_{n, m}$ . For the non-local transfers, on the other hand, the regions with $m \gt n$ are dominated by $\mathcal{H}^{b, j}_{n, m}$ whereas $\mathcal{H}^{b, b}_{n, m}$ becomes important for $m \lt n$ . A possible insight into the different natures of the transfer for the two channels can be obtained by expressing the transfers in terms of two-point correlators in physical space. To achieve this, we start with the two-point magnetic energy correlator $\mathscr{R}_b = \langle {\boldsymbol{b}}\boldsymbol{\cdot }{\boldsymbol{b}}' \rangle /2$ , where primed and unprimed quantities denote variables at positions ${\boldsymbol{x}} + \boldsymbol{r}$ and $\boldsymbol{x}$ , respectively, and $\langle (\boldsymbol{\cdot })\rangle$ denotes statistical average (equivalent to space average for homogeneous turbulence). Using (2.2), the total contribution to the magnetic energy transfer due to the Hall term can be written as $\mathscr{T}_{b\rightarrow b}^{H} + \mathscr{T}_{j\rightarrow b}^{H}$ where

(3.1) \begin{align} \mathscr{T}_{b\rightarrow b}^{\kern2.2pt H} &= \frac {d_i}{2}\langle {\boldsymbol{b}}\boldsymbol{\cdot }[({\boldsymbol{j}}'\boldsymbol{\cdot }\boldsymbol{\nabla }'){\boldsymbol{b}}']+{\boldsymbol{b}}'\boldsymbol{\cdot }\left [\left ({\boldsymbol{j}}\boldsymbol{\cdot }\boldsymbol{\nabla }\right ){\boldsymbol{b}}\right ] \rangle \;\; \text{and}\\[-12pt]\nonumber \end{align}

(3.2) \begin{align} \mathscr{T}_{j\rightarrow b}^{\kern2.2pt H} &= \frac {d_i}{2}\langle -{\boldsymbol{b}}\boldsymbol{\cdot }[({\boldsymbol{b}}'\boldsymbol{\cdot }\boldsymbol{\nabla }'){\boldsymbol{j}}']-{\boldsymbol{b}}'\boldsymbol{\cdot }[({\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{\nabla }){\boldsymbol{j}}]\rangle \nonumber \\[0pt] &= \frac {d_i}{4} \langle -{\boldsymbol{b}}'\boldsymbol{\cdot } [ ({\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{\nabla } ){\boldsymbol{j}} ]-{\boldsymbol{b}}\boldsymbol{\cdot } [ ({\boldsymbol{b}}'\boldsymbol{\cdot }\boldsymbol{\nabla }' ){\boldsymbol{j}}' ] -{\boldsymbol{j}}'\boldsymbol{\cdot } [ ({\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{\nabla } ){\boldsymbol{b}} ]-{\boldsymbol{j}}\boldsymbol{\cdot } [ ({\boldsymbol{b}}'\boldsymbol{\cdot }\boldsymbol{\nabla }' ){\boldsymbol{b}}' ] \rangle \nonumber \\[0pt] &-\frac {d_i}{4} \langle {\boldsymbol{b}}\boldsymbol{\cdot } [ ({\boldsymbol{j}}'\boldsymbol{\cdot }\boldsymbol{\nabla }' ){\boldsymbol{b}}' ]+{\boldsymbol{b}}'\boldsymbol{\cdot } [ ({\boldsymbol{j}}\boldsymbol{\cdot }\boldsymbol{\nabla } ){\boldsymbol{b}} ] \rangle \end{align}

are the transfer rates corresponding to the channels $\mathcal{BB}$ and $\mathcal{JB}$ , respectively. One can associate the net energy flux rates due to the Hall term $\sum _{|{\boldsymbol{k}}'| \lt k}\widehat {\mathscr{T}_{b\rightarrow b}^{\kern2.2pt H}}({\boldsymbol{k}}')$ and $\sum _{|{\boldsymbol{k}}'| \lt k}\widehat {\mathscr{T}_{j\rightarrow b}^{\kern2.2pt H}}({\boldsymbol{k}}')$ with $\sum _{l \gt n}\sum _{m \leqslant n} \mathcal{H}^{b, b}_{l, m}$ and $\sum _{l \gt n}\sum _{m \leqslant n} \mathcal{H}^{b, j}_{l, m}$ , respectively, if the wavenumber $k$ belongs to the intermediate shell $n$ (Teimurazov et al. Reference Teimurazov, Stepanov, Verma, Barman, Kumar and Sadhukhan2018). Whereas the channel $\mathcal{BB}$ consists of a single type of transfer (see (3.1)), the channel $\mathcal{JB}$ is the combination of various type of transfers (see (3.2)). The non-monotonic nature of the channel $\mathcal{JB}$ transfer is probably stemming from the interplay between those different type of transfer terms. However, a straightforward conclusion on this remains difficult at the moment.

3.3. Calculation of mediator-specific S2S transfers

The S2S transfer rates, discussed so far, examine the locality of transfer between a pair of giver–receiver shells $(m, n)$ without any information regarding the location of the mediator mode $(q)$ . However, the knowledge of $q$ is essential to distinguish between local transfers (comparable giver and receiver wavenumbers) mediated by local ( $k\simeq p \simeq q$ ) and elongated triads $(k \simeq p \gg q)$ . For HD and MHD, local transfers via non-local triads are found to play important role in the inertial energy cascade (Alexakis et al. Reference Alexakis, Mininni and Pouquet2005; Mininni et al. Reference Mininni, Alexakis and Pouquet2006, Reference Mininni, Alexakis and Pouquet2008; Cho Reference Cho2010). In order to investigate the same due to the Hall term, we calculate the energy transfer rates as a function of $m, n$ and mediator shell $o$

(3.3) \begin{align} \mathcal{H}^{b, b}_{n, m, o}&=-d_i\sum _{|{\boldsymbol{k}}|\in n}\sum _{|{\boldsymbol{q}}|\in o}\sum _{|{\boldsymbol{p}}|\in m}^{\varDelta }\Im [({\boldsymbol{p}}\boldsymbol{\cdot }{\boldsymbol{j}}_{{\boldsymbol{q}}})({\boldsymbol{b}}_{\kern-1pt{\boldsymbol{p}}}\boldsymbol{\cdot }{\boldsymbol{b}}_{{\boldsymbol{k}}})], \end{align}

(3.4) \begin{align} \mathcal{H}^{b, j}_{n, m, o}&=d_i\sum _{|{\boldsymbol{k}}|\in n}\sum _{|{\boldsymbol{q}}|\in o}\sum _{|{\boldsymbol{p}}|\in m}^{\varDelta }\Im [({\boldsymbol{p}}\boldsymbol{\cdot }{\boldsymbol{b}}_{{\boldsymbol{q}}})({\boldsymbol{j}}_{\kern-1.5pt{\boldsymbol{p}}}\boldsymbol{\cdot }{\boldsymbol{b}}_{{\boldsymbol{k}}})]. \end{align}

In the top and middle rows of figure 7, we plot $\mathcal{H}^{b, b}_{n, m, o}$ and $\mathcal{H}^{b, j}_{n, m, o}$ for $m = 5, 10$ and $19$ , respectively. Due to the logarithmic binning in the inertial range, these choices of giver shells approximately correspond to wavenumbers $\approx 10, 20, 70$ which capture the three regions of $kd_i \lt 1$ , $kd_i \sim 1$ and $kd_i \gt 1$ , respectively (see table 2). For $\mathcal{H}^{b, b}_{n, m, o}$ , regions with weak local transfer are observed for $m = 5$ and close to $o = 5$ , corresponding to local triads where $k\simeq p\simeq q$ (see bottom left of figure 7 a). Note that, for a fixed $(m, n)$ , we define the triads to be local if $|o-m| \leqslant 2$ or $|o-n| \leqslant 2$ and non-local otherwise. Another region with a weak but non-zero non-local transfer can be identified for $(n, o) \gg m$ , which naturally associates highly non-local triads with $k \simeq q \gg p$ (from centre to top right of figure 7 a). For $m = 10$ , the most intense region of local transfer occurs for $o = 2$ and $3$ , associating non-local triads with $k \simeq p \gg q$ (deep red and blue squares at the bottom of figure 7 b). In this case, two regions of substantial non-local transfers are also evident: (i) $m\simeq o \gg n$ corresponding to triads $p \simeq q\gg k$ (blue and red squares at the centre left) and (ii) $n\simeq o \gg m$ corresponding to triads $k\simeq q\gg p$ (elongated region at top right). Finally, for $ m = 19$ , a similar region of moderate local transfers through non-local triads $k\simeq p\gg q$ is observed up to $o \approx 12$ (moderate blue and red squares at bottom right of figure 7 c). Beyond this, a horizontal spread towards smaller $n$ values emerges, indicating non-local energy transfer involving non-local triads where $p\simeq q\gg k$ .

Figure 7.

Top row: $\mathcal{H}^{b, b}_{n, m, o}$ as a function of mediator and receiver shell numbers $(o, n)$ for a giver shell number $m$ . Middle row: $\mathcal{H}^{b, j}_{n, m, o}$ as a function of mediator and receiver shell numbers $(o, n)$ for a giver shell number $m$ . Bottom row: $|\mathcal{H}^{b, j}_{n, m, o}| - |\mathcal{H}^{b, b}_{n, m, o}|$ .

Figure 7. Long description

A heat map representing the distribution of values across different shell numbers for various parameters in plasma turbulence. The heat map is divided into three rows and three columns, each panel showing a different set of conditions. Panel A: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 5. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively. Panel B: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 10. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively. Panel C: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 19. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively. Panel D: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 5. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively. Panel E: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 10. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively. Panel F: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 19. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively. Panel G: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 5. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively. Panel H: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 10. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively. Panel I: Heat map showing the values of a parameter as a function of mediator and receiver shell numbers for a giver shell number of 19. The x-axis represents the receiver shell number, and the y-axis represents the mediator shell number. The color scale ranges from blue to red, indicating lower to higher values, respectively.

For $m = 5$ , unlike the previous case, local transfers of $\mathcal{H}^{b, j}_{n, m, o}$ with local triads become significant close to $o = 5$ . As with $\mathcal{H}^{b, b}_{n, m, o}$ , the non-local transfer for $n\simeq o \gg m$ remains extremely weak for this channel (see figure 7 d). For $m=10$ , similar regions of non-negligible local transfer via non-local triads $k\simeq p\gg q$ and non-local transfer with non-local triads $p\simeq q\gg k$ (dark red and blue squares in the centre left of figure 7 e) are also observed. However, in contrast to $\mathcal{H}^{b, b}_{n, m, o}$ , no substantial non-local transfer is evident for the $k\simeq q\gg p$ triads (elongated top right section). For $m = 19$ , a region of strong local transfer mediated by non-local triads persists up to $o\approx 12$ . Beyond this, a horizontal spread of diminishing intensity towards smaller $n$ values is observed, suggesting non-local energy transfer through triads with $p\simeq q\gg k$ (see figure 7 f).

Finally, to compare the relative strengths of $\mathcal{H}^{b, b}_{n, m, o}$ and $\mathcal{H}^{b, j}_{n, m, o}$ , we compute the difference $|\mathcal{H}^{b, j}_{n, m, o}| - |\mathcal{H}^{b, b}_{n, m, o}|$ for $m = 5, 10$ and $19$ (figure 7 bottom row). For each $(o, n)$ pair, positive values indicate the dominance of $|\mathcal{H}^{b, j}_{n, m, o}|$ whereas negative values indicate dominance of $|\mathcal{H}^{b, b}_{n, m, o}|$ . As is evident, for $m = 5$ , the strengths of the local transfers mediated by local triads are comparable for the two channels despite a slight dominance of $|\mathcal{H}^{b, b}_{n, m, o}|$ in the region governed by non-local transfers (elongated top right section of figure 7 g). For $m = 10$ , $|\mathcal{H}^{b, b}_{n, m, o}|$ is stronger for regions of local transfer via non-local triads (bottom middle of figure 7 h) as well as in the non-local transfer regime via non-local triads $k\simeq q\gg p$ (see elongated top right section). However, $|\mathcal{H}^{b, j}_{n, m, o}|$ takes over in the non-local transfer regime associated with $p\simeq q\gg k$ triads (see left middle section). Finally, for $m = 19$ , $|\mathcal{H}^{b, j}_{n, m, o}|$ clearly dominates in the region of local transfer through non-local triads (dark red squares at bottom right of figure 7 i). However, $|\mathcal{H}^{b, b}_{n, m, o}|$ slightly takes over $|\mathcal{H}^{b, j}_{n, m, o}|$ in the non-local energy transfer regime involving triads with $p\simeq q\gg k$ (see moderate blue squares at centre top).

3.4. Quantitative estimation using power laws

Figure 8.

(a) Schematic diagram of contributing triads involved in local and non-local transfers and compensated spectrum for (b) magnetic energy as a function of wavenumber $k$ .

In the preceding discussion, we have identified different types of triads that contribute to the local and non-local transfer of magnetic energy due to the Hall term. These are broadly classified into four categories: (i) $k\simeq p\simeq q$ , (ii) $k\simeq p\gg q$ , (iii)  $k\simeq q \gg p$ and (iv) $p\simeq q \gg k$ (see figure 8 a). Triads of type (i) and (ii) are primarily involved in local transfers whereas types (iii) and (iv) participate in non-local transfers. For a given triad $({\boldsymbol{k}}, {\boldsymbol{p}}, {\boldsymbol{q}})$ , we can define the ratio

(3.5) \begin{equation} \mathcal{R} = \frac {|\mathcal{H}^{b, b}_{n, m, o}|}{|\mathcal{H}^{b, j}_{n, m, o}|} \approx \frac {d_i p j_q b_p b_k}{d_i p b_q j_p b_k} = \frac {j_q b_p}{b_q j_p} , \end{equation}

where $\mathcal{R}$ can be taken as a measure of relative strength of the two channels. Assuming a power-law spectrum for the magnetic energy, one can write $E_b (k)\sim b_k^2 \sim k^{-\alpha } \implies b_k \sim k^{-\alpha /2}$ , where $\alpha = 5/3$ and $7/3$ for $k d_i \leqslant 1$ and $k d_i \gt 1$ , respectively (see figure 8 b). Due to the algebraic relation ${\boldsymbol{j}}_{{\boldsymbol{k}}} = i ({\boldsymbol{k}}\times {\boldsymbol{b}}_{{\boldsymbol{k}}}) \implies\! j_k^2 = k^2 b_k^2$ , leading to $j_k \sim k^{(1-\alpha /2)}$ . Using this in the expression (3.5), we get $\mathcal{R} = q/p$ irrespective of the wavenumber range. The local triads of type (i) lead to $\mathcal{R} \sim 1$ . Interestingly, this is similar to the patterns observed in the S2S transfer rates in figure 7 with $m = 10$ and $19$ . For $m = 5$ , the agreement is weaker, which may be due to the fact that the wavenumber region containing triads of type (i) is not completely free from the external forcing effect. In contrast, local transfers via non-local triads of type (ii) result in $\mathcal{R} \ll 1$ , indicating stronger transfer in the channel $\mathcal{JB}$ . This is also similar to the nature of S2S transfer rates in figures 7(c), 7(f) and 7(i). Triads of type (iii) give $\mathcal{R} \gg 1$ , leading to comparatively stronger transfer in the channel $\mathcal{BB}$ , consistent with observed enhanced values seen in the top right regions of all the sub-figures of figure 7 with $m = 5$ and $10$ . Finally, type (iv) triads give $\mathcal{R} \sim 1$ , similar to the patterns seen in the top left regions of figures 7(c), 7(f) and 7(i). Apart from the aforementioned triad configurations, another type of non-local triad with $k \simeq p \ll q$ is also possible, which leads to $\mathcal{R}\gg 1$ . However, due to the law of cosines mentioned earlier, such triads are practically small in number and thus contribute negligibly, as manifested by the negligible transfer rates observed in figure 7 for $m = 5$ (top left) and $10$ (top middle). For a better agreement between numerical observations and theoretical results we need a more rigorous analytical framework combined with high-resolution simulations.

3.5. Effect of the Hall wavenumber

Figure 9.

(a) Kinetic and magnetic energy spectra and (b) total energy flux rate, with and without the Hall contribution, as a function of $k$ for the simulation with $k_H = 50$ .

Figure 10.

Shell-to-shell transfer rates for the (a) $\mathcal{BB}$ and (b) $\mathcal{JB}$ channels for the run with $k_H = 50$ .

To assess the effect of the Hall wavenumber on the observed results, we run an additional simulation with parameters identical to those listed in table 1, except for $d_i = 0.02$ , corresponding to $k_H \approx 50$ . In the new simulation, we have found an extended inertial range for MHD associated with kinetic and magnetic energy spectral exponents of $-1.5$ and $-1.6$ , respectively (figure 9 a). This supports our interpretation that the previously observed deviations in the $k_H \approx 20$ run are primarily due to the limited separation between $k_f$ and $k_H$ . Despite a weaker $\varPi _{Hall}$ for the region $k\gt k_H$ , here, we also observe a range of intermediate wavenumbers where $\varPi _{Total}$ closely follows $\varPi _{inj}$ , thus confirming a Kolmogorov-like universal regime for the total energy cascade (figure 9 b). Finally, to examine the effect of $k_H$ on transfer rates, we plot S2S transfer rates for the $\mathcal{BB}$ and $\mathcal{JB}$ channels in figures 10(a) and 10(b), respectively. Similar to figure 4(a), the channel $\mathcal{BB}$ shows a consistent local inverse transfer for all wavenumbers while a direct non-local transfer is observed at small scales. For the channel $\mathcal{JB}$ , similar to figure 4(b), weak local and non-local inverse transfers are observed up to a certain shell region beyond which the transfer becomes considerably stronger and corresponds to direct magnetic energy transfer. This transition wavenumber (or the shell number) is now shifted towards a higher value with respect to the previous simulation. This analysis clearly shows that the main conclusion of the current work remains effectively unchanged even with the new Hall wavenumber.