4.1. Role of the magnetic guide field in the presence of $1/f$ spectrum

As previously mentioned, the simulations performed in this work were analysed only after a statistically stationary regime had been reached, in which the total energy and the total dissipation rate of the system (not shown here) exhibit a stable temporal evolution. In particular, we monitored the full time series of the kinetic and magnetic energies, $E_k(t)$ and $E_b(t)$ , for both magnetic guide field amplitudes. Figures 2(a) and 2(b) show the temporal evolution of these quantities during the simulation, after discarding the initial transient. In both cases, the energies fluctuate around well-defined mean values over the entire integration interval. Figure 2 also reveals a qualitative difference between the two magnetic guide field runs. For Run I (panel a), both $E_k$ and $E_b$ display comparatively faster and more irregular fluctuations, with shorter-time variations superimposed on the mean level. By contrast, for Run II (panel b), the global energies evolve more smoothly and exhibit broader, longer-time modulations.

Figure 2.

Temporal evolution of the kinetic and magnetic energies, $E_k(t)$ and $E_b(t)$ , for (a) Run I ( $B_0=1$ ) and (b) Run II ( $B_0=6$ ), after discarding the initial transient. Solid lines show the instantaneous energy time series, while dashed lines correspond to running averages computed over a fixed number of points and are included for visualisation purposes, highlighting the large-scale temporal trends in the energy evolution.

To quantify the energy partition, we computed the normalised residual energy over the statistically stationary interval as $\sigma _r=(\langle E_k\rangle -\langle E_b\rangle )/(\langle E_k\rangle +\langle E_b\rangle )$ . We obtain $\sigma _r=-0.28$ for Run I and $\sigma _r=0.02$ for Run II, indicating a magnetically dominated state in the former and a state close to kinetic-magnetic equipartition in the latter. Because the two runs are driven identically, this difference cannot be attributed to a change in the forcing scheme. However, with a single realisation for each guide-field amplitude, we regard it as a property of the statistically steady states reached in the present simulations rather than as a general consequence of increasing $B_0$ .

Figure 3.

Compensated magnetic energy spectra measured by the virtual probes for Run I ( $B_0=1$ , first row) and Run II ( $B_0=6$ , second row). Each column corresponds to a fixed probe velocity, normalised by the Alfvén speed, $V/V_A$ . The coloured curves represent different sampling angles between the probe trajectory and the magnetic guide field, from $0^\circ$ to $90^\circ$ , as indicated by the colourbar. For clarity, a vertical offset has been applied between curves. The grey shaded regions denote the frequency intervals used to identify the candidate $1/f$ ranges, while the black dashed and dotted vertical lines indicate the correlation frequency $f_c$ and the Taylor frequency $f_\lambda$ , respectively. The third row shows the locally fitted spectral slope $\alpha (f)$ for $\theta =0^\circ$ and $\theta =90^\circ$ , using the same colours as in the spectra. Dashed lines correspond to Run I and solid lines to Run II. The grey dashed and dotted horizontal lines indicate the reference slopes $\alpha =-1$ and $\alpha =-5/3$ , respectively. The vertical dashed and dotted lines indicate the corresponding $f_c$ and $f_\lambda$ , respectively, with lighter and darker grey lines referring to Run I and Run II. For the cases exhibiting the most extended $1/f$ ranges, corresponding to the highest probe velocity and perpendicular sampling, the locally fitted slopes within the candidate $1/f$ intervals are $\alpha \simeq -0.96$ for Run I and $\alpha \simeq -0.98$ for Run II.

Figure 3. Long description

Panel A: A line graph shows compensated magnetic energy spectra for Run I with varying probe velocities normalized by the Alfvén speed. Colored curves represent different sampling angles between the probe trajectory and the magnetic guide field, ranging from 0 to 90 degrees. Panel B: A similar line graph for Run I with a higher probe velocity. Panel C: A line graph for Run II with a lower probe velocity. Panel D: A line graph for Run II with a higher probe velocity. Panel E: A line graph for Run II with an even higher probe velocity. Panel F: A line graph for Run II with the highest probe velocity. Panel G: A line graph for Run II with the highest probe velocity and perpendicular sampling. Panel H: A line graph for Run II with the highest probe velocity and perpendicular sampling. Panel I: A line graph shows the locally fitted spectral slope for Run I and Run II, using the same colors as in the spectra. Dashed lines correspond to Run I and solid lines to Run II. The grey shaded regions denote the frequency intervals used to identify the candidate ranges, while the black dashed and dotted vertical lines indicate the correlation frequency and the Taylor frequency, respectively. The grey dashed and dotted horizontal lines indicate the reference slopes and -5/3, respectively. The vertical dashed and dotted lines indicate the corresponding and, respectively, with lighter and darker grey lines referring to Run I and Run II.

Figure 3 shows the average (in probes) temporal magnetic power spectral density (PSD) measured by the virtual probes for a range of angles and probe velocities, for Run I (panels a–d) and Run II (panels e–h). Each panel displays the compensated magnetic spectrum as a function of frequency for a fixed value of the probe velocity $V/V_A$ , normalised by the Alfvén speed $V_A$ . The different coloured curves correspond to different angles $\theta$ between the probe trajectory and the mean magnetic field, ranging from $0^\circ$ (light green) to $90^\circ$ (blue). For visualisation purposes, the spectra have been vertically offset, allowing the angular dependence of the spectral trends to be more clearly distinguished. The grey shaded region indicates the frequency interval considered as a candidate $1/f$ range, while the vertical dashed and dotted lines mark the correlation frequency $f_c$ and the Taylor frequency $f_\lambda$ , respectively. The correlation frequency $f_c$ is determined from a local spectral slope diagnostic. For each temporal spectrum, the local slope $\alpha (f)$ is computed within a centred moving frequency window by performing a linear least-squares fit of $\log _{10}E(f)$ as a function of $\log _{10}f$ . The resulting $\alpha (f)$ curve is then used to identify $f_c$ as the transition from the low-frequency, near- $1/f$ behaviour towards the steeper spectral range (e.g. Ghanbari & Florinski Reference Ghanbari and Florinski2023). For each fixed probe velocity, the value of $f_c$ shown in figure 3 corresponds to the median value over the different sampling angles. To quantify its angular variability, we also compute the normalised standard deviation of the $f_c(\theta )$ values, defined as the standard deviation divided by the median value of $f_c$ . For Run I, no angular dependence is found for $V/V_A=1$ , 2 and 4, while at $V/V_A=6$ , the normalised spread is approximately $21\,\%$ . For Run II, no angular variation is resolved for $V/V_A=0.7$ and 1, whereas the corresponding values are approximately $16\,\%$ and $21\,\%$ for $V/V_A=2$ and 3, respectively. The larger angular spread at the highest probe velocities is consistent with the stronger dependence of the temporal spectra on the sampling direction in these cases. Meanwhile, the Taylor frequency is computed (for each temporal spectrum) as $f_\lambda =(\int f^2 \textrm {PSD}(f)\,{\rm d}f/\int \textrm {PSD}(f)\,{\rm d}f )^{1/2}$ and the value shown corresponds to the median over sampling angles for each fixed probe velocity. This definition is the spectral counterpart of the Taylor scale estimated by Chuychai et al. (Reference Chuychai, Weygand, Matthaeus, Dasso, Smith and Kivelson2014). This frequency is therefore used as a temporal reference scale to indicate the transition towards smaller-scale fluctuations within the probe spectrum.

In this representation, a horizontal plateau indicates a spectral scaling close to $E(f)\propto f^{-1}$ , so that the extent of the plateau within the shaded region provides a direct measure of the presence of the $1/f$ range. The comparison between panels therefore allows us to evaluate how this low-frequency regime depends on the guide field amplitude, the probe speed and the sampling direction. In addition, in figure 3(i–l), the locally fitted spectral slope is shown as a function of frequency, which provides a complementary diagnostic of the frequency intervals where the spectra are consistent with a $1/f$ scaling or depart towards steeper slopes.

We first consider Run I, for which the spectra (panels a–d) exhibit only a limited low-frequency plateau and a weak dependence on the sampling direction, particularly at the lowest probe speeds. For $V/V_A=1$ , the spectra corresponding to different angles remain similar over a broad frequency interval. Even at higher relative probe speeds, however, the development of an extended $1/f$ range remains limited. Although some curves at larger angles display a flatter low-frequency behaviour, the plateau remains narrow and is followed by a transition towards steeper slopes. Moreover, in panels (i–l), the local slopes show only short intervals close to $\alpha =-1$ , followed by a rapid departure towards more negative values, and the separation between different sampling directions remains weak. Thus, in Run I, increasing the probe speed alone is not sufficient to produce a broad $1/f$ interval across all directions. This suggests that, for a weak guide field, the low-frequency spectral organisation is weaker and the $1/f$ regime is less developed. Under these conditions, the distinction between parallel and perpendicular sampling is reduced, indicating weaker anisotropy of the fluctuations.

A different behaviour is found in Run II, where the dependence on the sampling direction is much clearer. For all explored probe velocities (panels e–h), the spectra display a systematic ordering: as the trajectories become more perpendicular to the mean magnetic field, the compensated spectra become flatter at low frequencies and the corresponding plateau extends over a wider interval. This trend is already visible at the lowest velocity shown, $V/V_A=0.7$ (panel e), and becomes progressively more pronounced as the probe speed increases to $V/V_A=1$ , $2$ and $3$ (panels f–h). Thus, the extension of the candidate $1/f$ range increases not only with the degree of perpendicular sampling, but also with the relative probe velocity. The curves associated with nearly perpendicular trajectories show the widest plateaus, while more parallel samplings depart earlier from the low-frequency regime. This is further supported by the local spectral slopes (panels i–l) exhibiting an extended interval close to $\alpha =-1$ for the highest velocities and most perpendicular directions, followed by a transition to the inertial range. Therefore, in the presence of a strong guide field, the observed temporal spectrum is controlled by the combined effect of sampling geometry and probe speed, with faster perpendicular trajectories favouring the clearest development of a broad $1/f$ interval.

To quantify more explicitly the extent of the temporal $1/f$ behaviour, we divided the frequency axis into adjacent, non-overlapping intervals of equal logarithmic width and determined the widest contiguous interval compatible with an $f^{-1}$ spectrum. An interval was considered resolved when its global spectral slope remained within a prescribed tolerance around $\alpha =-1$ , the deviations from the corresponding power-law fit were sufficiently small and the slopes obtained over shorter subintervals remained locally stable. This procedure prevents an isolated crossing of $\alpha =-1$ from being interpreted as an extended $1/f$ range. For the most favourable sampling conditions, corresponding to the highest probe velocity and nearly perpendicular trajectories, the maximum logarithmic width is approximately $\simeq 0.28$ decade for Run I ( $B_0=1$ ) and $\simeq 0.65$ decade for Run II ( $B_0=6$ ). By contrast, no resolved $1/f$ interval is detected for the least favourable cases, corresponding to the lowest probe velocity and parallel sampling, in either run. In these cases, no contiguous frequency interval simultaneously satisfies the slope, fit-quality and local-stability conditions required by our diagnostic.

As an additional control, we also examined the temporal spectra measured by fixed probes ( $V=0$ ) in both simulations (not shown). The compensated spectra exhibit, at most, only a very narrow interval compatible with an $f^{-1}$ scaling, which does not satisfy our present criterion for a resolved $1/f$ range for either Run I or Run II. By contrast, the near- $1/f$ interval becomes progressively broader as the probe velocity increases, particularly for the strong guide field case. This comparison supports the interpretation that probe motion plays an important role in the development of the extended temporal $1/f$ ranges observed in the simulations. At the same time, these results do not exclude a possible contribution from the intrinsic long-time dynamics of the large-scale modes. In contrast, they highlight the relevance of probe motion in broadening and enhancing the observed $1/f$ signature.

Moreover, we analysed the display of the full angular dependence in figure 4, where $\alpha (\theta ,f)$ is shown for all sampled angles. The dashed contours delimit the regions in which the local spectral slope satisfies $|\alpha +1|\leqslant 0.2$ , corresponding to spectra locally compatible with an $f^{-1}$ scaling. For Run I, the contoured regions remain relatively narrow and show only a weak systematic dependence on $\theta$ . In contrast, for Run II, they broaden towards larger sampling angles and higher values of $V/V_A$ , further confirming that the clearest and most extended $1/f$ -like intervals are obtained for higher probe velocities and nearly perpendicular sampling.

Figure 4.

Angular dependence of the local spectral slope $\alpha (\theta ,f)$ , defined through $E(f)\propto f^{\alpha }$ . The top row corresponds to Run I ( $B_0=1$ ), with $V/V_A=1$ , $2$ , $4$ and $6$ , whereas the bottom row corresponds to Run II ( $B_0=6$ ), with $V/V_A=0.7$ , $1$ , $2$ and $3$ . The dashed contours delimit the regions in which the local spectral slope satisfies $|\alpha +1|\leqslant 0.2$ .

To complement the spectral analysis discussed previously, we constructed two-dimensional maps to quantify how clearly a $1/f$ range develops as a function of probe velocity and sampling angle. Although compensated spectra provide a direct visual indication of flat low-frequency plateaus, a quantitative criterion is useful to compare all sampled configurations in a systematic way. For this purpose, we define a $1/f$ score based on the compensated temporal magnetic spectrum, $Y(f)=f\kern-2.5pt E(f)$ , within a prescribed candidate frequency band.

The score is obtained by searching for locally $1/f$ -like intervals within a prescribed frequency band, hereafter referred to as the search band, through sliding window linear fits in log–log coordinates. These search bands are given by $0.7\leqslant f\leqslant 5.0$ and $1.8\leqslant f\leqslant 8.1$ for Run I and Run II, respectively. Local fits are performed using five spectral points, with consecutive windows displaced by one spectral point. A window is considered compatible with a candidate $1/f$ -like signature when its local slope satisfies $|m_{\textrm {loc}}|\leqslant 0.2$ , corresponding to $|\alpha +1|\leqslant 0.2$ for the underlying PSD. We additionally require $Y_{\textrm {med}}/Y_{\max }\geqslant 0.45$ , where $Y_{\textrm {med}}$ is the median compensated amplitude within the local window and $Y_{\max }$ is the maximum compensated amplitude within the search band. This condition prevents low-amplitude portions of the spectrum from being selected solely because they are locally flat. Finally, the r.m.s. residual scatter of the local fit is required to satisfy a prescribed threshold given by $\sigma _{\textrm {loc}}\leqslant 0.08$ and $\sigma _{\textrm {loc}}\leqslant 0.06$ for Run I and Run II, respectively. If several consecutive windows satisfy these conditions within the search band, the longest contiguous sequence defines the candidate interval $[f_1,f_2]$ . The frequency extent of the candidate interval is measured in logarithmic units as

(4.1) \begin{align} \mathrm{width}_{\textrm {dec}}=\log _{10}\left (\frac {f_2}{f_1}\right ). \end{align}

The final score is then defined as

(4.2) \begin{align} S=C^p F R, \end{align}

where

(4.3) \begin{align} C=\frac {\mathrm{width}_{\textrm {dec}}}{\log _{10}(f_{\textrm {max}}/f_{\textrm {min}})} \end{align}

is the relative coverage of the candidate band, while the flatness ( $F$ ) and smoothness ( $R$ ) factors are

(4.4) \begin{align} F= \frac {1}{1+\left (|m|/m_{\textrm {ref}}\right )^2} \end{align}

and

(4.5) \begin{align} R= \frac {1}{1+\left (\sigma /\sigma _{\textrm {ref}}\right )^2}. \end{align}

Here, $m$ and $\sigma$ are respectively the slope and r.m.s. residual obtained from a global log–log fit over the selected candidate interval. Since each sampled configuration $(V/V_A,\theta )$ may yield a different $[f_1,f_2]$ , the normalisation values $m_{\textrm {ref}}$ and $\sigma _{\textrm {ref}}$ are defined as the 75th percentiles of the distributions of $|m|$ and $\sigma$ obtained from all candidate intervals identified using the reference diagnostic parameters. These normalisation values are then kept fixed. The exponent $p$ controls the relative weight assigned to the spectral coverage and we retain the reference value $p=1.3$ . Thus, the score increases for candidate intervals that occupy a larger fraction of the search band, remain closer to an $f^{-1}$ scaling and exhibit smaller deviations from a power-law fit. A single valid local window is sufficient to produce a non-zero score. We refer to a sampled $(V/V_A,\theta )$ configuration with $S\gt 0$ as a score point. Therefore, the identification of a score point indicates the presence of a local candidate and does not, by itself, imply the existence of a resolved $1/f$ range, which is assessed independently using a more restrictive criterion described earlier.

Figure 5.

Two-dimensional maps of the $1/f$ score for (a) Run I ( $B_0=1$ ) and (b) Run II ( $B_0=6$ ), as a function of the sampling angle $\theta$ and the normalised probe velocity $V/V_A$ . For visualisation purposes, black contour lines indicate selected score levels.

The resulting score maps are shown in figures 5(a) and 5(b). To facilitate the visualisation of the global trends, the discrete score values were smoothed using a two-dimensional kernel filter and the resulting fields are displayed together with contour lines indicating selected score levels. In Run I (panel a), the score remains generally modest over most of the explored parameter space. The lowest values are found at the smallest probe velocities, where the spectra show only a very limited low-frequency flattening. As $V/V_A$ increases, the score also tends to increase, indicating that the detected $1/f$ -like interval becomes somewhat broader and better defined. However, the angular dependence remains weak: rather than showing a systematic enhancement towards nearly perpendicular sampling, the score is relatively similar across $\theta$ for a given velocity, consistent with a more isotropic distribution of fluctuations. A different pattern is observed in Run II (panel b), as the score map exhibits a clear and systematic increase with both $V/V_A$ and $\theta$ . The lowest scores are again associated with slow and nearly parallel sampling, but the score rises steadily as the trajectories become faster and more perpendicular to the guide field. This trend is clearly highlighted by the contour lines, which show a progressive displacement of higher score levels towards larger $V/V_A$ and $\theta$ . This behaviour indicates that, in the strong guide field regime, the detected $1/f$ -like intervals are not only flatter, but also more extended and smoother than in Run I. In other words, the score map quantitatively confirms that a strong field, combined with sufficiently fast and nearly perpendicular sampling, produces the clearest temporal signature of the $1/f$ range.

To assess the sensitivity of this diagnostic to its adjustable parameters, we varied each parameter independently around the reference configuration. Specifically, we considered coverage exponents $p=1.0$ and $1.6$ around the reference value $p=1.3$ , slope tolerances of $0.15$ and $0.25$ , amplitude thresholds $Y_{\textrm {med}}/Y_{\max }=0.35$ and $0.55$ , and residual thresholds varied by $\pm 25\,\%$ . We also displaced the candidate frequency interval by one spectral point while preserving the number of spectral points whenever permitted by the available frequency resolution.

For each set of diagnostic parameters $i$ , we define $\mathcal{C}_i$ as the set of score points. The overlap with the reference set $\mathcal{C}_0$ is quantified through the Jaccard index (Barnes et al. Reference Barnes, DeRosa, Jones, Arge, Henney and Cheung2023; Gonidakis et al. Reference Gonidakis2026; McClure et al. Reference McClure, Silva, Verth, Ballai and Fedun2026),

(4.6) \begin{align} J_i= \frac {|\mathcal{C}_0\cap \mathcal{C}_i|} {|\mathcal{C}_0\cup \mathcal{C}_i|}. \end{align}

The Jaccard index ranges from $0$ to $1$ , with $J=1$ indicating that the set of score points remains unchanged and $J=0$ corresponding to no overlap between the two sets. We additionally compute the Spearman rank correlation coefficient $\rho _{\textrm {S}}$ over score points common to the reference and modified cases to determine whether their relative ordering is preserved. In this case, $\rho _{\textrm {S}}=1$ indicates an identical ranking, $\rho _{\textrm {S}}=0$ indicates no monotonic relation and $\rho _{\textrm {S}}=-1$ corresponds to a completely reversed ordering. Changes associated with score points entering or leaving the set are therefore captured by $J$ , whereas $\rho _{\textrm {S}}$ measures the stability of the relative score distribution among the common points. When fewer than three common score points are available, no Spearman coefficient is reported. The results are summarised in table 1.

Table 1.

Sensitivity of the $1/f$ score to variations around the reference set of diagnostic parameters. $J$ is the Jaccard index relative to the reference case and $\rho _{\textrm {S}}$ is the Spearman rank correlation over common score points. A dash indicates that fewer than three common score points are available for a meaningful rank correlation. The low- $f$ and high- $f$ cases denote a one spectral point displacement of the complete reference search band towards lower and higher frequencies, respectively, while preserving the number of spectral points. For Run I, the low- $f$ displacement does not generate a distinct band and is therefore omitted.

Table 1. Long description

The table has two main sections labeled Run I and Run II, each with multiple rows and columns. The columns are labeled Variation, J, and ρS. Each run has a reference variation with p = 1.3 and several other variations with different parameters. The table lists the Jaccard index (J) and Spearman rank correlation coefficient (ρS) for each variation. For Run I, variations include changes in p, m_loc, Y_med/Y_max, σ_loc, and High-f band. For Run II, variations include changes in p, m_loc, Y_med/Y_max, σ_loc, Low-f band, and High-f band. The table shows the values of J and ρS for each variation, with some cells containing dashes indicating fewer than three common score points for meaningful rank correlation.

The sensitivity analysis reveals a clear difference between the two runs. For Run II ( $B_0=6$ ), the set of score points is highly stable against variations of the slope tolerance, amplitude threshold and residual threshold, with $J\gtrsim 0.93$ in all of these cases. The coverage exponent $p$ plays a different role because it enters only as a positive weighting of the coverage factor after the candidate interval has been identified. Therefore, varying $p$ cannot change whether a sampled configuration has $S\gt 0$ , and the corresponding $J=1$ values are expected by construction. Its non-trivial effect is instead assessed through the relative ordering of the score values. Changing $p$ from $1.3$ to $1.0$ or $1.6$ gives Spearman coefficients of $0.98$ and $0.99$ , respectively, showing that the score ranking is essentially unaffected by this choice. We retain the original choice $p=1.3$ in the present analysis. The candidate frequency interval produces the largest sensitivity, although the modified cases continue to overlap substantially with the reference set. Here, the low- $f$ and high- $f$ cases correspond to shifting the complete reference search band by one spectral point towards lower or higher frequencies, respectively, while preserving the number of spectral points. Importantly, these variations broaden or reduce the same overall portion of $(V/V_A,\theta )$ space rather than producing a qualitatively different trend, so the strongest $1/f$ -like signatures remain associated with sufficiently large $V/V_A$ and nearly perpendicular sampling.

Run I ( $B_0=1$ ) is more sensitive to the diagnostic parameters, particularly to the residual threshold and to the precise candidate frequency interval. This behaviour is consistent with the independently determined extent of the resolved $1/f$ -like interval. As discussed earlier, its maximum width is only approximately $0.28$ decade in Run I, so a modest modification of the accepted frequency range can strongly affect whether this narrow interval contributes to the score. By contrast, Run II develops a substantially broader interval, reaching approximately $0.65$ decade and, correspondingly, shows much greater stability under the same diagnostic variations. The sensitivity analysis therefore supports the distinction between the more marginal $1/f$ -like behaviour found in Run I and the substantially more developed signature obtained in Run II. In addition, we verified that the kernel filter used to display the score maps does not affect the conclusions of the analysis. The sensitivity metrics reported here are computed from the unsmoothed score values, and comparison with the filtered maps shows that the same overall distribution and trends in $(V/V_A,\theta )$ space are preserved.

Finally, we verified the robustness of these results with respect to numerical resolution by performing an analogous comparison using the corresponding $128^3$ simulations. The same overall trends with probe velocity and sampling angle are recovered, indicating that the conclusions drawn from the score analysis are not specific to the higher-resolution simulations.

4.2. Probe velocity and Taylor hypothesis

The results presented in the previous section suggest that the observed temporal spectra may be shaped, at least in part, by the way the probe samples the spatial structure of the turbulent fluctuations. We therefore examine to what extent the synthetic probe measurements are consistent with the Taylor hypothesis (Taylor Reference Taylor1938). In a few words, the frequency measured by a moving probe for an Alfvénic fluctuation is generally given by

(4.7) \begin{align} \omega _{\mathrm{sc}} = \pm k_{\parallel ,\mathrm{loc}}V_A + \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}, \end{align}

where $k_{\parallel ,\mathrm{loc}}$ is the component of the wavevector along the local magnetic field direction. Under the Taylor hypothesis, the temporal variations measured along the probe trajectory are assumed to be dominated by advection, so that the intrinsic temporal evolution of the turbulent structures can be neglected compared with the relative motion of the probe through the flow. Equation (4.7) then reduces to

(4.8) \begin{align} \omega _{\mathrm{sc}} \simeq \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}, \end{align}

which requires $\left |k_{\parallel ,\mathrm{loc}}\right |V_A \ll \left |\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{V}\right |$ . In this limit, the temporal spectrum measured by the probe can be directly associated with a reduced spatial spectrum along the sampling direction. This approximation is expected to perform better when the probe speed is sufficiently large compared with the characteristic propagation speed of the fluctuations. More generally, when the standard Taylor approximation is not strictly satisfied, the sampling angle relative to the local magnetic field and the spectral broadening associated with intrinsic plasma-frame dynamics become important for interpreting spacecraft-frame frequency spectra (Bourouaine & Perez Reference Bourouaine and Perez2020; Perez et al. Reference Perez, Bourouaine, Chen and Raouafi2021). In the present problem, this condition is naturally related to the ratio $V/V_A$ , since $V_A$ sets the relevant Alfvénic propagation speed of the system. A local magnetic field direction can be estimated along each probe trajectory. However, knowing $\widehat {\boldsymbol{B}}_{\mathrm{loc}}(t)$ does not determine the full wavevector $\boldsymbol{k}$ of the fluctuations contributing to the power measured at a given temporal frequency. In general, that power combines contributions from several wavevectors, orientations and intrinsic temporal evolutions, and therefore cannot be assigned a unique value of $k_{\parallel ,\mathrm{loc}}$ . Inferring such a spatial wavenumber from the probe series would require an additional correspondence between temporal and spatial scales, thereby reintroducing the same space–time mapping whose validity is being tested. For this reason, the following comparison is performed relative to the global guide field $\boldsymbol{B}_0=B_0\widehat {\boldsymbol{z}}$ , consistently with the definition of the available Eulerian spectra. In particular, the Eulerian spectrum denoted as parallel is a reduced spectrum in $k_{\parallel }=\boldsymbol{k}\boldsymbol{\cdot }\widehat {\boldsymbol{B}}_0$ , rather than a spectrum defined relative to the scale-dependent local magnetic field. Similarly, $k_\perp$ is defined with respect to $\boldsymbol{B}_0$ .

For probes moving parallel to the guide field, we approximate $k_{\parallel ,\mathrm{loc}}\simeq k_\parallel$ , so that (4.7) gives $\omega _{\mathrm{sc}}\simeq k_\parallel (V\pm V_A)$ . The two signs correspond to opposite Alfvénic propagation directions, which can be characterised through the Elsasser variables $\boldsymbol{z}^{\pm }=\boldsymbol{u}\pm \boldsymbol{b}$ . The corresponding spectral energies $E^\pm$ quantify the relative contribution of the two counter-propagating Alfvénic populations, so that a strong imbalance between $E^+$ and $E^-$ would provide a physical basis for emphasising one of the two branches. To assess this possibility, we performed an additional diagnostic using simultaneous velocity and magnetic field time series from a random subset of 100 probes in the fastest parallel sampling configuration of Run II. For each probe, we constructed $\boldsymbol{z}^{\pm }$ and estimated their spectral energies using the same procedure adopted for the probe spectra. The frequency-dependent Elsasser balance does not show a persistent dominance of either population, while the integrated energies give $E^+/E^-\simeq 1.08$ , corresponding to a normalised cross-helicity $\sigma _c=(E^+-E^-)/(E^++E^-)\simeq 0.04$ . These values indicate an approximately balanced state with respect to the two Elsasser populations and therefore do not provide a physical basis for uniquely selecting either propagation branch. Nevertheless, the $V+V_A$ branch produces a more systematic alignment of the spectra measured at different probe velocities and a better agreement with the Eulerian reference than the $|V-V_A|$ branch. We therefore adopt

(4.9) \begin{align} \frac {k_{\parallel }}{2\pi } \simeq \frac {f}{V+V_A}, \end{align}

as an empirical single-branch alignment for comparison with the Eulerian spectra, rather than as a uniquely selected physical Alfvénic mapping. For perpendicular sampling, the Alfvénic propagation contribution is reduced relative to the Doppler term for anisotropic fluctuations satisfying $\left |k_{\parallel ,\mathrm{loc}}\right |\ll k_\perp$ . We therefore retain the standard conversion

(4.10) \begin{align} \frac {k_\perp }{2\pi } \simeq \frac {f}{V}. \end{align}

The converted probe spectra are compared with the corresponding reduced Eulerian magnetic spectra. Here, the Eulerian reference corresponds to the isotropic spatial magnetic energy spectra parallel and perpendicular to the global guide field, obtained from Fourier space spectral outputs at fixed times and then averaged over all available snapshots. In this way, we obtain the mean Eulerian spectra $E(k_\parallel )$ and $E(k_\perp )$ , which characterise the average spatial distribution of magnetic fluctuations in the two directions. Unlike the probe spectra, which are constructed from temporal signals measured along moving trajectories, these Eulerian spectra contain only spatial information and therefore provide the natural reference for assessing the validity of the Taylor hypothesis.

Figure 6.

Comparison between temporal and spatial (Eulerian) compensated magnetic energy spectra. Panels (a) and (b) correspond to Run I ( $B_0=1$ ), while panels (c) and (d) correspond to Run II ( $B_0=6$ ). The left and right columns show spectra obtained from probes sampling parallel and perpendicular to the global guide field, respectively. The coloured curves represent the compensated temporal spectra measured at different normalised velocities $V/V_A$ , while the dashed black curves denote the corresponding compensated Eulerian magnetic spectra. Here, $k_{\parallel }$ and $k_\perp$ are defined relative to $\boldsymbol{B}_0$ .

Figure 6 shows the compensated magnetic energy spectra for Run I (panels a and b) and Run II (panels c and d), separated into parallel and perpendicular sampling with respect to the guide field. The coloured curves correspond to the spectra measured by virtual probes moving at different normalised velocities. For the parallel panels, temporal frequencies are converted using (4.9), whereas (4.10) is used for the perpendicular panels. In both cases, the wavenumber is plotted in units of $k/(2\pi )$ , consistent with the convention used for the Eulerian spectra. The dashed black curves represent the corresponding Eulerian compensated magnetic energy spectra. For Run I (panels a and b), increasing $V/V_A$ leads to a progressively better agreement between the probe spectra and the Eulerian spectra in both directions. At the lowest velocities, larger differences remain in their spectral shapes and characteristic scales, indicating a greater contribution from the intrinsic temporal dynamics of the fluctuations. As the velocity increases, the converted spectra recover the overall Eulerian shape more closely. The distinction between the parallel and perpendicular directions is relatively modest, consistent with the weaker anisotropy discussed in the previous section. Some discrepancies remain, particularly at the highest wavenumbers, where the global mean-field correction cannot account for variations of the local magnetic field direction.

For Run II (panels c and d), the agreement also improves systematically as $V/V_A$ increases. At intermediate and high wavenumbers, the spectra measured by the fastest probes recover the Eulerian spectral shape over an extended interval in both sampling directions. After including the global Alfvénic correction, the alignment of the parallel spectra becomes comparable to that obtained for perpendicular sampling, and their characteristic spectral scales are consistently recovered as the probe velocity increases. The main directional difference appears instead at the largest scales. For parallel sampling, the temporal spectra display a narrow $1/f$ interval and remain comparatively close to the corresponding Eulerian spectrum towards low wavenumbers. In contrast, the perpendicular probe spectra exhibit a substantially clearer temporal $1/f$ range, particularly for the fastest probes. This low-frequency contribution extends into a range of effective perpendicular wavenumbers where the Eulerian spectrum does not display a corresponding $1/k_\perp$ scaling. Consequently, the temporal and Eulerian spectra cease to overlap at the largest perpendicular scales. This mismatch shows that the temporal $1/f$ range cannot be interpreted as the direct Taylor-mapped image of a pre-existing Eulerian $1/k_\perp$ range, even though the temporal and spatial spectra agree well at intermediate and smaller scales. Together with the fixed-probe control discussed in § 4.1, these results show that probe motion substantially broadens the temporal near- $1/f$ range. However, the present analysis does not identify which specific aspect of the sampling process is responsible for this broadening.

As a complementary quantitative test, we independently estimated the Eulerian magnetic correlation length from the reduced spatial spectra as $L_{\textrm {corr}}=\int k^{-1}E(k)$ ${\rm d}k/\int E(k)\,{\rm d}k$ , separately for the global parallel and perpendicular directions. These spatial correlation scales were then converted into the predicted probe frame correlation frequencies, denoted by $f_{\kern-0.3pt\textrm{corr}}$ , using the same space–time mappings adopted previously (see (4.9) and (4.10)). For parallel sampling, $f_{\textrm {corr}}$ therefore represents the correlation frequency associated with the empirical $V+V_A$ single-branch mapping. In particular, we quantify the agreement between $f_c$ and $f_{\kern-0.3pt\textrm{corr}}$ through the relative difference $|f_c-f_{\textrm {corr}}|/f_{\textrm {corr}}$ . The correspondence generally improves as the probe velocity increases, with the directional differences reflecting the validity conditions of the parallel and perpendicular mappings. This behaviour is particularly clear for Run II. At the largest velocity, $V/V_A=3$ , the relative difference is approximately $15\,\%$ for parallel sampling and only $6\,\%$ for the perpendicular one. In this regime, the combination of strong spectral anisotropy and sufficiently large $V/V_A$ makes the Alfvénic contribution subdominant relative to the Doppler term. It is relevant to note that this agreement does not imply a direct correspondence between the temporal and Eulerian spectral shapes at the largest scales. In particular, for perpendicular sampling in Run II, the temporal $1/f$ range extends towards scales where the Eulerian spectrum does not exhibit a corresponding $1/k_\perp$ scaling, as previously discussed. At the lower velocity $V/V_A=1$ , the corresponding differences increase to approximately $36\,\%$ and $41\,\%$ for parallel and perpendicular sampling, respectively. For Run I, where the directional anisotropy is weaker, the particularly good agreement obtained for perpendicular sampling in Run II is no longer observed: at the largest velocity shown, $V/V_A=6$ , the differences are approximately $9\,\%$ and $17\,\%$ for parallel and perpendicular sampling, respectively. These results indicate that the agreement between $f_c$ and $f_{\textrm { corr}}$ depends not only on the relative sampling velocity, but also on the anisotropy of the fluctuations. We also compared $f_{\textrm {corr}}$ with the approximate probe-frame images of the box and forcing scales. For Run I, $f_{\textrm {corr}}$ lies close to the lower boundary of the mapped forcing interval in both sampling directions. For Run II, the parallel $f_{\textrm { corr}}$ lies between the mapped box and forcing frequencies, whereas the perpendicular $f_{\textrm {corr}}$ lies within the mapped forcing interval. Therefore, the present simulations do not provide sufficient scale separation to disentangle the inferred large-scale correlation from forcing and finite-box effects. Therefore, simulations with a larger separation between the box and forcing scales would be required, which is beyond the scope of the present work.

4.3. Spectral rescaling under Taylor hypothesis

The following rescaling analysis is restricted to perpendicular sampling ( $\theta =90^\circ$ ) in Run II. For these trajectories, the Alfvénic propagation term is expected to be subdominant relative to the Doppler term for anisotropic fluctuations. The dependence of the measured temporal spectra on the probe velocity can be understood, to first approximation, as a sampling effect under the Taylor hypothesis. Consider a probe moving at constant speed $V$ through a spatial field $u(x)$ , so that the recorded signal is

(4.11) \begin{align} s(t)=u(x=Vt). \end{align}

If the same field is instead sampled by a probe moving at speed $V'$ , the measured time series becomes

(4.12) \begin{align} s'(t)=u(V't)=s\!\left (\frac {V'}{V}t\right ). \end{align}

Thus, changing the probe speed does not modify the underlying spatial structure being sampled, but only changes the rate at which that structure is traversed in time. In this sense, the temporal signal is dilated or compressed depending on the ratio $V'/V$ . This simple time rescaling has a direct spectral consequence. If two probe signals correspond to the same spatial field sampled at different velocities, their PSDs are related by the standard scaling property

(4.13) \begin{align} \text{PSD}_{V\rightarrow V'}(f)=\frac {V}{V'}\,\text{PSD}_V\!\left (\frac {V}{V'}f\right ). \end{align}

Therefore, increasing the probe velocity shifts the temporal spectrum towards higher frequencies and rescales its amplitude accordingly. This relation provides a simple way to predict how the spectrum measured at one probe speed should transform when the same fluctuations are observed at another speed. In this framework, the dependence of the temporal spectrum on $V$ arises mainly from the change in sampling rate by a factor $V'/V$ , rather than from a modification of the underlying spatial fluctuations.

To test this idea, we take one measured spectrum as a reference and use the previous expression to predict its form at a different sampling velocity. The result is then compared with the spectrum obtained directly from probes moving at that target speed. Figure 7 shows the velocity rescaling procedure for Run II. The compensated temporal spectrum measured at $V/V_A=3$ is taken as the reference, while the spectra obtained at lower probe velocities, $V/V_A=0.7$ , $1$ and $2$ , are shown together with their rescaled counterparts. Each colour corresponds to a different probe velocity. Solid colour lines indicate the original spectra, dashed colour lines represent the rescaled spectra and the dotted black line denotes the reference spectrum at $V/V_A=3$ . For visualiastion purposes, a vertical offset has been applied to the original spectra, allowing the frequency displacement associated with the rescaling to be more clearly compared. Once the rescaling is applied, the remapped curves shift in frequency and collapse much more closely onto the reference spectrum over a broad frequency interval. The agreement is particularly good for the highest of the rescaled velocities, while larger discrepancies remain for the slowest probes. To measure the improvement of the spectral collapse, we compute the root-mean-square logarithmic distance between each rescaled compensated spectrum and the reference spectrum. Therefore, we introduce

(4.14) \begin{align} D_{\textit{rms}}(V) = \left [ \frac {1}{N} \sum _{i=1}^{N} \left ( \log _{10}Y_{\textrm {res}}(f_i) - \log _{10}Y_{\textrm {ref}}(f_i) \right )^2 \right ]^{1/2}, \end{align}

where $Y_{\textrm {res}}(f)$ is the rescaled compensated spectrum, $Y_{\textrm {ref}}(f)$ is the reference spectrum and $N$ is the number of frequency points considered. By construction, $D_{\textit{rms}}=0$ corresponds to a perfect collapse onto the reference spectrum. In particular, we obtain $D_{\textit{rms}}\simeq 0.065$ , $0.038$ and $0.034$ for $V/V_A=0.7$ , $1$ and $2$ , respectively. Thus, the decreasing spectral distance with increasing probe velocity quantitatively confirms the improved agreement observed in figure 7. This behaviour indicates that, when the probe velocity is sufficiently large, the effect of changing the sampling speed can be described to a good approximation as a dilation or compression of the temporal signal. Conversely, for slower probes, the rescaling is less successful, suggesting that the measured time series retains a stronger contribution from the intrinsic temporal dynamics of the fluctuations. This result is consistent with the previous analysis: the mapping between spatial and temporal scales becomes more reliable at larger probe velocities, while deviations from a simple Taylor-like interpretation become more important in the slow sampling regime.

Overall, this analysis shows that a substantial part of the velocity dependence of the measured temporal spectra can be interpreted as a consequence of the sampling procedure itself. In particular, the collapse obtained for the fastest probes supports the idea that, in that regime, the Taylor hypothesis provides a useful approximation for relating spectra measured at different probe velocities. At the same time, the residual differences observed at lower $V/V_A$ indicate that this approximation becomes less accurate, in agreement with the behaviour discussed earlier.

Figure 7.

Test of the sampling-rescaling relation for the perpendicular temporal magnetic spectra in Run II ( $B_0=6$ ). The compensated spectrum measured at $V/V_A=3$ is used as the reference and compared with the spectra measured at lower probe velocities, $V/V_A=0.7$ , $1$ and $2$ , together with their rescaled counterparts. Solid lines show the original spectra, dashed lines the rescaled spectra and the dotted black line the reference spectrum. Each colour corresponds to a different probe velocity. A vertical offset has been applied to the original spectra for visualisation purposes.