Scale-Dependent Statistics of Coherent Magnetic Structures in Early IMAP Observations

Introduction

Coherent magnetic structures, such as magnetic discontinuities [19, 57, 15, 11], current sheets [47, 28, 13, 52, 51] (Khaborova et al. 2015), and flux-tube-like structures [26, 55, 56, 39, 54], are frequently observed in the turbulent solar wind over a wide range of spatial scales [4, 20, 5, 41]. These structures are important because plasma heating [49, 3, 38, 43, 10], particle energization [17, 22, 37] (Khaborova et al. 2016), and transport [23, 26, 27] (Wu et al. 2025) can become concentrated in their vicinity and spatially organized by their presence.

A widely used diagnostic of coherent magnetic structures in spacecraft observations is the Partial Variance of Increments (PVI; see [12, 16, 13, 39, 40, 5], and references therein). Previous studies [14, 5] have shown that the temporal distribution of high-PVI structures is nonrandom, with power-law waiting times at shorter separations and approximately exponential behavior at larger separations; their occurrence and clustering also depend on the PVI lag. Magnetic helicity provides complementary information about coherent magnetic structure by characterizing the linkage and winding of magnetic flux [36, 31]. Previous studies have also shown that magnetic helicity exhibits systematic variations with spatial scale [8, 50] and heliospheric location [2, 4, 1]. These studies mainly describe the broader spatial or spectral organization of helicity. To identify helical structures locally in spacecraft observations, Pecora et al. [40] developed a real-space magnetic-helicity method based on the general theory provided by Matthaeus et al. [32] and combined it with PVI. In this framework, enhanced magnetic helicity indicates the presence of helical flux tubes, whereas strong PVI can identify sharp magnetic gradients near their internal or external boundaries.

Previous studies employing the PVI-helicity flux tube identification method have applied it mainly to specific intervals or events (see, e.g., Pecora et al. [41]). Here, we build on this work using several months of early IMAP observations from November 2025 to May 2026. We examine structures at six different scales, from 0.5 to 3 magnetic correlation lengths, and use phase-randomized surrogate data to set a significance threshold at each scale. This allows us to study how the occurrence, duration, and waiting times of helical structures change with scale, whether positive and negative helicity occur equally often, and how these structures relate to strong magnetic gradients. The statistical characterization of PVI-selected gradients and helical magnetic structures is important for understanding the local topology and dynamics of solar wind turbulence. Our contribution is therefore a statistical description of these properties across a multimonth sample, extending earlier applications of the PVI-helicity method.

Data and Methods

IMAP was launched in September 2025 and carries 10 instruments designed to investigate the interaction of the solar wind with the local interstellar medium and the processes responsible for particle acceleration and transport throughout the heliosphere [35]. We use magnetic field and solar wind velocity vector measurements from early IMAP observations spanning from November 1, 2025 through May 20, 2026. Magnetic field data are obtained from the magnetometer (MAG) instrument [18], while solar-wind velocity measurements are obtained from the Solar Wind and Pickup Ions (SWAPI) instrument [42]. We use magnetic field and velocity cadences of 16 sec and 60 sec, respectively.

We use PVI to identify intermittent magnetic field structures. PVI measures the magnitude of the magnetic field vector increment at a given time lag $\tau$ normalized by the rms amplitude of the increments [13]. It thus provides a pointwise measure of the strength of a magnetic field increment relative to the local level of magnetic fluctuations. For a magnetic field $\mathbf{B}(t)$, PVI is defined as [12, 5],

where $\Delta\mathbf{B}(t,\tau)=\mathbf{B}(t+\tau)-\mathbf{B}(t)$, $t$ is the spacecraft observation time, $\tau$ is the temporal lag, and $\langle\cdots\rangle$ denotes a local average. We use an averaging window of 6 hours for our analysis. Under Taylor’s hypothesis, the six-hour averaging interval corresponds to a spatial extent along the solar wind flow of $L_{\mathrm{avg}}\simeq V_{\mathrm{sw}}\Delta t_{\mathrm{avg}}$, where $\Delta t_{\mathrm{avg}}=6$ hr. For a typical solar wind speed of 400 km s$^{-1}$, this gives $L_{\mathrm{avg}}\simeq8.64\times10^{6}$ km. Compared with a representative magnetic correlation length of $\lambda_c\sim10^{6}$ km in our observations, the window spans about nine correlation lengths. This interval defines the local background used for PVI normalization. The PVI values are calculated at time lags of $\tau=16$, 60 and 120 s. Large PVI values ($>3$) correspond to fluctuations in the non-Gaussian tails of the increment distribution and are therefore associated with enhanced intermittency [34, 13, 5, 40].

To characterize helical magnetic structures, we estimate the local real-space magnetic helicity following the method of Pecora et al. [40]. For single-spacecraft measurements, a local estimate of the magnetic helicity at scale $\ell$ is obtained from the antisymmetric part of the magnetic field correlation,

where

and

is a Hann window. Here, $b_2$ and $b_3$ are the magnetic-field components perpendicular to the sampling direction $\hat{\mathbf{1}}$, where $\hat{\mathbf{1}}=\hat{\mathbf{2}}\times\hat{\mathbf{3}}$, and $w_0$ is an arbitrary interval of local integration. $x$ denotes the position along the one-dimensional spacecraft sampling direction at which the local magnetic helicity is evaluated, and $\xi$ is the dummy spatial coordinate used for integration over the local interval of width $w_0$ centered at $x$. We use $w_0 = 2\ell$. The variable $l$ is the spatial lag entering the local correlation, whereas $\ell$ denotes the maximum spatial scale included in the helicity integral. We calculate $H_m$ at six different scales, $\ell/\lambda_c = 0.5$, 1.0, 1.5, 2.0, 2.5, and 3.0, where $\lambda_c = V_{\mathrm{sw}}\tau_c$ is the magnetic correlation length. The correlation time $\tau_c$ is calculated as the time at which the autocorrelation falls to $1/e$ value ([33, 6, 9]).

The normalized magnetic helicity can be estimated as ([41]),

where $\langle\delta b^2\rangle$ is the locally averaged magnetic fluctuation energy. We estimate both $\langle\delta b^2\rangle$ and $\lambda_c$ within each six-hour analysis interval ($\Delta t_{\mathrm{avg}} = 6$ hr) because this is an appropriate duration (several correlation times at 1 au, [44, 21, 6]) to attain statistical homogeneity. The analysis is conducted over the first six months of continuous coverage of the IMAP mission at Lagrange point L1.

We determine a scale-dependent threshold for estimated magnetic helicity using phase-randomized surrogate data to distinguish significant helical structures from fluctuations expected in random phase magnetic field data, following approaches similar to those of Sahraoui [45]; Lion et al. [29]. For each 6-hr interval, we generate 100 surrogate magnetic field time series by randomizing the Fourier phases for each component while preserving the Fourier amplitudes. This procedure preserves the power spectrum of the original magnetic field fluctuations while removing the phase coherence associated with coherent structures. The normalized magnetic helicity is then calculated for each surrogate. For each scale $\ell/\lambda_c$, we construct the distribution of $|\widetilde{H}_m|$ from the phase-randomized realizations and define the threshold $H_{\mathrm{th}}$ as the $95^{\mathrm{th}}$ percentile of this distribution. Values of $|\widetilde{H}_m| > H_{\mathrm{th}}$ are therefore unlikely to arise from random phase fluctuations and are considered as significant helical structures.

We also calculate the waiting time (WT) for PVI events, where consecutive samples satisfying PVI $> 3$ are treated as a single event. For magnetic helicity, we estimate both the WT and event duration for consecutive samples satisfying $|\widetilde{H}_m| > H_{\mathrm{th}}$. The event duration is defined as the time interval over which the event remains continuously above the corresponding threshold. The WT between two successive events is defined as, $WT_i = t_{\mathrm{start},i+1} - t_{\mathrm{end},i}$, where $t_{\mathrm{end},i}$ is the end of the $i^{\mathrm{th}}$ event and $t_{\mathrm{start},i+1}$ is the beginning of the following event above threshold.

Results

In Figure 1, we present an example interval illustrating the identification of PVI and magnetic helicity in the IMAP observations from 2026 February 2–8. The panel (a) shows the three magnetic field components, and the panel (b) shows the PVI calculated at a time lag of $\tau= 60$ sec. The panel (c) shows the local magnetic helicity $H_m(\ell)$ estimated at six different scales, $\ell/\lambda_c = 0.5$, 1.0, 1.5, 2.0, 2.5, and 3.0. Increasing $\ell/\lambda_c$ provides contributions from larger-scale helical structures. The panel (d) shows the corresponding normalized magnetic helicity. The real-space helicity is normalized by the local magnetic field energy. Both positive- and negative-helicity structures are evident throughout the interval, and their characteristic amplitudes and temporal extents vary with $\ell/\lambda_c$. The shaded regions mark representative intervals identified as helical structures with normalized helicity values larger than the local threshold. Several of these regions are also associated with enhanced PVI activity, either at boundaries or in the interior.

Figure 2 shows the distributions of PVI calculated at different time lags of $\tau= 16$, 60, and 120 sec. All three distributions are dominated by low-PVI fluctuations and are nearly identical over the low-to-moderate PVI range, but begin to separate beyond PVI $\sim3$. The shortest lag shows the broadest high-PVI tail, suggesting a larger occurrence of strong intermittent fluctuations at smaller scales. Previous studies have associated PVI $> 3$ with non-Gaussian intermittent structures, PVI $> 6$ with current sheets, and PVI $> 8$ with possible magnetic reconnection sites (e.g., Matthaeus et al. [34]; Greco et al. [13]). Table 1 summarizes the occurrence of fluctuations above these thresholds. The number of events decreases gradually with increasing PVI threshold and time lag, with the strongest fluctuations occurring mostly at the shortest lag.

$\tau$	PVI > 3		PVI > 6		PVI > 8
(s)	$N$	Fraction (%)	$N$	Fraction (%)	$N$	Fraction (%)
16	17,725	1.73	1,436	0.14	284	0.03
60	16,367	1.60	535	0.05	57	0.01
120	13,540	1.32	222	0.02	5	< 0.001

In Figure 3(a-Figure 3c), we present the probability density functions (PDFs) of WT between successive PVI $> 3$ events for $\tau= 16$, 60, and 120 sec, respectively. For all three lags, the distributions exhibit a power-law behavior at shorter WT followed by an exponential decay at longer WT. This behavior suggests that closely spaced intermittent structures are temporally correlated and clustered, while structures separated by longer intervals are more randomly distributed, consistent with a Poisson noise process. The mean WT, which is measured as the first moment of the distribution, increases from $\langle\mathrm{WT}\rangle= 17.5$ min at $\tau= 16$ s to 35.3 min at 60 sec **Note—**The fractions represent the percentage of the total samples at each time lag satisfying the corresponding PVI threshold.

and 59.4 min at 120 sec, showing that strong intermittent events become less frequent with increasing lags.

Figure 4 shows the PDFs of the absolute normalized magnetic helicity $|\widetilde{H}_m|$ for $\ell/\lambda_c = 0.5$–3.0. At the smallest scale ($\ell/\lambda_c = 0.5$), the distribution is strongly concentrated at low values of $|\widetilde{H}_m|$ and decreases rapidly toward larger helicity. As $\ell/\lambda_c$ increases, the distributions become broader and develop more extended tails toward large $|\widetilde{H}_m|$. Thus, moderate and large normalized helicity values become gradually more common at larger scales. The vertical dashed lines indicate the scale-dependent thresholds obtained from the phase-randomized surrogate analysis. The threshold values also increase from 0.011 at $\ell/\lambda_c = 0.5$ to 0.037, 0.075, 0.123, 0.174, and 0.226 at $\ell/\lambda_c = 1$, 1.5, 2, 2.5, and 3, respectively. This increase indicates the scale dependence of the helicity fluctuations expected from the corresponding random-phase surrogate data and motivates the use of a scale-dependent threshold rather than a single fixed value.

The scale dependence evident in the PDFs is also reflected in the helicity statistics as shown in Table 2. The mean value of $|\widetilde{H}_m|$ increases from 0.016 at $\ell/\lambda_c = 0.5$ to 0.102 at $\ell/\lambda_c = 3$. The occurrence of moderately and strongly helical fluctuations also increases significantly with scale. For example, the fraction of samples with $|\widetilde{H}_m| > 0.25$ increases from 0.11% at $\ell/\lambda_c = 0.5$ to 9.69% at $\ell/\lambda_c = 3$, a factor of almost 100. These results demonstrate that large normalized-helicity fluctuations become gradually more common with increasing spatial scale. When the scale-dependent significance thresholds are applied, however, the fraction of samples identified as significant helical structures decreases with increasing scale. The above-threshold fraction decreases from 37.85% at $\ell/\lambda_c = 0.5$ to 11.64% at $\ell/\lambda_c = 3.0$, while the mean $|\widetilde{H}_m|$ of the above-threshold samples increases from 0.034 to 0.377. Thus, although large values of normalized helicity become more prevalent at larger scales, the corresponding significance criterion also becomes more restrictive because the surrogate-derived threshold increases with scale.

$\ell/\lambda_c$	$H_{\rm th}$	$\langle|\widetilde{H}_m|\rangle$	$f_{0.25}$ (%)	$f_{0.50}$ (%)	$f_{0.75}$ (%)	$\langle|\widetilde{H}_m|\rangle_{\rm th}$	$f_{\rm th}$ (%)
0.5	0.011	0.016	0.11	0.01	0.00	0.034	37.85
1.0	0.037	0.033	0.78	0.08	0.02	0.087	27.32
1.5	0.075	0.050	2.16	0.28	0.05	0.152	20.51
2.0	0.123	0.068	4.24	0.67	0.14	0.227	15.80
2.5	0.174	0.085	6.75	1.09	0.28	0.306	13.01
3.0	0.226	0.102	9.69	1.70	0.46	0.377	11.64

Figure 5 compares the statistical properties of positive and negative helicity structures as a function of $\ell/\lambda_c$. Panel (a) shows the fraction of positive helicity events, $N_+/(N_+ + N_-)$, with the horizontal dashed line marking 0.5 (equal occurrence of positive and negative helicity events). The fraction remains close to 0.5 across all scales, indicating no strong preference for either helicity sign, although a modest decrease in the positive helicity fraction is observed at the largest scales. Panels (b) and (c) compare the mean event durations and WTs, respectively. The mean durations of both helicity signs increase with scale and are comparable at smaller scales, whereas negative helicity structures tend to have slightly longer durations for $\ell/\lambda_c \gtrsim2$. The mean WT also increases with scale, with negative helicity structures generally showing longer WT than positive helicity structures at the larger scales.

Figure 6(a-f) shows the PDFs of the durations of helical structures at different scales ($\ell/\lambda_c = 0.5$–3). At all scales, the distributions exhibit a power-law behavior at shorter durations, followed by a more rapid, approximately exponential decrease at longer durations.

The fitted power-law indices show relatively little variation with scale, ranging from about $-0.60$ to $-0.67$. In contrast, the duration distributions extend progressively toward longer times as $\ell/\lambda_c$ increases. Accordingly, the mean event duration increases from $\langle T\rangle= 17.8$ min at $\ell/\lambda_c = 0.5$ to $64.5$ min at $\ell/\lambda_c = 3.0$. This scale dependence indicates that helical structures identified at larger spatial scales generally persist for longer periods. In a similar format, Figure 7 shows the PDFs of the WTs between successive helical structures. The WT distributions also exhibit similar fitting trends to the event duration distributions. The short WT power-law slope becomes gradually steeper with increasing $\ell/\lambda_c$, changing from approximately $-0.22$ at $\ell/\lambda_c = 0.5$ to $-0.96$ at

$\ell/\lambda_c = 3.0$. The distributions also extend toward longer waiting times as the scale increases. Accordingly, the mean waiting time increases from $\langle\mathrm{WT}\rangle= 183.8$ min at $\ell/\lambda_c = 0.5$ to $1431.7$ min at $\ell/\lambda_c = 3.0$. This increasing trend indicates that helical structures at larger scales are more widely separated in time and therefore occur less frequently. We note that although the duration and WT PDFs exhibit qualitatively similar behavior, our analysis of their joint probability distribution (not shown here) reveals no significant relationship between the duration of a helical structure and the WT between successive structures. Thus, longer-lived structures are not necessarily followed by longer or shorter waiting times.

We further examine the PVI–helicity relationship. In this framework, extrema of the local magnetic helicity tend to identify the cores of helical structures, whereas enhanced PVI is preferentially associated with sharp magnetic gradients near their boundaries. Thus, large values of PVI and $|\widetilde{H}_m|$ are not necessarily expected to occur simultaneously, and in fact their occurrences are more likely to be anti-correlated. We consider PVI at $\tau= 60$ sec and $|\widetilde{H}_m|$ at $\ell/\lambda_c = 1.5$ for this analysis. Figure 8 shows the joint probability distribution of PVI and $|\widetilde{H}_m|$. The vertical dashed line marks the helicity threshold $H_{\rm th} = 0.075$. The distribution is strongly concentrated at low PVI and low $|\widetilde{H}_m|$, while the probability decreases toward larger values of either quantity. Large PVI fluctuations occur mostly at relatively small $|\widetilde{H}_m|$, whereas samples with enhanced $|\widetilde{H}_m|$ are concentrated mainly at lower PVI. This behavior is qualitatively consistent with the picture of Pecora et al. [40], in which helicity maxima occur within relatively PVI-quiet regions and enhanced PVI marks nearby boundaries of the helical structures.

Discussion and Conclusions

We used early-mission IMAP observations to examine coherent magnetic structures from two complementary perspectives, (i) strong magnetic field gradients identified by PVI, and (ii) locally helical structures identified from the real-space magnetic helicity. A consistent picture emerging from the analysis is that both populations are strongly organized by scale. The high-PVI tail becomes less populated as the PVI lag increases, while the characteristic strength, duration, and separation of helical structures change systematically with $\ell/\lambda_{c}$. An important aspect of the helicity analysis is that the background level itself is scale dependent. Although large values of $\lvert\widetilde{H}_{m}\rvert$ become more common toward larger scales, the surrogate-derived significance threshold also increases with scale. Consequently, a single fixed helicity threshold would not provide an equivalent significance criterion at different $\ell/\lambda_{c}$ (as shown in Table 2).

The main results of the present analysis are summarized schematically in Figure 9. Panel (a) illustrates the physical picture in which a helical magnetic structure is bounded by sharp magnetic field gradients identified by enhanced PVI. The panel (b) summarizes the scale-dependent organization of the helical structures and the WTs between successive structures. Panel (c) shows a representative IMAP interval in which the helicity signatures at different spatial scales are viewed together with the PVI.

The observed PVI WT distributions are consistent with previous results near Earth and closer to the Sun [14, 5], showing power-law behavior at shorter separations and an approximately exponential tail at larger separations. The estimated mean WTs are comparable to near 1 au timescales (see Greco et al. [14], and references therein) and longer than those reported during Parker Solar Probe’s (PSP) first perihelion (Chhiber et al. [5]). The shorter WTs observed by PSP may reflect the smaller turbulence correlation scales and more rapidly varying magnetic structures in the young solar wind, which evolve and become more widely separated as the solar wind expands toward 1 au.

The duration statistics of magnetic helicity provide a connection with previous flux-rope studies. Small-scale magnetic flux ropes identified at 1 au have been found to span durations from several minutes to several hours and to exhibit scale-dependent occurrence statistics [58, 7]. More recently, Lynch et al. [30] applied the $H_{m}$–PVI technique to slow-to-moderate solar wind at 1 au and reported mean coherent structure durations of approximately one hour. This is comparable to the mean duration found here at the largest scales, where $\langle T\rangle$ approaches one hour.

The present analysis further shows that the characteristic duration of helical structures at scale $\ell$ changes with $\ell/\lambda_{c}$, while the short-duration power-law index remains relatively similar across the scales considered. This suggests that the characteristic size of the structures changes significantly with scale even though the statistical form of the short-duration population is comparatively similar. The WT statistics further show that the temporal spacing of the helicity-selected structures

also depends strongly on scale. This statistical behavior is represented conceptually in Figure 9(b), where structures identified at different spatial scales occupy different temporal extents and successive helical structures are separated by finite WTs. The increasing characteristic duration and WT with $\ell/\lambda_c$ therefore indicate a systematic scale-dependent organization of the coherent magnetic structures. The power-law behavior at shorter WT followed by an exponential tail resembles the statistical organization previously reported for magnetic discontinuities [14, 5]. An additional possible interpretation of these ensemble statistics is suggested by the superposition framework of Schwadron et al. [46]. In that framework, an observed distribution may arise from the superposition of individual stochastic realizations characterized by different local parameters or occurrence rates. In particular, a superposition of Poisson-like processes having different rates can produce power-law-like WT statistics even when the individual realizations have exponential WT distributions. The solar wind intervals sampled here likely represent a range of local turbulence and coherent-structure conditions, and therefore such a superposition may contribute to the observed duration and WT distributions. Establishing a quantitative connection, however, would require characterizing the distributions of the individual realizations and relating them to the corresponding ensemble statistics, which is beyond the scope of the present study. The joint PVI-helicity statistics are also consistent with the complementary interpretation proposed by Pecora et al. [40]. As summarized in Figure 9(a), enhanced helicity mainly characterizes the interior of the coherent structure, whereas large PVI values preferentially mark sharp gradients near its boundaries. The representative IMAP interval in Figure 9(c) provides an observational example of this organization, with scale-dependent helicity enhancements occurring predominantly within comparatively PVI-quiet intervals and strong PVI enhancements appearing near the boundaries of the helical structures.

We find nearly comparable occurrence rates of positive and negative helicity structures over the scales examined. This should not be interpreted as contradicting earlier observations of hemispheric helicity preferences [2, 4]. Those studies concern still larger-scale averaged helicity organized relative to heliographic hemisphere, whereas the present analysis characterizes local helical structures sampled near 1 au. Interestingly, although the occurrence of the two signs remains approximately balanced, differences in their durations and waiting times become more evident at larger scales. Whether this asymmetry is persistent or depends on solar wind type, magnetic sector, or position relative to the heliospheric current sheet will require a larger IMAP dataset that better samples solar magnetic polarities.

By extending the PVI-helicity method to several months of early IMAP observations, we show how the statistical properties of helical structures vary with scale when identified using surrogate-based significance thresholds. As the dataset grows, the same analysis can be applied to different solar wind types, magnetic sector, heliospheric current sheet proximity, transient activity, heliocentric distance, and solar activity phase to better understand the scale-dependent properties reported here. The broader IMAP instrument suite will also permit the identified helical structures and their PVI-rich boundaries, shown schematically in Figure 9, to be related to solar wind plasma properties, energetic particle populations, and composition. Such studies are particularly relevant in view of previous evidence that helical flux tubes can act as particle-transport barriers [48, 40].

Acknowledgements

This research was partially supported by the NASA IMAP project 80GSFC19C0027 at Princeton, under a subcontract SUB0000317 at the University of Delaware and in part by the PSP/ISOːIS project through subcontract SUB0000165 and from Princeton, as well as NSF SHINE grant AGS-2501387, and by NSF SHINE 2501387 at the University of Delaware. F.P. was partially supported by Agenzia Spaziale Italiana (ASI) in the framework of the project “Radial Evolution of large- and Kinetic-scale Processes in the Expanding Solar wind (REKIPES)” C83C25000900005. J.G. was supported by the Delaware NASA Space Grant program grant number 80NSSC20M0045 at the University of Delaware. R.B. acknowledges support from NSF FDSS Award No. AGS-2347952. IMAP magnetometer operations are funded by the UK Space Agency (grant UKRI12494). Tim Horbury is supported by STFC grant UKRI1202.