Turbulence Properties of Interplanetary Coronal Mass Ejections in the Inner Heliosphere: Dependence on Proton Beta and Flux Rope Structure

Alongside the continuous outflows of the fast and slow solar winds, interplanetary coronal mass ejections (ICMEs; Kilpua et al. 2017) represent a third, impulsive type of solar wind in the heliosphere. The plasma, magnetic field, and compositional properties of ICMEs differ to those of the other solar wind types in various significant ways (Zurbuchen & Richardson 2006), making ICMEs a distinctive environment for investigating a range of space plasma phenomena.

A characteristic signature of ICME wind at 1 au is the dominance of magnetic pressure over the ion component of the plasma pressure (i.e., β_i ≪ 1), a result of strong magnetic fields combined with low proton temperatures. Strong fields in ICMEs are often associated with the presence of large-scale, nearly force-free magnetic flux ropes (Goldstein 1983). An ICME is the interplanetary manifestation of an erupted coronal flux rope, although ICMEs are not always observed to have a flux rope structure in situ (e.g., Cane & Richardson 2003). ICMEs typically have field strengths in excess of the ambient solar wind field across a wide range of heliocentric distances (e.g., Wang et al. 2005). Low proton temperatures and the absence of a proton temperature–velocity correlation in ICMEs have been attributed to rapid expansion close to the Sun (e.g., Matthaeus et al. 2006) and to the dominance of magnetic forces in the magnetically closed structure of an ICME flux rope (Démoulin 2009). When modeled with polytropes, ICMEs show a nonadiabatic expansion with radial distance that suggests some local heating by turbulence (Liu et al. 2006) as in the solar wind generally. The radial evolution of the magnetic field strength, proton temperature, and density in ICMEs are such that low β_i is maintained within ICMEs from the Sun to 1 au and beyond. This contrasts with the fast and slow winds, in which β_i approaches unity well before reaching 1 au.

Like other types of solar wind, ICMEs contain fluctuations at all measurable scales (e.g., Leamon et al. 1998; Sorriso-Valvo et al. 2021; Márquez Rodríguez et al. 2023) though at relatively small amplitudes (Borovsky et al. 2019). The canonical power spectrum of fluctuations in the interplanetary magnetic field comprises distinctive power-law ranges (e.g., Verscharen et al. 2019). These include a k⁻¹ wavenumber spectrum ("1/f range" or "injection range") of noninteracting fluctuations at spatial scales exceeding the correlation length, thought to be an imprint from the wind's coronal sources. These large-scale fluctuations supply energy to a magnetohydrodynamic (MHD) cascade with a k^−3/2 or k^−5/3 spectrum between the correlation length and ion scales, the spectral indices in this "inertial range" being in agreement with various theories of Alfvénic MHD turbulence (Schekochihin 2022). Depending on the precise definition of compressibility used, typically 2%–10% of total fluctuation power is contained within compressive modes at MHD scales (Chen 2016). At ion scales, spectra with power-law indices significantly less than the −7/3 and −8/3 indices predicted for kinetic Alfvén wave turbulence may be observed; this range, which arises from a gradual transition between the MHD and kinetic regimes, is more consistently observed near the Sun than at 1 au (Bowen et al. 2020, and references therein). At still smaller scales, ion-kinetic effects become fully developed, and a k^−2.8 spectrum, most likely dominated by kinetic Alfvén wave turbulence, is present.

In this Letter, we present a spectral analysis of magnetic field fluctuations within ICMEs in terms of the phenomenology described above. For this analysis, ICMEs observed by the Solar Orbiter (SolO; Müller et al. 2020) and Parker Solar Probe (PSP; Fox et al. 2016) spacecraft have been selected, allowing the radial evolution of spectral properties in the inner heliosphere to be probed. A focus of the study has been to consider how certain well-known properties that distinguish ICME wind from other wind types—namely, background fields with a flux rope geometry and low proton β across a wide range of heliocentric distances—may affect various spectral properties relating to MHD turbulence. Properties analyzed include fluctuation power and compressibility, spectral indices and break scales, and correlation lengths in the magnetic field. The rotation of the background flux rope field in ICME intervals often has a strong spectral signature at large scales, and care has been taken to remove this field when analyzing properties of the underlying large-scale fluctuations. More broadly, we consider whether ICMEs support turbulence that is different in nature to the turbulence present in non-ICME wind occupying a similar parameter regime (e.g., the near-Sun wind at low β_i), and, if there is some difference, whether the ICME environment could provide a new regime for understanding aspects of the turbulence.

The Helio4Cast ICMECAT database (Möstl et al. 2017, 2020) provides a regularly updated list of ICMEs observed by SolO and PSP. From the current list, a total of 28 ICME intervals with good magnetic field and plasma data coverage were identified and selected for analysis. Details of these ICMEs are listed in the Appendix. Only the magnetic driver intervals of the ICMEs have been analyzed and not other ICME substructures, such as sheaths; all mentions of ICMEs in this work refer to the drivers only. The ICMEs were observed at heliocentric distances ranging from 0.25 to 0.95 au. All of the selected ICMEs had mean proton β less than 0.4 and at least some coherent, flux-rope-like rotation of the large-scale magnetic field, signatures characteristic of magnetic clouds. The mean value of proton β across all events was 0.16. Only the proton contribution to β_i has been considered, and β_i is hereafter equated with proton β. Magnetic field data from SolO's magnetometer (MAG; Horbury et al. 2020) and PSP's FIELDS instrument suite (Bale et al. 2016), and ion moments from SolO's Solar Wind Analyser (SWA; Owen et al. 2020) and PSP's Solar Wind Electrons Alphas and Protons—Solar Probe Cup (SWEAP-SPC; Kasper et al. 2016) have been used. The SolO/MAG data were at ∼0.127 s resolution and PSP/FIELDS data at 0.007 to 0.438 s resolution (typically 0.11 s), while data resolution from the plasma instruments varied between ∼1 and 28 s. Power spectral densities (PSDs) of magnetic field fluctuations in the ICME intervals have been determined using a multitapered fast-Fourier transform with bandwidth product NW = 5/2.

We begin this section by summarizing key features identified in the spectra, with the analysis described in detail in the following subsections. Figure 1 shows the smoothed, mean-subtracted, trace PSD (equivalent to the total power in fluctuations of the magnetic field vector) as a function of spacecraft-frame frequency, f_sc, for a selection of the ICMEs analyzed. The spectra are color-coded according to heliocentric distance, r. There is a general trend toward lower fluctuation power with increasing r at all frequencies. A similar trend can be seen in an analogous figure with solar wind intervals by Chen et al. (2020), but with PSD lower in the ICMEs at most scales (e.g., by a factor of ∼5 at f_sc = 10⁻² Hz) for a given r. The spectral indices are typically close to −5/3 at f_sc ≲ 10⁻¹ Hz in the inertial range, with steepening of the spectral slopes characteristic of a transition toward the kinetic range evident at higher frequencies (Section 3.1). At f_sc ≲ 10⁻³ Hz, slopes are highly variable and, in some cases, significantly steeper than −5/3 due to power in the flux rope fields (Sections 3.1 and 3.3); only irregular signatures of a 1/f range are present at low frequencies. The spectral breaks at the high-frequency end of the inertial range, marked with the red points in Figure 1, show an approximately linear tendency toward lower frequencies with r (Section 3.4).

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3.1. Spectral Breaks and Slopes

The top panels in Figure 2 show the trace PSD for two of the ICMEs, one observed by PSP at 0.39 au and the other by SolO at 0.89 au. Solid black lines represent linear fits to log(PSD) as a function of log(f_sc) in the inertial and transition ranges. These fits have been extrapolated until they intersect, with the intersection point giving the spectral break frequency, f_b. The fits were applied to linear regions of the logarithmically scaled spectra such that f_b approximately coincides with the midpoint of the frequency range where the spectral slope, α, steepens in the transition range. The bottom panels in Figure 2 show α calculated using a sliding window, with α values given by the gradients of linear fits across the sliding-window range and uncertainties determined from the linear fit quality. For both ICMEs, it can be seen that α mostly fluctuates between −1 and −2 at f_sc ≤ 10⁻² Hz, is approximately −5/3 at 10⁻² Hz ≤f_sc ≤ 10⁻¹ Hz, and steepens to values below −3 at higher frequencies. The spectra of magnetic compressibility, C, that are shown in the middle panels are discussed in Section 3.2.

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Spacecraft-frame frequencies associated with the proton gyroscale, ρ_i, and the proton inertial length, d_i, are marked in the figure. Applying Taylor's hypothesis, which gives wavenumber k = 2π f_sc/〈v〉 in the plasma frame, it may be shown that

and

where T_i, m_i, e, and v are the proton temperature, mass, charge, and speed, respectively, and angle brackets denote interval time averages. Thus spacecraft-frame frequencies and in Figure 2 correspond to the scales at which k ρ_i = 1 and kd_i = 1, respectively. ⁴ It can be seen that f_b is closer to than to in both of the examples shown.

Of the 28 ICME intervals analyzed, 16 had spectra without any significant non-power-law features (e.g., spikes, humps, or noise-floor flattening) near the spectral break. For these 16 spectra, f_b could be accurately determined. The top panel of Figure 3 shows the ratio of f_b to and for the 16 events, as a function of the mean β_i value in each ICME. Across the range of relatively low β_i examined, was nearer unity than , but with approaching unity as β_i increased. This trend is in agreement with the findings of Chen et al. (2014), who determined that d_i is systematically closer to the observed break scale at low β_i and ρ_i is closer at high β_i.

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In order to make comparisons in the same reference frame and in terms of more physically meaningful units (e.g., Wicks et al. 2010a; Sioulas et al. 2023a), all of the spectra are henceforth considered as a function of kd_i rather than f_sc, with the conversion from f_sc given by Equation (2). The bottom panel of Figure 3 shows the sliding-window profile of α across the inertial and ion-kinetic ranges (10⁻⁴ < kd_i ≤ 3), averaged across all events. Standard deviations are indicated by error bars. It can be seen that α is near −5/3 at kd_i ≲ 0.2 in the inertial range and steepens to approximately −3.5 at kd_i > 1 in the kinetic range, but with high variability between events in the latter case. Variability is similarly large at kd_i ≲ 3 × 10⁻³, where there is some indication of α being closer to −3/2 than −5/3. However, single fits at kd_i ≤ 10⁻³ that include points below the kd_i ≃ 10⁻⁴ cutoff in Figure 3 give a mean α ≃ −5/3. The nature of the spectrum at low kd_i is further considered in Section 3.3. We note also that none of the trends identified in Figure 3 showed a significant dependence on r. Radial evolution is considered explicitly in Section 3.4.

3.2. Compressibility Spectra

3.3. Correlation Lengths and Flux Rope Fields

We now turn to the low-frequency limit of the inertial range, which is often found at scales near the correlation length (e.g., Matthaeus & Goldstein 1982; Burlaga & Goldstein 1984). Magnetic field correlation lengths, λ, have been determined for 27 of the ICMEs using data at 1 minute resolution. The ICME with the shortest duration (197 minutes) has been excluded from this analysis. For each ICME, the autocorrelation of a 224 minute interval starting from the leading edge has been calculated, with lags, τ, ranging from 0 to 112 minutes. The interval length has been chosen to equal two-thirds the duration of the shortest-duration ICME analyzed. An autocorrelation curve for each radial–tangential–normal (RTN) field component j = {R, T, N} has been obtained and replotted as a function of spatial scale l = 〈v〉τ, where 〈v〉 is the mean proton speed in the ICME. The curves have then been fitted with exponentials of the form , with the average of the three λ_j values, λ, taken as the overall value for the ICME interval. A similar technique has been used by Wicks et al. (2010b).

The correlation length determined in this way is likely sensitive to the spatial variation of the ICME's flux rope structure. To reduce this sensitivity, and thus estimate the correlation length of the fluctuations rather than the background structure, correlation lengths have also been calculated with subtraction of the background field from the time series before performing the autocorrelation. The background field has been estimated with (i) the mean of each field component in the interval and (ii) with a Gold–Hoyle (GH) fit to the ICME flux rope (Farrugia et al. 1999), giving correlation lengths λ_M and λ_GH, respectively. The axial, poloidal, and radial components of the GH flux rope in cylindrical coordinates are given by

respectively, where fit-obtained constants B₀ and Γ are the field magnitude at the axis and the field-line twist per unit length, respectively, and ρ is the radial distance from the axis. Of the 27 ICMEs analyzed, 15 could be well-fitted with the GH rope model. The key parameters of the 15 fits are listed in the Appendix. Further details of the GH fitting procedure are described by Kilpua et al. (2019) and Good et al. (2019).

The top left panel of Figure 4 shows the magnetic field time series of an example ICME in RTN coordinates, with horizontal lines corresponding to the component mean values and smooth lines to the GH fit profile. The rotation of the magnetic field at the scale of the ICME duration is well captured by the GH fit and is a better approximation of the ICME's structure than the component mean values. The bottom left panel in Figure 4 shows the trace PSD for the same ICME with subtraction of the mean field (i.e., PSD calculated in the usual way) and with subtraction of the GH fit. While the mean-subtracted and GH-subtracted spectra overlap at kd_i ≳ 10⁻⁴, it can be seen that power at the lowest frequencies (equivalent to the largest spatial scales) is reduced in the GH-subtracted spectrum, with a corresponding increase in the spectral index from −1.36 to −1.21 at kd_i ≤ 10⁻³. Across all ICMEs fitted with the GH model, the average spectral index increased from −1.65 in the mean-subtracted spectra to −1.53 in the GH-subtracted spectra at kd_i ≤ 10⁻³. Also displayed in Figure 4 is the PSD of the GH fit profile (gold line, right panel), which shows the power contained within the low-frequency, large-amplitude rotation of the flux rope field. The significantly enhanced power and steep slopes seen at low frequencies in Figures 1 and 2 are due to such fields.

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The bottom left panel in Figure 4 shows the three ICME correlation lengths versus ICME radial width, L = 〈v〉T, where T is the ICME duration. The mean values of λ, λ_M, and λ_GH are 2.3, 1.8, and 1.4 × 10⁶ km, respectively, with λ also being larger than both λ_M and λ_GH in all ICMEs individually. All three ICME correlation lengths tend to increase with L, and in all cases are significantly less than L. Five-point running average lines indicate some weak leveling-off in the increase in correlation lengths at large L, consistent with a nonlinear trend.

3.4. Radial Evolution

Figure 5 displays a range of parameters in the ICME intervals plotted versus r, with points color-coded to β_i. Values in the top four panels are averages in the inertial range at 10⁻² ≤ kd_i ≤ 10⁻¹. The gray lines show five-point moving averages for each parameter. Furthermore, correlations of the parameters with r and β_i have been calculated using the weighted Kendall coefficient, τ_wK, as implemented by Lin et al. (2018). Similarly to other nonparametric correlation coefficients, perfect correlation is indicated by τ_wK = 1, perfect anticorrelation by τ_wK = −1, and progressively weaker correlations or anticorrelations as τ_wK approaches zero. A weighted coefficient has been used because the parameters are generally heteroscedastic (i.e., have variances that change) with respect to r and β_i.

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Power-law fits as functions of r are shown in Figure 5 for the two parameters that have the greatest r correlation as measured by ∣τ_wK∣, namely δ B² (τ_wK = −0.18) and f_b (τ_wK = −0.20); the logarithm of these two parameters also show the greatest r correlation (τ_wK = −0.24 and τ_wK = −0.21, respectively). The anticorrelation of δ B² with r and the δ B² ∝ r^−1.50 dependence reflect the falling PSD seen in Figure 1. The f_b ∝ r^−0.90 dependence is, within the fit uncertainty, comparable to the r^−1.09 dependence found by Bruno & Trenchi (2014) in fast wind and the r^−1.11 dependence found by Duan et al. (2020) in slow wind. There is very broadly an increase in λ_M with r, although the spread in values is large and the correlation with r relatively weak (τ_wK = 0.11). The mean of λ_M values at r > 0.9 au is in approximate agreement with the mean value of 2.33 × 10⁶ km found by Ruiz et al. (2014) in ICMEs at 1 au. The weakest correlations with r are shown by C and α, both having ∣τ_wK∣ ≲ 0.05. The correlation of β_i with r is negligible (τ_wK = 0.01), as expected for ICMEs.

The β_i correlations measured by ∣τ_wK∣ are generally weak across all parameters, in some cases (e.g., C) possibly due to the narrow range of β_i that has been examined. Of the parameters displayed in Figure 5, the fluctuation amplitude normalized to the mean field strength across the interval, , has the greatest correlation with β_i (τ_wK = 0.14, rising to τ_wK = 0.18 for the parameter logarithm). This correlation is consistent with a suppression of magnetic fluctuations at low β_i. As in Figure 3, the β_i color map indicates a qualitative correlation between C and β_i, although the quantified correlations are relatively low in this sample (τ_wK = 0.10 for the parameter logarithm).

At large scales (kd_i ≤ 10⁻³), removal of the flux rope fields rather than the mean fields gave less steep spectral slopes, with the mean index value rising to −3/2 from −5/3. Subtraction of the flux rope field is a form of detrending or frequency filtering that removes power in the spectrum associated with the rotation of the background field, a signature of the flux rope transit over the spacecraft. Time-varying background fields in ICME intervals contrast with the approximately constant, Parker-spiral background in the solar wind more generally. It has been implicitly assumed that the flux rope field is a passive background that does not participate in the turbulent cascade, with the residual time series representing waves, turbulence, or structures not related to the global flux rope. The shallower slopes found at large scales with subtraction of the flux rope field may represent a distinct subregion of the inertial range (e.g., Telloni 2022; Sioulas et al. 2023b) or a rollover toward a 1/f range. Sliding-window estimates of the spectral index in some cases do show α ≃ −1 values at large scales with or without flux rope subtraction, but only sporadically. The properties of the fluctuations at these scales merit further investigation given their potential role in supplying energy to the turbulent cascade at smaller scales.

Correlation lengths of ∼4 hr intervals were longer in the total ICME fields than in the residual fields (mean or flux-rope-subtracted). Longer correlation lengths in the total field were likely due to the dominant influence of the background field rotation, with strongly autocorrelated variations at timescales of order ∼1 hr. ICMEs not displaying simple rotations in the background field may also have ordered global structures with relatively long correlation lengths. The correlation lengths in the background-subtracted fields may be related to the outer scale of turbulence, or, alternatively, to some typical mesoscale length of ICME substructure (Lugaz et al. 2018). Taking the mean-subtracted correlation length, λ_M, as the turbulent outer scale and the high-frequency break scale calculated with Taylor's hypothesis, l_b, as the inner scale, then the span of the inertial range defined as λ_M/l_b was approximately invariant with r and β_i, at a constant value of 2.0 ± 1.2 × 10⁴ in the ICMEs analyzed. Correlation lengths increased with radial distance (as in the solar wind generally) and with the ICME radial width, L.

At kd_i ≳ 10⁻³, spectral indices in the ICMEs were the same in the mean and flux-rope-subtracted fields. The presence at inertial scales of a −5/3 index across a range of heliocentric distances suggests turbulent states that are already well developed close to the Sun, that develop independently of the specific energy injection processes occurring at larger scales, and that may be related to the low cross helicity commonly found in ICMEs (Good et al. 2020, 2022; Soljento et al. 2023). Low cross helicity and an associated −5/3 index (Podesta & Borovsky 2010) may, in turn, be general features of solar wind originating from closed-field regions in the corona (e.g., Borovsky et al. 2019). For example, solar wind near the heliospheric current sheet (HCS) has also been observed to have a −5/3 index close to the Sun (Chen et al. 2021), in contrast to the more typical −3/2 value seen further from the HCS at the same distances (e.g., Chen et al. 2020); the radial evolution from a −3/2 to −5/3 index seen in the solar wind away from the HCS is likely caused by an evolution in cross helicity from highly imbalanced to balanced values (Sioulas et al. 2023a; McIntyre et al. 2023). Like most ICMEs, the HCS forms at the closed-field streamer belt, and ICMEs often locally replace the HCS in situ (Crooker et al. 1998): thus some similarity between inertial-range spectral properties of near-HCS and ICME plasma might be expected. Steepening of the spectral index to values significantly below −2.8 in the transition range was observed consistently at all radial distances, unlike in the solar wind more generally. This behavior may be related to the ICMEs' low β_i, a dependency recently highlighted by Matteini et al. (2020).

The approximate invariance of magnetic compressibility with r is notable. It has been suggested by Verscharen et al. (2017) that the small amount of compressive fluctuation power in the solar wind is primarily due to the MHD slow mode rather than the kinetic slow mode. By assuming that the majority incompressible power is due to the MHD Alfvén mode, Chen et al. (2020) find that the magnetic compressibility is given by , where is the Alfvén-to-slow-mode amplitude ratio, γ is the adiabatic index, and θ_kB is the slow-mode propagation angle relative to the mean field. Making the further assumption that γ, θ_kB ∼ 90°, C, and β_i are all approximately invariant with r leads to the conclusion that ² is also invariant, such that ICMEs are a plasma environment in which the slow mode is frozen relative to the Alfvén mode with heliocentric distance. This contrasts with the rising slow-mode fraction found by Chen et al. (2020) in the solar wind more generally. Furthermore, while the strong expansion and closed magnetic structuring described in Section 1 are likely the dominant influences, the particularly low power in compressive modes, which are a source of ion heating at inertial scales (Schekochihin et al. 2019, and references therein), may also partly contribute to the low proton temperatures observed in ICMEs. A better understanding of turbulent heating in ICMEs could provide more stringent constraints for modeling their thermodynamic evolution with radial distance, and thus lead to more accurate modeling of ICME expansion and propagation for space weather prediction purposes.

We have analyzed the power spectra of magnetic field fluctuations within 28 ICMEs observed throughout the inner heliosphere by the PSP and SolO spacecraft. Low β_i across a wide range of heliocentric distances and a global magnetic flux rope structure make ICMEs a distinctive space plasma environment. At large spatial scales comparable to the correlation length (kd_i ≲ 10⁻⁴), a significant fraction of power was contained within the background flux rope fields of the ICMEs. Subtraction of these fields from the time series revealed shorter correlation lengths and shallower spectral slopes that, on average, increased from −5/3 to −3/2 at kd_i ≤ 10⁻³, with some sporadic signatures of a −1 slope also at these scales. The mean value of the spectral slope deep in the inertial range was −5/3 and did not show the −3/2 to −5/3 radial evolution observed in non-ICME wind away from the HCS. The inertial range terminated at scales closer to the proton inertial length than the proton gyroscale, consistent with previous analysis of low-β_i plasma. Steepening of the spectral slope to values below −3 in the transition range was observed at all radial distances. Magnetic compressibility in the inertial range held similar values to the non-ICME wind in the inner heliosphere but did not grow with radial distance to 1 au, likely mirroring very weak radial variations in β_i and the Alfvén-to-slow-mode fraction. Scaling of the Alfvénic turbulence in the inertial range appears to be independent of the global flux rope structure and low β_i that characterize ICMEs, suggestive of its universal, system-independent nature.

This work was funded by an Academy of Finland Research Fellowship (grants 338486 and 346612; INERTUM). C.H.K.C. is supported by UKRI Future Leaders Fellowship MR/W007657/1 and STFC Consolidated Grants ST/T00018X/1 and ST/X000974/1. C.M. is funded by the European Union (ERC, HELIO4CAST, 101042188). E.K.J.K. acknowledges support from Academy of Finland Centre of Excellence FORESAIL (grant 336807) and from the European Research Council under the European Union's Horizon 2020 research and innovation program, grant 724391 (SolMAG). Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. We thank the Solar Orbiter and Parker Solar Probe instrument teams for providing the data used in this work, and also the reviewers for the constructive comments on the manuscript. Open access to this work has been funded by Helsinki University Library.

The start and end times of the ICMEs analyzed in this study are listed in Table 1 and are taken from the Helio4Cast ICMECAT database. ⁵ For those ICMEs that have been fitted with the Gold–Hoyle flux rope model, key parameters of the fits are also listed in the table. The fit parameters include: the impact parameter of the spacecraft with the flux rope, p ("0" indicating an axis encounter, "1" an edge encounter); the angle between the rope axis and the R–T plane, θ₀; the angle between the projection of the rope axis onto the R–T plane and the R direction, ϕ₀; the magnetic field strength at the rope axis, B₀; and the field-line twist per unit length, Γ.