The theory of anisotropic magnetohydrodynamic (MHD) turbulence has been widely accepted and adopted for plasma systems ranging from clusters of galaxies, the interstellar medium and accretion disks to the heliosphere1,2,3. One of the most crucial predictions of the theory is an Alfvénic transition from weak to strong MHD turbulence when energy cascades from large to small scales4,5. The self-organized process from weak to strong MHD turbulence is the cornerstone of understanding the energy cascade in the complete picture of MHD turbulence.

The critical balance model is attractive for describing the physical behaviour of incompressible MHD turbulence4. When τAτnl (referred to as weak MHD turbulence), weak interactions among the counter-propagating wave packets transfer energy to higher k whereas no energy cascades to higher k, where τA = 1/(kVA) is the linear Alfvén wave time, τnl = 1/(kδV) is the nonlinear time, VA is the Alfvén speed, δV is the fluctuating velocity perpendicular to the background magnetic field (B0), and k and k are wavenumbers perpendicular and parallel to B0 (ref. 6). As turbulence cascades to smaller scales, nonlinearity strengthens until it reaches the critical balance (τA ≈ τnl) at the transition scale (λCB). On scales smaller than λCB, Alfvén wave packets are statistically destroyed in one τA. In addition to the first-order interactions of counter-propagating waves, all higher orders of interactions can contribute to create strong MHD turbulence4,7. In compressible MHD, small-amplitude fluctuations can be decomposed into three eigenmodes (namely, Alfvén, fast and slow modes) in a homogeneous plasma8,9,10,11,12. Alfvén modes decoupled from compressible MHD turbulence, which are linearly independent of fast and slow modes, show similar properties to those in incompressible MHD turbulence, for example, the Kolmogorov spectrum and the scale-dependent anisotropy8,13. Additionally, numerical simulations have confirmed that the Alfvénic weak-to-strong transition occurs in both incompressible MHD turbulence and Alfvén modes decomposed from compressible MHD turbulence9,14,15.

However, this transition has not been confirmed from observations. In this study, we present evidence for the Alfvénic weak-to-strong transition and estimate the transition scale λCB in Earth’s magnetosheath using data from the four Cluster spacecraft16. Earth’s magnetosheath offers a representative environment for studying plasma turbulence, given that most astrophysical and space plasmas with finite plasma β are compressible, where β is the ratio of the plasma to magnetic pressure.


Here we present an overview of fluctuations observed by Cluster-1 in geocentric-solar-ecliptic (GSE) coordinates during 23:00–10:00 Universal Time (UT) on 2–3 December in Fig. 1. During this period, the four Cluster spacecraft flew in a tetrahedral formation with relative separation dsc ≈ 200 km (around three proton inertial lengths di ≈ 74 km) on the flank of Earth’s magnetosheath around [1.2, 18.2, −5.7] RE (Earth radius). We chose this time interval to study the Alfvénic weak-to-strong transition because the fluctuations satisfy the following criteria. First, the background magnetic field (B) measured by the fluxgate magnetometer17 and the proton bulk velocity (Vp) measured by the Cluster Ion Spectrometer (CIS)18 were relatively stable, as shown in Fig. 1a,b. Second, we cross-verified the reliability of the plasma data based on the consistency between the proton density (Np) measured by CIS and the electron density (Ne) measured by the Waves of High frequency and Sounder for Probing of Electron density by Relaxation (WHISPER)19, as shown in Fig. 1c.

Fig. 1: Overview of fluctuations measured by Cluster-1 in Earth’s magnetosheath on 2–3 December 2003.
figure 1

af, Data are displayed in GSE coordinates. a, Magnetic field components (BX, BY and BZ). b, Proton bulk velocity (VX, VY and VZ). c, Proton and electron densities. d, Spectral slopes (α) of magnetic field and proton velocity fluctuations between 0.001 Hz and 0.1fci. The two horizontal lines represent α = −5/3 and −3/2. e, Proton plasma βp. f, Turbulent Alfvén Mach number (MA,turb = δVp/VA) and half of the relative amplitudes of the magnetic field (δB/(2B0)), where δVp and δB are the root mean square of the proton velocity and magnetic field fluctuations, respectively. The fluctuations analysed in detail were observed during 23:00–10:00 UT on 2–3 December and are marked between the two vertical dashed lines.

Source data

We set a moving time window with a length of 5 h and a moving step of 5 min. The selection of a 5 h length ensured that we obtained measurements at low frequencies (large scales) while the mean magnetic field (B0) within the moving time window was approaching the local mean field at the selected largest spatial scale. The uniformity of B0 was independent of the transformation between real and wavevector space; however, it differed from the theoretically expected local mean field at each scale. To assess such differences, Supplementary Fig. 1 shows that the local mean field at different scales is closely aligned with B0 most of the time, suggesting that using B0 to approximate the local mean field is acceptable. To further address this limitation of the mode decomposition method, which relies on a perturbative treatment of fluctuations in the presence of a uniform background magnetic field8, we provide results obtained using various time window lengths, all of which show similar conclusions (Supplementary Fig. 2).

Third, Fig. 1d shows the spectral slopes of the trace magnetic field and proton velocity power calculated by a fast Fourier transform with three-point centred smoothing in each time window. These spectral slopes at the spacecraft-frame frequency fsc ≈ [0.001 Hz, 0.1fci] are close to −5/3 or −3/2 (the proton gyro-frequency fci ≈ 0.24 Hz), suggesting that the turbulent fluctuations were in a fully developed state. The remaining magnetosheath fluctuations with spectra close to \({f}_\mathrm{sc}^{\;-1}\) are typically populated by uncorrelated fluctuations20,21 and are beyond the scope of the present paper. Figure 1e shows that the average proton plasma βp was around 1.4. Finally, Fig. 1f shows that the turbulent Alfvén number MA,turb ≡ δVp/VA ≈ δB/(2B0) ≈ 0.33, suggesting that the fluctuations include substantial Alfvénic components and satisfy the small-amplitude fluctuation assumption (the nonlinear terms \(\delta {V}_\mathrm{p}^{\;2}\) and δB2 are weaker than the linear terms VAδVp and B0δB). Nevertheless, the average magnetic compressibility \({C}_{\parallel}(\;{f}_\mathrm{sc})=\frac{| \delta {B}_{\parallel}(\;{f}_\mathrm{sc}){|}^{2}}{| \delta {B}_{\parallel}(\;{f}_\mathrm{sc}){|}^{2}+| \delta {B}_{\perp}(\;{f}_\mathrm{sc}){|}^{2}} \approx 0.34\), indicating that the fluctuations are a mixture of Alfvén and compressible magnetosonic modes (fast and slow)22, where δB and δB are the fluctuating magnetic fields parallel and perpendicular to B0.

Due to the homogeneous and stationary state of the turbulence (Supplementary Fig. 3), we can utilize frequency–wavenumber distributions of Alfvénic power, that is magnetic power \({P}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel },{f}_\mathrm{sc})\) and proton velocity power \({P}_{\mathrm{V}_\mathrm{A}}({k}_{\perp },{k}_{\parallel },{f}_\mathrm{sc})\), to investigate the structure of the turbulence. Alfvénic fluctuations were extracted in each time window based on their incompressibility and fluctuating directions perpendicular to B0 (Methods). To distinguish between spatial evolution and temporal evolution without relying on any spatio-temporal hypothesis, we determined the wavevectors by combining the singular value decomposition method23 (to obtain \({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\)) and a multispacecraft timing analysis24 (to obtain \({\hat{{{{\bf{k}}}}}}_\mathrm{A}\)). Note that \({\hat{{{{\bf{k}}}}}}_\mathrm{A}\) was not completely aligned with \({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\). Namely, \({\hat{{{{\bf{k}}}}}}_\mathrm{A}\) could deviate from \({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\) by angle η (Supplementary Fig. 4). Thus, we present results for η < 10°, η < 15°, η < 20°, η < 25° and η < 30°. Given the marginal impact of different choices of η (Supplementary Figs. 5 and 6), the spectral results are displayed by taking the dataset for η < 30° as an example without loss of generality. The fluctuations, in this event, count for 42% of the total Alfvénic fluctuations.

To ensure the reliability of the wavenumber determination, we established minimum thresholds k > 1/(100dsc) and k > 10−5 km−1. Consequently, our observations exclude the ideal two-dimensional (2D) case (k = 0). Here, τA is infinity, indicating persistent strong nonlinearity25,26. Nevertheless, quasi-2D (small k) modes are present, as k is much smaller than k at small wavenumbers (Fig. 2). These quasi-2D fluctuations satisfy τA < τnl, as shown in Fig. 3c, and exhibit weak nonlinearity. This weak turbulent state occurs since \(\delta {B}_\mathrm{A}^{2}({k}_{\perp },{k}_{\parallel })/{B}_{0}^{2}\) is very low, where the Alfvénic magnetic energy density at k and k is calculated with \(\delta {B}_\mathrm{A}^{2}({k}_{\perp},{k}_{\parallel})=\sum_{{k}_{\perp} = {k}_{\perp}}^{{k}_{\perp}\to \infty}\sum_{{k}_{\parallel} = {k}_{\parallel}}^{{k}_{\parallel}\to \infty}\int_{0}^{\infty}{P}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{k}_{\parallel},{f}_\mathrm{sc})\,\mathrm{d}\;{f}_\mathrm{sc}\).

Fig. 2: Comparison between wavenumber distributions of Alfvénic magnetic energy \({\hat{\boldsymbol{D}}}_{{\mathbf{B}}_{\mathbf{A}}}{\boldsymbol{(}}{\boldsymbol{k}}_{\perp},{\boldsymbol{k}}_{\boldsymbol{\parallel}}{\boldsymbol{)}}\) and the theoretical energy spectra \({\hat{\boldsymbol{I}}}_{\mathbf{A}}{\boldsymbol{(}}{\boldsymbol{k}}_{\perp},{\boldsymbol{k}}_{\parallel}{\boldsymbol{)}}\).
figure 2

a, 2D spectral image of \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) with a high resolution (400 × 400 bins). b, 2D filled contours of \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) with low-resolution binning (150 × 150 bins), to clarify the contours. In each panel, \({\hat{I}}_\mathrm{A}({k}_{\perp },{k}_{\parallel })\) at L0 ≈ 4.6 × 104 km is displayed by colour contours with black dashed curves with the same colour map as \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\). The black dotted curves mark \(k=\sqrt{{k}_{\parallel }^{2}+{k}_{\perp }^{2}}=0.01/{d}_{i}\) and 0.03/di. These figures utilize the dataset for η < 30°.

Source data

Fig. 3: Perpendicular wavenumber dependence of the compensated spectra, parallel wavenumber and nonlinearity parameter.
figure 3

a, \({k}_{\perp }^{\;5/3}{E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\) versus k. The dashed line represents the scaling \({k}_{\perp }^{\;5/3}{E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\propto {k}_{\perp }^{\;5/3-2}\). To facilitate comparison with proton velocity fluctuations, magnetic field fluctuations are in Alfvén speed units. b, Variation of k versus k. The dashed line represents the scaling \({k}_{\parallel }\propto {k}_{\perp }^{\;2/3}\). The horizontal dotted line marks k = 7 × 10−5 km−1. c, \({\chi }_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) spectrum calculated using the dataset for η < 30°. There are four zones: (1) 5 × 10−5 < k < 1.6 × 10−4 km−1, (2) 1.6 × 10−4 < k < 3 × 10−4 km−1, (3) 3 × 10−4 < k < 7 × 10−4 km−1 and (4) 7 × 10−4 < k < 1 × 10−3 km−1. The first, second and third vertical dotted lines are around the maximum of \({k}_{\perp }^{\;5/3}{E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\), the beginning and the end of flattened \({k}_{\perp }^{\;5/3}{E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\), respectively.

Source data

Evidence for the Alfvénic weak-to-strong transition

The 2D wavenumber distributions of magnetic energy are calculated by


\({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{k}_{\parallel})={D}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{k}_{\parallel})/{D}_{{B}_\mathrm{A},\max}\) is normalized by the maximum magnetic energy in all (k, k) bins, as displayed by the spectral image and contours in Fig. 2. Compared to the isotropic dotted curves, \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) is prominently distributed along the k direction, suggesting a faster perpendicular cascade. This anisotropic behaviour is more pronounced at higher wavenumbers, consistent with previous simulations and observations9,13,27,28.

Moreover, \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) is compared with the modelled 2D theoretical energy spectra based on strong turbulence4,29:

$${I}_\mathrm{A}\left({k}_{\perp},{k}_{\parallel}\right)\propto {k}_{\perp}^{-7/3}\exp \left(-\frac{{L}_{0}^{1/3}| {k}_{\parallel}|}{{M}_\mathrm{A,turb}^{\;4/3}{k}_{\perp}^{\;2/3}}\right),$$

where the injection scale L0 ≈ [4.6 × 104, 8.1 × 104] km is approximately estimated by the correlation time Tc ≈ [1,300, 2,300] s and root mean square of the perpendicular fluctuating velocity δVp ≈ MA,turbVA ≈ 35 km s−1. In Fig. 2, \({\hat{I}}_{A}({k}_{\perp },{k}_{\parallel })\) is normalized as IA(k, k) by a constant value (one-third of the maximum magnetic energy in all (k, k) bins), as displayed by the colour contours with black dashed curves. The 2D distribution \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) shows two different properties: (1) For k < 2 × 10−4 km−1, \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) is mainly concentrated at k < 7 × 10−5 km−1 and cascading along the k direction, suggesting that little energy cascades parallel to the background magnetic field, consistent with energy distributions in weak MHD turbulence6. (2) For k > 2 × 10−4 km−1, \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) starts to distribute to higher k, and both wavenumber distributions and intensity changes of \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) are almost consistent with \(\hat{I}_\mathrm{A}({k}_{\perp },{k}_{\parallel })\). This indicates that \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) captures some of the theoretical characteristics of strong MHD turbulence4. Besides, \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) is in good agreement with the Goldreich–Sridhar scaling \({k}_{\parallel }\propto {k}_{\perp }^{\;2/3}\) (ref. 4). This result further confirms that the properties of \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) at k > 2 × 10−4 km−1 are closer to those in strong MHD turbulence. The change in \({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) from purely stretching along the k direction to following the Goldreich–Sridhar scaling \({k}_{\parallel }\propto {k}_{\perp }^{\;2/3}\) reveals a possible transition in the energy cascade.

Figure 3a shows the compensated spectra (\({k}_{\perp }^{\;5/3}{E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\)). The magnetic energy spectral density is defined as \({E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })=\delta{B}_\mathrm{A}^{2}({k}_{\perp})/2{k}_{\perp}\), where \(\delta {B}_\mathrm{A}^{2}({k}_{\perp })\) is the magnetic energy density at k (Methods). In zone 2, \({k}_{\perp }^{\;5/3}{E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\) is roughly consistent with \({k}_{\perp }^{\;5/3-2}\) (the dashed line), indicating that the spectral slopes of \({E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\) are around −2. In zone 3, on the other hand, \({k}_{\perp }^{\;5/3}{E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\) is almost flat, suggesting that \({E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\) satisfies the Kolmogorov scaling (\({E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\propto {k}_{\perp }^{-5/3}\)). The sharp change in the spectral slopes of \({E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\) from −2 to −5/3 is apparent evidence for the transition of the turbulence regimes14,15. In addition, \({E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })\propto {k}_{\perp }^{-1}\) appears in a substantial portion of zone 1, indicating the weak turbulence forcing in action9,30.

Figure 3b shows the variation of k versus k given the same Alfvénic magnetic energy. In zone 1, as k increases, k is relatively stable at k ≈ 7 × 10−5 km−1. In zone 3, the variation of k versus k agrees with the Goldreich–Sridhar scaling \({k}_{\parallel }\propto {k}_{\perp }^{\;2/3}\) (the dashed line). Figure 3c shows k versus k distributions of the nonlinearity parameter \({\chi }_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\), which is one of the most critical parameters in distinguishing between weak and strong MHD turbulence5. Here, \({\chi }_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) is calculated from \(\frac{{k}_{\perp }\delta {B}_\mathrm{A}({k}_{\perp },{k}_{\parallel })}{{k}_{\parallel }{B}_{0}}\) (Methods). For the corresponding parallel and perpendicular wavenumbers in Fig. 3b, \({\chi }_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) is much less than unity at most wavenumbers in zone 1, whereas in zone 3, \({\chi }_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel })\) increases towards unity and follows the scaling \({k}_{\parallel }\propto {k}_{\perp }^{\;2/3}\). These results suggest that there is a transition from weak to strong nonlinear interactions, in agreement with theoretical expectations and simulations5,14,15.

We observe a similar Alfvénic weak-to-strong transition in the measured proton velocity fluctuations (Supplementary Fig. 7). The transition scale (λCB) is estimated from the smallest perpendicular wavenumber of strong turbulence (k,CB), where λCB ≈ 1/k,CB. For both magnetic field and proton velocity fluctuations, k,CB is around 3 × 10−4 km−1, marked by the second vertical lines in Fig. 3 and Supplementary Fig. 7. The consistency in the transition scales estimated by magnetic field and proton velocity measurements further confirms the reliability of our findings.

A notable perturbation is present in zone 2, as a result of local enhancements of the magnetic energy at k ≈ 1.8 × 10−4 km−1 (Fig. 2), leading to the simultaneous existence of strong nonlinearity (\({\chi }_{\mathrm{B}_\mathrm{A}} \approx 1\)  ) and weak nonlinearity (\({\chi }_{{B}_{A}}\ll 1\)    ) in the wave number range corresponding to those in Fig. 3b. Thus, the Alfvénic weak-to-strong transition more probably occurs within a ‘region’ rather than at a critical wavenumber. We do not discuss the fluctuations in zone 4. The deviations in the data for η < 10° and η < 15° in zone 3 of Fig. 3b are probably due to the limited number of data samples (Supplementary Fig. 6). The uncertainties mentioned above do not affect our main conclusions.

Figure 4 presents frest versus k distributions of the magnetic energy, where frest is the frequency in the plasma flow frame. At k < 5 × 10−5 km−1, the magnetic energy is concentrated at frest ≈ fA, where fA is the Alfvén frequency (horizontal dotted lines with error bars). At k > 1 × 10−4 km−1, the range of frest broadens, mostly deviating from fA. Nevertheless, the boundary of fluctuating frequencies is roughly consistent with the scaling \({f}_\mathrm{rest}\propto {k}_{\perp }^{\;2/3}\) (the dashed line), indicating that the magnetic energy at these wavenumbers satisfies the scaling \({k}_{\parallel }\propto {k}_{\perp }^{\;2/3}\) due to frestk for Alfvén modes. These results suggest that Alfvénic fluctuations with strong nonlinear interactions do not agree with linear dispersion relations but satisfy the wavenumber scaling of Alfvén modes. The change from single-frequency to broadening-frequency fluctuations with increasing k suggests that there is a possible transition of turbulence regimes.

Fig. 4: The k versus frest distributions of Alfvénic magnetic energy in the plasma flow frame.
figure 4

\({\hat{D}}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{f}_\mathrm{rest})={D}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{f}_\mathrm{rest})/{D}_{\mathrm{B}_\mathrm{A},\max}\) is normalized by the maximum magnetic energy in all (k, frest) bins. The horizontal dotted lines represent theoretical Alfvén frequencies fA = kVA/(2π), where k ≈ 7 × 10−5 km−1 in zone 1 of Fig. 3b, and k ≈ 1 × 10−4 km−1 in zone 1 of Supplementary Fig. 7b. The fA uncertainties are estimated by the standard deviation of VA (106 ± 11 km−1), illustrated by error bars on corresponding horizontal dotted lines. This figure utilizes the dataset for η < 30°.

Source data


The Alfvénic transition of weak to strong turbulence during cascades to smaller scales is one of the cornerstones of the modern MHD theory. Despite being proposed decades ago, evidence that confirms the existence of the Alfvénic transition is lacking. In this paper, we present direct evidence of the Alfvénic transition at different angles, for example, the transition of energy spectra (Fig. 3a), the Goldreich–Sridhar-type envelope for the nonlinear parameter (Fig. 3c) and the spread of frest on small scales (Fig. 4; see Table 1 for a summary). Our observation demonstrates that the Alfvénic transition to strong turbulence is bound to occur with an increase of the nonlinearity, even if fluctuations on large scales are considered as ‘small amplitude’ (MA,turb ≈ 0.33). Note that the plasma parameters in the analysed event are generic and that the Alfvénic weak-to-strong transition can occur in other astrophysical and space plasma systems. The impact of our findings goes beyond the study of turbulence to particle transport and acceleration31,32, magnetic reconnection33,34, star formation35,36 and all other relevant fields. See, for example, refs. 37,38.

Table 1 Transition wavenumbers are determined by magnetic field measurements


GSE coordinates

We use the GSE coordinates in this study. XGSE points towards the Sun from the Earth, ZGSE orients along the ecliptic north pole and YGSE completes a right-handed system.

Trace power spectral densities

The trace power spectral densities of the magnetic field and proton velocities (PB = PB,X + PB,Y + PB,Z and PV = PV,X + PV,Y + PV,Z) were calculated by applying a fast Fourier transform with three-point centred smoothing in GSE coordinates. We chose the intermediate instant of each time window as the time point at which the spectral slope varies with time.

Alfvén mode decomposition

We calculated the wavenumber–frequency distributions of the Alfvénic magnetic field and proton velocity power with an improved Alfvén mode decomposition method. This method combines the linear decomposition method8, singular value decomposition (SVD)23 and a multispacecraft timing analysis24. We performed the calculations in each moving time window with a length of 5 h and a moving step of 5 min. The window length selection (5 h) provides low-frequency (large-scale) measurements while ensuring that B0 approaches the local background magnetic field.

First, we obtained the wavelet coefficients (W) of the magnetic field and proton velocity using Morlet-wavelet transforms39. To eliminate the edge effect due to the finite length of the time series, we performed the wavelet transforms for twice the time window length and cutoff of the periods affected.

Second, the wavevector directions (kSVD(t, fsc)) were determined by SVD for the magnetic wavelet coefficients23. SVD was used to create a real matrix equation (\(\bf{S}\cdot {\hat{{{{\bf{k}}}}}}_\mathrm{SVD}=0\)) equivalent to the linearized Gauss’s law for magnetism (\({{{\bf{B}}}}\cdot {\hat{{{{\bf{k}}}}}}_\mathrm{SVD}=0\)). Notice that the minimum singular value of the real matrix S (6 × 3) is the best estimate of wavevector directions but cannot be used to determine the wavenumbers. Since the relative satellite separations were much shorter than the half-wavelength of MHD scales, the properties of the fluctuations simultaneously measured by the four Cluster spacecraft are similar. Thus, the average wavevector direction and background magnetic field are given by \({{{{\bf{k}}}}}_\mathrm{SVD}=\frac{1}{4}\sum_{\rm{i} = 1,2,3,4}{\hat{{{{\bf{k}}}}}}_{\mathrm{SVD},\mathrm{C_i}}\) and \({{{{\bf{B}}}}}_{0}=\frac{1}{4}\sum_{\rm{i} = 1,2,3,4}{{{{\bf{B}}}}}_{0,\mathrm{C_i}}\). Ci denotes the four Cluster spacecraft.

Third, we extracted Alfvénic components from proton velocity fluctuations based on their incompressibility (\({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\cdot \delta {{{{\bf{V}}}}}_\mathrm{p}=0\)) and perpendicular fluctuating directions (\({\hat{{{{\bf{b}}}}}}_{0}\cdot \delta {{{{\bf{V}}}}}_\mathrm{p}=0\)) in wavevector space, where δVp is expressed by vectors of velocity wavelet coefficients, \({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}={{{{\bf{k}}}}}_\mathrm{SVD}/| {{{{\bf{k}}}}}_\mathrm{SVD}|\), and \({\hat{{{{\bf{b}}}}}}_{0}={{{{\bf{B}}}}}_{0}/| {{{{\bf{B}}}}}_{0}|\). Similarly, Alfvénic magnetic field fluctuations were extracted from \({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\cdot \delta {{{\bf{B}}}}=0\) and \({\hat{{{{\bf{b}}}}}}_{0}\cdot \delta {{{\bf{B}}}}=0\), according to the linearized induction equation

$$\omega \delta {{{\bf{B}}}}={{{\bf{k}}}}\times \left({{{{\bf{B}}}}}_{0}\times \delta {{{{\bf{V}}}}}_\mathrm{p}\right) \approx | {{{\bf{k}}}}| {\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\times \left({{{{\bf{B}}}}}_{0}\times \delta {{{{\bf{V}}}}}_\mathrm{p}\right),$$

where k is the wavevector. Thus, Alfvénic proton velocity and magnetic field fluctuations are in the same direction \({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\times {\hat{{{{\bf{b}}}}}}_{0}/| {\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\times {\hat{{{{\bf{b}}}}}}_{0}|\) (see the schematic in Supplementary Fig. 4).

Fourth, the Alfvénic magnetic power at each time t and fsc was calculated as \({P}_{\mathrm{B}_\mathrm{A}}(t,{f}_\mathrm{sc})=\frac{1}{4}\sum_{\rm{i} = 1,2,3,4}{W}_{\mathrm{B}_\mathrm{A},\mathrm{C_i}}{W}_{\mathrm{B}_\mathrm{A},\mathrm{C_i}}^{\;*}\). The Alfvénic proton velocity power was calculated as \({P}_{\mathrm{V}_\mathrm{A}}(t,{f}_\mathrm{sc})={W}_{\mathrm{V}_\mathrm{A},\mathrm{C}_1}{W}_{\mathrm{V}_\mathrm{A},\mathrm{C}_1}^{\;* }\). This is because magnetic field data were available from the four Cluster spacecraft, whereas proton plasma data were available only from Cluster-1 during the period analysed.

Fifth, as SVD does not give the magnitude of the wavevectors, we calculated wavevectors (kA(t, fsc)) using the multispacecraft timing analysis based on phase differences between the Alfvénic magnetic field from the four spacecraft24. The magnetic field data were interpolated to a uniform time resolution of eight samples per second to give sufficient time resolution. We consider that the wavefront was moving in the direction \(\hat{{{{\bf{n}}}}}\) with velocity Vw. The wavevector kA = 2π fscm, where the vector \({{{\bf{m}}}}=\hat{{{{\bf{n}}}}}/{V}_\mathrm{w}\) and the subscript A represents the Alfvénic component.

$$\left(\begin{array}{c}{{{{\bf{r}}}}}_{2}-{{{{\bf{r}}}}}_{1}\\ {{{{\bf{r}}}}}_{3}-{{{{\bf{r}}}}}_{1}\\ {{{{\bf{r}}}}}_{4}-{{{{\bf{r}}}}}_{1}\end{array}\right){{{\bf{m}}}}=\left(\begin{array}{c}\delta {t}_{2}\\ \delta {t}_{3}\\ \delta {t}_{4}\end{array}\right),$$

where Cluster-1 has arbitrarily been taken as the reference. The left-hand side of equation (4) contains the relative spacecraft separations. The right-hand side of equation (4) represents the weighted average time delays, estimated by the ratio of six phase differences, \({\phi}_{\rm{ij}}=\arctan({{{\mathcal{S}}}}({W}_{\mathrm{B}_\mathrm{A}}^{\,{\rm{ij}}}),{{{\mathcal{R}}}}({W}_{\mathrm{B}_\mathrm{A}}^{\,{\rm{ij}}}))\), to the angular frequencies, ωsc = 2π fsc, where ϕij is from all spacecraft pairs (ij = 12, 13, 14, 23, 24, 34)). \({{{\mathcal{S}}}}\) and \({{{\mathcal{R}}}}\) are the imaginary and real parts of the cross-correlation coefficients, respectively. The four Cluster spacecraft provide six cross-correlation coefficients39, that is, \({W}_{\mathrm{B}_\mathrm{A}}^{\;12}=\langle {W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_1}{W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_2}^{\;*}\rangle\), \({W}_{\mathrm{B}_\mathrm{A}}^{13}=\langle {W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_1}{W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_3}^{\;*}\rangle\), \({W}_{\mathrm{B}_\mathrm{A}}^{14}=\) \(\langle {W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_1}{W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_4}^{\;*}\rangle\), \({W}_{\mathrm{B}_\mathrm{A}}^{23}=\langle {W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_2}{W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_3}^{\;*}\rangle\), \({W}_{\mathrm{B}_\mathrm{A}}^{\;24}=\langle {W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_2}{W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_4}^{\;*}\rangle\) and \({W}_{\mathrm{B}_\mathrm{A}}^{34}=\) \(\langle {W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_3}{W}_{\mathrm{B}_\mathrm{A},\mathrm{C}_4}^{\;*}\rangle\), where the angular brackets denote a time average over 256 s to ensure the reliability of the phase differences.

Note that a timing analysis was used to determine the actual wavevectors of the Alfvénic magnetic field. In contrast, SVD determines the best estimate of the wavevector sum in three magnetic field components23. Thus, kA is not completely aligned with \({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\). Besides, we restrict our analysis to fluctuations with small angle η between \({\hat{{{{\bf{k}}}}}}_\mathrm{SVD}\) and kA to ensure the reliability of the extraction process (the third step). By relaxing the η constraints, more sampling points are used so that the uncertainty from the limited measurements decreases. On the other hand, with relaxed η constraints, kA deviates more from kSVD, which may increase the uncertainty. This letter presents results from five datasets for η < 10°, η < 15°, η < 20°, η < 25° and η < 30° to investigate the effects of uncertainties introduced by the combination of SVD and a timing analysis.

Sixth, we constructed a set of 400 × 400 × 400 bins to obtain wavenumber–frequency distributions of the magnetic power \({P}_{\mathrm{B}_\mathrm{A}}({k}_{\perp },{k}_{\parallel },{f}_\mathrm{sc})\) and proton velocity power \({P}_{\mathrm{V}_\mathrm{A}}({k}_{\perp },{k}_{\parallel },{f}_\mathrm{sc})\), where the parallel wavenumber is \({k}_{\parallel }={{{{\bf{k}}}}}_\mathrm{A}\cdot {\hat{{{{\bf{b}}}}}}_{0}\) and the perpendicular wavenumber is \({k}_{\perp }=\sqrt{{{{{\bf{k}}}}}_\mathrm{A}^{2}-{k}_{\parallel }^{2}}\). Each bin subtended approximately the same k, k and fsc. To cover all MHD wavenumbers and ensure measurement reliability, we restricted our analysis to fluctuations with 1/(100dsc) < k < min(0.1/max(di, rcii), π/dsc) and 2/t* < frest < fci/2, and fluctuations beyond these wavenumber and frequency ranges were set to zero. Here, dsc is the relative satellite separation, min(*) and max(*) are the minimum and maximum, di is the proton inertial length, rci is the proton gyro-radius, t* is the duration studied, frest = fsc − kAVp/(2π) is the frequency in the plasma flow frame and Vp is the proton bulk velocity with the spacecraft velocity being negligible. This study utilizes the representation of absolute frequencies:

$$(\;{f}_\mathrm{rest},{{{{\bf{k}}}}}_\mathrm{A})=\begin{cases}(\;{f}_\mathrm{rest},{{{{\bf{k}}}}}_\mathrm{A}), &{f}_\mathrm{rest} > 0,\\ (-{f}_\mathrm{rest},-{{{{\bf{k}}}}}_\mathrm{A}), &{f}_\mathrm{rest} < 0.\end{cases}$$

\({P}_{{\epsilon }_\mathrm{A}}({k}_{\perp },{k}_{\parallel },{f}_\mathrm{sc})\) were obtained by averaging \({P}_{{\epsilon }_\mathrm{A}}({k}_{\perp },{k}_{\parallel },{f}_\mathrm{sc},t)\) over effective time points in all time windows at each fsc and each k, where ϵ = V or B represents the proton velocity (V) and magnetic field (B).

Alfvén speed units

For comparison, this study presents the fluctuating magnetic field in Alfvén speed units, which are normalized by \(\sqrt{{\mu }_{0}{m}_\mathrm{p}{N}_{0}}\), where μ0 is the vacuum permeability, mp the proton mass and N0 the mean proton density.

Magnetic energy spectral density

For this study, we define the energy spectral density of the magnetic field as \({E}_{\mathrm{B}_\mathrm{A}}({k}_{\perp })=\frac{1}{2}\frac{\delta {B}_\mathrm{A}^{2}({k}_{\perp })}{{k}_{\perp }}\), where the Alfvénic magnetic energy density is calculated as \(\delta {B}_\mathrm{A}^{2}({k}_{\perp})=2\sum_{{k}_{\perp} = {k}_{\perp}}^{{k}_{\perp}\to \infty}\sum_{{k}_{\parallel} = 0}^{{k}_{\parallel}\to \infty}\int_{0}^{\infty}{P}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{k}_{\parallel},{f}_\mathrm{sc})\,\mathrm{d}\;{f}_\mathrm{sc}\).

Nonlinearity parameter

The nonlinearity parameter was estimated with \({\chi}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{k}_{\parallel}) \approx\) \({k}_{\perp}\delta {B}_\mathrm{A}({k}_{\perp},{k}_{\parallel})/({k}_{\parallel}{B}_{0})\), where the Alfvénic magnetic energy density was calculated as \(\delta {B}_\mathrm{A}^{2}({k}_{\perp},{k}_{\parallel})=\sum_{{k}_{\perp} = {k}_{\perp}}^{{k}_{\perp}\to \infty}\sum_{{k}_{\parallel} = {k}_{\parallel}}^{{k}_{\parallel}\to \infty}\int_{0}^{\infty}{P}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{k}_{\parallel},{f}_\mathrm{sc})\,\mathrm{d}\;{f}_\mathrm{sc}\), where B0 in Alfvén speed units was around 106 km s−1.

Frequency–wavenumber distribution of the magnetic energy

The frequency–wavenumber distributions of the magnetic energy were approximately estimated by \({D}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{f}_\mathrm{sc}) \approx \sum_{{k}_{\parallel} = 0}^{{k}_{\parallel}\to \infty}{P}_{\mathrm{B}_\mathrm{A}}({k}_{\perp},{k}_{\parallel},{f}_\mathrm{sc}){{\Delta}}{f}_\mathrm{sc}\) and were transformed into the plasma flow frame by correcting for the Doppler shift frest = fsc − kAV/(2π).