Identification of the weak-to-strong transition in Alfv\'enic turbulence from space plasma

Plasma turbulence is a ubiquitous dynamical process that transfers energy across many spatial and temporal scales in astrophysical and space plasma systems. Although the theory of anisotropic magnetohydrodynamic (MHD) turbulence has successfully described phenomena in nature, its core prediction of an Alfvenic transition from weak to strong MHD turbulence when energy cascades from large to small scales has not been observationally confirmed. Here we report the first observational evidence for the Alfvenic weak-to-strong transition in MHD turbulence in the terrestrial magnetosheath using the four Cluster spacecraft. The observed transition indicates the universal existence of strong turbulence regardless of the initial level of MHD fluctuations. Moreover, the observations demonstrate that the nonlinear interactions of MHD turbulence play a crucial role in the energy cascade, widening the directions of the energy cascade and broadening the fluctuating frequencies. Our work takes a critical step toward understanding the complete picture of turbulence cascade, connecting the weak and strong MHD turbulence systems. It will have broad implications in star formation, energetic particle transport, turbulent dynamo, and solar corona or solar wind heating.


INTRODUCTION
The theory of anisotropic MHD turbulence has been widely accepted and adopted in plasma systems, ranging from clusters of galaxies, the interstellar medium, accretion disks, to the heliosphere Yan & Lazarian (2002); Brunetti & Lazarian (2007); Bruno & Carbone (2013).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 scales Goldreich & Sridhar (1995); Howes et al. (2011).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.
However, such a 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 λ CB in Earth's magnetosheath using data from four Cluster spacecraft Escoubet et al. (2001).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.

RESULTS
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, four Cluster spacecraft flew in a tetrahedral formation with relative separation d sc ≈ 200km (around 3 proton inertial length d i ≈ 74km) on the flank of Earth's magnetosheath around [1.2, 18.2, -5.7] R E (Earth radius).We choose this time interval to study the Alfvénic weak-to-strong transition because the fluctuations satisfy the following criteria.Firstly, the background magnetic field (B) measured by the Fluxgate Magnetometer (FGM) Balogh et al. (1997) and the proton bulk velocity (V p ) measured by the Cluster Ion Spectrometry (CIS) Rème et al. (2001) are relatively stable in Figs.1a-b.We cross-verify the reliability of plasma data, based on the consistency between the proton density (N p ) measured by CIS and electron density measured by the Waves of High frequency and Sounder for Probing of Electron density by Relaxation (WHISPER) Décréau et al. (1997) in Fig. 1c.
We set a moving time window with a five-hour length and a five-minute moving step.The selection of a five-hour length ensures that we obtain measurements at low frequencies (large scales) while the mean magnetic field (B 0 ) within the moving time window is approaching the local mean field at the selected largest spatial scale.The uniform B 0 is independent of the transformation between real and wavevector space; however, it differs 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 B 0 most of the time, suggesting that B 0 approximating 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 field Cho & Lazarian (2003), we provide results obtained using various time window lengths, all of which show similar conclusions (Supplementary Fig. 2).
Fig. 1d shows spectral slopes of the trace magnetic field and proton velocity power calculated by fast Fourier transform (FFT) with three-point centered smoothing in each time window.These spectral slopes at spacecraft-frame frequency f sc ≈ [0.001Hz, 0.1f ci ] are close to −5/3 or −3/2 (the proton gyro-frequency f ci ≈ 0.24Hz), suggesting turbulent fluctuations are in a fully-developed state.The remaining magnetosheath fluctuations with spectra close to f −1 sc are typically populated by uncorrelated fluctuations Hadid et al. (2015Hadid et al. ( , 2018) ) and are beyond the scope of the present paper.Fig. 1e shows the average proton plasma β p is around 1.4.Finally, Fig. 1f shows that the turbulent Alfvén number M A,turb ≡ δV p /V A ≈ δB/(2B 0 ) ≈ 0.33, suggesting that fluctuations include substantial Alfvénic components and satisfy the small-amplitude fluctuation assumption (the nonlinear terms (δV 2 p , δB 2 ) are weaker than the linear terms (V A δV p , B 0 δB)).Nevertheless, the average magnetic compressibility 34, indicating fluctuations are a mixture of Alfvén and compressible magnetosonic (fast and slow) modes Sahraoui et al. (2020), where δB ∥ and δB ⊥ are the fluctuating magnetic field parallel and perpendicular to B 0 .
Due to the homogeneous and stationary state of the turbulence (Supplementary Fig. 3), we can utilize frequencywavenumber distributions of Alfvénic power, i.e. magnetic power P B A (k ⊥ , k ∥ , f sc ) and proton velocity power P V A (k ⊥ , k ∥ , f sc ), to investigate the structure of turbulence.Alfvénic fluctuations are extracted based on their incompressibility and fluctuating directions perpendicular to B 0 (see Methods).The extraction is taken at each time window.To distinguish spatial and temporal evolutions without any spatiotemporal hypothesis, we determine wavevectors by combining the singular value decomposition method Santolík et al. (2003) (to obtain kSV D ) and multispacecraft timing analysis Pincon & Glassmeier (2008) (to obtain kA ).It is worth noting that kA is not completely aligned with kSV D .Namely, kA may deviate from kSV D by angle η (Supplementary Fig. 4).Thus, we present the results under 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), spectral results are displayed by taking the data set under η < 30 • as an example without loss of generality.The fluctuations, in this event, count for 42% of total Alfvénic fluctuations.
To ensure the reliability of wavenumber determination, we establish a minimum threshold of k > 1/(100d sc ) and k ∥ > 10 −5 km −1 .Consequently, our observations exclude the ideal two-dimensional (2D) case (k ∥ = 0), where τ A is infinity, indicating persistent strong nonlinearity Galtier et al. (2003); Nazarenko (2007).Nevertheless, quasi-2D (small k ∥ ) modes are present, as k ∥ is much smaller than k ⊥ at small wavenumbers (Fig. 2).These quasi-2D fluctuations satisfies τ A < τ nl , as shown in Fig. 3c, and exhibit weak nonlinearity.This weak turbulent state occurs since δB 2 A (k ⊥ , k ∥ )/B 2 0 is very low, where Alfvénic magnetic energy density at k ⊥ and k ∥ is calculated by Evidence for the Alfvénic weak-to-strong transition Two-dimensional wavenumber distributions of magnetic energy are calculated by where the injection scale L 0 ≈ [4.6 × 10 4 , 8.1 × 10 4 ]km is approximately estimated by the correlation time T c ≈ [1300, 2300]s and rms perpendicular fluctuating velocity suggesting that little energy cascade parallel to the background magnetic field, consistent with energy distributions in weak MHD turbulence Galtier et al. (2000).
(2) At k ⊥ > 2 × 10 −4 km −1 , DB A (k ⊥ , k ∥ ) starts to distribute to higher k ∥ , and both wavenumber distributions and intensity changes of DB A (k ⊥ , k ∥ ) are almost consistent with ÎA (k ⊥ , k ∥ ).It indicates that DB A (k ⊥ , k ∥ ) captures some theoretical characteristics of strong MHD turbulence Goldreich & Sridhar (1995).Besides, DB A (k ⊥ , k ∥ ) is in good agreement with the Goldreich-Sridhar scaling (1995).This result further confirms that the properties of DB ⊥ reveals a possible transition in the energy cascade.Fig. 3a shows the compensated spectra (k , where the magnetic energy spectral density is defined as , and ).The sharp change in spectral slopes of E B A (k ⊥ ) from −2 to −5/3 is apparent evidence for the transition of turbulence regimes Verdini & Grappin (2012); Meyrand et al. (2016).In addition, appears in a substantial portion of Zone (1), indicating the weak turbulence forcing in action Schekochihin et al. (2012); Makwana & Yan (2020).
Fig. 3b shows the variation of k ∥ versus k ⊥ given the same Alfvénic magnetic energy.As k ⊥ increases, k ∥ is relatively stable at k ∥ ≈ 7 × 10 −5 km −1 in Zone (1).In Zone (3), the variation of k ⊥ versus k ∥ agrees with the Goldreich-Sridhar , which is one of the most critical parameters in distinguishing between weak and strong MHD turbulence Howes et al. (2011), where (see Methods).At the corresponding parallel and perpendicular wavenumbers in Fig. 3b, χ B A (k ⊥ , k ∥ ) is much less than unity at most wavenumbers in Zone (1), whereas χ B A (k ⊥ , k ∥ ) increases towards unity and follows the scaling k ∥ ∝ k 2/3 ⊥ in Zone (3).These results suggest a transition from weak to strong nonlinear interactions, agreeing with theoretical expectations and simulations Howes et al. (2011); Verdini & Grappin (2012); Meyrand et al. (2016).
With the measurements of proton velocity fluctuations, we observe a similar Alfvénic weak-to-strong transition (Supplementary Fig. 7).The transition scale (λ CB ) is estimated by the smallest perpendicular wavenumber of strong turbulence (k ⊥,CB ), where λ CB ≈ 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 magnetic energy at k ⊥ ≈ 1.8 × 10 −4 km −1 (Fig. 2), leading to the simultaneous existence of strong nonlinearity (χ B A ≈ 1) and weak nonlinearity (χ B A ≪ 1) in the wave number range corresponding to those in Fig. 3b.Thus, the Alfvénic weak-to-strong transition more likely occurs within a 'region' rather than at a critical wavenumber.Besides, we do not discuss the fluctuations in Zone (4).The deviations of data sets under η < 10 • and η < 15 • in Zone (3) of Fig. 3b are likely due to the limited data samples (Supplementary Fig. 6).The uncertainties mentioned above do not affect our main conclusions.
Fig. 4 presents k ⊥ versus f rest distributions of magnetic energy, where f rest is the frequency in the plasma flow frame.At k ⊥ < 5 × 10 −5 km −1 , magnetic energy is concentrated at f rest ≈ f A , where f A is Alfvén frequency (horizontal dotted lines with error bars).At k ⊥ > 1 × 10 −4 km −1 , the range of f rest broadens, mostly deviating from f A .Nevertheless, the boundary of fluctuating frequencies is roughly consistent with the scaling f rest ∝ k 2/3 ⊥ (the dashed line), indicating that magnetic energy at these wavenumbers satisfies the scaling k ∥ ∝ k 2/3 ⊥ due to f rest ∝ k ∥ 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 a possible transition of turbulence regimes.

DISCUSSION
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 for confirming the existence of the Alfvénic transition is lacking.In this paper, we present direct evidence of the Alfvénic transition via different angles: e.g., the transition of energy spectra (Fig. 3a), Goldreich-Sridhar type envelope for the nonlinear parameter (Fig. 3c), and the spread of f rest on small scales (Fig. 4; See Tab. 1 for a summary).Our observation demonstrates that the Alfvénic transition to strong turbulence is bound to occur with the increase of nonlinearity even fluctuations on large scales are considered as "small amplitude" (M A,turb ≈ 0.33).We want to point out that plasma parameters in the analyzed event are generic, and 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 itself to particle transport and acceleration Schlickeiser (2002); Yan (2021), magnetic reconnection Matthaeus & Lamkin (1986); Lazarian & Vishniac (1999), star formation Crutcher (2012); Padoan et al. (2014), and all the other relevant fields (see, e.g.Zhang & Yan 2011;Hirashita & Yan 2009).

Method Geocentric-solar-ecliptic (GSE) coordinates
We use the GSE coordinates in this study.X GSE points towards the Sun from the Earth, Z GSE orients along the ecliptic north pole, and Y GSE completes a right-handed system.

Trace power spectral densities
The trace power spectral densities of magnetic field and proton velocity (P B = P B,X + P B,Y + P B,Z and P V = P V,X + P V,Y + P V,Z ) are calculated by applying the fast Fourier transform with three-point centered smoothing in GSE coordinates.We choose the intermediate instant of each time window as the time point where the spectral slope varies with time.

Alfvén mode decomposition method
We calculate wavenumber-frequency distributions of Alfvénic magnetic field and proton velocity power by an improved Alfvén mode decomposition method.This method combines the linear decomposition method Cho & Lazarian (2003), singular value decomposition (SVD) method Santolík et al. (2003), and multi-spacecraft timing analysis Pincon & Glassmeier (2008).We perform the calculations in each moving time window with a five-hour length and five-minute moving step.The window length selection (5 hours) provides low-frequency (large-scale) measurements while ensuring B 0 is approaching the local background magnetic field.First, we obtain wavelet coefficients (W ) of magnetic field and proton velocity using Morlet-wavelet transforms Grinsted et al. (2004).To eliminate the edge effect due to finite-length time series, we perform wavelet transforms twice the time window length and cut off the affected periods.Second, wavevector directions (k SV D (t, f sc )) are determined by SVD of magnetic wavelet coefficients Santolík et al. (2003).The SVD method creates a real matrix equation (S • kSV D = 0) equivalent to the linearized Gauss's law for magnetism (B • kSV D = 0).Notice that the minimum singular value of the real matrix S (6 × 3) is the best estimate of wavevector directions but cannot determine the wavenumbers.Since relative satellite separations are much shorter than the half-wavelength of MHD scales, the properties of fluctuations simultaneously measured by four Cluster spacecraft are similar.Thus, the average wavevector direction and background magnetic field are given by k SV D = 1 where k is the wavevector.Thus, Alfvénic proton velocity and magnetic field fluctuations are in the same direction kSV D × b0 /| kSV D × b0 | (see Schematic in Supplementary Fig. 4).
Fourth, Alfvénic magnetic power at each time t and f sc is calculated by . This is because magnetic field data are available on four Cluster spacecraft, whereas proton plasma data are only available on Cluster -1 during the analyzed period.Fifth, noticing that SVD does not give the magnitude of wavevectors, we calculate wavevectors (k A (t, f sc )) using the multispacecraft timing analysis based on phase differences between the Alfvénic magnetic field from four spacecraft Pincon & Glassmeier (2008).Magnetic field data are interpolated to a uniform time resolution of 8samples/s for sufficient time resolutions.We consider that the wave front is moving in the direction n with velocity V w .The wavevectors k A = 2πf sc m, where the vector m = n/V w , and the subscript A represent the Alfvénic component.
where Cluster -1 has arbitrarily been taken as the reference.The left side of Eq.( 4) is the relative spacecraft separations.The right side of Eq.(4) represents the weighted average time delays, estimated by the ratio of six phase differences ) to the angular frequencies (ω sc = 2πf sc ), where ϕ ij is from all spacecraft pairs (ij = 12, 13, 14, 23, 24, 34)).S and R are the imaginary and real parts of cross-correlation coefficients, respectively.Four Cluster spacecraft provide six cross-correlation coefficients Grinsted et al. (2004), i.e., W 12 ⟩, where ⟨...⟩ denotes a time average over 256s for the reliability of phase differences.It is worth noting that timing analysis determines the actual wavevectors of the Alfvénic magnetic field.In contrast, the SVD method determines the best estimate of the wavevector sum in three magnetic field components Santolík et al. (2003).Thus, k A is not completely aligned with kSV D .Besides, we restrict our analysis to fluctuations with small angle η between kSV D and k A , to ensure the reliability of the extraction process (the third step).With relaxed η constraints, more sampling points are involved; thus, the uncertainty from limited measurements decreases.On the other hand, with relaxed η constraints, k A deviates more from k SV D , which may increase the uncertainty.This letter presents results from five data sets under η < 10 • , η < 15 • , η < 20 • , η < 25 • , and η < 30 • to investigate the effects of uncertainties introduced by the combination of the SVD method and timing analysis.Sixth, we construct a set of 400 × 400 × 400 bins to obtain wavenumber-frequency distributions of magnetic power (P B A (k ⊥ , k ∥ , f sc )) and proton velocity power (P V A (k ⊥ , k ∥ , f sc )), where the parallel wavenumber is k ∥ = k A • b0 , and the perpendicular wavenumber is Each bin subtends approximately the same k ⊥ , k ∥ , and f sc .To cover all MHD wavenumbers and ensure measurement reliability, we restrict our analysis to fluctuations with 1/(100d sc ) < k < min(0.1/max(di , r ci ), π/d sc ) and 2/t * < f rest < f ci /2, and fluctuations beyond these wavenumber and frequency ranges are set to zero.Here, d sc is relative satellite separations, min( * ) and max( * ) are the minimum and maximum, d i is the proton inertial length, r ci is the proton gyro-radius, t * is the duration studied, f rest = f sc − k A • V p /(2π) is the frequency in the plasma flow frame, and V p is the proton bulk velocity with the spacecraft velocity being negligible.This study utilizes the representation of absolute frequencies: ) over effective time points in all time windows at each f sc and each k, where ϵ = V, 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 is normalized by µ 0 m p N 0 , where µ 0 is the vacuum permeability, m p is the proton mass, and N 0 is the mean proton density.magnetic energy spectral density This study defines the energy spectral density of magnetic field as , where the Alfvénic magnetic energy density is calculated by

Nonlinearity parameter
The nonlinearity parameter is estimated by , where the Alfvénic magnetic energy density is calculated by , and B 0 in Alfvén speed units is around 106km/s.

Frequency-wavenumber distribution of magnetic energy
The frequency-wavenumber distributions of magnetic energy is approximately estimated by Table 1.Transition wavenumbers are determined by magnetic field measurements.

Weak MHD turbulence
Strong MHD turbulence

A. DEVIATIONS BETWEEN THE MEAN MAGNETIC FIELD AND LOCAL FIELD
The anisotropy of Alfvénic fluctuations depends on the local background magnetic field.Although it would be better to use a scale-dependent mean magnetic field ideally, the mode decomposition method is based on a perturbative treatment of fluctuations in the presence of a uniform magnetic field (B 0 ).This method requires B 0 independent of the transformation between real and wavevector space.Nevertheless, Supplementary Fig. 1 shows the spacecraftframe frequency-time spectrum of the cosine of angle (|cos⟨B 0 , B local ⟩|) between B 0 and B local .The local mean field is calculated as , where τ is the timescale.The mean magnetic field (B 0 ) within a five-hour moving time window is closely aligned with B local , suggesting that B 0 approximating the local mean field is acceptable.

B. TWO-DIMENSIONAL ENERGY WAVENUMBER DISTRIBUTIONS AT DIFFERENT TIME WINDOW LENGTHS
To further address this limitation of the mode decomposition method, this study explores the variation of twodimensional (2D) wavenumber distributions of Alfvénic magnetic energy D B A (k ⊥ , k ∥ ) by adjusting the length of time windows, where (B1) We show magnetic energy spectra with different time window lengths in Supplementary Fig. 2. To simplify, the modeled theoretical energy spectra (I A (k ⊥ , k ∥ )) are estimated with the same parameters (M A,turb ≈ 0.33, L 0 ≈ 4.6 × 10 4 km).(i) The longer time window length provides more low-frequency (large-scale) measurements.(ii) Energy spectra with shorter time window lengths are more consistent with theoretical contours in the strong turbulence regime (at larger k ⊥ ).It is likely because the mean magnetic field in shorter time windows is closer to the local mean field of fluctuations with larger wavenumbers.(iii) The main changes in energy distributions are little affected by time window length: k ∥ distributions of magnetic energy start to broaden around k ⊥ > 2 × 10 −4 km −1 for all panels.
We show the results with a five-hour length in the main text for two reasons: (i) The length cannot be too long.The shorter the window length, the closer to the local background magnetic field.(ii) The length cannot be too short in order to ensure the measurements of low-frequency (large-scale) signals since the Alfvénic weak-to-strong transition is present on relatively large scales.The five-hour length selection provides the low-frequency (large-scale) measurements while ensuring B 0 is approaching the local background magnetic field.

C. DETAILS OF EXAMINATION OF THE TURBULENCE STATE
To examine the turbulent state, we calculate the normalized correlation function R(τ )/R(0), where the correlation function is defined as R(τ ) = ⟨δB(t)δB(t + τ )⟩, τ is the timescale, and angular brackets are a time average over the time window length (5 hours).Supplementary Fig. 3 shows R(τ )/R(0) for magnetic field δB ⊥1 and δB ⊥2 components in field-aligned coordinates.Fluctuations δB ⊥1 are in ( b0 × XGSE ) × b0 directions, and δB ⊥2 are in b0 × XGSE directions, where XGSE is the unit vector towards the Sun from the Earth.c, Proton and electron density.d, Spectral slopes (α) of magnetic field and proton velocity fluctuations between 0.001Hz and 0.1fci.The two horizontal lines represent α = −5/3 and −3/2.e, The proton plasma βp.f, The turbulent Alfvén Mach number (M A,turb = δVp/VA) and half of the relative amplitudes of the magnetic field (δB/(2B0)), where δVp and δB are rms proton velocity and magnetic field fluctuations, respectively.The fluctuations analyzed in detail are during 23:00-10:00 UT on 2-3 December, marked between the two vertical dashed lines.0)dτ .In Supplementary Fig. 3, T c ≈ [1300, 2300]s is much less than the time window length (5 hours), suggesting that fluctuations are approximately stationary.Moreover, R(τ )/R(0) profiles in all time windows are similar, suggesting that the starting time of the moving time window has a slight influence on R(τ )/R(0), and thus fluctuations are homogeneous.Above all, it is reasonable to describe structures of turbulent fluctuations using three-dimensional energy distributions.on Alfvénic magnetic field are not completely alighted with kSV D .Thus, we set the angle η between k A and kSV D as a threshold and only analyze the fluctuations inside the cone.

E. ONE-DIMENSIONAL AND TWO-DIMENSIONAL WAVENUMBER DISTRIBUTIONS OF ALFV ÉNIC ENERGY
One-dimensional (1D) wavenumber distributions of Alfvénic magnetic energy are calculated by In Supplementary Fig. 5, 1D wavenumber distributions of Alfvénic magnetic energy from data sets under different η limits nearly overlap both for D B A (k ⊥ ) and D B A (k ∥ ), where η is the angle between k SV D and k A (Supplementary Fig. 4).Due to the limited data samples, 1D wavenumber distributions from data sets with η < 10 • and η < 15 • show significant deviations from others in Supplementary Fig. 5, and more vacant bins exhibit in 2D wavenumber distributions under smaller η in Supplementary Fig. 6.More data samples are involved with the relaxation of η limits.On the whole, Alfvénic magnetic energy using data sets under different η limits shows similar distributions in Supplementary Figs. 5 and 6.

F. ENERGY SPECTRA AND NONLINEAR PARAMETERS WITH VELOCITY MEASUREMENTS
We observe a similar Alfvénic weak-to-strong transition with the measurements of proton velocity fluctuations.The energy spectral density of Alfvénic velocity is defined as 2k ⊥ , where the Alfvénic velocity energy density is calculated by ∞ 0 P V A (k ⊥ , k ∥ , f sc )df sc .Supplementary Fig. 7a shows the sharp change in spectral slopes of E V A (k ⊥ ) from wave-like (−2) to Kolmogorov-like (−5/3).In Supplementary Fig.
max is normalized by the maximum magnetic energy in all (k ⊥ , k ∥ ) bins, displayed by the spectral image and contours in Fig.2.Compared to the isotropic dotted curves, DB A (k ⊥ , k ∥ ) is prominently distributed along the k ⊥ direction, suggesting a faster perpendicular cascade.This anisotropic behavior is more pronounced at higher wavenumbers, consistent with previous simulations and observationsCho & Vishniac (2000);He et al. (2011);Makwana & Yan (2020);Zhao et al. (2022).Moreover, DB A (k ⊥ , k ∥ ) is compared with the modeled 2D theoretical energy spectra based on strong turbulenceYan & Lazarian (2008);Goldreich & Sridhar (1995)

Fig. 1 .
Fig.1.An overview of fluctuations measured by Cluster -1 in Earth's magnetosheath on 2-3 December 2003.The 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 density.d, Spectral slopes (α) of magnetic field and proton velocity fluctuations between 0.001Hz and 0.1fci.The two horizontal lines represent α = −5/3 and −3/2.e, The proton plasma βp.f, The turbulent Alfvén Mach number (M A,turb = δVp/VA) and half of the relative amplitudes of the magnetic field (δB/(2B0)), where δVp and δB are rms proton velocity and magnetic field fluctuations, respectively.The fluctuations analyzed in detail are during 23:00-10:00 UT on 2-3 December, marked between the two vertical dashed lines.