Pith. sign in

REVIEW 4 major objections 5 minor 54 references

An Intermittent Model for the $1/f$ Spectrum in the Pristine Solar Wind

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The 1/f range of the solar wind is not passive noise: it is a non-conservative, intermittent energy cascade whose rate falls as $1/\ell$.

desk verdict A genuinely new intermittent model for the 1/f range, but the observational support is weaker than the text suggests because the 1/τ fit includes the expansion term that the model does not describe. read the letter →

arxiv 2506.04366 v1 pith:KFP3ISGM submitted 2025-06-04 astro-ph.SR physics.plasm-phphysics.space-ph

Maia Brodiano , Fouad Sahraoui , Davide Manzini , Lina Z. Hadid , Facundo Pugliese , Pablo Dmitruk , Nahuel Andrés This is my paper
classification astro-ph.SRphysics.plasm-phphysics.space-ph
keywords solarwindturbulence1/fspectrumenergycascaderatethird-orderlawintermittencyParkerProbemagnetohydrodynamicsstructurefunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper uses Parker Solar Probe measurements to ask whether the mysterious $1/f$ part of the solar wind magnetic spectrum—where power falls with frequency instead of the steeper inertial-range falloff—is merely a collection of frozen fluctuations or an active turbulent cascade. The authors compute the MHD energy cascade rate from the exact third-order law, add expansion corrections, and find that in the $1/f$ range the cascade rate scales as $1/\tau$ rather than being constant. They then show, by dimensional analysis, that a scale-dependent cascade rate $\varepsilon_\ell \propto 1/\ell$ produces exactly a $k^{-1}$ magnetic spectrum. The conclusion is that the $1/f$ range is a non-conservative, intermittent energy-transfer region, with parallel field fluctuations carrying most of the transfer, and that this picture can explain why low-frequency spectral slopes vary in other space plasmas.

What carries the argument

The load-bearing object is the scale-dependent energy cascade rate $\varepsilon_\ell$, defined through the third-order mixed structure function in the exact MHD law. The argument works by extending the refined self-similarity hypothesis to the energy-containing scales: writing $\delta b_\ell \sim (\varepsilon_\ell \ell)^{1/3}$, assuming $\varepsilon_\ell \sim \ell^\alpha$, and requiring the spectrum to be $k^{-1}$ forces $\alpha = -1$, hence $\varepsilon_\ell \sim 1/\ell$. The other machinery is observational: magnetic-field and plasma data are interpolated to a 30-second cadence, Taylor's hypothesis $\ell = U_0 \tau$ converts time lags to spatial lags, and expansion terms from the solar wind's radial expansion are added to the cascade-rate estimate.

What would settle it

Measure the third-order structure function in the $1/f$ range using multi-spacecraft data that sample spatial separations directly, without Taylor's hypothesis, and check whether the relation $-(4/3)\varepsilon\ell = \rho_0\langle\dots\rangle$ holds as a linear function of $\ell$; if the third-order moment is not linear in $\ell$ at these scales due to non-local or expansion effects, the inferred $\varepsilon_\ell \propto 1/\ell$ cascade rate collapses. A controlled numerical MHD simulation with a forced $k^{-1}$ range could also test whether a $1/\ell$ cascade rate is actually produced under conditions where locality is known to hold.

Watch

Extended reading notes

Core claim

The central claim is that the $1/f$ range of the pristine solar wind is populated by fully developed, intermittent turbulence whose energy cascade is non-conservative. Using the exact third-order law of incompressible MHD turbulence, supplemented by solar-wind expansion terms, the authors measure the energy transfer rate in Parker Solar Probe data and observe $|\varepsilon(\tau)| \propto 1/\tau$ throughout the $1/f$ range before it flattens near the correlation scale. Assuming this exact law extends to those scales and adopting a scale-dependent dissipation rate in the refined self-similarity framework, they derive $\varepsilon_\ell \propto 1/\ell$, which yields a magnetic spectrum $b_k^2 \propto k^{-1}$ exactly. The probability density functions of magnetic increments show that parallel fluctuations are strongly non-Gaussian while perpendicular ones are quasi-Gaussian, identifying the parallel component as the driver of the $1/f$ transfers. The paper also finds slight departures from the self-similarity prediction for higher-order structure functions, pointing to a more complex intermittent model still to be built.

Load-bearing premise

Everything depends on assuming the exact third-order law of MHD turbulence, which is derived for scale-separated, local inertial-range dynamics, still holds at the largest scales of the $1/f$ range—an assumption the authors explicitly flag as uncertain.

Editorial extensions

If this is right

  • At scales in the $1/f$ range, energy transfer is not constant: the measured cascade rate falls as $1/\tau$, so the large-scale fluctuations are not frozen noise but actively participate in a non-conservative cascade.
  • Combining the inferred $\varepsilon_\ell \propto 1/\ell$ with Kolmogorov-style dimensional analysis reproduces the $k^{-1}$ magnetic spectrum, offering a dynamical explanation of the classic $1/f$ slope.
  • The intermittency of the $1/f$ range is carried mainly by the parallel magnetic fluctuations, whose probability density functions are strongly non-Gaussian, while the perpendicular component is quasi-Gaussian.
  • The slight failure of the predicted constants $C_m = C_3^{m/3}$ means the refined self-similarity is only approximate in the $1/f$ range, so a more complete intermittent model based on moments of the dissipation rate is still needed.
  • Because the cascade-rate slope is set by the ratio of integral to correlation scale, the $1/f$ spectral exponent need not be universal across environments such as planetary magnetosheaths.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to compute the moments of the local dissipation rate $\langle \varepsilon_i(\ell)^m \rangle$ in the $1/f$ range; the paper only measures the mean, and its own intermittency model predicts specific scaling for these moments if the cascade is lognormal or log-Poisson.
  • If the third-order law is invalid at the largest scales because locality fails, the dimensional link between $\varepsilon_\ell \propto 1/\ell$ and $k^{-1}$ remains mathematically correct, but the physical interpretation shifts from an active cascade to an artifact of non-local expansion effects; distinguishing these requires scale-by-scale flux budgets.
  • The dominance of the sub-dominant parallel component in driving $1/f$ transfers hints that compressible fluctuations are slaved to incompressible dynamics; extending this to a compressible intermittent model could predict density-fluctuation signatures at $1/f$ scales testable with proton data.
  • The non-universality speculation suggests a possible observational survey: measure low-frequency spectral indices across magnetosheaths with different $L_0/L_c$ and check for a systematic correlation with that ratio.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript analyzes Parker Solar Probe observations of the pristine solar wind to characterize the low-frequency 1/f range of the magnetic energy spectrum. Using the incompressible MHD third-order law with solar-wind expansion corrections, the authors estimate the energy cascade rate as a function of time lag and report a 1/tau scaling in the 1/f range, in contrast to the constant cascade rate in the inertial range. They propose an intermittent model in which a scale-dependent cascade rate epsilon_l ~ l^(-1) produces a k^(-1) magnetic spectrum, and they support this with structure-function and PDF analyses. The central claim is that the 1/f range is an active, non-conservative, intermittent cascade region rather than a passive or linear superposition range.

Significance. If the central claim holds, the paper provides a new physical picture of the 1/f range in solar wind turbulence, connecting spectral slope to a scale-dependent energy transfer rate and offering a testable, dimensional prediction. The use of the exact third-order law to obtain an independent estimate of the cascade rate is a strength, as is the explicit statement of the model's assumptions and caveats. The paper also makes a falsifiable prediction for higher-order structure functions, which is checked against data. However, the significance is currently limited by the small number of displayed intervals and by the incomplete separation of the expansion term from the nonlinear cascade rate in the central figure.

major comments (4)
  1. [Observational Results and an Intermittent Model, Fig. 1(a,b)] The 1/tau fits in Fig. 1 are applied to the full cascade rate, which is defined in the text as the sum of the nonlinear and expansion terms, while the expansion term F_exp is shown in gray but is not isolated in the fits. Because F_exp is second order in the fields and the authors state that it can become comparable to the turbulent cascade rate at large scales, the observed 1/tau scaling could in principle be dominated by F_exp rather than by a scale-dependent nonlinear cascade. The central conclusion epsilon_l ~ 1/l requires that the nonlinear third-order term alone exhibits 1/tau scaling in the 1/f range; please plot and fit the nonlinear contribution separately and report whether F_exp itself behaves as 1/tau.
  2. [Observational Results and an Intermittent Model, Eq. (6)] The model's prediction epsilon_l ~ l^(-1) is obtained by inserting the observed k^(-1) spectral slope into Eq. (6). As written, this is a consistency relation between two observables rather than an independent prediction of the model. The independent evidence for 1/l scaling comes from the third-order-law cascade-rate measurements, which is a genuinely separate input. Please reframe the derivation as a consistency check, propagate the uncertainty in the fitted spectral slope into the expected alpha, and show explicitly that the measured cascade-rate slope is consistent with that value.
  3. [Figure 1 and PSP Data Selection] Only two of the eighteen analyzed intervals are shown, and the fitted power-law slopes are quoted as -1.19 and -1.09 without error bars or a stated fitting range. For a claim that the cascade rate scales as 1/tau across the 1/f range, the paper should present the distribution of fitted slopes over all 18 intervals, including uncertainties and a sensitivity test to the choice of fitting boundaries. This is needed to establish that the two displayed cases are representative and that the scaling is not an artifact of a particular range selection.
  4. [Conclusions] The paper acknowledges that the extension of the Politano-Pouquet third-order law to the 1/f range assumes scale separation and locality that may not hold on the largest scales. This assumption is load-bearing for the central claim. The manuscript would be strengthened by a direct discussion of how a violation of locality would affect the magnitude and scaling of the measured cascade rate, or by a quantitative consistency test, such as checking whether the cascade-rate scaling is stable when the largest lags are excluded.
minor comments (5)
  1. [Third-order law section, Eq. (3)] The sentence 'These terms quantifies the expansion driven source term' contains a subject-verb agreement error; it should read 'These terms quantify'.
  2. [Observational Results and an Intermittent Model, footnote after Eq. (6)] The footnote says the power-law assumption allows obtaining Eq. (6) by integrating Eq. (1), but Eq. (6) follows directly from the dimensional relation delta_b ~ (epsilon_l l)^(1/3) and the assumed power law; the reference to integrating Eq. (1) is unclear and should be corrected.
  3. [Figure 2] The PDF panels for the parallel and perpendicular components share the same axis labels, but the Gaussian reference curve is only identified in one panel; adding a legend to each panel or noting the Gaussian in the caption would improve readability.
  4. [PSP Data Selection, Table I] The table column labeled delta_u_0 is described as the 'rms value of the outer-scale (energy-containing range) fluid velocity' in the caption; please define how this quantity is computed from the two-day intervals and clarify its relation to the fluctuation amplitude used in the cascade-rate calculation.
  5. [Observational Results and Intermittency in the 1/f Range] The notation for the structure functions uses superscripts for components inconsistently (e.g., 'delta_B_parallel' in Fig. 2 versus 'delta_b_ell' in Eq. (2)); unifying the notation for longitudinal and parallel/perpendicular components would reduce confusion.

Circularity Check

1 steps flagged · score 3.0 of 10

Model prediction of 1/ell cascade rate is inferred from the observed 1/f spectrum, but the empirical cascade-rate measurement is independent; partial circularity.

  1. fitted input called prediction [Section 'Observational Results and an Intermittent Model', Equations (5)-(7)]
    "From relation (6), it is straightforward to infer that if the magnetic energy spectrum scales as b 2 k ∼k −1, then α=−1, which in turn yields a scale-dependent energy dissipation rate: εℓ ∼ℓ −1.(7) This prediction is consistent with the estimated dissipation rates in Fig. 1."

    The model's central prediction ε_ℓ ∼ 1/ℓ is not derived from an independent dynamical principle; it is obtained by inserting the observed 1/f spectral slope into the same Kolmogorov-type dimensional relation used to define the model. Under the paper's own assumptions, a b_k^2 ∼ k^{-1} spectrum forces α = −1 by construction, so the 'prediction' is a rearrangement of its input rather than an independent result. The paper later acknowledges the constraint: 'the prediction ε_ℓ ∼ 1/ℓ was obtained from S_3, which thus acts as a stringent boundary condition.' However, the 1/τ scaling measured from the third-order law in Fig. 1 is an independent observable and is not fitted to force agreement, so the circularity is partial and affects the explanatory/model claim more than the empirical finding.

full rationale

The derivation chain contains one genuine but partial circular step: the model's predicted cascade-rate scaling is read off from the observed 1/f spectral slope via Eq. (6), so the model cannot independently explain the origin of the 1/f spectrum. The paper itself concedes that the prediction was obtained from the third-order quantity, calling it a 'stringent boundary condition.' This prevents the model from being a fully independent derivation. At the same time, the observational core is not circular: the 1/τ cascade-rate scaling in Fig. 1 comes from PSP data through the third-order law with expansion terms, separately from the model, and no free parameter is fitted to make the two agree. The conclusions explicitly flag the main caveat about extending the third-order law into the 1/f range and the possible failure of scale locality. The concern about the expansion term F_exp contaminating the fitted total cascade rate is a physical/statistical check, not a reduction-by-construction, and is therefore not scored as circularity. No load-bearing self-citation or imported uniqueness theorem is present; self-citations are used only for a secondary compressible-cascade estimate and standard third-order-law references. Overall, the partial circularity lies in the model claim, not in the measurement, supporting a low-moderate score rather than a fully circular verdict.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests primarily on two unproven extensions: applying the inertial-range exact law to the 1/f range and converting time lags to spatial lags via Taylor's hypothesis at those scales. The model itself adds a power-law ansatz and refined self-similarity for the scale-dependent rate. No new physical entities are postulated; the only fitted constants are observational exponents and a normalization used for self-consistency tests.

free parameters (4)
  • 1/f range cascade-rate power-law exponents = -1.19 (0.16 AU), -1.09 (0.44 AU)
    Power-law fits to |ε(τ)| in Fig. 1; the text calls this 1/τ, but the fitted values differ from -1 and no uncertainties are reported.
  • 1/f fitting range boundaries = not tabulated
    Selected per event by visual identification of 1/f scaling in the PSD (Fig. 1c,d); the fitted exponents depend on these hand-chosen boundaries.
  • C_3 normalization in the 1/f range = not quoted numerically
    Used in Eq. (8) and Fig. 4 to test the prediction C_m = C_3^(m/3); fixed from the observed S_3(ℓ) plateau rather than predicted from theory.
  • Sliding-window width for mean-field projection = unspecified (chosen to satisfy stationarity within a tolerance)
    Defines the parallel and perpendicular components in the PDF analysis (Fig. 2); the tolerance and exact window size are not given.
assumptions (6)
  • ad hoc to paper Validity of the incompressible MHD third-order law (Politano-Pouquet, Eqs. 1-2) in the 1/f range
    The authors state this is the key assumption and note that the locality and scale-separation conditions used to derive it may not hold at the largest scales (Conclusions).
  • domain assumption Taylor hypothesis ℓ = U0 τ at 1/f scales
    Single-spacecraft time series are converted to spatial lags assuming frozen-in flow (Section 'Third-order law'); this may be inaccurate at the largest, slowest scales.
  • ad hoc to paper Refined self-similarity extends to the 1/f range, S_m(ℓ) = (C_3 ε_ℓ ℓ)^(m/3) (Eq. 8)
    Used to generalize the third-order law to all orders and to predict plateau values C_m = C_3^(m/3); the paper later reports departures from this prediction.
  • ad hoc to paper Power-law ansatz ε_ℓ ∼ ℓ^α
    Introduced in 'Observational Results and an Intermittent Model' (footnote 44) to derive the spectral relation; no physical derivation is given for why the cascade rate must be a pure power law.
  • standard math Statistical stationarity, homogeneity, and ergodicity
    Needed to replace ensemble averages by time averages in Eq. (1); the paper checks correlation-time convergence but this remains an assumption.
  • domain assumption Expansion-term model F_exp (Eq. 3) captures solar wind expansion effects
    The expansion corrections are taken from cited works [33-35] and added to the exact law; their magnitude is not independently validated in this dataset.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An Intermittent Model for the $1/f$ Spectrum in the Pristine Solar Wind." pith.science (2026). https://pith.science/paper/KFP3ISGM

@misc{pith2026250604366,
  author       = {Pith},
  title        = {Pith review of: An Intermittent Model for the $1/f$ Spectrum in the Pristine Solar Wind},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFP3ISGM}},
  note         = {Machine review of arXiv:2506.04366}
}
abstract

We present a statistical, observational study of the $1/f$ range of solar wind turbulence, where $f$ denotes frequency, using in situ data from the Parker Solar Probe (PSP). We compute the energy cascade rate using the third order law of incompressible magnetohydrodynamic (MHD) turbulence, incorporating expansion terms to account for solar wind dynamics. Our results reveal a $1/\tau$ dependence of the energy cascade rate, where $\tau$ is the temporal lag, within the $1/f$ range, in contrast to the constant cascade rate in the inertial range. To explain this behavior, we propose a new intermittent model predicting a $1/\ell$ scaling of the cascade rate, where $\ell$ represents the spatial lag. The analysis of the probability density function (PDF) of magnetic field increments confirms the intermittent nature of the parallel fluctuation component, whereas the perpendicular fluctuations are found to be quasi Gaussian. These findings provide new insights into energy transfer processes in the $1/f$ range of solar wind turbulence, with potential applications in planetary magnetosheaths.

Figures

Figures reproduced from arXiv: 2506.04366 by the authors.

Figure 1
Figure 1. FIG. 1. The incompressible energy cascade rate [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. PDFs of the parallel and perpendicular magnetic [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Estimation of the average constant [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

54 extracted references · 50 canonical work pages

  1. [1]

    Bruno and V

    R. Bruno and V. Carbone, Living Reviews in Solar Physics10, 2 (2013)

    work page 2013

  2. [2]

    Verscharen, K

    D. Verscharen, K. G. Klein, and B. Maruca, inAGU Fall Meeting Abstracts, Vol. 2018 (2018) pp. SH53A–07

    work page 2018

  3. [3]

    A. W. Case, J. C. Kasper, M. L. Stevens, K. E. Ko- rreck, K. Paulson, P. Daigneau, D. Caldwell, M. Free- man, T. Henry, B. Klingensmith,et al., The Astrophysi- cal Journal Supplement Series246, 43 (2020)

    work page 2020

  4. [4]

    W. H. Matthaeus and M. L. Goldstein, Phys. Rev. Lett. 57, 495 (1986)

    work page 1986

  5. [5]

    Velli, R

    M. Velli, R. Grappin, and A. Mangeney, Phys. Rev. Lett. 63, 1807 (1989)

    work page 1989

  6. [6]

    Dmitruk and W

    P. Dmitruk and W. H. Matthaeus, Phys. Rev. E76, 036305 (2007)

    work page 2007

  7. [7]

    Bemporad, W

    A. Bemporad, W. H. Matthaeus, and G. Poletto, The Astrophysical Journal677, L137 (2008)

    work page 2008

  8. [8]

    Verdini, R

    A. Verdini, R. Grappin, R. Pinto, and M. Velli, The As- trophysical Journal Letters750, L33 (2012)

    work page 2012

Show all 54 references
  1. [9]

    Matteini, D

    L. Matteini, D. Stansby, T. S. Horbury, and C. H. K. Chen, The Astrophysical Journal Letters869, L32 (2018)

    2018

  2. [10]

    B. D. G. Chandran, Journal of Plasma Physics84, 905840106 (2018)

    2018

  3. [11]

    Magyar and T

    N. Magyar and T. V. Doorsselaere, The Astrophysical Journal938, 98 (2022)

    2022

  4. [12]

    Huang, N

    Z. Huang, N. Sioulas, C. Shi, M. Velli, T. Bowen, N. Davis, B. D. G. Chandran, L. Matteini, N. Kang, X. Shi, J. Huang, S. D. Bale, J. C. Kasper, D. E. Lar- son, R. Livi, P. L. Whittlesey, A. Rahmati, K. Paul- son, M. Stevens, A. W. Case, T. D. de Wit, D. M. Malaspina, J. W. Bo...

    2023

  5. [13]

    C. H. K. Chen, S. D. Bale, J. W. Bonnell, D. Borovikov, T. A. Bowen, D. Burgess, A. W. Case, B. D. G. Chan- dran, T. D. de Wit, K. Goetz, P. R. Harvey, J. C. Kasper, K. G. Klein, K. E. Korreck, D. Larson, R. Livi, R. J. Mac- Dowall, D. M. Malaspina, A. Mallet, M. D. McManus, M...

    2020

  6. [14]

    J. C. Perez and B. D. G. Chandran, The Astrophysical Journal776, 124 (2013)

    2013

  7. [15]

    N. Fox, M. Velli, S. Bale, R. Decker, A. Driesman, R. Howard, J. C. Kasper, J. Kinnison, M. Kusterer, D. Lario,et al., Space Science Reviews204, 7 (2016)

    2016

  8. [16]

    Politano and A

    H. Politano and A. Pouquet, Physical Review E57, R21 6 (1998)

    1998

  9. [17]

    Politano and A

    H. Politano and A. Pouquet, Geophysical Research Let- ters25, 273 (1998)

    1998

  10. [18]

    Sorriso-Valvo, V

    L. Sorriso-Valvo, V. Carbone, A. Noullez, H. Politano, A. Pouquet, and P. Veltri, Physics of Plasmas9, 89 (2002)

    2002

  11. [19]

    P. D. Mininni and A. Pouquet, Phys. Rev. E80, 025401 (2009)

    2009

  12. [20]

    Boldyrev, J

    S. Boldyrev, J. Mason, and F. Cattaneo, Astrophys. J. Lett.699, L39 (2009)

    2009

  13. [21]

    M. Wan, S. Servidio, S. Oughton, and W. H. Matthaeus, Phys. Plasmas17, 052307 (2010)

    2010

  14. [22]

    Verdini and R

    A. Verdini and R. Grappin, The Astrophysical Journal 808, L34 (2015)

    2015

  15. [23]

    Sorriso-Valvo, R

    L. Sorriso-Valvo, R. Marino, V. Carbone, A. Noullez, F. Lepreti, P. Veltri, R. Bruno, B. Bavassano, and E. Pietropaolo, Phys. Rev. Lett.99, 115001 (2007)

    2007

  16. [24]

    Sahraoui, Phys

    F. Sahraoui, Phys. Rev. E78, 026402 (2008)

    2008

  17. [25]

    J. T. Coburn, M. A. Forman, C. W. Smith, B. J. Vasquez, and J. E. Stawarz, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineer- ing Sciences373, 20140150 (2015)

    2015

  18. [26]

    A. N. Simakov and L. Chac´ on, Phys. Rev. Lett.101, 105003 (2008)

    2008

  19. [27]

    Masters, N

    A. Masters, N. Achilleos, M. K. Dougherty, J. A. Slavin, G. B. Hospodarsky, C. S. Arridge, and A. J. Coates, Jour- nal of Geophysical Research (Space Physics)113, A10210 (2008)

    2008

  20. [28]

    W. H. Matthaeus, G. P. Zank, C. W. Smith, and S. Oughton, Phys. Rev. Lett.82, 3444 (1999)

    1999

  21. [29]

    B. T. MacBride, C. W. Smith, and M. A. Forman, As- trophys. J.679, 1644 (2008)

    2008

  22. [30]

    B. T. MacBride, M. A. Forman, and C. W. Smith, inSo- lar Wind 11/SOHO 16, Connecting Sun and Heliosphere, ESA Special Publication, Vol. 592, edited by B. Fleck, T. H. Zurbuchen, and H. Lacoste (2005) p. 613

    2005

  23. [31]

    Biskamp,Magnetohydrodynamic turbulence(Cam- bridge University Press, 2003)

    D. Biskamp,Magnetohydrodynamic turbulence(Cam- bridge University Press, 2003)

    2003

  24. [32]

    Andr´ es and F

    N. Andr´ es and F. Sahraoui, Physical Review E96, 053205 (2017)

    2017

  25. [33]

    Verdini, A., Hellinger, Petr, Landi, Simone, Grap- pin, Roland, Montagud-Camps, Victor, and Papini, Emanuele, A&A690, A265 (2024)

    2024

  26. [34]

    Gogoberidze, S

    G. Gogoberidze, S. Perri, and V. Carbone, Astrophys. J. 769, 111 (2013), arXiv:1304.4505 [astro-ph.SR]

    2013 arXiv

  27. [35]

    Hellinger, Petr, Tr´ avn ´ ıˇ cek, Pavel M.,ˇStver´ ak,ˇStˇ ep´ an, Matteini, Lorenzo, and Velli, Marco, Journal of Geo- physical Research: Space Physics118, 1351 (2013), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/jgra.50107

    2013 doi

  28. [36]

    S. Bale, K. Goetz, P. Harvey, P. Turin, J. Bonnell, T. D. De Wit, R. Ergun, R. MacDowall, M. Pulupa, M. Andr´ e, et al., Space science reviews204, 49 (2016)

    2016

  29. [37]

    J. C. Kasper, R. Abiad, G. Austin, M. Balat-Pichelin, S. D. Bale, J. W. Belcher, P. Berg, H. Bergner, M. Berthomier, J. Bookbinder,et al., Space Science Re- views204, 131 (2016)

    2016

  30. [38]

    Wrench, T

    D. Wrench, T. N. Parashar, S. Oughton, K. de Lange, and M. Frean, The Astrophysical Journal961, 182 (2024)

    2024

  31. [39]

    W. H. Matthaeus, S. Dasso, J. M. Weygand, L. J. Milano, C. W. Smith, and M. G. Kivelson, Phys. Rev. Lett.95, 231101 (2005)

    2005

  32. [40]

    M. E. Cuesta, T. N. Parashar, R. Chhiber, and W. H. Matthaeus, The Astrophysical Journal Supplement Se- ries259, 23 (2022)

    2022

  33. [41]

    J. E. Stawarz, J. P. Eastwood, T. D. Phan, I. L. Gingell, P. S. Pyakurel, M. A. Shay, S. L. Robertson, C. T. Russell, and O. Le Contel, Physics of Plasmas 29, 012302 (2022), https://pubs.aip.org/aip/pop/article- pdf/doi/10.1063/5.0071106/16623691/012302 1 online.pdf

    2022 doi

  34. [42]

    A. M. Oboukhov, Journal of Fluid Mechanics13, 77–81 (1962)

    1962

  35. [43]

    Frisch,Turbulence: The Legacy of A

    U. Frisch,Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, 1995)

    1995

  36. [44]

    This assumption allows for obtaining the scaling law (6) directly by integrating Eq. (1)

  37. [45]

    L. Z. Hadid, F. Sahraoui, K. H. Kiyani, A. Retin` o, R. Modolo, P. Canu, A. Masters, and M. K. Dougherty, The Astrophysical Journal Letters813, L29 (2015)

    2015

  38. [46]

    S. Y. Huang, L. Z. Hadid, F. Sahraoui, Z. G. Yuan, and X. H. Deng, The Astrophysical Journal Letters836, L10 (2017)

    2017

  39. [47]

    S. Y. Huang, F. Sahraoui, X. H. Deng, J. S. He, Z. G. Yuan, M. Zhou, Y. Pang, and H. S. Fu, The Astrophysical Journal Letters789, L28 (2014)

    2014

  40. [48]

    W. H. Matthaeus, M. Wan, S. Servidio, A. Greco, K. T. Osman, S. Oughton, and P. Dmitruk, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences373, 20140154 (2015), https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2014.0154

    2015

  41. [49]

    K. H. Kiyani, S. C. Chapman, F. Sahraoui, B. Hnat, O. Fauvarque, and Y. V. Khotyaintsev, The Astrophysi- cal Journal763, 10 (2012)

    2012

  42. [50]

    G. G. Howes, K. G. Klein, and J. M. TenBarge, The Astrophysical Journal789, 106 (2014)

    2014

  43. [51]

    A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W. Hammett, G. G. Howes, E. Quataert, and T. Tatsuno, The Astrophysical Journal Supplement Series182, 310 (2009)

    2009

  44. [52]

    Andr´ es, F

    N. Andr´ es, F. Sahraoui, S. Galtier, L. Z. Hadid, P. Dmitruk, and P. D. Mininni, Journal of Plasma Physics84, 905840404 (2018)

    2018

  45. [53]

    Kolmogorov, Akademiia Nauk SSSR Doklady30, 301 (1941)

    A. Kolmogorov, Akademiia Nauk SSSR Doklady30, 301 (1941)

    1941

  46. [54]

    berkeley.edu/data/psp/

    NASA Parker Solar Probe,https://research.ssl. berkeley.edu/data/psp/

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.