Reconstructing plasma-frame turbulence spectra from Parker Solar Probe time series beyond the validated Taylor/sweeping regime: near-parallel sampling, large sweeping parameter, outer-scale validity, and single-point anisotropy diagnostics

Single-spacecraft measurements of solar-wind turbulence are interpreted as spatial spectra through Taylor's hypothesis (TH), which maps measured frequencies to wavenumbers via the advective Doppler shift and fails when the spacecraft speed relative to the plasma does not dominate fluctuation and Alfvénic speeds. Bourouaine & Perez (2019) proposed a sweeping-based spacetime-correlation model (BP19) that reconstructs the plasma-frame field-perpendicular spectrum $E(k_\perp)$ from spacecraft frequency spectra under oblique sampling, with a single control parameter $\epsilon=\delta u_0/(\sqrt{2}V_\perp)$; Perez, Bourouaine, Chen & Raouafi (2021) validated its use on four intervals from Parker Solar Probe's first four perihelia, but only for sampling angles above 30 degrees, $\epsilon\le0.2$, and inertial-range scales below roughly one quarter of the outer scale. The complementary regime — which contains most of what PSP actually measures — remains open and is the object of this task.

Background

A spacecraft immersed in the solar wind measures a time series of velocity and magnetic-field fluctuations, and nearly all knowledge of solar-wind turbulence spectra comes from converting such series into spatial information via Taylor's hypothesis (TH): if the plasma flow — or, for a fast spacecraft, the spacecraft velocity relative to the plasma frame — sweeps turbulent structures past the detector much faster than they evolve, the measured angular frequency $\omega$ maps to the wavenumber component along the sampling direction through $\omega=\mathbf{k}\cdot\mathbf{V}$, so a frequency power spectrum directly yields a one-dimensional spatial spectrum. The hypothesis requires the advective speed $V$ to dominate both the rms speed $\delta u_0$ of the energy-containing (outer-scale) eddies and the relevant wave phase speeds. The Parker Solar Probe (PSP) mission, which approaches the Sun to within tens of solar radii, breaks these conditions: near perihelion the spacecraft speed in the plasma frame is only a few times the Alfvén speed $V_A$, the sampling angle $\theta_{VB}$ between the spacecraft velocity in the plasma frame and the local magnetic field varies strongly in time and is frequently small, and outer-scale velocity fluctuations are no longer negligible. Solar-wind turbulence is also anisotropic: in the critically balanced picture of strong magnetohydrodynamic (MHD) turbulence, Alfvénic propagation along the field balances the nonlinear cascade time, so the parallel-to-perpendicular wavenumber ratio $k_\parallel/k_\perp$ is small in the inertial range (the range between the energy-containing scale and the dissipation scale) and the quantity of theoretical interest is the field-perpendicular reduced spectrum $E(k_\perp)$, the energy per unit $k_\perp$ after integrating over $k_\parallel$.

Bourouaine & Perez (2019) — "BP19" — proposed a way to interpret PSP turbulent signals beyond TH: extending the classical hydrodynamic random-sweeping picture to MHD, they modeled the Eulerian spacetime correlation of small-scale Alfvénic fluctuations as decorrelated by random advection through the outer-scale velocity, obtaining a kernel that relates the measured frequency spectrum to $E(k_\perp)$,

$P_{\rm sc}(\omega)=\frac{\Lambda_{\alpha,\epsilon}}{V_\perp}\,E\!\left(\frac{\omega}{V_\perp}\right),$

for sufficiently oblique sampling, where $\epsilon=\delta u_0/(\sqrt{2}V_\perp)$ compares the outer-scale rms velocity with the perpendicular spacecraft speed in the plasma frame and $\Lambda_{\alpha,\epsilon}$ is a computable broadening factor depending on the spectral index $\alpha$. TH is recovered as $\epsilon\to0$ and is expected to remain a good approximation for $\epsilon\lesssim0.5$ (a criterion established by Bourouaine & Perez 2020) irrespective of the Alfvén Mach number, provided the turbulence is strongly anisotropic and the sampling is sufficiently oblique ($\theta_{VB}$ much larger than a critical angle $\theta_c$ set by the anisotropy, $\tan\theta_c\sim k_\parallel/k_\perp$). The underlying spacetime-correlation model was validated against high-resolution simulations of strong reduced MHD (RMHD, the anisotropic limit of MHD for Alfvénic fluctuations) for inertial-range scales below roughly one quarter of the outer scale (Perez & Bourouaine 2020), and the reconstruction was demonstrated on Helios data at 0.6 au (Bourouaine & Perez 2020).

Perez, Bourouaine, Chen & Raouafi (2021) applied this methodology to four 24-hour intervals from PSP's first four close encounters (heliocentric distances 0.13–0.19 au). For sampling angles $\theta_{VB}\ge30^\circ$ they found $\epsilon\le0.2$, so TH and the $\epsilon\to0$ limit of BP19 hold there, spectral indices are faithfully recovered, and the reconstructed $E(k_\perp)$ is independent of sampling angle — the method's internal consistency check. However, the validated regime excludes most of PSP's data: the distribution of $\theta_{VB}$ peaks at small angles (a substantial portion of the signal lies at 10–20 degrees, nearly parallel to the field), the analysis neglects the parallel terms $k_\parallel V_\parallel$ in the Doppler relation, the critical angle $\theta_c$ is estimated only from a critically balanced closure ($\tan\theta_c\approx\delta u_0/V_A$, an acknowledged overestimate), the model's validity near the spectral break separating the $1/f$ and inertial ranges — where fluctuations are expected to be more isotropic — is unestablished, closer perihelia are expected to push $\epsilon$ toward and beyond the $\epsilon\gtrsim0.5$ threshold at which TH and its BP19 limit fail, and the sensitivity of reconstructed amplitudes and indices to plasma-moment uncertainties (between PSP's Solar Probe Cup, SPC, and SPAN-ion plasma instruments) is unquantified.

Problem Statement

Complete and delimit the program, initiated by Bourouaine & Perez (2019) and applied by Perez et al. (2021), of converting single-point Parker Solar Probe frequency spectra into plasma-frame field-perpendicular turbulence spectra $E(k_\perp)$: determine for which sampling angles $\theta_{VB}$, sweeping parameters $\epsilon=\delta u_0/(\sqrt{2}V_\perp)$, and scale ranges (relative to the outer scale and the $1/f$-to-inertial spectral break) the reconstruction is valid, and provide a reconstruction method covering PSP's actual measurement conditions near the Sun. The requested results are: (1) a generalized frequency–wavenumber reduction that retains the field-parallel contributions — the full $k_\parallel$ and $V_\parallel$ dependence of the Doppler relation and decorrelation kernel — so that near-parallel sampling ($\theta_{VB}\lesssim20^\circ$, below the critical angle $\theta_c$) can be interpreted, reducing to the published oblique, $\epsilon\to0$ kernel in the appropriate limits; (2) a theoretical and observational characterization of the large-sweeping regime $\epsilon\gtrsim0.5$ expected at PSP's closest perihelia, where both plain Taylor's hypothesis and its BP19 limit fail — in particular whether the sweeping kernel or any successor within the sweeping spacetime-correlation family (including the generalized kernel of result (1)) still reconstructs $E(k_\perp)$ and its spectral index there; (3) a quantitative validity boundary in scale — how close to the outer scale and the spectral break the sweeping phenomenology can be trusted as fluctuations become more isotropic; (4) empirical single-point diagnostics of the turbulence anisotropy and of the critical sampling angle $\theta_c$ that do not rely on the critically balanced estimate $\tan\theta_c\approx\delta u_0/V_A$, which is known to overestimate $\theta_c$; and (5) a quantified propagation of measurement uncertainties — plasma-moment discrepancies between the SPC and SPAN-ion instruments — into the reconstructed spectral amplitudes and indices. All five results are required and each is judged independently: establishing either outcome for a given result resolves that result, but a partial answer covering only a subset of the five does not complete the task. In particular, result (2) is a yes/no question, resolved by either a reconstruction that remains valid at $\epsilon\gtrsim0.5$ or a quantified demonstration that no kernel in the sweeping spacetime-correlation family reconstructs $E(k_\perp)$ there; for the latter outcome, the demand for a reconstruction method covering PSP's actual measurement conditions is met by the reconstruction method together with its explicitly delimited range of validity.

References

1. Jean C. Perez, Sofiane Bourouaine, Christopher H. K. Chen, Nour E. Raouafi, "Applicability of Taylor's hypothesis during Parker Solar Probe perihelia", Astronomy & Astrophysics 650, A22 (2021). DOI 10.1051/0004-6361/202039879, https://doi.org/10.1051/0004-6361/202039879; arXiv:2103.12022, https://arxiv.org/abs/2103.12022
2. Sofiane Bourouaine, Jean C. Perez, "On the Interpretation of Parker Solar Probe Turbulent Signals", The Astrophysical Journal Letters 879, L16 (2019). DOI 10.3847/2041-8213/ab288a, https://doi.org/10.3847/2041-8213/ab288a; arXiv:1906.05644, https://arxiv.org/abs/1906.05644 (BP19 sweeping model)
3. Sofiane Bourouaine, Jean C. Perez, "On the Limitations of Taylor's Hypothesis in Parker Solar Probe's Measurements near the Alfvén Critical Point", The Astrophysical Journal Letters 858, L20 (2018). DOI 10.3847/2041-8213/aabccf, https://doi.org/10.3847/2041-8213/aabccf
4. Jean C. Perez, Sofiane Bourouaine, "Eulerian space-time correlation of strong magnetohydrodynamic turbulence", Physical Review Research 2, 023357 (2020). DOI 10.1103/PhysRevResearch.2.023357, https://doi.org/10.1103/PhysRevResearch.2.023357; arXiv:2004.11458, https://arxiv.org/abs/2004.11458 (RMHD validation of the sweeping model)
5. Sofiane Bourouaine, Jean C. Perez, "Interpreting Solar Wind Turbulent Spectra beyond Taylor's Hypothesis", The Astrophysical Journal Letters 893, L32 (2020). DOI 10.3847/2041-8213/ab7fb1, https://doi.org/10.3847/2041-8213/ab7fb1; arXiv:2003.08924, https://arxiv.org/abs/2003.08924 (Helios 0.6 au application)
6. Sofiane Bourouaine, Jean C. Perez, Kristopher G. Klein, Christopher H. K. Chen, Mihailo M. Martinović, Stuart D. Bale, Justin C. Kasper, Nour E. Raouafi, "Turbulence Characteristics of Switchback and Nonswitchback Intervals Observed by Parker Solar Probe", The Astrophysical Journal Letters 904, L30 (2020). DOI 10.3847/2041-8213/abbd4a, https://doi.org/10.3847/2041-8213/abbd4a; arXiv:2010.00936, https://arxiv.org/abs/2010.00936
7. G. P. Zank, L.-L. Zhao, L. Adhikari, D. Telloni, J. C. Kasper, M. Stevens, A. Rahmati, S. D. Bale, "Turbulence in the Sub-Alfvénic Solar Wind", The Astrophysical Journal Letters 926, L16 (2022). DOI 10.3847/2041-8213/ac51da, https://doi.org/10.3847/2041-8213/ac51da; arXiv:2202.02563, https://arxiv.org/abs/2202.02563
8. L.-L. Zhao, G. P. Zank, H. Li, "Effects of Nonzero-frequency Fluctuations on Turbulence Spectral Observations", The Astrophysical Journal Letters 962, L14 (2024). DOI 10.3847/2041-8213/ad20e1, https://doi.org/10.3847/2041-8213/ad20e1
9. Mahendra K. Verma, "Taylor's frozen-in hypothesis for magnetohydrodynamic turbulence and solar wind", Physics of Plasmas 29, 082902 (2022). DOI 10.1063/5.0096743, https://doi.org/10.1063/5.0096743; arXiv:2204.12790, https://arxiv.org/abs/2204.12790
10. Norbert Magyar, Jaye Verniero, Adam Szabo, Jiyuan Zhang, Tom Van Doorsselaere, "Solar wind data analysis aided by synthetic modeling: a better understanding of plasma-frame variations from temporal data", Astronomy & Astrophysics 688, A74 (2024). DOI 10.1051/0004-6361/202348071, https://doi.org/10.1051/0004-6361/202348071; arXiv:2405.12547, https://arxiv.org/abs/2405.12547
11. Francesco Pecora, William H. Matthaeus, Antonella Greco, Pablo Dmitruk, Yan Yang, Vincenzo Carbone, Sergio Servidio, "Turbulence in the terrestrial magnetosheath: space-time correlation using the Magnetospheric Multiscale mission", Proceedings of the National Academy of Sciences 122, e2519635122 (2025). DOI 10.1073/pnas.2519635122, https://doi.org/10.1073/pnas.2519635122; arXiv:2511.15579, https://arxiv.org/abs/2511.15579
12. Zesen Huang, Marco Velli, B. D. G. Chandran, Chen Shi, Yuliang Ding, Lorenzo Matteini, Kyung-Eun Choi, "Two Types of 1/f Range in Solar Wind Turbulence", The Astrophysical Journal Letters 990, L34 (2025). DOI 10.3847/2041-8213/adfa13, https://doi.org/10.3847/2041-8213/adfa13; arXiv:2506.17523, https://arxiv.org/abs/2506.17523
13. Jean C. Perez, Sofiane Bourouaine, Mason Dorseth, "The Integral and Correlation Scales of Solar Wind Turbulence", The Astrophysical Journal Letters 1005, L4 (2026). DOI 10.3847/2041-8213/ae77f4, https://doi.org/10.3847/2041-8213/ae77f4; arXiv:2606.09750, https://arxiv.org/abs/2606.09750
14. Wenshuai Cheng, Ming Xiong, Yiming Jiao, Hao Ran, Liping Yang, Huidong Hu, Rui Wang, "Inertial-range Turbulence Anisotropy of the Young Solar Wind from Different Source Regions", The Astrophysical Journal Letters 988, L15 (2025). DOI 10.3847/2041-8213/adeb8a, https://doi.org/10.3847/2041-8213/adeb8a; arXiv:2507.04288, https://arxiv.org/abs/2507.04288
15. Y. Nariyuki, "Anomalous diffusion model in Fourier space describing time correlation of shear Alfvénic turbulence", Physics of Plasmas 31, 122303 (2024). DOI 10.1063/5.0220185, https://doi.org/10.1063/5.0220185
16. L.-L. Zhao, G. P. Zank, M. Opher, B. Zieger, H. Li, V. Florinski, L. Adhikari, X. Zhu, M. Nakanotani, "Turbulence, Waves, and Taylor's Hypothesis for Heliosheath Observations", The Astrophysical Journal 973, 26 (2024). DOI 10.3847/1538-4357/ad64c8, https://doi.org/10.3847/1538-4357/ad64c8; arXiv:2407.21673, https://arxiv.org/abs/2407.21673