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# High-frequency heating of the solar wind triggered by low-frequency turbulence

## Abstract

The fast solar wind’s high speeds and non-thermal features require that considerable heating occurs well above the Sun’s surface. Two leading theories seem incompatible: low-frequency ‘Alfvénic’ turbulence, which transports energy outwards and is observed ubiquitously by spacecraft but seems insufficient to explain the observed dominance of ion over electron heating; and high-frequency ion-cyclotron waves, which explain the non-thermal heating of ions but lack an obvious source. Here we argue that the recently proposed ‘helicity barrier’ effect, which limits electron heating by inhibiting the turbulent cascade of energy to the smallest scales, can unify these two paradigms. Our six-dimensional simulations show how the helicity barrier causes the large-scale energy to grow through time, generating small parallel scales and high-frequency ion-cyclotron-wave heating from low-frequency turbulence, while simultaneously explaining various other long-standing observational puzzles. The predicted causal link between plasma expansion and the ion-to-electron heating ratio suggests that the helicity barrier could contribute to key observed differences between fast and slow wind streams.

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## Data availability

The 6D simulations presented in this article generated approximately 30 TB of data. Interested parties are invited to contact the corresponding author to make arrangements for the transfer of those data.

## Code availability

All analysis scripts presented in this work are available on request from the corresponding author. The PEGASUS++ code will be made publicly available in the near future in conjunction with a detailed publication about its numerical methods. Readers can contact the corresponding author to get updates.

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## Acknowledgements

We thank B. Dorland, B. Chandran and A. Mallet for illuminating discussions. J.S. and R.M acknowledge support from the Royal Society Te Apārangi, New Zealand, through Marsden Fund grant number UOO1727 and Rutherford Discovery Fellowship RDF-U001804. M.W.K. and E.Q. were supported by the Department of Energy through the NSF/DOE Partnership in Basic Plasma Science and Engineering, award numbers DE-SC0019046 and DE-SC0019047, with additional support for E.Q. from a Simons Investigator Award from the Simons Foundation. L.A. acknowledges the support of the Institute for Advanced Study, and the work of A.A.S. was supported in part by UK EPSRC grant number EP/R034737/1. This research was part of the Frontera computing project at the Texas Advanced Computing Center, which is made possible by National Science Foundation award number OAC-1818253. Further computational support was provided by the New Zealand eScience Infrastructure (NeSI) high-performance computing facilities, funded jointly by NeSI’s collaborator institutions and the NZ MBIE, and through the PICSciE-OIT TIGRESS High Performance Computing Center and Visualization Laboratory at Princeton University. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

## Author information

Authors

### Contributions

J.S. and R.M. conceived the study. L.A., M.W.K. and J.S. developed the numerical methods and model, with J.S. and L.A. performing the simulations. Data analysis and visualization was carried out by J.S. and L.A., with all authors contributing to general understanding and interpretation of the results. The manuscript was written primarily by J.S. with M.W.K, A.A.S. and E.Q. leading revisions and editing.

### Corresponding author

Correspondence to Jonathan Squire.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Astronomy thanks Munehito Shoda, Christopher Chen and Philip Isenberg for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Electric field noise and numerical cooling.

Contributions to the energy budget per unit volume of the imbalanced simulation from the energy injection (ε/V; dashed red line), increase in thermal energy $${Q}_{i}={\partial }_{t}\langle {m}_{i}{v}_{{{{{{\rm{th}}}}}}}^{2}/2\rangle$$ (blue line), growth rate of mechanical energy $${\partial }_{t}\langle n{m}_{i}{u}_{i}^{2}/2+{B}^{2}/8\pi \rangle$$ (green line), and resistive dissipation εη/V (V is the volume and 〈 … 〉 denotes a box average). The black line shows the total energy budget $${{{{{\rm{Total}}}}}}=\varepsilon /V-{\varepsilon }_{\eta }/V-{\partial }_{t}\langle {m}_{i}{v}_{{{{{{\rm{th}}}}}}}^{2}/2\rangle -{\partial }_{t}\langle n{m}_{i}{u}_{i}^{2}/2+{B}^{2}/8\pi \rangle$$, which is constant and negative, indicating numerical cooling that is effectively independent of the turbulence or the heating of ions.

### Extended Data Fig. 2 The effect of particle noise on turbulence spectra.

Perpendicular (k) spectra of the magnetic field ($${{{{{{\mathcal{E}}}}}}}_{{{{{{\boldsymbol{B}}}}}}}$$), electric field $${{{{{{\mathcal{E}}}}}}}_{{{{{{\boldsymbol{E}}}}}}}$$, and KAW-normalized density $${{{{{{\mathcal{E}}}}}}}_{{n}_{{{{{{\rm{KAW}}}}}}}}={\beta }_{i}(1+2{\beta }_{i}){{{{{{\mathcal{E}}}}}}}_{n}$$ in the saturated state (solid lines) and at very early times (averaged over t≤0.2τA). The latter is from before the turbulence has developed and is thus a proxy for the noise floor in a given quantity. At the smallest scales, kρi 3, spectra are only modestly above the noise floor and therefore uncertain.

### Extended Data Fig. 3 Measurement of the parallel spectrum.

Two-dimensional perpendicular magnetic-field spectrum $${{{{{{\mathcal{E}}}}}}}_{{B}_{\perp }}({k}_{\perp },{k}_{\parallel })={{{{{{\mathcal{E}}}}}}}_{{B}_{x}}({k}_{\perp },{k}_{\parallel })+{{{{{{\mathcal{E}}}}}}}_{{B}_{y}}({k}_{\perp },{k}_{\parallel })$$ from the balanced simulation. The method recovers the large- and small-scale scalings of k with k measured using structure functions (see fig. 2 of ref. 19), as well as the predicted 2D spectrum in the kρi < 1 range (see appendix B of ref. 37).

### Extended Data Fig. 4 Assessment of the influence of stochastic heating.

We show perpendicular spectra of the electric potential Φ, computed from the curl free part of E. Colored lines show various times from the imbalanced simulation. The black line shows the equivalent balanced simulation, which is averaged over the early period of the simulation (between t = 3.5τA and t = 4.5τA) when stochastic-ion heating absorbs the majority of the turbulent energy flux19 Despite the larger turbulence amplitude in the imbalanced simulation, the electric-potential fluctuations around kρi ~ 1 – those important for stochastic heating – are smaller.

### Extended Data Fig. 5 Development of the ion beam.

We compare the rate of change parallel thermal energy (solid lines; see text) with the work done on particles by the parallel electric field ewEfi〉 (dotted lines). The thick dark-blue lines show the saturated state and the orange-pink lines show t = 7τA. The similarity of the magnitude and general shape of the two measures of heating suggests that Landau damping is responsible for the formation of the ion beam.

## Supplementary information

### Supplementary Video 1

A fly-through slice of the perpendicular electric field in the saturated state. A sampling of the magnetic-field-line structure is shown by the blue lines.

### Supplementary Video 2

Time evolution of the perpendicular electric field magnitude.

### Supplementary Video 3

Time evolution of the y component of the magnetic field.

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Squire, J., Meyrand, R., Kunz, M.W. et al. High-frequency heating of the solar wind triggered by low-frequency turbulence. Nat Astron 6, 715–723 (2022). https://doi.org/10.1038/s41550-022-01624-z

• Accepted:

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• DOI: https://doi.org/10.1038/s41550-022-01624-z

• ### A new way for turbulence to heat the corona

• Christopher H. K. Chen

Nature Astronomy (2022)