# On the Origin of High–Frequency Magnetic Fluctuations in the Interplanetary Medium: A Brownian–like Approach

^{1}Dipartimento di Fisica, Università Della Calabria, Rende, Italy^{2}Direzione Scientifica, Istituto Nazionale di Astrofisica, Roma, Italy^{3}Department of Astrophysics/IMAPP, Radboud University, Nijmegen, Netherlands^{4}LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, Meudon, France^{5}INAF-Istituto di Astrofisica e Planetologia Spaziali, Roma, Italy

Low–frequency fluctuations in the interplanetary medium have been extensively investigated and described in the framework of turbulence, and the observed universal scaling behavior represents a clear signature of the underlying energy cascade. On the contrary, the interpretation of observations of plasma fluctuations at high frequencies, where wave–wave coupling, collisionless dissipation, and anomalous plasma heating play a key role, still represents a challenge for theoretical modeling. In this paper the high frequency fluctuations occurring in the interplanetary space are described through a Brownian–like approach, where the plasma dynamics at small scales is described through a stochastic process. It is shown that a simple model based on this framework is able to successfully reproduce the main features of the spectrum of the observed magnetic fluctuations. Moreover, the Fluctuation-Dissipation Relation, derived by our model, leads to a power law between dissipation rate and temperature, which is compatible with the occurrence of Landau damping, interpreted thus as the main mechanism of dissipation in the solar wind plasma.

## 1 Introduction

Since the first measurements of magnetic fluctuations in the interplanetary space [1], showing that the spectral magnetic energy density decays with the frequency ω as

Contrary to the low frequency spectrum (

## 2 Model and Results

At small scales (high frequencies), smaller than the ion gyro–radius or inertial length, the plasma dynamics in the interplanetary space is extremely complex. More specifically, the linear mode waves become kinetic, exhibiting simultaneously a dispersive and dissipative character due to wave–particle interactions such as coherent scattering processes or incoherent processes (like pitch angle scattering). The collissionless damping mechanisms include cyclotron damping [26], Landau damping [27], energization of particles at current sheets, that can be spontaneously generated by an intermittent turbulent cascade [28–33], and stochastic heating [34–38].

It is generally agreed that the nonlinear energy cascade, which is surely active at the largest scales, transfers energy beyond the ion–cyclotron frequency (see e.g., [2] and Refs. therein), mainly exciting electric fluctuations [39], while the energy content in the magnetic fluctuations is lower (see e.g., [40]. At the same time, fluctuations are damped by plasma kinetic effects, thus providing a mechanism for heating in the collisionless plasma. The wave–particle mechanism involved in the dissipation acts as a feedback for fluctuations, as it generates particle beams which, in turn, are able to excite further fluctuations. The complex plasma dynamics at small scales, well documented in literature, involves a medium where random fluctuations and dissipation compete in generating magnetic fluctuations. In a range of scales where collisionless dissipation and plasma heating could take place and the presence of a lot of characteristic frequencies and lengths (e.g., cyclotron frequencies and inertial lenghts) breaks the scale–free behavior, the role of dispersion and dissipation is still poorly understood, and the origin of fluctuations is far from being clearly established. This framework is rather diffrent, even if compatible, from the “classical” turbulent dynamics where the nonlinear cascade operates within a scale–free range which is well separated from the smallest scales where dissipation occurs.

In order to provide a description of the high-frequency dynamics of magnetic fuctuations, a novel scenario, based on a stochastic Brownian approach, is introduced in the present work. This approach allows an interpretation of the observed high frequency magnetic spectra with no assumptions about dispersion relations from plasma turbulence theory. Based on the above considerations, we consider a simple framework where magnetic fuctuations

Here, without loss of generality, we consider only the time evolution of a single component of the fluctuations, but the model can be easily generalized to three-dimensional fluctuations or specific wavenumbers. In the simplest case, we assume that the dynamics of the fluctuations is due to two different contributions. The first contribution (first term in the right hand side) is due to the collisionless dissipative processes, which we parametrize with a linear damping term

Under the hypotheses described above, the Itô equation can be solved by Fourier transforms. This gives an obvious relation between the correlations of the Fourier modes of the forcing

where brackets denote time averaging and

so that we can immediately write again Eq. (2) in terms of the power spectrum

The spectral energy is, therefore, related to the spectral shape

As a further example, let us consider the case in which, close to the ion breakpoint, a variety of waves takes part in the process through wave–wave couplings, wave–particles interactions and dispersive effects. In this situation we can expect that the two–point correlations of the stochastic forcing term decay exponentially in time

where

In our framework, the values of the correlation time

**FIGURE 1**.

Equation (5) is compatible with the presence of two power law ranges, similarly to what is reported for observations in some previous works [17, 19]: the first one, between the two breakpoints, with a spectral slope

Since the power law index reported in observations for the range of scales between the ion and electron breaks varies in the interval

where

The spectum

When comparing the power spectra obtained from solar wind observations to those given by theoretical models, it is necessary, in general, to take into account the possible failure of the Taylor hypothesis. Measurements are obtained in the spacecraft reference frame, which is in relative motion with respect to the plasma frame of the solar wind. According to the Doppler shift formula, the measured frequency

### 2.1 Statistical Properties

The statistical properties of the fluctuations can be related to the properties of the macroscopic dissipation through the Sinai-Ruelle-Bowen (SRB) measure [43]. To this purpose, Eq. (1) can be reformulated as

where

(

According to the chaotic hypothesis [44, 45], in this case it can be shown [43] that there exists a SRB measure

where *A* is the contracting phase space and

and, as expected, the average of α defines the phase-space contraction rate and is proportional to the damping rate

For systems with reversible dynamics, as those described by Eq. (8), the chaotic hypothesis and the SRB measure generally imply the Onsager reciprocity and the fluctuation-dissipation relation [46]. We consider, from the Itô equation, an equation for the average energy of magnetic fluctuations

The relation between the magnetic fluctuations and the random forcing term can be formally obtained, from the Itô equation, in the following way

where we set

where

A nearly–stationary solution

where the unknown function

On the other hand, from the definition of the magnetic energy power spectrum and using Eq. (3), we obtain

At equilibrium

where

The last equation depends on the time correlations of the forcing. For example, by using Eq. (4), we obtain

where we defined

where

The FDR relation Eq. (18) is very interesting because it allows us to obtain information about the physical mechanism responsible for the dissipative term. If we conjecture that energy equipartition is present, as in standard statistical mechanics, we can interpret *temperature*

which corresponds to the classical scaling for the electron Landau damping. Therefore, according to our model the observed spectral properties of magnetic fluctuations at ionic scales are compatible with the occurrence of electron Landau damping. It is worthwhile to remark that according to our approach, the spectral properties of magnetic fluctuations are not necessarily the result of a turbulent cascade process. Rather the spectrum is a direct consequence of the FDR, which governs at the same time both fluctuations and dissipation, which represent the two ingredients of the same physical process. Of course, in a classical turbulent environment [2] the fluctuations generated by the cascade process are not subject to dissipation, which starts beyond the Kolmogorov microscale breakpoint. Our approach can be linked to kinetic turbulent cascades by using nonequilibrium ensembles in turbulence models [53, 54].

Note that assuming that the β parameter in Eq. (18) is the usual plasma-β parameter, the FDR (Eq. 18) suggests that the high-frequency spectral breakpoint shifts toward higher frequencies as the solar wind plasma-β decreases, in agreement with observations. The electron break can be hardly or no observable in the data because it can be located out of the instrumental range or hidden by the high-frequency instrumental noise (see e.g., [55]). Our approach enables, at least, to obtain an estimate of the break position even when it is out of the observable instrumental range. This is because the FDR has a predictive meaning, as the spectral properties of magnetic fluctuations depend on the parameter used to describe the dissipation, so that by measuring the parameter µ, through the magnetic power spectra at ionic scales, and the plasma-β parameter, we are able to investigate the frequency location of the electron break, even when it can not be observed.

## 3 Discussion

In this paper we introduce a framework to describe the high–frequency dynamics of magnetic fluctuations in the interplanetary space. Our description is rather different from the nonlinear energy cascade framework, successfully used to describe low–frequency fluctuations. By using a Brownian–like approach, we are able to describe the main properties of the magnetic energy spectra observed at high frequencies in the solar wind. We remark that the same kind of phenomenology was used by [56] to describe the susceptibility of fluctuations under the action of random forcing, within the Direct Interaction Approximation of the complex nonlinear mode couplings generated by the fluid turbulent cascade. Of course, our approach does not rule out the importance of all the complex dynamics coming from plasma physics. Kinetic plasma physics describes, indeed, all the microscopic features involved in the dynamics of fluctuations, namely the birth of the many modes involved, their nonlinear coupling, their dispersive properties, and the collissionless dissipative processes which lead to anomalous plasma–heating.

Using our approach, we describe, at the same time, both fluctuations and dissipation in the high-frequency range of solar wind plasmas, where high-frequency microphysical plasma effects are modelled as a stochastic source, whose details, in this framework, are unessential. Through the FDR, we evidence the relationship between fluctuations and dissipation in a way that, independently of the specific microphysical plasma dynamics, we can account for the main features of the spectral properties of high-frequency fluctuations in the interplanetary space. In fact, as usual in a Brownian-like approach [41], the FDR has a predictive meaning for some microphysical quantities. In our case, Eq. (18) opens a *window* on the high-frequency fluctuations, allowing us to estimate the position of the electron break as a function of fully measurable quantities in the solar wind, similarly to the Einstein’s approach to Brownian motion. Moreover, the scaling of the damping rate results compatible with the presence of electron Landau damping, which therefore can be identified as the main dissipation mechanism in the collisionless solar wind plasma.

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary material, further inquiries can be directed to the corresponding author/s.

## Author Contributions

VC conceived the study and the model. VC and FL prepared the article draft. All the authors contributed to scientific discussions, model refinement and writing of the article.

## Funding

VC, FL, and FC were supported by Italian MIUR-PRIN Grant No. 2017APKP7T on Circumterrestrial Environment: Impact of Sun-Earth Interaction.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## References

1. Coleman PJ. Jr. Turbulence, viscosity, and dissipation in the solar-wind plasma. *Astrophys J* (1968) 153:371. doi:10.1086/149674

2. Bruno R, Carbone V. *Turbulence in the solar wind*. Berlin, Germany: Springer (2016). doi:10.1007/978-3-319-43440-7

3. Burlaga LF. Intermittent turbulence in the solar wind. *J Geophys Res* (1991) 96:5847–51. doi:10.1029/91JA00087

4. Carbone V, Bruno R, Sorriso-Valvo L, Lepreti F. Intermittency of magnetic turbulence in slow solar wind. *Planet Space Sci* (2004) 52:953–6. doi:10.1016/j.pss.2004.02.005

5. Marsch E, Liu S. Structure functions and intermittency of velocity fluctuations in the inner solar wind. *Ann Geophysicae* (1993) 11:227–38.

6. Sorriso-Valvo L, Carbone V, Veltri P, Consolini G, Bruno R. Intermittency in the solar wind turbulence through probability distribution functions of fluctuations. *Geophys Res Lett* (1999) 26:1801–4. doi:10.1029/1999GL900270

7. Burlaga LF. Multifractal structure of the interplanetary magnetic field: voyager 2 observations near 25 AU, 1987-1988. *Geophys Res Lett* (1991) 18:69–72. doi:10.1029/90GL02596

8. Carbone V, Lepreti F, Sorriso-Valvo L, Veltri P, Antoni V, Bruno R. Scaling laws in plasma turbulence. *Nuovo Cimento Rivista Serie* (2004) 27:1–108. doi:10.1393/ncr/i2005-10003-1

9. Carbone V. Cascade model for intermittency in fully developed magnetohydrodynamic turbulence. *Phys Rev Lett* (1993) 71:1546–8. doi:10.1103/PhysRevLett.71.1546

10. Carbone V. Scaling exponents of the velocity structure functions in the interplanetary medium. *Ann Geophys* (1994) 12:585–90. doi:10.1007/s00585-994-0585-3

11. Marsch E, Tu C-Y, Rosenbauer H. Multifractal scaling of the kinetic energy flux in solar wind turbulence. *Ann Geophys* (1996) 14:259–69. doi:10.1007/s00585-996-0259-4

12. Banerjee S, Hadid LZ, Sahraoui F, Galtier S. Scaling of compressible magnetohydrodynamic turbulence in the fast solar wind. *Astrophys J* (2016) 829:L27. doi:10.3847/2041-8205/829/2/l27

13. Carbone V, Marino R, Sorriso-Valvo L, Noullez A, Bruno R. Scaling laws of turbulence and heating of fast solar wind: the role of density fluctuations. *Phys Rev Lett* (2009) 103:061102. doi:10.1103/PhysRevLett.103.061102

14. MacBride BT, Smith CW, Forman MA. The turbulent cascade at 1 AU: energy transfer and the third‐order scaling for MHD. *Astrophys J* (2008) 679:1644–60. doi:10.1086/529575

15. Sorriso-Valvo L, Marino R, Carbone V, Noullez A, Lepreti F, Veltri P, et al. Observation of inertial energy cascade in interplanetary space plasma. *Phys Rev Lett* (2007) 99:115001. doi:10.1103/PhysRevLett.99.115001

16. Leamon RJ, Smith CW, Ness NF, Matthaeus WH, Wong HK. Observational constraints on the dynamics of the interplanetary magnetic field dissipation range. *J Geophys Res* (1998) 103:4775–87. doi:10.1029/97JA03394

17. Sahraoui F, Goldstein ML, Robert P, Khotyaintsev YV. Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale. *Phys Rev Lett* (2009) 102:231102. doi:10.1103/PhysRevLett.102.231102

18. Alexandrova O, Lacombe C, Mangeney A, Grappin R, Maksimovic M. Solar wind turbulent spectrum at plasma kinetic scales. *Astrophys J* (2012) 760:121. doi:10.1088/0004-637x/760/2/121

19. Sahraoui F, Huang SY, Belmont G, Goldstein ML, Rétino A, Robert P, et al. Scaling of the electron dissipation range of solar wind turbulence. *Astrophys J* (2013) 777:15. doi:10.1088/0004-637X/777/1/15

20. Bruno R, Carbone V. The solar wind as a turbulence laboratory. *Living Rev Solar Phys* (2013) 10:2. doi:10.12942/lrsp-2013-2

21. Marsch E. Kinetic physics of the solar corona and solar wind. *Living Rev Solar Phys* (2006) 3:1. doi:10.12942/lrsp-2006-1

22. Parashar TN, Salem C, Wicks RT, Karimabadi H, Gary SP, Matthaeus WH. Turbulent dissipation challenge: a community-driven effort. *J Plasma Phys* (2015) 81:905810513. doi:10.1017/S0022377815000860

23. Narita Y, Gary SP, Saito S, Glassmeier K-H, Motschmann U. Dispersion relation analysis of solar wind turbulence. *Geophys Res Lett* (2011) 38:L05101. doi:10.1029/2010GL046588

24. Perschke C, Narita Y, Motschmann U, Glassmeier KH. Observational test for a random sweeping model in solar wind turbulence. *Phys Rev Lett* (2016) 116:125101. doi:10.1103/PhysRevLett.116.125101

25. Narita Y. Space-time structure and wavevector anisotropy in space plasma turbulence. *Living Rev Sol Phys* (2018) 15:2. doi:10.1007/s41116-017-0010-0

26. Hollweg JV, Isenberg PA. Generation of the fast solar wind: a review with emphasis on the resonant cyclotron interaction. *J Geophys Res* (2002) 107:1147. doi:10.1029/2001JA000270

27. Schekochihin AA, Cowley SC, Dorland W, Hammett GW, Howes GG, Quataert E, et al. Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. *Astrophys J Suppl Ser* (2009) 182:310–77. doi:10.1088/0067-0049/182/1/310

28. Dmitruk P, Matthaeus WH, Seenu N. Test particle energization by current sheets and nonuniform fields in magnetohydrodynamic turbulence. *Astrophys J* (2004) 617:667–79. doi:10.1086/425301

29. Karimabadi H, Roytershteyn V, Wan M, Matthaeus WH, Daughton W, Wu P, et al. Coherent structures, intermittent turbulence, and dissipation in high-temperature plasmas. *Phys Plasmas* (2013) 20:012303. doi:10.1063/1.4773205

30. Parashar TN, Servidio S, Shay MA, Breech B, Matthaeus WH. Effect of driving frequency on excitation of turbulence in a kinetic plasma. *Phys Plasmas* (2011) 18:092302. doi:10.1063/1.3630926

31. Parashar TN, Shay MA, Cassak PA, Matthaeus WH. Kinetic dissipation and anisotropic heating in a turbulent collisionless plasma. *Phys Plasmas* (2009) 16:032310. doi:10.1063/1.3094062

32. Sundkvist D, Retinò A, Vaivads A, Bale SD. Dissipation in turbulent plasma due to reconnection in thin current sheets. *Phys Rev Lett* (2007) 99:025004. doi:10.1103/PhysRevLett.99.025004

33. Wan M, Matthaeus WH, Karimabadi H, Roytershteyn V, Shay M, Wu P, et al. Intermittent dissipation at kinetic scales in collisionless plasma turbulence. *Phys Rev Lett* (2012) 109:195001. doi:10.1103/PhysRevLett.109.195001

34. Chandran BDG, Verscharen D, Quataert E, Kasper JC, Isenberg PA, Bourouaine S. Stochastic heating, differential flow, and the alpha-to-proton temperature ratio in the solar wind. *Astrophys J* (2013) 776:45. doi:10.1088/0004-637X/776/1/45

35. Chandran BDG. Alfvén-wave turbulence and perpendicular ion temperatures in coronal holes. *Astrophys J* (2010) 720:548–54. doi:10.1088/0004-637X/720/1/548

36. Chaston CC, Bonnell JW, Carlson CW, McFadden JP, Ergun RE, Strangeway RJ, et al. Auroral ion acceleration in dispersive Alfvén waves. *J Geophys Res* (2004) 109:A04205. doi:10.1029/2003JA010053

37. McChesney JM, Stern RA, Bellan PM. Observation of fast stochastic ion heating by drift waves. *Phys Rev Lett* (1987) 59:1436–9. doi:10.1103/PhysRevLett.59.1436

38. Xia Q, Perez JC, Chandran BDG, Quataert E. Perpendicular ion heating by reduced magnetohydrodynamic turbulence. *Astrophys J* (2013) 776:90. doi:10.1088/0004-637X/776/2/90

39. Bale SD, Kellogg PJ, Mozer FS, Horbury TS, Reme H. Measurement of the electric fluctuation spectrum of magnetohydrodynamic turbulence. *Phys Rev Lett* (2005) 94:215002. doi:10.1103/PhysRevLett.94.215002

40. Cerri SS, Califano F. Reconnection and small-scale fields in 2d-3v hybrid-kinetic driven turbulence simulations. *New J Phys* (2017) 19:025007. doi:10.1088/1367-2630/aa5c4a

42. Klein KG, Howes GG, TenBarge JM. The violation of the taylor hypothesis in measurements of solar wind turbulence. *Astrophys J* (2014) 790, L20. doi:10.1088/2041-8205/790/2/L20

43. Gallavotti G, Cohen EGD. Dynamical ensembles in nonequilibrium statistical mechanics. *Phys Rev Lett* (1995) 74:2694–7. doi:10.1103/PhysRevLett.74.2694

44. Ruelle D. A review of linear response theory for general differentiable dynamical systems. *Nonlinearity* (2009) 22:855–70. doi:10.1088/0951-7715/22/4/009

45. Ruelle D. General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. *Phys Lett A* (1998) 245:220–4. doi:10.1016/S0375-9601(98)00419-8

46. Gallavotti G. Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem. *J Stat Phys* (1996) 84:899–925. doi:10.1007/BF02174123

47. Callen HB, Welton TA. Irreversibility and generalized noise. *Phys Rev* (1951) 83:34–40. doi:10.1103/PhysRev.83.34

49. Araneda JA, Astudillo H, Marsch E. Interactions of alfvén-cyclotron waves with ions in the solar wind. *Space Sci Rev* (2012) 172:361–72. doi:10.1007/s11214-011-9773-0

50. Navarro R, Moya P, Muñoz V, Araneda J, Valdivia J, et al. Solar wind thermally induced magnetic fluctuations. *Phys Rev Lett* (2014) 112:245001. doi:10.1103/physrevlett.112.245001

51. Navarro RE, Araneda J, Muñoz V, Moya PS, F.-Viñas A, Valdivia JA. Theory of electromagnetic fluctuations for magnetized multi-species plasmas. *Phys Plasmas* (2014) 21:092902. doi:10.1063/1.4894700

52. Viñas AF, Moya PS, Navarro R, Araneda JA. The role of higher-order modes on the electromagnetic whistler-cyclotron wave fluctuations of thermal and non-thermal plasmas. *Phys Plasmas* (2014) 21:012902. doi:10.1063/1.4861865

53. Biferale L, Cencini M, De Pietro M, Gallavotti G, Lucarini V. Equivalence of nonequilibrium ensembles in turbulence models. *Phys Rev E* (2018) 98:012202. doi:10.1103/PhysRevE.98.012202

54. Gallavotti G. Chaotic principle: some applications to developed turbulence. *J Stat Phys* (1997) 86:907–34. doi:10.1007/BF02183608

55. Goldstein ML, Wicks RT, Perri S, Sahraoui F. Kinetic scale turbulence and dissipation in the solar wind: key observational results and future outlook. *Phil Trans R Soc A* (2015) 373:20140147. doi:10.1098/rsta.2014.0147

Keywords: Interplanetary medium, interplanetary turbulence, solar wind, magnetic fields, Heliosphere

Citation: Carbone V, Lepreti F, Vecchio A, Alberti T and Chiappetta F (2021) On the Origin of High–Frequency Magnetic Fluctuations in the Interplanetary Medium: A Brownian–like Approach. *Front. Phys.* 9:613759. doi: 10.3389/fphy.2021.613759

Received: 03 October 2020; Accepted: 08 January 2021;

Published: 12 April 2021.

Edited by:

Petr Hellinger, Astronomical Institute , CzechiaReviewed by:

Pablo S. Moya, University of Chile, ChileChristian L. Vásconez, National Polytechnic School, Ecuador

Copyright © 2021 Carbone, Lepreti, Vecchio, Alberti and Chiappetta. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Fabio Lepreti, fabio.lepreti@unical.it