BRIEF RESEARCH REPORT article
On the Origin of High–Frequency Magnetic Fluctuations in the Interplanetary Medium: A Brownian–like Approach
- 1Dipartimento di Fisica, Università Della Calabria, Rende, Italy
- 2Direzione Scientifica, Istituto Nazionale di Astrofisica, Roma, Italy
- 3Department of Astrophysics/IMAPP, Radboud University, Nijmegen, Netherlands
- 4LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, Meudon, France
- 5INAF-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.
Since the first measurements of magnetic fluctuations in the interplanetary space , 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 , Landau damping , 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.,  and Refs. therein), mainly exciting electric fluctuations , while the energy content in the magnetic fluctuations is lower (see e.g., . 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
In our framework, the values of the correlation time
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
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 . To this purpose, Eq. (1) can be reformulated as
According to the chaotic hypothesis [44, 45], in this case it can be shown  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 . 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
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
The last equation depends on the time correlations of the forcing. For example, by using Eq. (4), we obtain
where we defined
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
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  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., ). 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.
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  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 , 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.
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.
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.
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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 , Czechia
Reviewed by:Pablo S. Moya, University of Chile, Chile
Christian 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, email@example.com