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Exact law for compressible pressure-anisotropic magnetohydrodynamic turbulence: Toward linking energy cascade and instabilities
Abstract
We derive an exact law for compressible pressure-anisotropic magnetohydrodynamic turbulence. For a gyrotropic pressure tensor, we study the double-adiabatic case and show the presence of new flux and source terms in the exact law, reminiscent of the plasma instability conditions due to pressure anisotropy. The Hall term is shown to bring ion-scale corrections to the exact law without affecting explicitly the pressure terms. In the pressure isotropy limit we recover all known results obtained for isothermal and polytropic closures. The incompressible limit of the gyrotropic system leads to a generalization of the Politano and Pouquet's law where a new incompressible source term is revealed and reflects exchanges of the magnetic and kinetic energies with the no-longer-conserved internal energy. We highlight the possibilities offered by the new laws to investigate potential links between turbulence cascade and instabilities widely observed in laboratory and astrophysical plasmas.
... The impact of ion pressure anisotropy on the energy cascade rate of Hall-MHD turbulence with bi-adiabatic ions and isothermal electrons is evaluated in three-dimensional direct numerical simulations, using the exact law derived in Simon and Sahraoui [1]. It is shown that pressure anisotropy can enhance or reduce the cascade rate, depending on the scales, in comparison with the prediction of the exact law with isotropic pressure, by an amount that correlates well with pressure anisotropy a p = p ⊥ p ∥ ̸ = 1 developing in simulations initialized with an isotropic pressure (a p 0 = 1). ...
... Recently, Simon and Sahraoui [1,23] proposed a generic derivation of the exact law for the total energy, based on the internal-energy equation, and applied it to the Hall-MHD model with either isotropic or gyrotropic pressures. This work paves the road to more realistic studies of compressible turbulence in weakly collisional plasmas, such as those of the near-Earth space where the pressure is all but isotropic [34,41,42]. ...
... In the linear approximation, this model is known to permit the development of firehose and mirror-type instabilities [44,45]. In this paper, we propose to evaluate the cascade rate in driven-turbulence simulations of the CGL-Hall-MHD system for a proton-electron plasma, obtained by adding the Hall effect to the CGL model, with the aim to quantitatively estimate the effect of pressure anisotropy on the turbulent cascade, as described by the exact law proposed in Simon and Sahraoui [1]. The study is performed on the simulation data used in Ferrand et al. [39], complemented with a new simulation specifically designed for the present work. ...
The impact of ion pressure anisotropy on the energy cascade rate of Hall-MHD turbulence with bi-adiabatic ions and isothermal electrons is evaluated in three-dimensional direct numerical simulations, using the exact law derived in Simon and Sahraoui (2022). It is shown that pressure anisotropy can enhance or reduce the cascade rate, depending on the scales, in comparison with the prediction of the exact law with isotropic pressure, by an amount that correlates well with pressure anisotropy $a_p=\frac{p_\perp}{p_\parallel}\neq1$ developing in simulations initialized with an isotropic pressure (${a_p}_0=1$). A simulation with an initial pressure anisotropy, ${a_p}_0=4$, confirms this trend, yielding a stronger impact on the cascade rate, both in the inertial range and at larger scales, close to the forcing. Furthermore, a Fourier-based numerical method to compute the exact laws in numerical simulations in the full $(\ell_\perp,\ell_\parallel)$ scale separation plane is presented.
... The higher resolution observations provided by Cluster/ESA (Bale et al. 2005;Kiyani et al. 2015) have led us to propose new theories for plasma turbulence. As far as we are concerned, we can mention the generalization of the exact (MHD) Kolmogorov law (Kolmogorov 1941;Politano & Pouquet 1998;Galtier 2008) for compressible turbulence, first in the case of isothermal hydrodynamics (Galtier & Banerjee 2011) and then to MHD (with different closures and/or scales description) (Banerjee & Galtier 2013;Andrés et al. 2018;Ferrand et al. 2021;Simon & Sahraoui 2022). Second, the extensive use of these (compressible) laws as a solar wind model has led to a better estimate of the turbulent transfer and thus of the local heating, although we still do not know precisely by what mechanism this small scale heating occurs (Sorriso-Valvo et al. 2007;Osman et al. 2011;Banerjee et al. 2016;Hadid et al. 2017;Bandyopadhyay et al. 2020;Marino & Sorriso-Valvo 2023). ...
... As far as we are concerned, we can mention the generalization of the exact (MHD) Kolmogorov law (Kolmogorov 1941;Politano & Pouquet 1998;Galtier 2008) for compressible turbulence, first in the case of isothermal hydrodynamics (Galtier & Banerjee 2011) and then to MHD (with different closures and/or scales description) (Banerjee & Galtier 2013;Andrés et al. 2018;Ferrand et al. 2021;Simon & Sahraoui 2022). Second, the extensive use of these (compressible) laws as a solar wind model has led to a better estimate of the turbulent transfer and thus of the local heating, although we still do not know precisely by what mechanism this small scale heating occurs (Sorriso-Valvo et al. 2007;Osman et al. 2011;Banerjee et al. 2016;Hadid et al. 2017;Bandyopadhyay et al. 2020;Marino & Sorriso-Valvo 2023). ...
An analytical theory of wave turbulence is developed for pure compressible magnetohydrodynamics in the small $\beta$ limit. In contrast to previous works where the multiple scale method was not mentioned and slow magneto-acoustic waves were included, I present here a theory for fast magneto-acoustic waves only for which an asymptotic closure is possible in three dimensions. I introduce the compressible Elsasser fields (canonical variables) and show their linear relationship with the mass density and the compressible velocity. The kinetic equations of wave turbulence for three-wave interactions are obtained and the detailed conservation is shown for the two invariants, energy and momentum (cross-helicity). An exact stationary solution (Kolmogorov-Zakharov spectrum) exists only for the energy. I find a $k^{-3/2}$ energy spectrum compatible with the Iroshnikov-Kraichnan (IK) phenomenological prediction; this leads to a mass density spectrum with the same scaling. Despite the presence of a relatively strong uniform magnetic field, this turbulence is characterized by an energy spectrum with a power index that is independent of the angular direction; its amplitude, however, shows an angular dependence. I prove the existence of the IK solution using the locality condition, show that the energy flux is positive and hence the cascade direct, and find the Kolmogorov constant. This theory offers a plausible explanation for recent observations in the solar wind at small $\beta$ where isotropic spectra with a $-3/2$ power law index are found and associated with fast magneto-acoustic waves. This theory may also be used to explain the IK spectrum often observed near the Sun. Besides, it provides a rigorous theoretical basis for the well-known phenomenological IK spectrum, which coincides with the Zakharov-Sagdeev spectrum for acoustic wave turbulence.
... Several explanations of the steepening in the transition range have been proposed, but not all are consistent with, or directly linked to, each other. The physics of the transition range is indeed very complex because of, among other reasons, the presence of a variety of plasma instabilities that kick in near kρ i ∼ 1 whose effects on the background turbulence are still poorly understood (Sahraoui et al. 2004(Sahraoui et al. , 2006Kunz et al. 2014Kunz et al. , 2018Simon & Sahraoui 2022). In addition, theoretical predictions are generally obtained in asymptotic limits, typically kρ i = 1 or kρ i ? 1 and kρ e = 1 (Schekochihin et al. 2009), which excludes that range. ...
Large surveys of the power spectral density of the magnetic fluctuations in the solar wind have reported different slope distributions at MHD, sub-ion and sub-electron scales: the smaller the scale, the broader the distribution. Here, we review briefly some of the most relevant explanations of the broadening of the slopes at sub-ion scales. Then, we present a new one that has been overlooked in the literature, which is based on the relative importance of the dispersive effects with respect to the Doppler shift due to the mean flow speed. We build a toy model based on a dispersion relation of a linear mode that matches at high frequency ( ω ≳ ω ci ) the Alfvén (respectively whistler) mode at high oblique (respectively quasi-parallel) propagation angles θ kB . Starting with a double power-law spectrum of turbulence k ⊥ − 1.66 in the inertial range and k ⊥ − 2.8 at the sub-ion scales, the transformed spectrum (in frequency f ) as it would be measured in the spacecraft reference frame shows a broad range of slopes at the sub-ion scales that depend both on the angle θ kB and the flow speed V . Varying θ kB in the range 4°–106° and V in the range 400−800 km s ⁻¹ the resulting distribution of slopes at the sub-ion scales reproduces quite well the observed one in the solar wind. Fluctuations in the solar wind speed and the wavevector anisotropy of the turbulence may explain (or at least contribute to) the variability of the spectral slopes reported from spacecraft observations in the solar wind.
... Compressible turbulence is still an open problem. Relevant questions regard the existence of universal scaling laws; role and formation of shocklets; channels of energy dissipation and existence of exact laws; role of the forcing; nature of intermittency and locality of the energy cascades; and role in star formation [e.g., 35,2,9,44,10,131,3,116,4]. Turbulence transport models applicable beyond the HTS have not been developed yet [98], even though the first steps have been taken [47]. ...
Turbulence is ubiquitous in space plasmas. It is one of the most important subjects in heliospheric physics, as it plays a fundamental role in the solar wind - local interstellar medium interaction and in controlling energetic particle transport and acceleration processes. Understanding the properties of turbulence in various regions of the heliosphere with vastly different conditions can lead to answers to many unsolved questions opened up by observations of the magnetic field, plasma, pickup ions, energetic particles, radio and UV emissions, and so on. Several space missions have helped us gain preliminary knowledge on turbulence in the outer heliosphere and the very local interstellar medium. Among the past few missions, the Voyagers have paved the way for such investigations. This paper summarizes the open challenges and voices our support for the development of future missions dedicated to the study of turbulence throughout the heliosphere and beyond.
... This can reduce the available range of scales for the turbulence to develop with respect, for instance, to the solar wind as reported in the Earth's magnetosphere (Zimbardo et al., 2010). Practically, obtaining statistical observational results (e.g., power spectra, dissipation rates) that can be reliably compared to theoretical predictions based on stationary and homogenous theories (Andrés et al., 2018;Ferrand et al., 2019;Galtier, 2008;Simon & Sahraoui, 2022) requires having long time series without the crossings of boundary. Despite these difficulties, deep physical insight was obtained from previous observational studies in the terrestrial magnetosphere (Vörös et al., 2004(Vörös et al., , 2007Zimbardo et al., 2010). ...
The Kolmogorov scaling in the inertial range of scales is a distinct characteristic of fully developed turbulence, and studying it offers valuable insights into the evolution of turbulence. In this work, we perform a statistical survey of the power spectra with the Kolmogorov scaling in Saturn's magnetosphere using Cassini measurements. Two cases study show that both magnetic‐field and electron density spectra exhibit f −5/3 at the MHD scales. The statistical analysis reveals a wide‐ranging and abundant presence of Kolmogorov spectra throughout magnetosphere, observed across all local times. Interestingly, the occurrence rate of these Kolmogorov‐like events within Saturn's magnetosphere surpasses that observed in the planetary magnetosheath. The measurements of magnetic compressibility for the Kolmogorov‐like events show the dominance of incompressible Alfvénic turbulence (44.64%) with respect to magnetosonic‐like one (6.94%). In addition, the source and evolution of the turbulent fluctuations are further discussed.
... Similar terms emerge if one takes into account the compressive effects. 57,[59][60][61] At the intermediate scales, there is a significant overlap between the @S=@t term and the r l Á Y term for low guide field cases. This reduces the relative contribution of the r l Á Y term in the inertial range. ...
The effect of an external guide field on the turbulence-like properties of magnetic reconnection is studied using five different 2.5D kinetic particle-in-cell (PIC) simulations. The magnetic energy spectrum is found to exhibit a slope of approximately −5/3 in the inertial range, independent of the guide field. On the contrary, the electric field spectrum in the inertial range steepens more with the guide field and approaches a slope of −5/3. In addition, spectral analysis of the different terms of the generalized Ohm's law is performed and found to be consistent with PIC simulations of turbulence and MMS observations. Finally, the guide field effect on the energy transfer behavior is examined using the von Kármán–Howarth (vKH) equation based on incompressible Hall-MHD. The general characteristics of the vKH equation with constant rate of energy transfer in the inertial range are consistent in all the simulations. This suggests that the qualitative behavior of energy spectrum and energy transfer in reconnection are similar to that of turbulence, indicating that reconnection fundamentally involves an energy cascade.
... Second, in the present work, we only focus on the MHD scales. Recent exact relations could be used to estimate the transfer of energy as we reach the sub-ion scales (Andr es et al., 2018;Simon and Sahraoui, 2021;and Simon and Sahraoui, 2022). In forthcoming works, further statistical investigation of these topics will be carried out using in situ data in more compressible environments (like the Earth's magnetosheath). ...
We investigated incompressible and compressible magnetohydrodynamic (MHD) energy cascade rates in the solar wind at different heliocentric distances. We used in situ magnetic field and plasma observations provided by the Parker Solar Probe mission and exact relations in fully developed turbulence. To estimate the compressible cascade rate, we applied two recent exact relations for compressible isothermal and polytropic MHD turbulence, respectively. Our observational results show a clear increase in the absolute value of the compressible and incompressible cascade rates as we get closer to the Sun. Moreover, we obtained an increase in both isothermal and polytropic cascade rates with respect to the incompressible case as compressibility increases in the plasma. Further discussion about the relation between the compressibility and the heliocentric distance is carried out. Furthermore, we compared both exact relations as compressibility increases in the solar wind, and although we note a slight trend to observe larger cascades using a polytropic closure, we obtained essentially the same cascade rate in the range of compressibility observed. Finally, we investigated the signed incompressible and compressible energy cascade rates and its connection with the real cascade rate.
... Compressible turbulence is still an open problem. Relevant questions regard the existence of universal scaling laws; role and formation of shocklets; channels of energy dissipation and existence of exact laws; role of the forcing; nature of intermittency and locality of the energy cascades; and role in star formation (e.g., Carbone et al., 2009;Aluie et al., 2012;Banerjee and Galtier, 2013;Falgarone et al., 2015;Banerjee et al., 2016;Yang et al., 2016;Andrés et al., 2018;Andrés et al., 2021;Simon and Sahraoui, 2022). It is important to stress that turbulence transport models applicable beyond the HTS have not been developed yet (Oughton and Engelbrecht, 2021), even though the first steps have been taken (Fichtner et al., 2020). ...
Turbulence is ubiquitous in space plasmas. It is one of the most important subjects in heliospheric physics, as it plays a fundamental role in the solar wind—local interstellar medium interaction and in controlling energetic particle transport and acceleration processes. Understanding the properties of turbulence in various regions of the heliosphere with vastly different conditions can lead to answers to many unsolved questions opened up by observations of the magnetic field, plasma, pickup ions, energetic particles, radio and UV emissions, and so on. Several space missions have helped us gain preliminary knowledge on turbulence in the outer heliosphere and the very local interstellar medium. Among the past few missions, the Voyagers have paved the way for such investigations. This paper summarizes the open challenges and voices our support for the development of future missions dedicated to the study of turbulence throughout the heliosphere and beyond.
... MR in magnetized plasmas is a process that occurs in very thin and localized regions known as current sheets (CS) [28][29][30][31][32][33][34][35]. To quantify the local (in space) cross-scale energy transfer in reconnecting CS it is required to go beyond the popular Kolmogorov 4/5-law, which provides the turbulent transfer rate at scale after averaging over all spatial positions x [36][37][38][39][40][41][42]. To this end we employ the novel spatial coarse-graining (CG) model derived recently for incompressible Hall-MHD [43]. ...
The interplay between plasma turbulence and magnetic reconnection (MR) remains an unsettled question in astrophysical and laboratory plasmas. Here we report the first observational evidence that magnetic reconnection drives sub-ion scale turbulence in magnetospheric plasmas by transferring energy to small scales. We employ a spatial Coarse-Graining (CG) model of Hall magnetohydrodynamic, allowing us to measure the nonlinear energy transfer rate across scale $\ell$ at position $\mathbf{x}$. Its application to Magnetospheric Multiscale mission data shows that MR drives intense energy transfers to sub-ion scales even when little to no energy transfers is present at larger scales. This observational evidence is remarkably supported by the results from Hybrid Vlasov-Maxwell simulations of turbulence to which the CG model is also applied. These results can potentially answer some open questions on plasma turbulence in planetary magnetospheres.
... Second, in the present work, we only focus on the MHD scales. Recent exact relations could be used to estimate the transfer of energy as we reach the sub-ion scales (Andr es et al., 2018;Simon and Sahraoui, 2021;and Simon and Sahraoui, 2022). In forthcoming works, further statistical investigation of these topics will be carried out using in situ data in more compressible environments (like the Earth's magnetosheath). ...
We investigated the incompressible and compressible magnetohydrodynamic (MHD) energy cascade rates in the solar wind at different heliocentric distances. We used in situ magnetic field and plasma observations provided by the Parker Solar Probe (PSP) mission and exact relations in fully developed turbulence. To estimate the compressible cascade rate, we applied two recent exact relations for compressible isothermal and polytropic MHD turbulence, respectively. Our observational results show a clear increase of the compressible and incompressible cascade rates as we get closer to the Sun. Moreover, we obtained an increase in both isothermal and polytropic cascade rates with respect to the incompressible case as compressibility increases in the plasma. Further discussion about the relation between the compressibility and the heliocentric distance is carried out. Finally, we compared both exact relations as compressibility increases in the solar wind and although we note a slightly trend to observe larger cascades using a polytropic closure, we obtained essentially the same cascade rate in the range of compressibility observed.
The differential heating of electrons and ions by turbulence in weakly collisional magnetized plasmas and the scales at which such energy dissipation is most effective are still debated. Using a large data sample measured in Earth’s magnetosheath by the magnetospheric multiscale mission and the coarse-grained energy equations derived from the Vlasov-Maxwell system, we find evidence of a balance over two decades in scales between the energy cascade and dissipation rates. The decline of the cascade rate at kinetic scales (in contrast with a constant one in the inertial range), is balanced by an increasing ion and electron heating rates, estimated via the pressure strain. Ion scales are found to contribute most effectively to ion heating, while electron heating originates from both ion and electron scales. These results can potentially impact the current understanding of particle heating in turbulent magnetized plasmas as well as their theoretical and numerical modeling.
We investigate turbulence in magnetic reconnection jets in the Earth’s magnetotail using data from the Magnetospheric Multiscale spacecraft. We show that signatures of a limited inertial range are observed in many reconnection jets. The observed turbulence develops on the timescale of a few ion gyroperiods, resulting in intermittent multifractal energy cascade from the characteristic scale of the jet down to the ion scales. We show that at sub-ion scales, the fluctuations are close to monofractal and predominantly kinetic Alfvén waves. The observed energy transfer rate across the inertial range is ∼108 J kg−1 s−1, which is the largest reported for space plasmas so far.
Previously, using an incompressible von Kármán–Howarth formalism, the behavior of cross-scale energy transfer in magnetic reconnection and turbulence was found to be essentially identical to each other, independent of an external magnetic (guide) field, in the inertial and energy-containing ranges [Adhikari et al., Phys. Plasmas 30, 082904 (2023)]. However, this description did not account for the energy transfer in the dissipation range for kinetic plasmas. In this Letter, we adopt a scale-filtering approach to investigate this previously unaccounted-for energy transfer channel in reconnection. Using kinetic particle-in-cell simulations of antiparallel and component reconnection, we show that the pressure–strain interaction becomes important at scales smaller than the ion inertial length, where the nonlinear energy transfer term drops off. Also, the presence of a guide field makes a significant difference in the morphology of the scale-filtered energy transfer. These results are consistent with kinetic turbulence simulations, suggesting that the pressure strain interaction is the dominant energy transfer channel between electron scales and ion scales.
The interplay between plasma turbulence and magnetic reconnection remains an unsettled question in astrophysical and laboratory plasmas. Here, we report the first observational evidence that magnetic reconnection drives subion-scale turbulence in magnetospheric plasmas by transferring energy to small scales. We employ a spatial “coarse-grained” model of Hall magnetohydrodynamics, enabling us to measure the nonlinear energy transfer rate across scale ℓ at position x. Its application to Magnetospheric Multiscale mission data shows that magnetic reconnection drives intense energy transfer to subion-scales. This observational evidence is remarkably supported by the results from Hybrid Vlasov-Maxwell simulations of turbulence to which the coarse-grained model is also applied. These results can potentially answer some open questions on plasma turbulence in planetary environments.
An analytical theory of wave turbulence is developed for pure compressible magnetohydrodynamics in the small $\beta$ limit. In contrast to previous works where the multiple scale method was not mentioned and slow magneto-acoustic waves were included, we present here a theory for fast magneto-acoustic waves for which only an asymptotic closure is possible in three dimensions. We introduce the compressible Elsässer fields (canonical variables) and show their linear relationship with the mass density and the compressible velocity. The kinetic equations of wave turbulence for three-wave interactions are obtained and the detailed conservation is shown for the two invariants, energy and momentum (cross-helicity). An exact stationary solution (Kolmogorov-Zakharov spectrum) exists only for the energy. We find a $k^{-3/2}$ energy spectrum compatible with the Iroshnikov–Kraichnan (IK) phenomenological prediction; this leads to a mass density spectrum with the same scaling. Despite the presence of a relatively strong uniform magnetic field, this turbulence is characterized by an energy spectrum with a power index that is independent of the angular direction; its amplitude, however, shows an angular dependence. We prove the existence of the IK solution using the locality condition, show that the energy flux is positive and hence the cascade direct and find the Kolmogorov constant. This theory offers a plausible explanation for recent observations in the solar wind at small $\beta$ where isotropic spectra with a $-3/2$ power-law index are found and associated with fast magneto-acoustic waves. This theory may also be used to explain the IK spectrum often observed near the Sun. Besides, it provides a rigorous theoretical basis for the well-known phenomenological IK spectrum, which coincides with the Zakharov–Sagdeev spectrum for acoustic wave turbulence.
We present estimates of the turbulent energy-cascade rate derived from a Hall-magnetohydrodynamic (MHD) third-order law. We compute the contribution from the Hall term and the MHD term to the energy flux. Magnetospheric Multiscale (MMS) data accumulated in the magnetosheath and the solar wind are compared with previously established simulation results. Consistent with the simulations, we find that at large (MHD) scales, the MMS observations exhibit a clear inertial range dominated by the MHD flux. In the subion range, the cascade continues at a diminished level via the Hall term, and the change becomes more pronounced as the plasma beta increases. Additionally, the MHD contribution to interscale energy transfer remains important at smaller scales than previously thought. Possible reasons are offered for this unanticipated result.
We derive the exact relation for the energy transfer in three-dimensional compressible two-fluid plasma turbulence. In the long-time limit, we obtain an exact law which expresses the scale-to-scale average energy flux rate in terms of two point increments of the fluid variables of each species, electric and magnetic field and current density, and puts a strong constraint on the turbulent dynamics. The incompressible single fluid and two-fluid limits and the compressible single fluid limit are recovered under appropriate assumption. In the single fluid limits, analyses are done with and without neglecting the electron mass thereby making the exact relation suitable for a broader range of application. In the compressible two-fluid regime, the total energy flux rate, unlike the single fluid case, is found to be unaltered by the presence of a background magnetic field. The exact relation provides a way to test whether a range of scales in a plasma is inertial or dissipative and is essential to understand the nonlinear nature of both space and dilute astrophysical plasmas.
The first complete estimation of the compressible energy cascade rate |ϵ_{C}| at magnetohydrodynamic (MHD) and subion scales is obtained in Earth's magnetosheath using Magnetospheric MultiScale spacecraft data and an exact law derived recently for compressible Hall MHD turbulence. A multispacecraft technique is used to compute the velocity and magnetic gradients, and then all the correlation functions involved in the exact relation. It is shown that when the density fluctuations are relatively small, |ϵ_{C}| identifies well with its incompressible analog |ϵ_{I}| at MHD scales but becomes much larger than |ϵ_{I}| at subion scales. For larger density fluctuations, |ϵ_{C}| is larger than |ϵ_{I}| at every scale with a value significantly higher than for smaller density fluctuations. Our study reveals also that for both small and large density fluctuations, the nonflux terms remain always negligible with respect to the flux terms and that the major contribution to |ϵ_{C}| at subion scales comes from the compressible Hall flux.
We present a detailed guide to advanced collisionless fluid models that incorporate
kinetic effects into the fluid framework, and that are much closer to the collisionless
kinetic description than traditional magnetohydrodynamics. Such fluid models are
directly applicable to modelling the turbulent evolution of a vast array of astrophysical
plasmas, such as the solar corona and the solar wind, the interstellar medium, as well
as accretion disks and galaxy clusters. The text can be viewed as a detailed guide
to Landau fluid models and it is divided into two parts. Part 1 is dedicated to fluid
models that are obtained by closing the fluid hierarchy with simple (non-Landau
fluid) closures. Part 2 is dedicated to Landau fluid closures. Here in Part 1, we
discuss the fluid model of Chew–Goldberger–Low (CGL) in great detail, together
with fluid models that contain dispersive effects introduced by the Hall term and by
the finite Larmor radius corrections to the pressure tensor. We consider dispersive
effects introduced by the non-gyrotropic heat flux vectors. We investigate the parallel
and oblique firehose instability, and show that the non-gyrotropic heat flux strongly
influences the maximum growth rate of these instabilities. Furthermore, we discuss
fluid models that contain evolution equations for the gyrotropic heat flux fluctuations
and that are closed at the fourth-moment level by prescribing a specific form for the
distribution function. For the bi-Maxwellian distribution, such a closure is known as
the ‘normal’ closure. We also discuss a fluid closure for the bi-kappa distribution.
Finally, by considering one-dimensional Maxwellian fluid closures at higher-order
moments, we show that such fluid models are always unstable. The last possible non
Landau fluid closure is therefore the ‘normal’ closure, and beyond the fourth-order
moment, Landau fluid closures are required.
A comparison is made between several existing exact laws in incompressible Hall magnetohydrodynamic turbulence in order to show their equivalence, despite stemming from different mathematical derivations. Using statistical homogeneity, we revisit the law proposed by Hellinger et al. and show that it can be written, after being corrected by a multiplicative factor, in a more compact form implying only flux terms expressed as increments of the turbulent fields. The Hall contribution of this law is tested and compared to other exact laws derived by Galtier and Banerjee & Galtier using direct numerical simulations of three-dimensional electron MHD turbulence with a moderate mean magnetic field. We show that the studied laws are equivalent in the inertial range, thereby offering several choices on the formulation to use depending on the needs. The expressions that depend explicitly on a mean (guide) field may lead to residual errors in estimating the energy cascade rate; however, we demonstrate that this guide field can be removed from these laws after mathematical manipulation. Therefore, it is recommended to use an expression independent of the mean guide field to analyze numerical or in situ spacecraft data.
The description of the local turbulent energy transfer and the high-resolution ion distributions measured by the Magnetospheric Multiscale mission together provide a formidable tool to explore the cross-scale connection between the fluid-scale energy cascade and plasma processes at subion scales. When the small-scale energy transfer is dominated by Alfvénic, correlated velocity, and magnetic field fluctuations, beams of accelerated particles are more likely observed. Here, for the first time, we report observations suggesting the nonlinear wave-particle interaction as one possible mechanism for the energy dissipation in space plasmas.
A dynamical vectorial equation for homogeneous incompressible Hall-MHD turbulence together with the exact scaling law for third-order correlation tensors, analogous to that for the incompressible MHD, is rederived and applied to the results of two-dimensional hybrid simulations of plasma turbulence. At large (MHD) scales the simulations exhibits a clear inertial range where the MHD dynamic law is valid. In the sub-ion range the cascade continues via the Hall term but the dynamic law derived in the framework of incompressible Hall MHD equations is obtained only in a low plasma beta simulation. For a higher beta plasma the cascade rate decreases in the sub-ion range and the change becomes more pronounced as the plasma beta increases. This break in the cascade flux can be ascribed to non thermal (kinetic) features or to others terms in the dynamical equation that are not included in the Hall-MHD incompressible approximation.
Three-dimensional direct numerical simulations are used to study the energy cascade rate in isothermal compressible magnetohydrodynamic turbulence. Our analysis is guided by a two-point exact law derived recently for this problem in which flux, source, hybrid, and mixed terms are present. The relative importance of each term is studied for different initial subsonic Mach numbers $M_S$ and different magnetic guide fields ${\bf B}_0$. The dominant contribution to the energy cascade rate comes from the compressible flux, which depends weakly on the magnetic guide field ${\bf B}_0$, unlike the other terms whose modulus increase significantly with $M_S$ and ${\bf B}_0$. In particular, for strong ${\bf B}_0$ the source and hybrid terms are dominant at small scales with almost the same amplitude but with a different sign. A statistical analysis made with an isotropic decomposition based on the SO(3) rotation group is shown to generate spurious results in presence of ${\bf B}_0$, when compared with an axisymmetric decomposition better suited to the geometry of the problem. Our numerical results are eventually compared with previous analyses made with in-situ measurements in the solar wind and the terrestrial magnetosheath.
The first estimation of the energy cascade rate ${|\epsilon_C|}$ of magnetosheath turbulence is obtained using the CLUSTER and THEMIS spacecraft data and an exact law of compressible isothermal magnetohydrodynamics turbulence. ${|\epsilon_C|}$ is found to be of the order of ${10{^{-13}} J.m^{3}.s^{-1}}$, at least three orders of magnitude larger than its value in the solar wind. Two types of turbulence are evidenced and shown to be dominated either by incompressible Alfv\'enic or magnetosonic-like fluctuations. Density fluctuations are shown to amplify the cascade rate and its spatial anisotropy in comparison with incompressible Alfv\'enic turbulence. Furthermore, for compressible magnetosonic fluctuations, large cascade rates are found to lie mostly near the linear kinetic instability of the mirror mode. New empirical power-laws are evidenced and relate ${|\epsilon_C|}$ to the turbulent Mach number and the internal energy. These new finding have potential applications in distant astrophysical plasmas that are not accessible to in situ measurements.
We derive the exact law for three-dimensional (3D) homogeneous compressible isothermal Hall magnetohydrodynamics (CHMHD) turbulence, without the assumption of isotropy. The Hall current is shown to introduce new flux and sources terms that act at the small scales (comparable or smaller than the ion skin depth) to significantly impact the turbulence dynamics. The new law provides an accurate means to estimate for the first time the energy cascade rate over a broad range of scales covering the MHD inertial range and the sub-ion dispersive range in 3D numerical simulations and {\it in situ} spacecraft observations of compressible turbulence. This work is particularly relevant to astrophysical flows in which small scale density fluctuations cannot be ignored such as the solar wind, planetary magnetospheres and the interstellar medium (ISM).
The exact law for fully developed homogeneous compressible magnetohydrodynamics (CMHD) turbulence is derived. For an isothermal plasma, without the assumption of isotropy, the exact law is expressed as a function of the plasma velocity field, the compressible Alfv\'en velocity and the scalar density, instead of the Els\"asser variables used in previous works. The theoretical results show four different types of terms that are involved in the nonlinear cascade of the total energy in the inertial range. Each category is examined in detail, in particular those that can be written either as source or flux terms. Finally, the role of the background magnetic field $B_0$ is highlighted and comparison with the incompressible MHD (IMHD) model is discussed. This point is particularly important when testing the new exact law on numerical simulations and in situ observations in space plasmas.
Estimation of the energy cascade rate in the inertial range of solar wind turbulence has been done so far mostly within the incompressible magnetohydrodynamics (MHD) theory. Here, we go beyond that approximation to include plasma compressibility using a reduced form of a recently derived exact law for compressible, isothermal MHD turbulence. Using in-situ data from the THEMIS/ARTEMIS spacecraft in the fast and slow solar wind, we investigate in detail the role of the compressible fluctuations in modifying the energy cascade rate with respect to the prediction of the incompressible MHD model. In particular, we found that the energy cascade rate: i) is amplified particularly in the slow solar wind; ii) exhibits weaker fluctuations in spatial scales, which leads to a broader inertial range than the previous reported ones; iii) has a power law scaling with the turbulent Mach number; iv) has a lower level of spatial anisotropy. Other features of solar wind turbulence are discussed along with their comparison with previous studies that used incompressible or heuristic (non exact) compressible MHD models.
We propose an alternative formulation for the exact relations in three-dimensional homogeneous turbulence using two-point statistics. Our finding is illustrated with incompressible hydrodynamic, standard and Hall magnetohydrodynamic turbulence. In this formulation, the cascade rate of an inviscid invariant of turbulence can be expressed simply in terms of mixed second-order structure functions. Besides the usual variables like the velocity ${\bf u}$, vorticity $\omega$, magnetic field ${\bf b}$ and the current ${\bf j}$, the vectors ${\bf u } \times {\boldsymbol \omega}$, ${\bf u} \times {\bf b}$ and ${\bf j} \times {\bf b}$ are also found to play a key role in the turbulent cascades. The current methodology offers a simple algebraic form which is specially interesting to study anisotropic space plasmas like the solar wind, with in principle a faster statistical convergence than the classical laws written in terms of third-order correlators.
The first hybrid-kinetic numerical simulations of the firehose and mirror
instabilities in a collisionless plasma are performed in which pressure
anisotropy is driven as the magnetic field is changed by a persistent linear
shear S. For a decreasing field, it is found that mostly oblique firehose
fluctuations grow at Larmor scales and saturate with energies ~S^{1/2}; the
pressure anisotropy is pinned at the stability threshold by anomalous particle
scattering. In contrast, nonlinear mirror fluctuations are large compared to
the Larmor scale and grow secularly in time; marginality is maintained by an
increasing population of resonant particles trapped in magnetic mirrors. After
one shear time, saturated order-unity magnetic mirrors are formed and particles
scatter off their sharp edges. Both instabilities drive sub-Larmor-scale
fluctuations, which appear to be kinetic-Alfven-wave turbulence. Our results
impact theories of momentum and heat transport in astrophysical and space
plasmas, in which the stretching of a magnetic field by shear is a generic
process.
The first observed connection between kinetic instabilities driven by proton
temperature anisotropy and estimated energy cascade rates in the turbulent
solar wind is reported using measurements from the Wind spacecraft at 1 AU. We
find enhanced cascade rates are concentrated along the boundaries of the
($\beta_{\parallel}$, $T_{\perp}/T_{\parallel}$)-plane, which includes regions
theoretically unstable to the mirror and firehose instabilities. A strong
correlation is observed between the estimated cascade rate and kinetic effects
such as temperature anisotropy and plasma heating, resulting in protons 5-6
times hotter and 70-90% more anisotropic than under typical isotropic plasma
conditions. These results offer new insights into kinetic processes in a
turbulent regime.
Compressible isothermal magnetohydrodynamic turbulence is analyzed under the
assumption of statistical homogeneity and in the asymptotic limit of large
kinetic and magnetic Reynolds numbers. Following Kolmogorov we derive an exact
relation for some two-point correlation functions which generalizes the
expression recently found for hydrodynamics. We show that the magnetic field
brings new source and flux terms into the dynamics which may act on the
inertial range similarly as a source or a sink for the mean energy transfer
rate. The introduction of a uniform magnetic field simplifies significantly the
exact relation for which a simple phenomenology may be given. A prediction for
axisymmetric energy spectra is eventually proposed.
An expression for the internal energy of a fluid element in a weakly coupled,
magnetized, anisotropic plasma is derived from first principles. The result is
a function of entropy, particle density and magnetic field, and as such plays
the role of a thermodynamic potential: it determines in principle all
thermodynamic properties of the fluid element. In particular it provides
equations of state for the magnetized plasma. The derivation uses familiar
fluid equations, a few elements of kinetic theory, the MHD version of Faraday's
law, and certain familiar stability and regularity conditions.
Cambridge Core - Nonlinear Science and Fluid Dynamics - Turbulence - by Uriel Frisch
1] The proton resonant firehose instability may arise in collisionless plasmas in which the proton velocity distribution is approximately bi-Maxwellian with T kp /T ?p > 1, where ? and k denote directions relative to the background magnetic field B ° . Linear theory and one-dimensional simulations predict that enhanced field fluctuations from the proton resonant firehose instability impose a constraint on proton temperature anisotropies of the form 1 À T ?p /T kp = S p /b kp ap where b kp 8pn p k B T kp /B ° 2 , and the fitting parameters S p $ 1 and a p ' 0.7. Observations from the Wind spacecraft are reported here. These measurements show for the first time with a comprehensive plasma and magnetic field data set that this constraint is statistically satisfied in the solar wind near 1 AU, with best-fit values of S p = 1.21 ± 0.26 and a p = 0.76 ± 0.14.
The first direct determination of the inertial range energy cascade rate, using an anisotropic form of Yaglom's law for magnetohydrodynamic turbulence, is obtained in the solar wind with multispacecraft measurements. The two-point mixed third-order structure functions of Elsässer fluctuations are integrated over a sphere in magnetic field-aligned coordinates, and the result is consistent with a linear scaling. Therefore, volume integrated heating and cascade rates are obtained that, unlike previous studies, make only limited assumptions about the underlying spectral geometry of solar wind turbulence. These results confirm the turbulent nature of magnetic and velocity field fluctuations in the low frequency limit, and could supply the energy necessary to account for the nonadiabatic heating of the solar wind.
Compressible isothermal turbulence is analyzed under the assumption of homogeneity and in the asymptotic limit of a high Reynolds number. An exact relation is derived for some two-point correlation functions which reveals a fundamental difference with the incompressible case. The main difference resides in the presence of a new type of term which acts on the inertial range similarly as a source or a sink for the mean energy transfer rate. When isotropy is assumed, compressible turbulence may be described by the relation -2/3ε(eff)r = F(r)(r), where F(r) is the radial component of the two-point correlation functions and ε(eff) is an effective mean total energy injection rate. By dimensional arguments, we predict that a spectrum in k(-5/3) may still be preserved at small scales if the density-weighted fluid velocity ρ(1/3)u is used.
The proton temperature anisotropy in the solar wind is known to be constrained by the theoretical thresholds for pressure-anisotropy-driven instabilities. Here, we use approximately 1x10;{6} independent measurements of gyroscale magnetic fluctuations in the solar wind to show for the first time that these fluctuations are enhanced along the temperature anisotropy thresholds of the mirror, proton oblique firehose, and ion cyclotron instabilities. In addition, the measured magnetic compressibility is enhanced at high plasma beta (beta_{ parallel} greater, similar1) along the mirror instability threshold but small elsewhere, consistent with expectations of the mirror mode. We also show that the short wavelength magnetic fluctuation power is a strong function of collisionality, which relaxes the temperature anisotropy away from the instability conditions and reduces correspondingly the fluctuation power.
Temperature or pressure anisotropies are characteristic of space plasmas, standard magnetohydrodynamic (MHD) model for describing large-scale plasma phenomena however usually assumes isotropic pressure. In this paper we examine the characteristics of MHD waves, fire-hose and mirror instabilities in anisotropic homogeneous magnetized plasmas. The model equations are a set of gyrotropic MHD equations closed by the generalized Chew-Goldberger-Low (CGL) laws with two polytropic exponents representing various thermodynamic conditions. Both ions and electrons are allowed to have separate plasma beta, pressure anisotropy and energy equations. The properties of linear MHD waves and instability criteria are examined and numerical examples for the nonlinear evolutions of slow waves, fire-hose and mirror instabilities are shown. One significant result is that slow waves may develop not only mirror instability but also a new type of compressible fire-hose instability. Their corresponding nonlinear structures thus may exhibit anticorrelated density and magnetic field perturbations, a property used for identifying slow and mirror mode structures in the space plasma environment. The conditions for nonlinear saturation of both fire-hose and mirror instabilities are examined.
Various forms of exact laws governing magnetohydrodynamic (MHD) turbulence have been derived either in the incompressibility limit, or for isothermal compressible flows. Here we propose a more general method that allows us to obtain such laws for any turbulent isentropic flow (i.e., constant entropy). We demonstrate that the known MHD exact laws (incompressible and isothermal) and the new (polytropic) one can be obtained as specific cases of the general law when the corresponding closure equation is stated. We also recover all known exact laws of hydrodynamic (HD) turbulence (incompressible, isothermal, and polytropic) from this law in the limit B = 0. We furthermore show that the difference between the two forms (isothermal and polytropic) of the MHD exact laws of interest in this work resides in some of the source terms and in the explicit form of the flux term that depends on internal energy. Finally, we apply these two forms to Parker Solar Probe data taken in the inner heliosphere to highlight how the different closure equations affect the energy cascade rate estimates.
The properties of turbulence observed within the plasma originating from the magnetosheath and the magnetospheric boundary layer, which have been entrained within vortices driven by the Kelvin-Helmholtz Instability (KHI), are compared. The goal of such a study is to determine similarities and differences between the two different regions. In particular, we study spectra, intermittency and the third-order moment scaling, as well as the distribution of a local energy transfer rate proxy. The analysis is performed using the Magnetospheric Multiscale (MMS) data from a single satellite that crosses longitudinally the KHI. Two sets of regions, one set containing predominantly magnetosheath plasma and the other containing predominantly magnetospheric plasma, are analyzed separately, thus allowing us to explore turbulence properties in two portions of very different plasma samples. Results show that the turbulence in the two regions is different, with the boundary layer plasma including current structures that may not be originated by the turbulent cascade. This suggests that the observed turbulence is affected by the KHI.
The role of supersonic turbulence in structuring the interstellar medium (ISM) remains an unsettled question. Here, this problem is investigated using a new exact law of compressible isothermal hydrodynamic turbulence, which involves two-point correlations in physical space. The new law is shown to have a compact expression that contains a single flux term reminiscent of the incompressible case and a source term with a simple expression whose sign is given by the divergence of the velocity. The law is then used to investigate the properties of such a turbulence at integral Mach number 4 produced by a massive numerical simulation with a grid resolution of points. The flux (resp. source) term was found to have positive (resp. negative) contribution to the total energy cascade rate, which is interpreted as a direct cascade amplified by compression, while their sum is constant in the inertial range. Using a local (in space) analysis it is shown that the source is mainly driven by filamentary structures in which the flux is negligible. Taking positive defined correlations reveals the existence of different turbulent regimes separated by the sonic scale, which determines the scale over which the nonnegligible source modifies the scaling of the flux. Our study provides new insight into the dynamics and structures of supersonic interstellar turbulence.
Ninety-nine percent of ordinary matter in the Universe is in the form of ionized fluids, or plasmas. The study of the magnetic properties of such electrically conducting fluids, magnetohydrodynamics (MHD), has become a central theory in astrophysics, as well as in areas such as engineering and geophysics. This textbook offers a comprehensive introduction to MHD and its recent applications, in nature and in laboratory plasmas; from the machinery of the Sun and galaxies, to the cooling of nuclear reactors and the geodynamo. It exposes advanced undergraduate and graduate students to both classical and modern concepts, making them aware of current research and the ever-widening scope of MHD. Rigorous derivations within the text, supplemented by over 100 illustrations and followed by exercises and worked solutions at the end of each chapter, provide an engaging and practical introduction to the subject and an accessible route into this wide-ranging field.
Three-dimensional, compressible, magnetohydrodynamic turbulence of an isothermal, self-gravitating fluid is analyzed using two-point statistics in the asymptotic limit of large Reynolds numbers (both kinetic and magnetic). Following an alternative formulation proposed by S. Banerjee and S. Galtier (Phys. Rev. E,93, 033120, 2016) and S. Banerjee and S. Galtier (J. Phys. A, Math. and Theor.,50, 015501, 2017), an exact relation has been derived for the total energy transfer. This approach results in a simpler relation expressed entirely in terms of mixed second-order structure functions. The kinetic, thermodynamic, magnetic and gravitational contributions to the energy transfer rate can be easily separated in the present form. By construction, the new formalism includes such additional effects as global rotation, the Hall term in the induction equation, etc. The analysis shows that solid-body rotation cannot alter the energy flux rate of compressible turbulence. However, the contribution of a uniform background magnetic field to the flux is shown to be non-trivial unlike in the incompressible case. Finally, the compressible, turbulent energy flux rate does not vanish completely due to simple alignments, which leads to a zero turbulent energy flux rate in the incompressible case.
Self-gravitating isothermal supersonic turbulence is analyzed in the asymptotic limit of large Reynolds numbers. Based on the inviscid invariance of total energy, an exact relation is derived for homogeneous, (not necessarily isotropic) turbulence. A modified definition for the two-point energy correlation functions is used to comply with the requirement of detailed energy equipartition in the acoustic limit. In contrast to the previous relations (Galtier and Banerjee, Phys. Rev. Lett., 107, 134501, 2011; Banerjee and Galtier, Phys. Rev. E, 87, 013019, 2013), the current exact relation shows that the pressure dilatation terms plays practically no role in the energy cascade. Both the flux and source terms are written in terms of two-point differences. Sources enter the relation in a form of mixed second-order structure functions. Unlike kinetic and thermodynamic potential energy, gravitational contribution is absent from the flux term. An estimate shows that for the isotropic case, the correlation between density and gravitational acceleration may play an important role in modifying the energy transfer in self-gravitating turbulence. The exact relation is also written in an alternative form in terms of two-point correlation functions, which is then used to describe scale-by-scale energy budget in spectral space.
The role of compressible fluctuations in the energy cascade of fast solar wind turbulence is studied using a reduced form of an exact law derived recently (Banerjee and Galtier, PRE, 2013) for compressible isothermal magnetohydrodynamics and in-situ observations from the THEMIS B/ARTEMIS P1 spacecraft. A statistical survey of the data revealed a turbulent energy cascade over two decades of scales, which is broader than the previous estimates made from an exact incompressible law. A term-by-term analysis of the compressible model reveals new insight into the role played by the compressible fluctuations in the energy cascade. The compressible fluctuations are shown to amplify (2 to 4 times) the turbulent cascade rate with respect to the incompressible model in 10 % of the analyzed samples. This new estimated cascade rate is shown to provide the adequate energy dissipation required.
We derive exact scaling laws for a three-dimensional incompressible helical two-fluid plasma, without the assumption of isotropy. For each ideal invariant of the two-fluid model, i.e. the total energy, the electron helicity and the proton helicity, we derive simple scaling laws in terms of two-point increments correlation functions expressed in terms of the velocity field of each species and the magnetic field. These variables are appropriate for comparison with in-situ measurements in the solar wind at different spatial ranges and data from numerical simulations. Finally, with the exact scaling laws and dimensional analysis we predict the magnetic energy and electron helicity spectra for different ranges of scales.
We derive the von K\'arm\'an-Howarth equation for a full three dimensional incompressible two-fluid plasma. In the long-time limit and for very large Reynolds numbers we obtain the equivalent of the hydrodynamic "four-fifth" law. This exact law predicts the scaling of the third-order two-point correlation functions, and puts a strong constraint on the plasma turbulent dynamics. Finally, we derive a simple expression for the 4/5 law in terms of third-order structure functions, which is appropriate for comparison with in-situ measurements in the solar wind at different spatial ranges.
The idea of devoting a complete book to this topic was born at one of the Workshops on Nonlinear and Turbulent Processes in Physics taking place reg ularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors of this volume were participants at the third of these workshops. All of them were acquainted with each other and with each other's work. Yet it seemed to be somewhat of a discovery that all of them were and are trying to understand the same problem - the problem of integrability of dynamical systems, primarily Hamiltonian ones with an infinite number of degrees of freedom. No doubt that they (or to be more exact, we) were led to this by the logical process of scientific evolution which often leads to independent, almost simultaneous discoveries. Integrable, or, more accurately, exactly solvable equations are essential to theoretical and mathematical physics. One could say that they constitute the "mathematical nucleus" of theoretical physics whose goal is to describe real clas sical or quantum systems. For example, the kinetic gas theory may be considered to be a theory of a system which is trivially integrable: the system of classical noninteracting particles. One of the main tasks of quantum electrodynamics is the development of a theory of an integrable perturbed quantum system, namely, noninteracting electromagnetic and electron-positron fields.
In my note (Kolmogorov 1941 a ) I defined the notion of local isotropy and introduced the quantities B d d ( r ) = [ u d ( M ′ ) − u d ( M ) ] 2 , ¯ [ u n ( M ′ ) − u n ( M ) ¯ ] 2 , where r denotes the distance between the points M and M' , u d (M) and u d (M') are the velocity components in the direction MM' ¯¯ at the points M and M' , and u n (M) and u n (M') are the velocity components at the points M and M' in some direction, perpendicular to MM' .
§1. We shall denote by u α ( P ) = u α ( x 1 , x 2 , x 3 , t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 , x 2 , x 3 . In considering the turbulence it is natural to assume the components of the velocity u α ( P ) at every point P = ( x 1 , x 2 , x 3 , t ) of the considered domain G of the four-dimensional space ( x 1 , x 2 , x 3 , t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ ² α and (d u α /d x β ) ² ― are finite and bounded in every bounded subdomain of the domain G .
Compressible hydrodynamic turbulence is studied under the assumption of a polytropic closure. Following Kolmogorov, we derive an exact relation for some two-point correlation functions in the asymptotic limit of a high Reynolds number. The inertial range is characterized by: (i) a flux term implying in particular the enthalpy; and (ii) a purely compressible term l which may act as a source or a sink for the mean energy transfer rate. At subsonic scales, we predict dimensionally that the isotropic k(-513) energy spectrum for the density-weighted velocity field (rho(1/3)nu), previously obtained for isothermal turbulence, is modified by a polytropic contribution, whereas at supersonic scales 9 may impose another scaling depending on the polytropic index. In both cases, it is shown that the fluctuating sound speed is a key ingredient for understanding polytropic compressible turbulence.
Starting from the Boltzmann equation for a completely ionized dilute gas with no interparticle collision term but a strong Lorentz force, an attempt is made to obtain one-fluid hydromagnetic equations by expanding in the ion mass to charge ratio. It is shown that the electron degrees of freedom can be replaced by a macroscopic current, but true hydrodynamics still does not result unless some special circumstance suppresses the transport of pressure along magnetic lines of force. If the longitudinal transport of pressure is ignored, a set of self-contained one-fluid hydromagnetic equations can be found even though the pressure is not a scalar.
The electromagnetic proton cyclotron anisotropy instability may arise in collisionless plasmas in which the proton velocity distribution is approximately bi-Maxwellian with T⊥p/T||p>1, where ⊥ and || denote directions relative to the background magnetic field Bo. Theory and simulations predict that enhanced field fluctuations from this instability impose a constraint on proton temperature anisotropies of the form T⊥p/T||p-1=Sp/beta||palphap where beta||p≡8pinpkBT||p/Bo2, and the fitting parameters Sp
We derive two symmetric global scaling laws for third-order structure functions of magnetized fluids under the assumptions of full isotropy, homogeneity and incompressibility. The compatibility with previous laws involving both structure and correlation functions of only the longitudinal components of the fields is demonstrated. These new laws provide a better set of functions with which one can determine intermittency scaling of MHD turbulence, as in the Solar Wind.
A derivation in variable dimension of the scaling laws for mixed third-order longitudinal structure and correlation functions for incompressible magnetized flows is given for arbitrary correlation between the velocity and magnetic field with full isotropy, homogeneity, and incompressibility assumed. When close to equipartition between kinetic and magnetic energy, the scaling relations involve only structure functions in a manner similar to the ``45 law'' of Kolmogorov.
Measurements of parallel and perpendicular ion temperatures in the Large Experiment on Instabilities and Anisotropies (LEIA) space simulation chamber display an inverse correlation between the upper bound on the ion temperature anisotropy and the parallel ion beta (β = 8πnkT/B2). Fluctuation measurements indicate the presence of low frequency, transverse, electromagnetic waves with wave numbers and frequencies that are consistent with predictions for Alfvén Ion Cyclotron instabilities. These observations are also consistent with in situ spacecraft measurements in the Earth’s magnetosheath and with a theoretical/computational model that predicts that such an upper bound on the ion temperature anisotropy is imposed by scattering from enhanced fluctuations due to growth of the Alfvén ion cyclotron instability. © 2000 American Institute of Physics.
Anisotropic magnetohydrodynamics equations, which also capture the dynamics of quasi-transverse small scales obeying the gyrokinetic ordering, are derived using fourth-rank moment closures, based on a refined description of linear Landau damping and finite Larmor radius (FLR) corrections. This “FLR-Landau fluid model” reproduces the dispersion relation of low-frequency waves, up to scales that, in the case of quasi-transverse kinetic Alfvén waves, can be much smaller than the ion gyroradius. The mirror instability, which requires temperature anisotropy, is also captured, together with its quenching at small scales. This model that accurately reproduces the collisionless dissipation of low-frequency modes, should provide an efficient tool to simulate mesoscale turbulence in a magnetized collisionless plasma.
The relation, first written by Kolmogorov, between the
third-order moment of the longitudinal velocity increment
δu1 and the second-order moment of
δu1 is presented in a slightly more general
form relating the mean value of the product
δu1(δui)
2, where (δui)
2 is the sum of the square of the three velocity
increments, to the second-order moment of
δui. In this form, the relation is
similar to that derived by Yaglom for the mean value of the product
δu1(δθ)2, where
(δθ)2 is the square of the temperature
increment. Both equations reduce to a ‘four-thirds’
relation for inertial-range separations and differ only through the
appearance of the molecular Prandtl number for very small separations.
These results are confirmed by experiments in a turbulent wake, albeit
at relatively small values of the turbulence Reynolds number.
Hall magnetohydrodynamics (HMHD) is a mono-fluid approximation extending the validity domain of the ordinary MHD system to spatial scales down to a fraction of the ion skin depth or frequencies comparable to the ion gyrofrequency. In the paper by Galtier (2006 J. Plasma Physics), an incompressible limit of the HMHD system is used for developing a wave turbulence theory. Nevertheless, the possibility and the consequences of such an approximation are different in HMHD and in MHD. Here, we analyse these differences by investigating the properties of the HMHD equations in the incompressible limit: the existence of linear modes, their dispersion relations and polarizations. We discuss the possibility of replacing the fluid closure equation of a complete HMHD system by an incompressibility hypothesis and determine the validity range.
Solar wind
measurements at 1 AU during the recent solar minimum and previous studies of solar maximum provide an opportunity to study the effects of the changing solar cycle on in situ heating. Our interest is to compare the levels of activity associated with turbulence and proton heating. Large-scale shears in the flow caused by transient activity are a source that drives turbulence that heats the solar wind, but as the solar cycle progresses the dynamics that drive the turbulence and heat the medium are likely to change. The application of third-moment theory to Advanced Composition Explorer (ACE) data gives the turbulent energy cascade rate which is not seen to vary with the solar cycle. Likewise, an empirical heating rate shows no significan changes in proton heating over the cycle.
We investigate the nature of turbulent magnetic dissipation in the solar wind. We employ a database describing the spectra of over 800 intervals of interplanetary magnetic field and solar wind measurements recorded by the ACE spacecraft at 1 AU. We focus on the spectral properties of the dissipation range that forms at spacecraft frequencies ≥0.3 Hz and show that while the inertial range at lower frequencies displays a tightly constrained range of spectral indexes, the dissipation range exhibits a broad range of power-law indexes. We show that the explanation for this variation lies with the dependence of the dissipation range spectrum on the rate of energy cascade through the inertial range such that steeper spectral forms result from greater cascade rates.
We perform a test of MHD turbulent cascade theory in the solar wind and directly evaluate the contribution of local turbulence to heating the solar wind at 1 AU. We look at turbulent fluctuations in the solar wind velocity V and magnetic field B, using the vector Elsasser variables Z ± ≡ V ± B/(4πρ)1/2 as measured at the ACE spacecraft stationed at the Earth's L1 point. We combine the fluctuations δZ± over time lags in the inertial range, from 64 s to several hours, to form components of the mixed vector third moments, and we adopt the work of Politano & Pouquet, who derive an exact scaling law, similar to the Kolmogorov 4/5 law, but valid in anisotropic MHD turbulence, for these components. We demonstrate that the scaling is reasonably linear, as is expected for the inertial range. The total turbulent energy injection/dissipation rate that we derive this way agrees with the in situ heating of the solar wind that is inferred from the temperature gradient, whereas methods using the power spectra only seldom agree with the heating rates derived from gradients of the thermal proton distribution. We derive expressions of the third-order moments that are applicable to the spectral cascades parallel and perpendicular to the mean magnetic field. We apply these expressions to fast- and slow-wind subsets of the data, with additional subsetting for mean field direction. We find that both the fast wind and the slow wind exhibit an active energy cascade over the inertial range scales. Furthermore, we find that the energy flux in the parallel cascade is consistently smaller than in the perpendicular cascade.
We present a comparison between WIND/SWE observations [Kasper et al., 2006] of β p and T ⊥p /T p (where β p is the proton parallel beta and T ⊥p and T p are the perpendicular and parallel proton temperatures, respectively; here parallel and perpendicular indicate directions with respect to the ambient magnetic field) and predictions of the Vlasov linear theory. In the slow solar wind, the observed proton temperature anisotropy seems to be constrained by oblique instabilities, by the mirror one and the oblique fire hose, contrary to the results of the linear theory which predicts a domi-nance of the proton cyclotron instability and the parallel fire hose. The fast solar wind core protons exhibit an anticorrelation between β c and T ⊥c /T c (where β c is the core proton parallel beta and T ⊥c and T c are the perpendicular and parallel core proton temper-atures, respectively) similar to that observed in the HELIOS data [Marsch et al., 2004].
Beyond about 0.1 a.u. from the Sun, the density of the solar wind is so low that Coulomb collisions cannot keep the velocity distribution isotropic and Maxwellian. Analysis of the particle dynamics in an ideal spiral interplanetary magnetic field shows that if there are no collisions beyond 0.1 a.u., then at the Earth the temperature of the wind is very anisotropic, with T∥/T⊥ = 35; this is much greater than is observed. To study the effects of interactions with magnetic irregularities we solved numerically Boltzmann's equation with Krook's collision term; this shows that the anisotropy observed by the Vela satellite requires each particle to make an average of 2 or 3 collisions between 0.1 and 1 a.u. The temperature averaged over direction roughly follows an adiabatic law, with tends to increase with distance. The theory predicts an excess of high-velocity particles, as is observedby Vela, even when the collision frequency is independent of velocity, but to produce an effect as strong as that observed requires a fairly strong velocity-dependence of the collision frequency.