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Impact of pressure anisotropy on the cascade rate of Hall-MHD turbulence with bi-adiabatic ions
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Abstract
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.
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The third-order law links energy transfer rates in the inertial range of magneto- hydrodynamic (MHD) turbulence with third-order structure functions. Anisotropy, a typical property in the solar wind, challenges the applicability of the third-order law with the isotropic assumption. To shed light on the energy transfer process in the presence of anisotropy, we conducted direct numerical simulations of forced MHD turbulence with normal and hyper-viscosity under various strengths of the external magnetic field ( $B_0$ ), and calculated three forms of third-order structure function with or without averaging of the azimuthal or polar angles with respect to $B_0$ direction. Correspondingly, three estimated energy transfer rates were obtained. The result shows that the peak of normalized third-order structure function occurs at larger scales closer to the $B_0$ direction, and the maximum of longitudinal transfer rates shifts away from the $B_0$ direction at larger $B_0$ . Compared with normal viscous cases, hyper-viscous cases can attain better separated inertial range, thus facilitating the estimation of the energy cascade rates. We find that the widespread use of the isotropic form of the third-order law in estimating the energy transfer rates is questionable in some cases, especially when the anisotropy arising from the mean magnetic field is inevitable. In contrast, the direction-averaged third-order structure function properly accounts for the effect of anisotropy and predicts the energy transfer rates and inertial range accurately, even at very high $B_0$ . With limited statistics, the third-order structure function shows a stronger dependence on averaging of azimuthal angles than the time, especially for high $B_0$ cases. These findings provide insights into the anisotropic effect on the estimation of energy transfer rates.
We investigate properties of solar-wind-like plasma turbulence using direct numerical simulations. We analyze the transition from large, magnetohydrodynamic (MHD) scales to the ion characteristic ones using two-dimensional hybrid (fluid electrons and kinetic ions) simulations. To capture and quantify turbulence properties, we apply the Karman–Howarth–Monin (KHM) equation for compressible Hall–MHD (extended by considering the plasma pressure as a tensor quantity) to the numerical results. The KHM analysis indicates that the transition from MHD to ion scales (the so-called ion break in the power spectrum) results from a combination of an onset of Hall physics and an effective dissipation owing to the pressure–strain energy-exchange channel and resistivity. We discuss the simulation results in the context of the solar wind.
Various exact laws governing compressible magnetohydrodynamic (MHD) and compressible Hall-MHD (CHMHD) turbulence have been derived in recent years. Other than their fundamental theoretical interest, these laws are generally used to estimate the energy dissipation rate from spacecraft observations in order to address diverse problems related, e.g., to heating of the solar wind and magnetospheric plasmas. Here we use various 1024 ³ direct numerical simulation data of free-decay isothermal CHMHD turbulence obtained with the GHOST code (Geophysical High-Order Suite for Turbulence) to analyze two of the recently derived exact laws. The simulations reflect different intensities of the initial Mach number and the background magnetic field. The analysis demonstrates the equivalence of the two laws in the inertial range and relates the strength of the Hall effect to the amplitude of the cascade rate at sub-ion scales. When taken in their general form (i.e., not limited to the inertial range), some subtleties regarding the validity of the stationarity assumption or the absence of the forcing in the simulations are discussed. We show that the free-decay nature of the turbulence induces a shift from a large-scale forcing toward the presence of a scale-dependent reservoir of energy fueling the cascade or dissipation. The reduced form of the exact laws (valid in the inertial range) ultimately holds even if the stationarity assumption is not fully verified.
We derive two new forms of the Kármán–Howarth–Monin (KHM) equation for decaying compressible Hall magnetohydrodynamic (MHD) turbulence. We test them on results of a weakly compressible, 2D, moderate-Reynolds-number Hall MHD simulation and compare them with an isotropic spectral transfer (ST) equation. The KHM and ST equations are automatically satisfied during the whole simulation owing to the periodic boundary conditions and have complementary cumulative behavior. They are used here to analyze the onset of turbulence and its properties when it is fully developed. These approaches give equivalent results characterizing the decay of the kinetic + magnetic energy at large scales, the MHD and Hall cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. The Hall cascade appears when the MHD one brings the energy close to the ion inertial range and is related to the formation of reconnecting current sheets. At later times, the pressure dilatation energy exchange rate oscillates around zero, with no net effect on the cross-scale energy transfer when averaged over a period of its oscillations. A reduced 1D analysis suggests that all three methods may be useful to estimate the energy cascade rate from in situ observations.
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.
A brief analysis of the proton parallel and oblique firehose instability is presented from a fluid perspective and the results are compared to kinetic theory solutions obtained by the WHAMP code. It is shown that the classical CGL model very accurately describes the growth rate of these instabilities at sufficiently long spatial scales (small wavenumbers). The required stabilization of these instabilities at small spatial scales (high wavenumbers) naturally requires dispersive effects and the stabilization is due to the Hall term and finite Larmor radius (FLR) corrections to the pressure tensor. Even though the stabilization is not completely accurate since at small spatial scales a relatively strong collisionless damping comes into effect, we find that the main concepts of the maximum growth rate and the stabilization of these instabilities is indeed present in the fluid description. However, there are differences that are quite pronounced when close to the firehose threshold and that clarify the different profiles for marginally stable states with a prescribed maximum growth rate $\gamma_{max}$ in the simple fluid models considered here and the kinetic description.
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 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.
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.
Incompressible and isotropic magnetohydrodynamic turbulence in plasms can be
described by an exact relation for the energy flux through the scales. This
Yaglom-like scaling law has been recently observed in the solar wind above the
solar poles observed by the Ulysses spacecraft, where the turbulence is in an
Alfv\'enic state. An analogous phenomenological scaling law, suitably modified
to take into account compressible fluctuations, is observed more frequently in
the same dataset. Large scale density fluctuations, despite their low
amplitude, play thus a crucial role in the basic scaling properties of
turbulence. The turbulent cascade rate in the compressive case can moreover
supply the energy dissipation needed to account for the local heating of the
non-adiabatic solar wind.
We present an analysis of data stemming from numerical simulations of decaying magnetohydrodynamic (MHD) turbulence up to grid resolution of 1536(3) points and up to Taylor Reynolds number of approximately 1200 . The initial conditions are such that the initial velocity and magnetic fields are helical and in equipartition, while their correlation is negligible. Analyzing the data at the peak of dissipation, we show that the dissipation in MHD seems to asymptote to a constant as the Reynolds number increases, thereby strengthening the possibility of fast reconnection events in the solar environment for very large Reynolds numbers. Furthermore, intermittency of MHD flows, as determined by the spectrum of anomalous exponents of structure functions of the velocity and the magnetic field, is stronger than that of fluids, confirming earlier results; however, we also find that there is a measurable difference between the exponents of the velocity and those of the magnetic field, reminiscent of recent solar wind observations. Finally, we discuss the spectral scaling laws that arise in this flow.
We present numerical simulations of electron magnetohydrodynamic (EMHD) and electron reduced MHD (ERMHD) turbulence. Comparing scaling relations, we find that both EMHD and ERMHD turbulence show similar spectra and anisotropy. We develop new techniques to study anisotropy of EMHD turbulence. Our detailed study of anisotropy of EMHD turbulence supports our earlier result of k
|| ∝ k
1/3⊥ scaling, where k
|| and k
⊥ are wavenumbers parallel and perpendicular to local direction of magnetic field, respectively. We find that the high-order statistics show a scaling that is similar to the She-Leveque scaling for incompressible hydrodynamic turbulence and different from that of incompressible MHD turbulence. We observe that the bispectra, which characterize the interaction of different scales within the turbulence cascade, are very different for EMHD and MHD turbulence. We show that both decaying and driven EMHD turbulence have the same statistical properties. We calculate the probability distribution functions (PDFs) of MHD and EMHD turbulence and compare them with those of interplanetary turbulence. We find that, as in the case of the solar wind, the PDFs of the increments of magnetic field strength in MHD and EMHD turbulence are well described by the Tsallis distribution. We discuss implications of our results for astrophysical situations, including the advection-dominated accretion flows and magnetic reconnection.
Direct evidence for the presence of an inertial energy cascade, the most characteristic signature of hydromagnetic turbulence (MHD), is observed in the solar wind by the Ulysses spacecraft. After a brief rederivation of the equivalent of Yaglom's law for MHD turbulence, a linear relation is indeed observed for the scaling of mixed third-order structure functions involving Elsässer variables. This experimental result firmly establishes the turbulent character of low-frequency velocity and magnetic field fluctuations in the solar wind plasma.
The von Kármán-Howarth equations are derived for three-dimensional Hall magnetohydrodynamics in the case of a homogeneous and isotropic turbulence. From these equations, we derive exact scaling laws for the third-order correlation tensors. We show how these relations are compatible with previous heuristic and numerical results. These multiscale laws provide a relevant tool to investigate the nonlinear nature of the high-frequency magnetic field fluctuations in the solar wind or, more generally, in any plasma where the Hall effect is important.
The purpose of this work is to investigate whether a cascading process can be associated with the rotational motions of compressible three-dimensional turbulence. This question is examined through the lens of circulicity, a concept related to the angular momentum carried by large turbulent scales. By deriving a Monin-Yaglom relation for circulicity, we show that an “effective” cascade of this quantity exists, provided the flow is stirred with a force having a solenoidal component. This outcome is obtained independently from the expression of the equation of state. To supplement these results, a coarse-graining analysis of the flow is performed. This approach allows us to separate the contributions of the transfer and production terms of circulicity and to discuss their respective effects in the inertial range.
We derive the coarse-graining (CG) equations of incompressible Hall magnetohydrodynamic (HMHD) turbulence to investigate the local (in space) energy transfer rate as a function of the filtering scale ℓ. First, the CG equations are space averaged to obtain the analytical expression of the mean cascade rate. Its application to three-dimensional simulations of (weakly compressible) HMHD shows a cascade rate consistent with the value of the mean dissipation rate in the simulations and with the classical estimates based on the “third-order” law. Furthermore, we developed an anisotropic version of CG that allows us to study the magnitude of the cascade rate along different directions with respect to the mean magnetic field. Its implementation on the numerical data with moderate background magnetic field shows a weaker cascade along the magnetic field than in the perpendicular plane, while an isotropic cascade is recovered in the absence of a background field. The strength of the CG approach is further revealed when considering the local-in-space energy transfer, which is shown theoretically and numerically to match at a given position x, when locally averaged over a neighboring region, the (quasi-)local dissipation. Prospects of exploiting this model to investigate local dissipation in spacecraft data are discussed.
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.
Using an exact law for incompressible Hall magnetohydrodynamics (HMHD) turbulence, the energy cascade rate is computed from three-dimensional HMHD-CGL (biadiabatic ions and isothermal electrons) and Landau-fluid numerical simulations that feature different intensities of Landau damping over a broad range of wavenumbers, typically 0.05 ≲ k ⊥ d i ≲ 100. Using three sets of cross-scale simulations where turbulence is initiated at large, medium, and small scales, the ability of the fluid energy cascade to “sense” the kinetic Landau damping at different scales is tested. The cascade rate estimated from the exact law and the dissipation calculated directly from the simulation are shown to reflect the role of Landau damping in dissipating energy at all scales, with an emphasis on the kinetic ones. This result provides new prospects on using exact laws for simplified fluid models to analyze dissipation in kinetic simulations and spacecraft observations, and new insights into theoretical description of collisionless magnetized plasmas.
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.
We investigate the properties of the scale dependence and cross-scale transfer of kinetic energy in compressible three-dimensional hydrodynamic turbulence by means of two direct numerical simulations of decaying turbulence with initial Mach numbers M=1/3 and 1, and with moderate Reynolds numbers, Rλ∼100. The turbulent dynamics is analyzed using compressible and incompressible versions of the dynamic spectral transfer (ST) and the Kármán-Howarth-Monin (KHM) equations. We find that the nonlinear coupling leads to a flux of the kinetic energy to small scales where it is dissipated; at the same time, the reversible pressure-dilatation mechanism causes oscillatory exchanges between the kinetic and internal energies with an average zero net energy transfer. While the incompressible KHM and ST equations are not generally valid in the simulations, their compressible counterparts are well satisfied and describe, in a quantitatively similar way, the decay of the kinetic energy on large scales, the cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. There exists a simple relationship between the KHM and ST results through the inverse proportionality between the wave vector k and the spatial separation length l as kl≃3. For a given time, the dissipation and pressure-dilatation terms are strong on large scales in the KHM approach, whereas the ST terms become dominant on small scales; this is due to the complementary cumulative behavior of the two methods. The effect of pressure dilatation is weak when averaged over a period of its oscillations and may lead to a transfer of the kinetic energy from large to small scales without a net exchange between the kinetic and internal energies. Our results suggest that for large-enough systems, there exists an inertial range for the kinetic energy cascade. This transfer is partly due to the classical, nonlinear advection-driven cascade and partly due to the pressure dilatation-induced energy transfer. We also use the ST and KHM approaches to investigate the properties of the internal energy. The dynamic ST and KHM equations for the internal energy are well satisfied in the simulations but behave very differently with respect to the viscous dissipation. We conclude that ST and KHM approaches would better be used for the kinetic and internal energies separately.
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.
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 relationship between a decaying strong turbulence and the mirror instability in a slowly expanding plasma is investigated using two-dimensional hybrid expanding box simulations. We impose an initial ambient magnetic field perpendicular to the simulation box, and we start with a spectrum of large-scale, linearly-polarized, random-phase Alfvenic fluctuations which have energy equipartition between kinetic and magnetic fluctuations and vanishing correlation between the two fields. A turbulent cascade rapidly develops, magnetic field fluctuations exhibit a Kolmogorov-like power-law spectrum at large scales and a steeper spectrum at sub-ion scales. The imposed expansion (taking a strictly transverse ambient magnetic field) leads to generation of an important perpendicular proton temperature anisotropy that eventually drives the mirror instability. This instability generates large-amplitude, nonpropagating, compressible, pressure-balanced magnetic structures in a form of magnetic enhancements/humps that reduce the perpendicular temperature anisotropy.
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.
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' .
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.
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.
Supersonic turbulence plays an important role in a number of extreme
astrophysical and terrestrial environments, yet its understanding remains
rudimentary. We use data from a three-dimensional simulation of supersonic
isothermal turbulence to reconstruct an exact fourth-order relation derived
analytically from the Navier-Stokes equations (Galtier and Banerjee, Phys. Rev.
Lett., vol. 107, 2011, p. 134501). Our analysis supports a Kolmogorov-like
inertial energy cascade in supersonic turbulence previously discussed on a
phenomenological level. We show that two compressible analogues of the
four-fifths law exist describing fifth- and fourth-order correlations, but only
the fourth-order relation remains `universal' in a wide range of Mach numbers
from incompressible to highly compressible regimes. A new approximate relation
valid in the strongly supersonic regime is derived and verified. We also
briefly discuss the origin of bottleneck bumps in simulations of compressible
turbulence.
The different levels of description of fluid media [e.g., magnetohydrodynamics (MHD), Hall-magnetohydrodynamics, bi-fluid,…] are commonly known under the form of Newtonian systems of equations. Nevertheless, this form proves to be ill-suited to derive a fully analytical weak turbulence theory of these media, due to the well-known complexity of the calculations implied. For such studies, therefore, a more appropriate mathematical frame needs to be found and this is shown to be the Hamiltonian formalism, even though it can often appear difficult to handle. The goal of this paper is to look for Hamiltonian formulations for the different levels of the fluid description of a plasma using the variational principle. Starting from the bi-fluid system, it is shown that such a formulation can be obtained by combining the Lagrangians already used for describing: (i) the motion of a charged particle in an electromagnetic field; (ii) the evolution of an electromagnetic field in presence of sources; (iii) the motion of a neutral fluid (Clebsch variables). The equivalence of the obtained description in terms of the generalized-Clebsch variables to the familiar Newtonian formulation is discussed. It is shown that each solution of the Hamiltonian system is also a solution for the Newtonian one, but that the converse is not true. The origin and the implication of this restriction are discussed. Reducing the Hamiltonian formulation obtained for the bi-fluid system to lower orders of the fluid approximations is then shown to be mandatory when one tries to obtain analytical results for linear waves and nonlinear wave–wave couplings. It is shown that this goal can be reached in two steps. The first one leads to a “reduced bi-fluid” system, which is identical to the bi-fluid one when the displacement current is neglected but the electron inertia is still working. The number of linear modes then goes down from six to three. The second step, leading to the Hall-MHD system, consists in neglecting the electron mass. It is demonstrated that the only four generalized Clebsch variables are sufficient to describe the full Hall-MHD dynamics. Some future applications of such a powerful formalism are outlined. © 2003 American Institute of Physics.
We investigate experimentally the influence of a background rotation on the energy transfers in decaying grid turbulence. The anisotropic energy flux density F(r) = <δu(δu)²>, where δu is the vector velocity increment over separation r, is determined for the first time by using particle image velocimetry. We show that rotation induces an anisotropy of the energy flux ∇·F, which leads to an anisotropy growth of the energy distribution E(r) = <(δu)²>, in agreement with the von Kármán-Howarth-Monin equation. Surprisingly, our results prove that this anisotropy growth is essentially driven by a nearly radial, but orientation-dependent, energy flux density F(r).
- P Hellinger
- A Verdini
- S Landi
- L Franci
- E Papini
- L Matteini
P. Hellinger, A. Verdini, S. Landi, L. Franci, E. Papini, and L. Matteini, On cascade of kinetic
energy in compressible hydrodynamic turbulence, arXiv e-prints (2020), arXiv: 2004.02726.
Energy cascade rate in compressible fast and slow solar wind turbulence
- L Z Hadid
- F Sahraoui
- S Galtier
L. Z. Hadid, F. Sahraoui, and S. Galtier, Energy cascade rate in compressible fast and slow
solar wind turbulence, The Astrophysical Journal 838, 9 (2017), arXiv: 1612.02150.