Dissipation Range Spectra Research Articles

High-latitude Observations of Dissipation-range Turbulence by the Ulysses Spacecraft during the Solar Minimum of 1993–96: The Spectral Break and Anisotropy

We examine Ulysses magnetic field observations from 1993 to 1996 as the spacecraft made its first fast-latitude scan from the southern to the northern hemisphere. Most of the observations we use are representative of high-latitude solar minimum conditions. We examine magnetic field power spectra characteristics of interplanetary turbulence at high frequencies, where the spectrum breaks from an inertial range into the ion dissipation range. The onset and spectral index of the dissipation spectrum are consistent with low-latitude observations at 1 au. Both ranges have a ratio of power in perpendicular magnetic field components to parallel components near 3. The power spectrum ratio test developed by Bieber et al. for single-spacecraft analyses that determines the underlying anisotropy of the wave vectors yields only marginally more energy associated with field-aligned wave vectors than perpendicular wave vectors when comparing the inertial and dissipation-range spectra. The lack of significant change in the anisotropies between the inertial and dissipation ranges contrasts strongly with the turbulence found typically for 1 au near-ecliptic observations, where significant differences in both anisotropies are observed.

Nonlinear Coupling of Kinetic Alfvén Waves and Ion Acoustic Waves in the Inner Heliosphere

We study the nonlinear coupling of kinetic Alfvén waves with ion acoustic waves applicable to the Earth’s radiation belt and near-Sun streamer belt solar wind using dynamical equations in the form of modified Zakharov systems. Numerical simulations show the formation of magnetic field filamentary structures associated with density humps and dips which become turbulent at later times, redistributing the energy to higher wavenumbers. The magnetic power spectra exhibit an inertial range Kolmogorov-like spectral index value of −5/3 for k ⊥ ρ i < 1, followed by a steeper dissipation range spectra with indices ∼ −3 for the radiation belt case and ∼ −4 for the near-Sun streamer belt solar wind case, here k ⊥ and ρ i represent the wavevector component perpendicular to the background magnetic field and the ion thermal gyroradius, respectively. Applying quasilinear theory in terms of the Fokker–Planck equation in the region of wavenumber turbulent spectra, we find the particle distribution function flattening in the superthermal tail population which is the signature of particle energization and plasma heating.

THE VIOLATION OF THE TAYLOR HYPOTHESIS IN MEASUREMENTS OF SOLAR WIND TURBULENCE

Motivated by the upcoming Solar Orbiter and Solar Probe Plus missions, qualitative and quantitative predictions are made for the effects of the violation of the Taylor hypothesis on the magnetic energy frequency spectrum measured in the near-Sun environment. The synthetic spacecraft data method is used to predict observational signatures of the violation for critically balanced Alfv\'enic turbulence or parallel fast/whistler turbulence. The violation of the Taylor hypothesis can occur in the slow flow regime, leading to a shift of the entire spectrum to higher frequencies, or in the dispersive regime, in which the dissipation range spectrum flattens at high frequencies. It is found that Alfv\'enic turbulence will not significantly violate the Taylor hypothesis, but whistler turbulence will. The flattening of the frequency spectrum is therefore a key observational signature for fast/whistler turbulence.

Technique for measuring and correcting the Taylor microscale

AbstractWe discuss and develop methods to estimate and refine measurements of the Taylor microscale from discrete data sets. To study how well a method works, we construct a time series of discrete data with a known power spectrum and Taylor scale, but with various truncations of the resolution that eliminate higher frequencies in a controlled fashion. We compute the second‐order structure function and correlation function, assuming that the unresolved dissipation range spectrum has various values of spectral index. A series of Taylor scale estimates are obtained from parabolic fits to subsets of the correlation function data, and these are extrapolated to the limit of zero separation. The error in this procedure, for finite time resolution sampling, depends on the spectral index in the dissipation range. When the spectral form is known, we can compute a correction factor that improves the estimate of the Taylor microscale value determined from the extrapolation method and band‐limited data. Application of this technique to spacecraft observations of solar wind fluctuations is illustrated.

ON THE SPECTRAL HARDENING AT ≳300 keV IN SOLAR FLARES

It has been noted for a long time that the spectra of observed continuum emissions in many solar flares are consistent with double power laws with a hardening at energies $\sim > $ 300 keV. It is now largely believed that at least in electron-dominated events the hardening in photon spectrum reflects an intrinsic hardening in the source electron spectrum. In this paper, we point out that a power law spectrum of electron with a hardening at high energies can be explained by diffusive shock acceleration of electrons at a termination shock with a finite width. Our suggestion is based on an early analytical work by Drury et al., where the steady state transport equation at a shock with a tanh profile was solved for a $p$-independent diffusion coefficient. Numerical simulations with a $p$-dependent diffusion coefficient show hardenings in the accelerated electron spectrum which are comparable with observations. One necessary condition for our proposed scenario to work is that high energy electrons resonate with the inertial range of the MHD turbulence and low energy electrons resonate with the dissipation range of the MHD turbulence at the acceleration site, and the spectrum of the dissipation range $\sim k^{-2.7}$. A $\sim k^{-2.7}$ dissipation range spectrum is consistent with recent solar wind observations.

Magnetohydrodynamic dissipation range spectra for isotropic viscosity and resistivity

Dissipation range spectra for incompressible magnetohydrodynamic turbulence are derived for isotropic viscosity μ and resistivity η. The spectra are obtained from heuristic closures of spectral transfer correlations for cases with Pm=μ/η≤1, where Pm is the magnetic Prandtl number. Familiar inertial range power laws are modified by exponential factors that dominate spectral falloff in the dissipation range. Spectral forms are sensitive to alignment between flow and magnetic field. There are as many as four Kolmogorov wavenumbers that parametrize the transition between inertial and dissipative behavior and enter corresponding spectral forms. They depend on the values of the viscosity and resistivity and on the nature of alignment in inertial and dissipation ranges.

CASCADE AND DAMPING OF ALFVÉN-CYCLOTRON FLUCTUATIONS: APPLICATION TO SOLAR WIND TURBULENCE

It is well recognized that the presence of magnetic fields will lead to anisotropic energy cascade and dissipation of astrophysical turbulence. With the diffusion approximation and linear dissipation rates, we study the cascade and damping of Alfven-cyclotron fluctuations in solar plasmas numerically for two diagonal diffusion tensors, one (isotropic) with identical components for the parallel and perpendicular directions (with respect to the magnetic field) and one with different components (nonisotropic). It is found that for the isotropic case the steady-state turbulence spectra are nearly isotropic in the inertial range and can be fitted by a single power-law function with a spectral index of –3/2, similar to the Iroshnikov-Kraichnan phenomenology, while for the nonisotropic case the spectra vary greatly with the direction of propagation. The energy fluxes in both cases are much higher in the perpendicular direction than in the parallel direction due to the angular dependence (or inhomogeneity) of the components. In addition, beyond the MHD regime the kinetic effects make the spectrum softer at higher wavenumbers. In the dissipation range the turbulence spectrum cuts off at the wavenumber, where the damping rate becomes comparable to the cascade rate, and the cutoff wavenumber changes with the wave propagation direction. The angle-averaged turbulence spectrum of the isotropic model resembles a broken power law, which cuts off at the maximum of the cutoff wavenumbers or the 4He cyclotron frequency. Taking into account the Doppler effects, the model naturally reproduces the broken power-law turbulence spectra observed in the solar wind and predicts that a higher break frequency always comes along with a softer dissipation range spectrum that may be caused by the increase of the turbulence intensity, the reciprocal of the plasma βp, and/or the angle between the solar wind velocity and the mean magnetic field. These predictions can be tested by detailed comparisons with more accurate observations.

A model of turbulence in magnetized plasmas: Implications for the dissipation range in the solar wind

This paper studies the turbulent cascade of magnetic energy in weakly collisional magnetized plasmas. A cascade model is presented, based on the assumptions of local nonlinear energy transfer in wave number space, critical balance between linear propagation and nonlinear interaction times, and the applicability of linear dissipation rates for the nonlinearly turbulent plasma. The model follows the nonlinear cascade of energy from the driving scale in the MHD regime, through the transition at the ion Larmor radius into the kinetic Alfvén wave regime, in which the turbulence is dissipated by kinetic processes. The turbulent fluctuations remain at frequencies below the ion cyclotron frequency due to the strong anisotropy of the turbulent fluctuations, k∥ ≪ k⊥ (implied by critical balance). In this limit, the turbulence is optimally described by gyrokinetics; it is shown that the gyrokinetic approximation is well satisfied for typical slow solar wind parameters. Wave phase velocity measurements are consistent with a kinetic Alfvén wave cascade and not the onset of ion cyclotron damping. The conditions under which the gyrokinetic cascade reaches the ion cyclotron frequency are established. Cascade model solutions imply that collisionless damping provides a natural explanation for the observed range of spectral indices in the dissipation range of the solar wind. The dissipation range spectrum is predicted to be an exponential fall off; the power‐law behavior apparent in observations may be an artifact of limited instrumental sensitivity. The cascade model is motivated by a program of gyrokinetic simulations of turbulence and particle heating in the solar wind.

Dependence of the Dissipation Range Spectrum of Interplanetary Magnetic Fluctuationson the Rate of Energy Cascade

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.

Observational constraints on the dynamics of the interplanetary magnetic field dissipation range

The dissipation range for interplanetary magnetic field fluctuations is formed by those fluctuations with spatial scales comparable to the gyroradius or ion inertial length of a thermal ion. It is reasonable to assume that the dissipation range represents the final fate of magnetic energy that is transferred from the largest spatial scales via nonlinear processes until kinetic coupling with the background plasma removes the energy from the spectrum and heats the background distribution. Typically, the dissipation range at 1 AU sets in at spacecraft frame frequencies of a few tenths of a hertz. It is characterized by a steepening of the power spectrum and often demonstrates a bias of the polarization or magnetic helicity spectrum. We examine Wind observations of inertial and dissipation range spectra in an attempt to better understand the processes that form the dissipation range and how these processes depend on the ambient solar wind parameters (interplanetary magnetic field intensity, ambient proton density and temperature, etc.). We focus on stationary intervals with well‐defined inertial and dissipation range spectra. Our analysis shows that parallel‐propagating waves, such as Alfvén waves, are inconsistent with the data. MHD turbulence consisting of a partly slab and partly two‐dimensional (2‐D) composite geometry is consistent with the observations, while thermal paxticle interactions with the 2‐D component may be responsible for the formation of the dissipation range. Kinetic Alfvén waves propagating at large angles to the background magnetic field are also consistent with the observations and may form some portion of the 2‐D turbulence component.

Comment on the dissipation-range spectrum in turbulent flows

Recently, Smith and Reynolds [Phys. Fluids A 3, 992 (1991)] suggested that the turbulent energy spectrum with exponential decaying form E(k)=CKε2/3k−5/3 exp(−σk) (CK is Kolmogorov constant, ε is the energy dissipation rate, and σ is some constant) may be a poor model, because it does not give velocity derivative skewness in good agreement with experimental value. On the other hand, some direct numerical simulations of the Navier–Stokes equation demonstrate the above exponential form in the dissipation range. This disagreement is resolved by considering a model spectrum based on direct numerical simulation.

Cosmic-ray pitch angle scattering in isotropic turbulence. II - Sensitive dependence on the dissipation range spectrum

The leading-order quasi-linear expressions for cosmic-ray pitch-angle scattering in isotropic magnetic turbulence were analyzed using three separate spectral forms: the Gaussian, the exponential, and the power-law. It is shown that the low-energy limit for the pitch-angle diffusion coefficient did not universally yield the 'slab' results obtained with a Gaussian dissipation range by Bieber et al. (1988). Instead, it was found that the low-energy limit depends on the steepness of the dissipation range spectrum, and the asymptotic functional forms can range between the slab results and the Fisk et al. (1974) results. It is also shown that the 90-deg pitch-angle limit displays a sensitive dependence on the form of the dissipation range spectrum and can depart dramatically from either the slab result or the result of Fisk.

Numerical Integration of the Lagrangian Renormalized Approximation

The Lagrangian renormalized approximation (LRA) and the Modified Lagrangian renormalized approximation (MLRA) are integrated numerically for decaying homogeneous isotropic turbulences in two and three dimensions. At moderate Reynolds numbers the results are in good agreement with the direct numerical simulations by Herring et at . in two dimensions and with those by Orszag and Patterson in three dimensions. At high Reynolds numbers the one dimensional inertial and dissipation range spectra of the LRA and the MLRA agree well with the tidal channel data of Grant et al . The comparison with the DIA, TFM, ALHDIA and SBALHDIA shows that the MLRA behavior is very close to that of the TFM.

Turbulent magnetic reconnection

The effects of turbulence on magnetic reconnection are investigated by two-dimensional spectral method magnetohydrodynamic computations. The nonlinear evolution of the periodic sheet pinch configuration is studied as an initial value problem. Turbulence is initiated by including a low level of broadband fluctuations in the initial data. Nonlinear features of the evolution, appropriately described as turbulence, are seen early in the solutions and persist throughout the runs. Small-scale, unsteady coherent electric current and vorticity structures develop in the reconnection zone, resulting in enhanced viscous and resistive dissipation. Unsteady and often spatially asymmetric fluid flow develops. Large-scale magnetic islands produced by reconnection activity, undergo internal pulsations. Small-scale magnetic islands, or ‘‘bubbles’’ develop near the reconnection zone, prodcing multiple X points. Large-amplitude electric field fluctuations, often several times larger than the reconnection electric field, are produced by large island pulsations and by motion of magnetic bubbles. Spectral analysis of the fluctuations show development of broad band excitations, reminiscent of inertial and dissipation range spectra in homogeneous turbulence. Two-dimensional spectra indicate that the turbulence is broadband in both spatial directions. It is suggested that the turbulence that develops from the randomly perturbed sheet pinchbears a strong resemblence to homogeneous magnetohydrodynamic turbulence, and that analytical theories of reconnection must incorporate these effects.

Drag reduction in fibre suspension

Spectacular reductions in turbulent friction can be obtained by adding small amounts of either long-chain polymers1,2, or macroscopic fibres3,4 to a fluid. The polymer effect has been the more studied of the two. But one of the continuing puzzles is that measurements of the turbulent energy spectrum show no evidence of a direct eddy–polymer interaction, despite the fact that the influence of the polymers on the turbulence is so profound. Although some increase in energy may be observed at small wavenumbers (attributable to a decrease in the dissipation rate), inertial and dissipation range spectra in polymer solutions are unchanged from the newtonian result. In contrast with this situation, we have recently found very marked changes in energy spectra measured in fibre suspensions.