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DOE OSTI2014

Stochastic acceleration of electrons by fast magnetosonic waves in solar flares: the effects of anisotropy in velocity and wavenumber space

We develop a model for stochastic acceleration of electrons in solar flares. As in several previous models, the electrons are accelerated by turbulent fast magnetosonic waves ({sup f}ast waves{sup )} via transit-time-damping (TTD) interactions. (In TTD interactions, fast waves act like moving magnetic mirrors that push the electrons parallel or anti-parallel to the magnetic field). We also include the effects of Coulomb collisions and the waves' parallel electric fields. Unlike previous models, our model is two-dimensional in both momentum space and wavenumber space and takes into account the anisotropy of the wave power spectrum F{sub k} and electron distribution function f {sub e}. We use weak turbulence theory and quasilinear theory to obtain a set of equations that describes the coupled evolution of F{sub k} and f {sub e}. We solve these equations numerically and find that the electron distribution function develops a power-law-like non-thermal tail within a restricted range of energies E in (E {sub nt}, E {sub max}). We obtain approximate analytic expressions for E {sub nt} and E {sub max}, which describe how these minimum and maximum energies depend upon parameters such as the electron number density and the rate at which fast-wave energy is injected into the acceleration region at large scales. We contrast our results with previous studies that assume that F{sub k} and f {sub e} are isotropic, and we compare one of our numerical calculations with the time-dependent hard-X-ray spectrum observed during the 1980 June 27 flare. In our numerical calculations, the electron energy spectra are softer (steeper) than in models with isotropic F{sub k} and f {sub e} and closer to the values inferred from observations of solar flares.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2013

Kelvin-Helmholtz instability on the magnetopause, magnetohydrodynamic waveguide in the outer magnetosphere, and Alfven resonance deep in the magnetosphere

Oscillations of the 'magnetosphere-solar wind' system are studied analytically in the framework of a plane-stratified model of the medium. The properties of oscillations are determined by three phenomena: Kelvin-Helmholtz instability on the tangential discontinuity (magnetopause) separating the magnetosphere and the solar wind, the presence of a waveguide for fast magnetosonic waves in the magnetosphere, and the Alfven resonance-a sharp increase in the amplitude of oscillations having the properties of Alfven waves-in the inner magnetosphere. The oscillations of the system form a discrete spectrum of eigenmodes. Analytical expressions are obtained for the frequency and growth rate of instability of each mode, as well as for the functions describing the spatial structure of these modes. All these characteristics of the eigenmodes are shown to depend on the velocity of the solar wind as a parameter. The dependences of the main mode characteristics (such as the instability thresholds, the points of the maximum and minimum growth rate, and the spatial distributions of the oscillation energy) on this parameter are determined for each eigenmode.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2014

High-resolution observations of the shock wave behavior for sunspot oscillations with the interface region imaging spectrograph

We present the first results of sunspot oscillations from observations by the Interface Region Imaging Spectrograph. The strongly nonlinear oscillation is identified in both the slit-jaw images and the spectra of several emission lines formed in the transition region and chromosphere. We first apply a single Gaussian fit to the profiles of the Mg II 2796.35 Å, C II 1335.71 Å, and Si IV 1393.76 Å lines in the sunspot. The intensity change is ∼30%. The Doppler shift oscillation reveals a sawtooth pattern with an amplitude of ∼10 km s{sup –1} in Si IV. The Si IV oscillation lags those of C II and Mg II by ∼6 and ∼25 s, respectively. The line width suddenly increases as the Doppler shift changes from redshift to blueshift. However, we demonstrate that this increase is caused by the superposition of two emission components. We then perform detailed analysis of the line profiles at a few selected locations on the slit. The temporal evolution of the line core is dominated by the following behavior: a rapid excursion to the blue side, accompanied by an intensity increase, followed by a linear decrease of the velocity to the red side. The maximum intensity slightly lags the maximum blueshift in Si IV, whereas the intensity enhancement slightly precedes the maximum blueshift in Mg II. We find a positive correlation between the maximum velocity and deceleration, a result that is consistent with numerical simulations of upward propagating magnetoacoustic shock waves.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2012

The quantum dusty magnetosonic solitary wave in magnetized plasma

The effects of quantum statistic and quantum diffraction on the weakly two-dimensional fast quantum dusty magnetosonic wave propagating perpendicular to the external magnetic field are investigated by considering the inertialess electron, inertialess ion, and inertial cold dust in the low frequency limit. A Kadomstev-Petviashvili equation is derived for the magnetosonic solitary wave by using reductive perturbation method. The results show that the amplitude of soliton increases with the increasing of quantum diffraction effects of both electrons and ions, while the amplitude of the soliton decreases with the increasing of the quantum statistic effects. By using the numerical investigations, the interaction law of the nontrivial line-solitons with rich web structure in the interaction area among the line-solitons is studied by the Wronskian determinant method, which shows that there is no exchange of the energy, the momentum, and the angular momentum in the interaction.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2014

Longitudinal magnetohydrodynamics oscillations in dissipative, cooling coronal loops

This paper investigates the effect of cooling on standing slow magnetosonic waves in coronal magnetic loops. The damping mechanism taken into account is thermal conduction that is a viable candidate for dissipation of slow magnetosonic waves in coronal loops. In contrast to earlier studies, here we assume that the characteristic damping time due to thermal conduction is not small, but arbitrary, and can be of the order of the oscillation period, i.e., a temporally varying plasma is considered. The approximation of low-beta plasma enables us to neglect the magnetic field perturbation when studying longitudinal waves and consider, instead, a one-dimensional motion that allows a reliable first insight into the problem. The background plasma temperature is assumed to be decaying exponentially with time, with the characteristic cooling timescale much larger than the oscillation period. This assumption enables us to use the WKB method to study the evolution of the oscillation amplitude analytically. Using this method we obtain the equation governing the oscillation amplitude. The analytical expressions determining the wave properties are evaluated numerically to investigate the evolution of the oscillation frequency and amplitude with time. The results show that the oscillation period increases with time due to the effect of plasma cooling. The plasma cooling also amplifies the amplitude of oscillations in relatively cool coronal loops, whereas, for very hot coronal loop oscillations the damping rate is enhanced by the cooling. We find that the critical point for which the amplification becomes dominant over the damping is in the region of 4 MK. These theoretical results may serve as impetus for developing the tools of solar magneto-seismology in dynamic plasmas.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2014

Current generation by helicons and LH waves in modern tokamaks and reactors FNSF-AT, ITER and DEMO. Scenarios, modeling and antennae

The Innovative concept and 3D full wave code modeling Off-axis current drive by RF waves in large scale tokamaks, reactors FNSF-AT, ITER and DEMO for steady state operation with high efficiency was proposed [1] to overcome problems well known for LH method [2]. The scheme uses the helicons radiation (fast magnetosonic waves at high (20–40) IC frequency harmonics) at frequencies of 500–1000 MHz, propagating in the outer regions of the plasmas with a rotational transform. It is expected that the current generated by Helicons will help to have regimes with negative magnetic shear and internal transport barrier to ensure stability at high normalized plasma pressure β{sub N} > 3 (the so-called Advanced scenarios) of interest for FNSF and the commercial reactor. Modeling with full wave three-dimensional codes PSTELION and STELEC2 showed flexible control of the current profile in the reactor plasmas of ITER, FNSF-AT and DEMO [2,3], using multiple frequencies, the positions of the antennae and toroidal waves slow down. Also presented are the results of simulations of current generation by helicons in tokamaks DIII-D, T-15MD and JT-60SA [3]. In DEMO and Power Plant antenna is strongly simplified, being some analoge of mirrors based ECRF launcher, as will be shown. For spherical tokamaks the Helicons excitation scheme does not provide efficient Off-axis CD profile flexibility due to strong coupling of helicons with O-mode, also through the boundary conditions in low aspect machines, and intrinsic large amount of trapped electrons, as is shown by STELION modeling for the NSTX tokamak. Brief history of Helicons experimental and modeling exploration in straight plasmas, tokamaks and tokamak based fusion Reactors projects is given, including planned joint DIII-D – Kurchatov Institute experiment on helicons CD [1].

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2013

TURBULENCE AND RADIO MINI-HALOS IN THE SLOSHING CORES OF GALAXY CLUSTERS

A number of relaxed, cool-core galaxy clusters exhibit diffuse, steep-spectrum radio sources in their central regions, known as radio mini-halos. It has been proposed that the relativistic electrons responsible for the emission have been reaccelerated by turbulence generated by the sloshing of the cool core gas. We present a high-resolution MHD simulation of gas sloshing in a galaxy cluster coupled with subgrid simulations of relativistic electron acceleration to test this hypothesis. Our simulation shows that the sloshing motions generate turbulence on the order of {delta}v {approx} 50-200 km s{sup -1} on spatial scales of {approx}50-100 kpc and below in the cool core region within the envelope of the sloshing cold fronts, whereas outside the cold fronts, there is negligible turbulence. This turbulence is potentially strong enough to reaccelerate relativistic electron seeds (with initial {gamma} {approx} 100-500) to {gamma} {approx} 10{sup 4} via damping of magnetosonic waves and non-resonant compression. The seed electrons could remain in the cluster from, e.g., past active galactic nucleus activity. In combination with the magnetic field amplification in the core, these electrons then produce diffuse radio synchrotron emission that is coincident with the region bounded by the sloshing cold fronts, as indeed observed in X-rays and the radio. The result holds for different initial spatial distributions of pre-existing relativistic electrons. The power and the steep spectral index ({alpha} Almost-Equal-To 1-2) of the resulting radio emission are consistent with observations of mini-halos, though the theoretical uncertainties of the acceleration mechanisms are high. We also produce simulated maps of inverse-Compton hard X-ray emission from the same population of relativistic electrons.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2012

A GODUNOV METHOD FOR MULTIDIMENSIONAL RADIATION MAGNETOHYDRODYNAMICS BASED ON A VARIABLE EDDINGTON TENSOR

We describe a numerical algorithm to integrate the equations of radiation magnetohydrodynamics in multidimensions using Godunov methods. This algorithm solves the radiation moment equations in the mixed frame, without invoking any diffusion-like approximations. The moment equations are closed using a variable Eddington tensor whose components are calculated from a formal solution of the transfer equation at a large number of angles using the method of short characteristics. We use a comprehensive test suite to verify the algorithm, including convergence tests of radiation-modified linear acoustic and magnetosonic waves, the structure of radiation-modified shocks, and two-dimensional tests of photon bubble instability and the ablation of dense clouds by an intense radiation field. These tests cover a very wide range of regimes, including both optically thick and thin flows, and ratios of the radiation to gas pressure of at least 10{sup -4}-10{sup 4}. Across most of the parameter space, we find that the method is accurate. However, the tests also reveal there are regimes where the method needs improvement, for example when both the radiation pressure and absorption opacity are very large. We suggest modifications to the algorithm that will improve the accuracy in this case. We discuss the advantages of this method over those based on flux-limited diffusion. In particular, we find that the method is not only substantially more accurate, but often no more expensive than the diffusion approximation for our intended applications.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

Quantum dust magnetosonic waves with spin and exchange correlation effects

Dust magnetosonic waves are studied in degenerate dusty plasmas with spin and exchange correlation effects. Using the fluid equations of magnetoplasma with quantum corrections due to the Bohm potential, temperature degeneracy, spin magnetization energy, and exchange correlation, a generalized dispersion relation is derived. Spin effects are incorporated via spin force and macroscopic spin magnetization current. The exchange-correlation potentials are used, based on the adiabatic local-density approximation, and can be described as a function of the electron density. For three different values of angle, the dispersion relation is reduced to three different modes under the low frequency magnetohydrodynamic assumptions. It is found that the effects of quantum corrections in the presence of dust concentration significantly modify the dispersive properties of these modes. The results are useful for understanding numerous collective phenomena in quantum plasmas, such as those in compact astrophysical objects (e.g., the cores of white dwarf stars and giant planets) and in plasma-assisted nanotechnology (e.g., quantum diodes, quantum free-electron lasers, etc.)

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2013

NONLINEAR EVOLUTION OF THE RADIATION-DRIVEN MAGNETO-ACOUSTIC INSTABILITY

We examine the nonlinear development of unstable magnetosonic waves driven by a background radiative flux-the radiation-driven magneto-acoustic instability (RMI, a.k.a. the ''photon bubble'' instability). The RMI may serve as a persistent source of density, radiative flux, and magnetic field fluctuations in stably stratified, optically thick media. The conditions for instability are present in a variety of astrophysical environments and do not require the radiation pressure to dominate or the magnetic field to be strong. Here, we numerically study the saturation properties of the RMI, covering three orders of magnitude in the relative strength of radiation, magnetic field, and gas energies. Two-dimensional, time-dependent radiation-magnetohydrodynamic simulations of local, stably stratified domains are conducted with Zeus-MP in the optically thick, highly conducting limit. Our results confirm the theoretical expectations of Blaes and Socrates in that the RMI operates even in gas-pressure-dominated environments that are weakly magnetized. The saturation amplitude is a monotonically increasing function of the ratio of radiation to gas pressure. Keeping this ratio constant, we find that the saturation amplitude peaks when the magnetic pressure is comparable to the radiation pressure. We discuss the implications of our results for the dynamics of magnetized stellar envelopes, where the RMI should act as a source of sub-photospheric perturbations.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2012

The onset of impulsive bursty reconnection at a two-dimensional current layer

The sudden reconnection of a non-force free 2D current layer, embedded in a low-beta plasma, triggered by the onset of an anomalous resistivity, is studied in detail. The resulting behaviour consists of two main phases. First, a transient reconnection phase, in which the current in the layer is rapidly dispersed and some flux is reconnected. This dispersal of current launches a family of small amplitude magnetic and plasma perturbations, which propagate away from the null at the local fast and slow magnetosonic speeds. The vast majority of the magnetic energy released in this phase goes into internal energy of the plasma, and only a tiny amount is converted into kinetic energy. In the wake of the outwards propagating pulses, an imbalance of Lorentz and pressure forces creates a stagnation flow which drives a regime of impulsive bursty reconnection, in which fast reconnection is turned on and off in a turbulent manner as the current density exceeds and falls below a critical value. During this phase, the null current density is continuously built up above a certain critical level, then dissipated very rapidly, and built up again, in a stochastic manner. Interestingly, the magnetic energy converted during this quasi-steady phase is greater than that converted during the initial transient reconnection phase. Again essentially all the energy converted during this phase goes directly to internal energy. These results are of potential importance for solar flares and coronal heating, and set a conceptually important reference for future 3D studies.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2015

Simulation study of high-frequency energetic particle driven geodesic acoustic mode

High-frequency energetic particle driven geodesic acoustic modes (EGAM) observed in the large helical device plasmas are investigated using a hybrid simulation code for energetic particles and magnetohydrodynamics (MHD). Energetic particle inertia is incorporated in the MHD momentum equation for the simulation where the beam ion density is comparable to the bulk plasma density. Bump-on-tail type beam ion velocity distribution created by slowing down and charge exchange is considered. It is demonstrated that EGAMs have frequencies higher than the geodesic acoustic modes and the dependence on bulk plasma temperature is weak if (1) energetic particle density is comparable to the bulk plasma density and (2) charge exchange time (τ{sub cx}) is sufficiently shorter than the slowing down time (τ{sub s}) to create a bump-on-tail type distribution. The frequency of high-frequency EGAM rises as the energetic particle pressure increases under the condition of high energetic particle pressure. The frequency also increases as the energetic particle pitch angle distribution shifts to higher transit frequency. It is found that there are two kinds of particles resonant with EGAM: (1) trapped particles and (2) passing particles with transit frequency close to the mode frequency. The EGAMs investigated in this work are destabilized primarily by the passing particles whose transit frequencies are close to the EGAM frequency.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2012

QUANTIFYING SPICULES

Understanding the dynamic solar chromosphere is fundamental in solar physics. Spicules are an important feature of the chromosphere, connecting the photosphere to the corona, potentially mediating the transfer of energy and mass. The aim of this work is to study the properties of spicules over different regions of the Sun. Our goal is to investigate if there is more than one type of spicule, and how spicules behave in the quiet Sun, coronal holes, and active regions. We make use of high cadence and high spatial resolution Ca II H observations taken by Hinode/Solar Optical Telescope. Making use of a semi-automated detection algorithm, we self-consistently track and measure the properties of 519 spicules over different regions. We find clear evidence of two types of spicules. Type I spicules show a rise and fall and have typical lifetimes of 150-400 s and maximum ascending velocities of 15-40 km s{sup -1}, while type II spicules have shorter lifetimes of 50-150 s, faster velocities of 30-110 km s{sup -1}, and are not seen to fall down, but rather fade at around their maximum length. Type II spicules are the most common, seen in the quiet Sun and coronal holes. Type I spicules are seen mostly in active regions. There are regional differences between quiet-Sun and coronal hole spicules, likely attributable to the different field configurations. The properties of type II spicules are consistent with published results of rapid blueshifted events (RBEs), supporting the hypothesis that RBEs are their disk counterparts. For type I spicules we find the relations between their properties to be consistent with a magnetoacoustic shock wave driver, and with dynamic fibrils as their disk counterpart. The driver of type II spicules remains unclear from limb observations.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2012

THE SOURCE OF 3 MINUTE MAGNETOACOUSTIC OSCILLATIONS IN CORONAL FANS

We use images of high spatial, spectral, and temporal resolution, obtained using both ground- and space-based instrumentation, to investigate the coupling between wave phenomena observed at numerous heights in the solar atmosphere. Analysis of 4170 A continuum images reveals small-scale umbral intensity enhancements, with diameters {approx}0.''6, lasting in excess of 30 minutes. Intensity oscillations of Almost-Equal-To 3 minutes are observed to encompass these photospheric structures, with power at least three orders of magnitude higher than the surrounding umbra. Simultaneous chromospheric velocity and intensity time series reveal an 87 Degree-Sign {+-} 8 Degree-Sign out-of-phase behavior, implying the presence of standing modes created as a result of partial wave reflection at the transition region boundary. We find a maximum waveguide inclination angle of Almost-Equal-To 40 Degree-Sign between photospheric and chromospheric heights, combined with a radial expansion factor of <76%. An average blueshifted Doppler velocity of Almost-Equal-To 1.5 km s{sup -1}, in addition to a time lag between photospheric and chromospheric oscillatory phenomena, confirms the presence of upwardly propagating slow-mode waves in the lower solar atmosphere. Propagating oscillations in EUV intensity are detected in simultaneous coronal fan structures, with a periodicity of 172 {+-} 17 s and a propagation velocity of 45 {+-} 7 km s{sup -1}. Numerical simulations reveal that the damping of the magnetoacoustic wave trains is dominated by thermal conduction. The coronal fans are seen to anchor into the photosphere in locations where large-amplitude umbral dot (UD) oscillations manifest. Derived kinetic temperature and emission measure time series display prominent out-of-phase characteristics, and when combined with the previously established sub-sonic wave speeds, we conclude that the observed EUV waves are the coronal counterparts of the upwardly propagating magnetoacoustic slow modes detected in the lower solar atmosphere. Thus, for the first time, we reveal how the propagation of 3 minute magnetoacoustic waves in solar coronal structures is a direct result of amplitude enhancements occurring in photospheric UDs.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1993

Magnetohydrodynamic wave propagation in one-dimensional nonhomogeneous, self-gravitating clouds

We study the propagation of magnetohydrodynamic (MHD) waves through nonhomogeneous, selfgravitating, magnetic media representative of molecular cloud environments and focus on the issues of cloud support and line profiles. Since the general treatment of this topic is burdened by a complex mathematical formalism, we consider simplifying geometries which yield analytical solutions in the linear wave limit. In particular, we study both magnetoacoustic and Alfven wave propagation along the density gradient in a one-imensional slab. Of specific relevance to molecular clouds, we find that the back reaction of the Alfven waves can provide a pressure along the direction of the magnetic field lines; this pressure can help support a density enhancement against gravitational collapse. Furthermore, we find that the velocity amplitudes of these waves increase as the density decreases, in rough agreement with observational estimates of line width versus density relations.

Fatuzzo, Marco↗

DOE OSTI2016

OBSERVATIONS OF MAGNETIC FLUX-ROPE OSCILLATION DURING THE PRECURSOR PHASE OF A SOLAR ERUPTION

Based on combined observations from the Interface Region Imaging Spectrograph (IRIS) spectrometer with the coronal emission line of Fe xxi at 1354.08 Å and SDO /AIA images in multiple passbands, we report the finding of the precursor activity manifested as the transverse oscillation of a sigmoid, which is likely a pre-existing magnetic flux rope (MFR), that led to the onset of an X class flare and a fast halo coronal mass ejection (CME) on 2014 September 10. The IRIS slit is situated at a fixed position that is almost vertical to the main axis of the sigmoid structure that has a length of about 1.8 × 10{sup 5} km. This precursor oscillation lasts for about 13 minutes in the MFR and has velocities in the range of [−9, 11] km s{sup −1} and a period of ∼280 s. Our analysis, which is based on the temperature, density, length, and magnetic field strength of the observed sigmoid, indicates that the nature of the oscillation is a standing wave of fast magnetoacoustic kink mode. We further find that the precursor oscillation is excited by the energy released through an external magnetic reconnection between the unstable MFR and the ambient magnetic field. It is proposed that this precursor activity leads to the dynamic formation of a current sheet underneath the MFR that subsequently reconnects to trigger the onset of the main phase of the flare and the CME.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Novel internal measurements of ion cyclotron frequency range fast-ion driven modes

Abstract Novel internal measurements and analysis of ion cyclotron frequency range fast-ion driven modes in DIII-D are presented. Observations, including internal density fluctuation ( n ~ ) measurements obtained via Doppler backscattering, are presented for modes at low harmonics of the ion cyclotron frequency localized in the edge. The measurements indicate that these waves, identified as coherent ion cyclotron emission (ICE), have high wave number, k ⊥ ρ fast ≳ 1, consistent with the cyclotron harmonic wave branch of the magnetoacoustic cyclotron instability, or electrostatic instability mechanisms. Measurements show extended spatial structure (at least ∼1/6 the minor radius). These edge ICE modes undergo amplitude modulation correlated with edge localized modes (ELM) that is qualitatively consistent with expectations for ELM-induced fast-ion transport.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2017

Nonlinear Evolution of Short-wavelength Torsional Alfvén Waves

We analyze nonlinear evolution of torsional Alfvén waves in a straight magnetic flux tube filled in with a low- β plasma, and surrounded with a plasma of lower density. Such magnetic tubes model, in particular, a segment of a coronal loop or a polar plume. The wavelength is taken comparable to the tube radius. We perform a numerical simulation of the wave propagation using ideal magnetohydrodynamics. We find that a torsional wave nonlinearly induces three kinds of compressive flows: the parallel flow at the Alfvén speed, which constitutes a bulk plasma motion along the magnetic field, the tube wave, and also transverse flows in the radial direction, associated with sausage fast magnetoacoustic modes. In addition, the nonlinear torsional wave steepens and its propagation speed increases. The latter effect leads to the progressive distortion of the torsional wave front, i.e., nonlinear phase mixing. Because of the intrinsic non-uniformity of the torsional wave amplitude across the tube radius, the nonlinear effects are more pronounced in regions with higher wave amplitudes. They are always absent at the axes of the flux tube. In the case of a linear radial profile of the wave amplitude, the nonlinear effects are localized in an annulus region near the tube boundary. Thus, the parallel compressive flows driven by torsional Alfvén waves in the solar and stellar coronae, are essentially non-uniform in the perpendicular direction. The presence of additional sinks for the wave energy reduces the efficiency of the nonlinear parallel cascade in torsional Alfvén waves.

79 ASTRONOMY AND ASTROPHYSICS↗