Resonant Alfven Waves Research Articles

The Field of Shock‐Generated Alfven Oscillations Near the Plasmapause

AbstractA theoretical interpretation is given for magnetohydrodynamic (MHD) oscillations observed by the GOES and Cluster satellites on November 7, 2004 near the plasmapause (Zong et al., 2009, 2012, https://doi.org/10.1029/2009ja014393, https://doi.org/10.1029/2012ja018024), before and after an interplanetary shock passes through the magnetosphere (about 18:27 UT). It is shown that, after the shock front passage, the MHD oscillation spectra recorded by Cluster (C1) and GOES 12, located respectively inside and outside the plasmasphere, differ significantly. According to our interpretation, these differences are due to the shock wavefront affecting the plasmapause and exciting broadband resonant Alfven waves, thus significantly changing the MHD oscillation spectrum. A model in the form of a plasma cylinder was developed in order to study MHD oscillations in the dayside magnetosphere numerically. An analytical model is proposed for the source of resonant Alfven waves that has the form of two fast magnetosonic wave packets propagating along the plasmapause, resulting from the shock wave/plasmapause interaction. Numerically calculated parameters of Alfven waves excited by a source based on this model show a good agreement with the oscillation parameters observed by satellites.

ENSEMBLE SIMULATIONS OF PROTON HEATING IN THE SOLAR WIND VIA TURBULENCE AND ION CYCLOTRON RESONANCE

Protons in the solar corona and heliosphere exhibit anisotropic velocity distributions, violation of magnetic moment conservation, and a general lack of thermal equilibrium with the other particle species. There is no agreement about the identity of the physical processes that energize non-Maxwellian protons in the solar wind, but a traditional favorite has been the dissipation of ion cyclotron resonant Alfven waves. This paper presents kinetic models of how ion cyclotron waves heat protons on their journey from the corona to interplanetary space. It also derives a wide range of new solutions for the relevant dispersion relations, marginal stability boundaries, and nonresonant velocity-space diffusion rates. A phenomenological model containing both cyclotron damping and turbulent cascade is constructed to explain the suppression of proton heating at low alpha-proton differential flow speeds. These effects are implemented in a large-scale model of proton thermal evolution from the corona to 1 AU. A Monte Carlo ensemble of realistic wind speeds, densities, magnetic field strengths, and heating rates produces a filled region of parameter space (in a plane described by the parallel plasma beta and the proton temperature anisotropy ratio) similar to what is measured. The high-beta edges of this filled region are governed by plasma instabilities and strong heating rates. The low-beta edges correspond to weaker proton heating and a range of relative contributions from cyclotron resonance. On balance, the models are consistent with other studies that find only a small fraction of the turbulent power spectrum needs to consist of ion cyclotron waves.

Magnetosonic resonances in the magnetospheric plasma

A problem of coupling between fast and slow magnetosonic waves in Earth’s magnetosphere (magnetosonic resonance) is examined. Propagation both slow magnetosonic wave and Alfven wave can easily be canalized along the magnetic field line direction. The main difference between the two is that slow magnetosonic waves dissipate strongly due to their interaction with the background plasma ions, whose temperature is above the electron temperature. In Earth’s magnetosphere, however, there is a region where the dissipation of slow magnetosonic waves can be weak—the inner plasmasphere. The slow magnetosonic waves generated there can be registered directly. In other regions, with strong dissipation of slow magnetosonic waves, their signature may be detected through their impact on the Alfven resonance at frequencies for which the resonant Alfven and slow magnetosonic waves exist simultaneously in the magnetosphere. Owing to their strong coupling with the background plasma ions, resonant slow magnetosonic waves can transfer the energy and impulse from the solar wind to the magnetospheric plasma ions via fast magnetosonic waves penetrating into the tail lobes. A problem of resonant conversion of fast magnetosonic waves into slow magnetosonic oscillations in a magnetosphere with dipole-like magnetic field is also examined.

The Two Sources of Solar Energetic Particles

Evidence for two different physical mechanisms for acceleration of solar energetic particles (SEPs) arose 50 years ago with radio observations of type III bursts, produced by outward streaming electrons, and type II bursts from coronal and interplanetary shock waves. Since that time we have found that the former are related to "impulsive" SEP events from impulsive flares or jets. Here, resonant stochastic acceleration, related to magnetic reconnection involving open field lines, produces not only electrons but 1000-fold enhancements of 3He/4He and of (Z>50)/O. Alternatively, in "gradual" SEP events, shock waves, driven out from the Sun by coronal mass ejections (CMEs), more democratically sample ion abundances that are even used to measure the coronal abundances of the elements. Sometimes residual impulsive suprathermal ions contribute to the seed population for shock acceleration, complicating the abundance picture, but this process has now been modeled theoretically. Initially, impulsive events define a point source on the Sun, selectively filling few magnetic flux tubes, while gradual events show broad acceleration that can fill half of the inner heliosphere, beginning when the shock reaches ~2 solar radii. Shock acceleration occurs as ions are scattered back and forth across the shock by resonant Alfven waves amplified by the accelerated protons themselves. These waves also can produce a streaming-limited maximum SEP intensity and plateau region upstream of the shock. Behind the shock lies the large expanse of the "reservoir", a spatially extensive trapped volume of uniform SEP intensities with invariant energy-spectral shapes where overall intensities decrease with time as the enclosing "magnetic bottle" expands adiabatically. These reservoirs now explain the slow intensity decrease that defines gradual events, once erroneously attributed solely to slow outward diffusion.

SELF-CONSISTENT MODEL OF THE INTERSTELLAR PICKUP PROTONS, ALFVÉNIC TURBULENCE, AND CORE SOLAR WIND IN THE OUTER HELIOSPHERE

A self-consistent model of the interstellar pickup protons, the slab component of the Alfvenic turbulence, and core solar wind (SW) protons is presented for r ≥ 1 along with the initial results of and comparison with the Voyager 2 (V2) observations. Two kinetic equations are used for the pickup proton distribution and Alfvenic power spectral density, and a third equation governs SW temperature including source due to the Alfven wave energy dissipation. A fraction of the pickup proton free energy, fD , which is actually released in the waveform during isotropization, is taken from the quasi-linear consideration without preexisting turbulence, whereas we use observations to specify the strength of the large-scale driving, C sh, for turbulence. The main conclusions of our study can be summarized as follows. (1) For C sh 1-1.5 and fD 0.7-1, the model slab component agrees well with the V2 observations of the total transverse magnetic fluctuations starting from ~8 AU. This indicates that the slab component at low-latitudes makes up a majority of the transverse magnetic fluctuations beyond 8-10 AU. (2) The model core SW temperature agrees well with the V2 observations for r 20 AU if fD 0.7-1. (3) A combined effect of the Wentzel-Kramers-Brillouin attenuation, large-scale driving, and pickup proton generated waves results in the energy sink in the region r 10 AU, while wave energy is pumped in the turbulence beyond 10 AU. Without energy pumping, the nonlinear energy cascade is suppressed for r 10 AU, supplying only a small energy fraction into the k-region of dissipation by the core SW protons. A similar situation takes place for the two-dimensional turbulence. (4) The energy source due to the resonant Alfven wave damping by the core SW protons is small at heliocentric distances r 10 AU for both the slab and the two-dimensional turbulent components. As a result, adiabatic cooling mostly controls the model SW temperature in this region, and the model temperature disagrees with the V2 observations in the region r 20 AU.

Resonant Alfvén waves in partially ionized plasmas of the solar atmosphere

Context. Magnetohydrodynamic (MHD) waves are ubiquitous in the solar atmosphere. In magnetic waveguides resonant absorption due to plasma inhomogeneity naturally transfers wave energy from large-scale motions to small-scale motions. In the cooler parts of the solar atmosphere as, e.g., the chromosphere, effects due to partial ionization may be relevant for wave dynamics and heating. Aims. We study resonant Alfven waves in partially ionized plasmas. Methods. We use the multifluid equations in the cold plasma approximation. We investigate propagating resonant MHD waves in partially ionized flux tubes. We use approximate analytical theory based on normal modes in the thin tube and thin boundary approximations along with numerical eigenvalue computations. Results. We find that the jumps of the wave perturbations across the resonant layer are the same as in fully ionized plasmas. The damping length due to resonant absorption is inversely proportional to the frequency, while that due to ion-neutral collisions is inversely proportional to the square of the frequency. For observed frequencies in the solar atmosphere, the amplitude of MHD kink waves is more efficiently damped by resonant absorption than by ion-neutral collisions. Conclusions. Most of the energy carried by chromospheric kink waves is converted into localized azimuthal Alfven waves that can deposit energy in the coronal medium. The dissipation of wave energy in the chromosphere due to ion-neutral collisions is only effective for high-frequency waves. The chromosphere acts as a filter for kink waves with periods shorter than 10 s.

Transformation and absorption of MHD oscillations in plane-stratified models of the Earth’s magnetosphere

The process of resonant transformation of fast magnetosonic (FMS) waves originating from the solar wind in Alfven and slow magnetosonic (SMS) oscillations in the Earth’s magnetosphere is investigated. The study of this process in a one-dimensional inhomogeneous medium model shows that the presence of a resonant surface for SMS oscillations in the plasma configuration significantly increases the absorption of the energy of FMS waves incident on the magnetosphere. The spatial distribution of the absorption rate of an energy flux of FMS waves is examined. In numerical calculations, the Kolmogorov spectrum was used for FMS waves, which is typical of the waves in the transition layer between a shock wave and the magnetopause. It is shown that the rate of energy absorption of FMS waves due to the resonant excitation of SMS oscillations is by several orders of magnitude greater than the absorption of their energy associated with resonant Alfven waves at the same magnetic shells.

DETERMINATION OF NON-THERMAL VELOCITY DISTRIBUTIONS FROM SERTS LINEWIDTH OBSERVATIONS

Non-thermal velocities obtained from the measurement of coronal Extreme Ultraviolet (EUV) linewidths have been consistently observed in solar EUV spectral observations and have been theorized to result fro m many plausible scenarios including wave motions, turbulence, or magnetic reconnection. Constraining these velocities can provide a physical limit for the available energy resulting from unresolved motions in the corona. We statistically determine a series of non-thermal velocity distributions from linewidth measurements of 390 emission lines from a wide array of elements and ionization states observed during the Solar Extreme Ultraviolet Research Telescope and Spectrograph 1991-1997 flights covering the spectral range 174-418 A and a temperature range from 80,000 K to 12.6 MK. This sample includes 248 lines from active regions, 101 lines from quiet-Sun regions, and 41 lines were observed from plasma off the solar limb. We find a strongly peaked distribution corresponding to a non-thermal velocity of 19-22 km/s in all three of the quiet-Sun, active region, and off-limb distributions. For the possibility of Alfven wave resonance heating, we find thai velocities in the core of these distributions do not provide sufficient energy, given typical densities and magnetic field strengths for the coronal plasma, to overcome the estimated coronal energy losses required to maintain the corona at the typical temperatures working as the sole mechanism. We find that at perfect efficiency 50%-60% of the needed energy flux can be produced from the non-thermal velocities measured.

MHD Waves and Instabilities in Space Plasma Flows

Shear flow instabilities are an important aspect of hydrodynamic studies. The present review article discusses the role of an ambient magnetic field which both modifies the Kelvin-Helmholtz instability and may introduce new types of magnetohydrodynamic waves and instabilities. A brief overview of magnetospheric pulsations is presented with an emphasis on the long-period resonant Alfven waves associated with the high speed solar wind. The spatio-temporal evolution of magnetically modified shear flow instabilities in various space plasma structures is addressed. A distinction between convective and absolute instabilities is necessary for proper understanding of theory and correct interpretation of the observations. Finally, it is shown how incompressible Alfvenic disturbances may become unstable in a compressible flow in the absence of any shear. An application to coronal loops is presented.

STREAMING-LIMITED INTENSITIES OF SOLAR ENERGETIC PARTICLES ON THE INTENSITY PLATEAU

We examine the energy spectra of H, He, O, and Fe ions on the temporal intensity plateau region in large solar energetic-particle (SEP) events, where intensities may be ''streaming limited.'' Upstream of shock waves near the Sun, equilibrium may occur when outwardly streaming protons amplify resonant Alfven waves that then scatter subsequent protons sufficiently to reduce the streaming. In the largest SEP events, the so-called ground-level events (GLEs), we find proton energy spectra that are peaked near {approx}10 MeV with the energy of similar peaks decreasing for heavier ions and for smaller events. These spectra contrast sharply with spectra near the time of shock passage which rise monotonically above the plateau spectra with decreasing energy. We suggest that strong suppression of upstream ion intensities near {approx}1 MeV amu{sup -1} on the plateau occurs when those ions resonate with waves amplified earlier by streaming protons of {approx}10 MeV and above. GLEs with much lower intensities of 10-100 MeV protons on the plateau show spectra of ions that rise monotonically toward low energies with no peaking and no suppression of low-energy ions. Wave amplification by streaming protons and the pitch-angle dependence of the resonance condition are essential factors in our understanding ofmore » the limiting behavior.« less

NONLINEAR PROPAGATION OF ALFVÉN WAVES DRIVEN BY OBSERVED PHOTOSPHERIC MOTIONS: APPLICATION TO THE CORONAL HEATING AND SPICULE FORMATION

We have performed MHD simulations of Alfven wave propagation along an open flux tube in the solar atmosphere. In our numerical model, Alfven waves are generated by the photospheric granular motion. As the wave generator, we used a derived temporal spectrum of the photospheric granular motion from G-band movies of Hinode/SOT. It is shown that the total energy flux at the corona becomes larger and the transition region height becomes higher in the case when we use the observed spectrum rather than white/pink noise spectrum as the wave generator. This difference can be explained by the Alfven wave resonance between the photosphere and the transition region. After performing Fourier analysis on our numerical results, we have found that the region between the photosphere and the transition region becomes an Alfven wave resonant cavity. We have confirmed that there are at least three resonant frequencies, 1, 3 and 5 mHz, in our numerical model. Alfven wave resonance is one of the most effective mechanisms to explain the dynamics of the spicules and the sufficient energy flux to heat the corona.

Resonant absorption with 2D variation of field line eigenfrequencies

Context. Resonant absorption, also known as field line resonance, can be used to describe coupling between fast and Alfven waves in non-uniform plasmas. Since the conditions for resonant absorption occur widely in astrophysics, it is applicable in many different contexts, all of which are united by their common physics. For example, resonant absorption is known to play a major role in the excitation of ultra-low frequency pulsations in the terrestrial magnetosphere and is also a leading explanation for the decay of fast kink oscillations of coronal loops. The occurrence of non-axisymmetric conditions in the magnetosphere, and observational evidence that coronal loops may possess fine transverse structure, highlight a need to consider equilibria that vary in two dimensions across the background magnetic field. Aims. We investigate the properties of resonant absorption when field line eigenfrequencies vary in two dimensions across the background magnetic field. We aim to place the theory on a firm mathematical footing and explore some of its key features. Methods. Using cold, linear, ideal MHD with a straight, uniform background magnetic field, we systematically obtain a complete analytic solution for behaviour at late times. This provides a framework from which the features of resonant absorption may be understood. The time-dependent problem is solved numerically, reproducing key features of the analytic solution. Results. Energy is deposited from a monochromatic fast wave as a phase mixing Alfven wave, in the vicinity of the resonant surface, at which the local field line eigenfrequency matches the frequency of the driver. A generalisation of the one dimensional phase mixing length to higher dimensions is suggested, and shown to successfully estimate the finest lengthscales in time-dependent simulations. The resonant Alfven wave is driven by gradients of the field aligned magnetic field perturbation, which is associated with the fast wave pressure. This leads to amplitude variations of the Alfven wave that can be used to reveal the spatial form of the fast wave.

Simultaneous observations of the plasma density on the same field line by the CPMN ground magnetometers and the Cluster satellites

By applying the cross-phase method and the amplitude-ratio method to magnetic field data obtained from two ground stations located close to each other, we can determine the frequency of the field line resonance (FLR), or the field line eigenfrequency, for the field line running through the midpoint of the two stations. From thus identified FLR frequency we can estimate the equatorial plasma mass density ( ρ ) by using the T05s magnetospheric field model [Tsyganenko, N.A., Sitnov, M.I. Modeling the dynamics of the inner magnetosphere during strong geomagnetic storms, J. Geophys. Res. 110, A03208, 2005] and the equation of Singer et al. [Singer, H.J., Southwood, D.J., Walker, R.J., Kivelson, M.G. Alfven wave resonances in a realistic magnetospheric magnetic field geometry, J. Geophys. Res. 86 (A6) 4589–4596, 1981]. In this study we compare ρ estimated from magnetometer data at two stations in the CPMN (Circum-pan Pacific Magnetometer Network) chain, Tixie (TIK, geographic coordinates: 71.59°N, 128.78°E, L = 6.05 ) and Chokurdakh (CHD, geographic coordinates: 70.62°N, 147.89°E, L = 5.61 ), with the plasma electron number density ( Ne) observed by the WHISPER (Waves of HIgh frequency Sounder for Probing the Electron density by Relaxation) instrument onboard the Cluster satellites. For the interval of January 1, 2001–December 31, 2005, we have identified 19 events in which the Cluster spacecraft were located on the field line running through the midpoint of TIK and CHD when they observed FLR, and statistically compared the simultaneously observed ρ and Ne, although the number of events are limited (19). In 15 out of the 19 events the ratio of ρ to Ne falls into a realistic range. It is also suggested that the contribution of heavy ions tends to increase when the magnetosphere is disturbed.

Nonrelativistic and Relativistic Treatments for Propagation of Torsional Resonant Alfven Waves in Strongly Magnetized Neutron Stars

We study the propagation and transmission of torsional Alfven waves in magnetars. First, we consider a simple two-layer model composed of an outer crust and vaccum magnetosphere. In this model, we derive the classical formulation for the transmission coefficient of Alfven waves using an MHD approximation. We then include a displacement current in Maxwell’s equations, which can be shown to become important in the magnetically relativistic region of the crust, B 4πρc. The effects of strong magnetic fields of magnetars are specifically quantified and discussed. Second, we consider a three-layer model in order to investigate the propagation properties of Alfven waves excited in the deep crust. This model is adopted for the inner crust, outer crust, and vacuum magnetosphere. It is shown that the outer crust plays a significant role as a resonant cavity if a certain condition holds. Some nonrelativistic resonance of Alfven modes with low frequencies obtained in this work is probably associated with the observed QPOs from soft gamma-ray repeaters. The estimated energy flux carried by relativistic Alfven waves is in good agreement with the observed values, E = 10–10 erg s−1, of giant flares.

The coupling resonance of torsional alfven wave and fast wave in coronal loops

The method of Orthogonal Function Series Expansion (OFSE) is generalized and applied to the study of the evolution of the coupling of nondissipative torsional Alfven wave and fast wave in coronal loops. Using this method, the intrinsic angular frequency of the overall wave mode can be described mathematically and that of the Alfven waves along the magnetic lines in the coronal loop during the coupling of the Alfven and fast waves can be analyzed both theoretically and numerically. Also with this method, the relation between the coupling driven term and the Alfven wave resonance may be analyzed. Results of computation reveal the place of appearance of coupling resonance as well as the characteristics of the amplitudes of the Alfven and fast waves. As found by the calculations, if the footpoint driven angular frequency is not equal to the intrinsic angular frequency of the overall wave mode of the coronal loop and when a δ section appears at the place of coupled resonance, the radial gradient of the fast wave's amplitude is quite large. Sometimes it approximates to a discontinuity, and this is extremely favorable for the dissipation of the fast wave. If the footpoint driven angular frequency is equal to the intrinsic angular frequency of the overall wave mode and when a δ section occurs in the Alfven wave amplitude, abundant small-scale structures appear in the radial direction. Then the location of resonance approximately becomes a discontinuity, very favorable to the dissipation of the Alfven wave.

“Swing Absorption” of fast magnetosonic waves in inhomogeneous media

The recently suggested swing interaction between fast magnetosonic and Alfven waves (Zaqarashvili and Roberts 2002) is generalized to inhomogeneous media. We show that the fast magnetosonic waves propagating across an applied non-uniform magnetic field can parametrically amplify the Alfven waves propagating along the field through the periodical variation of the Alfven speed. The resonant Alfven waves have half the frequency and the perpendicular velocity polarization of the fast waves. The wave lengths of the resonant waves have different values across the magnetic field, due to the inhomogeneity in the Alfven speed. Therefore, if the medium is bounded along the magnetic field, then the harmonics of the Alfven waves, which satisfy the condition for onset of a standing pattern, have stronger growth rates. In these regions the fast magnetosonic waves can be strongly 'absorbed', their energy going in transversal Alfven waves. We refer to this phenomenon as 'Swing Absorption'. This mechanism can be of importance in various astrophysical situations.

Multi-instrument observations of ULF wave-driven discrete auroral arcs propagating sunward and equatorward from the poleward boundary of the duskside auroral oval

An investigation into how discrete frequency fast mode waves propagating sunward from the magnetotail may drive field line resonant Alfvén waves on closed field lines on the afternoon flank of the Earth’s magnetosphere is presented. Due to the morphology of the magnetic field lines on the flank, higher latitude poleward boundary auroral field lines map deeper into the magnetotail, implying that phase lags from mode coupling to sunward propagating fast mode waves in this stretched field geometry can overcome the standard poleward phase propagation of typical field line resonances, producing resonant Alfvén wave-driven discrete auroral arcs that propagate equatorward on closed field lines. A comparison is made between the predicted results and magnetometer, all sky camera and meridian scanning photometer observations of multiple east–west aligned discrete auroral arcs that propagate equatorward and sunward from the poleward boundary of the dayside auroral oval between 1400 and 1500 UT (approximately 1600 to 1700 MLT) on 3 January 1998. From this comparison, the conclusion is reached that the observations are consistent with the theoretical picture of discrete auroral arcs being driven by Alfvénic waves on closed field lines, the Alfvén waves being excited by discrete frequency fast mode waves propagating sunward from the magnetotail in the flank waveguide of the outer magnetosphere.

A mathematical description of the torsional Alfven wave resonance in coronal loopst

A method called complete orthogonal function series expression (OFSE) in Hilbert space is proposed to solve the non-dissipative torsional Alfven wave in coronal loops. Every base function corresponds to an intrinsic angular frequency ω n of each magnetic field line in the loop. Torsional Alfven wave resonance of the magnetic field lines arises when the driving angular frequency is equal to its intrinsic angular frequency. Using this method, we present a new form of the solution of torsional Alfven wave evolution under the boundary condition of being driven at the two foot-points. In the solution there exists a resonance term, from which it may be found that near the place of resonance with angular frequency ω a δ-type discontinuity appears at time t equal to odd multiples of π/(2ω). It is also found that the wave amplitude at the place of resonance linearly increases with time and the rate of increase is proportional to the Alfven wave speed, and inversely proportional to the length of the loop and is independent of the driving frequency.

Modeling Shock‐accelerated Solar Energetic Particles Coupled to Interplanetary Alfven Waves

We present an idealized model simulating the coupled evolution of the distributions of multispecies shock-accelerated energetic ions and interplanetary Alfven waves in gradual solar energetic particle (SEP) events. Particle pitch-angle diffusion coefficients are expressed in terms of wave intensities, and wave growth rates in terms of momentum gradients of SEP distributions, by the same quasilinear theory augmented with resonance broadening. The model takes into consideration various physical processes: for SEPs, particle motion, magnetic focusing, scattering by Alfven waves, solar wind convection, and adiabatic deceleration; for the waves, WKB transport and amplification by streaming SEPs. Shock acceleration is heuristically represented by continuous injection of prescribed spectra of SEPs at a moving shock front. We show the model predictions for two contrasting sets of SEP source spectra, fast weakening and softening in one case and long lasting and hard in the other. The results presented include concurrent time histories of multispecies SEP intensities and elemental abundance ratios, as well as sequential snapshots of the following: SEP intensity energy spectra, Alfven wave spectra, particle mean free paths as functions of rigidity, and spatial profiles of SEP intensities and mean free paths. Wave growth plays a key role in both cases, although the magnitude of the wave growth differs greatly, and quite different SEP abundance variations are obtained. In these simulations, the maximum wave growth rate is large, but small relative to the wave frequency, and everywhere the total wave magnetic energy density remains small relative to that of the background magnetic field. The simulations show that, as the energetic protons stream outward, they rapidly amplify the ambient Alfven waves, by several orders of magnitude in the inner heliosphere. Energetic minor ions find themselves traveling through resonant Alfven waves previously amplified by higher velocity protons. The nonuniformly growing wave spectra alter the rigidity dependence of particle scattering, resulting in complex time variations of SEP abundances at large distances from the Sun. The greatly amplified waves travel outward in an expanding and weakening shell, creating an expanding and falling reservoir of SEPs with flat spatial intensity profiles behind, while in and beyond the shell the intensities drop steeply. The wave-particle resonance relation dynamically links the evolving characteristics of the SEP and Alfven wave distributions in this new mode of SEP transport. We conclude that wave amplification, the counterpart to the scattering of streaming particles required by energy conservation, plays an essential role in the transport of SEPs in gradual SEP events. The steep proton-amplified wave spectra just upstream of the shock suggest that they may also be important in determining the elemental abundances of shock-accelerated SEP sources.

Fast and Alfvén waves driven by azimuthal footpoint motions

The excitation of Alfven and fast magneto-acoustic waves in coronal loops driven by footpoint motions is studied in linear, ideal MHD. The analysis is restricted to azimuthally polarized footpoint motions so that only Alfven waves are directly excited to couple to fast magneto-acoustic waves at later times. In the companion paper De Groof et al. (2002) (hereafter referred to as Paper I), the behaviour of the MHD waves is studied in case of a monochromatic driver. In the present study, the effects of a more realistic random driver are investigated. First, we consider loops of equal length and width in order to limit the number of quasi-modes in the frequency range of the driver so that the influence of quasi-modes in the system can easily be detected. In contrast to the single resonant surface which was found in case of a periodic driver (see Paper I), a random pulse train excites a variety of resonant Alfven waves and consequently the small length scales built up are spread over the whole width of the loop. The specific effects of the quasi-modes are not so easily recognized as for radial footpoint motions (De Groof & Goossens 2000) since the resonances corresponding to directly and indirectly excited Alfven waves are mixed together. In the second part of the paper, longer loops are considered. Since more quasi-modes are involved, the resonant surfaces are more numerous and widely spread throughout the whole loop volume. On the other hand, it takes more time for the MHD waves to cross the loop and to form standing waves. Nevertheless this negative effect does not have too much impact since the simulations show that after a small time interval, resonant surfaces are created all over the loop, with length scales which are short enough for effective dissipation.