Solar TErrestrial Relations Observatory-A (STEREO-A) and PRoject for On-Board Autonomy 2 (PROBA2) Quadrature Observations of Reflections of Three EUV Waves from a Coronal Hole

We investigate the interaction of three consecutive large-scale coronal waves with a polar coronal hole, simultaneously observed on-disk by the Solar TErrestrial Relations Observatory (STEREO)-A spacecraft and on the limb by the PRoject for On-Board Autonomy 2 (PROBA2) spacecraft on 27 January 2011. All three extreme ultraviolet (EUV) waves originate from the same active region, NOAA 11149, positioned at N30E15 in the STEREO-A field of view and on the limb in PROBA2. For the three primary EUV waves, we derive starting velocities in the range of ≈?310 km?s^?1 for the weakest up to ≈?500 km?s^?1 for the strongest event. Each large-scale wave is reflected at the border of the extended coronal hole at the southern polar region. The average velocities of the reflected waves are found to be smaller than the mean velocities of their associated direct waves. However, the kinematical study also reveals that in each case the ending velocity of the primary wave matches the initial velocity of the reflected wave. In all three events, the primary and reflected waves obey the Huygens–Fresnel principle, as the incident angle with ≈?10° to the normal is of the same magnitude as the angle of reflection. The correlation between the speed and the strength of the primary EUV waves, the homologous appearance of both the primary and the reflected waves, and in particular the EUV wave reflections themselves suggest that the observed EUV transients are indeed nonlinear large-amplitude MHD waves.     

A Global Model of MHD Wave Support of a Stratified Molecular Cloud

We perform numerical simulations of nonlinear MHD waves in a gravitationally stratified molecular cloud that is bounded by a hot and tenuous external medium, within a 1.5-dimensional approximation. Under the influence of a driving source of Alfvénic disturbances, the cloud is lifted up by the pressure of MHD waves and reaches a steady state characterized by oscillations about a new time-averaged equilibrium state. The nonlinear effect results in the generation of longitudinal motions and many shock waves. Models of an ensemble of clouds show that, for various strengths of the input energy, the velocity dispersion in the cloud σ ∝ Z ^0.5, where Z is a characteristic size of the cloud. Furthermore, σ is always comparable to the mean Alfvén velocity of the cloud, consistent with observational results.     

Generation and propagation of the ULF planetary-scale electromagnetic wavy structures in the ionosphere

In the present article, the results of theoretical investigation of the dynamics of generation and propagation of planetary (with wavelength 10³ km and more) ultra-low frequency (ULF) electromagnetic wave structures in the dissipative ionosphere are given. The physical mechanism of generation of the planetary electromagnetic waves is proposed. It is established, that the global factor, acting permanently in the ionosphere—inhomogeneity (latitude variation) of the geomagnetic field and angular velocity of the earth's rotation—generates the fast and slow planetary ULF electromagnetic waves. The waves propagate along the parallels to the east as well as to the west. In E-region the fast waves have phase velocities (2-20) km s⁻¹and frequencies (10⁻¹-10⁻⁴) s⁻¹; the slow waves propagate with local winds velocities and have frequencies (10⁻⁴-10⁻⁶) s⁻¹. In F-region the fast ULF electromagnetic waves propagate with phase velocities tens-hundreds km s⁻¹ and their frequencies are in the range of (10-10⁻³) s⁻¹. The slow mode is produced by the dynamoelectric field, it represents a generalization of the ordinary Rossby-type waves in the rotating ionosphere and is caused by the Hall effect in the E-layer. The fast disturbances are the new modes, which are associated with oscillations of the ionospheric electrons frozen in the geomagnetic field and are connected with the large-scale internal vortical electric field generation in the ionosphere. The large-scale waves are weakly damped. The features and the parameters of the theoretically investigated electromagnetic wave structures agree with those of large-scale ULF midlatitude long-period oscillations (MLO) and magnetoionospheric wave perturbations (MIWP), observed experimentally in the ionosphere. It is established, that because of relevance of Coriolis and electromagnetic forces, generation of slow planetary electromagnetic waves at the fixed latitude in the ionosphere can give rise to the reverse of local wind structures and to the direction change of general ionospheric circulation. It is considered one more class of the waves, called as the slow magnetohydrodinamic (MHD) waves, on which inhomogeneity of the Coriolis and Ampere forces do not influence. These waves appear as an admixture of the slow Alfven- and whistler-type perturbations. The waves generate the geomagnetic field from several tens to several hundreds nT and more. Nonlinear interaction of the considered waves with the local ionospheric zonal shear winds is studied. It is established, that planetary ULF electromagnetic waves, at their interaction with the local shear winds, can self-localize in the form of nonlinear solitary vortices, moving along the latitude circles westward as well as eastward with velocity, different from phase velocity of corresponding linear waves. The vortices are weakly damped and long lived. They cause the geomagnetic pulsations stronger than the linear waves by one order. The vortex structures transfer the trapped particles of medium and also energy and heat. That is why such nonlinear vortex structures can be the structural elements of strong macroturbulence of the ionosphere.     

Acceleration of the solar wind by whistler waves from coronal holes

We investigate the possibility of an additional acceleration of the high speed solar wind by whistler waves propagating outward from a coronal hole. We consider a stationary, spherically symmetric model and assume a radial wind flow as well as a radial magnetic field. The energy equation consists of (a) energy transfer of the electron beam which excites the whistler waves, and (b) energy transfer of the whistler waves described by conservation of wave action density. The momentum conservation equation includes the momentum transfer of two gases (a thermal gas and an electron beam). The variation of the temperature is described by a polytropic law. The variation of solar wind velocity with the radial distance is calculated for different values of energy density of the whistler waves. It is shown that the acceleration of high speed solar wind in the coronal hole due to the whistler waves is very important. We have calculated that the solar wind velocity at the earth's orbit is about equal to 670 km/sec (for wave energy density about 10^?4 erg cm^?3 at 1.1R⊙). It is in approximate agreement with the observed values.     

An analysis of the strong zonal circulation within the stratosphere of Venus

A dynamical model is presented for the observed strong zonal circulation within the stratosphere of Venus. The model neglects rotational effects and considers a compressible and radiating atmosphere. It is shown that diurnal radiative heating is negligible within the lower stratosphere, a region below 85km, while observational evidence for the strong zonal circulation pertains to the lower stratosphere within which a direct thermal driving for the circulation is absent. The analysis, however, suggests that propagating internal gravity waves generated by diurnal solar heating of the upper stratosphere induce mean zonal velocities within the upper and lower stratosphere.Considering the linearized equations of motion and energy, and following Stern's (1971) analysis for an analogous problem, it is shown that the zonal velocity induced by internal gravity waves is retrograde in direction, a result which is in agreement with observation. The nonlinear equations of motion and energy are then solved by an approximate analytical method to determine the magnitude of the zonal velocity. This velocity increases from zero at the tropopause to about 200 msec^?1 at the 85 km level. The velocity near the uv-cloud level compares favorably with the observed value of 100 msec^?1.     

Dynamics of a cloud of fast electrons travelling through the plasma

A simulation of normal type III radio bursts has been made in a whole frequency range of about 200 MHz to 30 kHz by the usage of the semi-analytical method as developed in previous papers for the plasma waves excited by a cloud of fast electrons. Three-dimensional plasma waves are computed, though the velocities of fast electrons are assumed to be one-dimensional. Many basic problems about type III radio bursts and associated solar electrons have been solved showing the following striking or unexpected results.Induced scattering of plasma waves, by thermal ions, into the plasma waves with opposite wave vectors is efficient even for a solar electron cloud of rather low number density. Therefore, the second harmonic radio emission as attributed to the coalescence of two plasma waves predominates in a whole range from meter waves to km waves. Fundamental radio emission as ascribed to the scattering of plasma waves by thermal ions is negligibly small almost in the whole range. On the other hand, third harmonic radio emission can be strong enough to be observed in a limited frequency range.If, however, the time integral of electron flux is, for example, 2 × 10¹³ cm^–2 (>5 keV) or more at the height of 4.3 × 10¹⁰ cm ( _p = 40 MHz) above the photosphere, the fundamental may be comparable with or greater than the second harmonic, but an effective area of cross-section of the electron beam is required to be very small, 10¹⁷ cm² or less, and hence much larger sizes of the observed radio sources must be attributed to the scattering alone of radio waves.The radio flux density expected at the Earth for the second harmonic can increase with decreasing frequencies giving high flux densities at low frequencies as observed, if x-dependence of the cross-sectional area of the electron beam is x ^1.5 or less instead of x ², at least at x 2 × 10¹² cm.The second harmonic radio waves are emitted predominantly into forward direction at first, but the direction of emission may reverse a few times in a course of a single burst showing a greater backward emission at the low frequencies.In a standard low frequency model, a total number of solar electrons above 18 keV arriving at the Earth orbit reduces to 12% of the initial value due mainly to the collisional decay of plasma waves before the waves are reabsorbed by the beam electrons arriving later. However, no deceleration of the apparent velocity of exciter appears. A change in the apparent velocity, if any, results from a change in growth rate of the plasma waves instead of the deceleration of individual electrons.Near the Earth, the peak of second harmonic radio flux as emitted from the local plasma appears well after the passage of a whole solar electron cloud through this layer. This is ascribed to the secondary and the third plasma waves as caused in non-resonant regions by the induced scattering of primary plasma waves in a resonant region.     

Wave systems in the chromosphere

Observations are reported of two, possibly three, distinct wave systems in the Hα chromosphere.

1. Velocity films show waves propagating predominantly outwards along mottles and fibrils from as close as 2000 km to the network axis at velocities of the order of 70 km s^-1. The line-of-sight component of the velocity amplitude is estimated to be typically 5 km s^-1. The velocities are accompanied by propagating intensity fluctuations. The system is interpreted as one of basically Alfvén waves. Similar waves are observed propagating predominantly outwards along superpenumbral fibrils radiating from a small sunspot.
2. The velocities in the chromospheric granulation undergo fluctuations of an oscillatory character but without any observable horizontal propagation. The intensities show a close correlation with the velocities, maximum intensity occurring about T/4 after maximum downward velocity. The period is variable across the surface (2.5 min upwards). The intensity-velocity correlation is characteristic of a standing compressional wave.
3. Intensity cinefilms at Hα line centre show in places a horizontal drift of the chromospheric granulation pattern at about 12 km s^-1 without any accompanying vertical velocity fluctuations. It is not known whether this is due to a gas stream at sonic velocities, or to a horizontally propagating sound wave.

The Alfvén wave system is shown to make a significant contribution to coronal heating. Whether the velocity fluctuations in the chromospheric granulation also make an important contribution depends on whether there are upwardly propagating or standing waves; this is not yet established despite the intensity-velocity correlation.     

Plasma emission due to isotropic fast electrons,and types I,II, and V solar radio bursts

The theory of plasma emission is developed under the assumption that the Langmuir waves are generated by an isotropic distribution of fast electrons. Emission from inverse power-law distributions tend to favor emission at the second harmonic with brightness temperatures up to about 10⁸ K at 100 MHz. The concept of a gap (in velocity space) distribution is developed. Very bright plasma emission can result from a gap distribution. For brightness temperatures between 10⁹ K and 10¹¹ K for the second harmonic the fundamental has a brightness temperature between 10⁸ K and 10⁹ K. For higher brightness temperatures the fundamental is amplified and can be very much brighter than the second harmonic. The maximum brightness temperatures for the fundamental and second harmonic at 100 MHz are about 10¹⁶ K and 10¹³ K respectively. Mechanisms by which a gap distribution might be formed are discussed and two effective mechanisms are identified. The theory is applied to the interpretation of radio bursts of types I, II, stationary IV and V. In each case the suggested mechanism appears to be favorable.     

Meridional circulation and turbulence in surface layers of the stars

The meridional circulation is considered in the surface layers of the stars where the optical depth τ?1. It is shown that the radial component of circulation velocity reaches its maximum value at τ≈1 and decreases at τ→0. The tangential velocity reverses its sign at τ≈1 — i.e., the meridional flows are closed in stellar atmospheres. The tangential velocities can be of the order of 10⁶–10⁷ cm s^?1 in atmospheres of O-B-A stars. Such hydrodynamical motions can result in the generation of turbulence in the surface layers. Characteristic turbulent velocities are of the order of 10⁵–10⁶ cm s^?1 in early-type stars. The small-scale turbulent motions generate the acoustic waves and the flux of such waves may be the source of energy to heat coronae in O and B stars.     

Modulational instability of nonlinear waves in the relativistic plasma with account of the nonlinear Landau damping

The modulational instability of the weakly nonlinear longitudinal Langmuir as well as the transverse electromagnetic waves, propagation in the relativistic plasma without the static fields is described. The nonlinear Schrödinger equation taking account of the nonlinear Landau damping for these waves has been derived by means of the relativistic Vlasov and Maxwell equations. The plasma with the weakly relativistic temperature and that with an ultrarelativistic one has been investigated. In the first case, for the electron-proton plasma with the temperature more than 2.3 KeV we found the regional change of the wave numbers for which the soliton of two types, subsonic and supersonic, can exist. The soliton of the transverse waves can exist when the group velocity of the waves is between the thermal velocity of the electron and ion and the length of the linear waves is less than 2c/ _pi .In the second case the regions of the wave numbers, with the solitons of the Langmuir and transverse waves have been determined.The nonlinear waves in the electron-positron plasma and the waves with the phase velocity, which is about the light one, are also considered in the following paper.     

Oscillations and waves in a sunspot

Observations have been made in H of the vertical velocity distribution in a sunspot. Over the umbra the pattern consists of structures of scale-size 2–3. The velocity distribution undergoes oscillations with a period of about 165 s and typical amplitude ±3 km s^–1, but the pattern breaks down after one or two cycles because the period of oscillation varies typically by ±20 s from place to place. Transverse waves develop in the outer 0.1 of the umbral radius and propagate outwards with a velocity of about 20 km s^–1, becoming gradually invisible by or before the outer penumbral boundary; the amplitude is about ±1 km s^–1 at the umbra-penumbra border.The penumbral waves are believed to be basically of the Alfvén type, with 3 × 10^–8 g cm^–3. The umbral oscillations presumably represent gravity waves. In both cases the fluxes are inadequate by two orders of magnitude to account for the sunspot energy deficit.     

Solar flares as resulting from the temporary interruption of energy flow to the corona: A case of hydromagnetic resonance

The hypothesis that solar flares may be caused by a choking off of the normal energy flux to the corona by the strong closed magnetic fields of a plage is examined. If the energy flux into a plage from the photosphere is of the order of 10⁸ ergs/cm² sec, and if a substantial fraction of this energy is carried in the form of Alfvén waves, then the rate of dissipation of the waves is slower than the rate at which energy is injected. Since the waves must propagate along the magnetic field and cannot reenter the photosphere, they must remain within the plage; hence, the magnetic and kinetic energy in a small-scale motion (either waves, turbulence, or high-energy particles) must increase with time, eventually causing disruption of the volume when the small-scale energy density exceeds the energy in the mean field. It is believed that the unusually broad wings in the emission lines represent evidence of this phenomenon. The accumulation of waves is manifested as a resonance which occurs initially only at discrete locations in the magnetic field, but later is expected to involve the whole flare volume. The response of a typical volume of flare dimensions due to a trapping of the normal wave supply to the corona is studied through use of the virial equation. For magnetic fields typical of a plage, the region expands in a time scale of 10²–10³ sec, with a velocity in the neighborhood of 10–20 km/sec. Small-scale velocities within the region, however, have reached 100–300 km/sec, indicating that almost all the energy in the flare resides in small-scale forms. The energy density of the flare region exhibits a behavior much more explosive than the expansion rate. There is a rapid rise to maximum in 10² sec or less, and a slow subsequent decline taking about 10³–10⁴ sec due to the dilution of energy caused by expansion of the region. The predicted temporal behavior of the energy density coincides qualitatively with the light curves observed during flares, and it is suggested that the rise and decline of the energy density is to be associated with the optical flare. The total flare is defined as the time required for the energy density of the chromosphere and corona to return to the pre-flare state. During this time (about one hour) a large flare can derive the necessary 10³² ergs from normal photospheric energy output.     

Coronal line-width variations

Line-width measurements of the coronal ion Siviii confirm earlier observations which show an increase in the non-thermal velocity above the solar limb. The present data, taken at the equatorial limb, show an increase from 24 km s^-1 at the limb to 28 km s^-1 some 25000 km above the limb. The electron density as measured from the Siviii line pair shows a decrease from 3.5 × 10⁸ cm^-3 to 1.8 × 10⁸ cm^-3 over the same distance. These data imply that the non-thermal velocity is inversely proportional to the quadratic root of the electron density, in excellent agreement with that predicted for undamped radially propagating Alfvén waves.     

On photospheric and chromospheric penumbral waves

The full magnetoatmospheric wave equation is numerically solved for the particular penumbral model of Nye and Thomas (1974). For chromospheric running penumbral waves it is found that the maximum vertical velocity occurs at the base of the Hα region (z ≈ 300 km) much lower than previous WKB estimates have suggested. The maximum ‘vertical’ kinetic energy occurs at z ≈ - 130 km. Very significant horizontal velocities are also found for these waves, and, in the absence of shear flow it appears that previous estimates of photospheric vertical velocities of order 10^?1 km s^?1 could be substantial underestimates. For the photospheric events of Musman et al. (1976), a high vertical velocity maximum is found in the corona, and the modes appear highly dispersive for periods ≥ 220 s. The effects of a sinusoidal shear flow profile on running penumbral waves are examined, and it is found that the eigenvalues (horizontal wavenumber k) are changed little, but the eigenmodes become significantly distorted; the position of the vertical velocity peak rises compared to the zero flow case, and the velocity below that peak drops significantly. This effect may well cancel the increased estimates based on zero flow.     

Impact ejecta exceeding lunar escape velocity

Microrater frequencies caused by fast (? 3 km s^?1) ejecta have been determined using secondary targets in impact experiments. A primary projectile (steel sphere, diam 1.58 mm, mass 1.64 × 10^?2 g) was shot in Duran glass with a velocity of 4.1 km s^?1 by means of a light gas gun. The angular distribution of the secondary crater number densities shows a primary maximum around 25°, and a secondary maximum at about 60° from the primary target surface. The fraction of mass ejected at velocities of ? 3 km s^?1 is only a factor of 7.5 × 10^?5 of the primary projectile mass. A conservative calculation shows that the contribution of secondary microcraters (caused by fast ejecta) to primary microcrater densities on lunar rock surfaces (caused by interplanetary particles) is on the statistical average below 1% for any lunar surface orientation. Calculation of the interplanetary dust flux enhancement caused by Moon ejecta turned out to be in good agreement with Lunar Explorer 35in situ measurements.     

Shock waves with large energy losses by direct radiation from the front

Highly nonadiabatic shock waves are formed at an early stage of a supernova explosion inside a stellar wind because of the large energy losses by direct radiation from the front. The properties of such waves are considered for velocities of (5?25)×10³km s^?1 and gas densities of 10^?17?10^?10 g cm^?3. A critical energy flux going to “infinity” that separates two modes is shown to exist. If the flux is lower than the critical one, then energy losses cause even an increase in the post-shock temperature. An excess of the flux over its critical value results in an abrupt cooling and in a strong compression of the gas. For the flux equal to the critical one, the post-shock gas velocity matches the isothermal speed of sound. Approximate formulas are given for estimating the degree of gas compression and the post-shock radiation-to-gas pressure ratio at energy losses equal to the critical ones and for the limiting compression.     

Stochastic wave growth, power balance, and beam evolution in type III solar radio sources

Energy-balance arguments are combined with the stochastic-growth theory of type III radio sources to determine the properties of the source in average dynamical equilibrium with the beam, and the beam's long-term evolution. Purely linear stochastic-growth theory has previously emphasized that the beam evolves to a state close to marginal stability. Small mean residual deviations from marginal stability are present at dynamical equilibrium and these lead to residual energy flows that feed the waves observed in situ and by remote receivers; consequently the beam energy is depleted. Here, dynamical equilibrium beam and wave levels are estimated for the first time and it is found that the main sink of beam-driven Langmuir waves is either via electrostatic decay into product Langmuir and ion-sound waves or via scattering by short-wavelength density fluctuations, depending on the conditions. Improved estimates of energy branching ratios imply that, at 1 AU from the Sun, typically 20% of the beam energy is converted to Langmuir waves that are scattered off low-frequency density fluctuations and then dissipated, with almost all the remaining waves undergoing electrostatic decay, although as little as one-third of the Langmuir waves may decay in atypical circumstances. Of order 10^–3 of the beam energy is converted into sound waves, which are mostly dissipated, and of order 10^–5 is converted into potentially observable electromagnetic waves. The mean lifetime of the Langmuir waves at 1 AU is 1–40 s, while that of the beam is of order 1000 s. The beam density decreases relative to that of the background as the beam propagates. For most parameters, analysis of energy losses from the beam to the waves shows that the beam velocity decreases at roughly the same rate as the thermal velocity of the background plasma. It is argued from these considerations, and from in situ observations at 1 AU, that these trends imply that only the densest and fastest type III beams will be able to penetrate much past 1 AU from the Sun. This implies a low-frequency cutoff to type III emission at roughly 10 kHz, in good agreement with recent Ulysses remote observations, showing their consistency with in situ measurements.     

Nonlinear excitation of small-scale Alfvén waves by fast waves and plasma heating in the solar atmosphere

We study a nonlinear mechanism for the excitation of kinetic Alfvén waves (KAWs) by fast magneto-acoustic waves (FWs) in the solar atmosphere. Our focus is on the excitation of KAWs that have very small wavelengths in the direction perpendicular to the background magnetic field. Because of their small perpendicular length scales, these waves are very efficient in the energy exchange with plasmas and other waves. We show that the nonlinear coupling of the energy of the finite-amplitude FWs to the small-scale KAWs can be much faster than other dissipation mechanisms for fast wave, such as electron viscous damping, Landau damping, and modulational instability. The nonlinear damping of the FWs due to decay FW = KAW + KAW places a limit on the amplitude of the magnetic field in the fast waves in the solar corona and solar-wind at the level B/B ₀∼10⁻². In turn, the nonlinearly excited small-scale KAWs undergo strong dissipation due to resistive or Landau damping and can provide coronal and solar-wind heating. The transient coronal heating observed by Yohkoh and SOHO may be produced by the kinetic Alfvén waves that are excited by parametric decay of fast waves propagating from the reconnection sites.     

The heating of the solar plasma due to microwave phenomena correlated with type II meter bursts

The heating of the solar plasma of those layers is considered where the microwave bursts are emitted. In a first step, we restrict ourselves to phenomena correlated with the so-called type II m bursts. Bursts of this kind are excited by shock-waves initiated near the optical flare region. These shock-waves spread out into the higher corona, and if the shock strength is sufficiently high, the microwave region is heated to 10⁷ K. But this temperature is too low to explain the burst radiation. In this paper, it is shown that at plasma temperatures about 10⁷ K a fairly high number of electrons is accelerated by Alfvén waves to equivalent kinetic temperatures of about 10⁸ K. We assume that the Alfvén waves are generated near the sunspots, and, therefore, the accelerated electrons run along the magnetic-field lines into the microwave source lying between the two spots of an assumed dipole field. Within this source, the considered electrons thermalize and, after a short time, the source reaches temperatures of 5 × 10⁷ K to 10⁸ K.A plasma of this temperature with an electron density about 5 × 10⁹ cm^–3 and a magnetic induction of 300 G is optically thick even at frequencies about 10 GHz, because the gyromagnetic absorption is very high.     

Plasma waves caused by transient heat conduction in a coronal loop and anomalous resistivity

Numerical simulation is made of the transient heat conduction during local heating in a model coronal magnetic loop with an axial electric current. It is assumed that a segment near the top of the normal coronal loop is heated to above 10⁷ K by a sufficiently small heat input as compared with the total flare energy. A hump appears in the velocity distribution of electrons moving down the temperature gradient with speeds slightly below the thermal one. Consequently, electron plasma waves are excited. The high intensity of the waves persists in the upper region of the loop for more than a second until the termination of the simulation. The energy density of the plasma waves normalized with respect to thermal density is 10^–3.5 at maximum. A theoretical estimate gives an anomalous resistivity 5 orders of magnitude greater than an initial value. Based on the above result, we propose a model for impulsive loop flares.