Comments on the proton whistler generation process

0032 0633’85S3 M)+o Cu I_” 1985 Percamon Press Ltd

COMMENTS

ON THE PROTON WHISTLER PROCESS

GENERATION

ROBERT J. STEFANT

Max-Planck

lnstitiit

fur Aeronomie,

341

I Katlenburg Lindau, Federal Republic of Germany

(Rrcrid

17 July 1984)

Abstract The problem of the propagation of an electromagnetic wave originating for instance in a lightning flash through the ionospheric medium is analysed in order to understand the formation at high ionospheric altitudes of the so-called proton whistler. II is shown that the accessibility the hydrodynamic (or kinetic) proton resonance at the satellite altitude requires that a mode conversion process must take place slightly above the transition region separating the one ion (0’) from the two ion (0 ’ + H ’ ) component plasmas. Moreover, the transformation conditions in the wave conversion region imply that the magnetic field should be (almost) perpendicular IO the density gradient. Otherwise, the incident electromagnetic wave will never reachthesatellitealtitudein thefrequencyrangeoftheproton whistler. However,someformcrproton whistler theories have postulated that the signal is the result of simple ionospheric propagation effects, in contradiction with the above results. These former proton whistler theories are reviewed and it is shown that the basic flaw in these theories lies in that the incident electromagnetic wave has been supposed from the beginning to have reached the high ionospheric altitudes where is located the satellite without being influenced by the lower ionospheric layers. Some various aspects, like the high variability of the.wave electric to magnetic held ratio and the harmonics bands as observed by Injun are analysed in the light of the obtained results. Finally, numerical solutions of the wave dispersion relation for both the fast hydrodynamic mode (the extraordinary mode) and the slow ion kinetic mode are presented which shows that a coupling process bctwcen the two modes may take place at various frequencies between the 0. and the H - gyrorrequencies.

of

1. INTRODUCTION

electromagnetic wave and the (hydrodynamic or kinetic) ion cyclotron mode. Indeed, it is now well known (Stefant, lY70a,b, 1977, 1978, 1981) that the torsional Alfven mode behaves like a sound wave and as such does not satisfy the free space propagation conditions. Consequently, some wave conversion process must take place somewhere along the ray path in order to convert the incident electromagnetic signal (which satisfies the free space propagation conditions) into the hydrodynamic mode which propagates only in dense plasmas. It is therefore extremely surprising that Gurnett PI ~11.(1965) Jones (1969), Wang (1971) Buddcn and Smith( 1974)claim that the proton whistler is the result of the most simple straightforward propagation of the whistler mode in the slowly varying ionospheric medium: no mode coupling, mode conversion or anything of the like seems to be required to explain the formation of the proton whistler. Moreover, various and contradictory versions of the early Gurnett theory have been published afterwards which require clarification. Indeed, depending on what one is looking for, sometimes the wave in question is an extraordinary (whistler) wave which of course does satisfy the fret space propagation conditions, sometimes the same wave is an ion cyclotron mode, i.e., has an hydrodynamic nature. For example, in order to explain the high variability of the E/If ratio with

Over thepast 10 years, there has beensomccontroversy (Gurnett eta/., 1965,196Y ;Shawhan, lY66;Jones, 1068, lY70; Wang, 1971; Chan et al., 1972; Buddcn and Smith, 1974;Terry, 1978;StCfant, 1970b, 1977,198l)as to whether the proton whistler which is supposed to be an ion cyclotron mode as recorded by satellites in the upper ionosphere is an electromagnetic wave or has a mixed (electromagnetic-electrostatic) nature. The proton whistler is thought to originate in a lightning flash at the ground. The controversy arose because some new features (cyclotron harmonics, highly variable electric to magnetic field ratio, see for instance Fig. 9 of Gurnctt er (I/.. 1969) were discovered later on (Gurnctt et al.. 1969) which favour a model in which cyclotron interactions arc important. These new featuressecm to invalidate themodel ofGurnett and his co-workers who claim that the proton whistler shape is due to simple dispersive cffccts of a hydrodynamic wave propagating in a cold plasma, while cyclotron interactions require the implication of a kinetic model (Stefant, 1970b. 198 1) in which tinitc gyroradius effects are of paramount importance. On the other hand. things were not clear from the very beginning since one should naturally expect some coupling process to take place between the incident 333

334

R. J. STEFANT

frequency (Gurnett et al., 1969, their Fig. 9) a model is constructed which takes account of the electrostatic nature of the extraordinary mode propagating at very large angles with the magnetic field Be. On the other hand, features like cyclotron harmonics which are known to affect only the slow mode (Stefant, 1978) are explained (Shaw and Gurnett, 1971) as two nearly simultaneous electron whistlers that beat with each other; in other words, destructive interference caused by multistroke lightnings in oacuo induces in the lightning Fourier spectrum an amplitude modulation which propagates as such in the whistler mode up to the satellite where the signal is recorded as a proton whistler. In addition, collisions, which usually lead to wave damping, are sometimes introduced (Jones, 1969) to explain some particular effects associated with the wave polarization. On the other hand, the same effects seem to be well understood if, instead of using a WKB (geometrical optics approximation) method, a full wave analysis is taken into consideration (Wang, 1971). In this paper, we present a systematic and detailed investigation on proton whistler theories. The paper is organized as follows : In the first part, the dispersion characteristics of the free space electromagnetic wave which has its origin in the lightning discharge through the ionospheric medium are thoroughly analysed using a full wave analysis when necessary. It is shown that the incident extraordinary wave impinging the critical region above 400 km separating the one-ion (O+) from the two-ion (O+ t Hi) component plasmas is totally reelected at frequencies lower than the local proton gyrofrequency ; in other words, there is transmission only for frequencies greater than the proton gyrofrequency. The weak proton concentration of the layer is the main reason which prevents the incident E mode to propagate upwards in the frequency range oftheproton whistler. In the second part, we try to explain why the hydrodynamic approach as proposed initially by Gurnett has been universally accepted without being criticized for almost two decades and the different models like those propounded by Jones (1969), Wang (1971) and Budden and Smith (1974) are discussed and compared along with the results and conclusions reached in paragraph 1. Moreover, the different arguments used by Gurnett and his co-workers to explain the numerous features discovered later on with Injure like, as already mentioned, the proton whistler cyclotron harmonics or the high variability of the wave electric to magnetic held ratio, are investigated in detail and discussed with simple cold plasma theory methods. Finally, the theory is extended to multi ion (Of + He+ +He+‘) component plasmas.

In the last part, we review some of the properties of the hydrodynamic and kinetic modes propagating in a warm plasma at large angles to the magnetic field. Numerical solutions ofthe wave dispersion relation for both modes are presented which show that a coupling process between the incident fast extraordinary mode and the ion kinetic mode may take place at various frequencies between the O+ and the H+ gyrofrequenties. One important coupling parameter is the ratio of the parallel (atong Be) wavelength to the heaviest ion (0 ‘) inertia length. Finally, we suggest as a test in order to confirm or invalidate the present statements on proton whistler theories to measure the ratio of the wave electric field to the wave magnetic field.

2.

TERMINOLOGY

It is well known that two modes and only two can propagate in a dense hot anisotropic plasma (see for instance Stefant, 1978). By dense plasma, we mean here a plasma the electron plasma frequency of which II, is greater than the electron gyrofrequency R,, which implies of course that the ion plasma frequency I& is (much) greater than the ion gyrofrequency Qi (Stefant, 1977,1978). By hot plasma, we mean a plasma in which the ion Larmor radius pi is of the order of (or greater than) the transverse (perpendicular to B,) wavelength of the perturbations ; in other words, the argument used by Gurnett and his co-workers that the ionospheric temperature is too low for the kinetic waves to exist does not hold here, since it has been shown (Stefant, 1970b) that one can always find a wave whose wavenumber is solution of the warm plasma dispersion relation and which satisfies the condition kp, 2 1. In the long wavelength regime (cold plasma approximation), the properties of the waves are described by a biquadratic AN4 + BN2 + C = 0, where N is the wave refractive index and the coefficients A, B, C should remain k independent. According to the frequency range of interest, these two modes are labelled one, the slow or shear or torsional or ordinary Alfvtn mode, the other, the fast or compressional or extraordinary or whistler mode. Sometimes and mainly in the space sciences, one often uses instead the terms left-handed and right-handed. However, one must stress that this definition based on the wave polarization is valid only for a one-ion component plasma. In a two-ion component plasma, each mode may acquire both types ofpolarization as shown by Fig. 1. Indeed, the proton branch of the slow mode has dispersion characteristics of the right handed type in the MHD regime, but of the left-handed type when the wave frequency approaches the proton gyrofrequency. In between, the polarization turns to be linear.

Comments

on the proton

whistler

generation

335

process

ka k, F1c.1. DISPERSION~HARACTERIST~CSOFTHETWO(SLOWANDFAST)HYDRODYNAMI~MODESPROPAGATINGINA TWO-IONCOMPONENTPLASMA.

The letters (R) and (L) refer to the polarization

of the waves, L for left-handed

Conversely, the fast whistler mode has a polarization of a left-handed sense near the cut-off (corresponding to the zeroes of the refractive index N), while having a polarization of a right-handed sense above the proton gyrofrequency. Consequently, it is better not to call them right-handed and left-handed modes. Therefore, as this paper is mostly concerned with waves propagating in the ion frequency range where one of the waves propagates faster than the other, we will term them slow and fast hydrodynamic modes, since their properties are derived from the hydrodynamic equations of motion combined with the Maxwell equations (Shafranov, 1967). When the wavelength decreases and becomes of the order of the ion gyroradius (hot plasma hypothesis), it has been shown (Stefant, 1970a,b, 1977, 1978) that the slow hydrodynamic mode goes over into a mixed (electrostatic-electromagnetic) mode which becomes purely electrostatic at large values of k such that k~c?/II~ >> 1 where c is the velocity of light, while the fast extraordinary (whistler) mode always keeps the properties of a cold mode and is not affected at all by the thermal motion of the particles (Stefant, 1977). Consequently, we will term this mode (the properties of which depend on particle gyroradius effects) magnetokinetic by opposition to the term magnetohydrodynamic designed to describe the long wavelength perturbations in fluids. This mode has ofcourse a phase velocity much lower than the slow hydrodynamic mode.

3. ANALYSIS IONOSPHERIC

and R for right-handed.

OF THE PROPAGATION MEDIUM

WAVE ORIGINATING ACCESSIBILITY

THROUGH

THE

OF AN ELECTROMAGNETIC IN A LIGHTNING

OF THE HYDRODYNAMIC

FLASH. RESONANCE

As already mentioned in the Introduction, the proton whistler is a signal detected in the upper ionosphere (>400 km) which exhibits on a frequencytime diagram dispersion characteristics similar to those of the slow hydrodynamic mode, i.e., a rapid rise in frequency followed by an asymptotic behaviour at the gyrofrequency for ions in the plasma surrounding the satellite. The new VLF phenomenon is therefore called an ion whistler, since it turns out to be generated by lightning in the same way as the electron whistler, but in the very low frequency range (lOt.%lOOO Hz). The proton whistler is thought to originate in a lightning flash at the ground. Therefore,any analysis dealing with the ion whistler generation process will have to question first the accessibility of this hydrodynamic resonance for the EM wave originating in the lightning flash. This EM wave will have to cross first the E and D regions where the plasma consists mainly of single ionized oxygen ions and electrons and next a region above the F layer (-400 km) where the major ion constituents are O+ and H+ (Taylor et al., 1968) and where is recorded the proton whistler. Next, since the slow hydrodynamic mode does not satisfy the free space propagation conditions, but is supposed to give rise to the proton whistler, a mechanism must take place along

336

R. J. STEFANT

the ray path in order to convert the incident electromagnetic energy into that of the hydrodynamic mode. This is the problem which we want to analyze. In the Earth ionosphere cavity, the vertical electric dipole radiates spherical waves. According to the formula : exp (ik - R) R

exp (ik,x + ik,y + ik,z)

zz s

dk, dk,

(1)

k,

the spherical wave can be expanded in plane waves. Assuming the outer surface of the cavity to be the E or the D layer, the refracted wave will be obviously represented as the superposition of plane waves obtained from the plane waves incident upon this outer surface into which the original spherical wave was decomposed. Next, one may ask whether the electromagnetic wave launched from the antenna system (here the electric dipole) will in fact reach the high density two-ion component layer where the proton whistler is effectively recorded or whether the waves will be reflected at some region of intermediate density. The wave in its upgoing path must first cross the outer spherical surface of the Earth ionosphere cavity (the D layer) where the conductivity is mainly determined by electron collisions. It is known that the Fresnel reflection coefficient of this outer layer is given by the expression exp

(-2/N,

cos a), where Ni = (~JJE~w)~/ exp (- i7r/4), gi being the conductivity of the layer and a the (complex) incidence angle. This formula yields for a ionospheric conductivity of 10-5-10m6 mho mm 1a refractive index of the order of 10 in the frequency range of the proton whistler and a transmission coefficient of a few per cent. Let us now assume that for the transmitted wave an accurate description of the propagation is given by the homogeneous plasma dispersion relation for a dilute plasma, i.e., N* = 1 + n,‘/(II, cos 0 - w)w where 0 is the propagation angle ; then a density of lo4 p/cc typical of the E layer yields a refractive index of 50 and a wavelength of 20 km which is about the scale height of the E layer at 150 km. Consequently, the wave propagation cannot be studied by using the plane wave solutions. The problem has to be again analysed by following a method exactly similar to the one used for the Earth ionosphere cavity (Fig. 2). The outer boundary of this spherical wave guide is determined by the frontier between the dilute ionospheric plasma (lI, < Q,) and the dense (n, > Q,) plasma where the plane wave dispersion relation N2 = 1+ lI:/(R,cos O-w)w applies. As the collisional conductivity of this second spherical layer is much less than the collisional conductivity of the D layer, one can predict a total transmission coefficient of 10m3 sufficient enough for some appreciable energy to be transmitted upwards.

Proton

200

km

D region

FIG.~. GROUP RAY PATHSFOR THESHORT FRACTIONALHOP WHISTLER AND THE PROTON WHISTLER. The wave. generated by the lightning flash crosses first the low density ionospheric layers before reaching the interaction region whereis generated the proton whistler. In B, the satellitemaydetect two kinds ofwaves: one is the proton whistler propagating along B, as indicated, the other is an electron whistler coming from some region D which should be located in the transition region. If there the magnetic field is perpendicular to the density gradient, the satellite in B will receive a signal polarized in the whistler mode in the proton whistler frequency range 0 - fuhd. (*bd being the minimum Buschbaum resonance along the path BD.

Comments on the proton whistler generation Next, above 200 km, the plasma is dense enough (ff, > Q,) so that the whistler dispersion relation applies and the waves are plane. In the foilowing,it will be assumed that the major ion constituent ofthe ionospheric plasma below 400 km is 0’. However, it is well known that other constituents like Oi, NO+, He+, etc. exist simultaneously with O+ and with a concentration sufficient enough to apply the criterion of dense plasma to these species. Therefore, one could make at once a propagation theory in a multi-ion component plasma to see if for instance an He + or NO + whistler with a shape similar to the wellknown proton whistler shape seen at high altitudes could be present at lower altitudes. However, as this paper is mainly concerned with the formation of the proton whistler, we will turn our attention first to this special case, but, next, we will analyse in detail if the heaviest ion plasma constituents of the lower ionosphere can generate by simple dispersive effects the same kind of wave shape as the proton whistler does in high altitudes. As is well known, the whistler mode propagates in a one-ion component plasma from 0 to Q cos @(Fig. 3). Propagating upwards, the Emode reaches the interface separating the single ion component (O.+) from the two (0’ -t H+) ion component plasma. We will see below that the scale height of the transition layer, i.e. the

process

337

length over which the plasma parameter I&/f& changes from 0 to 10 (say), is about 100 km. Consequently, the wave propagation can be studied again by using the plane wave solution. Then, the k of the wave will adjust itself to satisfy the iocal plasma dispersion relation which shows that in a two-ion component plasma, the whistler mode has a cut-off at a frequency given by :

where the indices iand o refer respectively to the H + and O+ components (Fig. 1). The dispersion characteristics change smoothly with distance in the transition from the O+ to the O+ + H+ plasma as indicated in Fig. 3. As at these altitudes, the electron and Of densities are much larger than the H+ density, it turns out that the cut-off frequency is always close to sZi in the transition region. Therefore, the frequency range in which the whistler mode propagates is now R, to R,COS 0 instead of 0 -0, cos 0 previously. It means that the transition region behaves like a high frequency band filter for the whistler mode. In fact, the energy contained in the Fourier components below Ri is partly absorbed and partlyreflected bytheionosphericlayer(Fig.4).But, the wave energy transmitted never contains Fourier components in the frequency range of the proton whistler.

incident extraordinary mode

FIG. 3. SPLITTINGOFTHE INCIDENTEXTRAORDINARY MOL)EIN ITSPATH THROUGH THETRANSITION REGION. The dashed traces indicate thedifferent dispersion characteristics shapes with increasing heights, the last one being indicated in full. Coupling takes placearound the proton gyrofrequency between the fast mode (labelled N-) and the slow hydrodynamic mode (labelled N,). The WKB approximation is assumed to be valid throughout.

338

R. J. 1 E refracted

Transition

E Absorbed below R,

STEFANT

However, the situation changes drastically, if the wave propagates almost perpendicular to the magnetic field. As well known, in that case, the E mode dispersion characteristics consist of two branches : one has a cutoff at a frequency given by expression (2), the other has an asymptote at a frequency given by :

CD;=

Incident wave

FIG.

E A

E Reflected below S12,

4. SKETCH SHOWING THE BEHAVIOUR OF THE E WAVE THROUGH THETRANSITION REGION.

This reasoning is valid for a homogeneous or slightly inhomogeneous plasma. In a slightly inhomogeneous plasma, the wave propagation is studied by using the WKB solutions of the governing differential equations. However, the method fails at somecritical points, called branch or singular points where two of the roots of the characteristic equation coalesce. At these singular points, a coupling may take place between two waves and an electromagnetic mode, say a whistler wave incident upon the transition sheath may be transformed into a slow hydrodynamic wave, provided that the two refractive indices are not too far from each other. This is the case for the incident electromagnetic wave and the slow hydrodynamic mode whose refractive indices are close enough near the proton gyrofrequency for the transformation to occur. However, the coupling process can take place only above the transition region. The explanation is as follows : it has been shown (Sttfant, 1977,1978) that the slow hydrodynamic wave propagates only in plasma whose density satisfies the relation IIi/Qi > 1; consequently, if any coupling process below Ri takes place along the path, it should occur in space regions above a critical density of - 1000 pp/cc (!& = 3000 s ‘, I&/Q = 10 yields the required density). As the E mode does not propagate below the proton gyrofrequency, the mode conversion process occurs in k space regions where the group velocity is very small so that the energy cannot escape. Consequently, a satellite at an altitude of 1000 km will never record a signal propagating either in the whistler mode or in the slow hydrodynamic mode in the frequency range below the proton gyrofrequency if the incident E wave propagates at an angle to the magnetic field. The particular case ofsmall propagation angles has been studied elsewhere (Sttfant, 1981) and will not be reproduced here.

II$:

co? a + (IIfs2f + I@!)

sin2 a

II,2cosZ~+(II~+II,2)sinZa

(3)

As shown by Fig. 5b, the upper branch exists from Qi to the lower hybrid frequency, the lower from 0 to the Buschbaum resonance whose value depends both on plasma density and the angle cx between the concentration gradient and the magnetic field. The Buschbaum resonance decreases with altitude and its farthest limits are the proton and the oxygen gyrofrequencies. Therefore, a frequency gap always exists between the cut-off and the Buschbaum resonance. We have not considered the model which involves the slow hydrodynamic wave since this wave does not satisfy the free space propagation conditions and is not consequently an electromagnetic wave. The distinction is of paramount importance in the following. In their theory, Gurnett and his co-workers used often the word mode conversion. Mode conversion here happens to mean polarization reversal which should not be confused with a real wave conversion (like the transformation of an ion acoustic wave into an electromagnetic wave). Indeed, a wave in a twocomponent plasma can suffer a polarization reversal without changing its name, i.e. its refractive index remains the same even if the plasma characteristics (density, temperature) change in time or space. 4. EXPERIMENTAL

RESULTS

BY HYDRODYNAMIC

AS EXPLAINED THEORIES

It is now obvious that the Fourier components below the gyrofrequency of a proton gyrating in the Earth magnetic field of the electromagnetic wave originating in a lightning flash, will never reach the satellite altitude of 1000 km (say). Some particular conditions should therefore be found in order to explain the characteristics of the noise which exhibits, as already diagram dispersion said, on a frequency-time characteristics similar to those of the slow hydrodynamic mode, i.e. a rapid rise in frequency followed by an asymptotic behaviour at the local proton gyrofrequency. Indeed, even the dispersion characteristics of the fast hydrodynamic mode propagating at very large angles to the magnetic field cannot explain the proton whistler features, as shown by Fig. 5b. However, several theories using the simple hydro-

Comments

on the proton

whistler

generation

transition

process

339

region

F1c.5. DISPERSIONCHARACTERISTICSOFTHEEMODEPROPA(;ATINGPERPENDICULARTOB,FORAONE-ION (0’) ~oMPo~~~TPLAS~~A (Fig.Sa)AND F~RATW~ION (O++H ~)~o~~~~~~ PI,ASMAI~ITSPATHTHR~UGHTHE TRANSITION REGION (Fig. 5b). The arrow means increasing

model in the W KB approximation have been published now for two decades without anyone raising objections to them. So it may be asked why this hydrodynamic theory has been universally accepted for so long, while simple straightforward propagation theory fails to explain the main characteristics of the phenomenon. This chapter is devoted to such an explanation. It should be stressed that the different versions of the Gurnett hydrodynamic interpretation ofthe proton whistler published thereafter may in some aspects look contradictory to the reader. Therefore, we will try to expose the facts as they have appeared in time in the literature. Most of the time, the signals received by the satellite antennaare Fourier analysed on aVarian spectrometer or a sonograph which yields the time frequency behaviour of the ionospheric noise. According to Gurnett et al. (1965), Shawhan (1966), Jones (1969, 1970), this spectrogram can be explained as follows : in Fig. 6, which is a sketch of the spectrogram as interprefated by these authors, the proton whistler trace has three important frequencies: the proton gyrofrequency F,, the crossover frequency F, and the cut-off frequency F,. The proton gyrofrequency is the frequency towards which the trace approaches asymptotically; the crossover frequency F, is the frequency where theelectron and proton whistler traces

dynamic

heights.

meet, and is the frequency at which polarization reversal takes place at the satellite altitude. The cut-off frequency in the Fig. 6 example, is greater than Fi and is the frequency below which the electron whistler trace is absent for part of its length. It is the greatest frequency for which polarization reversal occurs as the wave traverses the region below the satellite. However, sometimes on thesonagram, some whistler trace is observed below the proton gyrofrequency. Therefore, in order to explain the simultaneous presence (or absence) of both traces on the sonagrams, Jones (1969) introduces collisions in the polarization reversal process. He assumes that the EM waves are refracted so that the wave normals are vertical in the lower ionosphere and hence the angle @between the wave normal and the magnetic field depends on latitude. At higher latitudes, this angle is small and could for certain frequencies be less than a critical angle 8, whose value depends on the collision frequency and on the wave frequency, so that these frequencies do not suffer polarization reversal. Therefore, at high latitudes greater than a certain transition latitude, no reversal can occur and no proton whistler would be observed. In a certain range of latitudes less than the transition latitude, the angle 0 is less than 0, for some frequencies and greater than OCfor others so that proton whistlers are predicted to be only partially formed.

340

R. J.

STEFANT

___- Electron whistler

FC

(Ri

\

Anomalous component (requires mode coupling)

/

Proton whistler (no mode coupling required)

FIG. 6. SKETCH OF A SONAGRAMAS INTERPRETEDBY GURNBTT AND HIS CDLLEAGUES. For explanation, see text.

The Jones’ theory has been criticized by Wang (1971) who shows that these features can be explained without collisions; indeed a full wave treatment shows that a right-handed polarized electromagnetic wave incident on the interface separating the two plasma media gives rise not only to a refracted left-handed and righthanded wave, but also to a reflected wave with a polarization of a left-handed type. According to Wang (1971), this comes simply from the coupling boundary conditions and cannot be treated in the geometrical optics approximation. To summarize, the prior Gurnett treatment of the proton whistler has given rise to two models: one approach based on the geometrical optics approximation and followed by Jones (1969), Budden and Smith (1974) and Terry ( 1978) yields on sonagrams in addition to the well known whistler shape the kind of trace sketched on Fig. 6, i.e. a rapid rise in frequency starting at the crossover frequency F, followed by an asymptotic behaviour at the proton gyrofrequency. On the other hand, the full wave analysis of Wang (1971) yields in addition a left-handed trace starting from F, too, but having this time an asymptote at the cut-off frequency F, (see Fig. 1). The additional downgoing waves described by Wangin his theory are ofno interest in what follows. But the main point to emphasize here is that in both models, the proton whistler process is the result of the most simple straightforward propagation of an EM

wave through the slowly varying ionospheric medium (i.e. the WKB approximation). No mode coupling, mode conversion or anything of this type is required to explain the proton whistler shape. Coupling does play a part in the process not to explain the proton whistler, but rather to explain why in some cases the extraordinary trace is also observed in the same frequency range as the proton whistler (the dashed curve of Fig. 6). This last process, which involves a coupling between two waves, should require a more elaborate treatment than the one concerned by geometrical optics.

5.

CRITICAL

HYDRODYNAMlC

ANALYSIS

PROTON

OF THE

WHISTLER

THEORY

This proton whistler explanation has a basic flaw which lies in the misconception of the dispersion characteristics of the incident wave which should be by assumption a fast electromagnetic wave and which is termed improperly by Gurnett et al. (1965), Shawhan (1966), Jones (1969) and Budden and Smith (1974) a right-handed polarized wave. As already explained in paragraph 2, in a one-ion component plasma, one can term correctly a right-handed polarized signal an extraordinary or whistler wave. But, in a two-ion component plasma, the polarization has a hybrid nature for both modes (Fig. 1). Let us try now to understand how the Gurnett theory

Comments

on the proton

works. Assuming the satellite at an altitude of 1000 km, the correct dispersion characteristics are those plotted in Fig. 1.At a given frequency (indeed w is given and the k of the wave adjusts itself to satisfy the local plasma dispersion relation) below the so-called crossover frequency, two values of k exist: one labelled k, corresponds to the hydrodynamic fast mode (the extraordinary mode), the other labelled k, refers to the other mode, i.e. the slow hydrodynamic mode. The former has a cut-off at 0, and becomes the whistler mode at higher frequencies ; the latter has dispersion characteristics of the hydrodynamic fast type in the MHD regime, but a resonance at the proton gyrofrequency. With increasing heights, the mode whose wavenumber is represented by k, “sees” its polarization changing from left to right, while the mode whose wavenumber is represented by k, “sees” its polarization changing from right to left. If now we assume with Gurnett, Shawhan, Jones, Budden and Smith that the wave propagating upwards below the satellite altitude has a polarization of a righthanded sense, the only k which satisfies this criterion below the crossover Q, is k,. Then, with increasing heights, an observer moving with the wave will see effectively the wave polarization changing from right to left; in other words, the wave will suffer a polarization reversal at 0, (see also Fig. 6). As Gurnett and his supporters have termed (incorrectly) a wave with a right-handed polarization, a whistler or extraordinary wave (which of course satisfies the free space propagation conditions) and a wave with a left-handed polarization ion cyclotron wave, a reversal of polarization means effectively that the incoming wave becomes a proton whistler with increasing heights. This mechanism does not require of course any mode conversion or coupling process : the proton whistler is really in this polarization reversal mechanism, the result of the most simple straightforward propagation of an electromagnetic wave in the slowly ionospheric medium. Unfortunately, Gurnett and his colleagues have never questioned the accessibility of the hydrodynamic proton resonance for the incident EM wave originating in the lightning flash. If they had done so, they should have found that the EM wave could not reach the satellite altitude because of the presence between the two plasma media of a ionospheric layer of weak concentration in H+ which prevents the incident wave energy to have Fourier components in the frequency range of the proton whistler. A WKB theory would have yielded in that case a wave whose wavenumber is represented by k, but by no means by k,. There is consequently in these hydrodynamic theories a basic misinterpretation, the consequences of

whistler generation

process

341

which appear much clearer in what follows. Indeed, the sketches of spectrograms like those published by Gurnett ef a/. (1965, Fig. l), Shawhan (1966, his Fig. 1) Jones (1969, his Figs. 1 and 3) show a surprising discontinuity in the slopes df /dt of the traces above and below the crossover frequency F, (see Fig. 6) while the slow hydrodynamic branch which gives rise to the proton whistler trace is continuous in the whole frequency range from 0 to the proton gyrofrequency (Fig. 1). In lieu of the sketches plotted by Gurnett, Shawhan or Jones, one should have seen a diagram of the type plotted in Fig. 7 which does not show any discontinuity in the slopes of the curves at or near the crossover. Moreover, the low frequency cut-off attached to the fast hydrodynamic mode is not the consequence of collisions as proposed by the Jones’ theory, but only the result of the mode propagation characteristics (Fig. 1). This is the correct diagram that one should have expected from a hydrodynamic theory under the condition of course that both modes are excited (either directly or by coupling) in the ionospheric medium. In other words, one retrieves the full wave analysis results of Wang (1971) i.e. a dispersive whistler trace which shows an asymptotic behaviour at the cut-off frequency F, and the proton whistler trace (the trace in full of Fig. 7) which displays a nose at the “crossover” frequency F,. This proton whistler trace does not start at F, like in Fig. 1 of Gurnett et ul. (1965) but shows a continuous behaviour from the lowest frequencies to the proton gyrofrequency, in agreement with the dispersion curves of Fig. I As it was foreseeable. the sense of polarization has nothing to do in the basic process. Unfortunately, as shown before, the only upgoing mode which satisfies the free space propagation conditions is the fast hydrodynamic mode (the extraordinary mode) and this mode never reaches the high ionospheric altitudes where is recorded the proton whistler because, as already said, the presence between the two plasma media, i.e. here O+ and O+ + H+, of a layer of a weak concentration in protons. A hydrodynamic theory of the Gurnett, Jones, Budden and Smith type cannot consequently explain the formation at high ionospheric altitudes of a proton whistler. It is therefore extremely difhcult to explain the additional features discovered afterwards by Gurnett and his co-workers, like the ion cyclotron harmonics or the high variability of the E!H ratio. Should even the Gurnett theory have satisfied the necessary criterion of accessibility needed for the E mode to reach the satellite altitude, a correct explanation of these novel features would not have been easy to find. Indeed, the explanation given by Shaw and Gurnett (1971) of the harmonics bands observed in the proton whistler

342

R. J.

STEPANT

Anomalous component (requires mode coupling)

‘$>/s

required)

Anomalous component (requires mode coupling)

T FIG.~.CORR~CTSKETCHTHATONESHOULDHAVEEXPECTEDFROMAHYDRODYNAMICTHEORYIFBOTHMODESARE EXCITEDIYTHE 1O~oSP~ERIcMEDIUM. The differenceswith the previous sketch are huge. First, the proton whistler trace show-s a nose at the

“crossover”frequency.Next, thecurveinfull whichshowsanasymptoticbehaviouratlowfrequenciesdoesnot belong to the whistler mode as in the Crurnett theory(see Fig. 6), but to the proton whistler. Lastly, the electron whistler shows an asymptotic behaviour at the cut-off frequency F,.

structure in terms of destructive interference caused in wzcuoby multistroke lightnings refers obviously to the electromagnetic whistler wave, not to the proton whistler itself. As these two waves behave differently and as any mode conversion or coupling process is forbidden, it still remains to explain the presence of these harmonics bands in the proton whistler structure. To expound the high variability of the E/H ratio with frequency, a model is constructed which takes account of the electrostatic nature of the extraordinary mode propagating perpendicularly to the magnetic field. Here again, the trouble to deal with the Gurnett theory lies in that one does not know which mode is concerned and which properties shouid beattached to. Depending on what one is looking for, sometimes the wave is an extraordinary mode because its polarization is righthanded, sometimes the same mode is a hydrodynamic mode of the AlfvCn torsional type because the polarization has changed and is now left-handed. If one assumes with Gurnett that the incident mode propagates perpendicular to the magnetic field, the waves may indeed become electrostatic but, as shown by Fig. 5, its dispersion characteristics cannot account for the proton whistler features. On the other hand, if the incident wave is taken to be the ion cyclotron mode, it is known (Sttfant, 1970b, 1978) that this mode does

not propagate at large angles to B, in a cold plasma (its refractive index becomes infinite). Consequently, none of these ideas seem satisfactory to explain the high variability of the E/H ratio with frequency. One last word should be said about the model. Indeed, it has been assumed that the major ion constituent of the ionospheric plasma below 400 km is O’, while it is known that other constituents like NO +, He’, etc. exist at those altitudes. Consequently, one should expect to find an ion whistler whose shape would be similar to the proton whistler shape. The dispersion characteristics of the two (fast and slow) mod;?; capable to propagate in a plasma composed of oxygen and helium (He+ and He’ ‘) ions are plotted in Fig. 8. One can see that a coupiing mechanism similar to the one proposed here for the proton whistler is again necessary in order to feed energy into the ion cyclotron waves. First, a mode coupling process must take place in A in order to excite the slow He’ * mode. Once this slow wave is excited, a second mode conversion process between the He+ and the He’ ’ modes must occur in B in order to produce an He+ whistler. However, the mode which carries the energy originates in a lightning Aash and consequently has to cross first a layer of weak concentrationin He’ or He++ so that thesameprocess described earlier for H+ occurs, i.e. the mode

Comments

on

343

the proton whistler generation process

O-O

Slow -

Coupling ‘\

Whistler made A

7”:

FIG. 8. PLOTS OF THE DISPERSION CHARACTERISTICS A

OF THE HYDRODYNAMIC

conversion mechanism takes place in k space regions where the group velocity is so small that the energy cannot escape. Therefore, the incident electromagnetic wave can never trigger at low or high altitudes an ion whistler of the He+ or He++ type if the wave of course propagates in the whistler mode, i.e. at an angle with the magnetic field.

6.

(FAST AND SLOW) MODES iN A THREE

(O”, We+, HeZ+) ION COMPONENT DENSE PLASMA. coupling must take place first in A and next in B to generate an ion whistler of the He* or He’+ type.

KINETIC

THEORY

WHISTLER-AN

OF THE PROTON INTRODUCTION

As already said, the cold plasma properties do not allow us to give a satisfactory explanation of the proton whistler features. The solution of the problem which we have to solve is consequently linked to warm plasma characteristics. But first as soon as the accessibility problem has been cleared up, a process of wave conversion should be found which will convert the energy of the incident extraordinary wave into that of a kineticmode,i.e.amode thepropertiesofwhichdepend on particle gyroradius effects. The accessibility probiem is immediately solved by taking an incident E wave propagating almost perpendicular to the magnetic field. If this is the case, one already knows that the extraordinary mode has dispersion characteristics in the proton whistler frequency range (Fig. 5) : the upper branch has a cut-off at the proton gyrofrequency of the critical region, but

the lower one has an asymptote at the Buschbaum resonance which is very close to the proton gyrofrequency in the transition region. Consequently, the whole Fourier spectrum energy of the electroma~etic whistler wave goes through the critical transition region, except a tiny fraction around the proton gyrofrequency. Unfortunately, as explained earlier, the dispersion characteristics of the E mode propagating at very large angles to the magnetic field cannot account for the proton whistler properties. But these characteristics wit1 decide on the final structure of the wave shape seen on the sonagram, since, with the E wave propagating upwards, a larger and larger frequency gap will appear on the spectrogram because the Buschbaum resonance decreases with increasing altitudes (Fig. 5b). Therefore, an absence of a whistler trace will appear on the sonagram between a frequency corresponding to the gyrofrequency of a proton gyrating in the Earth magnetic field at 500 km and the local Buschbaum resonance at the satellite altitude. This frequency gap corresponds to the frequency range for which the incident E wavemay beentirely converted into a proton whistler. The wave transformation takes place when the relation wwhisller = c+, is satisfied somewhere along the density profile at or around the wavenumber k = EwJ&. As the properties ofthe cofd and warm plasma waves propagating in a hot plasma will be of paramount

344

R. J.

STEFANT

importance in what follows, it is now useful to recall some of the features of the (cold) hydrodynamic modes in a finite b plasma where /J is the ratio of the kinetic to the magnetic pressure. It has beenshown(StCfant, 1977) that thecold plasma approximation, i.e. the neglect of the finite ion gyroradius effects in the wave dispersion relation, is correct, when the perpendicular phase velocity of the hydrodynamic modes is greater than the ion thermal velocity by more than one order of magnitude [exactly (M/~z)‘/~] where M and m are respectively the ion and electron masses. It means that the introduction of temperature in the wave equation reduces the wavenumber range in which these waves are supposed to propagate. As should be expected, this wavenumber range shrinks with increas~ng~. Thisconclusion applies to the E mode for all propagation angles and to the slow hydrodynamic wave, but only if the propagation angle is not close to n/2. When the propagation angle lies inside the cone 7r/2 - O < m/M)“*, the refractive index of the slow hydrodynamic mode becomes infinite. This comes simply from the wave dispersion relation in a cold pIasma (Stefant, 1970a,b). Knowing that the refractive index of the ion acoustic wave in a cold plasma is also infinite, one may intuitively deduce that with the introduction of temperature, the slow hydrodynamic mode takes on the properties of this ion acoustic wave, i.e. becomes a kinetic mode in which the effects due to the ion gyroradius and the current should be very important.

7. REVIEW

OF THE KINETIC

OF THE ION ACOUSTIC

6.9 1 -expf-p)f, k2 Y2 L a

F

where I, is the first-order modified Bessel function of argument p = k&f. One immediately sees from this expression that the shear Alfven mode in a hot plasma is slightly affected by the thermal motion of electrons. The damping is maximum when the parallel phase velocity of the Alfven mode is nearly equal to the parallef phase velocity ofthe ion acoustic wave. This is indeed possible since the shear Alfven mode propagates in the long wavelength regimeat angles to themagneticfield, while the ion acoustic mode propagates at very large angles but in the short wavelength range (cos B very small). With increasing wavenumbers, the effect of the finite ion gyroradius has to be taken into account and the dispersion equation becomes :

MODE

An important aspect of the plasma oscillations described by (7) which is of paramount importance in our theoryofwaveconversion(St~fant, 198l)liesinthat the waves ofinterest here are by no means electrostatic ; they have a magnetic field created by electron currents flowing along the magnetic field whose magnitude is given by the expression :

e

%l-%Z E,, sin’ O+E,. cos2 @

where tensor For readily Alfvbn

] ---

PROPERTIES

The kinetic properties of the slow hydrodyn~i~ mode (Landau damping, influence of the temperature on the dispersive properties of the wave) have been first studied by Stefant (1970a) who shows that in the frequency range well below the ion gyrofrequency and for wavelengths greater than the ion gyroradius, the shear Alfven wave is affected by Landau damping. Indeed> the dispersion characteristics of the shear AlfvCn mode in the MHD regime are described by the following equation :

N2 z

where V, is the Alfven velocity. One immediately checks that the usual Alfvtn dispersion relation is valid for wavenum~rs satisfying the relation kfc’,FI,” e 1. In a finite temperature plasma, the Alfven velocity V, can take values less than the electron thermal velocity V,; this happens for values of B such that fl> 2mjM. The dispersion equation has then the following form (Sttfant, 1970a) :

(4)

E,, and F,,, are the components of the dielectric in Stix’s (1962) notation. values of 0 such that E,, sin’ 0 CCa,, cos’ 8, one obtains the dispersion relation for the shear mode in a cold plasma :

One immediately checks from expression (8) that a quasistatic analysis, i.e. an analysis which would assume from the very beginning that the oscillations are curlfree,implies that kfc’/Il: >> 1. One must stress that equation (7) is valid only for wavenumbers satisfying the conditions (Stefant, 1977, 1978): 112

-1

(5)

(9)

Comments on the proton whistler generation process and

where c is the ion thermal velocity and Ea small number of order (m/&f)“*. To summarize, when condition (10) is reversed, only cold plasma conditions prevail and two modes propagate in the plasma ; they are called the fast and slow hydrodynamic modes in the MHD regime and at higher frequencies the extraordinary and ordinary modes,. They are not affected by temperature and kinetic effects are not important for these modes. However, a correct study shows that the slow hydrodynamic wave is slightly affected by gyroradius effects in high /j’plasmas. When the wavenumber increases in such a way to satisfy (lo), the shear Alfven wave becomes a magnetokinetic mode in which finite gyroradius effects and currents are of paramount importance. Therefore, the frequency range in which this mode propagates widens and fundamental and ion cyclotron harmonics appear. When the wavelength becomes of the order of the electron gyroradius, only the interaction between the electric field of the wave and electrons become important and the oscillations are electrostatic. These electrostatic modes, which are usually called the Bernstein modes, propagate in the electron frequency range and only at very large angles to the magnetic field.

8. THE PROTON MECHANISM

WHISTLER,

BETWEEN

RESULT

THE

FAST

OF A COUPLING HYDRODYNAMIC

MODE AND THE ION KINETIC MODE?

It is known that for a conversion process to take place between two waves, their refractive indices should be close to each other. However, mathematically, the problem is quite complicated to solve for two essential reasons: firstly, because the approximation of geometrical optics does not apply in the singular region where coalescence occurs and secondly because the inclusion of temperature in the wave differential equation introduces finite gyroradius corrections in the dielectric tensor components which increases considerably the degree of the wave equation. Therefore, we have tried to solve this coalescence problem for a cold plasmain the WKB approximation(StCfant, 198 I). It is then easy to derive the solutions for such a onedimensional plasma and one can show that if the density gradient is perpendicular to the magnetic field (the easiest case), the k of the modes satisfies the well known dispersion relations for plane waves propagating at very large angles to the magnetic field, i.e. the

345

familiar equation for the extraordinary mode and equation (4) which refers to our so-called magnetokinetic mode. As at the present time, the correct differential wave equation (not WKB) for a hot plasma in which the density gradient makes an angle with the magnetic field has not been yet derived, we however will proceed further by giving the WKB solutions a trial, since they are correct solutions outside the critical coalescence region. The dispersion characteristics of the fast extraordinary wave are plotted in Fig. 9 for a two-ion component plasma with an equal concentration of protons and oxygen ions. On the same graph, we have also plotted the dispersion characteristics of the slow hydrodynamic mode for different values of the propagation angle 8. We recall that under real ionospheric propagation conditions, the branch on the extreme left which corresponds to the fast hydrodynamic wave, does not show a cut-off at w/Q, = 9, but at o/D, = 16, i.e. at the proton gyrofrequency. However, it does not really matter, since we are here concerned with the other branches which are not affected at all by the characteristics of the transition region. One notices that, with increasing 8, the wavenumber gap between the two values of the refractive index labelled here N + and N widens ; in other words, the two curves never intersect and one should not speak of a crossover point or of a crossover region as the former hydrodynamic theories do. As expected for large values of 0 such that (7c/2)- 0 < (m/M)“‘, the slow hydrodynamic branch disappears and a new (but still hydrodynamic) branch appears in the frequency range below the proton gyrofrequency which has the dispersion characteristics of the extraordinary mode. The transition occurs smoothly at the interface between the two plasma media Of and O+ + H+, since the two branches correspond to the same refractive index N _ (Fig. 5). We want to stress again that this is not so if one considers the case of small angles of propagation. Indeed, Fig. 3 shows that the incident E wave in its crossing through the transition region may give rise to two waves: one has dispersion characteristics of the extraordinary type and cannot propagate below a certain frequency OJ = Q, the other is hydrodynamic in nature and does not satisfy the free space propagation conditions. This newly generated wave can be excited only through a coupling process similar to the Z mode coupling between transverse waves at the electron plasma frequency (Stefant, 1981). The dispersion characteristics of -the magnetokinetic mode described by equation (4) are plotted in Fig. 10 for different values of the parameter i = kfc’/Fi,‘, where II, is the O+ plasma frequency. The curves are similar to those obtained by Crawford (1965)

346

R. J. STEFANT

16

I

IO-6

I

1o-5

1o-4

*

k cm-’

FIG.~.PLODOF THEDISPERSION CURVESOF HYDRODYNAMICMODES. The N- and N + labels refer respectively to the fast and slow hydrodynamic modes. For CDS0 less than (PI/M)“~, the mode is again extraordinary and shows a resonance at the Buschbaum resonance q,.

for the electron cyclotron harmonic waves. However, we want to stress that, unlike waves propagating exactly perpendicular to the magnetic field, these modes have dispersion characteristics below the O+ gyrofrequency (StCfant, 1970b). The following mechanism then takes place: the electromagnetic waves originating in the lightning flash propagate across the lower ionospheric layers (the E and D layers) in the whistler mode, i.e. their wave normals make some angle (different from 0 or n/2) with the magnetic field. Next, the waves reach the region between 300 and 600 km where the H’ and 0’ densities change widely in time and space and where it is known that strong latitudinal and azimuthal gradients exist (Misyura er al., 1971). Those gradients compel some of the incident waves to be refracted nearly at right angles to themagneticfield.The up-goingwhistler waves follow then the dispersion characteristics described by Fig. 5b which shows that the fast hydrodynamic mode (the E mode) not only propagates above but also below the proton gyrofrequency. The two ion (Buschbaum) resonance is close to the local proton gyrofrequency, i.e. roughly 650 cps, and a mode conversion takes place between the incident whistler mode and the ion kinetic mode at precisely that resonance (connect Fig. 9 to Fig. 10 to understand the whole process). The newly created wave is not a potential (electrostatic) wave, but, because of the averaging of the wave electric field over the Larmor circle, the mode has an appreciable magnetic field

w/R,

A i

FIG. 10. PLOTS OF THE DISPERSIONCURVES OF THE MAGNETOKINETIC MODE FOR DIFFERENT VALlJES OF THE PARAMETER i = k,“c”/II,$. Thecoupling

takes place between the two (hydrodynamic kinetic) modes at q,.

and

347

Comments on the proton whistler generation process created by electron currents along the field lines (Stefant, 1978). Unlike the EM wave, this non-potential mode energy propagates along B, and is received by the satellite in B but only in the frequency range where the mode conversion did take place, i.e. just around (but below) the local proton gyrofrequency. The remaining energy, i.e. the energy which has not been converted, propagates in the E mode, i.e. perpendicular to B, and is not received at the satellite. The newly created wave is now a proton whistler and because the mode conversion takes place close to the proton gyrofrequency, no cyclotron harmonics appear (see Fig. 2 of Stefant, 1970b and 1981 for explanations). However, the satellite can of course receive E mode energy from a different region D (Fig. 2) in the remaining frequency range, i.e. from 0 to roughly the proton gyrofrequency (one assumes the local plasma to behave identically everywhere); in other words, the Fourier spectrum energy received by the satellite in B will show an absence of a whistler trace between the frequency corresponding to the proton gyrofrequency at 500 km and the Buschbaum resonance at the satellite altitude. This frequency gap corresponds to the frequency range

for which the incident E wave is entirely converted into a proton whistler. As the E waves travel upwards, the two-ion resonance decreases and more and more wave packets are converted while the frequency gap in the whistler Fourier spectrum received at the satellite increases. As the magneto-kinetic modes become more and more electrostatic (see again Fig. 2 of Stefant, 1970b), O+ cyclotron harmonics appear mainly in the wave electric field since the mode is longitudinal in this frequency range. Of course, if the satellite antenna is sensitive enough to detect the magnetic field of the perturbations, the wave magnetic field should also be affected by harmonic interactions. Consequently, at the satellite altitude, a sonagram will look like the sketch of Fig. 11 with traces coming from : (a)

Whistlers originating from region C and propagating at angles to B,; as demonstrated earlier, these traces show a cut-off at the gyrofrequency of a proton gyrating in the Earth magnetic field at 500 km, i.e roughly 650 cps.

F (c.p.s.) A

Whistler

traces comlnq

from

C

o- : 651

F,

w>

wAc

d$?

Harmonics

I 3 FIG. 11. SONACRAMOF

T THE

POSSIBLESIGNALSRECEIVED BY THESATELLITEIN B(SEE

Fig.2).

The traces above Fi which is the proton gyrofrequency at the satellite altitude come from whistlers propagating at angles to B,. They show a cut-off at the gyrofrequency ofa proton gyrating in the Earth magnetic field at 500 km, i.e. roughly 650 cps. The upper trace below Fi is the proton whistler (for the explanation concerning the shape and the presence of harmonics, see Stefant, 1970b, 1981). The lower trace comes from whistlers propagating perpendicularly to the magnetic field originating from D.qd and m,, correspond respectively to the lowest Buschbaum resonance frequency along the paths BD and AC. In order not to overload the figure, we havejust shown below the proton gyrofrequency one proton whistler and one electron whistler traces as they should be recorded by the satellite in B.We have also figured the tangents at the “crossover”frequency which in our theory has no physical meaning.

348 @I

(cl

R. J, STEFANT Whistlers coming from region D and propagating perpendicular to B,. These traces display the well known time frequency of whistlers in the low frequency range, but have a high frequency cut-off corresponding to the lowest two-ion resonance along the path DB. Lastly the proton whistler traces triggered by the mode conversion process occurring in C which show a low frequency cut-off corresponding to the lowest Buschbaum resonance along the path CA.

AS both (the electron whistler and the proton whistler) traces are not triggered by the same electron whistler like in the Gurnett theory, it is possible that both traces or their extension in frequency space could cross each other and form a “crossover” point. At this crossover point, like in the sketches of spectrograms of Gurnettetal.(l965)orShawhan(1966), theslopesdfldt ofthe traces above and below the “crossover”should be effectively discontinuous. Lastly, to quote the observations, which after all must be the test of any theory, the sketch of the spectrogram plotted in Fig. 11 derived from our theoretical results seems well in agreement with the observations. If as claimed by Gurnett et al. (1965), Shawhan (1966) Jones (1969), Budden and Smith (1974), proton whistlers are always observed with their parent electron whistlers, our theory should then provide one other possibility of explanation of the “crossover” frequency which of course in our theory does not correspond to a change of polarization like in the Gurnett papers and has no physical meaning at all. One last word should be said about the coupling conditions. Indeed, one knows that in the conversion process, some boundary conditions between the two waves must be satisfied. These boundary conditions imply the continuity of both the wave magnetic field and of some longitudinal (along the dc magnetic held) component of the wave electric field, the electric field transverse component satisfying Gauss’ law. For large angles of propagation, the required space charge field is due on one side ofthe layer to the difference between the hydrodynamic ion velocities for the fast mode, on the other side to the finite Larmor radius effects for the kinetic mode. On the other hand, the continuity of the wave magnetic field is insured by the presence of currents in both modes which is not the case for the electron cyclotron harmonic modes. Indeed, in the electron frequency range, it has been proved (St&fan& 1977) that these short wavelength modes are purely electrostatic; so, the boundary conditions in that case are much more difficult to satisfy. Consequently, the presence of currents in both waves which implies the

continuity of the wave magnetic field is an important feature of the present mode conversion mechanism which should be pointed out. 9.CONCLUSIONS We have shown that the rising tone which has a resonance at the proton gyrofrequency and(by analogy with the atmospheric whistler) has been called the proton whistler, cannot be the result of simple propagation effects of an electromagnetic wave through the slowly varying ionospheric medium. The misunderstanding of Gurnett and co-workers, Jones, or Budden and Smith lies in that their theory implies from the very beginning that the formation of the proton whistler must be associated with a reversal of polarization from right-handed to left-handed. A wave is usually represented by its refractive index which is of course a function of the plasma characteristics and the question of polarization may be only examined, when the expression for the refractive index is derived. Next, the right question to ask is whether the wave which refers to the refractive index of interest satisfies or not the free space propagation conditions, i.e. is electromagnetic or acoustic in nature, the dispersion and the polarization should be studied afterwards. If one does so, since the phenomenon under study here is supposed to be a hydrodynamic wave which should take its energy from the lightning flash, a coupling process must necessarily take place between two waves of different nature. Then, the dispersion characteristics together with the currents study, since some boundary conditions are to be satisfied, should be analysed. But by no means, neither the polarization nor the collisions are of some importance in the basic process. The theory that we have proposed requires special propagation conditions, since a wave transformation implies theoretically a coalescence of two roots of the characteristic equation in k space. In our case, since the kinetic waves propagate at very large angles to the magnetic field, the density gradient should be almost perpendicular to B,. The study of the dispersion characteristics show then that both waves propagate perpendicular to each other, the kinetic mode along B, and the incident E mode perpendicular to it. This implies that on a At diagram the proton whistler and the short fractional hop whistler should never correspond to each other. Of course, both traces, i.e. a proton whistler + an electron whistler, but not the triggering one, can appear on the sonagram as the result of the principle of superposition (Fig. 11). It is the natural consequence of two different mechanisms which are present at the same time at the transition level: one, which gives rise to the electron whistler, is

Comments on the proton whistler generation related to dispersion, the other, the proton whistler, is due to the coupling process. It would be therefore extremely surprising that a one-to-one correspondence between the two traces could be found, since the transition region prevents the incident E mode to have Fourier components below the local gyrofrequency, except, as shown in this paper, ifthe extraordinary wave propagates at very large angles to B, (see also Fig. 2). If however, as claimed by Gurnett and his co-workers (1965,1969), Shawhan (1966),Jones( 1969), Budden and Smith ( 1974), a one-to-one correspondence is found, the explanation is given by the only satisfactory conclusions of the last paragraph which shows that both traces may indeed join at the “crossover” point. But this “crossover” frequency has nothing to do with our wave conversion mechanism or with a wave polarization process (Jones, 1969). To conclude, it would be worth computing the electric to magnetic field ratio, since Gurnett and his colleagues have measured with Injun the electric and magnetic fields of the waves. This ratio can be theoretically computed at different frequencies and yields : B, -Ic = NZ

=;x

for the slow hydrodynamic B

A=_: E,

N.. kZcZ

mode (Gurnett

for the magneto-kinetic

mechanism) mode

1 +-+ e

where both kz are respectively given by the dispersion relations for the slow hydrodynamic mode (Stix, 1962) and the ion kinetic mode (StCfant, 1970b). Orders of magnitude have been given elsewhere (St&ant, 1981). This would be the most relevant test which should be done to confirm or invalidate the present statements on proton whistler theories.

REFERENCES

Budden, K. G. and Smith, M. S. (1974) The coalescence coupling’s ionosphere.

points in the theory of radio Proc. Roy. Sot. A 341. 1.

waves

of in the

process

349

Chan, K. W., Burton, R. K., Holzer, R.E. and Smith, E.J.( 1972) Measurements of the wave normal vector of proton whisllers on OGO 6. J. yeophys. Res. 77. 635. Crawford, F. W. (1965) Cyclotron harmonics waves in warm plasmas. Rndio Sci. 69D, 789. Gurnett, D. A., Shawhan, S. D., Brice. N. M. and Smith, R. L. (1965) Ion cyclotron whistlers. J. geophgs. Res. 70, 1665. Gurnett, D. A.. Pfeiffer, G. W., Anderson. R. R., Mosier, S. R. and Cauffman, D. P. (1969) Initial observations of VLF electric and magnetic fields with the INJUN satellite. J. gwphys. Rex 74, 463 I. Jones, D. (1969) The effect of the latitudinal variation of the terrestrial magnetic field strength on ion cyclotron whistlers. J. uttnos. fzrr. Phys. 31, 971. Jones. D. (1970) The theory of the effect of collisions on ion cyclotron whistlers, in P/a.smrr War es in Space und in the Luhoratory, Vol. 7. p. 471. Edinburgh IJmversity Press, Edinburgh. Miayura, V. A. ef al. (1971) Ionospheric investigations on radio-wave propagation from space objects over a solar cycle. Spuc,ti Resetrrch Xl. Akademic, Berlin. Shafranov. V. D. (1967) In Re~Ccws ofP!asma Ph1sic.s (Edited by Leontovitch, M A.). Vol. 3. p. 1. Consultant Bureau Enterprises. New York. Shaw.R.R.andGurnett,D.A.( 1971) Whistlerswith harmonic bands caused by multiple stroke lightning. J. yeophys. Rex 76. 1851. Shawhan, S. D. (1966) Experimental observations of proton whistlers from lnjun 3 VLF data. J. geophvs. Res. 71, 29. St&pant, R. J. (I 970a) Alfvbn damping from finite gyroradius coupling to the ion acoustic mode. Phys. Fluids 13, 440. St&ant, R. J. (I 97Oh) Non potential ion harmonic waves. J. yroph~s. Rex 75. 7182. Stifant, R. J. (1977) Possible applications of the slow nonpotential modes to themagnetosphcre and consequences. J. yeophys. Res. 82. 1848. St&ant, R. J. (lY78) General dispersion for non-potential ion harmonic waves propagating in plasmas of arbitrary 8. Phys. Fluids 21, 55. St&ant, R. J. (1981) Linear wave conversion processes in the theory ofthe proton whistler. Astrophys. Space Sci. 76,301. Stix, 1‘. H. C1962) The Theory oJ’Plmma Waoes. McGraw-Hill, New York. Taylor. H. A.. Jr., Brinton, H. C., Pharo, M. W. III and Rahman, N. K. (1968) Thermal ion in the exosphere: evidence of solar and geomagnetic control. J. geophys. Res. 73, 5521. Terry, P. D. (1978) A complex ray tracing study of ion cyclotron whistlers in the ionosphere. Proc. Roy. Sot. A 363, 425. Wang, T. (1971) Intermode couphng at ion whistler frequencies in a stratified collisionless atmosphere. J. yrophys. Rex 76, 947.