Characteristic frequencies of a non-maxwellian plasma: A method for localizing the exact frequencies of magnetospheric intense natural waves near ƒpe

Planet. Printed

Space Sci. Vol. 29. No. in Great Britain.

II. pp. 1251-1266,

1981

00324l633/81/111251-16$02.00/O Pergamon Press Ltd.

CHARACTERISTIC FREQUENCIES OF A NON-MAXWELLIAN PLASMA: A METHOD FOR LOCALIZING THE EXACT FREQUENCIES OF MAGNETOSPHERIC INTENSE NATURAL WAVES NEAR fpe GtRARD

Centre de Recherches

BELMONT

en Physique de L’Environnement,

CNET, 92131 Issy-les-Moulineaux,

France

(Received 12 March 1981) Abstract-Intense natural waves are commonly observed, onboard satellites, in the outer Earth’s magnetosphere, inside a narrow frequency range, including the electron plasma and upper hybrid frequencies. In order to progress in the understanding of their emission processes, it is necessary to determine precisely the relationship which exists between their frequencies and the characteristic frequencies of the magnetospheric plasma. For this purpose, it is necessary to take into account the fact that some of these characteristic frequencies (f,,), which are provided by active sounding of the plasma, not only depend on the total density, but also on the shape of the distribution function (which has generally been assumed, in previous studies, to be maxwellian). A method providing a fine diagnosis of general non-maxwellian plasmas is developed. This method of analysis of the experimental data is based on a theoretical study, which points out the influence of the shape of the distribution function on the dispersion curves (for wave vectors perpendicular to the static magnetic field). 1. INTRODUCTION

seconds to several hours, very intense electrostatic waves, whose frequency is close to the local electron plasma frequency, can be observed in the outer Earth’s et al., 1978a,b). magnetosphere (Christansen Because of their large intensities, these waves are certainly the most important to be considered, in the VLF range, with regard to particle diffusion. However, they are presently much less studied, both experimentally and theoretically, than some waves which usually have weaker intensities, such as the so-called (n + l/2) fc. (Kennel et al., 1970; Scarf et al., 1973; Karpman et al., 1973, 197Sa, b; Young et al., 1973; Ashour-Abdalla et al., 1975, 1978, 1979, 1980; Christiansen et al., 1978b; Hubbard and Birmingham, 1978; Hubbard et al., 1979). Besides, it is actually impossible to check experimentally any theory about the nature of these intense waves, because the experimental accuracy did not allow, up to now, to state whether their frequency coincides exactly with the plasma frequency, or with the upper hybrid frequency, or with some other frequency in their close vicinity. So, it appears as a difficult but necessary step to obtain an unambiguous localization of this frequency with regard to the plasma characteristic frequencies. The present work, issued from data provided by the European Geostationary satellites GEOS-1 and -2, tries to give a method allowing to go in this direction. In this respect, we have been During

periods

from

a few

greatly helped by the presence, for the first time on satellites designed for magnetospheric studies, of a relaxation sounder. It allows observation of the natural spectrum superimposed, at regular intervals and during selected periods, with a number of by which most of the plasma “resonances”, characteristic frequencies can be determined (Muldrew, 1972a). The intense waves near f,. are very monochromatic (Af/f = 10m3), and they have rather sporadic intensities, often large enough to exceed the saturation threshold of the electronics (>3 mV mm’). Kurth et al. (1979, 1980) have reported on similar properties from observations made onboard IMP6, Hawkeye 1, and ISEE. They have shown, in particular, that there is no obvious correlation between the occurrence of such intense waves and other geophysical parameters, such as latitude or local time. The only unambiguous correlation which has been reported, to our knowledge, is the one pointed out by Horne et al. (1981)intense waves disappear in the presence of an electron beam parallel to the magnetostatic field B,,. These waves are thought to propagate in a direction quasi-perpendicular to B, (Kurth et al., 1979) but a more detailed polarization study is yet needed before being able to make a definite statement. In part 2 of this paper, we will describe the method that we have used to analyze the data of the relaxation sounder, in order to get the best 1251

1252

GERARD BELMONT

accuracy in the localization of the intense natural waves with regard to the plasma characteristic frequencies. The corresponding results will lead us to develop, in the third part, an extension of the theory of Bernstein modes (Bernstein, 1958) to non-maxwellian plasmas. In the last part, we will show that this theory allows to obtain, from the data shown in the first one, a diagnosis for the density of the colder population-and eventually of the total population-of the magnetospheric plasma, and consequently to localize the natural emissions with regard to the corresponding plasma frequencies.

fqN

--N

A

fee

10

,,I

I

I

I

I,

/

/

/

N.2 NI_3N.L N.5 Nz6 Nz7 N=EN9 N=lO N-11

2. DATA ANALYSIS

2.1. The relaxation sounder The GEOS relaxation sounder (Etcheto and Bloch, 1978; Higel, 1978, S300 Exp., 1979) operates according to principles which have been worked out, at first, for topside sounding (Proc. IEEE, 1969; Dougherty and Monaghan, 1966; McAfee, 1973). It is equipment which consists of transmitting and receiving antennas, both working in the frequency range of O-77 kHz. When used in the “passive mode” (transmitter OFF), it detects natural waves with a high sensitivity (= 5 x 10e9 V m-’ Hz-“‘), and a good frequency resolution (Af = 10 Hz). In the “active mode”, quasimonochromatic pulses are injected into the plasma by the transmitter during short times (At = 3 ms, Af = 300 Hz). Several milliseconds after each one, nothing more than natural waves can be detected, generally, by the receiver, except for some particular frequencies, for which echoing signals are received. These signals are much more monochromatic (Af = 20 Hz) than the emitted pulse, and they correspond to frequencies for which the group velocity is almost equal to zero. Afterwards, these frequencies will be called “resonance” frequencies. These frequencies are known to be the gyroharmonics Nf,,, the plasma cf,,) and upper hybrid (fUH) frequencies (not always observed), and the frequencies which correspond to maxima in the Bernstein mode dispersion curves, the SOcalled fqN (Warren and Hagg, 1968; Muldrew, 1972b). 2.2. Hamelin’s diagram As long as the electrons of the plasma can be described by a maxwellian distribution, the dispersion equation has the form which was first proposed by Bernstein (1958). For perpendicular propagation, the corresponding dispersion curves (Crawford, 1965) get an infinity of branches,

fpe =3.5fce

FIG.

fee

HAMELIN’S DIAGRAM.

The whole series of normalized f,Jf_ frequencies is given for each value of the normalized plasma frequency fP/fCP The ordinate determines the decimal part of f,JfC., while its integer part is determined by the number N of each curve.

separated by the gyroharmonics (e.g. Fig. 4), and they present maxima (fqN),in all branches above the upper hybrid frequency (the first of such branches, which contains fUH,is usually called the “transition branch”). The plasma frequency can be deduced, in these conditions, from the measurement of any fqNfrequency, because the values of f qN only depend on the ratio f,/f_.. For that purpose, it is especially convenient to use the diagram of Fig. 1, which was introduced by Hamelin (Hamelin, 1980; Pottelette et al., 1981). On this diagram, every curve N gives the normalized frequency separation between fq,., and its lower gyroharmonic (fqN - Nfc.)/fc. vs the normalized plasma frequency f,,/fc.,for fqN inside the interval N defined by Nf,, < fqN < (N + l)f_. 2.3. Method of analysis In order to be able to localize the maxima of the natural spectrum with the best accuracy with regard to the plasma characteristic frequencies (f,. f”“, fqw etc. which may be very close one to another), we tried to get out the information from the whole set of the observed resonances, by using the following method: we put one point on

Characteristic frequencies of a non-maxwellian plasma Hamelin’s diagram for each observed resonance frequency, except gyroharmonics, as if all of them were fqN. If the plasma is actually maxwellian, and if there is no additional effect, e.g. Doppler shifts (De Feraudy and Hamelin, 1978), all points corresponding to true fqN resonances will be aligned on a vertical line giving the plasma frequency in abscissa. Other resonances, if any, will give isolated points, anywhere else on the diagram. We shall see that the experimental diagrams are not, general, so easy to interpret. The main differences seem to be due to non-maxwellian distributions in the magnetosphere IS302 thermal and supra-thermal plasma detectors which are onboard GEOS confirm this point (Johnson et al., 1978)], and we shall analyze this effect in some detail, from a theoretical point of view. We shall also see that some observed discrepancies could be interpreted by Doppler effects, which we will not comment on in detail in this paper. Taking into account the accuracy of the frequency determination of each fqN resonance (Af = 10 Hz), the accuracy of the determination of f, is very good, provided that the plasma remains stationary over the period of time needed to make a full frequency sweep (Aht= 22 s). Furthermore, the accuracy of f, decreases when higher orders of N are used, because of the decrease of the slope of the different curves when frlN & f,. In Fig. 2 this effect is shown by “error bars”, which correspond to a constant and arbitrary value of the incertitude on each fqN determination of about 150Hz. (It corresponds to ~0.5% fluctuations in the characteristic frequencies.) 2.4. Results The results of four ache soundings of the plasma are presented in Fig. 2. They have been displayed on Hamelin’s diagrams, as explained previously, and their features can be considered as typical of those that we have obtained, up to now, by this method. Figure 2(a) seems to be the simplest, each resonance appears as a fqN, approximately giving a unique value of f,,. However, it must be noticed that this value is lower than the one obtained almost at the same by the other active wave experiment onboard GEOS: the mutual impedance probe S304 (Decreau et af,, 1978). Besides, the first resonance fq4(which is the highest point on the diagram) corresponds really to a slightly lower value of f, than the other fqN’s. The second case (Fig. 2(b)) is almost identical, but one additional resonance occurs. Its frequency, fx, cannot easily be related with the value of jPe

1253

issued from the fqN’s but it is very near the S304 value. Remember that the value fX = 7.47 t should be read from the ordinate, while the 5304 value of f,df_ is indicated on the abcissa axis). In the third case (Fig. 2c), a similar fX frequency can be observed, the values off, issued from each faN decreasing when N increases. This descrase cannot be attributed to a time variation of the ratio fdf=,, because the same feature is recurring during several successive active sweeps. In the final example (Fig. 2d), two resonances are found inside each branch, whose abscissas are decreasing for one and increasing for the other, when N increases. Figure 3 gives an example of the natural spectra which are obtained when the experiment is in a passive mode (just preceding, here, the active sweep which has been presented in Fig. 2b). These spectra almost continuously exhibit a few weak maxima, whose frequencies more or less correspond with the first fqN resonances (identified from the nearer active sweep by the previous method). We suggest, that these maxima can be due to thermal noise which should be sustained for frequencies for which the group velocity tends to be null. Sometimes, a stronger maximum occurs and its frequency, as in Fig. 3, seems to be very close to the resonance frequency fx which can be found from the active sweep. Generally, the exact identification of these two frequencies is very difficult to ensure by such comparisons, because of the possible fluctuations in the plasma characteristics, from a passive sweep to the next active one. In the future, we will try to make a definite statement about this point by investigating the spectrum of natural waves which is observed during the active sweep itself. If this identification was proved to be true, it would certainly be important in understanding the intense natural waves near f,. 3. EXTENSION

OF BERNSTEIN

NON-MAXWELLIAN

MODES TO

PLASMAS

In this theoretical part, we want to describe and discuss some results concerning the propagation in non-maxwellian plasmas. Before doing so, we will first briefly summarize the analytical and numerical methods that we have used in order to obtain and solve the dispersion equation. Then, we will give some numerical results, for distributions which consist of a maxwellian population mixed with (a) a “ring” population (delta distribution) and (b) another maxwellian population. The second case will appear as more consistent with experimental data, and it will be emphasized,

1254

G~RARDBELMONT

1.0 0.9

08

07 06 0.5 01 03 0.2 ot 0

08 07 06 05 06 03 02 0.1 0. 301

fee

WI

@I FIG. 2. ~XPERIM~NTALRESONANCE~EQU~NCI~SDIS~~AYE~O~ ~HAMELIN,SDIAGRAM. The arrow indicates the plasma frequency which is determined by the mutual impedance probe SM4.

1255

Characteristic frequencies of a non-maxwellian plasma 0.1 Ey(mV.

t

netoplasma

m-1)

can be written H(A, P) =

GEOS-

1

lIpp2.

(1)

H is a two variable function, depending only on the shape of the distribution function f(v). Its expression is, in the electrostatic approximation (e.g. Young et al., 1973)

JULY 5 ,1977 R =6.27R, ;T:g.,2 LT = 12150 fee = 2870 Hz natural spectrum --- S 301 resonances

Jn20,)

k, *

() Ji-,(A,) - -c+,(h,)

X(p-A,,-n)2+ xx

1 2

fq8 ’ fq9

’ fql0

’ fqll

‘fqrz

‘fq13

‘fqu

’

f

FIG. 3. NATURALSPECTRUM (SOLIDLINE). This passive sweep is must preceding the active one of Fig. 2(b) (At = 22s), from which the resonances fqN and Nf,, (dashed lines) are deduced.

by exploring a wide range of parameters. From this set of numerical results obtained for plasmas which are composed of two maxwellian components, we will try to draw out some general ideas on the propagation in general non-maxwellian plasmas. These ideas will be supported and develoned. in the last oart of this section, bv an analytical study of the hispersion relation. 3.1. Dispersion relation and numerical methods Let us first define the following notations (i) p is a normalized the frequency o

quantity

corresponding

to

I

(2)

where J,, are the Bessel functions of order n, and the subscripts /( and I refer to the direction of the static magnetic field. The plasma is supposed to be isotropic in the perpendicular plane. The distribution function f(v) is normalized to unity, the information about density being shifted into the second member I/~P2. It is important to note that, owing to its integral form, the function H is linear with respect to f(v), which allows to split it, eventually, into several terms corresponding to several electron populations, PP’H(A, P) = c /.~tiHi(A, CL). As we will take an interest in waves propagating in the perpendicular direction, we will ignore the contribution of the parallel term in relation (2), and we will keep only the second one, which gives the well-known Bernstein modes, when equation (1) is solved for a maxwellian plasma. These Bernstein modes will be considered as reference modes when discussing the dispersion curves obtained with other more complicated distributions. For most of numerical computations, we have used an expansion of H in power series which can be obtained by using the following identity,

(ii) A is a normalized quantity corresponding to the wave vector k, and depending on the thermal velocity V. Each component A, of A is defined by

kxVxlo,, = A,. where Small letters A will be used, when referring to dummy variables v, inside integrals upon particle velocities. Then, the dispersion relation of any mag-

P-AII-~

GERARDBELMONT

1256

This identity can be obtained by changing first the infinite sum of squared Bessel functions into a sum of two products (Tataronis and Crawford, 1970) “=+crJ’.-, -J’,+, n=--m p-n

c.

=J,+,J-,-,

- J,-IJ-,,I (sin prr)/ r

(6)

and developing then the products in power series (Petiau, 1955). From this form, and for any given distribution, one only has to calculate the integrals J v’“f(v)d3v, i.e. the even moments of the perpendicular distribution function, to obtain the corresponding dispersion equation, also developed in a power series. This is easy for most usual distribution functions (maxwellian, algebraic, etc.), and it generally gives a fast algorithm, cheap for computation time. Nevertheless, these series do not converge quickly enough to be used for too large values of Ai (typically A,> S-15, depending on the distribution). For large A,, these series, which are alternated do converge, but their maximum terms become too large with respect to the value of H. When the ratio between the maximum term and H exceeds the computer precision, this method must be forsaken. Then, it is often possible to use an asymptotic expansion, such as the one proposed by Karpman et al. (1973) for maxwellian plasmas,

equation (H = l/pp2) are not solutions (in general) of the dispersion equation of each population alone (Hi = l/p&). So, for determining the dispersion curves of a composite plasma, to know the H function of each component, for any couple of variables [A,, ~1, where this function may have any value, even negative. What we want to emphasize here, is that the H function of one population is completely described by diagrams of the type shown in Fig. 4. This diagram appears as a set of dispersion curves (Bernstein modes, in this example, since this population is supposed to be maxwellian), drawn for different values of the plasma frequency. Indeed, it can also be considered as a representation of the H function in the plane of its two variables A, and p, by considering the dispersion curves as the “iso-H” curves (H = l/pd). The only difference from usual set of dispersion curves is the presence (dashed lines) of negative values of H. Some characteristics of the H’function that are associated with a maxwellian plasma will be useful to remember: (i) For a fixed A,, H(p) monotically decreases H=

+006

6

and this one for a delta distribution, which can be obtained by using the asymptotic expansions of Bessel functions (Abramowitz and Stegun, 1964) in relation (6), H(A,,

p)=$$-p.

(8)

When H is calculated, equation (1) is solved, considering the frequency as the variable, and A, as a real parameter. This choice has no theoretical consequence on our work, because we only look for real solutions (k,, = 0). Besides, in the regions where the dispersion curves have maxima, it is more convenient than considering A, as the variable. 3.2. Dispersion curves and H function When we use relation (3), we have to remember that the solutions [A,, ~1 of the total dispersion

0

I

I

2

4

FIG. 4. All “iso-H”

.

“1

H FUNCTION.

curves with positive values are dispersion curves, for plasma frequencies given by H = l/p,,‘.

Characteristic

frequencies

+m to --co when g increases from a gyroharmonic to the next one. (ii) For a fixed p, H(A,) first increases from l&p* - 1) to a maximum value H,,,,, [the fixed frequency is a maximum (fqN) for the curve H(A,,p)= I&,,,]. Then, it decreases down to a negative minimum E?,,,;,,, before tending towards zero (as - l/A,*) as A, increases towards infinity. Furthermore, the quantity -(aH/+-‘, which can be considered as a temporal excitation coefficient for resonant instabilities, can be characterized, in Fig. 4, by the F spacing between the iso-H curves, on a vertical line A,. = Cst. in this respect, it can be seen that the most excitable modes lie around the transition branch, mainly for frequencies localized between the upper hybrid frequency and the first fqN_ We will no longer discuss this last point, since detailed instability studies are beydnd the scope of this paper.

from

3.3. Results for a maxwellian distribution mixed with a “ring” What we call here a “ring” is a popuIation whose distribution depends on the modulus of the perpendicular velocity as a Dirac one, 6(u,VR)/27rVR (the subscripts R refer to this ring population). Tataronis and Crawford (1970), in a theoretical study which was mainly devoted to pure1 y ring-, shell- or maxwellian-populations,

of a non-maxwellian

plasma

I251

have also introduced this kind of composite plasma, by just investigating, for it, the values of plasma characteristics giving non-resonant instabilities in the lower dispersion branches. Motivated by observations of intense waves near in order to f P, we did similar computations, analyse, in more detail, the changes in Bernstein modes when adding the ring population. The results of this numerical study are illustrated here by only two examples, displayed in Fig. 5, where V, is the velocity of the ring particles, and (2R= n,J(n, + nc) is the corresponding concentration (the subscript C refers to the maxwellian ‘population). The abscissa is normalized according to the maxwellian thermal velocity. The main conclusions are the following: (i) For very small concentrations (Ye of the ring population (Fig. 5a), the dispersion curves slightly oscillate around the Bernstein modes which correspond to the maxwellian distribution alone. (ii) When (Ye exceeds a value of the order of I%, these oscillations are sufficient to force some branches (around the transition branch) to cross the gyroharmonics, which induces collapses, very close to these gyroharmonics (Fig. Sb). In these cases, vertical tangents appear on the dispersion curves; between them, two real solutions are replaced by two conjugate complex solutions, whose real parts are only displayed on the figure.

a FIG. 5. DISPERSIONCURVESIN THE PRESENCE OF A RINGDISTRIBUTION.

Solid line: mixed distribution, maxwellian plus ring. Thin line: maxwellian component alone. aR and V, are the concentration and the velocity of the ring population, V, is the thermal velocity of the maxwellian population.

1258

CXRARDBELMONT

These regions correspond to non-resonant instabilities. (iii) For larger values of (Ye, the number of collapses increases, on both sides of the transition branch, and the structure of Bernstein modes becomes completely modified. When one tries to interpret the intense natural waves near fP as due to the non-resonant instabilities which occur in this model, one is necessarily led to consider small concentrations ((Ye= 1%) for the ring population, because of the following arguments: (i) There must be only one collapse, since multiple intense waves are not observed. (ii) The fqN frequencies must be quasi-unchanged with regard to those of the maxwellian component alone (they have been observed, during such events, to be very well aligned on Hamelin’s diagrams). (iii) We are not aware of any particle data, in the magnetosphere, showing a more important ring in the electron distribution function. Indeed, in contradiction with the predictions of this model for such small values of Q, the intense waves are not usually observed to be in the close vicinity of the gyroharmonics. So, we can conclude that these non-resonant instabilities seem to be irrelevant for explaining the strong emissions near fP. 3.4 Results plasma

for

a

two

maxwellian-components

As mentioned above, the dispersion equation for two maxwellian-component plasmas can simply be deduced from the dispersion equation of a pure maxwellian plasma, thanks to the linearity properties of H. It is possible to add, in this manner, many maxwellian components, in order to fit an experimental distribution, and to find numerically the frequencies and the growth rates of the electrostatic waves. There are many examples of this kind of computation in the literature about the (n + 1/2)fCe (Ashour-Abdalla et al., 1978, Hubbard and Birmingham, 1978, Rijnnmark et al., 1978). On the other hand, very few authors have tried to analyze theoretically the dispersion curves themselves, and the changes brought by the suprathermal components on their features. Only Caffey and Laquey (1976) have presented such a study, with a two-component plasma, the hottest component having the largest density, in agreement with the then prevailing ideas about the composition of the magnetospheric plasma in the regions where electrostatic emissions were obser-

ved. However the GEOS measurements have shown that the reverse situation is most commonly observed, at least in the day side and at geostationary altitudes, the cold density being of the order of 10 cme3 whereas the hot density does not exceed -1 cm-3 (Christiansen et al., 1978a, S300 experimenters, 1979). We present, in Fig. 6, a more complete set of curves, by varying two parameters: the concentration of the “hot” component, LYE= n&n, + n,), and its thermal velocity with regard to the “cold” one, V,lV,. For reference, we have also drawn the Bohm and Gross modes (Bohm and Gross, 1949) for each component alone, and the Bernstein mode for the cold population alone. For each situation, the main conclusion of Gaffey and Laquey can be verified, namely, for small wave numbers, the transition branch always starts at the total upper hybrid frequency (i.e. all the electrons act), while for large wave numbers, the dispersion curves merge with the Bernstein modes of the cold component alone (i.e. only the coldest electrons act). From the present study, we can also draw some other conclusions: (i) For large ratios V,lV, (Figs. 6a-c), the dispersion curves exhibit two distinct maxima in every branch above the transition branch, whose wave members are roughly given by the Bohm and Gross modes of each component. One of these series of maxima (which corresponds to the largest values of A,) is generally quasi-indistinct from the series of fqN corresponding to the Bernstein mode of the cold component alone, except for very large concentrations (Y” of the hot component. This series will be called “cold f+, series” from now on, while the other series will be called “hot fqN series”. (ii) For smaller ratios of VJ Vc (Figs. 6d-f), the two previous humps in the dispersion curves can merge together (a process which starts first just above the cold plasma frequency), and eventually give only one maximum. In Fig. 7, we study the relative locations of the frequencies of the maxima, by placing them on a Hamelin’s diagram, in the same way as we did with experimental data. Of course, the “cold fyN series” that remain unchanged with regard to thqse of the colder population, give series of points vertically aligned, their abscissa being the cold plasma frequency. For a large ratio V,lV, (Figs. 7a-c) and for increasing (Y”, it can be seen that the first fqN of this “cold series” is shifted left, and that the series itself is prolonged by new maxima, in branches

Characteristic frequencies of a non-maxwellian plasma corresponding to frequencies lower than the cold plasma frequency. The “hot frlN series” gives apparent plasma frequencies which tend towards the total plasma frequency in the upper part of the diagram, and towards the hot plasma frequency in the lower part. For smaller ratios V,/V, (Figs. 7d-f) the con-

5

1259

elusions concerning the unique maximum are almost identical to those concerning the previous “hot fqN series”, but the limit given in the lower part of the diagram, for the plasma frequency, may be closer to pPc or pPH, depending on whether the cold or the hot component has the largest density. Most of these conclusions are better understood

,/' Z'

hot

(a) 0 2

L

UC

0 2

L

2

L

\IC

6

; 0

(d) 2

1 FIG.

6.

\IC DISPERSION

0

(e)

CURVES

(f) Al,

FOR A TWO MAXWELLIAN-COMPONENT

0

2

1

PLASMA.

Solid line: two maxwellian-component plasma. Thin line: coldest maxwellian component alone (Bernstein modes). Dashed line: Bohm and Gross modes for each component alone. puHc is the normalized upper hybrid frequency corresponding to the cold component alone. puHT is the normalized upper hybrid frequency corresponding to the total plasma. All the curves are drawn for a constant cold density (pPc = 3.5).

UC

1260

GERARD BELMONT

with the help of some analytical considerations. Let us split H into two terms, as shown in relation (3): P&H(AK, and Ii

CL)+ CLZ~,JWA~

- 1,

Section 3.2). Taking into account that AIH and A,, vary proportionally to k, with A,JA,, = VH/Vc > 1, we can easily conclude: (i) For large and increasing k,, Hfh,,) tends towards zero before H(A,,), and equation (9) reduces to

(9)

remember the behaviour of the function associated with maxwellian p’.asmas (see

(a)

E

(b)

10

10

09

09

08

08

07

07

06

06

05

05

OL

OL

03

03

02

02

01

01

0

0

idI FIG.~.

THEORETICAL

06

(e) feN FREQUENCIES,DISPLAYEDON

HAMELIN'SDIAGRAMS,

~OMPONE~PLASMA.

Crosses correspond

to the “cold &N series”, full circles to the “hot

FORATWOMAXWELLIAN-

faMseries”. (Same as Fig. 6.)

Characteristic

frequencies

of a non-maxwellian

which is the dispersion equation of the colder population alone. This explains why the “cold jqN series” is usually indistinct from the fqN series of the cold component alone. (ii) For small and decreasing k,, H(ALC) tends towards l&p’- 1) before H(A,,), and equation (9) becomes

which is the dispersion equation that would be obtained with a fictive population having the hot thermal velocity, and a plasma frequency pLH defined by

(12) This apparent plasma frequency pLH depends on the frequency p, and it explains why the “hot fqN series” is not aligned on vertical lines on Hamelin’s diagrams. Furthermore,. this simple relation can easily be used in order to determine the respective concentrations of both populations in a two maxwellian-component plasma, from the set of the jqN frequencies. So, this may constitute a new method of plasma diagnosis from the resonance data. (iii) For k, small enough to have H(h,,) = H(ALH) = l/(p* - l), equation (9) is simply

and the only solution is which means that the phase than the thermal velocities and therefore the plasma completely cold.

H(A,, P) =

1Y(u&(A,,

cL)du,; Y(U~

Y(o,)=-97~~2af/atr, Z(Al, p) =

“;$j‘*;

G(AL) =

nJ.0,) [A,,2

* 1

.

(14) In these notations, both y and z are functions of vI, but the second one depends on u, through the quantity A, = k,u,/w,.. Then, when y and z are presented as functions of the same variable (Fig. 8), their respective positions depend on the value of the wave number k,. That is the way by which H, which is the integral upon uI of the product yz, also depends on k,. By just investigating Fig. 8, it can easily be understood why H = J yz de, tends towards zero when k, tends to infinity, why it tends towards a constant value when k, tends towards zero, and why it reaches a maximum (which corresponds to a jqN frequency) for an intermediate value of k,. This value can be roughly determined by assuming that H is maximum when the maxima of y and z occur at the same abscissa (A,,,, = k, VLmax/~,,, if

U-..u”H

(a)

obviously p = pLUHT, velocity is then larger of both components, can be considered as

I

I :

I, >>U”” b)

2

3.5. Generalization In the previous section, some numerical results have been presented for the particular case of a plasma composed of two maxwellian components. We will now try to insert these results in a more general frame; by performing an analytical study of the dispersion equation, without specifying the shape of the distribution function f(v). For that purpose, it is useful to re-write the function H in such a way that the respective roles of the distribution function f(v), the wave number A,, and the frequencv - ,u can easilv be senarated:

1261

plasma

1

6

.I&&% ,‘)C

8

increasing kl

(c) yI!!LLzz 0 FIG.&

2

L

6

8

.I,!XA ,,)C

Two CHARACTERISTlCFEATURESOFTHEZFUNCTION

(a AND b). WITHTHESAMEABSCISSA.THELOCATIONOFTHE~ FUNCTION

CC),(THEREFORE THE VALUE OF H=Jyzdu~, DEPENDSONTHEVALUE OFK~.

1262

GBRARDBELMONT

we define V,,,, and A,,,, by y(V,,,J ZbLmJ = zmx).

= Y,,,, and

Henceforth, it is also important to notice that all the information concerning the distribution function, when calculating H, appears through the function y(v,), which is normalized since J y(vJ du, = 1. By this way, it can be understood that the number of maxima of H, and therefore the number of fqN series, is fixed by the number of maxima of the y function. In particular, when two maxwellian populations, whose thermal velocities are sufficiently separated, are superimposed, the corresponding y function (which is the sum of the y functions of each population alone) has two peaks. That is the reason who two fqN series exist in such situations (Figs. 6d-f). Before being able to further analyze the influence of the shape of the distribution function upon the dispersion curves, we need to examine, in more detail, the properties of the function z(A,, p). This function is a linear combination of the functions z,(A,) (Fig. 9), with coefficients l/(~Z--*) which depend on the frequency. The maxima of z.(A,) decrease when n increases, while the coefficients have a maximum value for n = n,,, where n, is the nearest gyroharmonic from the frequency CL.In Fig. 8, two characteristic features of the function z(AJ are displayed, corresponding to two different values for the fixed p. In the first one (Fig. 8a), the term z, is predominant, and the maximum value of z, which is obtained for a small value of A,, is close to l/(p’I). Then, the positive part of this curve is almost rectangular, and it can be characterized by a large relative width AA,/A,. Figure 8(b) corresponds to a situation where the term z, is predominant: the initial value I/(p*- 1) is negligibly small with resZ”(.ll) 1 cl9 h

pect to the maximum value of z, which is obtained for a large value of A,. The relative width AAJA, decreases towards zero when n, increases towards infinity. We are now able to predict the behaviour of the fqN frequencies (on Hamelin’s diagrams), for any shape of the distribution function, and we will show it for three different situations. For this purpose, we just need to remember that the plasma frequency can be deduced from any fqN frequency by H,,,&.Q) = 1/pP2, and that, by using a Hamelin’s diagram, one implicitly uses the value corresponding to a maxwellian disof H,,, tribution, instead of the true one. This leads to an under-estimate of pLpwhen H,,, is over-estimated and vice versa. 3.5.1. “Compact” distribution. We will say that a distribution is “compact” when all electrons are included inside a velocity interval -2Av, around V lmax, AUI being the width of y at half its maximum value and V,,,, being the abscissa of ymax. So, the maxwellian distribution can be considered distribution with a good apas a “compact” proximation. With such distributions, when z looks like the one drawn in Fig. 8(a), the maximum value of H is obtained for a small k, value such that all electron velocities correspond to the positive part of z. Since this part is quasi-rectangular, and taking into account the normalization of y, H,,,,, is quasi-independent of the shape of f(v) and its value is close to 1/(p2 - 1). Therefore, /L = pUH, and this situation corresponds to the upper part of the Hamelin’s diagram. So, we can conclude that for “compact” the apparent plasma frequency distributions, determined in the upper part of the Hamelin’s diagram is equal to the true plasma frequency, with a good approximation. In the opposite situation, p % pUH, the fqN frequencies are very close and above the gyroharmonics, and z looks like Fig. 8(b). Then, H,,,,, is obtained for a large value k,, and its value tends towards H max= Ymax z(AJ dA, I

0.5 0.1 0.3

0.2 0.1 0 FIG.

2 9. Z.FUNCTIONS

6 (DEFINED

8 IN EQUATION

9).

(13

when n, tends towards infinity, because of the decrease of the relative width of the z function. Then, H,,, depends on the distribution function only through ymax, therefore the limiting value of the apparent plasma frequency, which can be obtained in the lower part of the Hamelin’s diagram, will be smaller than the true fP if y,,, is

Characteristic

frequencies

smaller than the one of a maxwellian and vice versa. Let us note that it is equivalent to speak about small ymax or to speak about large width AuJuI of y(ul), because of the normalization of y. 3.5.2. Multiple “compact” distribution. We will now consider another situation, where the plasma is composed of several “compact” populations, corresponding to clearly different velocities, so that y presents the same number of well-separated peaks. Then, by similar arguments as in the previous paragraph, we can conclude, for each fqN series: (i) The upper limit on the Hamelin’s diagram, when it can be determined, gives an apparent plasma frequency which includes the electrons belonging to the considered population and to all the colder populations. When the more energetic populations have not sufficiently higher velocities, their first effect, because of the strong negative part of z, is to diminish H,,,,,, therefore to diminish the apparent plasma frequency. It explains why the higher points are shifted left in Figs. 7(a)-(c). (ii) The lower limit gives exactly the same result as if the considered population was alone (its plasma frequency will be all the more under-estimated as its y function will be broader). 3.5.3. “Compact” distribution + tail. When the plasma contains a “compact” population, plus a high energy tail which does not give any other maximum in the y function, only one fqN series can be observed. Then, the upper limit of the diagram nearly gives the plasma frequency of the compact distribution alone. This result is actually under-estimated when the number of electrons of the tail that correspond to the negative part of z (typically until - 10 V,,,,) is not negligible. As previously, the lower part of the diagram only depends on the compact population alone: it allows to determine the real value of the corresponding plasma frequency only when this population is maxwellian. Finally, we would like to show that the simple analytical considerations which have been developed in this section are often sufficient to predict, in some detail, results concerning the f+. For that purpose, we will present two short illustrations: (i) All the examples presented in the previous section give, in the lower part of Hamelin’s diagram, apparent plasma frequencies smaller than the real total plasma frequency. The reason can be now easily understood: all these distributions, built as sums of maxwellian components, are neces-

of a non-maxwellian

plasma

1263

sarily “broader” than a single maxwellian, or more precisely, their corresponding y functions are broader (which means ymax smaller) than the one of a maxwellian. It is interesting to check that “thinner” distributions give really the opposite result. We did so, by solving numerically the dispersion equation with “truncated” distributions, such as rectangle, triangle or any distribution without a “tail”. Figure 10 (obtained from a rectangular distribution) shows that, in such cases, the apparent plasma frequency is actually larger than the real one, in the lower part of the diagram. (ii) We have already mentioned that the wave number k, which gives the maximum value of H, and consequently the fqN frequency, is roughly de.termined by identifying the abscissas of the maximas of y and z. Therefore we can deduce a simple asymptotic relation between the frequencies and the wave numbers of jqN, in the limit of large p (p 5 1 and p ti pUH). Then, the maximum of z is almost the one of znO(W = nOwC), and it is obtained for A,,,, = n, (Abramowitz and Stegun, 1964). When noting V,,,, the abscissa of the maximum of y, this relation is o = k, V,mx.

fq N ---N fee

t

1.0 -

FIG.

IO.

HAMELIN’S

Nz2 NJ

THEORETICAL DIAGRAM,

N.4 N,5 N.6 N3 N,8 N9

fqN FOR

FREQUENCIES, A

RECTANGULAR

FUNCTION.

i f N.10 N=ll

DISPLAYED

ON A

DISTRIBUTION

1264

GERARD BELMONT

This result can be checked by applying it to the particular case of a maxwellian plasma. In this case, the maximum value of y is obtained for a velocity V,,,, which is given by V:,,, = 3/2V:,,. Then, equation (11) is nothing but the asymptotic form of the Bohm and Gross equation that is consistent with our assumptions (W ti up): w2 = 3/2k,‘V:,,. 4. DISCUSSION ANDCONCLUSION

In order to determine precisely the frequencies of intense natural waves near fp, we have been led to develop an experimental method and a theoretical study, concerning the resonances associated with active sounding in a non-maxwellian plasma. In Section 2, we have first shown that the results of the relaxation sounder S301 are generally not consistent with the hypothesis of a single maxwellian distribution, by displaying the f resonance frequencies on Hamelin’s diagrams. ;F comparing them now, with the theoretical results developed in Section 3, where we have shown the relations between the location of the fqN frequencies and the shape of the distribution function, we can draw the following conclusions. For the experimental situations which have been investigated, up to now, and which all correspond to the day side of the outer magnetosphere, only one fqNseries is generally observed. [We will see that the situations such as Fig. 2(d) seem to be due to only one fqN series which would be shifted by two opposite Doppler effects.] So, in agreement with the previous theoretical study, we can conclude that this fqN series gives a diagnosis of the “compact” part of the distribution (which obviously corresponds, here, to the thennaf population), and that all other electrons are included in a much broader distribution which can be called a “high energy tail”, and which cannot be fitted by a maxwellian function. When all points o,f the fqNseries, on the Hamelin’s diagram, are aligned on a unique value off,, it means that the thermal population is maxwellian, its density being determined by this abscissa, and that there is no Doppler effect. On figures 2(a) and 2(b), this “cold” plasma frequency appears to be smaller than the S304 value, this last one being probably much closer to the total plasma frequency. When the points are not aligned (Fig. 2c), and if there is no Doppler shift, we can conclude that the thermal plasma is not maxwellian. Then, it is often possible to fit this thermal population with two maxwellian components, whose respective densities can be determined from the upper and lower limits of the

f qN series on Hamelin’s diagram. For situations such as Fig. 2(d), we think that the two series of resonances are due, in fact, to the Doppler shifts which correspond to two waves, propagating in the forward and backward modes, near and on both sides of fqN. The static electric field, measured simultaneously onboard GEOS (Pedersen et al., 1978) has been checked to be especially large (-5 mV m-‘) during this active sweep, so that the perpendicular drift which can be deduced from it ( = 40 km SC’) can adequately explain the observed frequency shifts. In the theoretical part, we have shown that, in long wavelength limit (k -+O), the mode of propagation merges with the cold plasma solution, which means that all electrons-even the more energetic ones-equally contribute to the propagation. It follows that a resonance frequency corresponding to the total density of the plasma always exists, which is fUHTif the wave number is perpendicular to BO, and which otherwise lies between fPTandfUHT,depending on the propagation angle. For every situation where only one fqN series exists (Figs. 6d-f), this resonance frequency is the only one which does not belong to the fqN series or to the Nfceseries. Thus, we can conclude that the fx resonance, when observed, is characteristic of the total density, and that its exact identification, between fPT and fUHT, can only be determined by an experimental polarization study. Notice that, as long as fPTti fee,which is a common situation at the geostationary orbit (except during night hours), the difference between fPT and fUHT remains very small,

@f&T. f “HT- fm = fPT(f:e/2fST-,

(17)

All these results, which have been deduced from data provided by an active wave experiment, have now to be compared with the results of the particle experiments. This study is being undertaken, in collaboration with the Mullard Space Laboratory Institute, and the preliminary results, concerning the non-maxwellian thermal populations (Fig. 2c), seem to be encouraging (Johnson and Wrenn, 1980, personal communication). In addition to these comparisons, this future work will contain a detailed experimental study of the intense natural waves near fpe,based on the methods which have been presented here, and including the necessary polarization studies. We will try to establish, on a statistical basis, the main results which seem to emerge from the few cases which we have looked at in this paper, in particular, the coincidence

Characteristic

frequencies

of a non-maxwellian

between the resonance fx and the frequency of the intense natural waves. We will also investigate the natural waves and the active resonances occurring in the night side of the magnetosphere.

Acknowledgements-1

wish to thank J. Johnson and G. Wrenn (Mullard Institute, U.K.) for communication of and comments about the thermal and suprathermal plasma, as observed onboard GEOS (S302 experiment), P. Decreau (CRPE, Orleans) for the results of the GEOS S304 mutual impedance experiment, and A. Pedersen (ESTEC, Netherlands) for the GEOS electric field data (S328 experiment). I would also like to express my gratitude to R. Gendrin and all members of his team (CRPE, Issy-les-Moulineaux), for their constant support during this study, in particular J. Etcheto and B. Higel, who have initiated me in sounder’s techniques, H. de Feraudy and B. Lembege, for fruitful theoretical discussions, as well as N. Cornilleau-Wehrlin and A. Roux, who have followed this study, at every step of its development. REFERENCES

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