by
Thomas Smid
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An equation of motion for a plasma electron is defined which takes the internal restoring field of the plasma additionally to the usual external electric and magnetic field terms into account.
The coupled and non-linear differential equations are numerically integrated for a constant magnetic and sinusoidally varying electric field, thus being representative for the interaction of E.M. (radio) waves with a collisionless, magnetized plasma.
A Fourier transformation of the perturbed component of the electron orbit is applied to show the associated spectra of the forced plasma oscillation. They prove to depend in a complicated way upon the relevant physical parameters. Non-linear effects (line structuring and shifting) show up if the radio wave is in resonance with one of the fundamental frequencies of the plasma. These are related to a limitation of the linear energy input to the plasma and a periodically modulated plasma oscillation.
Possible implications and consequences for the propagation of radio waves in plasmas are indicated.
1. Introduction
In most basic treatments of plasma physics, the motion of charged particles in electro-magnetic fields is considered only in an incomplete way, since the effect of internal fields produced by collective displacements of plasma electrons is neglected within the equation of motion. On the other hand, the equation for plasma oscillations is always derived only one dimensionally, therefore being inappropriate for anisotropic media as in the case of a magnetized plasma. The present treatment overcomes this flaw by incorporating the internal restoring field due to electron displacements in a consistent way into the according equation of motion. The latter is then specified for a constant external magnetic field and a time varying electric field, which is appropriate for instance for radio waves propagating through the ionosphere.
2. Equation of motion for a collisionfree plasma electron
The force on an electron subject to external (i.e. applied) electric and magnetic fields E(t), B(t) and the internal restoring electric field of the plasma can generally be written as (all quantities to be taken at r(t) (bold letters shall denote vectors here and in the following)) :
m.a(t) = -mwp2 .r(t) +e.E(t) +e/c.v(t)´B(t) , (1)
where m is the electron mass, e the electron charge (to be taken negatively), c the velocity of light,
wp = Ö (4p.Np . e2/m) (2)
is the plasma frequency at plasma density Np, whereby the fact has been used that the ion mass (M) is very much larger than the electron mass and therefore m can be used instead of the reduced mass m=mM/(m+M) to determine the eigenfrequency of oscillation of the electron and ion gas about their common center of mass (in the field free case).
Note that the total plasma density Np appears in the expression for the restoring frequency wp, reflecting the fact that by definition plasma oscillations are collective motions. A calculation of a single electron orbit according to Eq.(1) will thus be automatically representative for all other electrons as well. This means also that r(t) and r(t) may be treated as relative vectors describing the bounded orbit of the electron but not its absolute position.
The vector r(t) designates hereby the displacement of the electron from the ion background (or rather from the center of mass). For its specification it is necessary to consider the different physical situation for plasma oscillations along and perpendicular to the magnetic field:
Along the magnetic field, electrons are free to move within a homogeneous medium, so that any region of depletion of electrons caused by a displacement of a certain electron subvolume will tend to be filled up by electrons moving from the neighboring volume into this region. Any disturbance of the neutral plasma state will thus diffuse along the magnetic field line as a rarefaction region and consequently no local resonant build up of plasma oscillations along the magnetic field (or without any magnetic field) is possible (as long as the amplitude of the oscillation is considerably smaller than the distance travelled by an electron during one period of oscillation).
Perpendicular to the magnetic field, however, a rarefaction region of electrons cannot be filled up from neighboring volumes in a collisionless plasma, because each electron is fixed to its Larmor circle. Furthermore, any displacement of charges within a given volume has no physical effect outside this volume whenever the displacement is small compared to the spatial extension of the disturbed volume, because the local electric field is then only determined by the local displacement of electrons and thus vanishes outside the disturbed volume (see Mitchner and Kruger; 1973 (p. 137)). Therefore, no propagation of plasma disturbances across the magnetic field is possible, which enables then a local resonant excitation of plasma oscillations under the above restrictions (which, however, can be considered to hold for instance for all realistic cases of radio wave propagation and absorption in the ionosphere).
Within a quasi-neutral plasma, the undisturbed orbit of the electron perpendicular to the magnetic field is now given by its Larmor circle. Therefore the displacement vector cannot be defined with respect to a fixed point or plane, but must be taken as the radial distance from the initial Larmor circle, i.e.
r(t) = r(t).(1-rB/r(t)) , (3)
with
rB = cmv0,^ /e/B = v0,^ /wB , (4)
where v0,^ is the initial velocity of the electron perpendicular to the magnetic field, and
r(t) = r(t) , (5)
This form of r(t) means that the restoring electric field due to a collective electron displacement (relative to the ion background) perpendicular to the magnetic field acts always such as to pull the individual electron radially back to its initial Larmor circle (which defines the quasi-neutral state). (Note that quasi neutrality for the undisturbed plasma implicates a homogeneous initial distribution of the velocity vectors of all electrons in the plane of motion, i.e. an equal statistical distribution over all phase points on the Larmor circle, since otherwise a plasma oscillation (i.e. a deviation from charge neutrality) would already be present in the undisturbed state and the initial electron orbit would be different from a circle).
In the following, it is furthermore assumed that the magnetic field is independent of time (the magnetic component of any electromagnetic wave is therewith also neglected) and directed in the z-direction. If one neglects inhomogeneities along the magnetic field direction (see also below Eq.(15)) , only the electron motion in the x,y-plane needs to be considered because of the previous arguments concerning the conditions for the existence of plasma oscillations
Consequently, Eq.(1) can be written as
ax(t) = -wp2 . x(t).{ 1- rB/Ö(x2(t)+y2(t)) } + e/m.E(t) - wB.vy(t) (6)
ay(t) = -wp2 . y(t).{ 1- rB/Ö(x2(t)+y2(t)) } + wB.vx(t) , (7)
where the electric field E has been assumed to be linearly polarized in the x- direction (again, E, wB, rB and wp may in principle all depend on x and y ; see for instance Eq.(14) and comments below that).
Note that these second order equations of motion are both coupled and non-linear because of the Lorentz- force. This circumstance makes a general solution difficult and only a numerical integration is likely to determine the resultant electron orbit. The latter is of course only represented by Eqs.(6) and (7) as long as collisions can be neglected during the time interval being considered.
It should also again be emphasized that the use of the Larmor circle as the reference orbit for the charge displacement vector implicates necessarily an average over different starting positions on that circle. Unaveraged solutions could represent only a possible physical situation in a plasma which deviates initially from charge neutrality. In this case, however, a different form for the displacement vector would have to be defined (the discussion of the unaveraged results in Sect.4. is therefore somewhat academic but may illustrate the effect of the averaging procedure).
3. Method of solution
For the numerical integration scheme the most simple and straightforward approach has been chosen by setting the accelerations ax(t) and ay(t) acting upon the electron constant in a given small time interval Dt.
The velocity is then determined by
vx(t+Dt) = vx(t) + ax(t) .Dt (8)
vy(t+Dt) = vy(t) + ay(t) .Dt (9)
and the new position is found by integrating (8) and (9) with respect to Dt, to yield
x(t+Dt) = x(t) + vx(t).Dt + ax(t)/2 .Dt2 (10)
y(t+Dt) = y(t) + vy(t).Dt + ay(t)/2 .Dt2 . (11)
Repeating this procedure with the new variables taken as the old ones N times thus defines the electron orbit up to the time T=N .Dt .
The remaining problem is to define the starting position x(0),y(0) of the electron on its Larmor circle, i.e. to fix the initial phase of the orbiting electron on the circle with regard to the phase of the electric field E(t). According to the arguments in Sect.2. , an equal distribution over all possible values must be assumed in the case of an initially quasi neutral plasma, so that an according average has to be performed. This can be done simply by performing the whole integration independently for different starting positions on the circle (uniformly distributed) and then averaging the results, since by definition only collective motions are being considered and no care has to be taken about absolute positions. Any (inconsistent) non-collective oscillation components are thus automatically filtered out by the averaging procedure.
P
rx,P(t) = 1/P .årx,i(t) , (12)
i=1
with rx,i(t) being determined by Eqs.(10) , (11) and (3) for the initial values xi(0),yi(0) (each i representing a different location on the Larmor circle).
For a sufficient number of points one then obtains the actual displacement
rx(t) = lim rx,P(t) . (13)
P ®¥
4. Numerical results for a monoenergetic plasma
Eqs.(6) and (7) have been integrated numerically by means of the procedure outlined in Sect.3. . The electric field E(t) has hereby been taken as sinusoidal with allowance for a certain rise and fall time of the amplitude Tr and Tf., i.e.
E(t) = E0 .sin(w0t -k.r) · (1 -exp(-t/Tr)) . Foff (14)
where
Foff = { 1 for t=toff (15)
exp(-(t-toff)/Tf) for t>toff
for waves travelling perpendicular to the magnetic field, the phase term k.r is itself a sinusoidally varying function which becomes significant if the wavelength is comparable to or smaller than the Larmor radius;
waves propagating along B , on the other hand, will cause a Doppler shift of the apparent wave frequency by an amount kz.vz for an electron with a translational velocity vz along the magnetic field. Since under natural circumstances vz is distributed over a whole range of values, the frequency spectrum of the wave will be broadened accordingly , resulting in a reduction of the effective field strength of the wave in the case of plasma resonance if the spectral width of the latter (see Eq.(22)) is smaller than this Doppler broadening.
As the latter complication does also occur for the relatively long wavelengths considered here (corresponding to H.F. frequencies at or near the plasma resonance for ionospheric conditions), the results are thus strictly valid only for a radio wave propagating perpendicular to the magnetic field (with only the component of polarization perpendicular to B being relevant since the electric field component along B has no effect with regard to plasma resonance (see discussion in Sect.2 ) ). Even in this case one has to make the assumption, however, that the extension of the radio wave -plasma interaction region along the magnetic field is sufficient for an electron to stay in phase with the driving field during the time interval being considered (i.e. Dz = vz.DT, where DT may be given for instance by the oscillation period shown in Fig.6 as far as non-linear effects are concerned). For a wave propagating along B ,Eqs.(6) and (7) represent in any case of course only a linear polarization in the x- direction. Numerical test calculations with an additional y- component of the driving field (i.e. corresponding to circular polarization in the plane perpendicular to B ) indicate however that the resonant behaviour of the wave- plasma interaction is largely unaffected by this additional out-of- phase component (see also remarks below Eq.(21)).
For the plasma density a value of Np = 105 cm-3 and for the magnetic field B= 0.5 Gauss was taken. This leads to angular frequencies wp=1.78.107 rad/sec (Eq.(2)) and wB=8.79.106 rad/sec (Eq.(4)). The initial velocity perpendicular to the magnetic field was chosen as v0, = 2. 107 cm/sec, which correponds to an energy of 0.11 eV.
With these plasma parameters, the iteration procedure (Eqs.(10) and (11)) was then performed, whereby Dt=10-12 sec, the total integration time T= 4.10-5 sec, the rise time Tr=10-6 sec (Eq.(14)) and Tf also =10-6 sec with toff = 3.5.10-5 sec (Eq.(15)).
For each time step of 5.10-9 sec , the components of the displacement vector rx, ry and also the velocities vx, vy were stored for subsequent processing. A power spectrum of w =0 -6.28.108 rad/sec with a resolution Dw = 1.57 .105 rad/sec could thus be obtained.
In order to find the eigenfrequencies of the system represented by Eqs.(6) and (7), the integration was performed first for a free oscillation, i.e. the electron was displaced initially from its Larmor circle (here rx(0)= -0.5 cm) and the field strength E(t)º 0. The corresponding power spectrum of rx(t) is shown in Fig.1. Apparently, the resonance frequencies are given by
W- = wp,B -wB (16)
W+ = wp,B +wB , (17)
where
wp,B = Ö(wp2+wB2) . (18)
Higher harmonics occur at
W(n) = n. wp,B + wB ; (n ³ 2) . (19)
The latter are due to the increasing importance of anharmonic (i.e. non-linear) oscillation components if the displacement r becomes comparable to the Larmor radius (here rB =2.3 cm).
In order to observe both the resonant and non-resonant excitation of the plasma, two calculations were then carried out: one for a frequency of the exciting wave at w0 = W+ and another for a driving frequency well apart from the resonance points (here w0 = 6.07. 107 rad/sec. In either case, the peak electric field strength was taken as E0 = 2. 10-4 cgs-units (6 V/m) and the displacements rx(t) and their power spectra were both obtained unaveraged over the starting position on the Larmor circle and averaged, though the former do not represent a possible solution for a forced plasma oscillation on their own (see discussion in Sect.2. and 3. and Eqs.(12) and (13)).
For the off-resonant case (w0 = 6.07. 107 rad/sec) , the unaveraged oscillation and its spectrum is shown in Fig.2 ( a and b respectively). Two groups of lines are apparent from the spectrum: one is related to the driving frequency w0, namely w0 and w0-2wB , w0+2wB , whereas the weaker lines appear at the eigenfrequencies W- and W+ and at the Larmor frequency wB (the latter is however at least partially due to the finite numerical integration stepwidth, whereas the relative error of the rest of the spectrum is less than 5% for each line). It should be noted that the ratio of the line intensities is not fixed, but depends on the given parameters, in particular the initial velocity of the electron v0,^ ( see remarks below Figs.2d and 3 . Furthermore, for different initial parameters, additional lines may appear at w0-wB and w0+wB for the unaveraged spectra in the non-resonant case.
The averaged oscillation and its spectrum is depicted in Fig.2c and 2d. Apparently, the lines symmetrical to w0 in Fig.2b are due to a phase modulation and disappear when averaging over the phase of the starting position on the Larmor circle (convergence for the averaging process was achieved here with P=4 in Eqs.(12) and (13)). Thus, the only lines left in the spectrum representing the real physical state are at the driving frequency w0 itself and at the eigenfrequencies W-, W+. The ratio of the amplitudes of these oscillation components depends again strongly upon the asssumed initial values, as is evident from Fig.3, where w0 = 2.22. 107 rad/sec, v0,^ = 2. 108 cm/sec (e0,^ =11.2 eV) and E0 = 1. 10-4 cgs-units (3 V/m).
A completely different picture holds for excitation at one of the eigenfrequencies, namely here w0 = W+. Resonant conversion of wave energy leads to much larger oscillation amplitudes than non-resonant one (Fig.4a). On the other hand, the amplitude does not rise indefinitely despite the absence of any damping term in Eqs.(6) and (7). This behaviour is due to the appearance of non-linear terms in these equations, which limits the maximum amplitude and modulates it by a certain frequency. In the corresponding spectrum (Fig.4b), this leads to the splitting of the lines at W- and W+ and at the higher harmonics.
The width of the splitting and the relative importance of the higher harmonics depends directly on the field strength of the E.M. wave and inversely on the initial velocity of the electron (see Fig.5 for an initial velocity which is 10 times higher, i.e. v0, =2. 108 cm/sec ( e0,^ =11.2 eV)). An electron distribution function of a certain shape will thus be directly reflected in the spectrum of the oscillation.
An average over the possible initial electron positions leads, similar to the non-resonant case, to the disappearance of all lines apart from the group related to the driving frequency w0 , whereby the lines at W- disappear now as well (Fig.4d) (up to 8 points were used for averaging ; the resultant spectrum has then been furthermore filtered to yield Fig.4d, because convergence was not completely achieved with this number of points, i.e. small rests of the other lines were still present. A retransformation of the so corrected spectrum then yielded the corresponding oscillation amplitude (Fig.4c).
Note that in Fig.4c the amplitude of the (resonant) oscillation stays at rather large values, even after the driving field has nearly completely decayed to 0 at T= 4. 10-5 sec. This has to be seen in contrast to Fig.2c, where the dominant non-resonant part of the oscillation can not be maintained but decays together with the driving field.
It is furthermore interesting that even in the resonant case the energy absorbed from the wave does dominantly appear as kinetic energy of the electron (m/2.v2) compared to the potential energy of the plasma oscillation (m/2.wp2 r2) (the y- component of the oscillation has always an amplitude similar to the here considered x- component). This is evident for instance from a comparison of the energy associated with the oscillation in Fig.4c to the total energy during that resonant interaction process (Fig.6) (the time averaged energies are in this case Dekin= 3.3 .10-2 eV, epot= 1.3 .10-2 eV ; the averaging of the energy with regard to different starting positions on the Larmor circle was done analogously to the method described in Sect.3. for the case of the displacement vector) This increase of energy over the value given by the virial theorem for small oscillations (Dekin,osc= epot,osc) is due to the fact that Eqs.(6) and (7) describe a radial oscillator. Tangential force components therefore accelerate the electron without changing the potential energy of the oscillator.
De (w0 =W+,W-) » 5.5 .10-3 .E00.6 .e0,^ 0.6 .Np0.3 [eV] (20)
DT (w0 =W+,W-) » 5.9 .10-3 .E0-0.6 .e0,^ 0.4 .Np-0.4 [sec] , (21)
where E0 has to be taken in [V/m], e0,^ in [eV] and Np in [cm-3] . De has to be understood here as the total energy; the kinetic energy is in all cases smaller by about a factor 0.7 (see above). For an isotropic distribution of the velocity vector of the electron in space, Eq.(20) and (21) can be formulated as a function of the actual electron energy e0 through replacement of e0,^ by 2/3.e0 .
Although these relationships have been obtained from an equation of motion which implicates a driving electric field linearly polarized in the plane perpendicular to the magnetic field (Eqs.(6) and (7)), they appear also to be valid (at least approximately) for a circular polarized E- field (with the same amplitude), as is indicated by a test calculation with an accordingly modified equation of motion.
Apart from just giving the modulation period for the resonant, non-linear plasma oscillation, Eq.(21) determines also the frequency width of the plasma resonance lines through (roughly)
DwRes »± 1.5 . (2p / DT (w0 =W+,W-) ) . (22)
This expression can be considered to represent the width of a rectangular profile with De= De (w0 =W+,W-) , DT= DT (w0 =W+,W-) for |w0 - W+,W- |= DwRes and De= 0 , DT= ¥ for |w0 - W+,W- |> DwRes and emerges from some additional calculations with driving frequencies displaced from the resonance points. A more detailed analysis is necessary in order to reveal the exact frequency profile of the plasma resonance.
Experimental confirmation of Eqs.(20) -(22) can be found in Smid(1992), where the expressions prove to yield the observed enhancement of the ionospheric airglow intensity which is induced by high power H.F. radio waves.
5. Conclusions and Outlook
The above results demonstrate that a single consistent mechanical approach to plasma dynamics is capable to consider E.M-wave- plasma interaction without any formal need to distinguish a priori between linear and non-linear problems, i.e. the same set of equations can be used for any value of the parameters involved. Non-linearity is hereby observed to become significant if the frequency of the driving electric field is at one of the eigenfrequencies of the plasma. It limits the linear energy input to the plasma in this resonant case and leads to a periodically modulated plasma oscillation and -energy. This aspect is for instance of importance for the modification of plasmas (e.g. the ionosphere) by powerful radio waves. On the other hand, the (periodical) change of energy of the plasma due to the radio wave should also be related to a modulation of the latter, in order to have some kind of causal relationship. This can be achieved by taking the time dependent flux of the wave
Fw(s,t) = E2(s,t) . c/8p ,. (23)
where E(s,t) is its electric field strength, and relating it to the energy e of the individual plasma electron by postulating the non-trivial continuity equation
dFw(s,t) /ds = - Np(s) . de(s,t)/dt , (24)
where Np(s) is the local plasma density.
Any non-linear plasma oscillation as calculated in the present work should consequently also show up in a transmitted or reflected radio-wave as a certain line splitting (or up -and down-shifted sub-lines) according to the modulation of the field strength E(s,t). This may perhaps represent a possible explanation for certain features observed in the 'stimulated emission' associated with ionospheric heating experiments (Stubbe et al., 1984), which are presently only incompletely understood in terms of magnetoionic optics.
A quantitative proof , however, can only be given after integrating Eq. (24) over the whole path of the radio-wave through the plasma, i.e by solving the integral
s
Fw(s,t) = Fw(0,t) - ò ds' Np(s') .d(s',t)/dt , (25)
0
which is anything but trivial since e(s',t) (t may be identied with a certain phase or length of the exciting wavetrain) depends upon the plasma oscillation and thus the flux (or field strength ) of the wave for all previous times t' =0...t through Eqs.(6) and (7).. The latter must thus be solved simultaneously with Eq.(25) in order to arrive at a consistent solution for the modulation of the radio wave passing through the plasma. A possible method of solution would be to subdivide the integral in Eq.(25) into such small steps Ds that for each interval de/dt can be assumed as independent of s' and thus taken outside the integral. First test calculations show that even for the relatively short pulse lengths of the wave in the present calculations (i.e. tmax = 4. 10-5 sec) the spatial steps have to be taken so small ( even steps of the order of a few meters were not sufficient to yield a consistent recursion for the whole pulse length) that an enormous amount of computational effort is necessary, since for each step Eqs.(6) and (7) have to be solved again for the modified field strength of the wave E(t).
A further point which has to be taken into account when trying to obtain numerical results that can be compared directly to experimental data, is the fact that a realistic energetic distribution function for the electrons should be used instead of the monoenergetic one for the present calculations. This , however, is only a minor problem, since only a corresponding weighted average over computations for different initial electron energies has to be performed ( as already indicated at the end of section 3).
When this has been done, a comparison with experimental results (as indicated above) could also support a theoretical study which proved that 'thermalization' of ionospheric photoelectrons down to the order of 20000 K cannot take place by known laws of kinetics. The majority of ionospheric plasma electrons should have an energy of several eV, with the shape of the electron distribution function almost exclusively determined by inelastic collisions, i.e. electron impact ionization and excitation of molecular vibrational transitions (Smid, 1987). Because of the inverse relationship between non-linear line splitting and electron energy ( see Sect.4., Figs.4b and 5.), the frequency displacement of the related features in reflected ionospheric radio spectra from the original frequency should be much less here than for a Maxwell distribution of about 20000 K.
It should be noted that apart from this 'self-modulation' effect, Eqs.(6) and (7) are also capable to consider the combined effect of an arbitrary number of individual external fields, i.e. to compute the cross-modulation for an arbitrary number of radio waves (simply by specifying E(t) accordingly).
Once the difficulties indicated in this section have been overcome, the proposed non-linear equation of motion should thus represent a powerful tool for any radio propagation problem without the need for any further doubtful ad-hoc theories.
For problems involving high plasma- or neutral densities, i.e. if collisions become important (as for instance in the lower ionosphere), it might of course become necessary to modify the outlined computing procedure accordingly.
In any case, the present approach should represent a consistent formalism able to attack many other related problems in the field of plasma dynamics in a coherent and unified way. Valuable new insights into the true nature of several observational linear and non-linear plasma phenomena is then likely to be gained (a first quantitative application of the results obtained in this paper is presented in Smid (1992)).
References
Mitchner, M. and C.H. Kruger, Partially Ionized Gases, Wiley, New York, 1973.
Smid, T.S., Theoretische Aspekte der Ionosphärenphysik, Ph.D. Thesis, Bonn University, 1987.
Smid, T.S., Resonant excitation of airglow by high power radio waves, 1992.
Stubbe,P., H. Kopka, B. Thidé and H. Derblom, Stimulated Electromagnetic Emission: A New Technique to Study the Parametric Decay Instability in the Ionosphere, J.Geophys.Res. 89,7523, 1984.