Dispersion relation and group velocity for inhomogeneous waves in a hot magnetoplasma with application to an electron-Bernstein-wave propagation experiment in a laboratory plasma



[1] Lewis and Keller (1962) derive the dispersion relation for homogeneous waves propagating in a hot magnetoplasma. Homogeneous waves are ones for which the real and imaginary parts of the wave vector, kr and ki, are parallel. In this paper a generalization to Lewis and Keller is made for inhomogeneous waves, that is, waves for which kr and ki are not parallel. If ki is assumed to be in the same plane as kr and the magnetic field H0 in the Lewis and Keller generalization, comparison can be made with the dispersion relation of Stix (1992); good agreement is found with one exception. This generalization is applied to observations of electrostatic (ES) wave propagation in a laboratory plasma. Bernstein waves propagating perpendicular to H0 are undamped. The dispersion relation for homogeneous waves indicates that severe damping should occur for propagation slightly off perpendicular. Laboratory experiments indicate that severe damping does not occur. The laboratory results can be explained if inhomogeneous waves are considered. Muldrew and Gonfalone (1974) used the dispersion equation for homogeneous waves in a hot magnetoplasma to explain the signal maxima in the pattern when electron-Bernstein waves interfere with the electromagnetic field. Good agreement is obtained when ki is small compared to kr. However, when ki becomes significant, the pattern can no longer be explained. Different approaches to explaining the results using inhomogeneous waves are presented that are superior to the one using homogeneous waves. In one approach, plasma waves with a complex wave vector can propagate without large attenuation, and propagation characteristics can be determined, by choosing the direction of ki to be a free parameter that makes Im{k · vg} = 0, or have a minimum value; k = kr + iki, vg is the complex group velocity ∂ω/∂k and ω is the real angular wave frequency. When this condition is satisfied, good agreement in the signal maxima is obtained with the laboratory experiments if the direction of energy flow in the plasma is taken to be Re{vg}. This method of calculating the interference pattern is compared with the least damped method which calculates the potential of an oscillating point charge in a plasma. Good agreement between the two methods is found if an assumption is made regarding the wave(s) interfering with the ES Bernstein mode.

1. Introduction

[2] Fisher and Gould [1971] measured the field around a radio frequency antenna in a laboratory magnetoplasma and observed the resonance cone on which this field is a maximum. Fine structure on the edge of this cone was assumed to result from interference between the received signal due to an ES wave and an electromagnetic (EM) field. The EM field was evanescent and so, at a given time, had the same phase everywhere in the plasma. Gonfalone [1972] studied the influence of temperature on this structure.

[3] Kuehl [1973] derived the potential of an oscillating point charge in a magnetoplasma using inhomogeneous waves. The imaginary component of k, ki was taken to be in the direction perpendicular to H0. Thiel and Lembège [1982, hereafter TL] normalized the potential to the free space value and calculated the potential using the ES approximation to the tensor dielectric constant and obtained good agreement with their experimental results. They referred to this method as the least damped method in which the component of k in the direction of H0 is real and only two modes are considered. Using their technique good agreement is found here with the experimental results of Gonfalone [1972], although the method used by TL has been modified slightly.

[4] In this paper an attempt to explain the experimental results applying different approaches using inhomogeneous waves is carried out in order to see which is the most successful. In addition to the least damped method, another method is considered, called the “minimum-ki” method, for which the direction of the imaginary part of k is varied until ki obtains a minimum value. The minimum-ki method is more successful than the homogeneous method but cannot explain all the experimental results. A new approach, called the “Im{k · vg} = 0” method, for lack of a better name, is as successful as the least damped method in explaining the experimental results.

[5] There is no particular reason to assume homogeneous waves in propagation through an absorbing or nonabsorbing medium. Consider plane waves obliquely incident on an absorbing, bounded, horizontally stratified ionosphere. Below the ionosphere, ki can be considered to be in any direction since the amplitude does not change with distance; however, once in the ionosphere the amplitude decreases in a direction perpendicular to the horizontal stratification and kr is at some oblique angle determined by Snell's law. Hence the wave is inhomogeneous. Inside a plasma, plane waves can be assumed to exist at the source with equal amplitude in all directions; that is, all ki are possible just as all kr are possible. By using the dispersion relation for inhomogeneous waves, propagation characteristics for any direction of kr and ki can be determined.

[6] In the work of Muldrew and Gonfalone [1974, hereafter MG] the interference results of Gonfalone [1972, 1973] measured near the resonance cones and for Bernstein waves are presented and explained using the dispersion equation for homogeneous waves in a hot magnetoplasma. This dispersion equation [Lewis and Keller, 1962, hereafter LK] is valid for both ES and EM waves and was programmed by Muldrew and Estabrooks [1972]. Isotropic temperatures and approximate plasma frequencies were obtained from the experimental results at both the lower oblique resonance cone (associated with the whistler mode for which the wave frequency f is less than both the plasma frequency fp and the cyclotron frequency fc) and the upper oblique resonance cone (associated with the lower branch of the extraordinary or Z mode in which f > fp and f > fc but f < fuh, where fuh is the upper hybrid frequency). These temperatures were compared with the temperature obtained from an interference pattern resulting from the EM field and Bernstein waves with fc < f < 2fc and 2fc < fuh [Gonfalone, 1972, 1973; Gonfalone and Beghin, 1973] presented in MG. MG could not explain the complete interference pattern. The complete pattern is explained here using the dispersion relation for inhomogeneous waves for which Im{k · vg} = 0. This technique allows some understanding of wave propagation for waves with a complex wave vector. The results are close, but not identical, to those obtained using the least damped method.

[7] The reader is referred to MG for details of the experimental setup, the experimental results, and the dispersion curves obtained assuming the real and imaginary components of the wave vector are collinear or parallel. In this paper we present the modifications to LK that are necessary if the real and imaginary components of the wave vector are not parallel (see Appendix A). Dispersion curves are presented that can explain the experimental results. Under the assumption that ki is in the plane of kr and H0, the generalized LK dispersion curves are compared to those obtained with Stix [1992]. An unusual discrepancy is found that is not understood. The generalized LK theory is more general than the Stix theory since Stix takes the real and imaginary parts of the component of k perpendicular to H0 to be parallel [see Stix, 1992, equation (37), section 10].

2. Modification of the Dispersion Equation

[8] The dispersion relation developed by LK includes both ES and EM wave propagation and hence can be used to study the regions between them. LK consider homogeneous waves and take the wave vector to be

equation image

where k is complex, and β is real and is the angle between k and the magnetic field H0 (LK's nomenclature will be used here). H0 is in the 1 direction (LK use a (1, 2, 3) nomenclature rather than (x, y, z)) and k is in the (1, 2) plane. In this paper

equation image

where kr and ki are real vectors. The vector kr is perpendicular to planes of constant phase; the vector ki is perpendicular to planes of constant amplitude. These vectors are illustrated in Figure 1a. For inhomogeneous waves in a magnetoplasma, kr is chosen to be in the (1, 2) plane with H0 in the 1 direction; ki has three components.

Figure 1.

(a) Coordinate system for inhomogeneous waves. (b) Complex wave vector and group velocity angles for δ = 0 for inhomogeneous waves.

[9] Budden [1961] discusses inhomogeneous plane waves in which the wave vector is taken to be at a complex angle to the coordinate axes. He also shows that for the no-magnetic-field case the wave vector can be chosen to be in a plane containing only two of the coordinate axes. In more recent work [Caviglia and Morro, 1992] the wave vector is chosen to be of the form (2).

[10] For a complex k, the group velocity defined by

equation image

is also complex. For real k the group velocity is the velocity of the constructive interference maximum of a wave packet. Stix [1992] shows that for real k the group velocity is equal to the wave energy velocity, which is the energy flux divided by the energy density; this energy includes the nonelectomagnetic energy of the plasma.

[11] If the imaginary component of the complex group velocity is significant, it is difficult to attach a meaning to it. For real k, Stix [1992] shows that the energy velocity or vg is perpendicular to the dispersion curve. For complex k and inhomogeneous waves, good agreement with the experimental results is obtained by assuming the energy velocity is given by Re{vg} at kr values on the low-kr side of an inflection point of the real part of the dispersion curve (see below). Re{vg} is very nearly normal to the dispersion curve for the real part of the wave normal from the lowest values of k up to the inflection point in the dispersion curve, provided that the curvature of the dispersion curve is not excessively large. Good agreement with experiment on the low-kr side of an inflection point was found by MG, Lembège [1979], TL, and will be found again here. In the Im{k · vg} = 0 method there is no inflection point in the real dispersion curve; however, there is a small region of large curvature.

[12] LK take the field components to vary as exp i(k · xωt) where the position vector x = (x1, x2, x3), ω = 2πf and t is the time; x, ω, and t are real. In a study of inhomogeneous waves, Caviglia and Morro [1992] also take x to be real. Suppose x is replaced with vgt; the field components would then vary as exp i(k · vgtωt) and it might be supposed that the damping would be proportional to exp[− Im(k · vgt)]. However, it turns out that this is not the true damping, but significantly, if the inhomogeneous waves that satisfy

equation image

are found, the real part of the dispersion curve does not contain a point of inflection and all of the experimental results can be explained if the wave energy direction is taken to be in the direction of Re(vg). This result indicates that complex vg may have some physical meaning. Note that if (4) is satisfied, (k · vgtωt) is real in the field variation exp[i(k · xωt)] = exp[i(k · vgtωt)].

[13] The coordinate system used is shown in Figure 1a; Figure 1b shows the special two-dimensional case where δ, the angle between the plane containing ki and the 2 axis, and the (1, 2) plane, is zero. The three dimensional components of k are

equation image
equation image

[14] The purpose of the primes on α′ and γ′ is to prevent confusion with the α and γ used by LK. A value of α′ is chosen and the free parameters γ′ and δ are varied until (4) is satisfied, if this is possible. For the cases considered below, (4) can usually be satisfied with δ = 0.

[15] The required modifications to LK for inhomogeneous plane waves, which are relevant to solving the dispersion equation for a collisionless magnetoplasma, are given in Appendix A. Typographical errors in LK are also indicated. The dispersion equations of LK include collisions and collisions could also be included in the inhomogeneous dispersion equations; however, the complexity and length of Appendix A would be considerably greater. At the heights of the topside sounders in the ionosphere and for the laboratory plasma considered here collisions can be neglected. In Figure 2 for f/fc = 1.44 it can be seen that the amplitude of the first 7 fringes remains constant implying that collisional damping is not significant.

Figure 2.

Lower cone resonance for f/fc = 0.49 and 0.65 and the variation of Bernstein interference patterns in the early after glow [after Muldrew and Gonfalone, 1974].

[16] For a given angular frequency of propagation ω, direction of kr (i.e., α′), and direction of ki (i.e., γ′ and δ) the corresponding magnitude of kr, kr, and magnitude of ki, ki, can be determined from Appendix A. For fixed values of ω and α′, γ′ and δ can be considered to be free parameters; these free parameters are varied until Im{k · vg} = 0, if this is possible. The complex group velocity vg is given by (3). The determinant (A2) (see Appendix A) set equal to zero can be solved numerically in terms of (k1, k2, k3) for given α′, γ′ and δ using (5) and (6). Hence, making a small step in ω, while holding k2 and k3 constant, determines ∂ω/∂k1. Similarly, ∂ω/∂k2, ∂ω/∂k3 and hence vg can be determined. By iteration the condition Im{k · vg} = 0 or a minimum can be found. When this condition is satisfied, it is referred to as the Im{k · vg} = 0 method to distinguish it from the minimum-ki method and the least damped method.

3. Alternative Explanations of the Gonfalone Laboratory Plasma Experiment

3.1. Homogeneous Waves

[17] Most of the resonances observed on topside ionograms can be explained with homogeneous waves. This is because at frequencies near the resonances the waves responsible for the resonances have a real wave vector. A notable exception is the electron cyclotron resonance [Muldrew, 2006]. Similarly, the interference patterns observed in laboratory plasmas can be explained using homogeneous waves if the imaginary component of the wave vector is small. When the imaginary component of the wave vector is not small, experimental results can often be explained using inhomogeneous-wave dispersion equations.

[18] Figure 2 shows the lower cone resonance for f/fc = 0.49 and 0.65 and various Bernstein interference patterns in the early plasma afterglow. A careful scaling of the 26 maxima in the interference pattern for f/fc = 1.44 leads to 13 values of ∣β∣, where β is the angle between the line joining the transmitter and receiver, and the magnetic field direction. These ∣β∣ values are listed in Table 1, column 2, and are thought to be slightly more accurate than the values in MG. By measuring the distance between a − β maximum and the corresponding + β maximum value and also measuring the distance between the −90° and +90° markers on the bottom scale, surprisingly accurate values of ∣β∣ can be obtained. Double independent scalings resulted in an average absolute-value difference of only 0.22°.

Table 1. Calculated Values of ∣βc∣ for the Least Damped Case
FringeObserved ∣ββc∣ for ϕhr = minimum, Te = 5000 K, (fp/fc)2 = 10.3βc∣ for ϕhr = maximum, Te = 5000 K, (fp/fc)2 = 7.6
171.8 ±

[19] The homogeneous dispersion relation indicates that the waves with small ∣β∣ do not exist, assuming the wave energy velocity is given either by the normal to the dispersion curve or by Re{vg}. MG explain the first six of these maxima using homogeneous waves. Actually a seventh, the fringe maximum at ∣β∣ = 28.8° in Table 1, can also be explained. The dispersion curve for the real part of the wave vector for homogeneous waves is shown in Figure 3, taken from Figure 8 of MG. The length of the arrows shows the magnitude, and the orientation shows the direction, of vgr = Re{vg}. The additional value at ∣β∣ = 28.8° has been added to the curve of MG. As α′ continues to decrease from the value corresponding to ∣β∣ = 28.8°, ∣β∣ goes through a minimum value of about 24.8° and then increases so that the fringe maxima beyond number 7 cannot be explained. Note that the last value of ∣β∣ = 28.8° occurs just before the point of inflection in the dispersion curve. The numbers along the dispersion curve are the imaginary parts of . The electron cyclotron radius is

equation image

where vth is the thermal velocity given by (A10) and ∣λ∣ is the angular cyclotron frequency; λ is negative for electrons. The values 0.0002 and 0.0003 in Figure 3 are due to collision damping. The other values of kiρ are due to collisionless damping of homogeneous waves, and as soon as they become significant, the observations can no longer be explained.

Figure 3.

Polar plot of real dispersion curves for Bernstein interference data of Figure 2 for homogeneous waves [after Muldrew and Gonfalone, 1974].

3.2. Minimum-ki Inhomogeneous Waves

[20] Analyses based on homogeneous waves are not able to explain the experimental results if the imaginary part of is significant. However, there may be other techniques for explaining these results. In this section and sections 3.33.6 other possibilities are discussed. It might be thought that for a given α′ the minimum absorption might be obtained for values of γ′ and δ which give the minimum ki. This possibility is considered here. Other possibilities are discussed in sections 3.33.6.

[21] For homogeneous and inhomogeneous waves, the direction of energy flow is usually given to a good approximation by the perpendicular to the dispersion curve on the low-kr side of the inflection point. A dispersion curve can be determined by choosing α′ and then finding by iteration the value of γ′ (for δ = 0 in the present calculation) that gives the minimum value of Im(). This technique will be referred to as the minimum-ki solution. The minimum value of ki can be considerably smaller than the value of ki corresponding to the homogeneous solution for the same observed direction of energy flow. Figure 4 gives the real part of the minimum-ki dispersion curve corresponding to the Bernstein interference pattern of Figure 2 for f/fc = 1.44. The observed values of ∣β∣ up to about fringe number 9 can be explained. The slope of the curve then increases only slightly before going through an inflection point. The minimum ∣β∣ found by taking the normal to the curve is about 20°. Since the fringes beyond fringe number 9 cannot be explained, this possibility is not considered further.

Figure 4.

Real dispersion curve for the Bernstein interference data of Figure 2 for minimum-ki inhomogeneous waves.

3.3. Least Damped Inhomogeneous Waves

[22] TL studied propagation in a magnetoplasma at frequencies between the cyclotron frequency and the second harmonic of the cyclotron frequency when fp > 2fc. This is the same frequency regime studied by Gonfalone. TL calculated the scalar potential created by a point source using the “least damped” method proposed by Kuehl [1973] and compared their calculations to the experimental results. Using (7) and (24) of Kuehl and the nomenclature used in this paper (see below), the potential normalized to free space is

equation image

The distance from the point charge to the receiver is d, θ is the angle between the magnetic field direction and the distance vector from the point charge to the receiver, and k1 is the wave vector component in the direction of the magnetic field and is real. H0(1) is the first Hankel function of zero order, the summation is over all the modes m in the plasma, Dm is the tensor dielectric constant of the plasma for mode m, k2m is the complex wave normal perpendicular to the magnetic field for mode m and is found by solving the dispersion relation Dm = 0 for a given k1, and

equation image

TL used (8) and (9) successfully to explain their experimental results. These equations are used here to explain the results of Gonfalone, but a modification to the interfering mode(s) needs to be applied.

[23] In the least damped method, inhomogeneous waves are used for which the component of the wave vector is real in the direction of the magnetic field; that is, γ′ = 90°. TL found excellent agreement between theory and experiment. The reader is referred to their paper for details of this technique. Their equation (1) is the ES approximation to the dispersion equation for a hot magnetoplasma. (There is a typographical error in this equation; the symbol i should be dropped.) They state that the observed interference pattern results from interference between the Bernstein mode and the main mode, which is another solution to the dispersion equation (see TL for a discussion of this mode). They found the potential of the main mode to be negative and to change slowly with direction of the wave vector. Consequently the maxima of the total real potential ∣Φr∣ = ∣Φmr + Φhr∣, where Φmr is the real potential of the main mode and Φhr is the real potential of the Bernstein mode, is a maximum when Φhr is a minimum, i.e., −Φhr is a maximum (see Figure 4 of TL). Absolute values are used because Φ is the envelope of the rapid ωt oscillation.

[24] The potential Φhr is calculated from equations (1) and (2) of TL for Bernstein propagation for f/fc = 1.44, Te = 5000 K and (fp/fc)2 = 10.3; the value 10.3 was chosen to get good agreement with fringe number 2. Good agreement with fringe number 2 could also be obtained with (fp/fc)2 = 6.0 but then only 12 fringe values are obtained. The minima of Φr are listed in Table 1, column 3. Although the number of fringe maxima agrees with the observed number, most of the values lie between the observed values. Varying the values of Te and (fp/fc)2 does not improve the agreement. However, significantly, assuming that the maximum values of ∣Φhr∣ are obtained with Φhr maxima greater than zero, very good agreement with observations is obtained as shown in Table 1, column 4 for Te = 5000 K and (fp/fc)2 = 7.6; here 7.6 was chosen to get good agreement with fringe number 2. If a somewhat different value of Te had been chosen, the results of column 4 would not change significantly. The good agreement between column 2 and column 4 implies that the positive values of Φhr should be used to calculate the fringe maxima.

[25] The maxima of the interference patterns in Figure 2 for f/fc > 1 result from interference between Bernstein waves and another mode (or other modes) of propagation. MG assumed the other mode was an evanescent EM mode; TL assumed it was the main mode. In calculating the potential of the main mode, TL used the ES approximation when, in fact, the full dispersion relation of the generalized LK or Stix [1992] theory should be used. Using the ES approximation, k2i goes to zero as k1r goes to zero. Using the generalized LK or Stix, k1r has a minimum value as illustrated in Figure 5. Figure 5 shows both the ES approximation and the generalized LK or Stix solution for the main mode. Most of the contribution to the integral in (8) comes from small k1r values. The value of the integral for the ES solution from the minimum in the LK solution to infinity is roughly half of that using the ES approximation from zero to infinity. In addition, the upper portion of the main mode curve needs to be considered. In addition to the main mode and the EM X and O modes, at least one more mode needs to be considered. Using the generalized LK, a strange least damped mode (γ′ = 90°) is found that satisfies to a good approximation

equation image

This mode is shown by the straight line in Figure 6, which is an asymptote to the upper branch of the main mode. However, this solution is not found using the full dispersion relation given by Stix. It is not certain if the discrepancy is due to a problem with the generalized LK dispersion relation, the Stix dispersion relation, or a computer programming error. This is the only example where the author has found a discrepancy between the two dispersion relations. Using Stix and/or the generalized LK dispersion equations, a least damped dispersion curve is found which joins the maximum k1ρ of the EM solution with the minimum k1ρ of the main mode. This mode will be called the “spit mode” solution and k2ρ has a small real part that is shown along the top of the curve at the dots. It should be pointed out that the strange mode and the spit mode do not intersect since the 2 component of the spit mode has a real part whereas the strange mode does not. In any case, the sum of the potentials of all the modes other than the Bernstein mode may not be negative as given by TL. The fact that good agreement between the theory and experiment is found assuming the potential to be positive and slowly varying with θ indicates that this is probably the case. The calculated values of ∣β∣, ∣βc∣, corresponding to the maximum real potentials for the Bernstein mode are shown in Table 1, column 4. The good agreement indicates that to calculate the interference between the Bernstein mode and all the other modes, only the potential maxima of the real Bernstein mode need be considered.

Figure 5.

Hot dispersion curves for the least damped modes (γ′ = 90°). A dispersion curve is shown for the main mode using the electrostatic approximation. Dispersion curves for the main mode and for the O and X electromagnetic modes are shown using the hot dispersion relations of both the generalized LK and Stix. The “strange” dispersion curve is only obtained using the generalized LK dispersion relation. The strange solution is an asymptote to the upper branch of the main mode. Re(k2ρ) = 0 for all curves except the “spit” dispersion curve which has a small real part (see values along curve).

Figure 6.

The real part of the dispersion curve for the Im{k · vg} = 0 case for (fp/fc)2 = 8.8, Te = 5000 K. The arrows give the direction of, and their lengths are proportional to, Re{vg}.

[26] If the group velocity is calculated corresponding to the least damped case using generalized LK, (γ′ = 90°), only the first seven fringes can be explained before a point of inflection occurs in the real part of the dispersion curve. The reason the least damped method explains all of the maxima while using Re(vg) for the generalized LK with γ′ = 90° does not, must be that the imaginary part of the dispersion relation cannot be ignored for most inhomogeneous waves; the Im{k · vg} = 0 case is an exception.

3.4. Inhomogeneous Waves With Im{k · vg} = 0

[27] Using the homogeneous theory of LK, seven of the 13 ∣β∣ values in Table 2, column 2 for f/fc = 1.44 can be explained. It is shown below that by using the theory for inhomogeneous waves with Im{k · vg} = 0, all of the maxima can be explained.

Table 2. Calculated Values of ∣βc∣ for the Im{k · vg} = 0 Case
FringeObserved ∣βαγβc∣, Te = 5000 K, (fp/fc)2 = 8.8
171.887.89 (1.071, 0.000)69.7
259.586.85 (1.081, 0.000)59.5
351.786.10 (1.092, 0.000)51.7
445.085.5242.(1.104, 0.000)45.2
539.185.0646.(1.115, 0.001)39.3
634.184.6849.9(1.127, 0.004)33.9
728.884.3653.7(1.141, 0.008)28.9
824.184.0957.5(1.156, 0.016)24.1
919.583.8661.52(1.174, 0.030)19.4
1015.4.83.7065.96(1.195, 0.053)15.0
1110.383.63570.98(1.221, 0.098)10.6
126.083.70976.72(1.241, 0.203)6.3
132.382.96083.22(0.987, 0.570)1.8

[28] It will be assumed that the sum of the phases of all the modes other than the Bernstein mode is positive and changes slowly with direction in the plasma. Hence the maxima of the interference pattern can be assumed to be due only to the Bernstein mode for which the phase is a maximum and changes rapidly with direction. If the phase of the Bernstein wave is assumed to be krd cos θ where the real part of the wavenormal is kr, d is the distance between transmitter and receiver equal to 0.10 m in the Gonfalone experiment, and θ is the angle between the real wave vector kr and real part of the group velocity vgr, then the interference maxima are given by

equation image

The angle between kr and vgr is θ = α′ − βc where βc, the calculated value of β, is given by βc = tan−1[Re(vg2)/Re(vg1)]. It can be seen from Figure 3 that the Bernstein waves are backward waves, which means that the angle between kr and vgr is greater than 90°. The procedure to obtain βc in Table 2 is as follows: A value of α′ is chosen (for δ = 0) and γ′ is varied until (4) is satisfied; the value of krρ is then known from the dispersion relation ((A2) set equal to zero); kr is determined from the temperature and (7); then vgr, βc and θ are calculated. A new value of α′ is chosen and the procedure is repeated until (11) is satisfied. For given α′ and γ′, the 1 and 2 components of the group velocity, vg1 and vg2, are calculated from the dispersion relation and equation (3). In MG the observed values of ∣β∣ led to an electron temperature of 5000 K for an assumed value of (fp/fc)2 = 6.25. Here, in a more thorough study, the values of (fp/fc)2 and Te were varied and the best agreement between β and βc for fringe number 2 was found for Te = 5000 K and (fp/fc)2 = 8.8. For these values the real part of the dispersion curve for Im{k · vg} = 0 is shown in Figure 6. It was found that (4) could be satisfied with δ = 0°. The two-dimensional geometry for δ = 0° is shown in Figure 1b. A curve was also calculated for δ = 45° that was slightly different than the curve for δ = 0° but if plotted in Figure 6, the two curves would be indistinguishable. Note that the value of α used in MG (see Figure 3) is consistent with the value of α′ used in this paper.

[29] The arrows along the curve in Figure 6 show the direction of Re{vg} and their lengths are proportional to the magnitude of Re{vg} corresponding to the 13 values of βc that satisfy (11). The direction of Re{vg} is very close to perpendicular to the curve except for number 12 near the maximum curvature. The curve has no inflection point and all of the observed values of β can be explained. In Table 2 the columns for α′, γ′ and correspond to the (fp/fc)2 = 8.8, Te = 5000 K values which satisfy (4) and (11). It can be seen that not only do all the observed values of ∣β∣ have a corresponding calculated value but the agreement is excellent.

[30] The complex group velocity vg does not normally have a clear interpretation. For complex k and real x where the wave field components vary as exp[i(k · xωt)], Suchy [1972] concluded that in a moderately absorbing medium the direction of energy flow is given by Re{vg} but that a correction factor had to be applied to the magnitude; this factor approaches unity as the absorption approaches zero. The agreement between experiment and observation suggests that when (4) is satisfied the direction of energy flow is given by Re{vg} even when Im{vg} is quite large.

[31] Here, we have a complex k and complex vg and if it is assumed that the absorption is proportional to exp[−Im{k · vgt}], then growing waves will exist in the plasma since if (4) is satisfied, Im{k · vgt} can become negative. This is not possible in a Maxwellian plasma. However, the value of exp(−ki · vgrt) always indicates a decreasing signal and may approximate the true damping. Assuming it does give the damping yields (in dB)

equation image

where E/E0 is the received electric field, normalized to the field if there were no absorption, and d = 0.1 m is the distance from transmitter to receiver. In (12) the damping rate 8.6(ki · vgr) in dB/s is divided by the group velocity ∣vgr∣ in m/s to get the damping rate in dB/m. For f/fc = 1.44 the amplitudes of fringes 1–8 in Figure 2 are reasonably constant. Using the dB scale in Figure 2, the difference between the maximum and minimum amplitudes is about 5.2 dB. From this value it can be determined that the signal(s) interfering with the Bernstein mode is (are) about 5.3 dB above the Bernstein signal. From the strength of the interfering signal(s) and by measuring the amplitude of fringe 13, it can be determined that the Bernstein waves suffer about 7 dB of attenuation in propagating between transmitter and receiver. In the Im{k · vg} = 0 method, from (12) the 8th fringe has about 1.5 dB and fringes 1 to 7 have less than 0.7 dB of attenuation, which agrees with the observed fringes in Figure 2. However, the calculated attenuation for fringe 13 is about 29 dB, much more than the observed 7 dB of attenuation. Also, for the least damped case the calculated attenuation of fringe 8 can be calculated to be about 3 dB and for fringe 13 about 30 dB.

[32] Figure 2 shows two patterns for the lower cone resonances for f/fc = 0.49 and 0.65. The f/fc = 0.65 case will be considered. For small values of (whistler EM mode) the calculated values of ∣β∣, ∣βc∣, increase as increases and reach a maximum value of 41.86° at = (0.0497, 0.0000) and α′ = 47.76°. Above this value of , ∣βc∣ decreases and the corresponding mode can be considered to be the ES mode. At the maximum value of ∣βc∣ the phases of the whistler and ES modes are equal and a maximum in the signal is to be expected near this value of ∣βc∣. The observed maximum signal occurs at a value of ∣β∣ = 39.5°. It is reasonable that the observed signal maximum occurs at a slightly smaller value of ∣β∣ since no signal can propagate at ∣β∣ > 41.86°. Signal maxima are also expected when the phase difference between the ES and whistler modes are 2π and 4π. For (fp/fc)2 = 8.8 and T = 5000 K, equation (4) can be satisfied corresponding to a phase difference of 2π for ∣βc∣ = 18.5°. The observed secondary maximum for f/fc = 0.65 occurs at ∣β∣ = 20.0° reasonably close to the ∣βc∣ value; this value is thought to be slightly more accurate than the value of 20.5° found by MG. Equation (4) cannot be satisfied corresponding to a phase difference of 4π. There appears to be a small maximum for β > 0 near β = 6.5° but perhaps this feature is just a variation in the noise background.

[33] Figure 2 of MG is reproduced in Figure 7 of this paper. It shows experimental results for both the lower and upper resonance cones. Figure 4 of MG gives the homogeneous wave dispersion curve for the lower resonance cone. Part of Figure 4 of MG is shown in Figure 8 here. The homogeneous curve for the lower resonance cone is reproduced on the left in Figure 8. The inhomogeneous curve for the Im{k · vg} = 0 case is shown on the right in Figure 8. The interference pattern for f/fc = 0.49 shows three interference maxima at ∣β∣ values of 18°, 23°, and 29°. The corresponding values of krρd cos θ for the Im{k · vg} = 0 solution are 0.60, 0.30, and 0.19 cm. These agree with the homogeneous wave values of MG even though the dispersion curve for the inhomogeneous wave solution is significantly different than the homogeneous wave curve. Note that the krρ value for the Im{k · vg} = 0 case reaches a maximum value near 0.24. From krρ = 0.24 to about 0.33 equation (4) cannot be satisfied, the minimum value of Im{k · vg} is shown by a dashed line. The krρ value then decrease slightly before Im{k · vg} becomes very large.

Figure 7.

The lower resonance cone for f/fc = 0.49 and the variation of the upper resonance cone with frequency in the late afterglow [after Muldrew and Gonfalone, 1974].

Figure 8.

Real dispersion curves corresponding to Figure 7 for the lower resonance cone with f/fc = 0.49. The homogeneous case is on the left, and the Im{k · vg} = 0 case is on the right shown in solid line. The dashed line is for Im{k · vg} equal to a minimum value. After Muldrew and Gonfalone [1974, Figure 4].

[34] Figure 6 of MG gives the homogeneous wave dispersion curve for the upper resonance cone, and Figure 9 gives the homogeneous (on the left) and Im{k · vg} = 0 (on the right) dispersion curves for the upper cone resonance at f/fc = 1.45. In Figure 7 for f/fc = 1.45, two interference maxima can be seen at ∣β∣ = 76.5° and 86°. The values of krρd cos θ in MG for the homogeneous case are 0.40 and 0.20 cm, respectively. The Im{k · vg} = 0 values are 0.375 and 0.194 cm (the third decimal place is not too significant), respectively, yielding an electron temperature of about 490 K in the late afterglow (see MG).

Figure 9.

Real dispersion curves corresponding to Figure 7 for the upper resonance cone with f/fc = 1.45. The homogeneous case is on the left, and the Im{k · vg} = 0 case is on the right. After Muldrew and Gonfalone [1974, Figure 6].

[35] It should be pointed out that (4) cannot be satisfied for values of ω close to c where n is an integer. As ω approaches c the minimum value of Im{k · vg} for inhomogeneous waves approaches the value for homogeneous waves.

3.5. Poynting Velocity for Homogeneous Waves

[36] It is possible to explain the electron cyclotron resonance as observed on topside ionograms using the Poynting vector and wave energy density in a plasma [Muldrew, 2006]. In general thermal kinetic terms must be considered in calculating the Poynting vector and energy density in a hot plasma [Allis et al., 1963]. However, for wave frequencies near the electron cyclotron frequency, these thermal terms can be neglected and the wave energy velocity, which is real, can be calculated and enables the resonance to be understood.

[37] For Bernstein-wave propagation these thermal terms cannot be neglected. Including them results in a wave energy velocity equal to the group velocity [Allis et al., 1963], which is complex for α′ ≠ 90°. With the exception of Im{k · vg} = 0 inhomogeneous waves, a complex group velocity is difficult or impossible to interpret and the experimental results cannot be explained.

[38] At frequencies just below fc, equation (4) cannot be satisfied. At somewhat lower frequencies, say about f/fc ≤ 0.97, equation (4) can be satisfied. However, when (4) is satisfied Re(vg) does not agree with the Poynting velocity which is found by dividing the Poynting vector by the energy density of the wave [Muldrew, 2006]. Hence, although the direction of Re(vg) appears to give the correct energy velocity for the electrostatic Bernstein waves, it does not give the correct energy velocity for the electromagnetic waves with complex wave number at frequencies somewhat below fc.

3.6. Radiation From Plane Containing the Dipole Antenna and Magnetic Field

[39] In a study of proton echoes on topside ionograms [Muldrew, 1998], it was discovered that the Bernstein waves responsible for the echoes were radiated from the plane containing the Earth's magnetic field and the linear dipole antenna. Protons close to the antenna are energized by the short transmitter pulse and circle the magnetic field with various cyclotron radii. They all return to this plane at integer multiples of the proton cyclotron period and reproduce the transmitter pulse, here generating a Bernstein wave.

[40] A similar effect occurs with electrons. Electrons energized along but close to the antenna circle the magnetic field and return to the plane at integral multiples of the electron cyclotron period reproducing the transmitter pulse in this plane. EM waves with extremely slow wave-energy velocity are radiated and received by the satellite. These waves are responsible for generating the electron cyclotron resonance observed on ionograms [Muldrew, 2006].

[41] It is possible that a similar effect could occur in the laboratory plasma. Electrons energized by the antenna would circle the magnetic field and return to the magnetic field lines through the short antenna. Bernstein waves would radiate from these field lines and be received by the receiving antenna. At all positions of the receiving antenna Bernstein waves would be received with α′ ≈ 90° and these waves would be undamped. For reasonable values of Te and ωp2/ωc2, the observed values of β corresponding to the small calculated values of β are too high.

4. Discussion and Conclusions

[42] In MG only the first seven values of ∣β∣ in Table 2 are explained using the plasma dispersion relation for homogeneous waves and no solutions exist beyond the inflection point in the real part of the dispersion curve. For the least damped inhomogeneous waves (the wave normal real in the direction of the magnetic field) all values can be explained. Modifications have been made to the dispersion equations of LK to calculate the dispersion relation for inhomogeneous waves for which the imaginary component of the wave vector can have any direction. Calculations based on the modified equations with Im{k · vg} = 0 allow all of the observed values in Table 2 to be explained and there is no inflection point in the real part of the dispersion curve. Exceptionally good agreement is obtained between experiment and both the least damped method and the Im{k · vg} = 0 method.

[43] The complex wave vector consists of two vectors, the real wave vector and the imaginary wave vector. The group velocity is then also complex and consists of two vectors. If the direction of the imaginary wave vector is taken to be a free parameter then the condition

equation image

can often be satisfied. If (13) is satisfied, (k · xωt) is real in the variation of the field components given by exp [i(k · xωt)] if x is taken to be vgt. If it is assumed that Im{k · vgt} is proportional to the absorption in dB, then as Im{k · vgt} passes through the condition where equation (13) is satisfied, the propagation goes from damped to growing and growing propagation is not possible in a Maxwellian plasma. However, for the Bernstein waves considered here, exp(ki · vgrt) is always less than one and may approximate the absorption. The calculated point at which absorption becomes significant (near fringe number 8) agrees with the observed point for both the least damped case and the Im{k · vg} = 0 case. However, beyond fringe number 8 the calculated absorption is much greater than the observed for both cases. The agreement between the observed and calculated values of β in the Im{k · vg} = 0 case indicates that the direction of the wave energy velocity is given by the direction of vgr. The magnitude of vgr may also give the magnitude of the wave energy velocity; however, this cannot be concluded from the agreement between the observations and calculations. Calculations based on the generalized LK equations and on (13) allow all of the observed values in Table 2 to be explained. This result suggests that equation (13) in conjunction with the dispersion relation for inhomogeneous waves is fundamental to understanding the propagation of waves with a complex wave vector in a magnetoplasma. The least damped method determines the field at a point in the plasma by integrating over all the significant plane waves arriving at that point from the source. The Im{k · vg} = 0 method determines the field by calculating the group-velocity direction.

[44] For real k, Stix [1992] shows that the direction of vg is perpendicular to the dispersion curve. Numerical calculations of the angle between the dispersion curve and the group velocity corresponding to Figure 6 for complex k show that except for the region of high curvature near k1rρ = 1.23, vgr is approximately perpendicular to the dispersion curve. At some point near maximum curvature vg becomes parallel to the dispersion curve.

Appendix A: Modifications to Lewis and Keller [1962] for Inhomogeneous Plane Waves

[45] The order of the modifications below is the same as the order of the equations found in LK; LK should be used as a reference in the following since the details are not discussed. The symbols used here are the same as those in LK except for k1, k2, k3, α′, γ′ and δ.

[46] In the line following LK(2.16), the second H0 should be H0. In LK(2.20) the first and fourth terms on the left hand side of the equation should be −(df/dθ) and −k · vf. LK(3.1) becomes (2), (5) and (6). LK(3.3) becomes

equation image

[47] The components of the determinant in LK(3.16) become

equation image

Setting this determinant equal to zero, using equation (5), and solving for a given ω, a given direction of kr (which fixes α′) and a given direction of ki (which fixes γ′ and δ) determines k = kr + iki. See Muldrew and Estabrooks [1972] for details on solving the determinant by computation.

[48] A simplification is made here by setting the electron collision frequency ν = 0, although no difficulty results if collisions are included; LK(3.4) to LK(3.11) are then simplified by setting f0 = 0. In LK(3.17) there is no change but one must be careful in evaluating k2; it is k · k, not k × k. LK(3.19) is simply

equation image

LK(3.22) becomes

equation image

[49] There are two typographical errors in LK(3.25). In element (1, 1) t should be t2 and in element (3,3) sin β should be sin2β. The nine modified elements of LK(3.25) are as follows:Element (1,1)

equation image

Elements (1,2) and (2,1)

equation image

Elements (1,3) and (3,1)

equation image

Element (2,2)

equation image

Elements (2,3) and (3,2)

equation image

Element (3,3)

equation image

where the top sign of ± refers to the first element listed just above and the bottom sign refers to the second. The Φ in LK(3.23) is given in LK(3.26) and is the Φ in LK(3.22) without the terms −γ(W2 + V2). LK(3.26) becomes, using the modified Appendix A of LK (see below)

equation image


equation image
equation image
equation image
equation image

where m is the electron mass, κ is Boltzmann's constant, vth is the thermal electron velocity and λ is negative for electrons.

[50] The first − sign in LK(3.30) should be +. The Qij become

equation image

[51] LK state that equations (A11) are valid if Re(z2) > 0, i.e., if Re(z · z) > 0 (see equation (A8)) with no restrictions on y or the direction of k. LK discuss solutions for g(y) in their Appendix C, but there are presently better ways of determining g(y), which is defined by

equation image

where Φ is given by (A6). The integral g(y) can be obtained from a combination of the plasma function and an infinite series of modified Bessel functions [see Muldrew and Estabrooks, 1972]. The plasma function uses only y and z1(y and z cos β in LK) and the modified Bessel functions use only z22z32 (z2 sin2β in LK).

[52] Appendix A of LK indicates how three of the integrals in LK(3.21) can be solved. The modifications to their Appendix A for inhomogeneous waves are given below.

[53] LK(A2) is the same as (A4). LK(A8) becomes

equation image

LK(A11) becomes

equation image

LK(A12) become

equation image

LK(A14) becomes

equation image

LK(A15) and LK(A16) become

equation image

where the top sign of ± is for v and the bottom sign is for u. The three components of the vectors v and u are separated by commas.


[54] I am grateful to H. G. James and L. R. O. Storey for helpful discussions and correspondence.