On the problem of quasi-resonant wave trapping in the LHR waveguide

F. Jiricek

Institute of Atmospheric Physics, Academy Science of the Czech Republic, Prague, Czech Republic

D. R. Shklyar

Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Troitsk, Moscow Region, Russia

Contents

Abstract

Introduction

Lower hybrid resonance (LHR) waves [Brice and Smith, 1965] are commonly found in VLF wave data obtained from satellites in the outer ionosphere and the magnetosphere (e.g., Laaspere and Taylor, 1970]). LHR noise observed in early satellite experiments was interpreted by Smith et al.[1966] to result from the trapping of whistler energy in the valley of LHR frequency altitude profile [Kimura,1985], or in the ionospheric duct, as it was originally called [Smith et al.,1966]. Later in this paper, we use the term LHR waveguide for the corresponding profile of LHR frequency.

Later on, other mechanisms of LHR wave generation were suggested, namely, wave excitation by energetic electron streams (e.g., [Horita and Watanabe,1969; Horita,1972]) and linear and nonlinear VLF wave scattering on small-scale plasma turbulence (see, for instance, Bell and Ngo[1990, and references therein] and Trakhtengerts and Hayakawa[1993], and references therein). We will not further discuss these processes here, since in this paper we deal only with the above mentioned mechanism suggested by Smith et al.[1966]. Since these ideas had been put forward, the knowledge of LHR waveguide has been widely used in interpreting VLF data from satellite measurements (e.g., [Jirícek and Tríska,1976; Nishino and Tanaka,1987]).

Although wave propagation inside the waveguide was considered in many papers beginning with the pioneering work by Smith et al.[1966], the mechanism of the waveguide feeding remained unclear. That question is dealt with in this paper. To formulate the problem in more detail, we recall the well-known features of quasi-resonant LHR waves [Stix, 1962]: large (close to p/2 ) wave normal angle q with respect to an ambient magnetic field B₀ ; short wavelength corresponding to large refractive index (more precisely, k² c²/w_p² 1, where k is the wave number, w_p is electron plasma frequency, and c is the speed of light); and the quasi-electrostatic character of the wave field. Such a wave cannot propagate in the region where its frequency w is essentially smaller than the local LHR frequency w_LH (see dispersion equation (5) and relation (7) below). A typical LHR frequency profile in the outer ionosphere is shown in Figure 1. According to the tradition of representing this profile, the altitude is along the vertical axis, while the frequency w_LH is along the horizontal one. In the plane ( w_LH, altitude), the wave trajectory corresponds to a vertical line w = const. If the wave moves towards increasing values of w_LH, its group velocity decreases, and the wave is reflected in the vicinity of the point where this line crosses the LHR frequency profile. Thus waves with frequencies (w_LH)_min < w< (w_LH)_max, if they exist in the waveguide (that is, below h_max ), would be trapped inside the waveguide. But, at first sight, the waveguide could not be fed either from above or from below, and the quasi-resonant LHR waves described above should be excited directly inside the waveguide. As was mentioned above, the corresponding mechanisms of local excitation of LHR waves were suggested, namely, scattering of upgoing waves on small-scale turbulence or plasma stream instability. However, these local mechanisms require some additional conditions to be fulfilled in the waveguide region, namely, the existence of small-scale plasma turbulence or density irregularities in the case of wave scattering or the existence of energetic electron streams in the case of instability. At the same time, the most natural way of waveguide feeding would be wave penetration from higher altitudes, where quasi-resonant VLF waves propagating in a nonducted mode normally exist. In this paper we suggest a mechanism of LHR waveguide feeding by those waves and give evidence for this mechanism derived from ray tracing calculations and satellite observations. The analysis is based on the solution of the equations of geometrical optics, with the account of a realistic two-dimensional shape of the LHR frequency profile. We should emphasize underline that we consider wave propagation only in the upper ionosphere and the magnetosphere, while wave propagation in the lower ionosphere, where the LHR frequency essentially decreases owing to exhaustion of the ionized plasma component, is beyond the scope of the present work. Accordingly, in Figure 1 and Figure 4 below, we do not show LHR profiles in this region of the lower ionosphere (heights h 100-200 km).

2. Quasi-Resonant Whistler Mode Wave Propagation in the LHR Frequency Band

We remind the reader that the LHR frequency in the outer ionosphere (from, say, 500 km up to a few thousand kilometers) ranges from a few kHz up to above 10 kHz, and usually it has the characteristic shape of its profile as shown in Figure 1. Thus waves in the LHR frequency range belong to the VLF frequency band (1-30 kHz). As for the mode of propagation in the magnetosphere, these waves correspond to whistler mode waves, as a few tens of kHz is a characteristic value of the equatorial electron gyrofrequency in the plasmasphere ( L shells from about 2 to 4, where L is McIlwain's parameter). Owing to essential increases of electron gyrofrequency w_H with decreasing heights, the same waves fall in the LHR frequency band in the ionosphere.

Using a general equation for the wave refractive index in a cold, magnetized plasma [Stix, 1962], one can find the dispersion relation for whistler mode waves in the frequency range w_Hi ww_H ( w_Hi is the ion gyrofrequency):

(1)

where LHR frequency w_LH is determined by

(2)

( n_e and m_e are electron concentration and mass, respectively, and n_a and m_a are the same for ions of species a ), and

(3)

From (1) one can see that the characteristic value of the wave number for whistler mode waves is k w_p/c. For the so-called quasi-longitudinal waves [Ratcliffe,1959; Helliwell,1965] k k, while the inequality k k corresponds to quasi-resonant waves (see, for example, Shklyar and Washimi [1994]). In general, whistler mode waves exist when w > w_LH as well as when w< w_LH ; however, the last range is accessible only for waves in the quasi-longitudinal mode of propagation. At the same time, quasi-resonant waves having k² k² and wave normal angles q close to p/2 are very sensitive to the LHR frequency profile.

To consider the propagation of quasi-resonant waves in the LHR frequency range, we simplify the dispersion equation (1), assuming

(4)

which gives

(5)

where k and k are the longitudinal and transversal components of the wave vector, respectively, and

(6)

We should mention that the last inequality in (4) is in fact implied by the previous ones. From (5) it is easy to see that for the waves under discussion

(7)

where (4) and (2) have been taken into account. Equation (5) constitutes the basis for analysis of the quasi-resonant wave propagation in the LHR frequency band.

We will restrict ourselves to consideration of wave propagation in the meridional plane and will suppose the Earth's magnetic field to be dipolar. As is well known (e.g., [Fermi,1968]), the equations of geometrical optics can be written in Hamiltonian form with the components of coordinates r and wave vector k being canonically conjugate variables. The Hamiltonian from which these equations should be derived is the wave frequency w expressed through the variables r, k according to the dispersion relation (in our case, equation (5)). Since the quantities k and k which enter (5) are expressed through Cartesian coordinates and wave vector components in a rather complicated way, the Cartesian variables are not the proper ones for our problem. That is why we will use dipolar coordinates L, M, connected with Cartesian coordinates (x, z) by the relations

(8)

Here R_E is the Earth's radius; the origin of the Cartesian coordinates is supposed to be at the Earth's center, the z axis pointing along the dipolar axis from S to N. The coordinate L is McIlwain's parameter, which is constant along the field line; the coordinate M is measured along the field line from the equator (see Figure 2). The Lamé coefficients corresponding to the coordinates L, M are

(9)

where l is the geomagnetic latitude: tan l = z/x.

The momenta P_L, P_M canonically conjugate to the coordinates L, M are related to the transverse and longitudinal wave vector components k, k by

(10)

In new variables (L, M, P_L, P_M), the expression for the wave frequency (5) playing the role of the Hamiltonian from which the equations of geometrical optics can be derived takes the form

(11)

where the following notation has been introduced

(12)

and it is supposed that the quantities w_LH, W, x, and h are expressed as the functions of coordinates L, M. Using (4) and notation (12) and taking into account (w_LH² /w_H²) 1, we obtain the relations between different quantities entering (11):

(13)

We now write the equations of geometrical optics which follow from (11):

(14)

While deriving equations (14) for momenta P_L and P_M, we have made use of the fact that, under conditions (13), the derivatives of the quantities x² and h² can be neglected.

The physical meaning of the derivatives dL/dt and dM/dt is revealed by the relations which connect these quantities with transversal and longitudinal components of the group velocity. Since h_L dL is equal to the length across the geomagnetic field line corresponding to the variation dL of the coordinate L at constant M and h_M dM is the length along the field line corresponding to the variation dM at constant L, then

(15)

which, of course, is equivalent to the first two equations in (14) in view of the relations (10).

We proceed with the analysis of the wave propagation near the LHR frequency by comparing the longitudinal and transversal group velocities (15). From (14) it follows that the ratio between these two quantities depends on the magnitude of the longitudinal momentum P_M. We will relate the value of P_M² to the quantity x², keeping in mind that in view of limitations (13) the relation between P_M² and x² may be arbitrary. We divide the range of the permitted values of the quantity P_M² into two domains: P_M² x² and P_M² < x². From (13) and (14) it is easy to see that for P_M² x², |h_MdM/dt| |h_LdL/dt| ; thus the wave group velocity is almost aligned with the ambient magnetic field. For P_M² < x², | v_g | is still larger than | v_g | unless P_M² is as small as h_L² x⁴/h_M² P_L², and for smaller P_M², | v_g | > | v_g |. Thus on the greater part of the wave trajectory, | v_g | is larger than |v_g |, which permits, in some cases, the reduction of the problem to a one-dimensional one. This has been done in the papers by Bosková et al. [1988, 1992]. On this basis, the explanation of the lower hybrid resonance whistler traces has been attacked. However, for the problem discussed here, such an approach is not pertinent. Thus in this study, we will abandon the limitation of one-dimensional wave propagation.

It follows from (14) that the sign of v_g depends on the sign of P_L ; namely, they have the opposite signs for (P_M² - x²) > 0 and the same signs for (P_M² - x²) < 0. As was shown by ray tracing calculations [Walter and Angerami, 1969], the wave normal vector of the nonducted whistler wave turns antiearthward while propagating in the magnetosphere (for realistic models of plasma distribution). Thus the wave reaches the outer ionosphere after propagating from one hemisphere to another with P_L > 0 (see (10)). For these waves, which we consider later on, v_g < 0 on the greater part of a trajectory, so that the wave moves predominantly toward lower L shells. As for v_g , it has the same sign as P_M, and both change sign at the wave reflection point (over the longitudinal coordinate M ).

3. The Mechanism of Wave Trapping in the Waveguide

No matter that the wave group velocity is almost aligned with the ambient magnetic field, even slow motion of the wave across the field lines is essential to our problem. Thus we shall consider a two-dimensional problem, although the dependence of the quantities upon the coordinate L may be considered as a slow dependence on the time scale of the wave longitudinal motion.

We now rewrite the equations (14) in the form

(16)

where

(17)

The qualitative physical mechanism of the wave trapping into the waveguide can be understood from the well-known analogy between geometrical optics and mechanics [Fermi,1968], where mass point corresponds to the wave packet, and particle trajectory corresponds to the ray path. Equations (16) for M and P_M resemble the equations of particle motion in the potentional of the kind (17) which has a maximum and a minimum over the M coordinate, corresponding to the maximum and minimum of the LHR frequency profile, and a potential wall, corresponding to the bottom part of the profile (see Figure 1). Since the problem is two dimensional, the effective potential also depends on the coordinate L as a parameter. Once passing near the potential maximum, the particle continues to move over both M and L coordinates. Then if the potential maximum increases, the particle can be trapped after reflection from the potential wall. A monotonic increase of the potential maximum, connected with a monotonic variation of the LHR maximum with L coordinate, guarantees that the particle will stay in the trap and, correspondingly, the wave will remain in the waveguide.

For a precise description of the wave trapping, a more strict consideration is needed. Toward this direction, we proceed in the following way. Since the Hamiltonian (11) is independent of time and, thus, is a constant of motion, the problem can be reduced to a one-dimensional one in a standard way (see, for example, Arnold [1974]). We suppose the quantity U / L to be sign determined and choose P_L as a new independent variable. Dividing the equations (14) for M, and P_M by (dP_L/dt) and excluding the quantity L with the help of the integral (11), we arrive at two equations for the variables M and P_M as functions of P_L. These equations are derived from the Hamiltonian H = L(M, P_M; P_L, w), which (up to arbitrary constant) is nothing but the coordinate L expressed from (11) as a function of M, P_M; P_L, and parameter w. To find the function L(M, P_M; P_L, w), we rewrite the relation (11) in the form

(18)

For the sake of simplicity, we neglect the dependence of the quantities x² and h² on coordinate L. As was shown above, the variation of coordinate L is small; thus we may expand the expression on the left-hand side in (18) around initial coordinate L₀. It is convenient to write this expansion in the form

(19)

Putting this expansion into (18), we find the quantity (L - L₀) which plays the role of the Hamiltonian in the problem

(20)

In our case, the role of independent variable is played by the quantity P_L, whereas the usual independent variable in the equations of motion is time. To restore the habitual notation, we denote

(21)

The Hamiltonian (20) depends on t and, thus, varies along the wave trajectory. At the same time, according to the definition of H as L - L₀, its initial value at L = L₀ is equal to zero.

We proceed with the analysis of the equations derived from (20), keeping in mind that the equations of geometrical optics describe the wave propagation in real space with a group velocity, the role of momentum being played by a wave vector. Positive direction of the M axis in the northern hemisphere is from the equator to the Earth (see Figure 2), so that increasing M values correspond to decreasing heights. We will consider waves having positive P_M above the LHR maximum and, thus, propagating to the Earth.

To get an idea about the shape of potential V(M), we notice that according to (19)

(22)

It is natural to assume that a(M)/W² is a slowly varying (as compared to w_LH² ), sign-determined function. Thus roughly speaking, the shape of potential V(M), up to the sign of a(M), is similar to the shape of the LHR frequency profile. The qualitative shape of the potential V(M) is schematically illustrated in Figure 3a and Figure 3b for a > 0 and a< 0, respectively. We should mention that a > 0 corresponds to an increase of LHR frequency with decreasing L shell, and vice versa.

We can now give a simple, but strict, interpretation of the wave trapping into the wave-guide. The equations for M, P_M as functions of the independent variable t which follow from (20) have the form

(23)

At the same time, the Hamiltonian (20) itself varies with t according to the equation

(24)

Let us first consider the case a > 0, when the potential V(M) has the shape represented in Figure 3a. Using the definitions of the quantities V (equation (22)) and x² (equation (12)) and inequalities (13), one can easily show that DV V(M) ax²/(t² + h²), where DV is the variation of potential across the LHR profile. From (20) it then follows that the wave cannot penetrate deeply under the potential wall. At point M_max, corresponding to the maximum of the potential V(M), let the quantity H (equation (20)) be larger, but close to V_max. (We remind the reader that initially H = 0, while V_max (w_LH² - w²)_max, so that such a situation appears when w (w_LH)_max.) In this case the wave passes over the potential maximum. Then the quantity H varies according to (24), and as was shown above, for a > 0 it decreases over the greater part of the profile. Thus the wave becomes trapped between the potential wall and the potential maximum. In this consideration we assumed t P_L >0, in agreement with general properties of nonducted whistler wave propagation (see the end of section 2).

Let us turn to the case a< 0. One should now be cautious in using a similarity with particle motion in a potential, since the case a< 0 formally corresponds to the motion of a particle with negative mass. As before, the initial value of the Hamiltonian H (equation (20)) is equal to zero, while the shape of potential V (Figure 3b) is reversed as compared with the case a > 0. Using similar arguments as above, one can show that in the case a< 0, the wave cannot penetrate far above the potential minimum V_min, while trapped waves correspond to H > V_min. Since initially H < V_min, for a wave to be trapped, an increase of the quantity H along the wave trajectory would be necessary. However, it is not the case when a< 0. Indeed, from (16) and (22) it follows that for a< 0, dP_L/dt dt/dt < 0, so that the value of t decreases along the wave path. Equation (24) then shows that during the time of wave crossing of the LHR profile, the quantity H decreases on the whole. Thus the wave cannot be trapped in the waveguide in the case a< 0, that is, when the LHR frequency decreases with a decrease of L shell. We should stress that, independently of the sign of a, the quantity P_L t is large (as compared with P_M and h ) and positive and that, on the average, P_M² x². These facts have been assumed in the analysis of trapping conditions.

To summarize, the possibility for LHR waveguide feeding by nonducted VLF waves propagating in the magnetosphere appears when the two-dimensional LHR frequency profile satisfies the following requirements: the profile of the LHR frequency along a given geomagnetic field line (i.e., the line L = const) has a valley lying between local maximum, local minimum, and the bottom wall, and the value of local maximum along the field line increases with decreasing L shells.

Figure 4 shows model LHR frequency profiles which satisfy these requirements. The LHR frequency (horizontal axis) is plotted as a function of height from the Earth (vertical axis) along the geomagnetic field lines L = 2, 3, and 4. Similar to Figure 1, the lower ionospheric heights are not shown in Figure 4.

Examples of wave trajectories corresponding to wave trapping in the LHR waveguide, obtained from the ray tracing calculations using the profiles of Figure 4, are shown in Figure 5. We should emphasize that they are model ray tracing calculations, since they use analytical LHR profiles shown in Figure 4 that resemble the calculated profile in Figure 1 and they satisfy the above formulated conditions for the wave trapping. Figure 5a shows the ray paths in Cartesian coordinates (x, z) where the trapping is manifested as line broadenings in the near-Earth region; Figure 5b displays the same wave trajectories in (L, M) coordinates where the wave trapping is most evident. When the wave becomes trapped in the LHR waveguide, the longitudinal coordinate M of the wave trajectory starts oscillating, while the transversal coordinate L decreases on the average, in agreement with the analytical results of section 3. We should stress that the quasi-resonant mode of propagation and proper features of LHR profiles do not guarantee wave trapping: the wave may well be reflected either above h_max or even from the bottom of the profile, but still leave the waveguide. For a wave to be trapped, it should have a frequency sufficiently close to the local maximum of the LHR frequency along the wave path. That is why the wave may be trapped after one or several hops in the magnetosphere, when appropriate conditions for trapping arise.

The trajectory marked "1 hop trapping" gives an example of wave trapping in the LHR waveguide after one hop between conjugate hemispheres. The wave propagates in the magnetosphere in a nonducted mode, enters a quasi-resonant regime of propagation, and crosses the LHR maximum in the opposite hemisphere having a frequency slightly higher than (w_LH)_max. While the wave continues moving toward the bottom of the profile, at the same time it shifts to lower L shells at which the LHR maximum increases. After reflection from the bottom of the profile, the wave approaches the maximum of the LHR profile with w< (w_LH)_max at which quasi-resonant waves cannot propagate. The wave is reflected from the LHR maximum and, thus, becomes trapped in the waveguide.

The trajectories marked "2 hop trapping" and "3 hop trapping" give examples of more complicated trajectories where the trapping takes place after two and three hops in the magnetosphere. The case of a two hop trapping has been found by B. V. Lundin (private communication, 1998) in ray tracing calculations using the model of diffusive equilibrium [Angerami and Thomas, 1964] for evaluation of LHR profiles. These rather specific wave trajectories shown in Figure 5 become clear in view of the present study.

4. Comparison With Experimental Data

In this section we discuss the experimental results which show the possibility of LHR waveguide feeding by the mechanism suggested above. Since satellite measurements are always local in space, some evidence of the fact that a satellite is inside the wave-guide should be provided. In fact, for a more profound discussion of the wave trapping, we need simultaneous measurements of the electromagnetic field in a wide band and cold plasma parameters, namely, ion content and temperature. Such complex data were available from the measurements performed on board the Intercosmos 24 satellite [Bosková et al.,1993]. To give a reference in light of which the waveguide feeding could be discussed, it is convenient to consider it together with other well-known phenomena, the so-called LHR whistlers [Bosková et al., 1992; Charcosset et al.,1973; Morgan et al.,1977]. That is why we discuss here the same experimental data which have previously been used in the study of LHR whistlers [Bosková et al., 1992]. It is all the more reasonable since, according to the results of Bosková et al.[1992], which we will use in the following discussion, LHR whistlers should be observed just inside the LHR waveguide.

An example of a spectrogram in the 10-kHz frequency band is shown in the upper panel of Figure 6. The spectrogram is obtained from electric field measurements and is most appropriate for discussing quasi-electrostatic wave phenomena. During the time interval shown in the figure, the satellite height was about 600 km, while the McIlwain parameter increased from L = 2.2 to L = 2.4. The main VLF wave phenomena seen in the picture are as follows: usual ducted whistlers whose traces expand in the whole frequency band; LHR whistlers seen in the upper part of the panel, which have a clear cutoff frequency; and pronounced noise close to the local LHR frequency. The lower cutoff frequency of LHR whistlers gives a good estimation of the maximum LHR frequency above the satellite. This frequency is shown by the dashed line in the lower panel of Figure 6, while the solid line shows the local LHR frequency along the satellite path calculated from the measurements of cyclotron frequency and ion composition.

The LHR frequency profile shown in Figure 1 corresponds to the point denoted by arrows in Figure 6. That profile has been calculated with the help of the diffusive equilibrium model [Angerami and Thomas, 1964], with the use of ion density and temperature measurements at satellite altitudes. Four ions (H ⁺, He ⁺, O ⁺, N ⁺ ) have been included in the model, and a temperature gradient of 0.6^o K km ^-1 has been used. The maximum frequency in the LHR profile shown by the bold dot in the lower panel of Figure 6 is quite close to the LHR whistler cutoff frequency, in agreement with the theory developed in the paper by Bosková et al. [1992]. From the satellite position shown in Figure 1 we see that during this time the satellite was inside the waveguide. As we believe that the LHR whistler cutoff frequency is close to the maximum frequency in the LHR profile, the noise band below that frequency clearly seen on the spectrogram in Figure 6 indicates that the LHR waveguide is fed by quasi-electrostatic waves. The fact that the noise band is more pronounced in the lower part of the waveguide frequency band is quite consistent with the mechanism of the waveguide feeding discussed in section 3. Indeed, these waves should mostly have been trapped early. Since the parameter k²c²/w_p² increases while the wave propagates in the duct, it becomes more electrostatic and, thus, more intense.

We should point out that in the same frequency band that corresponds to the frequencies in the waveguide, the parts of usual whistler traces are stressed, which are weaker (or not seen at all) out of that band.

In conclusion, in this paper we have theoretically studied the LHR waveguide feeding by quasi-resonant VLF waves propagating in the magnetosphere in a nonducted mode. The analysis of wave trapping is performed in the framework of two-dimensional geometrical optics. It is shown that the trapping of downward propagating quasi-resonant whistler mode waves into the LHR waveguide takes place when the values of local maxima of LHR frequencies along geomagnetic field lines increase with decreasing L shells. Analytical consideration is illustrated and supplemented by model ray tracing calculations. The theoretical model is tested against experimental data, and qualitative consistency is found.

Acknowledgments

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