Polarization properties of Gendrin mode waves observed in the Earth’s magnetosphere: observations and theory

We show a case of an outer zone magnetospheric electromagnetic wave propagating at the Gendrin angle, within uncertainty of the measurements. The chorus event occurred in a “minimum B pocket”. For the illustrated example, the measured angle of wave propagation relative to the ambient magnetic field θkB was 58 ±4. For this event the theoretical Gendrin angle was 62 . Cold plasma model is used to demonstrate that Gendrin mode waves are right-hand circularly polarized, in excellent agreement with the observations.

to determine wave polarization is based on multidimensional spectral analyses (Santolik et al., 2003).Their approach provides comprehensive analysis but invokes both magnetic and electric field measurements.In the current study the focus is on the magnetic polarization properties of the waves and thus the study will be restricted to magnetic field measurements alone.Thus, we chose the MVA method to analyze GEOTAIL data.

Observations
Electromagnetic chorus have been studied extensively in the Earth's magnetosphere (see Tsurutani andSmith, 1974, 1977;Anderson and Maeda, 1977;Koons and Roeder, 1990;Meredith et al., 2001Meredith et al., , 2003, for chorus observational statistics).These waves are believed to be responsible for the creation of relativistic electrons in the radiation belts (Horne and Thorne, 1998) and may define the structure of the belts themselves (Horne et al., 2005).Here we focus on the basic physical properties of a distinct wave mode within the chorus frequency range.
In this paper, GEOTAIL observations are used to illustrate the property of waves in the dayside outer zone region of the magnetosphere.High-resolution magnetic field measurements from the PWI and WFC instruments (Matsumoto et al., 1994;Nagano et al., 1996) are analyzed.The PWI (plasma wave instrument) contains a 3-component searchcoil with sensitivity of 1.5×10 −5 nT/Hz 1/2 .The WFC (wave form capture) receiver samples 8.7 s snapshots every 5 min between 10 Hz and 4 kHz.
Intense electromagnetic emissions were observed around 23:25:51 UT on 29 April 1993.The GEOTAIL position was (6.5, −4.3, 0.6 R E ) in GSE coordinates.GEOTAIL was at roughly noon local time in a minimum-B pocket.The Tsyganenko model (Tsyganenko, 2002) was used for reference.Chorus was observed in a low-band (ω < 0.5ω ce ) range, with Published by Copernicus Publications on behalf of the European Geosciences Union.a wave frequency f ≈ 700 Hz.In the above, ω ce is the local electron cyclotron frequency.Ambient magnetic field, B 0 , in GSE coordinates was (21.1, 57.4, 107.5 nT).Details on the GEOTAIL data analysis are presented in Verkhoglyadova et al. (2009).
Figure 1 shows the wave event.The top panels are 3 cycles of the wave and the B 1 , B 2 and B 3 variations.The bottom two panels are the wave hodograms.The left-hand panel is the B 1 -B 2 hodogram and the right-hand panel is the B 1 -B 3 hodogram.The former shows that the wave is circularly polarized (the ratio λ 1 /λ 2 is 1.1).The magnetic field direction is out-of-the-paper and the sense of the wave rotation is right-handed.The right-hand panel shows that the wave is plane-polarized.The wave was propagating at ∼ 58 • relative to B 0 .
There are several sources of errors in the MVA determinations.The first one is associated with the presence of random statistical noise within the magnetosphere.Tsurutani et al. (2009) estimated this isotropic noise to be ∼20 pT.For the above chorus subelement examples, the peak wave am-plitudes were ∼300 pT.The noise superposed on the chorus would give an angular error of ∼ 4 • .Another source of error is instrument noise (0.02 pT/Hz −1/2 , Matsumoto et al., 1994).The lack of full wave cycles (360 • of rotation) used in the analyses is a third source.These latter two errors are small compared to that of the presence of noise, so we will assume an error of ∼ 4 • for the specific events analyzed.
It is theoretically well-known that whistler waves are circularly polarized if they propagate parallel to B 0 (Stix, 1962;Helliwell, 1965).However it is not so clear for obliquely propagating electromagnetic waves.In the next section we investigate electromagnetic whistler waves propagating in the Gendrin mode and compare their properties with the above observations.

Whistler mode waves and Gendrin modes
We follow the standard approach for a two-component cold plasma (Landau and Lifshitz, 1960;Krall and Trivelpiece, 1973) to study electromagnetic waves in the whistler wave frequency range between the ion cyclotron frequency (ω ci ) and the electron cyclotron frequency: ω ci ω ω ce .We assume that ω ce ω pe , where ω pe is the electron plasma frequency.The dispersion relation for electromagnetic waves in this frequency range is: where θ is the propagation angle relative to B 0 .We introduce the electron inertial length, a e = c ω pe , and consider different limiting cases of Eq. ( 1).In the long-wave limit of (ka e ) 2 1, we obtain the classic whistler wave dispersion relation: ω = ω ce c 2 k 2 cosθ/ω 2 pe .The electron cyclotron mode ω = ω ce cosθ corresponds to a short-wave limit, or (ka e ) 2 1.
The Gendrin mode is a special mode of electromagnetic whistler waves (1) with ω = ω G = ω ce cosθ/2 that exists strictly at ka e = 1.It has unique propagation properties, i.e., it propagates at an angle θ = θ G , which is called the Gendrin angle, corresponding to a minimum value of the refractive index parallel to B 0 (Gendrin, 1961).Its phase velocity is maximum among whistler waves with different k.The mode phase and group velocities are equal: V g = V ph = ω G a e (see also Sauer et al., 2002;Dubinin et al., 2003).The Gendrin mode is "magnetically guided" in the sense that its group velocity orthogonal to B 0 is zero.Parallel group velocity of whistler waves (1) is highest if they propagate under the Gendrin angle.Here we should note that the result by Storey (Storey, 1953;Stix, 1962) on the maximum angle of ∼ 19 • 28 between V g and B 0 was obtained for the long-wave limit of Eq. (1) (i.e.classic whistler only) and is not applicable for Gendrin modes.According to Gendrin (1961), Gendrin modes with all frequencies ω G (defined by θ G ) propagate with the same phase and group velocities along B 0 (V g|| = V ph|| = ω ce a e 2 ).

Fig. 2. Wave diagram for Gendrin modes.
There is unique relationship between the propagation angle and (local) wave frequency for each of the discrete modes.Note that the mode frequency is bounded by lower-hybrid (not shown) and half-electron cyclotron frequencies.
Since the Gendrin mode frequency is defined by the propagation angle at fixed k, the mode is therefore non-dispersive.This point is illustrated by the relationship between the Gendrin mode frequency and propagation angle shown in Fig. 2.There is one mode for each value of θ G .It should also be noted that Gendrin modes exist only in the ω ≤ ω ce /2 frequency range.For a lower-frequency part of the range of ω < 0.3ω ce , the Gendrin modes can be highly oblique with θ G > 50 • .
Since we are restricting ourselves to electron waves only, ion contributions are ignored in the dispersion (1).In other words, our results are valid for relatively fast wave processes that do not involve ion motions and the lower-hybrid frequency is a lower limit for the waves/modes considered in this paper.Namely, the frequency range for Gendrin modes is from ω > ω ce √ m e /m i ≈ 0.02ω ce to 0.5 ω ce .The Gendrin angle (θ G ) calculated for the above observational example from GEOTAIL is 62 • whereas the measured θ kB was 58 • ±4 • .Thus, it is possible that the electromagnetic wave was propagating at the Gendrin angle within measurement uncertainties, i.e., it was a Gendrin mode wave.

Polarization analysis
Below, we analyze the polarization of the magnetic component B j of the Gendrin mode from a theoretical point of view.A standard coordinate system is assumed: the background magnetic field B 0 is directed along the Z-axis and the electromagnetic wave propagates in the (XZ) plane (axes shown in Fig. 3 in black).The index j denotes components along X, Y and Z directions.Assuming ω ci ω ω ce , the nonzero components of the Hermitian tensor of the dielectric permittivity for cold magnetized plasma (Krall and Trivelpiece, 1973;Stix, 1962) take the form: Using Maxwell's equations and Eq. ( 2) it is straightforward to find the corresponding polarization relations for a plane wave with dispersion (1): where Here we introduce the parallel and perpendicular (relative to B 0 ) components of the wave vector, k ⊥ = k sinθ, k || = k cosθ.For Gendrin modes, we modify Eqs. ( 4) and (3) by using ω = ω G and k = 1/a e explicitly: We examine the wave polarization in a corresponding MVA frame, where the new Z -axis is aligned in the direction of wave propagation k.To find this coordinate frame, we perform a linear transformation of the coordinate system (XY Z) → X Y Z by rotating it through an angle −θ G (or anti-clockwise) about the Y-axis (see Fig. 3).Following Korn and Korn (1961;Eqs. 14.10-18b), we perform this transformation of the wave magnetic field B = T B with the matrix: For the wave components we obtain with Eqs. ( 5) and ( 6): Thus the Gendrin mode is right-hand circularly polarized (B x = iB y , B z = 0) in a plane normal to the wave propagation direction.Note that the mode propagation can be highly oblique and the polarization plane is generally not orthogonal to B 0 .The Gendrin mode polarization was derived for a cold plasma model.Kinetic effects are negligible for this frequency range wave for spatial scales larger than electron gyro-radius.This condition is satisfied for Gendrin modes (spatial scale ∼ a e ), assuming typical plasma parameters in the region of GEOTAIL observations.

Discussion and conclusion
Low-band frequency (f < 0.5ω ce ) electromagnetic waves observed in the outer region of the Earth's dayside magnetosphere can propagate at highly oblique angles to B 0 (Goldstein and Tsurutani, 1984).We presented GEOTAIL observations showing an example of a right-hand highly oblique circularly polarized electromagnetic wave propagating at Gendrin angle within measurement uncertainties (see also Tsurutani et al., 2009).We suggest that it is a Gendrin mode wave.
The Gendrin mode is a distinct mode of electromagnetic whistler waves (Eq.1).These modes exist only at frequencies below ω ce /2 and above the lower-hybrid frequency.Based on a cold plasma model, we have demonstrated theoretically for the first time that the wave magnetic field for the Gendrin mode is right-handed and circularly polarized.This theoretical result is in excellent agreement with results of the data analysis presented here, in Tsurutani et al. (2009) and in Verkhoglyadova et al. (2009).
Gendrin modes have unique propagation properties.They are non-dispersive and are "magnetically guided" so that their group velocity along background magnetic field is maximum if the wave propagates at the Gendrin angle.According to Gendrin (1961): "Thus for each frequency there is an emission angle θ = 0, such that the beam is strictly propagated along the line of magnetic force, at a velocity independent of the frequency."Thus, these modes could be responsible for non-ducted electromagnetic wave propagation in the low-band range (Lauben et al., 2002;Helliwell, 1995).There is observational evidence that electromagnetic waves originated in the outer region of the Earth's dayside magnetosphere can propagate to the ground (Spasojevic et al., 2008).It is possible that Gendrin mode waves may be responsible for these ground observations and we suggest that further modeling efforts be undertaken to determine if this is the correct interpretation or not.However this effort is beyond the scope of the present paper.

Fig. 1 .
Fig. 1.Minimum variance analyses of a section of a subelement/packet.The event occurred at 23:25:51 UT on 29 April 1993.In the hodogram the direction of B 0 is indicated, which in this case is out-of-the paper.The wave is noted to be right-hand circularly polarized, planar, and propagating at ∼ 58 • relative to B 0 .

Fig. 3 .
Fig. 3. Coordinate systems used.In a standard coordinate system (axes shown in black), B 0 is directed along the Z-axis and an electromagnetic wave propagates in the (XZ) plane.The MVA frame axes X Y Z , where the new Z -axis is aligned in the direction of wave propagation k, are shown in blue.The wave magnetic field is polarized in the X Y plane.