Giant pulsations as modes of a transverse Alfvénic resonator on the plasmapause
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences;TERRAPUB. 2012
Received: 21 March 2012
Accepted: 2 October 2012
Published: 10 June 2013
The paper assumes that the giant pulsations are oscillations trapped within a resonator resulting from finite plasma pressure on the outer edge of the plasmapause. This resonator is bounded, across the L-shells, by two turning points allowing the wave energy to be channeled azimuthally. This assumption can explain the basic properties of the giant pulsations: strong localization across magnetic shells, poloidal polarization, presence of a significant compressional component in the Pg magnetic field, the fact that their frequency does not depend on the radial coordinate. The wave field structure both across the L-shells and along the field lines is studied. In order to explain the amplitude modulation it is sufficient to suppose that the resonator is excited by some non-stationary process. Generation by a moving source comprised of substorm-injected particles is considered.
Key wordsULF waves giant pulsations poloidal modes finite ³ moving source plasmapause
First identified by Kristian Birkeland in 1901, the giant pulsations (Pg) were probably the first class of ULF waves identified in the terrestrial magnetosphere. They gained their name after Bruno Rolf, who exclaimed “True giant amongst dwarfs!” when studying one of those events (Rolf, 1931). Indeed, it had an amplitude of several tens of gammas, while most of the micropulsations found by then had dozens of times smaller amplitudes. Although later the giant pulsations lost their amplitude champion title, this term was assigned to waves similar to those observed by Birkeland and Rolf featuring predominant polarization in the D-component, long duration, quasi-sinusoidal shape with some amplitude modulation. Simulation of the Pg pulsations began with the papers by Kato and Watanabe (1955, 1956) and Lehnert (1956), who identified them with MHD waves in space plasmas. Now it is well established that Pgs represent a variety of guided poloidal Alfvén waves in the terrestrial magnetosphere, that is, waves with high az-imuthal wave numbers, m ≫ 1. Besides, the term “giant pulsations” is by no means improper: being high-m waves, they are heavily attenuated by the atmosphere and must reach large enough amplitudes in space in order to be detected at the ground.
1.1 Observable features of the giant pulsations: A synopsis
Giant pulsations are usually observed in the Pc4–5 range, with amplitudes of several tens of nT at low geomagnetic activity (Ol’, 1963; Annexstad and Wilson, 1968; Brekke et al., 1987; Taylor et al., 1989). They are characterized by a sinusoidal shape with some amplitude modulation, and by a long duration of wave packets. The major part of their power is, according to data from ground magnetometers, in the D -component, which means that these waves are poloidally polarized in the magnetosphere (the field lines oscillate in the meridional direction). Spacecraft observations of Pgs often detect a considerable compressional magnetic field component as well (Hughes et al., 1979; Kokubun et al., 1989). They are usually registered in the auroral regions or just outside the plasmapause (Rostoker et al., 1979; Chisham etal., 1992; Takahashi etal., 2011), and sometimes within the plasmasphere (Green, 1985). Other kinds of poloidal Alfvén waves have been observed in the vicinity of the outer edge of the plasmapause (Takahashi and Anderson, 1992; Schäfer et al., 2008).
A necessary (although not sufficient: Klimushkin et al., 2004) condition for the Alfvén waves to be poloidally polarized is a high azimuthal wave number value, ǀmǀ ≫ 1. Indeed, direct measurements confirm that Pgs are moderately high-m waves, ǀmǀ = 15–40. The azimuthal phase velocity is westward (negative m values), that is, it coincides with the ion gradient-curvature drift direction. Some studies found that their localization region is only 600–700 km wide in the azimuthal (east-west) direction slowly drifting westward at a velocity typical of the 10–20 keV ion drift velocities (Glassmeier, 1980; Chisham et al., 1992). However, other studies failed to detect such a drift (Glassmeier et al., 1999).
Many studies also revealed that the Pgs are strongly localized across magnetic shells. Projected onto the Earth’s surface, their localization region is only 200–300 km wide in the north-south direction, which corresponds to ~1 R E in the magnetosphere (e.g., Rostoker et al, 1979; Glassmeier, 1980; Chisham et al, 1992). The frequency does not change with the L-shell (Takahashi et al, 2011). As Chisham et al. (1997) found, the maximum of the D-component is observed on the same positions as the dip of the H-component. The latitudinal profile of Pg pulsations in the D -component can be described by a Gaussian function.
The parallel (field-aligned) structure of giant pulsations is a subject of major controversy. Comparing the observed Pgs’ frequencies to those calculated in the quasi-dipole models relying on a power-law radial distribution of mass density, many authors concluded that Pgs represent the second (N = 2, or even) harmonic of the standing wave which has one node (Glassmeir, 1980; Poulter et al, 1983; Chisham and Orr, 1991; Chisham et al., 1992). On the other hand, observations at conjugate ground stations have detected the fundamental (N = 1, or odd) parallel harmonic (Green, 1979; Thompson and Kivelson, 2001). Satellite observations usually confirm the latter case (Kokubun et al, 1989; Takahashi et al., 1992, 2011; Glassmeier et al, 1999). We also prefer the latter interpretation because a simple power-law radial distribution of mass density does not adequately models the Alfvén speed variation across the plasmapause and the fact the observable Pgs are usually found just outside the plasmapause. It should be noted that Green (1985) observed three Pg events within the plasmapause and inferred the fundamental parallel harmonic by comparing the observed vs. calculated frequencies.
1.2 Modeling and the purpose of this paper
There is a general consensus that the high-m waves are generated due to various kinetic instabilities in hot components of the magnetospheric plasma. Regarding Pgs, this opinion has some observational basis. Pgs usually have the same (westward) direction of the azimuthal phase velocity as the ion gradient-curvature drift. The Pgs localization region sometimes shows a westward drift at velocities of the same order as the 10–20 keV ion drift velocities. Moreover, Chisham et al. (1992) and Wright et al. (2001) found some association between several Pg events and substorm-injected 10–20 keV particles. Another hint at the role of energetic particles in Pg generation is a close association between Pgs and Pc1 pearl pulsations (Kurazhkovskaya et al., 2004). Wave-particle interaction was also invoked in explaining Pg amplitude modulation (Pokhotelov et al, 2000a).
However, it is not clear which kind of kinetic instability may be responsible for their generation. Two kinds of instabilities are usually suggested (Southwood, 1976). The first is the bump-on-tail instability, an instability due to the nonmonotonic dependence of the ion distribution function on energy. Such a distribution can result from substorm injection, since faster protons reach a given point on the azimuthal coordinate earlier than the lower energy ones. Therefore high-energy particles are added to the local background plasma at a higher rate than low-energy particles (Glassmeier et al, 1999; Wright et al., 2001). The bump-on-tail distribution can also appear under steady state conditions due to the existence of a global magnetospheric electric field causing energy independent drift (Chisham, 1996; Ozeke and Mann, 2001). Chisham (1996) suggested that the second mechanism may provide an explanation for the rarity of Pgs and their occurrence during geomagnet-ically quiet times. Several studies reported simultaneous observations of the bump-on-tail ion distribution functions and the poloidal pulsations very similar or identical to Pgs (Hughes et al., 1979; Baddeley et al., 2005).
The bump-on-tail instability is a natural candidate if Pgs represent the second parallel harmonic (Chisham and Orr, 1991; Chisham et al., 1992). Glassmeier et al. (1999) suggested that the bump-on-tail instability may also be a source of the fundamental harmonic Pg event, but they assumed the wave to be highly asymmetric due to different ionospheric conductivities at opposite magneto-conjugated points. This suggestion caused some controversy (Glass-meier, 2000; Mann and Chisham, 2000), so this point has not been finally established.
Another possible source of energy is an instability associated with an inward ion density gradient, which can occur in the ring current region. That suggestion was usually made in studies supposing Pgs to be the fundamental parallel harmonic of standing wave (Takahashi et al., 1992, 2011).
Uncertainty in our understanding of the Pgs parallel structure leads to difficulties when attempting to determine their generation mechanism. There are also some additional difficulties. As Mager and Klimushkin (2005) showed, the instability growth rate only weakly depends on the m number. Therefore the instability cannot select a narrow range of the m numbers, although the observed waves have well-defined m values. Therefore, the instability can generate waves propagating in both azimuthal directions, thus even the direction of the azimuthal phase velocity cannot be explained (Mager and Klimushkin, 2005).
Furthermore, there are some theoretical difficulties with the instability. First, instabilities associated with the bump-on-tail distributions and plasma density gradients should not be considered as the only possible candidates for the Pg generation mechanism. Such factors as temperature gradients, pressure anisotropy can also generate the Alfvén waves (Pokhotelov et al., 1985; Klimushkin and Mager, 2011, 2012). Second, the entire instability theory has been established for the monochromatic waves (stationary wave field structure) only, although it is evident that wave generation is an impulsive process.
It is worth noting that the impulsive character of the wave generation assume that the wave is launched by a source switched on at some instant. The wave was absent before it was generated by the source, but it does not have to disappear with the termination of the source activity. If dissipation in the ionosphere-magnetosphere system is not too high, the wave generated by an impulsive source can last for quite a long time. In the case of Pgs, it can be two or more consecutive days (Rostoker et al., 1979).
A problem associated with the impulsive character of wave generation is a phase mixing phenomenon. At the initial onset of the perturbation all field lines oscillate with about the same phase and the wave field is characterized by a predominantly poloidal polarization. However, since each field line oscillates with its own eigenfrequency, the oscillations on neighboring magnetic shells rapidly acquire significant phase differences. As a consequence the wave acquires a very small spatial scale across magnetic field shells and, hence, becomes toroidally polarized to preserve the source-free nature of the magnetic field (e.g., Mann et al., 1997; Leonovich and Mazur, 1998; Klimushkin et al., 2012). Inclusion of instability into this picture only further complicates the problem. Even though the wave amplification rate decreases in the course of poloidal-to-toroidal transformation, it remains positive. Therefore the most amplified oscillations should be the toroidally polarized ones and the larger the instability, the larger the amplitude of the toroidal oscillation (Klimushkin, 2000; Klimushkin and Mager, 2004a; Klimushkin et al., 2007).
Although the phase mixing phenomenon is a firm output of the MHD theory, the expected change of the wave polarization has not been observed for the Pgs, which constitutes yet another peculiarity of these pulsations (Chisham et al., 1997). The only factor which can prevent the transformation is wave damping due to the finite ionospheric conductivity. If the wave does not have enough time for the transformation and remain poloidal in the course of all period or its existence, then the damping must be stronger than the instability. Thus, it is the ionospheric damping rather than the instability which is favorable to the poloidal polarization (Klimushkin, 2007). In this case, the role of the instability in the poloidal wave generation is not clear.
Based on earlier ideas of Zolotukhina (1974) and Guglielmi and Zolotukhina (1980), Mager and Klimushkin (2007, 2008) discussed another generation mechanism of high-m waves. This involved emission by azimuthally drifting proton clouds injected during substorms, similar to the Cerenkov emission. Other analogies include the Kelvin ship waves and lee (mountain) waves in the atmosphere. This approach avoids some difficulties of the instability theory and has a firm observational basis (Zolotukhina et al., 2008; Mager et al., 2009a; Yeoman et al., 2010, 2012). Several cases were observed when the giant pulsations appeared at some azimuthal location at the same time as did a cloud of particles injected during substorm arrival at the same spot (Chisham et al., 1992; Wright et al., 2001).
Another peculiar feature of Pg pulsations, their narrow localization across magnetic shells, is usually postulated in theoretical studies but not explained (Chisham et al., 1997). The high localization across the L-shells is not unusual for the field line resonances (Tamao, 1965), but these low-m resonances have toroidal rather than poloidal polarization (the H-component on the ground), while Pgs are poloidally polarized waves. Other kinds of poloidal waves also exhibit sharp localization across the L-shells (Singer et al., 1982; Engebretson et al., 1992; Takahashi and Anderson, 1992; Cramm et al., 2000; Schäfer et al., 2008). Besides, the maximum of the poloidal component in the field line resonance occurs at the same location as the maximum of the toroidal component, while for the giant pulsations the maximum of the poloidal component corresponds to the dip of the toroidal component (Chisham et al., 1997). A possible solution is suggested by the strong compressional magnetic field component of giant pulsations which is an indicator of the role of finite plasma pressure in ULF wave formation (Klimushkin et al., 2004). Indeed, Pgs are a subclass of the poloidal Alfvén wave, but the poloidal eigenfrequency is very sensitive to plasma pressure (e.g., Klimushkin, 1997; Mager and Klimushkin, 2002; Agapitov and Cheremnykh, 2011; Mazur et al., 2012). A combined effect of the field line curvature and finite plasma pressure can result in transverse resonators forming in some magnetospheric regions (plasmapause, ring current). The wave energy in these resonators is trapped between the conjugate ionospheres and cut-off shells, and is channeled along azimuth (Vetoulis and Chen, 1994; Denton and Vetoulis, 1998; Klimushkin, 1998). These resonator modes have a discrete spectrum determined by the field line curvature, plasma pressure, and the equilibrium current. The eigenfrequencies do not depend on the radial coordinate. Therefore, the resonator modes are not subject of the phase mixing phenomenon (Mager and Klimushkin, 2006).
Trapping of energy across magnetic shells may provide a natural explanation for the strong latitudinal localization of the poloidal modes. Under very general assumptions, the resonator can be situated on the outer edge of the plasma-pause (Klimushkin et al., 2004), which may explain the location of the giant pulsations in that region. The purpose of this paper is to explore this suggestion, consider a possible generation mechanism of the wave in the resonator, and compare with observations.
2. Main Equations
Both the toroidal and the poloidal frequencies can be plotted as functions of the radial coordinate. The equation k x (ω, x1) = 0 at fixed ω has a solution which can be found graphically as the intersection point of the function Ω PN (x1) curve and the horizontal line for the frequency ω (see Fig. 2). The magnetic shell with a radial coordinate will be referred to as the poloidal surface. Similarly, a toroidal (or resonance) surface with a radial coordinate can be introduced as a solution of the equation k x (ω, x1) = oo for fixed ω. If Ω P (x1) > Ω T (x1) and both functions are decreasing with x1, then x P (ω) > x T (ω). As can be seen from Eq. (11) and Fig. 2, the transparent region (where the mode is localized), where , generally lies between the poloidal and the toroidal surfaces (Leonovich and Mazur, 1993; Mager and Klimushkin, 2002).
3. Transverse Resonator on the Plasmapause
If the poloidal frequency Ω PN (x1) has extremes as a function of the radial coordinate, there can exist a transverse resonator (or azimuthal waveguide), that is a transparent region bounded on both sides by cut-off surfaces. In this case the wave is channeled along the azimuth and trapped between the cut-off shells, being a standing wave in the direction transverse to the magnetic shells (Vetoulis and Chen, 1994; Leonovich and Mazur, 1995; Denton and Vetoulis, 1998; Klimushkin, 1998). As is evident from Fig. 2, one such region is located in the vicinity of the plasmapause and the other can be located in the vicinity of the ring current. The eigenfrequency of the resonator is determined from the condition that the number of half-waves between the cutoff surfaces should be an integer. The radial wave vector k1 depending on the wave frequency ω leads to frequency quantization.
4. Wave Field Structure in a Resonator Excited by a Nonstationary Source
Let us further examine two different kinds of non-stationary sources: the impulsive source and the moving source comprised of drifting substorm-injected particles.
4.1 Impulse source
4.2 Moving source
5. The Wave Structure in the Direction of the Ambient Magnetic Field
In a cold plasma, when H = 0, an Alfvén wave has no turning point, that is everywhere. Thus the entire field line between the conjugate ionospheres is transparent for the shear Alfvén wave and the oscillation has a familiar sinusoidal structure.
As follows from Eq. (38), the function in this case can reverse its sign along the field line. The point l0, where , is the turning point for a poloidal Alfvén wave. The region where , is an opaque region, where the wave becomes evanescent, and the region where is transparent for waves.
When H has a maximum at the magnetospheric equator, which is usually the case, the opaque region is located in the vicinity of the magnetospheric equator, where the field line curvature reaches its highest value (Mager et al., 2009b; Mazur et al., 2012). Thus, two sub-resonators (regions I and II in Fig. 9) form bounded by the ionosphere and the turning point near the equator, ±l0. However, due to the tunneling effect part of the wave energy leak out the ionospheric sub-resonators and forms wave field with small, but finite amplitude even in the opaque region. If this region is wide enough, this amplitude is exponentially small in its center and there is practically no coupling between the sub-resonators in the Northern and Southern hemispheres. Otherwise, the oscillations are coupled and the total wave functions Φ N are composed of symmetric and antisymmetric combinations of the harmonics in regions I and II. The coupling of these two sub-resonators influences also the wave eigenfrequency.
The suggestion that the giant pulsations are oscillations trapped within a resonator on the outer edge of the plasma-pause is a plausible explanation given the narrow radial localization of these waves. Since the resonator is bounded across the L-shells by two turning points and the vector of the Alfvén wave’s electric field is proportional to the transverse wave vector, the wave must be poloidally polarized, especially near the center of the resonator. The maximum of the poloidal component should be located at the same position in the resonator as the dip of the toroidal component. The eigenfrequencies of the resonator are determined by its global properties (height, width, difference between the poloidal and toroidal frequencies inside the resonator), providing a possible explanation for the Pgs frequency being independent of the radial coordinate.
The resonator on the outer edge of the plasmapause owes its existence to the fact that the poloidal eigenfrequency is very sensitive to plasma pressure. An additional hint at the role of the pressure is the considerable compressional component of the wave magnetic field. When geomagnetic activity decreases, the plasmapause shifts away from the Earth, into higher-β regions (O’Brien and Moldwin, 2003), which offers an explanation for the occurrence of the giant pulsations in geomagnetically quiet periods.
The principal transverse harmonic of the resonator must be close to the Gaussian function, which offers a satisfactory explanation of the radial shape of the Pgs wave field. However, if the resonator is excited by some external source, its structure must be the superposition of all transverse harmonics. In order to explain the amplitude modulation it is sufficient to suppose that the resonator is excited by some non-stationary process. Both the non-stationary sources considered lead to similar temporal structures of the oscillation shown in Figs. 6 and 8. Such structure was observed earlier with radars for a moderately high-m pulsation (Wright and Yeoman, 1999), and was interpreted in terms of transverse resonator theory (Yeoman et al., 2012).
The assumption on the resonator can explain also absence of the transformation from the poloidally to toroidally polarized wave expected from the theory of the phase mixing of the Alfvén wave with the continuous spectrum (Chisham et al., 1997). As is seen from Figs. 6 and 8, there is no such transformation of the discrete resonator modes. Thus, we may conclude that the presence of a transverse resonator can at least explain the basic properties of the giant pulsations.
However, some peculiarities of Pgs cannot be directly explained in this model. Among them are localization to the dawn sector, occurrence at the equinox and at the solar minimum (Brekke et al., 1987). To include these features in the general picture, it would be necessary to advance the theory, that is, to find the dependence of the resonator properties on the geomagnetic indexes and to generalize it for the case of the azimuthally-inhomogeneous magnetosphere.
Finally, the close vicinity of Pg location to the auroral regions raises questions of the role of the ionosphere in Pg generation by means of, e.g., feedback instability (e.g., Watanabe, 2010) or changing ionospheric conductivity (Maltsev et al., 1974), although both the mechanisms are suggested as explanations of a quite different kind of ULF waves, the irregular pulsations Pi2. The field aligned currents in the auroral regions further complicate the mode structure by, e.g., influencing the poloidal eigenfrequency (Klimushkin and Mager, 2004b). The impact of this factor is unclear.
The paper assumes that giant pulsations are oscillations trapped within a resonator appearing, due to finite plasma pressure, on the outer edge of the plasmapause. The general features of this resonator that explain the basic properties of the giant pulsations.
1) The wave in the resonator is bounded across the L-shells by two turning points. The giant pulsations also have narrow radial localizations and are usually observed on the outer edge of the plasmapause where such a transversal resonator can exist.
2) The wave is predominantly poloidally polarized. At the center of the resonator, it has the poloidal component of the wave field only, while the toroidal component becomes significant closer to its edge. A similar behavior of polarization across the L-shells is observed for the giant pulsations.
3) In regions where toroidal and poloidal eigenfrequency profiles are monotonic, oscillation frequencies depend on the L-shells. Conversely, the resonator eigenfrequencies are determined by its global properties. This is a possible explanation for the Pgs frequency being independent of the radial coordinate. Constant wave frequencies over the L-shells mean that the resonator modes are not subject of the phase mixing phenomenon which explains absence of the transformation from the poloidally to toroidally polarized for the Pg waves.
4) The wave in the resonator is subject to amplitude modulation due to superposition of harmonics with close eigenfrequencies. Such modulation is observed for the giant pulsations.
5) The resonator on the outer edge of the plasmapause can exist when β is high enough and the poloidal eigenfrequency becomes higher than the toroidal eigenfrequency. The giant pulsations are usually observed in geomagneti-cally quiet periods when the plasmapause is shifted from the Earth into the higher-β regions. Moreover, they have a considerable compressional component of the wave magnetic field, which also indicates significant plasma pressure.
6) As seen from observations, the giant pulsations often represent the fundamental (N = 1) harmonic of the standing wave. In this case, plasma pressure has a strong influence on the parallel structure of the poloidal wave. The magnetic field of the fundamental harmonic must have three nodes, rather than one as in the case of cold plasma. Such a structure has not yet been observed, possibly due to difficulties of spacecraft observations along magnetic field lines.
The work was supported by RFBR grants 12-05-00121-a and 12-05-98522-r-vostok-a, Program #22 of the Presidium of the Russian Academy of Sciences.
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