A numerical simulation of the Pi2 pulsations associated with the substorm current wedge

JOURNA OF GEOPHYSICA RESEARCH, VO. 17, NO. A3, 134, 1.129/21JA9137, 22 A numerical simulation of the Pi2 pulsations associated with the substorm current wedge S. Fujita, 1,2 H. Nakata, 3 M. Itonaga, 4 A. Yoshikawa, and T. Mizuta 6 Received 12 January 21; revised September 21; accepted September 21; published 9 March 22. [1] The present paper deals with the transient behavior of MHD perturbations in the inner magnetosphere induced by an impulsive localized eastward current (source current) as a model of Pi2 pulsations in the magnetosphere. The magnetospheric model consists of a dipole magnetic field, plasmasphere, with Pedersen conductivity, and a free outer boundary. The source current is an impulsive magnetospheric current at the onset of the substorm current wedge and is distributed around the ial plane of = 1 with ±2 hour longitudinal extent around midnight. The numerical results allow us to track variation in the expected Pi2 pulsation signals in both local and. The poloidal-mode wave exhibits plasmasphere virtual resonance, resulting in large amplitudes around midnight, weakening toward dayside. The toroidal-mode wave is excited as a field line resonance immediately after the wave front of the poloidal-mode wave crosses regions where the radial gradient of V A is steep. The toroidal-mode wave has largest amplitude at the local of the east/west edge of the source current. The duration of this wave is min. In the middle plasmasphere where the radial gradient of the V A is smaller, the poloidal-mode wave tends to predominate over the toroidal-mode wave. These numerical results are consistent with satellite observations, in so far as the day-night asymmetry of Pi2 pulsations and the observation of transient toroidal waves. INDEX TERMS: 272 Magnetospheric Physics: MHD waves and instabilities; 273 Magnetospheric Physics: Numerical modeling; 2768 Magnetospheric Physics: Plasmasphere; KEYWORDS: Pi2 Pulsation, Current Wedge, Plasmasphere Cavity Resonance, Numerical Model, Coupled Waves 1. Introduction [2] As Pi2 pulsations are associated with substorm onset, they are regarded as the magnetohydrodynamic (MHD) waves generated by an impulse due to a sudden change in the magnetic configuration in the near-earth magnetotail [e.g., Yumoto et al., 1989, 199; Yumoto, 199]. Since the substorm onset is synchronized with development of the current wedge, the phenomenological current wedge model [McPherron et al., 1973; Pashin et al., 1982; Sakurai and McPherron, 1983; Yumoto, 199] proposes that a MHD impulse launches the Pi2 pulsations. Takahashi et al. [199] presented a qualitative model of Pi2 pulsations in which the crosstail current disruption (associated with the substorm onset) launches fast-mode disturbances and (they propagate earthward) excites cavity-mode oscillations in the plasmasphere. It is important to investigate whether numerical simulation of the disturbances produces oscillations similar to the observed Pi2 pulsations. [3] et us summarize the observed features of the Pi2 pulsations. Recent study of the Pi2 pulsations reveals that the highlatitude Pi2 pulsations and the middle- and low-latitude ones behave differently. Namely, Pi2 pulsations at high latitudes, in 1 Meteorological College, Kashiwa, Japan. 2 Department of Geophysics and Planetary Sciences, Kyoto University, Kyoto, Japan. 3 Solar-Terrestrial Environment aboratory, Nagoya University, Toyokawa, Japan. 4 Faculty of Education, Yamaguchi University, Yamaguchi, Japan. Department of Geophysics and Planetary Sciences, Kyushu University, Fukuoka, Japan. 6 Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan. Copyright 22 by the American Geophysical Union. 148-227/2/21JA9137 general, have lower frequencies (1 ), and those at middle and low latitudes tend to have higher frequencies (1 2 ) [i et al., 1998]. It has also been noted that the frequency varies with latitude at auroral latitudes but not at middle and low latitudes. The latter behavior is explained by the cavity-mode oscillation, or the plasmasphere virtual resonance (PVR) according to the terminology of ee and Kim [1999]. Here the term virtual is used because the plasmapause is not a perfect barrier to MHD waves [Fujita and Glassmeier, 199]. This idea is supported by the following observational facts: (1) Pi2 pulsations with a constant frequency at latitudes corresponding to the plasmasphere [Yumoto et al., 199; Sutcliffe and Yumoto, 1991], (2) a spatial wave field structure standing in the plasmasphere [Takahashi et al., 1992; Takahashi, 1994], (3) a 18 H component phase shift across the plasmasphere [Yeoman and Orr, 1989], and (4) agreement between the observed frequency and theoretically obtained eigenfrequencies of the plasmaspheric cavity mode [in et al., 1991; Nosé, 1999]. Recently, Takahashi et al. [199] observed the cavity-mode like behavior of the Pi2 signals in the inner magnetosphere. In addition, local variation of Pi2 pulsations has been observed [Takahashi et al., 1996; i et al., 1998]. The Pi2 pulsations are mainly observed at night at middle and low latitudes [e.g., i et al., 1998] and show characteristic local dependence in the magnetosphere (i.e., the transient toroidal wave discussed by Takahashi et al. [1996]). Consequently, the Pi2 pulsation current wedge model should explain the formation of PVR, latitudinal variation of Pi2 pulsation behavior, and its local variation. [4] et us review the previous theoretical and modeling works. The previous works mainly considered formation of PVR by an impulse [Allan et al., 1986; ee, 1996, 1998; Itonaga et al., 1997a, 1997b; Pekrides et al., 1997; ee and Kim, 1999; ee and ysak, 1999]. These studies indicated that an impulsive electric field change occurring somewhere in the magnetosphere induces PVR because the magnetosphere has a plasmaspheric structure. SMP 2-1

SMP 2-2 FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS Earth Jfield-aligned 18h - fast wave Jpolarization wave front source current Alfven wave (field line) + 6h Figure 1. An illustration of the generation mechanism of a Pi2 pulsation. The eastward source current is closed via the polarization current associated with the fast-mode wave that spreads toward the inner magnetosphere as well as toward the magnetospheric tail (not shown). The polarization charge associated with the source current also generates a field-aligned current flowing into the (the Alfvén wave). This field-aligned current is partially reflected by the. The fast-mode wave also produces the Alfvén wave, owing to the nonuniformity of the magnetosphere. Recently, ee and ysak [1999], using the dipole magnetosphere model with a plasmasphere and realistic boundary conditions at the magnetopause, calculated magnetospheric response to an electric field impulse in the near-tail magnetosphere. They found in their numerical results that the compressional disturbances in the nightside plasmasphere are likely causes of Pi2 pulsations. However, their PVR has a significantly large amplitude on the dayside, contradicting observations of nightside localization of Pi2 pulsations in the inner magnetosphere [Takahashi et al., 199]; their numerical simulation gave PVR that had significantly large amplitude in the day. They attributed this discrepancy to an unrealistic boundary condition at the in their model. To summarize, previous theoretical works showed that an impulse could produce PVR, but they did not explain the latitude-dependent features of Pi2 pulsations nor consider explicitly the source of the impulse. [] It is important to consider explicitly the source of the impulse not only to see if it would be consist with the current wedge model but also to see if it gives the correct signature for the Pi2 pulsation sources. For example, seismic waves can be used to estimate the location and nature of the wave source, and in the same way, Pi2 pulsations can be used to identify the location and nature of the source. The first attempt to do this was by Fujita et al. [2], who employed a longitudinally uniform source current model instead of the more realistic current wedge model. The source current in this study was taken to be the eastward magnetospheric current that is associated with cross-tail current disruption. They studied the transient behavior of the response to the impulse of the inner magnetosphere with its dipole magnetic field and plasmaspheric structure. This magnetosphere has a wave-absorbing layer in the outer parts as a boundary condition. On the basis of this model, they found that PVR with a typical Pi2 waveform is produced when the source current is located at 1 and has a characteristic scale as small as several tens of seconds. Thus it was found that the current wedge location could be estimated from the waveforms of the Pi2 pulsations. [6] Fujita et al. [21] (hereinafter referred to as Paper 1) developed a two-dimensional (2-D) model in which the source current and the wave field are assumed to vary as e imj, where j and m are the longitudinal angle and the longitudinal wave number, respectively. The longitudinally nonuniform structure of the wave field and radial nonuniformity of the model magnetosphere lead to coupling between the fast-mode and the Alfvén waves. Paper 1 used a model that has only the Pedersen conductivity. They found that the ionospheric conductivity controls not only the damping of the field line resonance oscillation (FR) but also the eigenfrequencies. The numerical results also show that behavior of the Alfvén wave depends on latitude; this wave is a standing wave along field lines in the plasmasphere but is a bouncing wave between the at high latitudes. [7] We extend here the previous 1-D and 2-D models to a full three dimensions. Indeed, the day-night asymmetry of Pi2 pulsations [e.g., i et al., 1998] and peculiar local dependence of the transient toroidal wave [Takahashi et al., 1996] require a 3-D model. [8] This paper is organized as follows. Section 2 explains the model, basic equations, and the numerical technique. In section 3 the numerical results are presented and compared with observation. We then address possible future improvements to this Pi2 model. Section 4 summarizes our major findings. 2. Model and Basic Equations [9] The numerical technique and magnetosphere- models used in this paper are the same as those used in Paper 1 except for localization of the source current. This section provides a brief description of the model and basic equations. Then we discuss the characteristics of the numerical scheme employed.

FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS SMP 2-3 6 outer boundary z (Re) 4 2 inner boundary x (Re) 1 ial 1 plane Figure 2. Grid point distribution in a meridional plane. Value z is the coordinate from the South Pole to the North Pole, and x is perpendicular to z. The inner and outer boundaries are located at = 3 and 16.2. The is located at 2. The solid lines trace dipole magnetic field lines, and the grid points trace coordinates orthogonal to these lines. 2.1. Model [1] Figure 1 illustrates the generation mechanism of Pi2 pulsation considered here. The eastward current generates the Pi2 pulsation. This current has a scale (approximately a few tens of seconds) much smaller than that of a substorm (approximately a couple of hours). This source current would produce a polarization charge in the east/west wedge of the current, which is neutralized by the polarization current. The source current also produces the fast-mode wave front that propagates into the inner magnetosphere as well as toward the magnetospheric tail (not shown here). This fast-mode wave is partially trapped in the plasmasphere as PVR. The fast-mode wave also excites the Alfvén wave, owing to the nonuniformity of the magnetospheric plasma. At the same, dj is generated in the east/west edge of the source current. Namely, the Alfvén wave is generated there. This Alfvén wave is reflected by the and bounces between the s of the two hemispheres. Note that the evolution of the source current represents only the beginning part of what happens in a substorm. [11] The model configuration and grid point distribution at a fixed j are shown in Figure 2. The model magnetosphere has the plasmasphere and the with only the Pedersen conductivity (see Figure 1 of Paper 1). The ial distribution of V A is adopted after Pekrides et al. [1997]. The is assumed to be perpendicular to the dipole magnetic field lines. The outer boundary of the magnetosphere ( = 16.2) is a free boundary for the wave. This assumption is somewhat oversimplified because the dayside magnetosphere has the magnetopause boundary. (This will be discussed in section 3.) The inner boundary is a rigid boundary. The inner boundary was placed at = 3 because of the comparatively small volume of the region inside and because of the increase in V A there should have relative little effect on the PVR. The nodes were positioned on the grid points in the m n space, where m and n are dipole coordinates, m = cosq/r 2, n =sin 2 q/r, and q is the colatitude from the North Pole. We used 3 (radial) 2 (field-aligned, in one hemisphere) 24 (longitudinal) nodes. We confirmed that the use of finer radial grid points does not significantly affect the numerical results. 2.2. Basic Equations and the Numerical Scheme [12] The height-integrated Pedersen conductivity ( P ) is given day-night asymmetry as follows: X PðjÞ ¼ X P;day 1 þ cosj þ X P;night 2 1 cosj ; ð1þ 2 where m V P,day =1( P,day = 2 mho for V = km s 1 ) and m V P,night =1( P,night = 2 mho). (V is V A in the ial plane at = 1.) No latitudinal variation is assumed. We assume de j = at the and no downward Poynting flux below the as discussed in section 3. The Joule loss ( P de 2 n ) dissipates field-aligned Poynting flux toward the. Energy balance derived from the cold MHD equations and the boundary conditions is given by Z ms Z dvðrde? ÞðrdE? Þþ ms dv 1 @ 2 de? VA 2 de? @t 2 Z þ ds X Z @de n @J S ðr; tþ PdE n ¼ m is @t dvde? : ð2þ ms @t The third term on the right-hand side comes from the ionospheric Joule dissipation due to the Pedersen current. A Galerkin formulation of (2) is employed to facilitate further numerical analysis [Fujita and Patel, 1992]. Value J S in (2) is the source current, which is expressed as @J S @t ¼ ^j @J j @t ¼ ^jj S j ðm; n; jþtt ðþ; where j is the longitudinal angle from 12 T and J is the intensity of the source current. S j (m, n) and T(t) are S j ðm; n; jþ ¼ e m2 =m 2 e ð n ncent ð3þ Þ2 =n 2 e ð j j centþ 2 =j 2 ; ð4þ Tt ðþ¼ t T 2 e t=t : ðþ

SMP 2-4 FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS 7.4 6.2.64 (a) h.1 7.4 6.2.64 (b).1 7.4 6.2.64 1 1 2 2 3 3 (c) 18h.1 7.4 6.2.64 1 1 2 2 3 3 (d).1 7.4 6.2.64 1 1 2 2 3 3 (e).1 1 1 2 2 3 3 1 1 2 2 3 3 Figure 3. Waveforms for de j in the ial planes of = 3.44 at (a), (b) 21, (c) 18, (d) 1, and (e) 12 T. The vertical bar in the upper right corner of each panel indicates an amplitude of.1. The positive deviation in this figure signifies a westward change in de j. We use n cent =.1 ( cent = 1), j cent = 18 (midnight), m =.1 (field-aligned extent 2 R E ), n =.1 (radial extent 2 R E ), j =3 (2 hours), and T =. ( 6 s). The meridional cross section of S j (m,n,j = 18 ) is shown in Figure 3 of Fujita et al. [2]. The longitudinal extent of the source current is 1 R E. [13] All physical values are normalized using R E = 63 km and V (V A in the ial plane at = 1) = km s 1. Electric and magnetic fields are normalized with respect to m V R E J and m R E 2 J /V, respectively. Value m J is fixed as 1 when presenting the numerical results. [14] An implicit scheme (Watson s q method) was used instead of the modal analysis [Fujita et al., 2] to calculate the temporal evolution of de? (r,t) because the modal analysis does not yield adequate results when coupling resonance occurs as described in Paper 1. In Watson s q code, de? (r, t i + 1 )(t i + 1 = t + t) is obtained from de? (r,t i ) and de? (r, t i + qt), where t is step of numerical integration, q = 1.4, and t =.1(.13 s). This scheme filters out perturbations with scales shorter than t. 2.3. Relation Between the Poloidal (Toroidal) Mode and the Global (ocalized) Mode [1] We make the distinction between toroidal and poloidal modes in discussing the simulated MHD oscillations. The toroidal mode refers to azimuthal magnetic field line perturbations, whereas the poloidal mode refers to radial magnetic field line perturbations. Because the perturbed electric field is perpendicular to both the mean magnetic field and the bulk velocity perturbations, the electric field of the toroidal (poloidal) mode is directed radially (azimuthally). Since our numerical code yields the toroidal- and poloidal-mode perturbations of electric fields, we need to examine

FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS SMP 2-7.4 6.2.64 7.4 6.2.64 7.4 6.2.64 (a) h 1 1 2 2 3 3 (c) 18h 1 1 2 2 3 3 (e) 1 1 2 2 3 3 (b).1.1 7.4 6.2.64 1 1 2 2 3 3 (d).1.1 7.4 6.2.64.1 1 1 2 2 3 3 Figure 4. Waveforms for de n in the ial planes of = 3.44 at (a), (b) 21, (c) 18, (d) 1, and (e) 12 T. The vertical bar in the upper right corner of each panel indicates an amplitude of.1. The positive deviation in this figure signifies an outward change in de n. how the poloidal- (toroidal-) mode wave relates to the global- (localized-) mode wave usually used in coupled oscillation theory. [16] et us first explain pure modes. The axisymmetric (azimuthally uniform) perturbations in the Earth s dipole magnetosphere are decoupled into two wave modes; the fast-mode wave (compressional wave) and the Alfvén wave (transverse wave). The fastmode wave has poloidal and compressional (db ) perturbations. The Alfvén wave has toroidal perturbation. The fast-mode wave is called the global-mode wave. Its frequency is quantized when it is trapped in a closed system such as the Earth s plasmasphere (PVR). The Alfvén wave is called the localized-mode wave. Its frequency is the eigenfrequencies of field lines, which depends on latitude. [17] When the perturbations have azimuthal structures, the fastmode wave and the Alfvén wave are coupled. These two waves then consist of both the poloidal and the toroidal perturbations. Coupled oscillation theory [e.g., Southwood, 1974] indicates that the monochromatic global-mode wave has enhanced poloidalmode perturbation on the resonant field line where the frequency of the localized-mode wave matches that of the global-mode wave. Strictly speaking, de n (db j ) has 1 singularity, and de j (db n ) has singularity at the resonant field line for a monochromatic wave without wave damping. On the resonant field line both the toroidaland poloidal-mode waves behave as localized-mode waves. Note that the first singularity is stronger than the second. In addition, Fujita and Patel [1992] demonstrated that the latitudinal width of the enhanced poloidal-mode wave is narrow. Their numerical analysis showed that the poloidal-mode perturbation of the monochromatic coupled wave exhibits global-mode behavior except

SMP 2-6 FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS =.1.8.6.4.2 2 3 4 h 18h.T..1.8.6.4.2 =7.4 2 3 4 h 18h.T..1.8.6.4.2 =.64 2 3 4 h 18h.T..1.8.6.4.2 =3.44 2 3 4 h 18h.T. Figure. ocal variations of the power spectra for de j in the ial plane at = (source region), 7.4 (outside the plasmasphere),.64 (plasmapause), and 3.44 (inner plasmasphere). The vertical axis ranges from to.1, as shown. along the resonant field line and that the toroidal-mode wave appears only on the resonant field line. A wave-damping mechanism, such as ionospheric Joule dissipation, eliminates the singularity [Tamao, 196; Southwood, 1974]. This means that the perturbation is not enhanced on the resonant field line when ionospheric dissipation is included. [18] As for a nonmonotonic input, numerical treatment of the coupled wave done by ee and ysak [1989, 1999] and Pekrides et al. [1997] showed that the toroidal-mode wave basically exhibits localized-mode behavior and that the poloidal-mode wave exhibits global-mode behavior. Indeed, the power spectral analysis presented by ee and ysak [1989] shows that B radial (db n in the present paper) exhibits the global-mode behavior with only small traces of localized-mode behavior. However, ee and ysak [1989] did not assume a wave-damping mechanism. When a wave-damping mechanism is included, B radial in Figure 3 of ee and ysak [1999] exhibits almost pure global-mode behavior with no traces of localized-mode behavior. The numerical results of Pekrides et al. [1997] also demonstrated, from the waveform and power spectra of the toroidal- and poloidal-mode waves, that the toroidal- (poloidal-) mode wave exhibits the localized- (global-) mode behavior, respectively. Therefore, on the basis of these previous numerical results, it is possible to conclude that the toroidal-mode wave and the poloidal-mode wave represent basically the localized-mode wave (the Alfvén wave) and the globalmode wave (the fast-mode wave), respectively. 3. Results and Discussion [19] First, we describe the numerical results and then compare them with observations. Possible future improvements are addressed at the end. 3.1. ocal Time Variation of DE? [2] The essential features of the de? waveform are similar in the 2-D and 3-D models with the exception that the latter can track the variation in local. Accordingly, we discuss just this variation. 3.1.1. ocal variation of the waveforms for DE?. [21] Figures 3 and 4 illustrate the de j and de n waveforms in the ial plane from = to 3.44 from hours

FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS SMP 2-7.1.1. = h =7.4.1. h 2 3 4 18h.T. 2 3 4 18h.T. =.64 =3.44.2.2.1 h.1 h 2 3 4 18h.T. 2 3 4 18h.T. Figure 6. ocal variations of the power spectra for de n in the ial planes at = (source region), 7.4 (outside the plasmasphere),.64 (plasmapause), and 3.44 (inner plasmasphere). (midnight) to 12 hours (noon) at 3-hour intervals (duskside) based on the 3-D model. [22] The de j waveforms at midnight (Figure 3a) are essentially the same as those shown in Figure 2 of Paper 1. It is evident from Figure 3 that the de j wave front mainly propagates toward the Earth at midnight ( T) in the ial plane and that the oscillation of de j in the inner magnetosphere ( < 6, plasmasphere) is sinusoidal. This is a PVR. (Strictly speaking, the PVR of de j extended beyond the plasmapause from 21 T of Figure 3.) Appearance of poloidal-mode PVR in the plasmasphere is consistent with the work of Pekrides et al. [1997] and ee and ysak [1999]. In our calculations the intensity of the wave is greatest at midnight, and the smallest amplitude in the inner plasmasphere occurs at noon. ater, we shall show the local variation of the power spectra. [23] As only de j is discussed in the present paper, we will also describe the magnetic field perturbation of the PVR. The variation of db n is essentially the same as that of de j near the ial plane. Note that it disappears in the ial plane because the present model assumes an electric perturbation symmetric with respect to the. As found by Fujita et al. [2], db has a node in the plasmapause region. [24] et us examine de n in Figure 4. The toroidal-mode wave exhibits FR in the plasmapause ( < < 6) and near the inner plasmasphere ( < 4), where the gradient of V A is relatively steep. This result is also consistent with the work of Pekrides et al. [1997] and ee and ysak [1999]. Its duration is min. Since the source current is assumed to be symmetrical with respect to the noonmidnight meridian and the model magnetosphere is azimuthally uniform, de n is zero at midnight and at noon. Since the polarization current (Figure 1) flows in just the east-west direction at the midnight meridian, the perturbation is regarded as uniform in the azimuthal direction. Therefore we expect that de n disappears there. The situation is the same at noon. Among the local s shown in Figure 4, de n has the largest amplitude at 21 T. This local zone is near to the east/west edge of the source current. Note that the waveforms for de n at 21 T are similar to those in Figure 2 of Paper 1. As noted in Paper 1, FR in the plasmasphere is stimulated almost simultaneously with the passing of the wave front of the initial impulse. 3.1.2. ocal variation of the power spectra for DE?. [2] We calculated the power spectra for de? using the fast Fourier transform (FFT) technique at every hour. Figures and 6 illustrate local variations in the power spectra for de j and de n in the ial plane, respectively. Data from t =to s without any filters were used for the power spectrum analysis. Note that the power spectra for de j ranges from to

SMP 2-8 FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS.6.4.2 =.4.3.2.1 =7.4 2 3 4 2 3 4.2.1 =.64.8.6.4.2 =3.44 2 3 4 2 3 4 Figure 7. Variations of the power spectra for de j in the midnight meridian at = (source region), 7.4 (outside the plasmasphere),.64 (plasmapause), and 3.44 (inner plasmasphere) along magnetic field lines..1 in Figure and that the precision of the spectral analysis may not be sufficient, owing to the damped waveform of the initial impulse. [26] It is evident from Figure that de j is confined around midnight in the source region ( = ) and outside the plasmapause ( = 7.4). At =.64 (the plasmapause) the power is enhanced at 14. This enhanced component extends from midnight to 18 T. There appears to be a small spectral enhancement at 18 in the dayside plasmasphere. In the inner plasmasphere ( = 3.44) the spectral peak at 2 expands to all local regions. Note that the strongest power appears at midnight, even in the inner plasmasphere. The MHD signal emitted by the source current tends to pass around the Earth toward the dayside in the inner plasmasphere. Eventually, PVR is induced in the plasmasphere. Therefore the poloidal-mode wave has a comparatively flat spectral power distribution in local. Probably, day-night asymmetry in the amplitude of de j in the plasmasphere depends on the longitudinal extent of the source current. When the source current is more confined than the present example, de j in the day may be hardly detectable. The frequency of PVR in the plasmasphere (2 ) is near to the observed upper limit of the middle- and low-latitude Pi2 pulsations [i et al., 1998]. Note that this frequency is smaller when the inner boundary is located at smaller. [27] The power spectra for de n in Figure 6 exhibits the largest amplitude in the local region away from midnight. In the source region ( = ) there are three clear spectral peaks (4, 12, and 2 ) with maximum amplitude in the edge of the source current (22 T). The two components with the lowest frequencies are largely enhanced in the night region, whereas the third harmonic component spreads to all local regions. Note that de n at the source latitude shows a bouncing wave between the s in two hemispheres (Figure 3 of Paper 1). When the behavior of the wave is investigated by using only the power spectra, we only obtain the spectral peaks and not the dynamical behavior (a wave bouncing between s). Therefore it is essential to look at the waveform itself for phenomena such as Pi2 pulsations. Outside the plasmasphere ( = 7.4) there appears the fundamental and second harmonic components (8 and 26 ) with frequencies larger than those at =. This is attributed to an increase in the eigenfrequencies of FR. There is only one spectral peak (22 ) in the plasmapause region (at =.64). This feature is consistent with the sinusoidal variation of de n shown in Figure 4. In the inner plasmasphere ( = 3.44) there are two spectral powers at 14 and 2. The latter comes from PVR at this frequency (see de j at = 3.44 in Figure ). [28] Figure 7 shows field-aligned variations of the power spectra for de j in the midnight meridian. PVR with 2, which is evident in the inner plasmasphere ( = 3.44), is the fundamental mode along field lines. In addition, Figure 3 shows that PVR has the fundamental mode structure and period of s (=2 ) at T in the region of 3 < < 6. Therefore these two results reveal that PVR with a period of 2 is the fundamental mode (lowest frequency) oscillation. [29] Figure 8 illustrates the field-aligned variations of the power spectra for de n in the 21 T meridian. It is clear that there are four spectral peaks (4, 12, 2, and 28 ) at the source latitude ( = ). The field-aligned profiles of these four components show fundamental, second, third, and fourth harmonic oscillations, respectively. The fourth harmonic wave is not clear in Figure 6, being weak in the ial plane. It is characteristic that the second and higher harmonic oscillations of de n tend to have enhanced intensity near the at the higher latitudes ( = and 7.4). Additionally, the fundamental harmonic oscillation has an almost constant intensity along the field lines. [3] From the spectral analysis the poloidal- and toroidal-mode waves represent basically the global- and localized-mode waves, respectively, in our numerical analysis. This is consistent with

FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS SMP 2-9.4.3.2.1 =.1. =7.4 2 3 4 2 3 4 =.64 =3.44.2.1.2.1 2 3 4 2 3 4 Figure 8. Variations of the power spectra for de n in the 21 T meridian at = (source region), 7.4 (outside the plasmasphere),.64 (plasmapause), and 3.44 (inner plasmasphere) along magnetic field lines. previous numerical analyses [Pekrides et al., 1997; ee and ysak, 1999]. 3.1.3. Ionospheric effect. [31] Paper 1 showed that P controls the behavior of the MHD waves. However, Paper 1 did not investigate whether P affects local variation of the MHF waves. Therefore, let us check it. [32] We show the power spectra of de j by using m V P,night =.1 and m V P,day = 1 in Figure 9, although this is smaller than a typical night value. As for PVR in the plasmasphere, it can be seen from Figures and 9 that amplitude of PVR is reduced in the day when the conductivity is small. Figure 1 depicts the local variations of the power spectra for de n for the case of lower conductivity. Please note the difference in the vertical axes shown in Figures 6 and 1. We notice that the relative variation against the local is not so influenced by P, although the power is reduced when the conductivity is smaller. This is due to severe ionospheric dissipation, as shown in Figure 2 of Paper 1. Note that the variation in the FR s eigenfrequency with P is not so evident in the present numerical simulation as compared with Paper 1 and that the eigenfrequency of FR is smaller in association with a decrease in P. These discrepancies may be due to higher dayside P and coarse mesh distribution along a field line in the present simulation. 3.1.4. Polarization. [33] Figure 11 depicts local variations in the polarization of the electric field perturbations in the ial plane from = to 3.44. We use the data in the s interval. Electric field perturbations polarized linearly in the east-west direction are observed at midnight. At 23 and 1 T the first large impulse invokes an east-west polarized perturbation. After this large initial impulse the electric field perturbations in the region outside the plasmapause show almost circular polarization. In the local region away from midnight the major axis of the polarization ellipse is oriented in the radial direction. Polarization features are somewhat different in the plasmapause and plasmasphere regions. In the plasmapause region where FR is enhanced (6 < < ), radially polarized electric field perturbations are evident. Enhanced generation of FR in the region of steep radial gradient of V A (the plasmapause and inner plasmasphere) is consistent with the work of Pekrides et al. [1997]. In the middle plasmasphere where the spatial gradient of V A is small ( < < 4), FR is not generated effectively by wave coupling. Thus a comparatively large PVR induces a polarization in the east-west direction. [34] ocal variations of the polarization for @db? /@t in the plane of colatitude 7 from = to 3.44 are shown in Figure 12. Since @db? /@t vanishes in the ial plane, the polarization is shown off the ial plane. Essentially, the polarization direction is perpendicular to that for the electric field perturbation. Other features are the same as those for the electric field perturbations. 3.2. Geophysical Implications of the Present Numerical Results 3.2.1. ocal dependence of Pi2 occurrence in the inner magnetosphere. [3] Takahashi et al. [199] reported that the compressional component observed in the plasmasphere is highly correlated with Pi2 pulsations on the ground, whereas the compressional component detected in the dayside plasmasphere is less correlated with ground Pi2 pulsations. They concluded from this analysis that the nightside Pi2 pulsations do not propagate to the dayside plasmasphere. Further, they also concluded that Pi2 pulsations in the plasmasphere are likely a plasmaspheric cavity mode (denoted PVR in this paper). It is noteworthy that the day UF waves with the frequency of the Pi2 pulsations possibly have both the Pi2 pulsation component and the day Pc3 and Pc4 components. The latter masks the day Pi2 pulsations.

SMP 2-1 FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS = =7.4.1.8.6.4.2 2. 3. h 18h.T. 4..1.8.6.4.2 2. 3. h 18h.T. 4. =.64.1.8.6.4.2 2. 3. h 18h.T. 4..1.8.6.4.2 =3.44 2. 3. h 18h.T. 4. Figure 9. ocal variations of the power spectra for de j in the ial planes at = (source region), 7.4 (outside the plasmasphere),.64 (plasmapause), and 3.44 (inner plasmasphere). The vertical axis ranges from to.1. Value m V P,night =.1, and value m V P,day =1. [36] Our numerical simulation shows that de j has a large amplitude in the nightside and a small amplitude in the dayside outer plasmasphere. In addition, de j becomes comparatively large in the inner plasmasphere in the day. If the power of poloidal mode is below the detection threshold of the satellite instrumentation in the dayside plasmasphere, the present results are consistent with the observations [Takahashi et al., 199]. In addition, when the longitudinal extent of the source current is narrower than the present example, the Pi2 pulsation in our simulation may be more localized in the local. This may lead to the rare detection of a Pi2 signal in the dayside plasmasphere. It should be noted that the poloidal mode has a reduced intensity in the day when the ionospheric conductivity is low. Therefore the smaller longitudinal extent of the source current and the lower ionospheric conductivity may provide the conditions favorable to have day-night asymmetry in Pi2 pulsations in the plasmasphere. 3.2.2. Transient toroidal waves associated with the substorm. [37] Takahashi et al. [1996] reported transient toroidal waves (TTW) in the plasmapause region (Figure 8 of that paper). TTW is associated with the substorm. The same results were obtained by Nosé et al.[1998]. Approximately 3% of the Pi2 pulsations accompany a toroidal component in the magnetosphere [Takahashi et al., 1996]. On the other hand, the numerical simulation presented in this paper indicates that the toroidal component (de n in our calculation) with a duration of min always appears immediately after the passage of the poloidalmode wave front in the plasmapause and inner plasmasphere. The same results were obtained using other numerical simulations [Pekrides et al., 1997; ee and ysak, 1999; Paper 1]. The toroidal-mode wave is generated by the poloidal-mode wave by mode coupling in the nonuniform magnetosphere. It is noteworthy that the toroidal wave tends to be weaker than the poloidal wave in the middle plasmasphere where the radial gradient of V A is small. Therefore, if the radial gradient of V A in the real plasmasphere is small, a toroidal wave may be observable in the plasmapause region. Thus our numerical simulation gives results that are consistent with the observations. In addition, the smaller ionospheric conductivity (i.e., the case of m V P =.1) induces de n with a smaller amplitude and a shorter duration (see Figure 2 of Paper 1). This toroidal wave is quite similar to TTW. 3.2.3. ocal dependence of polarization. [38] Saka et al. [1996] reported on the quasiperiodic field line oscillations (QPO) in the geosynchronous orbit. This type of oscillation is probably the same as the transient toroidal wave found by

FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS SMP 2-11.1.1. = 2. 3. h 18h phi 4..1.1. =7.4 2. 3. h 18h phi 4. =.64 =3.44.1.1. 2. 3. h 18h phi 4..1.7..2 2. 3. h 18h phi 4. Figure 1. ocal variations of the power spectra for de n in the ial planes at = (source region), 7.4 (outside the plasmasphere),.64 (plasmapause), and 3.44 (inner plasmasphere). Value m V P,night =.1, and value m V P,day =1. Takahashi et al. [1996]. After Saka et al. [1996], QPO is dominant in postmidnight and premidnight regions and rare in the midnight region. This observation is consistent with the numerical simulation presented here, which shows dominant poloidal-mode waves in the midnight region and dominant toroidal -mode waves in the local s away from midnight. [39] Recently, Uozumi et al. [2] estimated the polarization of Pi2 pulsations in the magnetospheric ial plane from groundbased magnetic data. They found that magnetic field perturbations of the Pi2 pulsations in the magnetospheric ial plane are longitudinally polarized in the dawn region. This result is consistent with our numerical model. On the other hand, Uozumi et al. [2] also reported that the duskside Pi2 pulsations exhibit radially polarized magnetic field perturbations. As the plasmasphere structure is complicated in the dusk region (for example, the plasmasphere dusk bulge), longitudinal variations of V A and/or the background pressure may deform the wave front of the fast-mode wave, which results in deflection of the wave front. A longitudinally nonuniform plasmasphere model needs to be considered in the future. 3.2.4. Appearance of PVR and FR. [4] Observed Pi2 pulsations show FR behavior (latitude-dependent period) at high latitudes and PVR behavior (constant period over latitudes) at middle and low latitudes. Figures 11 and 12 indicate that the poloidal mode with the PVR feature is dominant in the plasmasphere, whereas the toroidal mode with the FR feature is dominant at the plasmapause latitude and at the source latitudes. Therefore our numerical results and observed ones on the ground [Yumoto et al., 199; Sutcliffe and Yumoto, 1991] are consistent if the dominant mode pattern is conversed from the ial plane to the. To confirm this, we need to improve the model by using the correct ionospheric boundary conditions as discussed below. 3.3. On the Assumptions to be Improved 3.3.1. Ionospheric boundary condition. [41] In this section the simplifications to the boundary conditions will be discussed, namely, taking de j = in the and ignoring the Hall conductivity. [42] The former is assumed because the fast-mode wave tends to vanish at the, owing to evanescent propagation and the high conductivity of the Earth as explained by Paper 1. This boundary condition is robust when the Alfvén wave and the fastmode wave are decoupled. However, bearing in mind that de j has both FR (Alfvén-mode) and PVR (fast-mode) components, the condition of de j = at the does not hold rigorously. On

SMP 2-12 FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS 1(Re) 18h 1(Re) h Earth 1(Re) 1 6h 1(Re) Figure 11. Spatial variation of the polarization of the electric field in the ial plane at = to 3.44. The horizontal bar in the left bottom corner indicates m V R E J. the other hand, as shown in Figure, the longitudinal wave number (m number) of de j is smaller in the plasmasphere because power spectra of de j is comparatively flat in local. This means that the coupling between the fast-mode wave and the Alfvén wave weakens. Therefore the condition that de j = in the may not have such a significant effect in the magnetosphere. However, it may reduce the magnetospheric de j at the sourcecurrent latitude because the m number is larger. [43] et us consider the consequence of no Hall conductivity in the present model. Hall conductivity has a possible influence on ionospheric reflection of the Alfvén wave [Yoshikawa et al., 1999; Nakata et al., 2]. In addition, the ground magnetic field perturbations cannot be obtained directly from the simulation when Hall conductivity is ignored. Therefore rigorous consideration of the ionospheric boundary conditions may be an important issue. These shortcomings should be improved in a future model. [44] The no-hall-conductivity assumption also results in no westward ionospheric current associated with the current wedge in the present model. Therefore the present model does not yield a closed current circuit of the current wedge through the at auroral latitudes. This may initially appear unrealistic; however, if the source current is a short-lived impulsive current flowing at the onset of the substorm current wedge as discussed previously, the source current constitutes a closed current circuit with the magnetospheric polarization current. The MHD signal in the plasmasphere, which is the main subject in the present paper, is derived from this magnetospheric polarization current (the propagating fast-mode wave). Therefore the behavior of PVR is not so affected by the incomplete current circuit in the high-latitude region. On the other hand, the field-aligned current bouncing between the s may be adversely affected by this unrealistic assumption. We do not regard the magnitude of the fieldaligned current to be realistic. However, the -varying part of the field-aligned current, which is denoted by de n in the present paper, seems to give qualitatively realistic behavior of the MHD signal at the auroral latitudes. Therefore, as long as PVR and the qualitative behavior of the toroidal-mode wave at the auroral latitudes are considered, the proposed model shown in Figure 1 is plausible. Note that the present simulation is not designed to reproduce the entire current system associated with the current wedge. Further research is needed to develop a physically reasonable and quantitative model of Pi2 pulsations. We expect that a nonlinear global simulation of the solar wind magnetosphere system will yield more accurate results.

FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS SMP 2-13 1(Re) 18h 1(Re) Earth h 1(Re).1 6h 1(Re) Figure 12. Spatial variation of the polarization of the magnetic field (db? ) in the off-ial plane (the tenth node from the ial plane: colatitude 7 ) at = to 3.44. The horizontal bar in the left bottom corner indicates.1m R 2 E J /V. 3.3.2. Cold plasma approximation and linear wave treatment. [4] The cold plasma approximation used in the present study is an unrealistic assumption as for the source current. The plasmas are hot in the current disruption region [ui, 1996]. In addition, Saka et al. [1996] suggested that the injection of hot plasmas in the current disruption region may trigger the UF oscillations observed in the geosynchronous orbit. This observation also requires further modification to the present cold plasma model. [46] The electromagnetic perturbations associated with Pi2 pulsations should be treated as nonlinear perturbations in the real substorm process in the current disruption region because the variations associated with the dipolarization of the magnetospheric magnetic field are as large as the background magnetic field. Therefore a quantitative comparison between the observed and simulated Pi2 pulsations may be difficult in the current disruption region. These deficiencies may be deviated by using a global MHD simulation. 3.3.3. Axial symmetry of the plasmasphere. [47] The current wedge is not actually symmetric about the noon-midnight meridian. Usually, the source of a Pi2 pulsation is regarded to be located around 23 22 T [Singer et al., 1983]. We need to reconsider the T location of the source current. However, as long as we use the longitudinally uniform magnetosphere and plasmasphere, results do not depend on the longitudinal location of the source current. Actually, the magnetosphere is not longitudinally uniform. Consequently, we need to use a realistic inner magnetosphere model with a longitudinally nonuniform structure and a realistic local of the current wedge in order to explain the dawn-dusk asymmetry of Pi2 pulsations. 3.3.4. Boundary condition at the magnetopause. [48] We assume that the outer boundary is a free boundary. This assumption is based on geometrical optics of MHD wave propagation done by McKenzie [197], who showed that only MHD waves with a small cone angle can be transmitted from the magnetosheath to the magnetosphere. This result indicates that a MHD wave is transmitted almost freely across the magnetopause to the magnetosheath. On the other hand, Fujita et al. [1996] demonstrated that the magnetospheric eigenmode oscillation can be evanescent in the magnetosheath unless the solar wind speed is

SMP 2-14 FUJITA ET A.: NUMERICA SIMUATION OF PI2 PUSATIONS not so small. This result proposes that the wave energy in the magnetosphere does not escape from the magnetosphere. As for the Pi2 pulsation simulation, ee and ysak [1999] assumed partial reflection of the MHD wave at the dayside magnetopause. This boundary condition seems to be more realistic. [49] However, ee and ysak [1999] indicated that the impulsestimulated wave continues for as long as 16 s, which is too long compared with typical Pi2 pulsation observations. They thought that this discrepancy might go away if ionospheric dissipation is considered. Then it is noteworthy that the duration of the pure PVR in the same model as the present one is approximately several wave periods [Fujita et al., 2], which appears to be consistent with the observations. This pure PVR is not affected by ionospheric dissipation, because it does not have de j at the [Fujita et al., 2]. Therefore this result may indicate that the effect is not so significant for the duration of PVR. Further research of the outer boundary for the MHD waves in the magnetosphere is therefore needed. 4. Conclusion [] We investigated the propagation of Pi2 pulsations using a numerical model characterized by a localized and impulsive source current for MHD waves, a dipole magnetic field, a plasmasphere, an with finite Pedersen conductivity, and a free outer boundary. The source current is located at midnight and on the magnetic, with a radial and field-aligned extent of 2R E and a longitudinal extent of 1R E. Our simulations produced Pi2 pulsations and transient toroidal waves in the plasmasphere, with spatial variations consistent with satellite observations. The simulated wave fields have the following features: 1. The poloidal-mode electric field (de j ) shows a clear PVR behavior at midnight but only a weak perturbation at noon. This is consistent with observations in the plasmasphere [Takahashi et al., 1996]. 2. The toroidal-mode electric field (de n ) has zero amplitude at both midnight and noon and a maximum amplitude at the east/west edge of the source current. Immediately after the passage of the de j wave front, de n in the plasmasphere exhibits an FR behavior lasting min. In the region of small V A gradient (middle plasmasphere), de n may become smaller than de j, a feature consistent with transient toroidal waves observed in the plasmasphere [Takahashi et al., 1996; Saka et al., 1996]. 3. When the Pedersen conductivity is low, de j in the plasmasphere is much smaller at noon than at midnight, suggesting that the conductivity is a factor contributing to the reduced Pi2 amplitude on the dayside as reported from ground and satellite observations [Takahashi et al., 1996; i et al., 1998]. A low conductivity also reduces the amplitude and duration of de n. Note that the longitudinal extent of the source current is also a possible factor limiting the propagation of Pi2 to the dayside. 4. The major axis of polarization of the electric field is in the east-west direction at midnight and in the radial direction away from midnight. Correspondingly, the major axis for the magnetic field is radially oriented at midnight and azimuthally oriented at dawn and dusk.. The perturbed field shows a latitude-independent frequency within the plasmasphere (a PVR behavior) but a latitude-dependent frequency at the plasmapause gradient and within the latitude of the source current. [1] Acknowledgments. The authors are very grateful to B. Hausman and F. C. Michel of STE for careful reading of the manuscript. We also acknowledge discussions with K. Yumoto and T. Uozumi of Kyushu Univ., T. Araki of Kyoto Univ., and T. Saito, Professor Emeritus of Tohoku Univ. 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