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EMIC waves around the plasma-pause region
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Planetary and Space Science 56 (2008) 1023–1029 www.elsevier.com/locate/pss
EMIC waves around the plasma-pause region P. Varmaa, G. Ahirwarb, M.S. Tiwaria, a
Department of Physics & Electronics, Dr. H. S. Gour University, Sagar, M.P. 470003, India b School of Studies in Physics, Vikram University, Ujjain, M.P., India
Received 6 September 2007; received in revised form 18 January 2008; accepted 22 January 2008 Available online 30 January 2008
Abstract Electromagnetic ion-cyclotron (EMIC) instability has been studied using the general loss-cone distribution function by investigating the trajectories of charged particles and using the method of particle aspect analysis. A low b (ratio of plasma pressure to magnetic pressure) plasma consisting of resonant and non-resonant particles has been considered. It is assumed that the resonant particles participate in energy exchange with the wave, whereas non-resonant particles support the oscillatory motion of the wave. The wave is assumed to propagate parallel to the static magnetic field. The effects of steepness of loss-cone distribution with thermal anisotropy are discussed. The growth rate, perpendicular and parallel resonant energies of the particles and marginal instability condition are derived. The effect of general loss-cone distribution function is to enhance the growth rate of EMIC waves. The results are interpreted for the space plasma parameters appropriate to the plasma-pause region of the earth’s magnetoplasma. The results of the work is consistent for EMIC emissions observation by SAMPEX and CRRES satellite around the plasma-pause region as reported by Bortnik et al. [Bortnik, J., Thorne, R.M., O’Brien, T.P., Green, J.C., Strongeway, R.J., Shprits, Y.Y., Baker, D.N., 2006. Observation of two distinct, rapid loss mechanisms during the 20 November 2003 radiation belt dropout event. J. Geophys. Res. 111, A12216, doi:10.1029/2006JA011802] and Xinlin et al. [Xinlin, Li., Baker, D.N., O’Brien, T.P., Xie, L., Zong, Q.G., 2006. Correlation between the inner edge of outer radiation belt electrons and the innermost plasmapause location. Geophys. Res. Lett. 33, L14107, doi:10.1029/2006GL026294]. r 2008 Elsevier Ltd. All rights reserved. Keywords: Electromagnetic ion-cyclotron instability; Loss-cone distribution; Thermal anisotropy; Plasma-pause region; Particle aspect analysis
1. Introduction The outer boundary of the plasmasphere is referred to as the plasma-pause region where the plasma density has a steep gradient. Carpenter et al. (1971) reported the position of the plasma-pause related to the region of enhanced trapped energetic electrons. Xinlin et al. (2006) have renewed interesting features about the plasma-pause region. Moldwin et al. (2002) using CRRES satellite observations reported clear and sharp ‘classic’ isolated steep density gradients and that plasma-pause observations with significant structure or density cavities were more common. During storm times, the pitch angle scattering of the radiation belt electrons is more prominent near and just
Corresponding author.
E-mail addresses: [email protected] (P. Varma), [email protected] (M.S. Tiwari). 0032-0633/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2008.01.007
outside the plasma-pause where electromagnetic ioncyclotron (EMIC) waves are also noticed Albert (2003). EMIC waves play an important role in the overall dynamics of space plasmas. They not only act as a useful diagnostic of the relevant physical processes, but they can also affect the macroscopic plasma state by influencing the transport properties. These waves preferentially generated in high-density region. Horne and Thorne (1993) provided even faster loss of MeV electrons on the scale of hours which affects only relativistic electrons since resonant energies for interaction with EMIC waves are above 0.5 MeV (Summers and Thorne, 2003; Albert, 2003). Bortnik et al. (2006) have reported that rapid losses of the outer radiation belt electrons can be attributed to resonant wave particle interaction between a variety of plasma waves and energetic electrons (Kennel and Engelmann, 1966), which drive pitch angle scattering into the drift and bounce loss cone. The evidence of the presence of EMIC waves and associated thermal electron
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heating (Cornwall et al., 1970) at low latitudes is found indirectly through studies of thermosphere heating. Spasojevic et al. (2004) reported that the sunward transport of ring current ions during distributed periods produces anisotropic (T ? 4T P ) particle distributions through adiabatic heating. Solar wind compressions of the magnetosphere can also increase the ion anisotropy (Anderson and Hamilton, 1993). The EMIC instability is derived by the thermal anisotropy of the hot ring current proton distribution in the outer magnetosphere. The thermal anisotropy is the free energy source for the EMIC instability. This instability has maximum growth rate at propagation parallel and anti parallel to the background magnetospheric magnetic field, so that protons are predominantly pitch angle scattered by the wave. Wave particle interactions strongly scatter particles only when the anisotropy is sufficiently strong to excite the instability. This pitch angle scattering acts to reduce the free energy source. The observational study by Xinlin et al. (2006), Bortnik et al. (2006), Fuselier et al. (2004) and Spasojevic et al. (2004) provides the authensity to our theoretical model which describe the EMIC wave structure in plasma-pause region. The present analysis is based upon (Dawson’s, 1961) theory of Landau damping which was further extended by Terashima (1967), Misra and Tiwari (1979), Varma and Tiwari (1992, 1993), Dwivedi et al. (2001, 2002), Shandilya et al. (2003, 2004), Mishra and Tiwari (2006), Ahirwar et al. (2006, 2007) to the analysis of electrostatic and electromagnetic instabilities. The whole plasma is considered to consist of resonant and nonresonant particles. Non-resonant particles support the oscillatory motion of the EMIC waves whereas, the resonant particles participate in the energy exchange with the wave. An EMIC wave starts at t=0, when the resonant particles are not disturbed. Using the particle trajectory in the presence of EMIC wave, the dispersion relation and growth rate are derived and discussed for different distribution indices and the thermal anisotropy around the plasma-pause. In the present study, we focus our attention to explain the EMIC wave generation and energy transfer in the plasma-pause region accordance to the observations by Solar, Anomalous, and Magnetospheric Particle Explorer [SAMPEX] and Combined Radiation and Release Experiment Satellite [CRRES] satellite. 2. Basic assumptions The left-handed circularly polarized EMIC wave having angular frequency o is defined by Ahirwar et al. (2006, 2007) as, Bx ¼ B cosðkz otÞ;
By ¼ B sinðkz otÞ.
(1)
When the system is co-moving with the wave, the electric field vanishes. Thus, the wave magnetic field has the form, B ¼ Bx ðcos kzÞx þ By ðsin kzÞy,
(2)
where the following conditions are valid, o t, (3a) zwave ¼ zlab k o vwave ¼ vlab . (3b) k Since ck/oc1, the magnetic field amplitude may be regarded identical in both systems. Considering the equation of motion of ions and the general loss-cone distribution function the energy calculations and derivation of growth rate are performed by Ahirwar et al. (2006, 2007). The general loss-cone distribution function is expressed as (Ahirwar et al., 2006, 2007), NðV¯ Þ ¼ N 0 f ? ðV ? Þf P ðV P Þ,
(4)
where "
#
V 2? f ? ðV ? Þ ¼ , exp 2 V T? pV 2ðJþ1Þ J! T? V 2J ?
and f P ðV P Þ is defined by (Varma and Tiwari, 1992) 1 V 2P p ffiffiffi f P ðV P Þ ¼ exp , pV TP V 2TP
(5)
(6)
where J is the distribution index and measures the steepness of the loss-cone feature (Tiwari and Varma, 1991, 1993; Varma and Tiwari, 1992). In the case of J ¼ 0 this represents a bi-Maxwellian distribution and for J ¼ N this reduces to the Dirac delta function (Tiwari and Varma, 1993). V 2TP ¼ 2T P =m and V 2T? ¼ ðJ þ 1Þ1 ð2T ? =mÞ are the squares of parallel and perpendicular thermal velocities with respect to the external magnetic field. 3. Dispersion relation and particles energy We consider the cold plasma dispersion relation for the EMIC wave as (Misra and Tiwari, 1979; Ahirwar et al., 2006, 2007), ! o2pi c2 k 2 o1 ¼ , (7) 1 O o2 O2 where o2pi ¼ ð4pN 0 e2 Þ=mi is the square of plasma frequency for the ions and O=qB0/mc is the ion-cyclotron frequency, m is the mass of the ion and q is the charge of ion and c is the velocity of light. The wave energy density Ww per unit wavelength is the sum of the pure field energy and the changes in the energy of the non-resonant particles, i.e. the total energy per unit wavelength is given as W w ¼ U þ Wi,
(8)
where U is the energy of electromagnetic wave as defined by the expression (Misra and Tiwari, 1979) as 1 d n 2 U¼ ðoik ÞE 1 E k þ jBj , (9) 16p do
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where eik is the dielectric tensor. After the calculation, the electromagnetic wave energy per unit wavelength is given by 2 lB 2O o U¼ , (10) Oo 8p
4. Growth rate
whereas the parallel non-resonant energy is given as (Ahirwar et al., 2006, 2007)
the growth/damping rate g is derived as (Misra and Tiwari, 1979; Ahirwar et al., 2006, 2007)
Using the law of conservation of energy d ðW r þ W w Þ ¼ 0, dt
g ðO=kq V q Þ½ððO oÞ=OÞððJ þ 1ÞV 2T? =ðV 2Tq ÞÞ 1 exp½ð1=V 2Tq Þððo OÞ=kq Þ2 ¼ , Oi ðckq =opi Þ2 ðð2O oÞ=ðO oÞÞ þ 12o2 =ðO oÞ2
W iP
2 lB2 C J opi 1 oO Z 1 ðzÞ þ ¼ Z 2 ðzÞ , kP V TP 8p V 2TP c2 k2P 2
(11)
The perpendicular (transverse) energy and the parallel resonant energy of the resonant ions are calculated as
W rP
O2i C J o Oi ¼ þ DJ Oi C 2 k2P o k2P V TP V 2TP " ## 1 k2 ðo Oi Þ2 þ C J exp 2 2 (13) 2 O2 kP V TP p3=2 B2 o2pi
" # O2i C J o Oi 2 ¼ Oi C 2 k2P o k2P V TP V 2TP " # ðo Oi Þ2 exp 2 2 , kP V TP p3=2 B2 o2pi
where CJ ¼
DJ ¼
Z
2p V 2ðJþ1Þ J! T?
0
Z
2p
1
V 2ðJþ1Þ J! T?
V 2? dV 2? V 2ðJþ1Þ exp , ? V 2T?
1
0
dV 2? V 2J ?
V 2? exp 2 , V T?
and 1 Z n ðxÞ ¼ pffiffiffi P p
x¼
oO . kP V TP
Z
expðxÞ2 dx, nþ1 1 ðx xÞ 1
(16)
where
and perpendicular non-resonant energy as " ! 2 l B2 opi 2O O2 W i? ¼ DJ 1 ZðzÞ þ 2 2 Z 1 ðzÞ 2 8p c2 k2P kP V TP kP V TP 2C J O þ 2 Z 1 ðzÞ Z 2 ðzÞ . (12) kP V TP V TP
W r?
(15)
V 2T? ¼ ðJ þ 1Þ1
2T ? 2T q and V 2Tq ¼ . m m
Those particles with velocities near the phase velocity of the wave give up an energy 2U to the wave. Half of this goes to potential energy and the other half goes into kinetic energy of oscillation of the bulk of the particles. Here it is noticed that J has affected the growth rate through the thermal anisotropy as discussed by Summers and Thorne (1995) for the electromagnetic waves propagating parallel to the magnetic field with general loss-cone distribution function. Due to the ion energy anisotropy the growth is possible only when ðV T? =V TP Þ41. Thus, we are interested in the behavior of those particles for which ðV T? =V TP Þ41. All the results obtained by using a loss-cone distribution function can be obtained from a bi-Maxwellian distribution function by defining an effective thermal anisotropy which depends on the loss-cone index J (Gomberoff and Cuperman, 1981; Summers and Thorne, 1995). 5. Marginal instability criteria
(14)
For the marginal instability condition g ¼ 0, the maximum stable frequency is obtained 1 V Tq o¼O 1 . (17) J þ 1 V T? In the past, it has been predicted that to determine the growth rate of the EMIC waves the hot proton temperature anisotropy, the hot plasma bPh (the ratio of parallel hot particle thermal energy to the magnetic field energy), and the cold plasma density nc are needed (Fuselier et al., 2004 and references therein). In the present analysis it is assumed that the EMIC instability is mainly driven by the thermal anisotropy of the hot ring current proton distribution in the outer magnetosphere. The nc/nh ratio (nh is the hot proton density) changes the growth rate by changing the cyclotron resonance velocity of the hot protons. As discussed by Fuselier et al. (2004), the cold plasma density decreases at the plasma-pause. Further from the earth the cold plasma density profile might cause the EMIC instability to be driven above marginal stability.
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At intermediate distances from the Earth and not further from the earth the temperature anisotropy in the dayside outer magnetosphere increases with increasing radial distance from the Earth. This anisotropy is the most important quantity in determining the EMIC growth rate. However, the cold plasma density associated with plasmaspheric plume may be of the importance to explain subauroral proton arcs (Spasojevic et al., 2004). Thus, in view of decreased cold plasma density at the plasma-pause and further, we have not considered the cold plasma density contribution in our analysis as nc/nho0.1 has little effect on the maximum growth rate (Fuselier et al., 2004).
1.6
-0.6
-1.2
V 2T q
2.6
2.8
3
J=0
J=1
A = 25 J=2
-2
g ¼ 6 107 Wb=m2 ;
Fig. 2. Variation of perpendicular resonant energy Wr? erg cm versus wave vector kP cm1 for different values of distribution indices J, and A ¼ 25.
.
Eqs. (13), (14) and (16) are solved numerically and results are presented in Figs. 1–6. Fig. 1 shows the variation of growth rate versus wave number kq for different values of distribution indices J. The enhancement of growth rate at lower values of kq is observed. The corresponding loss of perpendicular resonant energies of ions is exhibited by Fig. 2. It is noticed that the growth rate of ion-cyclotron wave occurs by extracting energy from particles moving perpendicular to 0.09 A = 25
0.08
-1.8
0.7 A = 25
0.6 0.5 WrII x 10-4 erg cm
A¼
2.4
-1
-1.6
ri ¼ 170 m;
2.2
-0.8
We have evaluated the growth rate, transverse energies and dispersion relation using the following plasma-pause parameters. (Hasegawa, 1971; Kintner and Gurnett, 1978; Varma and Tiwari, 1991; Tiwari and Varma, 1993). V 2T ?
2
-0.4
-1.4
B0 ¼ 600;
1.8
-0.2
6. Results
n0 ¼ 500 cm1 ;
KII x 10-08 cm-1
0
Wr⊥ x10-3 erg cm
1026
0.4
J=2
0.3
J=1
0.2 0.07
J=0 0.1
0.06 J=2
γ/Ωi
0.05
0 1.6
1.8
2
2.2 KII x
0.04 J=1
J=0
0.01 0 1.6
1.8
2
2.2 KII x
10-08
2.4
2.6
2.8
2.4
2.6
2.8
3
cm-1
Fig. 3. Variation of parallel resonant energy W rP erg cm1 versus wave vector kP cm1 for different values of distribution indices J, and A ¼ 25.
0.03 0.02
10-08
3
cm-1
Fig. 1. Variation of growth rate (g/Oi) versus wave vector kP cm1 for different values of distribution indices J, and A ¼ 25.
the magnetic filed. Thus, the closed and conversing dipolar magnetic filed lines at plasma-pause may permit the step loss-cone distribution function and the EMIC waves are excited by extracting perpendicular energy of the ions. The increase in parallel component of energy of ions with distribution indices J is noticed in Fig. 3. The energy transfer from perpendicular to the parallel direction is noticed along with the generation of EMIC waves at higher loss-cone indices.
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0.09
0.7
0.07
γ/Ωi
0.06 0.05 0.04
J = 0, A= 5
0.6
J = 0, A= 15 J = 0, A= 25
WrII x 10-4 erg cm
J=0, A=5 J=0, A=15 J=0, A=25 J=1, A=5 J=1, A=15 J=1, A=25 J=2, A=5 J=2, A=15 J=2, A=25
0.08
0.5
J = 1, A= 5 J = 1, A= 15
0.4
J = 1, A= 25 J = 2, A= 5
0.3
0.03
0.2
0.02
0.1
0.01
0
J = 2, A= 15 J = 2, A= 25
1.6
0 1.6
2.1
3.1
2.6
Fig. 4. Variation of growth rate (g/Oi) versus wave vector kP cm1 for different values of thermal anisotropies A, and distribution index J ¼ 0, 1 and 2.
KII x 10-08 cm-1
0 -0.2
1.6
2.1
2.6
3.1 J=0, A=5
-0.4 -0.6 -0.8
J=0, A=15 J=0, A=25 J=1, A=5
-1
J=1, A=15
-1.2
J=1, A=25
-1.4
J=2, A=5
-1.6
J=2, A=15
-1.8
J=2, A=25
-2 Fig. 5. Variation of perpendicular resonant energy W rP erg cm1 versus wave vector kP cm1 for different values of thermal anisotropies A, and distribution index J ¼ 0, 1 and 2.
The effectiveness of thermal anisotropy at different distribution indices (J ¼ 0, 1 and 2) on the enhancement of growth rate is predicted by Fig. 4. It is seen that the thermal anisotropy enhances the growth rate towards lower wave number kq at all the distribution indices, but more effective as the distribution indices increase. Thus, the lower values of thermal anisotropy effectively excite the wave in steep loss-cone distributions. Fig. 5 predicts the loss of perpendicular resonant energy Wr? at the distribution indices J ¼ 0, 1 and 2, respectively. The reduction in Wr? with thermal anisotropy is noted in all the cases, which becomes more effective with the increase of distribution index J. The corresponding gain in
1.9
2.2
2.5
2.8
3.1
3.4
KII x 10-08 cm-1
3.6
KII x 10-08 cm-1
Wr⊥ x 10-3 erg cm
1027
Fig. 6. Variation of parallel resonant energy W rP erg cm1 versus wave vector kP cm1 for different values of thermal anisotropies A, and distribution index J ¼ 0, 1 and 2.
parallel resonant energy W rq with thermal anisotropy at different distribution indices J is noted in Fig. 6. The increase of W rq with thermal anisotropy is predicted at all the distribution indices J. However, the lower anisotropy with higher distribution index J effectively enhances W rq . Thus, we may conclude that thermal anisotropy increases the growth rate by extracting energy from perpendicular component velocity of ions, as well as enhances the parallel component of velocity which becomes more effective at higher distribution indices J. 7. Conclusion 1. In view of the observations of EMIC wave around the plasma-pause our theoretical investigations indicated that the EMIC wave emissions and the related phenomena can be suitably described considering the general loss-cone distribution function in the anisotropic plasma which may be the cause of ring current destabilization and pitch angle scattering. 2. In the description of scattering of ring current protons by EMIC waves (Jordanova et al., 1999) has indicated that a region of strong ion-cyclotron instability forms just inside and along the plasma-pause that can be explained more effectively incorporating general losscone distribution function in anisotropic plasma. 3. The EMIC waves themselves are most likely to be generated by anisotropic distribution of low energy (keV) protons which should be also precipitated along with the MeV electrons (Nigel et al., 2003). These EMIC waves observed by CRRES and SAMPEX satellites around the plasma-pause may be suitably described in our model. 4. The occurrence frequency of EMIC waves also peaks close to the magnetopause (Anderson et al., 1992) as does the ring current proton anisotropy (e.g. Sibeck et al., 1987).
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Scattering by the EMIC waves is particularly promising because these waves are common in the outer magnetosphere near noon and near dawn, even in the absence of a compression of the magnetosphere (Anderson et al., 1992). The ring current proton distribution can be near the threshold of the instability (Anderson et al., 1996) and the waves are readily generated during compression events. The growth of the instability is suppressed by rapid pitch angle scattering, reducing the anisotropy of ring current protons. This rapid pitch angle scattering will force ring current protons into the atmospheric loss cone (Fuselier et al., 2004) that could excite EMIC. 5. In spite of cold ion density there may be various minor ion species drifting and non-drifting relative to the main proton component at the plasma-pause. In the present analysis, we have considered only one ion species. The presence of various drifting ion species affects the cold plasma dispersion relation and growth rate in substantial way (Gomberoff and Neira, 1983; Gomberoff and Elgueta, 1991; Gomberoff et al., 1996), which may be the matter of further analysis for the plasma-pause. However, the present investigation may be useful to explain some of the observations qualitatively.
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Varma, P., Tiwari, M.S., 1992. Ion and electron beam effects on drift wave instability with different distribution function—particle aspect analysis. Phys. Scripta 45, 275. Varma, P., Tiwari, M.S., 1993. Drift wave in the presence of an AC electric field with different distribution function—particle aspect analysis. Ind. J. Pure Appl. Phys. 31, 616. Xinlin, Li., Baker, D.N., O’Brien, T.P., Xie, L., Zong, Q.G., 2006. Correlation between the inner edge of outer radiation belt electrons and the innermost plasmapause location. Geophys. Res. Lett. 33, L14107.
Planetary and Space Science 56 (2008) 1023–1029 www.elsevier.com/locate/pss
EMIC waves around the plasma-pause region P. Varmaa, G. Ahirwarb, M.S. Tiwaria, a
Department of Physics & Electronics, Dr. H. S. Gour University, Sagar, M.P. 470003, India b School of Studies in Physics, Vikram University, Ujjain, M.P., India
Received 6 September 2007; received in revised form 18 January 2008; accepted 22 January 2008 Available online 30 January 2008
Abstract Electromagnetic ion-cyclotron (EMIC) instability has been studied using the general loss-cone distribution function by investigating the trajectories of charged particles and using the method of particle aspect analysis. A low b (ratio of plasma pressure to magnetic pressure) plasma consisting of resonant and non-resonant particles has been considered. It is assumed that the resonant particles participate in energy exchange with the wave, whereas non-resonant particles support the oscillatory motion of the wave. The wave is assumed to propagate parallel to the static magnetic field. The effects of steepness of loss-cone distribution with thermal anisotropy are discussed. The growth rate, perpendicular and parallel resonant energies of the particles and marginal instability condition are derived. The effect of general loss-cone distribution function is to enhance the growth rate of EMIC waves. The results are interpreted for the space plasma parameters appropriate to the plasma-pause region of the earth’s magnetoplasma. The results of the work is consistent for EMIC emissions observation by SAMPEX and CRRES satellite around the plasma-pause region as reported by Bortnik et al. [Bortnik, J., Thorne, R.M., O’Brien, T.P., Green, J.C., Strongeway, R.J., Shprits, Y.Y., Baker, D.N., 2006. Observation of two distinct, rapid loss mechanisms during the 20 November 2003 radiation belt dropout event. J. Geophys. Res. 111, A12216, doi:10.1029/2006JA011802] and Xinlin et al. [Xinlin, Li., Baker, D.N., O’Brien, T.P., Xie, L., Zong, Q.G., 2006. Correlation between the inner edge of outer radiation belt electrons and the innermost plasmapause location. Geophys. Res. Lett. 33, L14107, doi:10.1029/2006GL026294]. r 2008 Elsevier Ltd. All rights reserved. Keywords: Electromagnetic ion-cyclotron instability; Loss-cone distribution; Thermal anisotropy; Plasma-pause region; Particle aspect analysis
1. Introduction The outer boundary of the plasmasphere is referred to as the plasma-pause region where the plasma density has a steep gradient. Carpenter et al. (1971) reported the position of the plasma-pause related to the region of enhanced trapped energetic electrons. Xinlin et al. (2006) have renewed interesting features about the plasma-pause region. Moldwin et al. (2002) using CRRES satellite observations reported clear and sharp ‘classic’ isolated steep density gradients and that plasma-pause observations with significant structure or density cavities were more common. During storm times, the pitch angle scattering of the radiation belt electrons is more prominent near and just
Corresponding author.
E-mail addresses: [email protected] (P. Varma), [email protected] (M.S. Tiwari). 0032-0633/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2008.01.007
outside the plasma-pause where electromagnetic ioncyclotron (EMIC) waves are also noticed Albert (2003). EMIC waves play an important role in the overall dynamics of space plasmas. They not only act as a useful diagnostic of the relevant physical processes, but they can also affect the macroscopic plasma state by influencing the transport properties. These waves preferentially generated in high-density region. Horne and Thorne (1993) provided even faster loss of MeV electrons on the scale of hours which affects only relativistic electrons since resonant energies for interaction with EMIC waves are above 0.5 MeV (Summers and Thorne, 2003; Albert, 2003). Bortnik et al. (2006) have reported that rapid losses of the outer radiation belt electrons can be attributed to resonant wave particle interaction between a variety of plasma waves and energetic electrons (Kennel and Engelmann, 1966), which drive pitch angle scattering into the drift and bounce loss cone. The evidence of the presence of EMIC waves and associated thermal electron
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heating (Cornwall et al., 1970) at low latitudes is found indirectly through studies of thermosphere heating. Spasojevic et al. (2004) reported that the sunward transport of ring current ions during distributed periods produces anisotropic (T ? 4T P ) particle distributions through adiabatic heating. Solar wind compressions of the magnetosphere can also increase the ion anisotropy (Anderson and Hamilton, 1993). The EMIC instability is derived by the thermal anisotropy of the hot ring current proton distribution in the outer magnetosphere. The thermal anisotropy is the free energy source for the EMIC instability. This instability has maximum growth rate at propagation parallel and anti parallel to the background magnetospheric magnetic field, so that protons are predominantly pitch angle scattered by the wave. Wave particle interactions strongly scatter particles only when the anisotropy is sufficiently strong to excite the instability. This pitch angle scattering acts to reduce the free energy source. The observational study by Xinlin et al. (2006), Bortnik et al. (2006), Fuselier et al. (2004) and Spasojevic et al. (2004) provides the authensity to our theoretical model which describe the EMIC wave structure in plasma-pause region. The present analysis is based upon (Dawson’s, 1961) theory of Landau damping which was further extended by Terashima (1967), Misra and Tiwari (1979), Varma and Tiwari (1992, 1993), Dwivedi et al. (2001, 2002), Shandilya et al. (2003, 2004), Mishra and Tiwari (2006), Ahirwar et al. (2006, 2007) to the analysis of electrostatic and electromagnetic instabilities. The whole plasma is considered to consist of resonant and nonresonant particles. Non-resonant particles support the oscillatory motion of the EMIC waves whereas, the resonant particles participate in the energy exchange with the wave. An EMIC wave starts at t=0, when the resonant particles are not disturbed. Using the particle trajectory in the presence of EMIC wave, the dispersion relation and growth rate are derived and discussed for different distribution indices and the thermal anisotropy around the plasma-pause. In the present study, we focus our attention to explain the EMIC wave generation and energy transfer in the plasma-pause region accordance to the observations by Solar, Anomalous, and Magnetospheric Particle Explorer [SAMPEX] and Combined Radiation and Release Experiment Satellite [CRRES] satellite. 2. Basic assumptions The left-handed circularly polarized EMIC wave having angular frequency o is defined by Ahirwar et al. (2006, 2007) as, Bx ¼ B cosðkz otÞ;
By ¼ B sinðkz otÞ.
(1)
When the system is co-moving with the wave, the electric field vanishes. Thus, the wave magnetic field has the form, B ¼ Bx ðcos kzÞx þ By ðsin kzÞy,
(2)
where the following conditions are valid, o t, (3a) zwave ¼ zlab k o vwave ¼ vlab . (3b) k Since ck/oc1, the magnetic field amplitude may be regarded identical in both systems. Considering the equation of motion of ions and the general loss-cone distribution function the energy calculations and derivation of growth rate are performed by Ahirwar et al. (2006, 2007). The general loss-cone distribution function is expressed as (Ahirwar et al., 2006, 2007), NðV¯ Þ ¼ N 0 f ? ðV ? Þf P ðV P Þ,
(4)
where "
#
V 2? f ? ðV ? Þ ¼ , exp 2 V T? pV 2ðJþ1Þ J! T? V 2J ?
and f P ðV P Þ is defined by (Varma and Tiwari, 1992) 1 V 2P p ffiffiffi f P ðV P Þ ¼ exp , pV TP V 2TP
(5)
(6)
where J is the distribution index and measures the steepness of the loss-cone feature (Tiwari and Varma, 1991, 1993; Varma and Tiwari, 1992). In the case of J ¼ 0 this represents a bi-Maxwellian distribution and for J ¼ N this reduces to the Dirac delta function (Tiwari and Varma, 1993). V 2TP ¼ 2T P =m and V 2T? ¼ ðJ þ 1Þ1 ð2T ? =mÞ are the squares of parallel and perpendicular thermal velocities with respect to the external magnetic field. 3. Dispersion relation and particles energy We consider the cold plasma dispersion relation for the EMIC wave as (Misra and Tiwari, 1979; Ahirwar et al., 2006, 2007), ! o2pi c2 k 2 o1 ¼ , (7) 1 O o2 O2 where o2pi ¼ ð4pN 0 e2 Þ=mi is the square of plasma frequency for the ions and O=qB0/mc is the ion-cyclotron frequency, m is the mass of the ion and q is the charge of ion and c is the velocity of light. The wave energy density Ww per unit wavelength is the sum of the pure field energy and the changes in the energy of the non-resonant particles, i.e. the total energy per unit wavelength is given as W w ¼ U þ Wi,
(8)
where U is the energy of electromagnetic wave as defined by the expression (Misra and Tiwari, 1979) as 1 d n 2 U¼ ðoik ÞE 1 E k þ jBj , (9) 16p do
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where eik is the dielectric tensor. After the calculation, the electromagnetic wave energy per unit wavelength is given by 2 lB 2O o U¼ , (10) Oo 8p
4. Growth rate
whereas the parallel non-resonant energy is given as (Ahirwar et al., 2006, 2007)
the growth/damping rate g is derived as (Misra and Tiwari, 1979; Ahirwar et al., 2006, 2007)
Using the law of conservation of energy d ðW r þ W w Þ ¼ 0, dt
g ðO=kq V q Þ½ððO oÞ=OÞððJ þ 1ÞV 2T? =ðV 2Tq ÞÞ 1 exp½ð1=V 2Tq Þððo OÞ=kq Þ2 ¼ , Oi ðckq =opi Þ2 ðð2O oÞ=ðO oÞÞ þ 12o2 =ðO oÞ2
W iP
2 lB2 C J opi 1 oO Z 1 ðzÞ þ ¼ Z 2 ðzÞ , kP V TP 8p V 2TP c2 k2P 2
(11)
The perpendicular (transverse) energy and the parallel resonant energy of the resonant ions are calculated as
W rP
O2i C J o Oi ¼ þ DJ Oi C 2 k2P o k2P V TP V 2TP " ## 1 k2 ðo Oi Þ2 þ C J exp 2 2 (13) 2 O2 kP V TP p3=2 B2 o2pi
" # O2i C J o Oi 2 ¼ Oi C 2 k2P o k2P V TP V 2TP " # ðo Oi Þ2 exp 2 2 , kP V TP p3=2 B2 o2pi
where CJ ¼
DJ ¼
Z
2p V 2ðJþ1Þ J! T?
0
Z
2p
1
V 2ðJþ1Þ J! T?
V 2? dV 2? V 2ðJþ1Þ exp , ? V 2T?
1
0
dV 2? V 2J ?
V 2? exp 2 , V T?
and 1 Z n ðxÞ ¼ pffiffiffi P p
x¼
oO . kP V TP
Z
expðxÞ2 dx, nþ1 1 ðx xÞ 1
(16)
where
and perpendicular non-resonant energy as " ! 2 l B2 opi 2O O2 W i? ¼ DJ 1 ZðzÞ þ 2 2 Z 1 ðzÞ 2 8p c2 k2P kP V TP kP V TP 2C J O þ 2 Z 1 ðzÞ Z 2 ðzÞ . (12) kP V TP V TP
W r?
(15)
V 2T? ¼ ðJ þ 1Þ1
2T ? 2T q and V 2Tq ¼ . m m
Those particles with velocities near the phase velocity of the wave give up an energy 2U to the wave. Half of this goes to potential energy and the other half goes into kinetic energy of oscillation of the bulk of the particles. Here it is noticed that J has affected the growth rate through the thermal anisotropy as discussed by Summers and Thorne (1995) for the electromagnetic waves propagating parallel to the magnetic field with general loss-cone distribution function. Due to the ion energy anisotropy the growth is possible only when ðV T? =V TP Þ41. Thus, we are interested in the behavior of those particles for which ðV T? =V TP Þ41. All the results obtained by using a loss-cone distribution function can be obtained from a bi-Maxwellian distribution function by defining an effective thermal anisotropy which depends on the loss-cone index J (Gomberoff and Cuperman, 1981; Summers and Thorne, 1995). 5. Marginal instability criteria
(14)
For the marginal instability condition g ¼ 0, the maximum stable frequency is obtained 1 V Tq o¼O 1 . (17) J þ 1 V T? In the past, it has been predicted that to determine the growth rate of the EMIC waves the hot proton temperature anisotropy, the hot plasma bPh (the ratio of parallel hot particle thermal energy to the magnetic field energy), and the cold plasma density nc are needed (Fuselier et al., 2004 and references therein). In the present analysis it is assumed that the EMIC instability is mainly driven by the thermal anisotropy of the hot ring current proton distribution in the outer magnetosphere. The nc/nh ratio (nh is the hot proton density) changes the growth rate by changing the cyclotron resonance velocity of the hot protons. As discussed by Fuselier et al. (2004), the cold plasma density decreases at the plasma-pause. Further from the earth the cold plasma density profile might cause the EMIC instability to be driven above marginal stability.
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At intermediate distances from the Earth and not further from the earth the temperature anisotropy in the dayside outer magnetosphere increases with increasing radial distance from the Earth. This anisotropy is the most important quantity in determining the EMIC growth rate. However, the cold plasma density associated with plasmaspheric plume may be of the importance to explain subauroral proton arcs (Spasojevic et al., 2004). Thus, in view of decreased cold plasma density at the plasma-pause and further, we have not considered the cold plasma density contribution in our analysis as nc/nho0.1 has little effect on the maximum growth rate (Fuselier et al., 2004).
1.6
-0.6
-1.2
V 2T q
2.6
2.8
3
J=0
J=1
A = 25 J=2
-2
g ¼ 6 107 Wb=m2 ;
Fig. 2. Variation of perpendicular resonant energy Wr? erg cm versus wave vector kP cm1 for different values of distribution indices J, and A ¼ 25.
.
Eqs. (13), (14) and (16) are solved numerically and results are presented in Figs. 1–6. Fig. 1 shows the variation of growth rate versus wave number kq for different values of distribution indices J. The enhancement of growth rate at lower values of kq is observed. The corresponding loss of perpendicular resonant energies of ions is exhibited by Fig. 2. It is noticed that the growth rate of ion-cyclotron wave occurs by extracting energy from particles moving perpendicular to 0.09 A = 25
0.08
-1.8
0.7 A = 25
0.6 0.5 WrII x 10-4 erg cm
A¼
2.4
-1
-1.6
ri ¼ 170 m;
2.2
-0.8
We have evaluated the growth rate, transverse energies and dispersion relation using the following plasma-pause parameters. (Hasegawa, 1971; Kintner and Gurnett, 1978; Varma and Tiwari, 1991; Tiwari and Varma, 1993). V 2T ?
2
-0.4
-1.4
B0 ¼ 600;
1.8
-0.2
6. Results
n0 ¼ 500 cm1 ;
KII x 10-08 cm-1
0
Wr⊥ x10-3 erg cm
1026
0.4
J=2
0.3
J=1
0.2 0.07
J=0 0.1
0.06 J=2
γ/Ωi
0.05
0 1.6
1.8
2
2.2 KII x
0.04 J=1
J=0
0.01 0 1.6
1.8
2
2.2 KII x
10-08
2.4
2.6
2.8
2.4
2.6
2.8
3
cm-1
Fig. 3. Variation of parallel resonant energy W rP erg cm1 versus wave vector kP cm1 for different values of distribution indices J, and A ¼ 25.
0.03 0.02
10-08
3
cm-1
Fig. 1. Variation of growth rate (g/Oi) versus wave vector kP cm1 for different values of distribution indices J, and A ¼ 25.
the magnetic filed. Thus, the closed and conversing dipolar magnetic filed lines at plasma-pause may permit the step loss-cone distribution function and the EMIC waves are excited by extracting perpendicular energy of the ions. The increase in parallel component of energy of ions with distribution indices J is noticed in Fig. 3. The energy transfer from perpendicular to the parallel direction is noticed along with the generation of EMIC waves at higher loss-cone indices.
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0.09
0.7
0.07
γ/Ωi
0.06 0.05 0.04
J = 0, A= 5
0.6
J = 0, A= 15 J = 0, A= 25
WrII x 10-4 erg cm
J=0, A=5 J=0, A=15 J=0, A=25 J=1, A=5 J=1, A=15 J=1, A=25 J=2, A=5 J=2, A=15 J=2, A=25
0.08
0.5
J = 1, A= 5 J = 1, A= 15
0.4
J = 1, A= 25 J = 2, A= 5
0.3
0.03
0.2
0.02
0.1
0.01
0
J = 2, A= 15 J = 2, A= 25
1.6
0 1.6
2.1
3.1
2.6
Fig. 4. Variation of growth rate (g/Oi) versus wave vector kP cm1 for different values of thermal anisotropies A, and distribution index J ¼ 0, 1 and 2.
KII x 10-08 cm-1
0 -0.2
1.6
2.1
2.6
3.1 J=0, A=5
-0.4 -0.6 -0.8
J=0, A=15 J=0, A=25 J=1, A=5
-1
J=1, A=15
-1.2
J=1, A=25
-1.4
J=2, A=5
-1.6
J=2, A=15
-1.8
J=2, A=25
-2 Fig. 5. Variation of perpendicular resonant energy W rP erg cm1 versus wave vector kP cm1 for different values of thermal anisotropies A, and distribution index J ¼ 0, 1 and 2.
The effectiveness of thermal anisotropy at different distribution indices (J ¼ 0, 1 and 2) on the enhancement of growth rate is predicted by Fig. 4. It is seen that the thermal anisotropy enhances the growth rate towards lower wave number kq at all the distribution indices, but more effective as the distribution indices increase. Thus, the lower values of thermal anisotropy effectively excite the wave in steep loss-cone distributions. Fig. 5 predicts the loss of perpendicular resonant energy Wr? at the distribution indices J ¼ 0, 1 and 2, respectively. The reduction in Wr? with thermal anisotropy is noted in all the cases, which becomes more effective with the increase of distribution index J. The corresponding gain in
1.9
2.2
2.5
2.8
3.1
3.4
KII x 10-08 cm-1
3.6
KII x 10-08 cm-1
Wr⊥ x 10-3 erg cm
1027
Fig. 6. Variation of parallel resonant energy W rP erg cm1 versus wave vector kP cm1 for different values of thermal anisotropies A, and distribution index J ¼ 0, 1 and 2.
parallel resonant energy W rq with thermal anisotropy at different distribution indices J is noted in Fig. 6. The increase of W rq with thermal anisotropy is predicted at all the distribution indices J. However, the lower anisotropy with higher distribution index J effectively enhances W rq . Thus, we may conclude that thermal anisotropy increases the growth rate by extracting energy from perpendicular component velocity of ions, as well as enhances the parallel component of velocity which becomes more effective at higher distribution indices J. 7. Conclusion 1. In view of the observations of EMIC wave around the plasma-pause our theoretical investigations indicated that the EMIC wave emissions and the related phenomena can be suitably described considering the general loss-cone distribution function in the anisotropic plasma which may be the cause of ring current destabilization and pitch angle scattering. 2. In the description of scattering of ring current protons by EMIC waves (Jordanova et al., 1999) has indicated that a region of strong ion-cyclotron instability forms just inside and along the plasma-pause that can be explained more effectively incorporating general losscone distribution function in anisotropic plasma. 3. The EMIC waves themselves are most likely to be generated by anisotropic distribution of low energy (keV) protons which should be also precipitated along with the MeV electrons (Nigel et al., 2003). These EMIC waves observed by CRRES and SAMPEX satellites around the plasma-pause may be suitably described in our model. 4. The occurrence frequency of EMIC waves also peaks close to the magnetopause (Anderson et al., 1992) as does the ring current proton anisotropy (e.g. Sibeck et al., 1987).
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Scattering by the EMIC waves is particularly promising because these waves are common in the outer magnetosphere near noon and near dawn, even in the absence of a compression of the magnetosphere (Anderson et al., 1992). The ring current proton distribution can be near the threshold of the instability (Anderson et al., 1996) and the waves are readily generated during compression events. The growth of the instability is suppressed by rapid pitch angle scattering, reducing the anisotropy of ring current protons. This rapid pitch angle scattering will force ring current protons into the atmospheric loss cone (Fuselier et al., 2004) that could excite EMIC. 5. In spite of cold ion density there may be various minor ion species drifting and non-drifting relative to the main proton component at the plasma-pause. In the present analysis, we have considered only one ion species. The presence of various drifting ion species affects the cold plasma dispersion relation and growth rate in substantial way (Gomberoff and Neira, 1983; Gomberoff and Elgueta, 1991; Gomberoff et al., 1996), which may be the matter of further analysis for the plasma-pause. However, the present investigation may be useful to explain some of the observations qualitatively.
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