Diffuse auroral precipitation by resonant interaction with electron cyclotron harmonic and whistler mode waves

Journal of Atmospheric and Solar-Terrestrial Physics 97 (2013) 125–134

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Diffuse auroral precipitation by resonant interaction with electron cyclotron harmonic and whistler mode waves A.K. Tripathi a,b,n, R.P. Singhal a, K.P. Singh b, O.N. Singh IIa a b

Department of Applied Physics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India Department of Electronics Engineering, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India

a r t i c l e i n f o

abstract

Article history: Received 18 May 2012 Received in revised form 7 January 2013 Accepted 20 January 2013 Available online 13 February 2013

Bounce-averaged pitch angle diffusion coefficients of electrons due to resonant interaction with electrostatic electron cyclotron harmonic (ECH) and whistler mode waves have been calculated. Temporal growth rates obtained by solving the appropriate dispersion relation have been used to represent the distribution of wave energy with frequency. Calculations have been performed at two spatial locations L ¼4.6 and L ¼ 6.8. The results obtained suggest that ECH waves can put electrons on strong pitch angle diffusion at both spatial locations. However, at L ¼ 4.6, electrons with energy o 100 eV and at L¼ 6.8 electrons with energy up to  500 eV can be put on strong diffusion contributing to diffuse auroral precipitation. Whistler mode waves can put electrons of energy r 5 keV on strong pitch angle diffusion at L ¼ 6.8 whereas at L ¼ 4.6 observed wave fields are insufficient to put electrons on strong diffusion. ECH waves contribute up to 17% of the total electron energy precipitation flux due to both ECH and whistler mode waves. A case study has been performed to calculate pitch angle diffusion coefficients using Gaussian function to represent wave energy distribution with frequency. It is found that, for electron energy o500 eV, the calculated diffusion coefficients using Gaussian function to represent ECH wave energy distribution are several orders of magnitude smaller or negligible as compared to diffusion coefficients calculated by temporal growth rates. However, the calculated pitch angle diffusion coefficients using Gaussian function for whistler mode wave energy distribution are in very good agreement with diffusion coefficients calculated by temporal growth rates. It is concluded that representing the ECH wave energy distribution with frequency by a Gaussian function grossly underestimates the low energy (o500 eV) electron precipitation flux due to ECH waves. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Electrostatic electron cyclotron harmonic wave Whistler mode wave Resonant interaction Pitch-angle and strong diffusion Diffuse auroral precipitation

1. Introduction The diffuse aurora is a low-level light emission from auroral ionosphere. There is evidence to show that the diffuse aurora is the result of keV electron precipitation from the magnetosphere, with a smaller contribution from proton precipitation (Lui et al., 1977). Unlike the discrete aurora, the diffuse aurora is relatively unstructured and is an almost permanent feature of the auroral ionosphere. This semipermanent property is important since, when integrated over time, it means that the electron precipitation is major source of loss of magnetospheric particles at keV energies (Horne et al., 2003; Meredith et al., 2009; Thorne et al., 2010). A precise and definitive relation between diffuse auroras and their particle source region was determined by Meng et al. (1979). These authors examined simultaneously the particle observations at the geosynchronous orbit and the auroral display as well as the auroral electron precipitation by the polar-orbiting DMSP 32 satellite. n Corresponding author at: Department of Applied Physics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India. Tel.: þ 91 542 6702006. E-mail address: [email protected] (A.K. Tripathi).

1364-6826/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jastp.2013.01.013

It was found that the spectral shape and differential fluxes of precipitated auroral electrons of diffuse auroras are very similar, and sometimes almost identical, to those of the trapped plasma sheet electrons. Diffusion in pitch angle is important in scattering trapped magnetospheric particles into the terrestrial atmosphere. Cyclotron resonance wave-particle interaction is the most commonly discussed mechanism (Kennel and Petschek, 1966; Roberts, 1969). The pitch angle diffusion based on efficiency has both weak-diffusion and strong diffusion limits. For weak pitch angle diffusion the fluxes within the loss-cone are much smaller than those outside whereas in strong pitch angle diffusion the fluxes within the loss-cone are nearly equal to those of the trapped component. The nearly identical differential spectrum and intensity between the electron precipitations of diffuse auroras and trapped electrons of the plasma sheet indicate that the strong pitch angle diffusion is the process of scattering plasma sheet electrons into the polar atmosphere. Schumaker et al. (1989) studied the relationship between diffuse aurora and plasma sheet electron distributions in the energy range 50–20 keV using data from P78-1 and SCATHA satellites. They found that the electron distributions were isotropic inside the loss cone and anisotropic outside. This requires

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the scattering process to operate more efficiently within and up to the edge of the loss cone than at other pitch angles, i.e., that the diffusion coefficient (Daa) is large from a ¼0 to the edge of the loss cone and then abruptly decreases at the edge of the loss cone. There are two main theories for the origin of the diffuse auroral electron precipitation: the first is pitch angle scattering by electrostatic electron cyclotron harmonic (ECH) waves and second, by whistler mode waves. ECH emissions, also known as ‘‘n þ1/2’’ waves, occur within bands between the harmonics of the electron gyrofrequency, fce. Whistler mode waves exist below fce. Both ECH and whistler mode waves resonate with the electrons in  keV energy range. However, which of these two mechanisms is more influential in the production of the diffuse aurora remains a subject of controversy in the magnetospheric physics (e.g., Belmont et al., 1983; Lyons, 1984; Roeder and Koons, 1989; Johnstone et al., 1993; Villalon and Birke, 1995; Meredith et al., 1999, 2000; Horne and Thorne, 2000; Horne et al., 2003; Ni et al., 2008; Meredith et al., 2009; Thorne et al., 2010). It was first suggested by Kennel et al. (1970) that ECH waves could provide a mechanism for pitch angle diffusion and energization of auroral zone electrons. This suggestion was based on observations of large amplitudes (typically between 1 and 10 mV m  1 and occasionally up to 100 mV m  1) for ECH emissions from the OGO 5 satellites. Lyons (1974a) quantitatively evaluated the bounce-averaged diffusion coefficients for scattering in both energy and pitch angle. The results indicated that the ECH emissions reported by Kennel et al. (1970) could cause pitch angle scattering approaching the strong diffusion rate over the energy range from hundreds of eV to tens of keV. These energies are typical of particles responsible for the diffuse aurora. Belmont et al. (1983) have calculated the pitch angle diffusion coefficients by ECH waves and conclude that ECH waves could be partly responsible for diffuse auroral precipitation, but not entirely by themselves. Roeder and Koons (1989) present a survey of ECH waves using data from both the SCATHA and (AMPTE) IRM plasma wave instruments. They find that the occurrence of intensity of ECH wave emissions is too small to account for the continuous precipitation of magnetospheric electrons in the diffuse aurora. Schumaker et al. (1989) also support the conclusion that ECH waves are not the principal mechanism which scatters lowenergy plasma sheet electrons. Horne and Thorne (2000) evaluated the bounce-averaged rates of pitch angle diffusion for representative ECH wave frequencies in each harmonic band. They found that ECH waves have sufficient power to cause scattering near the loss cone at a rate comparable to strong diffusion limit for electrons below 500 eV. However, the magnitudes of diffusion coefficients and their variations with equatorial pitch angle are strongly dependent on adopted wave frequency and wave normal angle distribution. Further, these authors also studied the origin of pancake distributions. Highly anisotropic electron velocity distributions, peaked at 901 to the magnetic field are called pancakes from their appearance in velocity space. They conclude that ECH waves can produce pancake distributions from an initially isotropic distribution on a time scale of the order of a few hours for wave amplitude of 1 mV m  1. Horne et al. (2003) have analyzed the role of ECH and whistler mode waves during an isolated substorm. They find that ECH waves resonate with electrons in the energy range from few hundred eV to a few keV and can scatter electrons with pitch angles of up to 801 into the loss-cone. The observed hiss and chorus were only able to resonate with electrons below and above this range respectively, suggesting that ECH waves are probably the main component of diffuse auroral precipitation during this event. Another type of wave that can cause precipitation of plasma sheet electrons is whistler mode chorus. These waves, which are electromagnetic in nature, are observed in the frequency range

from 0.1 to 0.8fce (Tsurutani and Smith, 1977; Koons and Roeder, 1990). The emissions are often observed in two distinct bands, referred to as upper (0.5fce of ofce) and lower (0.1fce ofo0.5fce) band chorus. There are distinct differences in the characteristics of nightside chorus and dayside chorus. Nightside chorus waves can be excited by cyclotron resonance with anisotropic 1–100 keV electrons injected from the plasma sheet (Kennel and Petschek, 1966). In contrast, occurrence of dayside chorus could be attributed to the natural enhancement of electron anisotropy in the noon sector (Li et al., 2009). Nonlinear wave growth theories have also been developed to better understand the generation process and wave characteristics of chorus emissions (Nunn et al., 1997, 2009; Omura et al., 2008, 2009). While the interactions between whistler mode chorus and energetic magnetospheric electrons (4100 keV) have been well investigated, much less attention has been paid to the interactions between chorus waves and magnetospheric electrons below  100 keV. Wave particle interaction involving whistler mode waves and tens of keV electrons has been the subject of several early studies (Lyons et al., 1972; Inan et al., 1992; Johnstone et al., 1993; Villalon and Birke, 1995). Inan et al. (1992) and Horne et al. (2003) examined pitch angle diffusion by whistler mode waves involving electron energies 430 keV. A quantitative analysis of the resonant scattering of plasma sheet electrons  100 eV–20 keV at L¼ 6 due to resonant interaction with whistler mode chorus has been presented by Ni et al. (2008). It is demonstrated that the rate of pitch angle scattering can exceed the level of strong diffusion over a broad energy range (200 eV–10 keV). Upper band chorus is the dominant scattering process for electrons below 5 keV while lower band chorus is more effective at higher energies. Su et al. (2009) in their study find that the upper band chorus can effectively scatter the electrons with energies 0.1–2 keV into loss cone. The lower band chorus can only cause precipitation loss of the electrons with energies above 1 keV. Further, Su et al. (2010) studied the dependence of pitch angle diffusion rates on spatial position and properties of chorus waves. Their numerical results suggest that the resonant scattering of electrons by the whistler mode chorus can be a substantial mechanism responsible for the formation of a pancake distribution outside L¼6 and more effective in the production of diffuse aurora at larger L shells. Recently, Thorne et al. (2010) report an analysis of satellite wave data and the Fokker–Planck diffusion calculations and conclude that scattering by chorus waves is the dominant cause of the most intense diffuse auroral precipitation. More recently Ni et al. (2011a,b) have performed a very comprehensive analysis on the scattering of plasma sheet electrons by ECH and whistler mode chorus waves at Lr6. These authors used statistical wave power spectral profiles obtained from the Combined Release and Radiation Effects Satellite (CRRES) under different levels of geomagnetic activity. It was concluded that pitch angle scattering by ECH waves is negligible for electrons having energies o100 eV. Further, ECH wave induced resonant diffusion coefficients are at least one order of magnitude smaller compared to the effects of chorus waves and are negligible under any geomagnetic condition. Meredith et al. (1999) examined the electron pitch angle distributions over the energy range 100 eV–30 keV measured by Low Energy Plasma Analyzer on board the CRRES. Their results suggest that whistler mode waves play a dominant role in the formation of pancake distributions outside L¼6, whereas inside L¼6 and in particular, in the vicinity of plasmapause, the ECH waves also play a significant role. Consequently, both types of wave should be considered in any attempt to explain the diffuse aurora (Meredith et al., 2009) and the variation with L should be taken into account. The terrestrial magnetosphere produces various plasma instabilities which lead to the emission of plasma waves propagating in

A.K. Tripathi et al. / Journal of Atmospheric and Solar-Terrestrial Physics 97 (2013) 125–134

various modes. Most of these instabilities are due to anisotropic electron distribution, such as beams, loss-cone feature and temperature anisotropies. Instability analysis shows that the temperature anisotropy excites whistler mode waves and the ECH waves are driven unstable by the loss-cone. We have simulated whistler mode waves in different frequency ranges by changing the temperature anisotropy. In situ measurements by ISEE 1 and CRRES found a well developed non-Maxwellian and suprathermal tail of energetic electrons in terrestrial magnetosphere (Christon et al., 1988; Meredith et al., 1999, 2009). The electron distributions are a mixture of both Maxwellian, as a cold component, and nonMaxwellian suprathermal tail, as a hot component. In the studies of pitch angle diffusion due to resonant scattering by ECH and whistler mode waves the waves are assumed to have a Gaussian distribution of wave energy with frequency. In the present study we solve the appropriate dispersion relation to obtain the temporal growth rate profiles. These are used to represent the wave energy distribution with frequency to calculate the pitch angle diffusion coefficients. The spectral intensity of the wave is an important input for the calculation of diffusion coefficients. It is therefore necessary to make a quantitative analysis to explore whether or not different spectral intensity profiles will result in different physical conclusions. The objectives of the present study are: (1) to compare the pitch angle diffusion coefficients calculated by using Gaussian frequency distribution and by using the profile obtained by solving the appropriate dispersion relation (2) to examine the role of ECH and whistler mode waves inside L¼6 and outside L¼ 6 (3) to estimate the relative contribution of ECH and whistler mode waves towards production of diffuse aurora in the terrestrial atmosphere (4) to compare the local and bounceaveraged pitch angle diffusion coefficients. The paper is arranged in the following manner: In Section 2 we briefly present the plasma wave observations of ECH and whistler mode waves. Section 3 describes the dispersion relation for each wave mode. This section also presents the expressions used for calculating the local and bounce-averaged pitch angle diffusion coefficients by resonant interaction with ECH and whistler mode waves. Expressions for strong diffusion limit and precipitation flux are given in Section 4. Section 5 presents the observations of plasma parameters used in the present study. Section 6 summarizes the results and discussion. Finally, Section 7 gives the conclusions.

2. Wave observations Analysis of the occurrence of whistler mode and ECH waves has been provided by the PWE on board the CRRES during a magnetically quiet period (Meredith et al., 1999, 2009). Weak ECH emissions typically less than 0.1 mV m  1, are seen during quiet conditions but only in the region L44.5. These emissions became more enhanced during moderate conditions and extend to slightly lower L with peak amplitudes in the range (0.4–0.8) mV m  1. The most enhanced emissions are seen during active conditions in the region 3.5 oLo7.0 where wave amplitudes exceed 1 mV m  1, particularly in the region outside L¼ 6. ECH waves are seen in the first three harmonic bands. In the first harmonic band the emissions are double-banded with peak emissions near 1.1fce and 1.5fce. In the second harmonic band the wave emissions peak in the center of the band, whereas in the third harmonic band the emissions maximize at the lower end in the band. ECH waves are also present over a wide range of frequencies in the first harmonic band. Frequencies typically maximize in the range 1.4fce of o1.8fce but also extend to higher and lower frequencies.

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Weak plasmaspheric hiss is observed below 2 kHz inside the plasmapause. Weaker upper band chorus amplitudes  0.01 mV m  1 are seen during quiet conditions in the region L44.5. These emissions intensify during moderate conditions and extend to slightly lower L. The waves are most intense during active conditions in the region L44. Here the wave amplitudes exceed 1 mV m  1. Upper band chorus waves are typically observed from 0.5fce–0.6fce (Horne et al., 2003; Meredith et al., 2009).

3. Expressions for dispersion relation and pitch-angle diffusion coefficients 3.1. Dispersion relation We consider a plasma consisting of a cold electron component represented by a Maxwellian distribution, fM and a hot component given by the generalized Lorentzian (Kappa) loss-cone distribution, fk. The Maxwellian distribution is given by   1 v2 v2 exp  2O  2l , f M ¼ 3=2 2 ð1Þ vcl p vcl vcO vcO with associated perpendicular and parallel thermal speeds v2cl,O ¼ 2T cl,O =me

ð2Þ

where Tc and me are the temperature and mass of electron, respectively. The bi-Lorentzian loss-cone distribution is given by  2‘ p3=2 Gðk þ ‘ þ 1Þ vl fk ¼ 2 ‘ þ 3=2 k yl yO Gðk1=2Þ Gð‘ þ 1Þ yl ( )ðk þ ‘ þ 1Þ v2O v2l  exp 1 þ þ , ð3Þ 2 2

kyO

kyl

where yl and yO are thermal speeds given by     2k3 1=2 1 T hl 1=2 yl ¼ , k ð‘ þ 1Þ1=2 me

yO ¼



ð4Þ

   2k3 1=2 T hO 1=2 k me

ð5Þ

Dispersion relation for electrostatic waves in magnetized plasma given by Summers and Thorne (1995) has been used in our study. The angle between wave normal and ambient magnetic field is set equal to 891. For cold electron population temperatures Tcl ¼TcO ¼Tc and for the hot component temperatures Thl ¼ ThO ¼Th. The spectral index parameter k ¼2 and the loss-cone index ‘ ¼ 1 have been taken for our computation. For whistler mode waves the dispersion relation for parallel propagating R-mode is used (Summers and Thorne, 1995; Mace, 1998). The cold electrons are assumed isotropic and hot electrons are anisotropic with temperature anisotropy A¼

T hl 1 T hO

ð6Þ

Dispersion relation is broken into real and imaginary parts and a numerical technique is used to find the real and imaginary components of the angular frequency (o ¼ or þig). 3.2. Pitch angle diffusion coefficients for ECH waves The pitch-angle diffusion coefficient for electrostatic ECH waves, in units of per second, is given by Lyons (1974a) 2 !2 3 Z 1 X ðnOe =ok Þsin2 a 5 , ð7Þ kl dkl 4cn,k Daa ¼ sinacosa n ¼ 1 kO ¼ kres

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where

and

 2 e2 Ek  cn,k ¼ 4pm2e V

o  k

!2

k

J 2n ðkl vl =Oe Þ   v4 vO @ k =@kO 

o

ð8Þ

andkl and kO are the components of the wave vector perpendicular and parallel to the ambient magnetic field Bo, respectively, k  res ¼(ok  nOe)/vO is the resonant parallel wave number, Oe ¼ eBo =me  is the (angular) electron gyro-frequency, ok is the wave frequency as a function of k, Ek is the wave electric field at each k. a and v are the particle pitch-angle and velocity respectively. V is plasma volume, e/me is the electron  2 charge to mass ratio, and Jn is the Bessel function of order n. Ek  is expressed as  2 Ek  ¼ ðV=NðoÞÞE2 ðoÞg o ðyÞ ð9Þ where NðoÞ ¼ 2=ð2p2 Þ

Z 1

0

   kl ,kO  dx, g o ðcos1 xÞkl J o,x 

ð10Þ

The distribution of wave energy with frequency and wave normal angle is parameterized as follows (Tripathi and Singhal, 2009) f ðoÞ f ðoÞdo

ð12Þ

and  2 g o ðyÞ pexp x=xo

ð13Þ

here x ¼cos y and xo is set equal to 0.01. The function f(o) determines the distribution of wave energy with wave frequency. We have considered two profiles for f(o). The first profile is given by the temporal growth rate calculated by solving the dispersion relation. It is assumed that wave energy is proportional to the linear temporal growth rate (g). The constant of proportionality does not appear in the expression of diffusion coefficients (Lyons, 1974b; Tripathi and Singhal, 2009). The linear temporal growth rate profiles, therefore, represent the distribution of wave energy with frequency. The second profile is a Gaussian frequency distribution given by 

oom 2 f ðoÞpexp  ðolc o o o ouc Þ ð14Þ

do

where om and do are the frequency of maximum wave power and bandwidth, respectively. olc and ouc are lower and upper cutoff frequency to the wave spectrum outside which the wave power is zero. 3.2.1. Bounce-averaged diffusion coefficients The bounce-averaged diffusion coefficient is given by (Lyons et al., 1972) Z tB hDaa i ¼ 1=tB Daa ðtÞð@aeq =@aÞ2 dt ð15Þ 0

where aeq is the equatorial pitch angle and tB is the particle bounce time for one complete cycle between the particle mirror points given by (Orlova and Shprits, 2011). Here   tB ¼ 4RE L=v T ðyÞ ð16Þ

ð17Þ

with y¼sin aeq where RE is Earth’s radius and v is particle velocity. Since ECH waves are generally confined to within a few degrees of the magnetic equator l  731 (Gough et al., 1979; Meredith et al., 2009), pitch angle diffusion is effectively zero at higher latitudes. Assuming that the local diffusion coefficient is approximately constant over this narrow latitude region and neglecting any variations due to changes in pitch angle, we can approximate (Horne and Thorne, 2000) Z Daa lint hDaa i ¼ ð2=vcosaeq Þds,

tB

lint

¼ T f rac Daa

and x¼cos y, E2(o) is the wave electric field intensity squared per unit frequency and go(y) gives the variation of wave electric field energy with wave normal angle for each frequency. The Jacobian in N(o) can be expressed as (Lyons, 1974a)     kl ,kO @kl  @kO  ¼ J ð11Þ o,x @x o @o kl

E2 ðoÞ ¼ E2wave R

T ðyÞ ¼ 1:38090:1851y0:5 0:4559y0:863 ,

ð18Þ

where T f rac ¼ ð4RE Llint Þ=ðvcosaeq tB Þ

ð19Þ

We set Tfrac ¼1 for particles whose mirror point is less than lint. 3.3. Pitch angle diffusion coefficients for whistler mode waves Following Lyons (1974b), the normalized diffusion matrix D is defined as " #1 B2wave 2 D ¼ D Oe 2 v ð20Þ B0 where v is electron speed, D is diffusion matrix and Bwave is the wave magnetic field. Each element of the matrix is written as a sum over all resonances n and an integral over x Z xmax 1 X D¼ xdxDnx ð21Þ n ¼ 1

0

where x ¼tan y and y is the wave normal angle (the angle between Bo and k). Dnx is given by   pcos5 y Oe ðsin2 anOe =ok Þ2 Fn,k 2 nx Daa ¼  3 3=2  2C 1 c 1 þ nOe =ok  Iðok Þ    1 @ok  ð22Þ f ðoÞg o ðxÞ 1  V @k  ok  ok  O

O x

Oe

¼

Oe

res

where f(o) is the wave spectral density and g o ðxÞ gives the wave normal distribution. For g o ðxÞ we use for x r 1 ð23Þ for x Z 1 R In Eq. (22) C 1 ¼ f ðoÞdo, a is pitch angle. Expressions for    2 Fn,k  , c, I (ok), (ok/Oe)res, and 1 1 @ok9 are lengthy and not V O @kO x g o ðxÞpexpðx2 Þ 0

being given here for the sake of brevity. These can, however, be found in the papers of Lyons (1974b) and also Singhal and Tripathi (2006). As for ECH waves, we have considered two profiles for f (o). The first profile is given by the temporal growth rate calculated by solving the appropriate dispersion relation. The second profile is a Gaussian frequency distribution given by Eq. (14). 3.3.1. Bounce-averaged diffusion coefficients Following Lyons et al. (1972) bounce-averaged pitch angle diffusion coefficient in a dipole magnetic field is given by Z lm cosðaðlÞÞcos7 l hDaa i ¼ 1=TðyÞ dl Daa ðaðlÞÞ ð24Þ cos2 ðaeq Þ 0

A.K. Tripathi et al. / Journal of Atmospheric and Solar-Terrestrial Physics 97 (2013) 125–134

where l is the latitude and a(l) is the local pitch angle which is related to the equatorial pitch angle aeq by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 3sin2 l sin2 aeq ð25Þ sin2 a ¼ cos6 l The mirror latitude lm is found by solving the sixth order equation (Shprits et al., 2006a) X 6 þð3sin4 ðaeq ÞÞx4sin4 ðaeq Þ ¼ 0,

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Table 1 Plasma parameters. Spatial location L¼ 4.6 L¼ 6.8

Case A Case B

nc (cm  3)

nh (cm  3)

100.0 21.5

20.0 4.3

4.0 4.0

800 800

12.0

6.0

10.0

1000

Tc (eV)

Th (eV)

Bo (nT) 312 312 92.5

ð26Þ

where x¼cos2(lm) In the present calculations we assume that the whistler mode waves are confined within 151 of the magnetic equator (Meredith et al., 2009).

4. Strong diffusion and precipitation flux Pitch angle diffusion is weak when particles are unable to diffuse significantly across the loss cone aLC within a bounce time tB. At opposite extreme, the wave intensity may be sufficiently intense so that particles can readily diffuse across the dimensions of the loss cone within a bounce period. In this case the bounce averaged precipitation time approaches an asymptotic limit value, controlled only by the particle bounce time and the geometric size of the loss cone. The strong diffusion limit DSD is given by (Lyons, 1974a) DSD ¼ 4a2LC =tB

ð27Þ

Under the dipole approximation, loss cone is written as h i1=4 sinaLC ¼ L5 ð4L3Þ

ð28Þ

Pitch angle scattering of particles, during wave particle interaction, can violate the first and second adiabatic invariants and transport the particles into the loss cone. Here they are removed by collisions with the atmospheric particles. Shprits et al. (2006b) show that the electron’s lifetime are most sensitive to the value of the pitch angle scattering rate near the edge of the loss cone. Electron precipitation flux Jp(E) due to resonant interaction with ECH and whistler mode waves is calculated by using the relation (Thorne, 1983) J p ðEÞ ¼ ðhDaa i=DSD ÞJ T ðEÞ

1

at L ¼ 4:6

ð30Þ

1

at L ¼ 6:8

ð31Þ

cm2 s1 sr1 keV

and J T ðEÞ ¼ ð4:60  105 Þ=E

location of the plasmapause is quite variable and often difficult to determine. It generally lies between  L¼2.5 and 5.5 (Carpenter and Anderson, 1992). The location L¼4.6 may therefore be inside or outside of the plasmapause. Electron density in the plasmasphere is high and drops by about a factor of 5 outside the plasmapause. We have therefore considered two electron densities for L¼4.6: (i) case A (nc ¼100 cm  3) and (ii) case B (nc ¼21.5 cm  3). The plasma parameters used in this study are given in Table 1.

ð29Þ

where hDaa iis the bounce-averaged diffusion coefficient at the edge of the loss cone and JT(E) is the trapped electron flux given by Summers et al. (2009) J T ðEÞ ¼ ð2:25  106 Þ=E

Fig. 1. Normalized temporal growth rates gð ¼ g=Oce Þ for ECH waves as function of normalized frequency or ¼ 1þ Z at L ¼4.6 (nc ¼ 100 cm  3 and nc ¼ 21.5 cm  3) and L¼ 6.8.

cm2 s1 sr1 keV

E is the electron energy in keV. Energy precipitation flux is calculated by the expression Z Emax ep ¼ p EJ p ðEÞdE ð32Þ Emin

5. Plasma parameters Meredith et al. (1999) have presented the plasma wave observations and plasma parameters at the equatorial crossing from the PWE on board CRRES spacecraft. DeForest and McIlwain (1971) have reported the cold electron densities and temperatures at synchronous altitude. Similar electron densities have been deduced at synchronous orbit from electron fluxes measured by ATS6 (Meng et al., 1979) and SCATHA satellites (Schumaker et al., 1989). Numerical calculations have been performed at two spatial locations L¼4.6 and L¼6.8. The

6. Results and discussion Temporal growth rates obtained from the solution of appropriate dispersion relation will be denoted as standard profiles. As noted earlier, these profiles directly represent the wave energy profiles. In Fig. 1 we show the temporal growth rates for ECH waves in the first harmonic band at L¼4.6 and 6.8. Two profiles are shown for location 4.6 corresponding to electron densities 100 cm  3 and 21.5 cm  3. At first location 4.6 growth rate peaks around normalized frequency (o/Oc)E1.85 and covers a wide range of frequencies. At second location 6.8 double banded ECH waves are obtained having peaks around o/Oc E1.2 and 1.8. These are generally in agreement with ECH wave observations (Meredith et al., 2009). In Figs. 2 and 3 we present the bounce-averaged pitch angle diffusion coefficients due to ECH waves using the standard profiles shown in Fig. 1. The wave electric field 1 mV m  1 is used in the calculation of diffusion coefficients. From Fig. 2, representing the location 4.6 corresponding to nc ¼100 cm  3, it is noted that the diffusion coefficients drop by about two orders of magnitude within equatorial pitch angles  351–401. Further, coefficients for electron energies r200 eV remain constant inside the loss cone  4.31 and after that fall off at higher pitch angles. At higher electron energies diffusion coefficients decrease from the edge of the loss cone towards smaller pitch angles. Diffusion coefficients for energies 42 keV are an order of magnitude smaller as

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Table 2 Diffusion rates of ECH waves for several electron energies. Spatial op/Oe location

aLC

EK (deg.) (eV)

/DaaS(s  1) DSD (s  1)

L¼ 4.6 10.3 Case A

4.3

50 100

3.4  10  4 5.8  10  5

6.4  10  4 1.4 9.0  10  4 (0.02–1.4) 3.9

L¼ 6.8

2.4

50 100 200 500 1000

6.3  10  4 1.6  10  4 1.6  10  4 1.3  10  4 2.5  10  5

1.3  10  4 0.45 1.8  10  4 1.1 2.5  10  4 (0.003–2.6) 1.3 3.9  10  4 1.7 5.7  10  4 4.8

12.0

n

EWobs EWreq (mV m  1) (mV m  1)

aLC is loss-cone angle at particular height; n

EWobs is the range of magnitude of observed electric field at particular location EWreq is magnitude of required electric field for strong diffusion at particular location. Fig. 2. Bounce-averaged electron pitch angle diffusion coefficients versus equatorial pitch angle for ECH waves at L¼4.6 (nc ¼ 100 cm  3) using temporal growth rate profile from Fig. 1 for various electron energies.

Fig. 3. Same as in Fig. 2 but at L¼ 6.8.

compared to values at lower electron energies. Thus the scattering for energies r200 eV is efficient within and up to the edge of the loss cone. Diffusion coefficients at location 4.6 for nc ¼21.5 cm  3 (not shown) are smaller up to a factor of 5 as compared for values of nc ¼100 cm  3. It is noticed from Fig. 3 (L¼6.8) that the diffusion coefficients remain constant up to about 101 and then drop down by two orders of magnitude within 301. Thus ECH wave scattering cannot account for the formation of pancake distribution which exhibits a depleted electron population for pitch angle o701. Lyons (1974a) has calculated bounce-averaged pitch angle diffusion coefficients at L¼ 7 for electrons of energies 10 eV to few keV for ECH waves of single frequency o ¼ 1.5Oc. Similar calculations have been performed by Horne and Thorne (2000) at several frequencies in first and higher harmonic bands for electrons of energies 200 eV to 2 keV. The results obtained in the present work are in general agreement with these works. It may, however, be noted that the diffusion rates calculated by Lyons (1974a) and Horne and Thorne (2000) cover a wider range of pitch angles than obtained in the present work. Recently, Ni et al. (2011a) have calculated diffusion coefficients using Gaussian distribution of ECH wave energy with frequency. It is found that pitch angle scattering by ECH waves is negligible for electrons with energies o100 eV. Further, the diffusion coefficients for energies 200 eV and 500 eV reported by these authors are smaller than calculated in the present work. Comparing the diffusion coefficients at the edge of loss cone with strong diffusion limit, we get the wave field

Fig. 4. Normalized temporal growth rates gð ¼ g=Oce Þ versus normalized frequency o r ð ¼ or =Oc Þ for whistler mode waves at L¼ 4.6 (nc ¼100 cm  3) corresponding to three values of temperature anisotropy (A).

required for strong pitch angle diffusion. The observed (Meredith et al., 1999) and required wave fields for ECH waves are given in Table 2. It is found that the observed wave fields at location 4.6 for nc ¼ 21.5 cm  3 are insufficient to put electrons on strong diffusion for any electron energies. However, for nc ¼100 cm  3 electrons of energy  50 eV can be put on strong diffusion. For location 6.8 electrons of energy up to E500 eV can be put on strong diffusion. Thus electrons of energies less than 500 eV can be rapidly scattered by ECH waves into the atmosphere and contribute to diffuse aurora. In Figs. 4 and 5 we present the temporal growth rates for whistler mode waves at L¼4.6 (nc ¼100 cm  3) and 6.8, respectively. Three curves are shown in each figure corresponding to temperature anisotropies A¼0.1, 1.1, and 6.0. As the anisotropy is increased, frequency of whistler mode waves shifts to higher frequency range. The anisotropies A¼0.1, 1.1 and 6.0 give rise to whistler mode waves in the normalized frequency ranges of o/Oc E0.03–0.06, 0.1–0.5, and 0.2–0.8, respectively. Figs. 6 and 7 represent the bounce-averaged diffusion coefficients due to whistler mode waves at L¼4.6 (nc ¼100 cm  3) for temperature anisotropies 1.1 and 6.0, respectively. Standard profiles shown in Fig. 4 have been used and a wave magnetic field 10 pT is taken in calculations. It is noted from Fig. 6 that the diffusion coefficients for electron energies 1 keV, 5 keV, 10 keV, and 30 keV cover pitch angles up to about 401, 661, 721, and 741 respectively. From Fig. 7 it is seen that the diffusion coefficients cover a wide range of pitch angles up to about 881. As the electron energy increases from 1 to

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Fig. 5. Same as in Fig. 4 but at L ¼6.8.

Fig. 6. Bounce-averaged electron pitch angle diffusion coefficients versus equatorial pitch angle for whistler mode waves at L ¼4.6 (nc ¼ 100 cm  3) using temporal growth rate profile from Fig.4 (A ¼1.1) for various electron energies.

Fig. 7. Same as in Fig. 6 but at A ¼6.0.

30 keV, diffusion coefficients drop by two orders of magnitude. The coefficients due to whistler mode waves have also been calculated for L¼4.6 (nc ¼21.5 cm  3) (not shown). It is found that diffusion coefficients are increased by about 3 to 10 as compared

131

Fig. 8. Same as in Fig. 6 but at L¼ 6.8 using temporal growth rate from Fig.5 (A¼1.1).

Fig. 9. Same as in Fig. 8 but for A ¼ 6.0.

to values for case nc ¼100 cm  3. Other characteristics of diffusion coefficients are same as for nc ¼100 cm  3. In Figs. 8 and 9 we present the diffusion coefficients due to whistler mode waves at L¼6.8 for temperature anisotropies 1.1 and 6.0, respectively. Standard profiles shown in Fig. 5 have been used with wave magnetic field 10 pT. It is seen in Fig. 8 that the coefficients cover a range of pitch angles up toE581 for electron energy of 1 keV. As electron energy is increased to 30 keV the range of angles also increases to E781. In Fig. 9 the diffusion coefficients cover a wide range of pitch angles up to E861 at all energies. Thus whistler mode waves can account for the formation of pancake distributions at both locations L¼ 4.6 and 6.8. Resonance scattering of plasma sheet electrons by whistler mode chorus waves has been the subject of many studies (Ni et al., 2008; Su et al., 2009, 2010; Ni et al., 2011b). The values of pitch angle diffusion coefficients obtained in the present work are generally in agreement with values reported by these authors. The observed (Meredith et al., 1999) and required wave fields for strong diffusion due to whistler mode waves are given in Table 3. Meredith et al. (1999) have reported the wave electric field component for whistler mode waves. It is converted to wave magnetic field using the relation BW ¼EW/vg, where the group velocity vg is obtained by solving the dispersion relation. From Table 3 it may be noted that the observed wave magnetic field at L¼4.6 is insufficient to put 1–30 keV electrons on strong diffusion. However, at L¼6.8 electrons of energies r5 keV may be put on strong diffusion.

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Table 3 Diffusion rates of whistler mode waves for several electron energies. Spatial op/ location Oe

aLC

L ¼4.6 Case A

4.3

L ¼4.6

(deg.)

10.3

4.8

4.3

A

DSD (s  1)

EK /DaaS (keV) (s  1)

n

BWobs (pT)

BWreq (pT)

3.9  10  5

2.1  10  2

232.0

1.1 1.0 5.0 6.0 1.0 5.0 0.1 –

5

4.2  10 1.0  10  5 2.9  10  5 3.6  10  6 –

3

2.8  10 6.4  10  3 2.8  10  3 6.4  10  3 (0.45–7.6) –

82.0 253.0 99.0 422.0 –

1.1 5.0 10.0 6.0 1.0 5.0 0.1 30.0 50.0

5.4  10  5 2.6  10  5 1.2  10  4 1.6  10  5 6.0  10  5 3.1  10  5

6.4  10  3 9.1  10  3 (0.45–7.6) 2.8  10  3 6.4  10  3 3.1  10  4 4.0  10  3

109.0 187.0 49.0 200.0 72.0 114.0

1.1 1.0 5.0 10.0 6.0 1.0 5.0 10.0

9.6  10  5 3.0  10  5 8.7  10  6 6.0  10  5 1.8  10  5 4.8  10  6

5.7  10  4 (0.14–64) 1.3  10  3 1.8  10  3 5.7  10  4 1.3  10  3 1.8  10  3

24.4 65.8 144.0 30.8 85.0 194.0

0.1 50

Case B

L ¼6.8

12.0

2.4

Fig. 10. Bounce-averaged electron pitch angle diffusion coefficients versus equatorial pitch angle for ECH waves at L¼ 4.6 (nc ¼21.5 cm  3) using temporal growth rate profile from Fig. 1 (standard profile) and Gaussian profile with bandwidth do ¼ 0:15 for various electron energies. The scaling factors by which diffusion coefficients have been multiplied are given in the parentheses in the figure.

aLC is loss-cone angle at particular height; n

BWobs is the range of magnitude of observed magnetic field at particular location BWreq is magnitude of required magnetic field for strong diffusion at particular location.

Table 4 Electron energy precipitation flux (ergs cm  2 s  1). Spatial location

Whistler mode waves A

L ¼4.6

L ¼4.6

L ¼6.8

Case A

Case B

ECH waves

eP

eP 4

0.1 1.1 6.0

6.0  10 1.4  10  3 8.0  10  4

5.9  10  4

0.1 1.1 6.0

– 2.1  10  3 3.4  10  3

2.5  10  4

0.1 1.1 6.0

1.8  10  3 2.6  10  3 2.0  10  3

9.6  10  4

Fig. 11. Same as in Fig.10 but with Gaussian profile having bandwidthdo ¼ 0:25.

In Table 4 we present the electron energy precipitation flux due to ECH and whistler mode waves. The wave electric field for ECH waves 1.0 mV m  1 and wave magnetic field for whistler mode waves 30 pT is used. It is found that ECH waves at L¼4.6 contribute 4–17% electron energy fluxes and at L¼6.8 contribution is 13% of the total precipitation flux (due to both waves). It may be remarked that these estimates are based on quasilinear diffusion theory. Magnetospheric wave–particle interaction processes are, however, necessarily nonlinear phenomena. Dominant waves such as whistler mode chorus emissions are due to strongly nonlinear processes. The present calculations based on the linear growth rates may not represent the real magnetospheric wave effects on precipitation with sufficient accuracy. 6.1. Gaussian distribution of wave energy with frequency We have also calculated bounce-averaged pitch angle diffusion coefficients due to ECH waves at L¼4.6 (nc ¼ 21.5 cm  3) using a Gaussian function for the ECH wave energy distribution with frequency. Parameters of the Gaussian profile are: o m ¼ 1:85,

do ¼ 0:15(Ni et al., 2011a), o ‘c ¼ 1:125 and o uc ¼ 1:90 (bar denotes normalized quantities). In Fig. 10 we compare the diffusion coefficients by Gaussian distribution (do ¼ 0:15) with coefficients calculated by temporal growth rates shown in Fig. 1 (standard profile of L¼4.6 for nc ¼21.5 cm  3). It is observed from Fig. 10 that the Gaussian profile produces negligible diffusion coefficients ( 10  11 s  1) at electron energy of 50 eV. At energy 100 eV the magnitude of coefficients due to Gaussian profile are 3–4 orders of magnitude smaller as compared to values using standard profile. For energy 500 eV the magnitude of Gaussian diffusion coefficients are found again an order of magnitude smaller inside the loss cone as compared to values for standard profile. Fig. 11 shows the same variation as in Fig. 10 but in this case bandwidth do ¼ 0:25has been considered. It may be noticed from Fig. 11 that for electron energies 50 and 100 eV, the Gaussian diffusion coefficients are two orders of magnitude smaller than the values for standard profile. For energy 200 eV Gaussian diffusion coefficients are an order of magnitude smaller as compared to values of Standard diffusion coefficients (not shown). Thus it may be concluded that the low energy (o500 eV) electron precipitation flux for ECH Gaussian function is grossly underestimated. In Fig. 12 we present the bounce-averaged pitch angle diffusion coefficients due to whistler mode waves at L¼6.8 for both Gaussian

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Fig. 12. Bounce-averaged electron pitch angle diffusion coefficients versus equatorial pitch angle for whistler mode waves at L ¼ 6.8 using temporal growth rate profile from Fig. 5 (A¼ 1.1) (standard profile) and Gaussian profile with bandwidth do ¼ 0:1 for various electron energies. The scaling factors by which diffusion coefficients have been multiplied are given in the parentheses in the figure.

and standard profile shown in Fig. 5 (standard profile for A¼ 1.1 at L¼6.8). Parameters of the Gaussian profile are: om ¼ 0:3, do ¼ 0:1, o‘c ¼ 0:003 and ouc ¼ 0:53 (Ni et al., 2011 b). It may be noted from Fig. 12 that the diffusion coefficients of both profiles are within a factor of 2–3 for electron energies r10 keV. For energy 30 keV, Gaussian diffusion coefficients are an order of magnitude smaller as compared to values for standard profile. Further, for a Gaussian profile for bandwidth do ¼ 0:15 there is very good agreement in the diffusion coefficients calculated by the standard and Gaussian profile (diffusion coefficients not shown).

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2. The observed ECH wave field amplitudes are sufficient to cause strong diffusion of electrons of energy o500 eV. However, at L¼4.6 electrons of energy o100 eV can be put on strong diffusion whereas at L¼6.8 electrons of energy up to  500 eV can be put on strong diffusion. Observed whistler mode wave amplitudes at L¼4.6 are insufficient to cause strong diffusion of electrons of energy 1–30 keV. At L¼6.8 whistler mode waves can put electrons of energy r5 keV on strong diffusion. 3. ECH waves contribute 4–17% energy precipitation flux to diffuse aurora at location L¼4.6. Further, ECH wave contribution to diffuse aurora at L¼6.8 is about 13% of the energy flux due to both waves. 4. Using Gaussian function to represent ECH wave energy distribution with frequency calculated pitch-angle diffusion coefficients, for electrons of energy o500eV, are several orders of magnitude smaller or negligible as compared to diffusion coefficients using temporal growth rate profile. Thus, a Gaussian ECH wave energy distribution grossly underestimates low energy ( o500 eV) electron precipitation flux. 5. Use of a Gaussian function for whistler mode wave energy distribution with frequency produces pitch-angle diffusion coefficients which are in very good agreement with diffusion coefficients calculated using the temporal growth rate profile.

Acknowledgment This work was supported by financial assistance provided by the Planetary Sciences and Exploration Programme, Indian Space Research Organization (ISRO), PRL, Ahemdabad under the sanctioned project scheme P-32-17. Calculations reported in the present work were carried out at the Computer Centre, Banaras Hindu University.

6.2. Local versus bounce-averaged pitch angle diffusion coefficients. We have calculated the local pitch angle diffusion coefficients and compared with bounce-averaged diffusion coefficients. It is found that for ECH waves local diffusion coefficients are up to factor of 25 larger than the bounce-averaged coefficients. Thus, using the local diffusion coefficients the electron precipitation flux would be overestimated and even be equal to the trapped electron flux. In case of whistler mode waves local diffusion coefficients are up to a factor of 5 larger than the bounce-averaged diffusion coefficients. Electron precipitation flux due to whistler mode waves would be overestimated using the local pitch angle diffusion coefficients.

7. Conclusions In the present study we have calculated bounce-averaged electron pitch angle diffusion coefficients due to resonant interaction with ECH and whistler mode waves. Temporal growth rate profiles obtained by solving the appropriate dispersion relation have been used to represent the wave energy distribution with frequency. Calculations of diffusion coefficients have also been performed using a Gaussian function to represent the wave energy distribution with frequency. Numerical calculations have been carried out at two spatial locations, L¼4.6 and 6.8. Our main conclusions are summarized as follows:

1. ECH wave scattering cannot account for the formation of pancake distribution at any spatial location. Whistler mode waves can account for the formation of pancake distribution both at L¼4.6 and 6.8.

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