Regular and turbulent mechanisms of relativistic electron acceleration in the

magnetosphere if the Earth: Theoretical treatment and results of experimental

observations

E.E. Antonova

M.F. Bahareva, I.P. Kirpichev, I.N. Myagkova., I.L. Ovchinnikov, K.G. Orlova, M.V. Stepanova, M.O.Riazantseva, V.V. Vovchenko

Dedicated to the memory of S. N. Kuznetsov

•Problem of acceleration of relativistic electrons in the magnetosphere of the Earth. • Inner magnetosphere source of relativistic electrons. •Role of regular and turbulent mechanisms of relativistic electron acceleration.•Magnetospheric topology and magnetic storms.•Local plasma traps in the magnetosphere of the Earth and their formation.

Outer radiation belt (ORB) was discovered in 1958 by Vernov, Chudakov, Gorchakov, Logachev (discovery No 23 in the list of USSR discovery registration) in SINP MSU and was studied on satellites Electron, SAMPEX, Polar, GPS, CRRES, LANL, GOES, HEO ets. Real progress in ORB study was achieved due to CORONAS satellite measurements.

Results of Image satellite observations

Acceleration of relativistic electrons takes place in most cases during storm recovery phase. The level of magnetospheric turbulence is greatly increased during magnetic storm, the plasmasphere is decreased and the slot between external and internal radiation belts disappears.

The process of acceleration can be connected with the high level of turbulence and with the change of magnetic field.

Two main approaches of the solution of the problem of acceleration of relativistic

electrons:

Stochastic acceleration by fluctuating electric fields.

Injection of “seed” population in the region of depressed magnetic field inside the magnetosphere and regular betatron acceleration due to magnetic field increase during storm time recovery phase.

Stochastic mechanisms of acceleration are most popular now. Interactions with ULF (Fujimoto and Nishida, 1990; Liu et al., 1999; Hudson et al., 1999; Elkington et al., 1999; Liu et al., 1999; Summers and Ma, 2000; Dmitriev et al., 2001; Bahareva and Dmitriev, 2002; Kozyreva et al., 2007; Degeling et al., 2008; Ozeke and Mann, 2008 ets.) and whistler mode waves (Horne and Thorne, 1998; Summers et al., 1998, 2002, 2004; Summers and Ma, 2000; Horne et al., 2005; Varotsou et al., 2005; Demihov et al., 2006; Summers et al., 2007; Thorne et al., 2007; Meredith et al., 2007 ets.), AKR ets. are analyzed. However the effectiveness of stochastic mechanisms can be clarified only excluding action of regular mechanism.

Schematic diagram showing the MLT distributionof plasma waves capable of resonant interactions withrelativistic electrons (Shprits et al., 2006).

,...2,1,0 где ,/ nnvk BIIII

A composite sketch of electron acceleration.(top). The inner magnetosphere for Kp 4–6. (bottom). Aschematic of how and where ULF and VLF/ELF wavesaccelerate and transport electrons (O’Brien et al, 2003).

Wave-particle interactions in the resonant conditions leads to simultaneous electron acceleration and pitch-angle scattering losses.

esct

f

p

fpDp

ppt

f

)(1 2

2

cp

21 p

- dimensionless momentum

- dimensionless full energy

The diffusion coefficient is determined by the temp of acceleration and time of acceleration

)(2 pDptac ( )D p

1act

act

0( ) qD p D p 2q - in the case of power low spectra of turbulence

D0 show the affectivity of acceleration

Definite features of stochastic acceleration can be studied analyzing nonstationary diffusion equation with losses (Bahareva and Orlova, 2008)

Cyclotron resonance with whistler turbulence

30)( pDpD B

B

M

m

B

BD

5

0

2)1(2

22

2

0 )4(

)1( 42

pDpD 0

~)(

BB

M

m

B

BD

3

0

212

0 8

)1(~

21

20

minmin

2 1)( h

k

k

kdk

dhk

2

02 8)( hB

min

)(20

k

dkkh0min 2 Lk where

Dimensionless time: 0D t Dimensionless parameter: 1

0( )esca D t q

esc

ac

ac

esc apt

t

t

t

21

1

Nonstationary equation of diffusion with losses in such a case has the form:

22

1 qf fp af

p p p

Temporal dynamics of relativistic electron fluxes in the case of finite injection of seed particles was analyzed and gives the possibility to describe the formation of plato in fluxes of relativistic electrons.

0

22/32

0 )4

exp( dyayy

pypA

dE

dJ

0

)4

exp(2

2/320 dyay

y

pypA

dE

dJ

)( 0

)( 0

1000 )4( mcDQA Q0=const is power of seed particles

Numerical calculations for the duration of injection: τ0=D0 t0=1 and 4

The dependence of differential fluxes dJ/dE of electrons with energy 1 MeV on dimensionless time τ and fixed D0 in the case of instantaneous injection

of seed population if τ0=D0t0=4 (firm lines) and τ0 =1 (dotted lines) and

a=(D0 tesc )-1 =0, 0.5, 2 and 4. Dotted line is the flux of electrons with energy 0.5

MeV if а=4. Upper horizontal axes is real t=τ/D0 if D0=10-5 с-1.

Popular mechanism of relativistic electron acceleration has definite problems connected with the time of acceleration as amplitudes of observed waves are comparatively small. Very large amplitude whistler-mode waves ~240 mV/m in Earth’s radiation belts were observed by Cattell et al. (2008) in STEREO experiment. However such large amplitudes are not observed typically. Second problem is the localization of the region of acceleration.

Li and Temerin, (2001)

July 1990 Orbit Number Oct 1991

Relativistic electrons as a rule appear during recovery phase of magnetic storm. The localization of the place of there appearance is connected with the minimum value of Dst variation by relation of Tverskaya (1986), Tverskaya et al. (2003).Sometimes it is possible to observe the increase of relativistic electrons at low L without increase at geostationary satellites (Tverskaya et al., 2005).

Appearance of relativistic electrons is deeply connect with magnetospheric substorms at the time of magnetic storms. Substorms produce the “seed” population of electrons with energies ~100-200 keV which later is accelerated till MeV energies. Strong substorms can accelerate electrons till relativistic energies (Ingraham et al., 2001). However ordinary substorms can not be considered as powerful souse of relativistic electrons. High and long lasting substorm activity on a storm recovery phase and high velocity of the solar wind are the necessary condition of the highest electron output.

Obtained by Tverskaya (1986), Tverskaya et al. (2003) relation was checked by Slivka, Kudela and Kuznetsovz (2006).

Among studied 22 cases the best correspondence between the observed minimum value of Lmax and the extreme value of Dst is observed for the events with large value of IMF module (B> 40 nT).

The radial profiles of relativistic 0.6-1.5 MeVelectrons fluxes during the period includingthe geomagnetic storm on November 5, 2001

The dependence nTl 1075.2 4max

4

max

LDst

obtains the theoretical explanation (see Tverskoy, 1997) including the value of coefficient (Antonova, 2005) in azimuthally symmetric case:

nTl 140

27

70

27 472

470* inex

eq

lobeinexex

eq

critst LL

B

BLLp

BD

It was taken into account that upper limit of the inner magnetospheric particle feeling is determined by the stability of the distribution of the plasma pressure. This limit exists in spite of the action of different acceleration and transport mechanisms of plasma particles.

Comparison of theory predictions with the results of experimental observations

pL-s

Tverskaya et al. (2005)

Antonova and Stepanova (2005)

However symmetric ring current does not produce great magnetic field distortion and plasma parameter <1. An order of magnitude larger with >1 magnetic field distortions are connected with asymmetric ring current

                 

                                                                                            

Brandt and Mitchell, 2006

Ring current Dynamics for the magnetic storm 15 May, 2005

Skoug et al., 2003

Magnetic storm 31 March 2001

(Dst)min = -350nT

At the geostationary orbit Bz ~0,B~300 nT

The restore of magnetic field during storm recovery phase

can produce considerable betatron

acceleration.

To evaluate the contribution of regular mechanism of relativistic electron acceleration it is necessary to have comparatively good model of the near Earth magnetic field. Most existing models of magnetic field suggest that the main sources of the magnetospheric magnetic field are magnetopause currents, tail current, ring current and partial ring current, field-aligned currents. Proper selection of magnetospheric current system is rather important as neglecting the contribution of some comparatively powerful systems can lead to real uncertainties during analysis of the experimental data. Storm time Tsyganenko-2004 model meets with definite difficulties. As, for example Kudela and Bucik (2005) show the discrepancies comparing Tsyganenko-04 (Tsyganenko and Sitnov, 2005) model predictions of the positions of the geomagnetic cutoffs with the results of observations.

Magnetic configuration for the event 1-8 January 2008 (Dstmin=-211 nT) in accordance with Tsyganenko-2004 model.

The latest version of Tsyganenko model TS07 (Tsyganenko and Sitnov, 2007; Sitnov and Tsyganenko, 2008) does not postulate the existence of definite current systems. The magnetic field of magnetospheric equatorial currents is expanded into a sum of orthogonal basis functions with different scales. The set of functions is built on the basis of the general solution of Ampere’s equation for infinitely thin equatorial current layer.

TS07 model is based on the suggestion of all transverse current flow at the equatorial plane.

Example of calculation of isolines B=const (for 100 nT, 90 nT, …) in accordance with Tsyganenko-2004 model at pdin=2 нПа, IMF Bz=-5нТ, By= Bx=0.

Daytime magnetospheric compression leads to distortion of magnetic field. Minimal values of magnetic field takes place far from the equatorial plane. Analysis of the configuration of B=const isolines leads to the conclusion that most part of daytime transverse current can be concentrated far from the equatorial plane.

It is possible to determine transverse current density in the condition of magnetostatic equilibrium using measured plasma pressure gradient and magnetic field as [jB]=p. Global distribution of plasma pressure at L<9RE was obtained by DeMichelis et al. (1998).

DeMichelis et al. (1998)

Obtained by DeMichelis et al. (1998) picture of current densities does not take into account the daytime magnetic field configuration in which minimal values of magnetic field on the field line take place far from the equatorial plane. Therefore the distribution of current densities at the equatorial plane does not show the distribution of integral current in the flux tube.

DeMichelis et al. (1998) obtained current densities at the equatorial plane

First results of the calculation of current densities on dayside auroral field lines using obtained by DeMichelis et al. (1998) radial plasma pressure profile and Tsyganenko-2001 magnetic field model (Antonova et al., in press) leads to values comparable with nighttime current densities at the same geocentric distances. The distribution of plasma pressure from 9 till 10RE was approximated by exponential dependence. Integral daytime current between 7.5 and 10 RE constitute ~105-106 A. The direction of daytime transverse current is opposite to the direction of magnetopause current.

Considerable part of nighttime transverse current at geocentric distances ~7-10RE can be closed inside the magnetosphere.

Themis multisatellite project measurements near the equatorial plane (see http://www.nasa.gov/missions_pages/themis/) give possibility to obtain radial plasma pressure distribution at geocentric distances from 7 till 10RE.

All Themis-B observations for the period 02.06-29.10.2007 at the geocentric distances 7 < r < 12 RE with 20 longitude deviation were analyzed. The position of the magnetopause was obtained using Bz IMF and Psw measured by Wind satellite.

Distribution of maximal current densities at the daytime field lines obtained using averaged distribution of plasma pressure and Tsyganenko-2001 magnetic field model for solar wind parameters Bz=-5 nT, By=0, solar wind dynamic pressure equal to 2.5 nPa and Dst=-5 nT. Integral current in both hemispheres constitute in studied case 5.8105 A.

The center of transverse currents.

Transverse daytime current is connected with the existence of radial plasma pressure gradients and is closed by transverse currents inside the magnetosphere. It is the high latitude continuation of the ordinary ring current which was not included in the existing models of magnetospheric current systems. Therefore it is necessary to modify existing models of magnetospheric transverse currents and include the high latitude continuation of ordinary ring current – cut ring current (CRC)

Results of CORONAS-F and METEOR-3M simultaneous observations show the existence of comparatively stable regions of increased fluxes of relativistic electrons to the pole from the external boundary of the external radiation belt (Myagkova et al., 2008, in press). 412 crossings of polar region at altitude 400-450 km were analyzed using low altitude 500 km polar orbiting Russian satellite CORONAS-F data. Electron precipitations at L>8 were observed in 248 cases.

8 10 12 14 16 18 20L

1E+0

1E+1

1E+2

Je 3

00-6

00 k

eV, 1

/(cm

**2-

s-sr

)

1 - 2003/07/15 23:45 M LT 10:152 - 2003/07/16 01:20 M LT 11:003 - 2003/07/16 02:55 M LT 11:40

Electron precipitations observed during 4.5 hours (3 orbits) 15-16 July, 2003

3.75 3.80 3.85 3.90 3.95 4.00U T, h (01/01/2004)

1E -1

1E +0

1E+1

1E+2

Je (30

0-60

0 ke

V), 1

/(cm

**2-

s-sr

)

0

4

8

12

16

20

24

28

L

L=16MLT= 10:20

84 polar crossings with electron precipitations observed during December 2003 and January 2004 were compared with auroral oval position using Meteor-3M data. The time of observations, coordinates and MLT- sectors were suitable for 32 of them. The position only 1 of electron precipitation could be identified as polar cap, the other 31 ones are situated in auroral oval.

Practically all of observed increases of relativistic electron fluxes were localized inside the auroral oval.

Two possible explanations of observed phenomena can be suggested. First traditional connects observed features of polar outer radiation belt boundary with regime of pitch-angle diffusion. Second (from our point of view more probable) – the appearance of localized quasistationary regions of quasitrapping due to magnetic field distortion. The lust effect is connected with inhomogenuity of plasma pressure distribution.

Results of Interball/Tail probe observations (Kirpichev, 2007)

Results of CRRES observations (Kozelova et al., 2008)

Asymmetric ring current is observed in the high latitude magnetosphere. Its intensity is greatly increased during magnetic storms.

Results of IMAGE observations

Plasma pressure gradients in the asymmetric ring current are directed to its center Eastward transverse current is developed in accordance with the condition of magnetostatic equilibrium

[jB]=p

Current loops and B=const loops can be formed at definite conditions.

Results of self consistent modeling of distribution plasma pressure, electric and magnetic fields in the magnetosphere

Closed local B=const loops appear near midnight at definite conditions.

Isolines B=const at the equatorial plane (Vovchenko et al., 2008)

Rice Convection Model is modified including the distortion of magnetic field due to the increase of plasma pressure (diamagnetic plasma effect). The problem of magnetosphere-ionosphere interactions is solved including determination of Region 2 field-aligned currents, self-consistentelectric fields and particle distribution function in the modeled region.

Conclusions:

•Regular and stochastic acceleration mechanisms can produce definite contribution in the acceleration of relativistic electrons. The contribution of stochastic mechanisms can be obtained only after extraction of the contribution of regular mechanisms.•The solution of the problem of acceleration of relativistic electrons requires proper models of magnetic field taking into account the distribution of plasma pressure inside the magnetosphere.•Dayside part of inner magnetosphere transverse currents flow at high latitudes. Such currents are closed inside the magnetosphere by nighttime transverse currents and constitute the high latitude continuation of ordinary ring current.•The development of asymmetric ring current can lead to the appearance of local plasma traps for relativistic electrons.

Thank you for your attention

The existence of maximal possible plasma pressure profile and sharp equatorial RC boundary give the possibility to obtain the dependence of Dst on the

position of plasma pressure maximumDessler and Parker [1959]; Sckopke [1966] relation has the form

DeqT BB 3/2/

022 3/4 EED RB nT102.3 4eqB

ER

prdrdz 3 p(L)=pex(Lex/L)7

4473

2 ininexexE LconstLLpRA

Lin is the position of the internal boundary of the ring current.

02 2/ lobeex Bp

Tverskoy [1997] relation ex

in

W

WE

critst dW

L

W

dW

dWp

RD

4

3 0*

Win and Wex are the values of flux tube volume at the inward

and outward boundaries of the ring current 4

04 35/32 LWBLRW eqE in the dipole magnetic field

In case of a quasidipole magnetic field accuracy of the Tverskoy [1997] relation reaches 20% when the equatorial magnetic field differs from the dipole one in less than 100%

If p(L)=pex(Lex/L)7 and 02 2/ lobeex Bp

nTl 140

27

70

27 472

470* inex

eq

lobeinexex

eq

critst LL

B

BLLp

BD