Kinetic theory for the ion humps at the foot of the Earth’s bow shock

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Kinetic theory for the ion humps at the foot of the

Earth’s bow shock

D. Jovanović, V. Krasnoselskikh

To cite this version:

Phys. Plasmas 16, 102902 (2009); https://doi.org/10.1063/1.3240342 16, 102902

© 2009 American Institute of Physics.

Kinetic theory for the ion humps at the foot

of the Earth’s bow shock

Cite as: Phys. Plasmas 16, 102902 (2009); https://doi.org/10.1063/1.3240342

Submitted: 03 August 2009 . Accepted: 04 September 2009 . Published Online: 05 October 2009 D. Jovanović, and V. V. Krasnoselskikh

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Kinetic theory for the ion humps at the foot of the Earth’s bow shock

D. Jovanović1,a兲and V. V. Krasnoselskikh2,b兲

1

Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia

2

LPCE/CNRS, 3A, Avenue de la Recherche Scientifique, 45071 Orléans Cedex 2, France

共Received 3 August 2009; accepted 4 September 2009; published online 5 October 2009兲

The nonlinear kinetic theory is presented for the ion acoustic perturbations at the foot of the Earth’s quasiperpendicular bow shock, that is characterized by weakly magnetized electrons and unmagnetized ions. The streaming ions, due to the reflection of the solar wind ions from the shock, provide the free energy source for the linear instability of the acoustic wave. In the fully nonlinear regime, a coherent localized solution is found in the form of a stationary ion hump, which is traveling with the velocity close to the phase velocity of the linear mode. The structure is supported by the nonlinearities coming from the increased population of the resonant beam ions, trapped in the self-consistent potential. As their size in the direction perpendicular to the local magnetic field is somewhat smaller that the electron Larmor radius and much larger that the Debye length, their spatial properties are determined by the effects of the magnetic field on weakly magnetized electrons. These coherent structures provide a theoretical explanation for the bipolar electric pulses, observed upstream of the shock by Polar and Cluster satellite missions. © 2009 American Institute

of Physics. 关doi:10.1063/1.3240342兴

I. INTRODUCTION

Low frequency electrostatic perturbations in the fre-quency range of 102_{– 10}3 _{Hz, which lies both below and}

above the local electron cyclotron frequency, are commonly observed in the vicinity of a quasiperpendicular shock front. In the classical study,1it was shown that these can be attrib-uted to either ion-sound or whistler waves. Usually, waves above the electron cyclotron frequency are thought to be the Doppler shifted ion-sound waves, and those below are whis-tler waves. At the front of a supercritical, quasiperpendicular shock, various sources of free energy have been identified, that give rise to different plasma instabilities. The generation of the ion acoustic waves is commonly attributed either to instabilities related with the electron temperature gradient, or to a strongly inhomogeneous electric current within the ramp. Other possible candidates for the electrostatic turbu-lence in this frequency range include the lower-hybrid and electron acoustic waves that can be excited respectively by the Bunemann instability, in the presence of an electron beam, and by the fan instability, in the presence of a hot electron population.

The analysis by Balikhin et al.2of the data collected by the Cluster 3 satellite on February 26, 2002, revealed the existence of the ion-sound turbulence within the Earth’s bow shock, upstream of the ramp. As there are no strong electron-electron temperature gradients in the foot of the shock, it was concluded that these ion acoustic waves are probably gener-ated by electric currents. The appropriate short lived, small scale current layers were not detected by the mission Ref.2, but they were predicted theoretically, within the nonstation-ary model of the shock front.3,4 Similar observations at the Earth’s bow shock, under extreme solar wind conditions,

were obtained also by the Polar spacecraft.5Besides the large amplitude coherent wave packets, with the frequency in the range of 100–4000 Hz and lasting 10–30 cycles, that were the dominant feature of the ion-sound turbulence at the mag-netic ramp, a number of coherent solitary structures was also observed. The same kind of electrostatic structures have been identified recently6 in the Cluster 3 data of February 26, 2002, in the foot region of Earth’s quasiperpendicular bow shock共the angle between the shock normal and the magnetic field was⬃55°兲. These structures feature a characteristic bi-polar pulse in the electric field, corresponding to a localized negative potential of⫺4 to ⫺5 V.6The amplitude is 10–20 mV/m, which is several times larger than that of the sound waves and packets. They propagate mostly in the solar wind direction toward the shock, at a significant angle to the am-bient magnetic field, with the velocity at the acoustic range. The velocity in the solar wind frame was estimated at 620 km/s, which is faster than the ion bulk velocity and the local ion-sound velocity共cs= 40 km/s兲 but slower than the

elec-tron thermal velocity 共vT_e= 2400 km/s兲, for Te= 17 eV.

Their scale perpendicular to the magnetic field is a few hun-dred meters, which in the plasma parameters scales as ⬃20␭D⬃0.5␳L_e. The parallel size is larger, within one order

of magnitude, with the typical aspect ratio 0.26, which was interpreted as the propagation at the angle⬃75° to the mag-netic field.6

The actual source of the ion acoustic turbulence at the foot of the shock still remains elusive. There are no strong electron-electron temperature gradients in this region and the existence of the appropriate short lived localized electric cur-rents has not been confirmed. The latter was inferred2 only from the properties of the observed sound waves. An alter-native mechanism has been suggested, based on the observa-tions of an ion beam at the foot region, see, e.g., Refs.7–9 and references therein. Such beam arises from the reflection a兲_{Electronic mail: djovanov@phy.bg.ac.yu.}

b兲_{Electronic mail: vkrasnos@cnrs0orleans.fr.}

PHYSICS OF PLASMAS 16, 102902共2009兲

of the ions of the solar wind from the shock, and the ions propagating roughly along the magnetic field direction are observed at the onset of the event. Later on, the angular distribution often becomes gyrophase-bunched, which is in-terpreted as the specular7 or nonspecular8 reflection. Such ion beams can efficiently produce linearly unstable ion acoustic waves.

In the present paper, we present the nonlinear kinetic theory of electrostatic perturbations associated with acoustic perturbation in an ion beam-plasma system in a weakly mag-netized plasma. Using the standard Schamel’s model for the ion distribution in the stationary state,10 we find a coherent localized structure in the form of an ion hump, that is sup-ported by the increased population of the beam ions that are trapped in the self-consistent potential. In contrast to the standard ion holes and humps in unmagnetized plasmas, whose spatial dispersion is governed by the ion thermal ef-fects and the charge separation, in weakly magnetized plasma in front of the shock, that is characterized by small Debye length and large electron Larmor radius,␭DⰆ␳L_e, the

spatial properties of the ion humps are determined by the 共large兲 electron Larmor radius effects. In Secs. I and II we derive the appropriate nonlinear descriptions of the ions and electrons, respectively. In Sec. III we summarize the linear properties of the unstable ion acoustic waves and in Sec. IV we present a numerical solution of the full nonlinear system, in the form of a coherent ion hump. The conclusions are given in Sec. V.

II. THE DESCRIPTION OF IONS

We study coherent nonlinear kinetic plasma structures at the foot of the Earth’s bow shock, whose typical size is somewhat larger than the electron Larmor radius and the ions are essentially unmagnetized. The ion velocity distribution is typically non-Maxwellian. As a simple model, we take that the ion population consists of stationary and streaming共i.e., beam兲 ions. The ion beam originates from the reflection of the solar wind from the bow shock. The unperturbed distri-bution is adopted as fi共0兲= 共1 −␤兲n0

冑

2␲vT_iS 2 exp

冉

− v2 2vT_iS 2

冊

+ ␤n0

冑

2␲v_T iB 2 exp

冋

− 共vជ−vជB兲2 2vT_iB 2

册

, 共1兲

where the subscripts S and B denote the stationary and beam ions, andvជB is the beam velocity. The unperturbed ion

den-sities and temperatures are taken to be homogeneous, since we study the phenomena whose typical size is much shorter than the characteristic length of inhomogeneity. In the pres-ence of the electrostatic potential␾ that is stationary in the reference frame propagating with the velocity uជ, one easily finds an integral of motion of the ion Vlasov equation, which is the ion energy W, W = mi共vជ− uជ兲2/2+e␾. In the simple case

when the potential is strictly one-dimensional共1D兲, the en-ergy is the only conserved quantity. Taking that the potential vanishes in the infinity, that it was adiabatically switched on at some distant past, and that the entire evolution was

adia-batic, we can express the initial ion velocityvជ共0兲as

vជ共0兲= uជ+ vជ− uជ

兩vជ− uជ兩

冑

共vជ− uជ兲

2₊2e

mi

␾. 共2兲

Obviously, negative potentials may trap ions, i.e., the “initial velocity” is a complex quantity for the particles whose ve-locities satisfy 共vជ− uជ兲2_{+ 2e␾/m}

i⬍0. Then, for the ions we

adopt the standard Schamel’s 1D model distribution function10 fi=共1 −␤兲n0

冑

2␲vT2_iS exp

冋

−共Re vជ 共0兲_兲2_−␣ S共Im vជ共0兲兲2 2vT2_iS

册

+ ␤n0

冑

2␲vT2_iB exp

冋

− 共Re vជ 共0兲_−vជ B兲2−␣B共Im vជ共0兲兲2 2vT2_iB

册

. 共3兲 Integrating the Schamel’s distribution, Eq.共3兲and taking, for simplicity, that the potential propagates in the direction of the beam, we obtain the ion density perturbation that, in the small amplitude regime 兩e␾/Ti兩Ⰶ1, with the accuracy to

⬃O共兩e␾/Ti兩3/2兲 can be written as10 ␦ni n0 = − aS e␾ Ti S − bS 4 3

冏

e␾ Ti S

冏

3/2 − aB e␾ Ti B − bB 4 3

冏

e␾ Ti B

冏

3/2 , 共4兲

where the coefficients aS, bS, aB, and bB are given by aS= 1 −␤

冑

2␲vT_i,S 2

冕

−⬁ ⬁ _vdv v − uexp

冉

− v2 2vT2_iS

冊

, 共5兲 bS= 1 −␤

冑

␲

冉

1 −␣S− u2 vT_iS 2

冊

exp

冉

− u2 2vT_iS 2

冊

, 共6兲 aB= ␤

冑

2␲vT2_iB

冕

−⬁ ⬁ _vdv v − u + vBexp

冉

− v2 2v_T iB 2

冊

, 共7兲 bB=

_冑

␤ ␲

冋

1 −␣B− 共vB− u兲2 vT2_iB

册

exp

冋

−共2vB− u兲 2 2vT2_iB

册

, 共8兲

and␣S and␣B are the appropriate Schamel’s trapping

coef-ficients for the stationary and moving ion populations. The ion particle trapping, which produces the nonlinear terms ⬀兩e␾/Ti兩3/2exist only if␾⬍0.

The velocity of the solar wind, relative to the Earth, is supersonic. Consequently, the velocity of the reflected ions in the solar wind frame, denoted byvBin Eqs.共7兲and共8兲, is of

the same order, i.e., it is共much兲 bigger than the ion thermal speed. Thus, for the structures that propagate with the veloc-ity close to that of the ion beam, and thus may be in reso-nance with the beam, viz.,兩u−vB兩ⱗvT_iB, the nonmoving

共sta-tionary兲 fraction of the unperturbed ion population can be regarded as cold, uⰇvT_iS. Thus, we take that the particle

trapping occurs only in the ion beam, and consequently

aS= −共1 −␤兲

vT_iS

2

u2 , bS= 0, 共9兲

so that we may write ␦ni n0 = −

冉

aB+ aS Ti B Ti S

冊

e␾ Ti B − bB 4 3

冏

e␾ Ti B

冏

3/2 . 共10兲

III. THE DESCRIPTION OF ELECTRONS

The typical size of the coherent nonlinear structures ob-served at the foot of the Earth’s bow shock is somewhat bigger than the electron Larmor radius.5,6 Thus, the simple model of Boltzmann distributed electrons, that is used in the classical theory of ion holes in an unmagnetized plasma, may not be appropriate. Noting that for relatively small negative potentials 0⬎e␾/TiⰇ−1 and in the regime when the

elec-tron temperature is comparable with the ion temperature, within the accuracy of Eq.共4兲 共i.e., when the electron trap-ping does not take place and the quadratic nonlinearities are negligible兲, we can use the linear description for electrons. Using the cylindrical coordinates in the velocity space and solving the Vlasov equation, assuming that the magnetic field is constant, one readily obtains the textbook expression for the Fourier component␦fe共␻, kជ,vជ兲 of the perturbation of

the electron distribution function, as the series expansion in terms of the modified Bessel functions, see, e.g., Ref.11,

␦fe= fe共0兲 e␾ Te_m=−

兺

_⬁ ⬁ im_J m

冉

k_⬜v_⬜ ⍀e

冊

exp

冉

ik⬜v⬜ ⍀e cos␪

冊

⫻

兺

l=−⬁ ⬁ l⍀e− k储v储 ␻+ l⍀e− k储v储 i−lJl

冉

k⬜v⬜ ⍀e

冊

, 共11兲

where Jl is the Bessel function of the order l, k⬜and k储are

the components of the wave vector that are perpendicular and parallel to the magnetic field Bជ,⍀eis the electron

gyrof-requency,⍀e= −eB/me, and the cylindrical coordinates of the

electron velocity are given byv储=vជ· Bជ/B, v_⬜=兩vជ⫻Bជ兩/B, and

␪= arccos关eជy·共vជ⫻Bជ兲/兩vជ⫻Bជ兩兴. Taking that the unperturbed

electron distribution function is Maxwellian, viz.,

fe共0兲= n₀ 共2␲vT_e 2 _兲3/2exp

冉

− v2 2vT_e 2

冊

, 共12兲

and making use of the recursive properties and orthogonality of the Bessel function, we can integrate the electron distri-bution function, Eq.共11兲, with respect to the perpendicular electron velocity components, yielding

␦f储e= e␾ Te n0

冑

2␲vT2_e exp

冉

− v储 2 2vT_e 2 − k⬜ 2_␳ L_e 2

冊

⫻

兺

l=−⬁ ⬁ l⍀e− k储v储 ␻+ l⍀e− k储v储 Il共k_⬜2␳L_e 2 _{兲, 共13兲} or, equivalently, ␦f储e= e␾ Te n0

冑

2␲vT2_e exp

冉

− v储 2 2vT_e 2

冊

⫻

冋

1 − exp共− k_⬜2_␳ L_e 2_兲

兺

l=−⬁ ⬁ ␻ ␻+ l⍀e− k储v储 Il共k_⬜2␳L_e 2_兲

册

, 共14兲 where ␦f储e=兰02␲d␪兰⬁0v⬜dv⬜␦fe and ␳L_e is the electron

Lar-mor radius,␳L_e=vT_e/兩⍀e兩.

A. Small Larmor radius or perpendicular propagation

Using the symmetry property of the Bessel functions

I_−l= Il, we conveniently rewrite Eq.共14兲as

␦f储e= e␾ Te n0

冑

2␲vT2_e exp

冉

− v储 2 2vT_e 2

冊

⫻

冋

1 − ␻ ␻− k储v储 I0共k⬜2␳L_e 2_{兲exp共− k} ⬜ 2_␳ L_e 2_兲 − exp共− k_⬜2␳L2_e兲

兺

l=1 ⬁ 2␻共␻− k储v储兲 共␻− k储v储兲2− l2⍀e 2Il共k_⬜2␳L2_e兲

册

. 共15兲 We note that the last term in Eq. 共15兲, proportional to the infinite sum, is small and that it can be neglected with the accuracy to the leading order, if either k_⬜␳L_eⰆ1 共i.e., the

electron Larmor radius is small compared with the character-istic perpendicular wavelength兲 or 兩␻− k储v储兩Ⰶ兩⍀e兩 共i.e., in the

drift regime兲. Taking ␻ⱗk储v储, the latter condition can be

rewritten as k_⬜␳L_e⬍共k⬜/k储兲共vT_e/v储兲, which allows also large

values of k_⬜␳L_e if the propagation is quasiperpendicular to

the magnetic field, k_⬜Ⰷk储.

Dropping the infinite series in Eq. 共15兲 and integrating forv储, we readily obtain the expression for the electron

den-sity, including the Larmor radius corrections, viz.,

␦ne⬇e␾ Te n0

冑

2␲vT2_e

冕

−⬁ ⬁ dv储exp

冉

− v储2 2v_T e 2

冊

⫻

冋

1 − ␻ ␻− k储v储 I0共k⬜2␳L_e 2_{兲exp共− k} ⬜ 2_␳ L_e 2_兲

册

. 共16兲

For the parallel phase velocities that exceed the electron ther-mal speed, ␻/k储ⰇvT_e, our Eq. 共16兲 yields the standard

ex-pression for the density perturbation, e.g., the one used in Ref.12, in the context of drift-wave electron holes. For prac-tical purposes, Eq.共16兲can be further simplified by the Padé approximation, viz., e−␨I0共␨兲⬇1/共1+␨兲. Finally, for the

per-turbations whose parallel phase velocity is smaller than the electron thermal speed uⰆvT_e and with the leading order

accuracy, the perturbation of the electron density ␦ne= ne

− n0can be written as

␦ne= n0 e␾ Te

冉

1 + u2 vT_e 2 1 1 +␨

冊

, 共17兲

where␨= k_⬜2␳L2_e. Using the inverse Laplace transform and for

the modes propagating in the acoustic range of velocities

冑

me/mivT_eⱗuⰆvT_e, the perturbation of the electron density

can finally be written in the form ␦ne

n0

=共1 +␦aˆe₁兲 e␾

Te, 共18兲

where the operator␦aˆe

1is given by ␦aˆe₁=

u2

vT_e

2 共1 −␳L2_eⵜ_⬜2兲−1. 共19兲

B. Large Larmor radius and oblique propagation

As already mentioned, the plasma at the foot of the bow shock is relatively weakly magnetized. Coherent nonlinear structures observed in this region have the typical size per-pendicular to the magnetic field of a few hundred meters, which scales as⬃20␭D⬃0.5␳L_e,5where␭iis the ion Debye

length, and␳L_eis the electron gyroradius. Their parallel

char-acteristic size may be somewhat larger共within one order of magnitude兲. Such aspect ratio corresponds to an oblique propagation, k_⬜/k储ⱗ10, as suggested in Ref. 6, where the

propagation at the angle⬃75° was found, which corresponds to the aspect ratio 0.26. Thus, the approximative expression 共16兲or its Padé version, Eq.共18兲, may not be applicable.

In the series expansion 共14兲, for the Bessel functions with relatively small orders and large arguments, lⰆ␨, we use an asymptotic form similar to Eq.共9.2.1兲, p. 364, of Ref. 13, improved by including the higher order terms in␨−1_that

are relevant for the spatial dispersion共the detailed derivation is given in the Appendix Sec. 1兲

Il共␨兲 = exp共␨− l2_/2␨兲

冑

2␲␨

冋

1 + 1 8␨

冉

1 − 2l2 ␨ + l4 3␨2

冊

册

. 共20兲

The above differs by the factor exp共−l2_{/2␨兲 from the}

stan-dard handbook expression, see, e.g., Eq.共9.7.1兲, p. 377, Ref. 13. This factor arises from the term兺_n=0K 共1/n!兲共−l2/2␨兲n in our Eq.共A14兲, which is often neglected for␨Ⰷ1. However, it is important in the study of the linear waves in weakly magnetized plasmas, since it produces the appropriate con-tribution of perpendicular particle dynamics to the electron density, that is well known in the case of an unmagnetized plasma, see Eq.共22兲.

Conversely, in the expansion 共14兲, for the Bessel func-tions with large orders, lⰇ␨Ⰷ1, we use the principal asymptotic expression共9.3.1兲, p. 365 of Ref.13,

Il共␨兲 =

_冑

1 2␲l

冉

e␨ 2l

冊

l exp

冉

␨ 2 4l

冊

. 共21兲

The derivation of Eq. 共21兲 from the more general uniform

asymptotic expansion for large orders and the outline of its

validity region are given in the Appendix Sec. 2.

In a further approximation, we will use these asymptotic forms on the entire range of the variable l, switching between the expressions共20兲and共21兲at l =␨. As the discontinuity of two asymptotic forms at this point does not exceed 30% in the region of interest, 3⬍␨ⱗ10, see Eq.共27兲, such simple approximation may be used to obtain a reasonable order-of-magnitude estimate for the corrections to the density in the limit of large FLR. Using Eq.共13兲, the electron density can now be written as␦ne=␦n_e共0兲+␦n_e共0兲, where␦n_e共0兲is the stan-dard result for an unmagnetized plasma共␳L_e→⬁兲, viz.,

␦ne共0兲= e␾ Te n0 2␲vT_e

冑

␨l=−

兺

⬁ ⬁

冕

−⬁ ⬁ dv储⫻ l⍀e− k储v储 ␻+ l⍀e− k储v储 ⫻exp

冉

− v储 2 2vT_e 2 − l2 2␨

冊

. 共22兲

For the phase velocities comparable with the ion beam speed 共i.e., smaller that the electron thermal speed兲, u =␻/

冑

k_⬜2 + k2_储ⰆvT_e, and for␨⬎1,␦ne 共0兲_{reduces to} ␦ne共0兲= n0 e␾ Te

冉

1 + u 2 vT2_e

冊

. 共23兲

It is worth noting that, if we replace the infinite sum by the integral 兺_l=−⬁ _⬁→兰_−⬁⬁ dl and introduce the new variable v_⬜

= −l⍀e/k⬜, Eq.共22兲exactly recovers the standard linear

ex-pression for the density perturbation in an unmagnetized plasma.

Conversely,␦n_e共1兲is the correction due to a large Larmor

radius. In the limit␻ⰆkvT_e, it has the form ␦ne共1兲= e␾ Te n0

冑

2␲vT2_e

冕

−⬁ ⬁ dv储exp

冉

− v储2 2vT_e 2

冊

⫻

再

兺

l=−␨ ␨ exp共− l2_/2␨兲

冑

2␨8␨

冉

− 2l2 ␨ + l4 3␨2

冊

+ 2

兺

l=␨ ⬁

冋

exp共−␨+␨2/4l兲

冑

2l

冉

e␨ 2l

冊

l −exp共− l 2_/2␨兲

冑

2␲␨

册

冎

. 共24兲 Approximating in Eq.共24兲the infinite sum by an integral, it is simplified to ␦ne共1兲= n0 e␾ Te

冋

G共␨兲 − ␨− 3 12

冑

2␲␨exp

冉

− ␨ 2

冊

− erfc

冑

␨ 2

册

, 共25兲 where erfc is the complementary error function erfc共z兲 = 2␲−1/2兰z⬁dt exp共−t2兲 with erfc共z兲⬇exp共−z2兲/共z

冑

␲兲 for zⰇ1, while the function G共␨兲 is given by

G共␨兲 = 2e−␨

冕

␨ ⬁ _dl

冑

2␲l

冉

e␨ 2l

冊

l exp

冉

␨ 2 4l

冊

⬇0.653 277 ␨0.424 381 e−0.444 262␨. 共26兲

For simplicity, we express the large Larmor radius correction

to the electron density ␦n_e共1兲, Eqs. 共25兲 and共26兲 in a Padé approximation. A simple expression, which provides a rea-sonable accuracy if␨⬎3.5, has the form

␦ne共1兲⬃ n0

e␾ Te

− 0.005

1 −共2/3兲2␨+共1/2兲4␨2. 共27兲 Using ␨= k_⬜2␳L2_e and the inverse Laplace transform, for the

modes propagating in the acoustic range of velocities

冑

me/mivT

eⱗuⰆvTe, the electron density can finally be writ-ten in the form

␦ne

n₀ =共1 +␦aˆe2兲 e␾

Te

, 共28兲

where the operator␦aˆe

2is given by ␦aˆe 2= − 0.005

冋

1 +

冉

2␳L_e 3

冊

2 ⵜ_⬜2 +

冉

␳Le 2

冊

4 ⵜ_⬜4

册

−1 . 共29兲

IV. LINEAR SOLUTION AND ITS INSTABILITIES

In the regime when either the finite Larmor radius共FLR兲 corrections are small, or the propagation is in the quasiper-pendicular direction, using Eqs. 共10兲, 共18兲, and 共28兲 in the Poisson’s equation, 共␦ni−␦ne兲/n0= −␭i

2_ⵜ2_e␾/T

iB, where ␭i

=vT

iB共⑀0mi/n0e

2_兲1/2 _{is the Debye length calculated with the}

temperature of the beam ions, we have 共1 −␳L2_eⵜ⬜2兲

冉

a␸− 4 3bB␸ 3/2_−␭ i 2_ⵜ2_␸

冊

₋ u 2 vT_e 2 TiB Te␸= 0, 共30兲 while for the large FLR corrections and oblique propagation, we obtain

冋

1 +

冉

2␳Le 3

冊

2 ⵜ_⬜2 ₊

冉

␳Le 2

冊

4 ⵜ_⬜4

册

⫻

冉

a␸−4 3bB␸ 3/2_−␭ i 2_ⵜ2_␸

冊

_{− 0.005}TiB Te␸= 0. 共31兲

Here␸= −e␾/TiBand we consider the regime␸⬎0 共␾⬍0兲

in which the ion trapping takes place. Using Eq. 共9兲, the parameter a is defined as a = aB−共1 −␤兲 vT_iB 2 u2 + TiB Te. 共32兲

In the linear case, ␸3/2→0, Eqs. 共30兲 and 共31兲 describe acoustic waves that are dispersive due to the thermal motion of ions and electrons, if the wavelength is close to the Debye length or to the electron Larmor radius. In an unmagnetized plasma, ␳L_e→⬁, on the scale much longer than the Debye

length, ␭i→0, and for modes propagating in the acoustic range of velocities,vT

eⲏuⲏ

冑

me/mivTe, Eqs.共30兲 and共31兲 describe nondispersive waves, whose phase velocity is deter-mined from aB−共1 −␤兲 vT2_iB u2 + TiB Te = 0. 共33兲

For linear waves which propagate with the beam u⬇vB, we

have aB=␤and consequently u2_=v

T_iB

2 1 −␤

␤+ TiB/Te

, 共34兲

which is positive because 0⬍␤⬍1. Such wave is stable. It can be identified as the acoustic wave, whose beam ions are thermalized and do not participate in the wave dynamics.

Conversely, when the phase velocity is sufficiently dif-ferent from that of the beam 兩u−vB兩ⰇvT_iB, we may regard

the ion beam as cold, TiB→0, and we have aB= −␤vT_iB

2 _/共u

−vB兲2. Thus, Eq. 共33兲reduces to

1 −共1 −␤兲cs 2 u2− ␤cs2 共u − vB兲2 = 0, 共35兲 or equivalently 共u2_{− c} s 2_{兲共u − v} B兲2+␤cs 2_v B共vB− 2u兲 = 0, 共36兲

where cs=

冑

Te/miis the acoustic speed. In the absence of a

beam, ␤→0, this dispersion relation describes the standard acoustic waves. In the presence of an ion beam, Eq.共36兲is of the fourth order and may have complex roots, which corre-spond to the linearly unstable modes. The growth rate of such beam driven acoustic instability is displayed in Fig.1. The boundary of the unstable region is found as the locus, in the␤−vBplane, of the double root of the dispersion relation, yielding

vB6− 3vB4+ 3vB2关9␤共␤− 1兲 + 1兴 − 1 = 0.

Furthermore, modes that are stable with respect to such hy-drodynamic beam instability, can be destabilized by the in-verse Landau damping共kinetic bump-on-tail instability兲. Us-ing Re aB= ␤

冑

2␲vT2_iB P

冕

−⬁ ⬁ _vdv v − u + vBexp

冉

− v2 2vT2_iB

冊

, 共37兲 0.0 0.5 1.0 0 1 2 3 0.0 0.1 0.2 0.3 β vB __ c_s Im ω _____ c k_{s ⊥}

FIG. 1. 共Color online兲 The dimensionless growth rate Im␻/csk⬜ of the acoustic mode driven by the ion beam.

Im aB= ␤␲共u − vB兲

冑

2␲vT_iB 2 exp

冋

− 共u − vB兲2 2vT_iB 2

册

, 共38兲

where P denotes the principal value of the integral, and as-suming that the number of resonant ions is relatively small, 兩Re aB兩Ⰷ兩Im aB兩, setting u→u+iw, where u and w are real

quantities, we calculate the real part u from the dispersion relation共33兲兲 in which we substituted aB by Re aB, and the

imaginary part w is found as

w = Im aB

共d/du兲关Re aB−共1 −␤兲共vT_iB

2 _/u2_兲兴. 共39兲

Obviously, under the condition兩Re aB兩Ⰷ兩Im aB兩, the growth

rate of the bump-on-tail instability, Eq.共39兲, is much smaller than that of the hydrodynamic beam-driven instability.

V. COHERENT NONLINEAR SOLUTION

We seek a solution whose spatial scale is comparable or smaller that the electron Larmor radius␳Leand much larger

than the Debye length␭i. First, in the regime of small FLR

corrections or quasiperpendicular propagation, setting ␭i →0, we rewrite Eq.共30兲as 共1 −␳L_e 2_ⵜ ⬜ 2_兲

共

_a␸−4 3bB␸ 3/2

兲

_−共u2_/v T_e 2_兲共T iB/Te兲␸= 0. 共40兲

Using the variables␸

⬘

=␸共4bB/3a兲2 and rជ

⬘

= rជ/␳L_e, Eq.共40兲

obtains a dimensionless form 共1 − ⵜ_⬜2_兲共␸

− s␸3/2兲 −␰␸= 0, s = sign bB, 共41兲 which a relatively simple equation of second order with one physical parameter, ␰=共u2/avT_e

2 _兲共T

i_B/Te兲. However, we did

not find any localized nonlinear solutions.

Localized solutions are found in the regime of large FLR corrections and oblique propagation. We use Eq.共31兲, which is rewritten as

冉

␳L_e 2 4 ⵜ⬜ 2 _{+ q}

冊冉

␳Le 2 4 ⵜ⬜ 2 ₊1 −␥ q

冊

冉

a␸− 4 3bB␸ 3/2

冊

−␥4 3bB␸ 3/2_{= 0, 共42兲}

where the notations are

q = 8/9 +

冑

共8/9兲2+␥− 1,

共43兲 ␥= 0.005TiB/aTe.

We note that the parameter q is complex if ␥⬍1−共8/9兲2, when in the linear case 共␸3/2→0兲 Eq. 共42兲 describes two wave modes with complex wave numbers.

Conversely, for ␥⬎1, we have q⬎0 and 共1−␥兲/q⬍0, i.e., the first mode is sinusoidal and the second is evanescent. In the nonlinear regime,␸=O共1兲, the latter may give rise to a localized coherent structure, whose tails are exponentially decreasing in space, and smoothly connected in the center by the variation in the effective 共nonlinear兲 wave number, and/or by the coupling with the sinusoidal mode q. Obvi-ously, such coherent structure may be possible only if we have the scaling 0⬍a⬃0.005Ti_B/Te⬃兩bB␸1/2兩Ⰶ1, i.e., if its

phase velocity is very close to that of the nondispersive lin-ear mode, determined by a = 0, Eqs.共33兲–共35兲.

Using the variables ␸

⬘

=␸/⌽ and rជ

⬘

= rជ/L, where ⌽ =关3a共␥− 1兲/4␥bB兴2_{and L =共␳}

L_e/2兲

冑

q/共␥− 1兲 from Eq.共42兲,

we obtain the following dimensionless equation for the elec-trostatic potential which, by virtue of Eq.共43兲, contains only one physical parameter, viz.,

冉

␥− 1

q2 ⵜ⬜

2

+ 1

冊

共ⵜ_⬜2 − 1兲

冉

␸−␥− 1

␥ s␸3/2

冊

= s␸3/2, 共44兲 where s = sign bB= bB/兩bB兩. Equation共44兲possesses localized

solutions if bB⬍0 共i.e., when there exists the surplus of trapped ions, ␣B⬎1兲, which corresponds to an ion hump.

The existence of a localized solution is easily verified in the long wavelength limit. Namely, for␥− 1Ⰶ1 共q⬇16/9兲, Eq. 共44兲reduces to the standard equation for small amplitude ion or electron holes that has been studied in details, see, e.g., Ref.10共one needs to be careful here, because in the limit␥

→1, the condition of reasonably large Larmor radii in Eq.

共27兲,␨⬎3.5, may be violated兲, and it is readily solved as ␸共x兲 = 共25/16兲sech4_{共x/4兲. 共45兲}

In the general case, ␥=O共1兲, we have found a 1D 共ⵜ_⬜ = eជyd/dy兲 localized solution of Eq. 共44兲for ␥= 1.5 It has a similar shape as the long wavelength solution共45兲, but with a somewhat larger amplitude, see Fig.2.

VI. CONCLUSIONS

We have developed the nonlinear kinetic theory of ion acoustic perturbations, in the ion beam-plasma system in the presence of a weak magnetic field, under the conditions that are typical for the region at the foot of the Earth’s quasiper-pendicular bow shock, whose spatial dispersion comes from the effects of the weak magnetic field on the electrons. In the fully nonlinear regime, we found a time stationary, coherent, localized 1D solution in the form of an ion hump, supported by the nonlinearities coming from the increased population

-10 -5 5 10 1 2 3

ϕ'

x’

FIG. 2.共Color online兲 Dimensionless electrostatic potential of a slab 1D ion hump, obtained as the solution of Eq.共44兲 when␥= 1.5 and s = −1. The normalizations are defined by ␾=共−e⌽/Ti_B兲␸⬘ and rជ= eជxLx⬘, where ⌽

=关3a共␥− 1兲/4b␥兴2_{, L =共␳}

L_e/2兲

冑

q/共␥− 1兲, and a,␥, and q are defined in Eqs.

共32兲and共43兲.

of the beam ions that are in resonance with the structure, and trapped in the self-consistent potential. These ion humps are somewhat smaller that the local electron Larmor radius and much larger that the Debye length. In contrast to the usual ion holes/humps in an unmagnetized plasma, their spatial properties are determined by the large ␳L_e effects and they

provide an adequate explanation for the observations of the bipolar pulses upstream of the shock by several satellite missions2,5,6

We may envisage two possible scenarios for the creation of the ion humps by the saturation of the linear instabilities of the ion acoustic waves. First, we may imagine that the kinetic, bump-on-tail instability, Eq.共39兲, associated with the resonant ions, gets saturated by the deformation of the ion distribution function in the resonant region, in the form of a

hump 共or maximum兲, rather than a plateau. However, the

growth rate of such instability is not very large. Conversely, the faster growing, hydrodynamic beam-plasma instability, see Fig.1, corresponds to the complex values of the phase velocity u found from the dispersion relation a = 0. As the phase velocity of such linear mode features Re u⬃Im u, its nonlinear stabilization by the ion trapping must involve also the acceleration/deceleration, to the velocity of a linearly stable共or a marginally unstable兲 mode.

ACKNOWLEDGMENTS

This work was financed in part 共D.J.兲 by the Serbian MNTR Grant No. 141031 and by the Pavle Savic program of scientific collaboration between Serbia and France.

APPENDIX: ASYMPTOTIC EXPRESSIONS FOR BESSEL FUNCTIONS

1. Bessel functions with large arguments

In the limit ␨Ⰷ1, the asymptotic expansion for Bessel functions has the form given in p. 269 of Ref.14, viz.,

Il共␨兲 = e ␨

冑

2␲␨

冋

兺

k=0 K−1 共−␨兲−k_{Ak共l兲 +␦} K

册

+ e −␨

冑

2␲␨

冋

兺

k=0 K−1 ␨−k_{Ak共l兲 +␥} K

册

, 共A1兲 where A0共l兲=1 and Ak共l兲 = 1 k ! 8k

兿

_n=1 k 关4l2_{−共2n − 1兲}2_{兴, k ⱖ 1. 共A2兲}

The remainders 兩␦K兩 and 兩␥K兩 are comparable with the first

term in the expansion that has been omitted, 兩␦K兩, 兩␥K兩

⬃兩␨−K_{AK共l兲兩. The limit in the summation of Eq.共A1兲cannot}

be adopted as K→⬁, because such infinite series would di-verge. The limit K is adopted so that the remainders are smaller than the last term retained in the sum, 兩␨−KAK共l兲兩

⬍兩␨1−K_A

K−1共l兲兩, which is realized if either

␨⬎ K − 1/2 ⬎ 兩l兩, 共A3兲

or

兩l兩 ⬎ K − 1/2 ⬎

冑

l2−␨+␨2−␨. 共A4兲 In the case兩l兩Ⰷ␨Ⰷ1 the condition Eq.共A3兲 cannot be real-ized, while the second condition, Eq. 共A4兲, simplifies to 兩l兩 ⬎K⬎兩l兩−␨. Conversely, in the opposite limit 兩l兩Ⰶ␨, these two conditions reduce to␨⬎K⬎l2_{/2␨. Thus, the summation}

in Eq.共A1兲can be truncated at a relatively small upper limit,

K⬃O共1兲, only for the Bessel modes whose order is not very

high,兩l兩ⱗ

冑

2␨.

We proceed by rewriting the coefficients Ak共l兲 as

poly-nomials in l2_{, viz.,} Ak共l兲 = 1 k ! 8km=0

兺

k Pk,m共2l兲2m, 共A5兲 where Pk,m= lim l→0 1 共2m兲 ! 22m ⳵2m ⳵l2m

兿

n=1 k 关4l2_{−共2n − 1兲}2_{兴, 共A6兲} which yields Pk,0=共− 4兲k

冤

⌫

冉

1 2+ k

冊

⌫

冉

1 2

冊

冥

2 =共− 1兲k关共2k − 1兲 ! !兴2, 共A7兲 Pk,1= − Pk,0 2 ! 22

冋

␲ 2_{− 2␺}共1兲

冉

_{k +}1 2

冊

册

, 共A8兲 Pk,2= Pk,0 4 ! 24

再

␲ 4_{+ 12␺}共1兲

冉

1 2+ k

冊

冋

␺ 共1兲

冉

1 2+ k

冊

−␲ 2

册

+ 2␺共3兲

冉

1 2+ k

冊

冎

, ... Pk,k−3= 1 3!

再

冋

−

兺

_n=1 k 共2n − 1兲2

册

3 + 3

兺

n=1 k 共2n − 1兲2 ⫻

兺

n=1 k 关共2n − 1兲2_兴2_{− 2}

兺

n=1 k 关共2n − 1兲2_兴3

冎

= − k 945共2240k 8_{− 12 096k}7_{+ 15 600k}6_{+ 13 104k}5 − 29 820k4− 4284k3+ 17 605k2+ 441k − 2790兲, 共A9兲 Pk,k−2= 1 2!

再

冋

_n=1

兺

k 共2n − 1兲2

册

2 −

兺

n=1 k 关共2n − 1兲2_兴2

冎

= k 90共80k 5_{− 144k}4_{− 40k}3_{+ 120k}2_{+ 5k − 21兲,} 共A10兲 Pk,k−1= −

兺

n=1 k 共2n − 1兲2_{= −}k 3共4k 2_{− 1兲, 共A11兲}

Pk,k= 1 共A12兲

where␺共z兲 is the digamma function, ␺共z兲=⌫

⬘

共z兲/⌫共z兲, and ␺共n兲_共z兲=dm_{␺共z兲/dz}m

.

Neglecting exponentially small terms ⬃exp共−␨兲 in Eq. 共A1兲, the asymptotic expansion, for␨Ⰷ1, of the Bessel func-tion Iltakes the form

Il共␨兲 = e ␨

冑

2␲␨

冋

1 +

兺

k=1 K 共− 1兲k k !共8␨兲k

兺

m=0 k Pk,m共4l2兲m

册

. 共A13兲

Separating the term with m = k from the second sum, and changing the order of the summations, we can rewrite this expression as Il共␨兲 = e ␨

冑

2␲␨

冋

兺

n=0 K 1 n!

冉

− l2 2␨

冊

n +

兺

m=0 K−1

兺

k=m+1 K Pk,m共4l2兲m k !共− 8␨兲k

册

. 共A14兲 For the Bessel functions whose order is not too large, viz., ␲2_l2_{ⱕ4␨, with the accuracy to leading order in ␨}−1_{, in Eq.}

共A14兲we may keep only one term in the sum with respect to

k, i.e., we take 兺_k=m+1K ⬇兺_k=m+1m+1 . Two other sums in Eq. 共A14兲 converge, and we can take their upper limits to be arbitrarily large, viz.,兺_n=0K →兺_n=0⬁ and 兺_m=0K−1→兺_m=0⬁ , which finally yields Il共␨兲 =exp共␨− l 2_/2␨兲

冑

2␲␨

冋

1 + 1 8␨

冉

1 − 2l2 ␨ + l4 3␨2

冊

册

, 共A15兲 which is applicable if lⰆ␨. We note that Eq.共A15兲differs by the factor exp共−l2_{/2␨兲 from the often used expression}

共9.7.1兲, p. 377, Ref.13. This factor arises from the first term, 兺n=0

K

in our Eq.共A14兲, which is usually neglected for␨Ⰷ1, but it is of a crucial importance for the waves in plasmas in the limit of weak magnetic fields.

2. Bessel functions with large orders

For the Bessel functions whose order is larger than their argument lⰇ␨, we use the uniform asymptotic expansion for large orders, Eq.共9.7.7兲, p. 378, Ref.13

Il共lz兲 =

_冑

1 2␲l l␩ 共1 + z2_兲1/4

兺

k=0 K l−kuk共t兲. 共A16兲 Here ␩=

冑

1 + z2_{+ log关z/共1+}

冑

_{1 + z}2_{兲兴, t=1/}

冑

_{1 + z}2_{, u} 0共t兲=1, u₁共t兲=共3t−5t2_{兲/24, and} u_k+1共t兲 =1 2t 2_{共1 − t}2_兲u k

⬘

共t兲 +1 8

冕

₀ t 共1 −␶2_兲u k共t兲dt.

For very large orders and large arguments, lⰇ␨Ⰷ1 共z=␨/l Ⰶ1兲, with the accuracy to first order, we may truncate the summation in Eq.共A16兲using K = 1 as the upper boundary, which yields Eq.共9.3.1兲 in Ref.13, viz.,

Il共␨兲 =

_冑

1 2␲l

冉

e␨ 2l

冊

l exp

冉

␨ 2 4l

冊

. 共A17兲

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