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Nonlinear mirror waves in non-Maxwellian space plasmas
O. A. Pokhotelov,1 R. Z. Sagdeev,2 M. A. Balikhin,1 O. G. Onishchenko,3
and V. N. Fedun4
Received 16 July 2007; revised 14 December 2007; accepted 15 January 2008; published 29 April 2008.
[1] A theory of finite-amplitude mirror type waves in non-Maxwellian space plasmas isdeveloped. The collisionless kinetic theory in a guiding center approximation, modifiedfor accounting of the finite ion Larmor radius effects, is used as the starting point. Themodel equation governing the nonlinear dynamics of mirror waves near instabilitythreshold is derived. In the linear approximation it describes the classical mirror instabilitythat is valid for a wide class of the velocity distribution functions. In the nonlinear regimethe mirror waves form solitary structures that have the shape of magnetic holes. Theformation of such structures and their nonlinear dynamics has been analyzed bothanalytically and numerically. It is suggested that the main nonlinear mechanismresponsible for mirror instability saturation is associated with modification (flattening) ofthe shape of the background ion distribution function in the region of small parallelparticle velocities. The width of this region is of the order of the particle trapping zone inthe mirror hole. Near the mirror instability threshold the saturation arises before its widthreaches the ion thermal velocity. The nonlinear mode coupling effects in thisapproximation are smaller and unable to take control over evolution of the space profile ofsaturated mirror waves or lead to their magnetic collapse. This results in the appearance ofquasi-stable solitary mirror structures having the form of deep magnetic depressions. Aphenomenological description of this process is formulated. The relevance of thetheoretical results to recent satellite observations is stressed.
Citation: Pokhotelov, O. A., R. Z. Sagdeev, M. A. Balikhin, O. G. Onishchenko, and V. N. Fedun (2008), Nonlinear mirror waves in
non-Maxwellian space plasmas, J. Geophys. Res., 113, A04225, doi:10.1029/2007JA012642.
1. Introduction
[2] The classical mirror instability (MI) theoreticallyidentified in the end of the 50th [Vedenov and Sagdeev,1958; Rudakov and Sagdeev, 1958; Chandrasekhar et al.,1958] is a common feature in planetary and cometarymagnetosheaths. The early theories were limited by purelylinear analysis and thus could not provide a comprehensivecomparison with in situ observations. Interest to the MI wassubstantially reinforced by its possible relevance to space-craft observations of the so-called magnetic holes - deepdropouts of the magnetic field in high-b plasmas. The termmagnetic hole was first suggested by Turner et al. [1977] todescribe decreases in the solar wind magnetic field strength.They were found in numerous planetary magnetosheaths[Kaufmann et al., 1970; Tsurutani et al., 1982; Luhr andKlocker, 1987; Treumann et al., 1990; Bavassano Cattaneoet al., 1998; Joy et al., 2006] in the solar wind [Turner et al.,
1977; Fitzenreiter and Burlaga, 1978; Winterhalter et al.,1994; Stevens and Kasper, 2007] and in the vicinity ofcomets [Russell et al., 1987; Glassmeier et al., 1993]. Themirror wave structures were found also in the sheath regionsbetween fast interplanetary coronal mass injections and theirpreceding shocks [Liu et al., 2006]. Ultimately they repre-sent a nonlinear phenomenon. It should be noted thatbesides the magnetic holes some observations reveal otherforms of the mirror structures as quasi-sinusoidal profiles oras magnetic humps. For example, they have been found inthe Jovian magnetosheath [Joy et al., 2006] and in numer-ical simulations [Baumgartel et al., 2003].[3] The cornerstone of the instability mechanism is the
pressure anisotropy that provides the free energy necessaryfor its onset. As the solar wind slams into the magneto-sphere it abruptly slows down, and a bow shock and amagnetosheath are formed. In the magnetosheath the pres-sure anisotropy is generated when the plasma flow crossesthe bow shock. The passage of the solar wind through thisregion leads to an increase in the ion temperature, withthe larger proportion of the heating going preferentially intothe perpendicular component. This results in the appearanceof anisotropic ion populations with stronger pressure an-isotropy than in the solar wind. It should be noted that thepressure anisotropy is also generated in the magnetosheathitself as the plasma flows around the magnetopause.
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, A04225, doi:10.1029/2007JA012642, 2008
1Automatic Control Systems Engineering, University of Sheffield,Sheffield, United Kingdom.
2Department of Physics, University of Maryland, College Park,Maryland, USA.
3Institute of Physics of the Earth, Moscow, Russia.4Department of Applied Mathematics, University of Sheffield, Shef-
field, United Kingdom.
Copyright 2008 by the American Geophysical Union.0148-0227/08/2007JA012642
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[4] The MI has a number of peculiarities that makes it apossible candidate to fit into a scenario of magnetic holeformation, among these are the nonpropagating nature ofthe mode and the anti-correlation of plasma density andmagnetic field.[5] The MI corresponds to a resonant type instability that
arises due to the interaction of the zero-frequency modewith the ions possessing very small velocities along theexternal magnetic field. The physical mechanism of thelinear MI has been reviewed by Southwood and Kivelson[1993]. The linear mirror mode theory has been extensivelydeveloped in a number of previous papers [e.g., Pokhotelovet al., 2002 and references therein]. Those papers includedthe effects of a finite electron temperature, a nonvanishingelectron temperature anisotropy, multi-ion plasmas, thefinite ion Larmor radius (FLR) effect and can be appliedto a wide class of the ion and electron distribution functionsof the form f(m,W), where m andW are the particle magneticmoment and energy, respectively. A very general compactexpression for the linear marginal mirror-mode stabilitycondition and instability growth rate suited for application tosuch class of velocity distribution functions has been derivedby Leubner and Schupfer [2000, 2001] and Pokhotelov et al.[2002] and has then been applied to bi-Maxwellian, Dory-Guest-Harris (DGH), Kennel-Ashour-Abdalla (KA) andk -distributions. However, all these papers have beenrestricted by the linear approximation.[6] One of the first attempts of a nonlinear treatment of
the MI was made more than four decades ago by Shapiroand Shevchenko [1964]. These authors using the randomphase approximation have reduced the problem of nonlinearsaturation of the MI to the study of a quasilinear diffusionequation for the ion distribution function in which all ionswere considered to be fully adiabatic. Indeed, the back-ground ion distribution function is shown to be modifiedwhich leads to saturation of MI. However, such a scenariothat assumes the adiabaticity of particle trajectories does nottake into account the effects of resonant trapping. Below wewill incorporate this effect for the study of nonlinearsaturation of the MI. We note that Shapiro and Shevchenko[1964] came to an important conclusion on the special roleof ions having small parallel velocities. In what follows, ingoing beyond the random phase approximation, this featurewill remain intact.[7] It should be noted that some effects related to non-
linear saturation of mirror waves in the magnetosheath havebeen already discussed by Pantellini [1998] and Kivelsonand Southwood [1996]. Recently Kuznetsov et al. [2007]suggested a new nonlinear theory of MI in bi-Maxwellianplasmas that describes the formation of magnetic holes interms of a process known under the name of wave collapse[Kuznetsov, 1996].[8] However, in real conditions such a scenario can face
some problems which we will describe in what follows. Inthis connection a few comments are in order.[9] First, it should be noted that measured particle dis-
tributions in near-Earth space do, in the overwhelmingmajority of cases, considerably deviate from the bi-Max-wellian shape. They frequently exhibit long suprathermaltails on the distribution function or, in other cases, possessloss cones. In collisionless plasmas, such as planetarymagnetosheaths, the decisive role in determining the shape
of the particle velocity distribution functions may belong towave-particle interactions. In the range of resonance theseinteractions may lead to the formation of diffusion plateaus.For example, such plateaus were recently observed in thevelocity distribution of fast solar wind protons [Heuer andMarsch, 2007].[10] Second, the wave collapse can be arrested in the
presence of particle trapping in the mirror hole that arises inthe resonance region. The latter is generally much strongerthan the mode coupling effects that lead to the wave collapse.[11] The main goal of the present paper is to develop a
comprehensive nonlinear theory of the mirror instability fora wide class of the plasma velocity distribution functionsand incorporate the effects due to the particle trapping.[12] The paper is organized as follows: section 2 formu-
lates the derivation of the nonlinear plasma pressure balancecondition for the MI in terms of an arbitrary ion distributionfunction. The generalization of this condition to the casewhen the FLR effect is included is carried out in section 3.The linear growth rate of the MI accounting for the FLReffect is derived in section 4. The mirror mode nonlinearequation is obtained in section 5. The effects of particletrapping in the mirror holes are discussed in section 6. Ourconclusions and outlook for future research are found insection 7.
2. Plasma Pressure Balance Condition
[13] We consider a collisionless plasma composed of ionsand electrons having the total mass density r and macro-scopic velocity v, and embedded in a magnetic field B. Ouranalysis will be limited to the case of most importance whenthe ion temperature is much larger than the electron tem-perature. The momentum equation governing the timeevolution of this plasma is
rdv
dt¼ J� B�r � P; ð1Þ
where J is the electric current and d/dt � @/@ t + v � r is theLagrangian time derivative. The pressure tensor representsthe anisotropic pressure tensor P given by
P ¼ p?Iþ pk � p?� �
zz: ð2Þ
[14] Here I is the unit dyadic, pk(?) is the parallel(perpendicular) plasma pressure and z is the unit vector, z =B/B, along the magnetic field [e.g., Krall and Trivelpiece,1973].[15] We make use of a Cartesian coordinate system in
which the unperturbed magnetic field B0 is directed alongthe z axis. The wave fields exhibit only the y component ofthe electric field Ey and the z and the x components of themagnetic field, d Bz and dBx, respectively. The latter areconnected through the property of solenoidality of themagnetic field, i.e. r � B � @Bx/@x + @Bz/@ z = 0. Asregards the fluid velocity v = (E � B) /B2, only its xcomponent survives in the leading order, i.e. v � vx. The dBy
component corresponds to the so-called non-coplanar mag-netic component, and does not enter our basic equations andcan thus be set to zero.
A04225 POKHOTELOV ET AL.: MIRROR WAVES
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[16] Similarly, equation (1) shows that the only nonzerocomponent of the electric current is the y component Jy � Jgiven by
J ¼ rB
@v
@tþ v
@v
@x
� �þ 1
B
@p?@x
þ @
B@z
pk � p?
B
@
@z
ZdBzdx: ð3Þ
[17] The physical meaning of the different terms on theright-hand side of equation (3) is as follows: The first one,containing the Lagrangian time derivative, is due to plasmainertia. This term is important for the fast magnetosonicsolitons (MS) in which w 2 ’ k2vA
2(1 + b?), where w and kare the characteristic wave frequency and wave number,respectively, and vA is the Alfven velocity. We note, that in ahigh-b plasma the Alfven velocity is of the order of the ionthermal velocity vT. The propagation of MS solitons nearlyperpendicular to the external magnetic field in non-Maxwellian plasmas has recently been considered byPokhotelov et al. [2007a, 2007b]. However, for mirrorwaves, which are the subject of our study, the situation isquite different. These waves are found in the so-calledmirror approximation, w kk(n
�1hvk2Fi)12 [cf. Southwood
and Kivelson, 1993; Pokhotelov et al., 2002]. Here kk is thetypical parallel wave number, n is the plasma number density,vk is the parallel particle velocity and F is the undisturbedparticle distribution function. In this limit the inertial term inequation (3) is small as w2/k2vT
2 1, relative to the secondterm and must thus be neglected. The differences in themathematical description of the fast MS and mirror waves inthe high-b plasmas have been extensively discussed recentlyby Ferriere and Andre [2002]. Finally the last term inequation (3) describes the effect of magnetic field linebending.We note that since this term already contains a smallparameter kk
2/k?2 / dBz/B0/ e 1 one may set here the value
of the magnetic field B as well as the difference pk � p? equalto their unperturbed values.[18] Thus, in the mirror limit which is the subject of our
study the inertial term can be neglected and the electriccurrent J reduces to
J ’ 1
B
@dp?@x
�b? � bk
2m0
@2
@z2
ZdBz x
0ð Þdx0; ð4Þ
where b?(k)= 2m0p?(k)/B02 is the perpendicular (parallel)
plasma beta, and m 0 is the permeability of free space anddelta denotes a perturbation.[19] Furthermore, the Ampere’s law reads
@dBx
@z� @dBz
@x¼ m0J : ð5Þ
[20] Substituting (4) into (5) and integrating over x oneobtains a nonlinear equation for the perpendicular plasmabalance condition
dp? þ B0dBz
m0
þ dB2z
2m0
¼ � 1þb? � bk
2
� �@2
@z2
Z ZB0dBz
m0
dx0dx00; ð6Þ
where the variation of the plasma perpendicular pressurecan be found from the Vlasov equation for the particle dis-tribution function. In the linear approximation equation (6)coincides with the perpendicular plasma balance conditionthat was usually used for the study of linear MI [cf.Pokhotelov et al., 2002].
3. Incorporation of FLR Effect
[21] The pressure balance condition (6) can easily begeneralized to the case when the FLR effect is taken intoconsideration. This effect has been extensively studied formirror waves by many authors [e.g., Hasegawa, 1969; Hall,1979; Pokhotelov et al., 2004; Hellinger, 2007]. It wasshown that when the transverse scale size of the mirrorwave becomes comparable to the ion Larmor radius, theactual impact of the wave magnetic and electric fields isreduced due to partial cancelation of it integrated over theion Larmor orbit. This leads to modification of the expres-
sion for Fourier transform (dp?(k?) =Rdp?(r?)e
�ik?�r?
dr?) of the variation of the perpendicular plasma pressure dp? ! (1 + 3
2ri2r?
2 ) dp? with ri being the ion Larmor radiusand r?
2 � @2/@x2. This effect was first introduced byRosenbluth et al. [1962] for stabilization of flute instability.Later it was widely applied by many authors to otherproblems of plasma physics. In particular, this effect issimilar to the FLR suppression of the perpendicular electriccurrent described by Kadomtsev [1965] [cf. Cheng, 1991]for kinetic Alfven waves. Using these considerationsequation (6) reduces to
dp?þB0dBz
m0
� 3
2r2i r2
?B0dBz
m0
þ dB2z
2m0
¼ � 1þb? � bk
2
� �r�2
?@2
@z2B0dBz
m0
: ð7Þ
[22] Here and in what follows we make use of thefollowing hierarchy of small parameters
r2i r2? ’ r�2
?@2
@z2’ dBz
B0
’ e: ð8Þ
[23] According to this hierarchy the FLR corrections in thenonlinear term and in the term containing the z derivative areneglected as corrections of higher order on a small parameter e.
4. Mirror Instability With FLR Effect
[24] In order to derive the equation governing the dynamicsof the MI it is necessary to calculate the variation of theplasma pressure. For that purpose we make use of the driftkinetic equation. Since we are only interested in waves withfrequencies much lower than the ion cyclotron frequency weuse the distribution function averaged over the ion gyromo-tion, f(t, rg, v?, vk), where t is the time, rg is the guidingcenter position vector, v? is the gyration speed, and vk is thevelocity along the field lines. The distribution functionobeys the Vlasov equation
@f
@tþ _rg � rf þ _v?
@f
@v?þ _vk
@f
@vk¼ 0; ð9Þ
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A04225
where _rg = vE + vk z, vE is the E � B velocity, and _v? isgiven by the expression for the conservation of the magneticmoment, dm/dt = 0, which implies
_v? ¼ 1
2
_B
Bv?; ð10Þ
and _vk is given by the parallel component of the equation ofthe ion motion [cf. Northrop, 1963]
m _vk ¼ �mz � rB; ð11Þ
where m is the ion mass.[25] Let us first analyze MI in the linear approximation.
To the first order in the perturbations we have _v? ’ (v?vk/2)@/@ z(dBz/B0) and _vk ’ � (v?
2 /2) @/@z(dBz/B0). Substitutingthese expressions into equation (9), after Fourier transfor-mation and division by the factor (w � kk vk) in equation (9)one finds
df 1ð Þ ¼ �m@F
@mdBz
B0
� wmB0
w� kkvk
@F
@W
dBz
B0
; ð12Þ
where F is the ion unperturbed distribution function anddf (1) is the perturbation of the distribution function in thelinear approximation, respectively. In (12) we passed to thenew variables, particle magnetic moment and total energy
m = mv?2 /2B0 and W = m(vk
2 + v?2 )/2. Expression (12) agrees
with all previous calculations of the distribution functionbased on Leuville’s theorem [cf. Southwood and Kivelson,1993; Pantellini and Schwartz, 1995; Pokhotelov et al.,2000; Ferriere and Andre, 2002]. We note, however, thatexpression for df(1) obtained by Kuznetsov et al. [2007]cannot be reduced to that given by equation (12).[26] The physical content of the different terms in
equation (12) is as follows: The first one on the rightcorresponds to the mirror effect, i.e. it represents the changein the distribution function that is associated with theexpulsion of particles from the regions of increased mag-netic field by the quasi-static compressive magnetic fieldperturbation dBz. The second term describes the resonantwave-particle interactions contributing to the mirror mode.Note that if w is small as we expect, at least for marginalstability, then the second term in equation (12) is negligibleexcept for particles with vk = 0. These particles, which weterm resonant particles, do not move along the magneticfield and their contribution can be comparable with thehydromagnetic term corresponding to the mirror effect.[27] Using equation (12) one easily finds the variation of
the plasma pressure in the linear approximation
dp 1ð Þ? ¼ hmB0df 1ð Þi �
Xs¼�1
2p
m32
Z10
dW
ZW=B0
0
mdmB20df
1ð Þ
2 W � mB0ð Þ½ �12
;
ð13Þ
where the parentheses h. . .i denote the averaging processover the full velocity space, the parameter s has the values±1 that corresponds to the direction of particle velocityalong (s = 1) or opposite (s = �1) to the ambient magneticfield.
[28] The substitution of (12) into (13) gives
dp 1ð Þ? ¼ � m2B0
@F
@m
� dBz
B0
� w mB0ð Þ2
w� kkvk
@F
@W
* +dBz
B0
: ð14Þ
[29] As was already mentioned the mirror modes are foundin the mirror approximation, w kk[n
�1hvk2Fi]12. In this case
the term containing resonant wave-particle interaction inequation (14) can be further simplified, i.e.
�w mB0ð Þ2
w� kkvk
@F
@W
’ � 2p2i
m
wkk�� ��
Z10
W 2 @Fres
@WdW
� 4p
m12
w2
k2k
Z10
dWP
ZW=B0
0
m2B30
2 W � mB0ð Þ½ �32
@F
@Wdm
8<:
9=;; ð15Þ
where
@Fres
@W� @F
@W
� �m¼W=B0
: ð16Þ
[30] In equation (15) P{. . .} denotes the principal value ofthe integral. The first term on the right-hand side of thisequation describes the contribution of resonant particleswith velocity vk = 0. These particles do not move alongthe magnetic field line, and thus for them the condition m =W/B0 holds. The second term corresponds to the adiabaticresponse of the nonresonant particles. In the limit w kk(n�1hvk2Fi)
12 this term can be neglected. In this case (14)
reduces to
dp 1ð Þ? ¼ �bp?
hm2@F=@mihmFi � b
2p2iwDm kk�� �� ; ð17Þ
where p? denotes unperturbed plasma pressure, b = d Bz/B0
is the dimensionless wave amplitude and we haveintroduced the notation
D ¼ �Z10
W 2 @Fres
@WdW : ð18Þ
[31] Under normal conditions in a plasma the particledistribution function in the range of resonance decreaseswith increasing energy such that @Fres/@W < 0,corresponding to D > 0. The analysis of the particular casewhen @Fres/@W > 0 and hence D < 0 requires specialconsiderations. It has been extensively studied by Pokhotelovet al. [2005] and corresponds to the case of the so-called haloinstability. This, however, is beyond the scope of the presentstudy.[32] Expression (17) can also be re-written in the form
given by Pokhotelov et al. [2002]
dp 1ð Þ? ¼ �2bAp? � b
2p2iwDm kk�� �� ; ð19Þ
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A04225
where the parameter A appearing in equation (17) is defined as
A ¼ �hm2B20@F=@W i2p?0
� 1: ð20Þ
[33] The physical content of this parameter can easily beunderstood when inserting a specific velocity distributionfunction into equation (20). For a bi-Maxwellian plasmawith parallel and perpendicular temperatures Tk, T? relation(20) reduces to the familiar anisotropy factor, A = T?/Tk �1.Hence A is a generalized anisotropy factor.[34] Substituting (19) into the Fourier transformed pres-
sure balance condition (7), to the first order in perturbationsone obtains
D� 3
2b?k2?r
2i þ
ip2Dwmp? kk
�� �� ¼ 0; ð21Þ
where the parameters D and K are given by
D ¼ K �k2k
b?k2?
1þb? � bk
2
� �; ð22Þ
K ¼ A� 1
b?: ð23Þ
[35] From equation (21) one finds that Re w = 0 (zero-frequency mode) and
Im w ¼ gL ¼mp? kk
�� ��p2D
K �k2k
b?k2?
1þb? � bk
2
� �� 3
2b?k2?r
2i
" #:
ð24Þ
[36] Let us now obtain the condition of maximum growthof the mirror mode as this for certain problems is of utmostimportance. For a fixed perpendicular wave number k?,maximum growth occurs at a ratio of parallel toperpendicular wave numbers given by
kk
k?
� �2
max
¼ b?3
K � 32b?
k2?r2i
1þ 12
b? � bk
� � : ð25Þ
[37] Substituting (25) into (24) and maximizing the growthrate over k? we find that it attains its maximum value at
k?rið Þ2max¼ b?K
6and kkri
� �2max
¼ b2?
1þ 12
b? � bk
� � K2
24:
ð26Þ
[38] Using equation (26) from (24) we obtain the expres-sion for the maximum growth rate
gmax ¼g04ffiffiffi6
p ð27Þ
where
g0 ¼mp?b?p2riD
K2
1þ b?�bk2
� �12
: ð28Þ
[39] For a bi-Maxwellian distribution function we have
g0 ¼ wci
2
p
� �12
T32
?T�3
2
kK2
1þ b?�bk2
� �12
: ð29Þ
5. Mirror Mode Nonlinear Equation
[40] Let us now consider how the linear theory is mod-ified in the presence of nonlinear effects. For that we re-write equation (7) in the form
dp 1ð Þ? þ dp 2ð Þ
?2p?
þ 1
b?1� 3
2r2i r2
?
� �bþ b2
2b?
¼ � 1
b?1þ
b? � bk
2
� �r�2
?@2
@z2b; ð30Þ
where the variation of the perpendicular plasma pressure isdecomposed into its linear d p?
(1) and nonlinear dp?(2) parts.
Here dp?(2) is the variation of the plasma pressure of the
second order on the dimensionless wave amplitude b = dBz/B0.[41] In the second order in the parameter b we have
_v? ’v?vk
2
@b
@z� b
@b
@zþ dBx
B0
@b
@x
� �; ð31Þ
_vk ¼ � v2?2
@b
@z� b
@b
@zþ dBx
B0
@b
@x
� �: ð32Þ
[42] Substituting (31) and (32) into the Vlasov equation(9) one obtains
@df 2ð Þ
@z¼ � dBx
B0
@df 1ð Þ
@x� @b
@zm@df 1ð Þ
@mþ b
@b
@z� dBx
B0
@b
@x
� �m@F
@m:
ð33Þ
[43] Equation (33) is too difficult for analysis since it cannotbe integrated over the spatial coordinates z and x. We note thatin the similar analysis of Kuznetsov et al. [2007] the nonlinearterms containing d Bx were not taken into consideration.This has led to the erroneously conclusion that differentialequation for d f (2) can be fully integrated in the general case.[44] Here, we limit our consideration to the case when all
perturbed quantities depend solely on one variable z = x +az, where a is the angle of the wave propagation. Since df (1) ’ b = dBz/B0 and the perturbed magnetic field dB obeysthe property of solenoidality, @dBx/@x + @dBz/@z = 0, inequation (33) one can make the following replacementsdBx@df
(1)/@x ! �adBz@df(1)/@z and b@b/@ z � (dBx/B0)@b/
@x ! 2ab@b/@z. Then equation (33) can easily be inte-grated over & and d f (2) reduces to
df 2ð Þ ¼ b2 m@F
@mþ 1
2m2 @
2F
@m2
� �: ð34Þ
[45] Taking the second moment of the distribution func-tion (34) one finds
dp 2ð Þ? ¼ b2 mB0 m
@F
@mþ 1
2m2 @
2F
@m2
� �� ; ð35Þ
A04225 POKHOTELOV ET AL.: MIRROR WAVES
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A04225
or
dp 2ð Þ?
p?¼ 2b2Aþ b2
hm3@2F=@m2i2hmFi � 2Lb2; ð36Þ
where A is the generalized anisotropy factor given byequation (20) and r is
L ¼ Aþ hm3@2F=@m2i4hmFi : ð37Þ
[46] For a bi-Maxwellian distribution function F / exp(mB0A/T? �W/Tk) andr = A + (3/2)A2, where A = T?/Tk �1. We note that r vanishes in pure Maxwellian plasmaswhen the ion distribution depends solely on particle energyW.[47] Substituting (19) and (36) into equation (30) and
making an inverse Fourier transformation we find
Dþ 3
2b?r2i
@2
@&2
� �b� Lþ 1
2b?
� �b2 � p2D
mp?ak�1@tb ¼ 0;
ð38Þ
where D is
D ¼ K � a2
b?1þ
b? � bk
2
� �: ð39Þ
[48] Furthermore, the operator k = H@/@z is a positivedefinite integral operator and H is the Hilbert transform[e.g., Bracewell, 1986].
Hf zð Þ ¼ 1
pP
Z1�1
f z 0ð Þz � z 0
dz 0: ð40Þ
[49] It is convenient to pass to the dimensionless variablesx and t according to
x ¼ zri
b?Kð Þ12 and t ¼ g0t: ð41Þ
where g 0 is given by (28).[50] Then we have
Dþ 3
2b?r2i
@2
@z2¼ K 1� c2 þ 3
2
@2
@x2
� �; ð42Þ
where
c ¼ a 1þb? � bk
2
� �12
b?Kð Þ�12: ð43Þ
[51] With the help of expression (42) equation (38)reduces to
@h
@t¼ ckx 1� c2 þ 3
2
@2
@x2
� �h� h2
� �; ð44Þ
where kx = H@/@x and the normalized amplitude h is
h ¼ Lþ 2b?ð Þ�1
Kb: ð45Þ
[52] Maximizing equation (44) over c we find that thefastest growing mode corresponds to c = 3�
12 and thus the
equation governing the nonlinear dynamics of this mode is
@h
@t¼ 2kx
332
1þ 9
4
@2
@x2
� �h� 3
2h2
� �: ð46Þ
[53] After rescaling h = (3/2)h0, t = (332/2) t0 and x = (3/2)
x0 equation (46) reduces to
@h
@t¼ kx 1þ @2
@x2
� �h� h2
� �; ð47Þ
where for the sake of convenience all the primes areomitted. Equation (47) describes the nonlinear dynamics ofmirror waves in the case when the ion distribution functionF does not vary in time. The case when it varies due to theparticle trapping will be considered in the next section.Equation (47) has been solved numerically by usingpseudo-spectral scheme [e.g., Boyd, 2000]. The spatial partof the function h in equation (47) has been decomposed intosparse grid by pseudo-spectral method and the timeevolution has been given by the fourth-order Runge-Kuttacode. The results of numerical simulations are shown inFigures 1 and 2 where the dimensionless amplitude h isplotted as a function of the spatial and temporal variables xand t, respectively. At the initial moment, t = 0, the waveamplitude has been chosen in the form of a normal Gaussiandistribution h(t= 0) = Aexp (�x2/2s2), where A = �0.2 isthe wave amplitude and s = 0.05 stands for the standarddeviation. The latter is connected with the full width at halfthe maximum amplitude (FWHMA) Dx through theordinary relation Dx = 2.35s. In order to stress the roleof nonlinear effects this dependence is depicted in twodifferent cases: when the nonlinear term in equation (47) isdisregarded (Figure 1) and when both the linear andnonlinear terms are included (see Figure 2). Comparison ofthese two figures shows that the nonlinearity dramaticallychanges the evolution of mirror waves. One sees that theexponential increase in the wave amplitude that is inherentto the linear stage of the MI is then replaced by an evenfaster increase resulting in the formation of a magnetictrough or magnetic hole at the origin of x. Figure 2 showsthat in a finite time the wave solution blows up. Figure 3shows the evolution of the mirror mode in case when initialperturbation has the sinusoidal (single-mode) form with thewave amplitude A = 0.2. Similar to the previous case thenonlinear evolution possesses the blow-up behavior.[54] The physical reason for such a nontrivial evolution of
the magnetic perturbations can easily be understood if onere-writes equation (47) in the Hamiltonian form
@h
@t¼ �kx
dLdh
; ð48Þ
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where
L ¼Z
� h2
2þ r?hð Þ2
2þ h3
3
" #dx: ð49Þ
[55] The functional L has the meaning of the free energyor the Lyapunov functional.[56] The remarkable feature of this functional is that it
cannot grow but solely monotonically decays in time, i.e.dL/dt � 0. This becomes clear from Figure 4 where dL/dtis plotted as a function of time. The formation of singularityis connected with the fact that L becomes more and morenegative. This effect is produced by the last term in theexpression for the Lyapunov functional (49), i.e. by
R(h3/
3)dx < 0. Physically the blow up behavior arises due to thefast decrease in the free energy shown in Figure 4. We note
that in Figures 1–3 the dimensionless amplitude h introducedhere for numerical simulations can take large negative valuesbecause according to our scaling the dimensionless waveamplitude h introduced here for numerical simulations is infact/ (dBz/B0)/K. Since we have limited our consideration bythe marginal conditions, i.e. K 1 then even for dBz/B0 1we have finite values of h.[57] It should be noted that our numerical modeling
shows the appearance of solely magnetic holes. When themagnetic humps as an initial perturbation have been select-ed they rapidly relax and disappear within relatively shorttime. This agrees with the particle-in-cell simulations pro-vided by Baumgartel et al. [2003].[58] The above consideration is valid only for the mirror
mode. For the large amplitude fast MS waves the nonlinearscenario is different. These waves can propagate nearlyperpendicular to the ambient magnetic field in the form of
Figure 1. A plot of h as a function of x and t calculated with the help of equation (47) for the case ofthe Gaussian initial distribution when the nonlinear term is disregarded. The initial amplitude is A = �0.2.
Figure 2. Same as in Figure 1 but when the nonlinear term is retained. One sees that nonlinearevolution of the MI is accompanied by the formation of a finite-time singularity in the magnetic field.
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stable MS solitons [e.g., Pokhotelov et al., 2007a, 2007b].Their shapes are sensitive to the details of the ion distribu-tion function. For example, in a bi-Maxwellian plasma thedispersion of the fast MS waves is negative, i.e. the phasevelocity decreases with an increase of the wavenumber. Thisassumes that the solitary solution in this case has the form ofa bright soliton with the magnetic field increased. On thecontrary, in some non-Maxwellian plasmas, such as thosewith ring-type ion distributions or Dory-Guest-Harris plas-mas they can have a positive dispersion and thus solitarysolutions may have the form of a magnetic hole (darksolitons). We note that analytical and numerical studies ofnonlinear structures involving MS waves have also been thesubject of previous research [e.g., Kaufman and Stenflo,1975; Sharma and Shukla, 1983].
6. Incorporation of the Particle Trapping.Collapse Break-up
[59] In the scenario described above it was assumed thatunperturbed ion distribution function does not vary in time.This is valid as long as the effects due to particle trappinginto the magnetic troughs do not play a role. The effects ofparticle trapping in the nonlinear mirror modes has beendiscussed by Pantellini et al. [1995]. These authors arguedthat the main mechanism that ends the linear phase of theMI has to be particle trapping into the mirror holes. Itshould be mentioned that in the formation of collapse thenonlinear corrections to the plasma pressure are quadratic inwave amplitude, i.e. d p?
NL / b2. However, the nonlinearcorrections to the variation of the particle pressure due tothe particle trapping can be as high as jbj3/2. Thus, thechange in the particle distribution function can developfaster than the wave collapse evolves. As will be shownbelow this effect can terminate the wave collapse and canresult in a collapse break-up. It should be noted that
nonlinear FLR effect that has been used by Kuznetsov et al.[2007] for similar purposes is of the higher order in smallparameter and thus is less important.[60] In what follows we adopt the following model:[61] . Due to the rapid motion of resonant particles we
assume that in the vicinity of small parallel velocities thebackground ion distribution function would flatten andtakes the shape of a quasi-plateau. This to happen doesnot require the assumption of random phases and is valideven in the single-mode (sinusoidal) regime:[62] . The width of such flattened region is of the order of
the trapping zone in the mirror hole, i.e. ’ D vk ’ 2vk*
232(mjdBzj/m)
12 = 2v?(jdBzj/B0)
12 = 2v?jbj
12 (see Figure 5).
[63] . The particle trapping is irreversible, i.e. accompa-nied by bifurcation and violation of adiabatic invariancesince the depth of the mirror hole increases more slowly forthe trapped particles that possess the variation of the
Figure 3. Same as in Figure 2, but for the case single-mode (sinusoidal) initial perturbation withA = �0.2.
Figure 4. A plot of the time derivative of the Lyapunovfunctional dL/dt as a function of the dimensionlesstime t.
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characteristic velocities dvk ’vTjbj12 whereas for the
nonresonant particles dvk ’ dv? ’ vTjbj.[64] In order to construct a self-consistent model describ-
ing the evolution and final saturation of mirror waves weinterpolate the background ion distribution function as arigorous plateau stretched along vk from �v?jbj
12 to v?jbj
12
(Figure 5). Outside this region the ion distribution functiontakes a bi-Maxwellian form. Clearly, such interpolationcould be valid up to the numerical coefficient of the order ofunity. Its relevant value can be found only by numericalsimulation of the time dependent process of particletrapping by the emerging magnetic holes prior to the MIsaturation. In somewhat similar problem of saturation of a
single mode electrostatic beam instability due to thetrapping of the electrons into electrostatic potential wellssuch computational analysis established that equivalentnumerical coefficient appeared to be close to unity [cf.Fried et al., 1970].[65] In our case with the assumed form of the background
distribution function, the contribution of the perpendicularplasma pressure variation due to the resonant particlesentering the plasma pressure balance condition reduces to
dpres? ¼ �pm@b
@t
Z10
v5?dv? limn!0
Z1v�k
dvkn
n2 þ k2kv2k
@F
vk@vk; ð50Þ
Figure 6. Same as in Figure 2 but for the case when particle trapping is incorporated.
Figure 5. The ion distribution function with the plateau stretched along the parallel velocities.
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where vkR= v?jbj
12 and F is given by
F ¼ n
p3=2v2T?vTke�
v2?v2T?
�v2k
v2Tk : ð51Þ
[66] Here n is the unperturbed plasma number density,and vT? and vTk are the thermal perpendicular and parallelvelocities, respectively.[67] When vk* vTk, i.e. jbj 1, from (50) and (51) one
obtains
dpres? ¼ 2nm
p1=2 kk�� ��v2T?v3Tk
@b
@tlimn!0
Z10
v5?e�
v2?v2T?
� dv?p2� arctan
v?jkkjjbj12
n
!ð52Þ
[68] If the wave amplitude is small and the plateau is verynarrow, Dvk = 2vk* v/jkkj, the perturbation of the plasmapressure of the resonant particle reduces to its linear(Landau) value, i.e.
dpres? ¼ p?2p1=2
kk�� ��vTk
T?
Tk
@b
@t: ð53Þ
[69] In this case the substitution of dp?res into the
perpendicular plasma pressure condition recovers thestandard expression for the linear MI growth rate whenthe undisturbed ion distribution function is bi-Maxwellian[cf. Vedenov and Sagdeev, 1958].[70] On the contrary, when the plateau is saturated, i.e.
jkkj vk* � n, we expand arctan (v?jkkjjbj12/v) ! p/2 � v/
v?jkkjjbj12 and thus dp?
res ! 0, i.e. the wave growth isterminated. These features of the resonant particle dynamicsshould be taken into consideration in the course of the
derivation of the corresponding nonlinear equation (47).Taking into account the above considerations we concludethat the left-hand side of this equation should be replaced bya more complex expression that takes into account theeffects due to the particle trapping. In the general case it hasa quite complex form. For the qualitative conclusions wecan use a simplified (model) form for this equation. Thesimplest extrapolation for the nonlinear equation that takesinto account the basic features of the particle trapping effectcan be phenomenologically written as
1� 2
parctan
hj j1=2
l
!@h
@t¼ kx 1þ @2
@x2
� �h� h2
� �; ð54Þ
where l is a numerical coefficient of the order of unity.[71] Equation (54) has been solved numerically. The
results are depicted in Figures 6 and 7 for the case of
Figure 7. Same as in Figure 6, but for the one-mode initial regime.
Figure 8. A plot of maximum depth of the mirror hole asthe function of time for single-mode regime.
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Gaussian and single-mode regimes, respectively. Figure 8shows the evolution of the maximum depth of the magnetichole. One sees that the collapse is arrested when h’ �0.8,then the magnetic hole depth possesses small oscillationsand finally is saturated at this value.[72] We note that numerical simulation of the mirror
mode nonlinear dynamics has been recently carried out byBorgogno et al. [2007] in the framework of a fluid modelthat incorporates linear Landau damping and FLR correc-tions. However, the effects due to nonlinear Landau inter-actions considered above were not taken into account inthis study.
7. Discussion and Conclusions
[73] We have presented a local analysis of the MI in ahigh-b non-Maxwellian plasma in the framework of astandard mixed magnetohydrodynamic-kinetic theoryaccounting for the FLR effect and nonlinearity of the mirrorwave perturbations near the instability threshold. It has beenshown that nonlinear effects substantially modify the MIdynamics and can lead to the formation of magnetic holessimilar to those observed in the satellite data. The charac-teristic scales of these holes are of the order of a few ionLarmor radii. It has been shown that the resonant ionshaving small parallel velocities play an important role inthe MI nonlinear dynamics. It has been shown that theeffect of flattening of the ion distribution function in theresonant region can lead to saturation of the magnetic holedepth and results in the appearance of quasi-stationary mirrormode structures. Our paper generalizes previous studiesrelated to the linear MI and provides some additionalphysical bases for better understanding of large ampli-tude nonlinear mirror waves in space and astrophysicalplasmas.[74] In addition to the previous analysis provided by
Kuznetsov et al. [2007] the new kinetic nonlinear effectsof the order of jbj 3/2 associated with modification of theshape of the background ion distribution function in theregion of small parallel particle velocities are taken intoaccount. These effects are much stronger than those non-linear MHD effects considered by Kuznetsov et al. [2007](/b2) and thus should play an important role in nonlineardynamics of the mirror mode. It has been suggested that thiseffect can efficiently suppress or even arrest the mirrorcollapse (see Figures 6–8). The corresponding phenome-nological equation (see equation (54)) has been formulated.The nonlinear FLR effect that has been used by Kuznetsovet al. [2007] for similar purposes is of the higher orderin small parameter and thus is of minor importance. Ac-cordingly the nonlinear FLR effect has been disregarded inour study.[75] Furthermore, the analysis presented above shows that
the effects due to the magnetic bending are important notonly in the linear approximation but in the nonlinear regimeas well. These effect was not included in the analysis ofKuznetsov et al. [2007]. The importance of these additionalterms (the terms including dBx) becomes clear from equa-tion (33). Using this equation one readily obtains that theratio of the terms containing dBz and those with dBx that
correspond to the effect of magnetic field line bending is ofthe order of
R ¼ dBz@dBz=@z
dBx@dBz=@x’
dBzkk
dBxk?: ð55Þ
[76] Since the magnetic field perturbations are diver-gence free, r � dB = 0, one finds that dBz/dBx ’ k?/kk andthus R ’ 1. The latter means that the terms neglected byKuznetsov et al. [2007] are exactly of the same order asthose retained.[77] It is worth mentioning that the model presented in
this paper remains oversimplified. So far our considerationis valid solely for the marginal conditions when the param-eter K = A � b?
�1 is small. An additional analysis is requiredwhen the system is far from this state. Furthermore, in thepresent paper we have considered the case of cold electrons.Actually this is quite common for the magnetosheath wherethe ion temperature is always larger than the electrontemperature. The incorporation of the finite electrontemperature in the MI linear theory has been the subjectin some previous studies [e.g., Pantellini and Schwartz,1995; Pokhotelov et al., 2000]. In these papers it has beenshown that when the electron temperature becomes of theorder of the ion temperature the growth rate of the ionmirror mode is reduced by the presence of the field-alignedelectric field. The origin of the electric field is the electronpressure gradient set up as electrons are dragged by thenonresonant ions that have been accelerated as they passfrom regions of high magnetic flux into lower flux regions.All these factors have to be taken into account in a morecomprehensive nonlinear theory of the MI.[78] The major and so far unresolved question of the
problem at hand regarding the appearance of the MI inspace plasmas focuses on the relation between the mirrorlike and ion-cyclotron instabilities (ICI). Both instabilitiescompete for the pressure anisotropy. In the anisotropic caseit is believed that the latter should grow at faster rate andshould therefore appear before the MI can set on. Thisquestion was urged by Gary [1992] and by Gary et al.[1995], who presented the results of numerical simulationsof the fully kinetic dispersion relation describing bothinstabilities, the MI and ICI. The key to the understandingthe problem was offered in some previous publications [e.g.,Russell et al., 1999; Huddleston et al., 1999] in the courseof the analysis of the Galileo magnetic field data on theedges of the cold Io wake. They showed that the multispe-cies content of the Io plasma may create more favorableconditions for the excitation of mirror like instabilities. Themore species present in the plasma, the more ion cyclotronmodes are possible in a system. However, the growth rate ofeach individual mode is now smaller than that of a singlemode arising in an electron-proton plasma. On the otherhand, the mirror like instability exhausts the free energystored in all components of a multispecies plasma.[79] Generally the wave particle interaction at cyclotron
resonance is touching particles with energies higher thanthermal, unlike the MI which deals mainly with ions belowthermal energies. In this respect those two modes couldcoexist if the background distribution is capable to supportboth of them.
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[80] The intention of the present approach is to providedeeper insight into the physics of the MI. Hence this papercan be considered as an extension of our previous approachto the study of this instability which was limited by theconsideration of a solely linear theory. The results of ourstudy might be useful for a better understanding of the MIproperties, as well as for the interpretation of recent satelliteobservations provided by the Cluster fleet [e.g., Narita etal., 2006; Hobara et al., 2007].
[81] Acknowledgments. This research was supported by PPARCthrough grant PP/D002087/1, the Russian Fund for Basic Research grantsNo 08-05-00617 and 07-05-00774, by ISTC project No 3520 and by theProgram of the Russian Academy of Sciences No 16 ‘‘Solar activity andphysical processes in the Solar-Earth system’’. The authors are grateful toV. V. Krasnoselskikh, E. A. Kuznetsov and M. S. Ruderman for valuablediscussions.[82] Amitava Bhattacharjee thanks the reviewer for his assistance in
evaluating this paper.
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�����������������������M. A. Balikhin and O. A. Pokhotelov, Department of Automatic Control
and Systems Engineering, Sheffield University, Mappin Street, SheffieldS1 3JD,UnitedKingdom. ([email protected]; [email protected])V. N. Fedun, Department of Applied Mathematics, University of
Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom. ([email protected])O. G. Onishchenko, Institute of Physics of the Earth, 10 B. Gruzinskaya
Street, 123995 Moscow, Russia. ([email protected])R. Z. Sagdeev, Department of Physics, The University of Maryland,
College Park, MD 20740, USA. ([email protected])
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