Nonlinear mirror waves in non-maxwellian space plasmas

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi:1.19/7ja164, 8 Nonlinear mirror waves in non-maxwellian space plasmas O. A. Pokhotelov, 1 R. Z. Sagdeev, M. A. Balikhin, 1 O. G. Onishchenko, 3 and V. N. Fedun 4 Received 16 July 7; revised 14 December 7; accepted 15 January 8; published 9 April 8. [1] A theory of finite-amplitude mirror type waves in non-maxwellian space plasmas is developed. The collisionless kinetic theory in a guiding center approximation, modified for accounting of the finite ion Larmor radius effects, is used as the starting point. The model equation governing the nonlinear dynamics of mirror waves near instability threshold is derived. In the linear approximation it describes the classical mirror instability that is valid for a wide class of the velocity distribution functions. In the nonlinear regime the mirror waves form solitary structures that have the shape of magnetic holes. The formation of such structures and their nonlinear dynamics has been analyzed both analytically and numerically. It is suggested that the main nonlinear mechanism responsible for mirror instability saturation is associated with modification (flattening) of the shape of the background ion distribution function in the region of small parallel particle velocities. The width of this region is of the order of the particle trapping zone in the mirror hole. Near the mirror instability threshold the saturation arises before its width reaches the ion thermal velocity. The nonlinear mode coupling effects in this approximation are smaller and unable to take control over evolution of the space profile of saturated mirror waves or lead to their magnetic collapse. This results in the appearance of quasi-stable solitary mirror structures having the form of deep magnetic depressions. A phenomenological description of this process is formulated. The relevance of the theoretical results to recent satellite observations is stressed. Citation: Pokhotelov, O. A., R. Z. Sagdeev, M. A. Balikhin, O. G. Onishchenko, and V. N. Fedun (8), Nonlinear mirror waves in non-maxwellian space plasmas, J. Geophys. Res., 113,, doi:1.19/7ja Introduction [] The classical mirror instability (MI) theoretically identified in the end of the 5th [Vedenov and Sagdeev, 1958; Rudakov and Sagdeev, 1958; Chandrasekhar et al., 1958] is a common feature in planetary and cometary magnetosheaths. The early theories were limited by purely linear analysis and thus could not provide a comprehensive comparison with in situ observations. Interest to the MI was substantially reinforced by its possible relevance to spacecraft observations of the so-called magnetic holes - deep dropouts of the magnetic field in high-b plasmas. The term magnetic hole was first suggested by Turner et al. [1977] to describe decreases in the solar wind magnetic field strength. They were found in numerous planetary magnetosheaths [Kaufmann et al., 197; Tsurutani et al., 198; Lühr and Klöcker, 1987; Treumann et al., 199; Bavassano Cattaneo et al., 1998; Joy et al., 6] in the solar wind [Turner et al., 1 Automatic Control Systems Engineering, University of Sheffield, Sheffield, United Kingdom. Department of Physics, University of Maryland, College Park, Maryland, USA. 3 Institute of Physics of the Earth, Moscow, Russia. 4 Department of Applied Mathematics, University of Sheffield, Sheffield, United Kingdom. Copyright 8 by the American Geophysical Union /8/7JA ; Fitzenreiter and Burlaga, 1978; Winterhalter et al., 1994; Stevens and Kasper, 7] and in the vicinity of comets [Russell et al., 1987; Glassmeier et al., 1993]. The mirror wave structures were found also in the sheath regions between fast interplanetary coronal mass injections and their preceding shocks [Liu et al., 6]. Ultimately they represent a nonlinear phenomenon. It should be noted that besides the magnetic holes some observations reveal other forms of the mirror structures as quasi-sinusoidal profiles or as magnetic humps. For example, they have been found in the Jovian magnetosheath [Joy et al., 6] and in numerical simulations [Baumgartel et al., 3]. [3] The cornerstone of the instability mechanism is the pressure anisotropy that provides the free energy necessary for its onset. As the solar wind slams into the magnetosphere it abruptly slows down, and a bow shock and a magnetosheath are formed. In the magnetosheath the pressure anisotropy is generated when the plasma flow crosses the bow shock. The passage of the solar wind through this region leads to an increase in the ion temperature, with the larger proportion of the heating going preferentially into the perpendicular component. This results in the appearance of anisotropic ion populations with stronger pressure anisotropy than in the solar wind. It should be noted that the pressure anisotropy is also generated in the magnetosheath itself as the plasma flows around the magnetopause. 1of13

2 [4] The MI has a number of peculiarities that makes it a possible candidate to fit into a scenario of magnetic hole formation, among these are the nonpropagating nature of the mode and the anti-correlation of plasma density and magnetic field. [5] The MI corresponds to a resonant type instability that arises due to the interaction of the zero-frequency mode with the ions possessing very small velocities along the external magnetic field. The physical mechanism of the linear MI has been reviewed by Southwood and Kivelson [1993]. The linear mirror mode theory has been extensively developed in a number of previous papers [e.g., Pokhotelov et al., and references therein]. Those papers included the effects of a finite electron temperature, a nonvanishing electron temperature anisotropy, multi-ion plasmas, the finite ion Larmor radius (FLR) effect and can be applied to a wide class of the ion and electron distribution functions of the form f(m, W), where m and W are the particle magnetic moment and energy, respectively. A very general compact expression for the linear marginal mirror-mode stability condition and instability growth rate suited for application to such class of velocity distribution functions has been derived by Leubner and Schupfer [, 1] and Pokhotelov et al. [] and has then been applied to bi-maxwellian, Dory- Guest-Harris (DGH), Kennel-Ashour-Abdalla (KA) and k -distributions. However, all these papers have been restricted 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 Shapiro and Shevchenko [1964]. These authors using the random phase approximation have reduced the problem of nonlinear saturation of the MI to the study of a quasilinear diffusion equation for the ion distribution function in which all ions were considered to be fully adiabatic. Indeed, the background ion distribution function is shown to be modified which leads to saturation of MI. However, such a scenario that assumes the adiabaticity of particle trajectories does not take into account the effects of resonant trapping. Below we will incorporate this effect for the study of nonlinear saturation of the MI. We note that Shapiro and Shevchenko [1964] came to an important conclusion on the special role of ions having small parallel velocities. In what follows, in going beyond the random phase approximation, this feature will remain intact. [7] It should be noted that some effects related to nonlinear saturation of mirror waves in the magnetosheath have been already discussed by Pantellini [1998] and Kivelson and Southwood [1996]. Recently Kuznetsov et al. [7] suggested a new nonlinear theory of MI in bi-maxwellian plasmas that describes the formation of magnetic holes in terms 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. In this connection a few comments are in order. [9] First, it should be noted that measured particle distributions in near-earth space do, in the overwhelming majority of cases, considerably deviate from the bi-maxwellian shape. They frequently exhibit long suprathermal tails on the distribution function or, in other cases, possess loss cones. In collisionless plasmas, such as planetary magnetosheaths, the decisive role in determining the shape of the particle velocity distribution functions may belong to wave-particle interactions. In the range of resonance these interactions may lead to the formation of diffusion plateaus. For example, such plateaus were recently observed in the velocity distribution of fast solar wind protons [Heuer and Marsch, 7]. [1] Second, the wave collapse can be arrested in the presence of particle trapping in the mirror hole that arises in the resonance region. The latter is generally much stronger than 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 for a wide class of the plasma velocity distribution functions and incorporate the effects due to the particle trapping. [1] The paper is organized as follows: section formulates the derivation of the nonlinear plasma pressure balance condition for the MI in terms of an arbitrary ion distribution function. The generalization of this condition to the case when the FLR effect is included is carried out in section 3. The linear growth rate of the MI accounting for the FLR effect is derived in section 4. The mirror mode nonlinear equation is obtained in section 5. The effects of particle trapping in the mirror holes are discussed in section 6. Our conclusions and outlook for future research are found in section 7.. Plasma Pressure Balance Condition [13] We consider a collisionless plasma composed of ions and electrons having the total mass density r and macroscopic velocity v, and embedded in a magnetic field B. Our analysis will be limited to the case of most importance when the ion temperature is much larger than the electron temperature. The momentum equation governing the time evolution of this plasma is r dv dt ¼ J B r^p; where J is the electric current and t + v ris the Lagrangian time derivative. The pressure tensor represents the anisotropic pressure tensor ^P given by ð1þ ^P ¼ p ^I þ p k p ^z^z: ðþ [14] Here ^I is the unit dyadic, p k () 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 B is directed along the z axis. The wave fields exhibit only the y component of the electric field E y and the z and the x components of the magnetic field, d B z and db x, respectively. The latter are connected through the property of solenoidality of the magnetic field, i.e. x /@x z /@ z =.As regards the fluid velocity v = (E B) /B, only its x component survives in the leading order, i.e. v v x. The db y component corresponds to the so-called non-coplanar magnetic component, and does not enter our basic equations and can thus be set to zero. of13

3 [16] Similarly, equation (1) shows that the only nonzero component of the electric current is the y component J y J given by J ¼ þ þ B p k p B@z db z dx: [17] The physical meaning of the different terms on the right-hand side of equation (3) is as follows: The first one, containing the Lagrangian time derivative, is due to plasma inertia. This term is important for the fast magnetosonic solitons (MS) in which w k v A (1 + b ), where w and k are the characteristic wave frequency and wave number, respectively, and v A is the Alfvén velocity. We note, that in a high-b plasma the Alfvén velocity is of the order of the ion thermal velocity v T. The propagation of MS solitons nearly perpendicular to the external magnetic field in non- Maxwellian plasmas has recently been considered by Pokhotelov et al. [7a, 7b]. However, for mirror waves, which are the subject of our study, the situation is quite different. These waves are found in the so-called mirror approximation, w k k (n 1 hv k Fi) 1 [cf. Southwood and Kivelson, 1993; Pokhotelov et al., ]. Here k k is the typical parallel wave number, n is the plasma number density, v k is the parallel particle velocity and F is the undisturbed particle distribution function. In this limit the inertial term in equation (3) is small as w /k v T 1, relative to the second term and must thus be neglected. The differences in the mathematical description of the fast MS and mirror waves in the high-b plasmas have been extensively discussed recently by Ferrière and André []. Finally the last term in equation (3) describes the effect of magnetic field line bending. We note that since this term already contains a small parameter k k /k / db z /B / e 1 one may set here the value of the magnetic field B as well as the difference p k p equal to their unperturbed values. [18] Thus, in the mirror limit which is the subject of our study the inertial term can be neglected and the electric current J reduces to J b b Z db z ðx Þdx ; where b (k) = m p (k) /B is the perpendicular (parallel) plasma beta, and m is the permeability of free space and delta denotes a perturbation. [19] Furthermore, the Ampére s law reads ð3þ x ¼ m J: ð5þ [] Substituting (4) into (5) and integrating over x one obtains a nonlinear equation for the perpendicular plasma balance condition dp þ B db z þ db z m m ¼ 1 þ b Z Z B db z m dx dx ; ð6þ where the variation of the plasma perpendicular pressure can be found from the Vlasov equation for the particle distribution function. In the linear approximation equation (6) coincides with the perpendicular plasma balance condition that was usually used for the study of linear MI [cf. Pokhotelov et al., ]. 3. Incorporation of FLR Effect [1] The pressure balance condition (6) can easily be generalized to the case when the FLR effect is taken into consideration. This effect has been extensively studied for mirror waves by many authors [e.g., Hasegawa, 1969; Hall, 1979; Pokhotelov et al., 4; Hellinger, 7]. It was shown that when the transverse scale size of the mirror wave becomes comparable to the ion Larmor radius, the actual impact of the wave magnetic and electric fields is reduced due to partial cancelation of it integrated over the ion Larmor orbit. This leads to modification of the expression for Fourier transform (dp (k ) = R dp (r )e ik r dr ) of the variation of the perpendicular plasma pressure d p! (1 + 3 r i r ) dp with r i being the ion Larmor radius and /@x. This effect was first introduced by Rosenbluth et al. [196] for stabilization of flute instability. Later it was widely applied by many authors to other problems of plasma physics. In particular, this effect is similar to the FLR suppression of the perpendicular electric current described by Kadomtsev [1965] [cf. Cheng, 1991] for kinetic Alfvén waves. Using these considerations equation (6) reduces to dp þ B db z 3 B db z m r i r m ¼ 1 þ b b k þ db z r B db : ð7þ m [] Here and in what follows we make use of the following hierarchy of small parameters r i db z e: B [3] According to this hierarchy the FLR corrections in the nonlinear term and in the term containing the z derivative are neglected as corrections of higher order on a small parameter e. 4. Mirror Instability With FLR Effect [4] In order to derive the equation governing the dynamics of the MI it is necessary to calculate the variation of the plasma pressure. For that purpose we make use of the drift kinetic equation. Since we are only interested in waves with frequencies much lower than the ion cyclotron frequency we use the distribution function averaged over the ion gyromotion, f(t, r g, v, v k ), where t is the time, r g is the guiding center position vector, v is the gyration speed, and v k is the velocity along the field lines. The distribution function obeys the g rf þ _v þ _v k k ð8þ ð9þ 3of13

4 where _r g = v E + v k ^z, v E is the E B velocity, and _v is given by the expression for the conservation of the magnetic moment, dm/dt =, which implies _v ¼ 1 _B B v ; ð1þ and _v k is given by the parallel component of the equation of the ion motion [cf. Northrop, 1963] m_v k ¼ m^z rb; ð11þ where m is the ion mass. [5] Let us first analyze MI in the linear approximation. To the first order in the perturbations we have _v (v v k z(db z /B ) and _v k (v z /B ). Substituting these expressions into equation (9), after Fourier transformation and division by the factor (w k k v k ) in equation (9) one finds df ðþ 1 ¼ db z db z B w k k v B ð1þ where F is the ion unperturbed distribution function and df (1) is the perturbation of the distribution function in the linear approximation, respectively. In (1) we passed to the new variables, particle magnetic moment and total energy m = mv /B and W = m(v k + v )/. Expression (1) agrees with all previous calculations of the distribution function based on Leuville s theorem [cf. Southwood and Kivelson, 1993; Pantellini and Schwartz, 1995; Pokhotelov et al., ; Ferrière and André, ]. We note, however, that expression for df (1) obtained by Kuznetsov et al. [7] cannot be reduced to that given by equation (1). [6] The physical content of the different terms in equation (1) is as follows: The first one on the right corresponds to the mirror effect, i.e. it represents the change in the distribution function that is associated with the expulsion of particles from the regions of increased magnetic field by the quasi-static compressive magnetic field perturbation db z. The second term describes the resonant wave-particle interactions contributing to the mirror mode. Note that if w is small as we expect, at least for marginal stability, then the second term in equation (1) is negligible except for particles with v k =. These particles, which we term resonant particles, do not move along the magnetic field and their contribution can be comparable with the hydromagnetic term corresponding to the mirror effect. [7] Using equation (1) one easily finds the variation of the plasma pressure in the linear approximation dp ðþ 1 ¼hmB df ðþ 1 i X p Z 1 s¼1 m 3 dw ZW=B B mdm df ðþ 1 ; ½ðW mb ÞŠ 1 ð13þ where the parentheses h...i denote the averaging process over the full velocity space, the parameter s has the values ±1 that corresponds to the direction of particle velocity along (s = 1) or opposite (s = 1) to the ambient magnetic field. [8] The substitution of (1) into (13) gives * + dp ðþ 1 dbz m B wmb B w k k v db z B : ð14þ [9] As was already mentioned the mirror modes are found in the mirror approximation, w k k [n 1 hv k Fi] 1. In this case the term containing resonant wave-particle interaction in equation (14) can be further simplified, i.e. wmb ð p i w k k v m 4p w m 1 kk where Z 1 8 < dw P : ZW=B w k k Z 1 dw 9 m B = ½ðW mb ÞŠ dm ; ; res m¼w=b ð16þ [3] In equation (15) P{...} denotes the principal value of the integral. The first term on the right-hand side of this equation describes the contribution of resonant particles with velocity v k =. These particles do not move along the magnetic field line, and thus for them the condition m = W/B holds. The second term corresponds to the adiabatic response of the nonresonant particles. In the limit w k k (n 1 hv k Fi) 1 this term can be neglected. In this case (14) reduces to dp ðþ 1 ¼ bp hmfi b p iwd mk k ; ð17þ where p denotes unperturbed plasma pressure, b = d B z /B is the dimensionless wave amplitude and we have introduced the notation D ¼ Z 1 dw: ð18þ [31] Under normal conditions in a plasma the particle distribution function in the range of resonance decreases with increasing energy such res /@W <, corresponding to D >. The analysis of the particular case res /@W > and hence D < requires special considerations. It has been extensively studied by Pokhotelov et al. [5] and corresponds to the case of the so-called halo instability. This, however, is beyond the scope of the present study. [3] Expression (17) can also be re-written in the form given by Pokhotelov et al. [] dp ð1þ ¼ bap b p iwd mk k ; ð19þ 4of13

5 where the parameter A appearing in equation (17) is defined as A ¼ hm p 1: ðþ [33] The physical content of this parameter can easily be understood when inserting a specific velocity distribution function into equation (). For a bi-maxwellian plasma with parallel and perpendicular temperatures T k, T relation () reduces to the familiar anisotropy factor, A = T /T k 1. Hence A is a generalized anisotropy factor. [34] Substituting (19) into the Fourier transformed pressure balance condition (7), to the first order in perturbations one obtains D 3 k b r i þ ip Dw mp k k ¼ ; where the parameters D and K are given by ð1þ D ¼ K k k b k 1 þ b b k ; ðþ K ¼ A 1 b : ð3þ [35] From equation (1) one finds that Re w = (zerofrequency mode) and Im w ¼ g L ¼ mp k k " p K k k D b k 1 þ b b # k 3 k b r i : ð4þ [36] Let us now obtain the condition of maximum growth of the mirror mode as this for certain problems is of utmost importance. For a fixed perpendicular wave number k, maximum growth occurs at a ratio of parallel to perpendicular wave numbers given by k k k ¼ b max 3 K 3 b k r i : ð5þ 1 þ 1 b b k [37] Substituting (5) into (4) and maximizing the growth rate over k we find that it attains its maximum value at ðk r i Þ max ¼ b K 6 and k k r i max¼ b 1 þ 1 b b k K 4 : ð6þ [38] Using equation (6) from (4) we obtain the expression for the maximum growth rate where g ¼ mp b p r i D g max ¼ g p 4 ffiffiffi 6 K ð7þ 1 þ b 1 : ð8þ b k [39] For a bi-maxwellian distribution function we have 1 T 3 g ¼ w ci p T 3 k K 5. Mirror Mode Nonlinear Equation 1 þ b 1 : ð9þ b k [4] Let us now consider how the linear theory is modified in the presence of nonlinear effects. For that we rewrite equation (7) in the form þ dp ðþ þ p b r i r b þ b b ¼ 1 1 þ b b k b; dp ðþ 1 ð3þ where the variation of the perpendicular plasma pressure is decomposed into its linear d p (1) and nonlinear dp () parts. Here dp () is the variation of the plasma pressure of the second order on the dimensionless wave amplitude b = db z / B. [41] In the second order in the parameter b we have _v v v k _v k @z þ db ; þ db : ð3þ [4] Substituting (31) and (3) into the Vlasov equation (9) one ðþ ¼ db ðþ 1 B ðþ db : B ð33þ [43] Equation (33) is too difficult for analysis since it cannot be integrated over the spatial coordinates z and x.wenotethat in the similar analysis of Kuznetsov et al. [7] the nonlinear terms containing d B x were not taken into consideration. This has led to the erroneously conclusion that differential equation for d f () 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 d f (1) b = db z /B and the perturbed magnetic field db obeys the property of x /@x z /@z =,in equation (33) one can make the following replacements db (1) /@x! adb (1) /@z and b@b/@ z (db x /B ab@b/@z. Then equation (33) can easily be integrated over & and d f () reduces to df ðþ ¼ þ 1 : ð34þ [45] Taking the second moment of the distribution function (34) one finds dp ðþ ¼ b þ 1 ; ð35þ 5of13

6 or where ^kx = ^H@/@x and the normalized amplitude h is dp ðþ p ¼ b A þ b F=@m i Lb ; hmfi ð36þ h ¼ L þ ð b K Þ 1 b: ð45þ where A is the generalized anisotropy factor given by equation () and r is L ¼ A þ F=@m i : ð37þ 4hmFi [46] For a bi-maxwellian distribution function F / exp (mb A/T W/T k ) and r = A + (3/)A, where A = T /T k 1. We note that r vanishes in pure Maxwellian plasmas when the ion distribution depends solely on particle energy W. [47] Substituting (19) and (36) into equation (3) and making an inverse Fourier transformation we find D þ 3 i b L þ 1 b p D b mp a ^k t b ¼ ; ð38þ where D is D ¼ K a 1 þ b b k : ð39þ b [48] Furthermore, the operator ^k = ^H@/@z is a positive definite integral operator and ^H is the Hilbert transform [e.g., Bracewell, 1986]. ^HfðÞ¼ z 1 Z 1 p P 1 fðz Þ z z dz : ð4þ [49] It is convenient to pass to the dimensionless variables x and t according to x ¼ z ðb r KÞ 1 and t ¼ g t: ð41þ i where g is given by (8). [5] Then we have where D þ 3 b ¼ K 1 c ; ð4þ c ¼ a 1 þ b 1 b k ð b KÞ 1 : ð43þ [51] With the help of expression (4) equation (38) ¼ c^k x 1 c h h ; ð44þ [5] Maximizing equation (44) over c we find that the fastest growing mode corresponds to c =3 1 and thus the equation governing the nonlinear dynamics of this ¼ ^k x þ h 3 h : ð46þ [53] After rescaling h = (3/)h, t =(3 3 /) t and x = (3/) x equation (46) ¼ ^k x h h ; ð47þ where for the sake of convenience all the primes are omitted. Equation (47) describes the nonlinear dynamics of mirror waves in the case when the ion distribution function F does not vary in time. The case when it varies due to the particle trapping will be considered in the next section. Equation (47) has been solved numerically by using pseudo-spectral scheme [e.g., Boyd, ]. The spatial part of the function h in equation (47) has been decomposed into sparse grid by pseudo-spectral method and the time evolution has been given by the fourth-order Runge-Kutta code. The results of numerical simulations are shown in Figures 1 and where the dimensionless amplitude h is plotted as a function of the spatial and temporal variables x and t, respectively. At the initial moment, t =, the wave amplitude has been chosen in the form of a normal Gaussian distribution h(t= )=Aexp ( x /s ), where A =. is the wave amplitude and s =.5 stands for the standard deviation. The latter is connected with the full width at half the maximum amplitude (FWHMA) Dx through the ordinary relation Dx =.35s. In order to stress the role of nonlinear effects this dependence is depicted in two different cases: when the nonlinear term in equation (47) is disregarded (Figure 1) and when both the linear and nonlinear terms are included (see Figure ). Comparison of these two figures shows that the nonlinearity dramatically changes the evolution of mirror waves. One sees that the exponential increase in the wave amplitude that is inherent to the linear stage of the MI is then replaced by an even faster increase resulting in the formation of a magnetic trough or magnetic hole at the origin of x. Figure shows that in a finite time the wave solution blows up. Figure 3 shows the evolution of the mirror mode in case when initial perturbation has the sinusoidal (single-mode) form with the wave amplitude A =.. Similar to the previous case the nonlinear evolution possesses the blow-up behavior. [54] The physical reason for such a nontrivial evolution of the magnetic perturbations can easily be understood if one re-writes equation (47) in the ¼ ^k dl x dh ; ð48þ 6of13

7 Figure 1. A plot of h as a function of x and t calculated with the help of equation (47) for the case of the Gaussian initial distribution when the nonlinear term is disregarded. The initial amplitude is A =.. where Z L ¼ " # h þ ð r hþ þ h3 dx: 3 ð49þ [55] The functional L has the meaning of the free energy or 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. This becomes clear from Figure 4 where dl/dt is plotted as a function of time. The formation of singularity is connected with the fact that L becomes more and more negative. This effect is produced by the last term in the expression for the Lyapunov functional (49), i.e. by R (h 3 / 3)dx <. Physically the blow up behavior arises due to the fast decrease in the free energy shown in Figure 4. We note that in Figures 1 3 the dimensionless amplitude h introduced here for numerical simulations can take large negative values because according to our scaling the dimensionless wave amplitude h introduced here for numerical simulations is in fact / (db z /B )/K. Since we have limited our consideration by the marginal conditions, i.e. K 1 then even for db z /B 1 we have finite values of h. [57] It should be noted that our numerical modeling shows the appearance of solely magnetic holes. When the magnetic humps as an initial perturbation have been selected they rapidly relax and disappear within relatively short time. This agrees with the particle-in-cell simulations provided by Baumgartel et al. [3]. [58] The above consideration is valid only for the mirror mode. For the large amplitude fast MS waves the nonlinear scenario is different. These waves can propagate nearly perpendicular to the ambient magnetic field in the form of Figure. Same as in Figure 1 but when the nonlinear term is retained. One sees that nonlinear evolution of the MI is accompanied by the formation of a finite-time singularity in the magnetic field. 7of13

8 Figure 3. Same as in Figure, but for the case single-mode (sinusoidal) initial perturbation with A =.. stable MS solitons [e.g., Pokhotelov et al., 7a, 7b]. Their shapes are sensitive to the details of the ion distribution function. For example, in a bi-maxwellian plasma the dispersion of the fast MS waves is negative, i.e. the phase velocity decreases with an increase of the wavenumber. This assumes that the solitary solution in this case has the form of a bright soliton with the magnetic field increased. On the contrary, in some non-maxwellian plasmas, such as those with ring-type ion distributions or Dory-Guest-Harris plasmas they can have a positive dispersion and thus solitary solutions may have the form of a magnetic hole (dark solitons). We note that analytical and numerical studies of nonlinear structures involving MS waves have also been the subject 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 that unperturbed ion distribution function does not vary in time. This is valid as long as the effects due to particle trapping into the magnetic troughs do not play a role. The effects of particle trapping in the nonlinear mirror modes has been discussed by Pantellini et al. [1995]. These authors argued that the main mechanism that ends the linear phase of the MI has to be particle trapping into the mirror holes. It should be mentioned that in the formation of collapse the nonlinear corrections to the plasma pressure are quadratic in wave amplitude, i.e. d p NL / b. However, the nonlinear corrections to the variation of the particle pressure due to the particle trapping can be as high as jbj 3/. Thus, the change in the particle distribution function can develop faster than the wave collapse evolves. As will be shown below this effect can terminate the wave collapse and can result in a collapse break-up. It should be noted that nonlinear FLR effect that has been used by Kuznetsov et al. [7] for similar purposes is of the higher order in small parameter and thus is less important. [6] 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 the background ion distribution function would flatten and takes the shape of a quasi-plateau. This to happen does not require the assumption of random phases and is valid even in the single-mode (sinusoidal) regime: [6]. The width of such flattened region is of the order of the trapping zone in the mirror hole, i.e. D v k v k * 3 (mjdbz j/m) 1 =v (jdb z j/b ) 1 =v jbj 1 (see Figure 5). [63]. The particle trapping is irreversible, i.e. accompanied by bifurcation and violation of adiabatic invariance since the depth of the mirror hole increases more slowly for the trapped particles that possess the variation of the Figure 4. A plot of the time derivative of the Lyapunov functional dl/dt as a function of the dimensionless time t. 8of13

9 Figure 5. The ion distribution function with the plateau stretched along the parallel velocities. characteristic velocities dv k v T jbj 1 whereas for the nonresonant particles dv k dv v T jbj. [64] In order to construct a self-consistent model describing the evolution and final saturation of mirror waves we interpolate the background ion distribution function as a rigorous plateau stretched along v k from v jbj 1 to v jbj 1 (Figure 5). Outside this region the ion distribution function takes a bi-maxwellian form. Clearly, such interpolation could be valid up to the numerical coefficient of the order of unity. Its relevant value can be found only by numerical simulation of the time dependent process of particle trapping by the emerging magnetic holes prior to the MI saturation. In somewhat similar problem of saturation of a single mode electrostatic beam instability due to the trapping of the electrons into electrostatic potential wells such computational analysis established that equivalent numerical coefficient appeared to be close to unity [cf. Fried et al., 197]. [65] In our case with the assumed form of the background distribution function, the contribution of the perpendicular plasma pressure variation due to the resonant particles entering the plasma pressure balance condition reduces to dp res Z 1 v 5 dv lim n! Z1 v k dv k n þ kk ; v v k k ð5þ Figure 6. Same as in Figure but for the case when particle trapping is incorporated. 9of13

10 Figure 7. Same as in Figure 6, but for the one-mode initial regime. where v k R = v jbj 1 and F is given by n F ¼ p 3= v e T v Tk v k v v T T k : v ð51þ [66] Here n is the unperturbed plasma number density, and v T and v Tk are the thermal perpendicular and parallel velocities, respectively. [67] When v k * v Tk, i.e. jbj 1, from (5) and (51) one obtains dp res Z 1 ¼ p 1= k k v T v 3 T lim n! p dv arctan v 1! jk k jjbj n v v 5 e v T ð5þ derivation of the corresponding nonlinear equation (47). Taking into account the above considerations we conclude that the left-hand side of this equation should be replaced by a more complex expression that takes into account the effects due to the particle trapping. In the general case it has a quite complex form. For the qualitative conclusions we can use a simplified (model) form for this equation. The simplest extrapolation for the nonlinear equation that takes into account the basic features of the particle trapping effect can be phenomenologically written as j j1= 1 p arctan ¼ ^k x h h ; 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 [68] If the wave amplitude is small and the plateau is very narrow, Dv k =v k * v/jk k j, the perturbation of the plasma pressure of the resonant particle reduces to its linear (Landau) value, i.e. ¼ p dp res p 1= k k vtk T : ð53þ res [69] In this case the substitution of dp into the perpendicular plasma pressure condition recovers the standard expression for the linear MI growth rate when the undisturbed ion distribution function is bi-maxwellian [cf. Vedenov and Sagdeev, 1958]. [7] On the contrary, when the plateau is saturated, i.e. jk k j v k * n, we expand arctan (v jk k jjbj 1 /v)! p/ v/ v jk k jjbj 1 and thus dp res!, i.e. the wave growth is terminated. These features of the resonant particle dynamics should be taken into consideration in the course of the Figure 8. A plot of maximum depth of the mirror hole as the function of time for single-mode regime. 1 of 13

11 Gaussian and single-mode regimes, respectively. Figure 8 shows the evolution of the maximum depth of the magnetic hole. One sees that the collapse is arrested when h.8, then the magnetic hole depth possesses small oscillations and finally is saturated at this value. [7] We note that numerical simulation of the mirror mode nonlinear dynamics has been recently carried out by Borgogno et al. [7] in the framework of a fluid model that incorporates linear Landau damping and FLR corrections. However, the effects due to nonlinear Landau interactions considered above were not taken into account in this study. 7. Discussion and Conclusions [73] We have presented a local analysis of the MI in a high-b non-maxwellian plasma in the framework of a standard mixed magnetohydrodynamic-kinetic theory accounting for the FLR effect and nonlinearity of the mirror wave perturbations near the instability threshold. It has been shown that nonlinear effects substantially modify the MI dynamics and can lead to the formation of magnetic holes similar to those observed in the satellite data. The characteristic scales of these holes are of the order of a few ion Larmor radii. It has been shown that the resonant ions having small parallel velocities play an important role in the MI nonlinear dynamics. It has been shown that the effect of flattening of the ion distribution function in the resonant region can lead to saturation of the magnetic hole depth and results in the appearance of quasi-stationary mirror mode structures. Our paper generalizes previous studies related to the linear MI and provides some additional physical bases for better understanding of large amplitude nonlinear mirror waves in space and astrophysical plasmas. [74] In addition to the previous analysis provided by Kuznetsov et al. [7] the new kinetic nonlinear effects of the order of jbj 3/ associated with modification of the shape of the background ion distribution function in the region of small parallel particle velocities are taken into account. These effects are much stronger than those nonlinear MHD effects considered by Kuznetsov et al. [7] (/b ) and thus should play an important role in nonlinear dynamics of the mirror mode. It has been suggested that this effect can efficiently suppress or even arrest the mirror collapse (see Figures 6 8). The corresponding phenomenological equation (see equation (54)) has been formulated. The nonlinear FLR effect that has been used by Kuznetsov et al. [7] for similar purposes is of the higher order in small parameter and thus is of minor importance. Accordingly the nonlinear FLR effect has been disregarded in our study. [75] Furthermore, the analysis presented above shows that the effects due to the magnetic bending are important not only in the linear approximation but in the nonlinear regime as well. These effect was not included in the analysis of Kuznetsov et al. [7]. The importance of these additional terms (the terms including db x ) becomes clear from equation (33). Using this equation one readily obtains that the ratio of the terms containing db z and those with db x that correspond to the effect of magnetic field line bending is of the order of R ¼ db z@db z =@z db z =@x db zk k db x k : ð55þ [76] Since the magnetic field perturbations are divergence free, rdb =, one finds that db z /db x k /k k and thus R 1. The latter means that the terms neglected by Kuznetsov et al. [7] are exactly of the same order as those retained. [77] It is worth mentioning that the model presented in this paper remains oversimplified. So far our consideration is valid solely for the marginal conditions when the parameter K = A b 1 is small. An additional analysis is required when the system is far from this state. Furthermore, in the present paper we have considered the case of cold electrons. Actually this is quite common for the magnetosheath where the ion temperature is always larger than the electron temperature. The incorporation of the finite electron temperature in the MI linear theory has been the subject in some previous studies [e.g., Pantellini and Schwartz, 1995; Pokhotelov et al., ]. In these papers it has been shown that when the electron temperature becomes of the order of the ion temperature the growth rate of the ion mirror mode is reduced by the presence of the field-aligned electric field. The origin of the electric field is the electron pressure gradient set up as electrons are dragged by the nonresonant ions that have been accelerated as they pass from regions of high magnetic flux into lower flux regions. All these factors have to be taken into account in a more comprehensive nonlinear theory of the MI. [78] The major and so far unresolved question of the problem at hand regarding the appearance of the MI in space plasmas focuses on the relation between the mirror like and ion-cyclotron instabilities (ICI). Both instabilities compete for the pressure anisotropy. In the anisotropic case it is believed that the latter should grow at faster rate and should therefore appear before the MI can set on. This question was urged by Gary [199] and by Gary et al. [1995], who presented the results of numerical simulations of the fully kinetic dispersion relation describing both instabilities, the MI and ICI. The key to the understanding the problem was offered in some previous publications [e.g., Russell et al., 1999; Huddleston et al., 1999] in the course of the analysis of the Galileo magnetic field data on the edges of the cold Io wake. They showed that the multispecies content of the Io plasma may create more favorable conditions for the excitation of mirror like instabilities. The more species present in the plasma, the more ion cyclotron modes are possible in a system. However, the growth rate of each individual mode is now smaller than that of a single mode arising in an electron-proton plasma. On the other hand, the mirror like instability exhausts the free energy stored in all components of a multispecies plasma. [79] Generally the wave particle interaction at cyclotron resonance is touching particles with energies higher than thermal, unlike the MI which deals mainly with ions below thermal energies. In this respect those two modes could coexist if the background distribution is capable to support both of them. 11 of 13

12 [8] The intention of the present approach is to provide deeper insight into the physics of the MI. Hence this paper can be considered as an extension of our previous approach to the study of this instability which was limited by the consideration of a solely linear theory. The results of our study might be useful for a better understanding of the MI properties, as well as for the interpretation of recent satellite observations provided by the Cluster fleet [e.g., Narita et al., 6; Hobara et al., 7]. [81] Acknowledgments. This research was supported by PPARC through grant PP/D87/1, the Russian Fund for Basic Research grants No and , by ISTC project No 35 and by the Program of the Russian Academy of Sciences No 16 Solar activity and physical processes in the Solar-Earth system. The authors are grateful to V. V. Krasnoselskikh, E. A. Kuznetsov and M. S. Ruderman for valuable discussions. [8] Amitava Bhattacharjee thanks the reviewer for his assistance in evaluating this paper. 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