Gyrokinetic theory of magnetic structures in high-\beta plasmas of the Earth’s magnetopause and of the slow solar wind

Gyrokinetic theory of magnetic structures in high- plasmas of the Earth’s magnetopause and of the slow solar wind

Dušan Jovanović dusan.jovanovic@ipb.ac.rs Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Zemun), Serbia    Olga Alexandrova olga.alexandrova@obspm.fr Observatoire de Paris–Meudon, Laboratoire d’Etudes Spatiales et d’Instrumentation en Astrophysique (LESIA), Centre National de la Recherche Scientifique (CNRS), Meudon, France    Milan Maksimović milan.maksimovic@obspm.fr Observatoire de Paris–Meudon, Laboratoire d’Etudes Spatiales et d’Instrumentation en Astrophysique (LESIA), Centre National de la Recherche Scientifique (CNRS), Meudon, France    Milivoj Belić milivoj.belic@qatar.tamu.edu Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar
July 6, 2019
Abstract

Nonlinear effects of the trapping of resonant particles by the combined action of the electric field and the magnetic mirror force is studied using a gyrokinetic description that includes the finite Larmor radius effects. A general nonlinear solution is found that is supported by the nonlinearity arising from the resonant particles, trapped by the combined action of the parallel electric field and the magnetic mirror force. Applying these results to the space plasma conditions, we demonstrate that in the magnetosheath plasma, coherent nonlinear magnetic depression may be created associated with the nonlinear mirror mode and supported by the population of trapped ions forming a hump in the distribution function. These objects may appear either isolated or as the train of weakly correlated structures (the cnoidal wave). In the Solar wind and in the Earth’s magnetopause, characterized with anisotropic electron and ion temperatures that are of the same order of magnitude, we find coherent magnetic holes of the same form that are attributed to the two branches of the nonlinear magnetosonic mode, the electron mirror and the field swelling mode, including also the kinetic Alfvén mode, and supported by the population of trapped electrons. The localized magnetic holes may have the form of a moving oblique slab or of an ellipsoid parallel to the magnetic field and strongly elongated along it, that propagates along the magnetic field and may be convected in the perpendicular direction by a plasma flow. While the ion mirror structures are purely compressional magnetic, featuring negligible magnetic torsion and electric field, the magnetosonic and kinetic Alfvén structures possess a finite electrostatic potential, magnetic compression, and magnetic torsion, but the ratio of the perpendicular and parallel magnetic fields remains small.

pacs:
52.30.Gz, 52.35.Sb, 94.05.Fg, 94.05.Lk, 94.30.cj,

I Introduction

Coherent magnetic structures are ubiquitous in the space plasma of the solar system, where they have been observed over the full range of distances and latitudes relative to the Sun. They were detected in the solar wind Perrone et al. (2016); Winterhalter et al. (1994); Chisham et al. (2000), in the Earth’s magnetosphere, i.e. in the magnetotail Rae et al. (2007) and in the magnetopause Stasiewicz et al. (2003, 2001); Gershman et al. (2017), in the magnetospheres of the Mars, Saturn, and Jupiter Bertucci et al. (2004); Tsurutani et al. (1982); Erdős and Balogh (1996), and also in the induced magnetospheres of Venus, Io, and comets Volwerk et al. (2008); Russell et al. (1999, 1987); Glassmeier et al. (1993). The Voyager mission detected magnetic structures in the heliosheath, beyond the heliospheric termination shock Avinash and Zank (2007). Magnetic structures mostly have the form of solitary magnetic depressions (holes) or the trains of magnetic holes Stasiewicz et al. (2001). Solitary magnetic humps were detected less frequently, e.g. in the Earth and Jovian magnetosheaths Lucek et al. (1999); Joy et al. (2006), while trains of humps and the combinations of humps and holes were observed in the Earth’s magnetosheath Burlaga et al. (2006). Magnetic structures often feature a large perturbation of the intensity of the magnetic field (10–50%, sometimes Stasiewicz et al. (2001) as large as 98%), and very little bending. They are pressure balanced, i.e. exhibit the anticorrelation between the magnetic and thermal pressures. Their perpendicular scale is several, to several tens proton gyroradii, but holes of several hundreds gyroradii have also also detected Stasiewicz et al. (2001). Their pitch angle to the magnetic field is close to , yielding the aspect ratio of 7–10.

In the sheath plasmas the thermal pressure exceeds the magnetic pressure and we often have , where is the ratio of thermal and magnetic pressures. The ion temperature is usually both anisotropic and much larger than the electron temperature, . Under such conditions, several linear modes are unstable. The thermal anisotropy in a high- plasma drives both the ion mirror mode Hasegawa (1969); Southwood and Kivelson (1993), whose parallel phase speed is much smaller than the ion thermal speed, and the ion cyclotron mode Price et al. (1986); Southwood and Kivelson (1993). In the same range of phase speeds, the halo in the tail of the distribution function drives the halo instability Pokhotelov et al. (2005). Conversely, in the magnetopause the electron and ion temperatures are close to each other and the temperature anisotropy is usually not very large, but there exist a strong current that yields the magnetic reconnection, contributing also to the creation of magnetic structures Stasiewicz et al. (2001). Moreover, the coexistent inhomogeneity of the pressure and magnetic field, via the Hall instability, destabilizes the kinetic Alfvén wave Duan and Li (2005) that propagates faster than the ion acoustic speed and slower than the electron thermal speed. The gradient and the anisotropy of the electron temperature, for certain combinations of the plasma and of the anisotropy , can also excite the instabilities Basu and Coppi (1982) of the magnetosonic mode, whose parallel phase velocity lies between the parallel electron and ion thermal speeds. In the literature, the unstable fast magnetosonic mode is referred to as the field swelling mode and the unstable slow magnetosonic mode as the electron mirror instability, for details see e.g. Pokhotelov et al. (2003). Particularly important is the short wavelength limit of the ideal MHD fast magnetosonic mode, in which the spatial scale of disturbances is close to the ion Larmor radius or to the electron inertial length, and the Alfvén mode acquires an electric field parallel to the background magnetic field. Such mode is usually referred to as the kinetic Alfvén mode. It often occurs in space physics where it is responsible for the acceleration and energization of particles as well as for the exchange of energy between waves and particles. Linear kinetic Alfvén waves are unstable in the presence of inhomogeneities of the density and magnetic field Duan and Li (2005) and of the electron temperature anisotropy Chen and Wu (2010).

Because of such richness of linear instabilities, that presumably saturate into the magnetic structures Stasiewicz et al. (2001); Bertucci et al. (2004); Tsurutani et al. (1982); Erdős and Balogh (1996); Volwerk et al. (2008); Russell et al. (1999, 1987); Glassmeier et al. (1993); Rae et al. (2007); Stasiewicz et al. (2003); Gershman et al. (2017); Winterhalter et al. (1994); Chisham et al. (2000); Avinash and Zank (2007); Lucek et al. (1999); Joy et al. (2006); Burlaga et al. (2006), the nature of the latter still remains elusive. The prevailing theory relates them with the nonlinear mirror mode, but other models have also been proposed, based on the magnetic reconnection Baumgärtel et al. (2003), magnetohydrodynamic (MHD) beam microinstabilities Vasquez and Hollweg (1999), Hall MHD of charge-exchange processes Avinash and Zank (2007), and on magnetosonic solitons Stasiewicz et al. (2001, 2003); Baumgärtel (1999).

The linear mirror mode in a spatially uniform, bi-Maxwellian () plasma is weakly dispersive due to the finite ion Larmor radius effects. Under magnetosheath conditions, when the electrons are cold and massless, the mirror mode is purely growing Southwood and Kivelson (1993) due to the resonant contribution of the particles with the zero parallel velocity. However, 1-D particle simulations Qu et al. (2008); Pokhotelov et al. (2008) revealed that the saturation of the mirror instability produced humps rather than holes, if the linear drive was strong enough. In weakly unstable configurations periodic structures with moderate humps and holes were obtained, while under linearly stable conditions initially imposed holes persisted for very long times. Accordingly, mirror-mode humps are observed in the middle of the magnetosheath, while the holes are observed close to the magnetopause Génot et al. (2009), where the mirror mode is marginally stable. In most cases, the trains of humps are created rather than isolated humps, and on a very long timescale these are inverted to become holes Califano et al. (2008). The saturation mechanism, for a strong drive, comes from the trapping of the resonant ions by the mirror force, producing vortices in the phase space, which actually dominates the mirror mode dynamics in the case of a weak drive Califano et al. (2008); Qu et al. (2008).

KdV-type magnetosonic solitons exist in the case of propagation at sufficiently large angles to the magnetic field Kawahara (1969); Ohsawa (1986). Conversely, for a quasiparallel propagation, envelope solitons become possible Baumgärtel (1999) that are essentially Alfvén wave packets modulated by zero-frequency acoustic perturbations and described by the derivative nonlinear Schrödinger equation (DNSE). However, both the KdV and the DNSE equations describe the dynamics of finite (but small!) amplitude perturbations of the compressional magnetic field that are strictly 1-D (slab) structures, unstable in the transverse direction. The soliton theory has been criticized Avinash and Zank (2007), because it requires a quasiparallel propagation, in a sharp disagreement with the observed large aspect ratios. Conversely, 1-D slow magnetosonic solitons propagate at close to to the magnetic field Stasiewicz et al. (2003). On the proton scale there may also exist electrostatically charged magnetic structures, whose self-organization comes from the nonlinear effects associated with trapped electrons and the magnetic hole or a hump is created by the current of the drift of trapped electrons Treumann and Baumjohann (2012). Such magnetized electron phase-space holes have been observed in the plasma sheath Sundberg et al. (2015) and in 2-D PIC simulations Haynes et al. (2015). Moreover, phase-space structures can be driven also by the grad-B and currents of ions that are trapped in a self-consistent magnetic bottle Jovanović and Shukla (2009).

Magnetic bubbles were observed by the Polar satellite Stasiewicz et al. (2001) in the high-latitude magnetopause boundary and in the presence of strong magnetopause currents (i.e. near a possible reconnection site). They featured strong depressions (up to 98%) of the ambient magnetic field and were filled with heated solar wind plasma and immersed in a broadband turbulent spectrum of kinetic Alfvén waves. Numerical simulations Stasiewicz et al. (2001) indicated that the bubbles could be produced by the magnetic reconnection, with the accompanying kinetic Alfvén fluctuations coming from the Hall instability driven by the macroscopic gradients of pressure and magnetic field. A similar situation has been recently revisited by the NASA’s Magnetospheric Multiscale (MMS) mission Gershman et al. (2017) that enabled 3-d measurements of both the charged particles and the electromagnetic fields, with a sufficiently high resolution to resolve the ion kinetic scale (i.e. the scale of the ion Larmor radius). The MMS mission observed compressive fluctuations featuring anti-correlated perturbations of the electron density and the magnetic field magnitude, in the vicinity of a recent magnetic reconnection that produced a plasma jet flowing nearly anti-parallel to the background magnetic field with a speed , where and are the acoustic and the Alfvén speeds, respectively. These magnetic field fluctuations and bursts of electron phase space holes appeared together with the kinetic Alfvén wave in the locations of strong electron pressure gradients. The magnetic structure had the form of a kinetic Alfvén wave packet, propagating at the pitch angle to the ambient magnetic field, that exhibited spatial structure in the transverse direction, of the order of an ion gyroradius. The close examination of the electron velocity distribution function in the wave packet revealed that besides the isotropic thermal core and two suprathermal beams counterstreaming along the magnetic field, commonly observed in the magnetopause boundary layer, there existed also a population of trapped particles which accounted for of the density fluctuations and a increase in the electron temperature within the KAW. However, the latter was not indicative of heating but rather of a nonlinear capture process that may have provided the nonlinear saturation of Landau and transit-time damping. These electrons were trapped within adjacent wave peaks by the combined effects of the parallel electric field and the magnetic mirror force. Their distribution function unmistakably exhibited the loss-cone features, since it contained only the particle velocities with near magnetic pitch angles. In the magnetic hole recorded by Gershman et al. (2017), the ratio of the minimum to maximum magnetic field magnitude was , and the resulting magnetic mirror force was sufficient to trap electrons with magnetic pitch angles between and .

In the present paper, we study the effects of particle trapping in a high- plasma with anisotropic temperature, using the Chew–Goldber–Law gyrokinetic theory and including the Dippolito–Davidson treatment of higher-order corrections Frieman et al. (1966); Davidson (1967); Dippolito and Davidson (1975). We derive the nonlinear equation for the compressional magnetic field, allowing also for a finite parallel electric field (the latter is short-circuited only when the electrons are cold), including also the convection of both particle species by the grad-B drift. In the stationary regime, the appropriate expressions for the energy, magnetic moment, and canonical momentum for both species are found, and used to construct their distribution functions Luque and Schamel (2005). In appropriate limits, our equations reduce to the nonlinear ion mirror Southwood and Kivelson (1993); Kuznetsov et al. (2007a); Jovanović and Shukla (2009), kinetic Alfvén Gershman et al. (2017), electron mirror-, and the field swelling modes Pokhotelov et al. (2003), as well as to the magnetized electrostatic electron and ion holes Treumann and Baumjohann (2012). We demonstrate that in the general case, all perturbations whose characteristic perpendicular scale exceeds the ion scales (i.e. the ion plasma length, the ion Larmor radius, or the ion acoustic radius) are described by the same generic nonlinear equation (62), which possesses two distinct coherent solutions in the form of a slab that is oblique to the magnetic field and propagates perpendicularly to it, or of a finite length filament (’cigar’) parallel to the magnetic and propagating along the latter. A propagating, infinitely long, oblique filament, i.e. a cylinder with ellipsoidal cross section, is also possible but its description requires the solution of a 2-D nonlinear equation and it has been left out from our present study. Our oblique slab is, actually, the limiting case of the well known periodic cnoidal wave solution Luque and Schamel (2005) that can fully reproduce the properties of the Ref. Gershman et al. (2017) structures. Conversely, our filaments are fundamentally different from the high- MHD (quasi)monopolar vortices, governed by the fluid convective nonlinearity, which are prohibited in the kinetic Alfvén regime Jovanović et al. (2017).

Ii Gyrokinetic description of perturbations in a warm plasma, somewhat bigger than the ion-scale

In order to study the effects of the mirror force on plasma particles, we use the classical Chew–Goldber–Law gyrokinetic theory, including the Dippolito–Davidson treatment of higher-order corrections Frieman et al. (1966); Davidson (1967); Dippolito and Davidson (1975). The latter is obtained by the integration of the Vlasov equation for the particles’ gyroangle, taking that the dynamics of the particles’ guiding centers is slow on the temporal scale of the their cyclotron gyrations, that the dynamics of magnetic field lines belongs to the same slow temporal scale and that their curvature is relatively small. It includes the terms of the zeroth and of the first order in the small parameter pertinent to the drift scaling and to the small corrections coming from the finite Larmor radius and from the displacement current, viz.

(1)

where and are the characteristic frequency and characteristic perpendicular wavenumber. The Dippolito–Davidson theory was developed under the ordering and which resulted in a rather complicated gyrokinetic equation (1) of Ref. Davidson (1967). The latter is considerably simplified if we relax their ordering between the parallel and perpendicular wavenumbers as well as for parallel and perpendicular electric fields. Here, in addition to the constraints Eq. (1), we assume a weak -dependence, an electric field that is mostly perpendicular to the magnetic fiel, and small perturbations of the density and of the magnetic field, viz.

(2)

and the gyrokinetic equation of Refs. Frieman et al. (1966); Davidson (1967); Dippolito and Davidson (1975) obtains an elegant form that is accurate to the leading order in , viz.

(3)

where is the particle velocity, while and are the magnitudes of its components parallel and perpendicular to the magnetic field, respectively. The guiding-centers’ distribution function is obtained by the integration of the particle distribution function for the gyroangle , defined as , where is a unit vector in the direction of the bi-normal of the magnetic field line, . Here is the drift velocity, while , , and are the kinetic counterparts of the grad-B, polarization, and parallel drift velocities. The parallel acceleration comes from the electric field and from the mirror force, while the perpendicular acceleration is equal to the divergence of the guiding center velocity, viz.

(4)
(5)

Here is the gyrofrequency, . Velocities and accelerations given in Eqs. (3)-(5) have been calculated with the accuracy to second order in the small parameter introduced in the ordering of Eq. (2), viz.

(6)

In Ref. Jovanović and Shukla (2009), a gyrokinetic equation has been derived that permits also large perturbations of the compressional magnetic field, if the curvature of the magnetic field lines is sufficiently small so that magnetic curvature and helicity can be neglected in the (small) terms coming from the ion polarization by grad-B drift. In other words, when the unperturbed magnetic field is oriented along the -axis, viz. , the results of Ref. Jovanović and Shukla (2009) are applicable when , but . Note that the scaling of Eq. (6) does not set a strong constraint on the Larmor radius, since it gives

(7)

where appears to be of arbitrary order. However, although it is not self-evident, our gyrokinetic equation may be valid when . To demonstrate this, we deduce from Eq. (3) the corresponding hydrodynamic equations of continuity and parallel momentum, and compare them with the hydrodynamic equations that exist in the literature, whose domain of validity and accuracy with respect to the small parameters and are known. Integrating the gyrokinetic equation (3) in velocity space and with appropriate weight functions and and after some tedious but straightforward algebra, carefully keeping the leading terms in the small parameter , we arrive at

(8)
(9)

for the notations, see Eq.s (46) and (47). Equations (8) and (9) include the effects of particles’ gyromotion, through the convection by the grad- drift and the acceleration by the mirror force, which makes them more general than the standard Strauss’s equations of reduced MHD Strauss (1976, 1977) in a moderately cold plasma, , from which the mirror force is absent. More accurate fluid calculations (see e.g. Ref. Jovanović et al. (2015)) include also finite Larmor radius corrections to the convective derivative , and the diamagnetic and grad-B contributions to the plasma polarization, viz.

(10)
(11)

where . Obviously, our moment equations (8) and (9) agree with the more accurate fluid equations (10) and (11) in the regime of small Larmor radius corrections, , when both the density perturbations are sufficiently small and the nonlinear convection by grad-B drift in the polarization term can be neglected. The latter is possible not only when , that is realized in low- plasmas, but also for in 1-D slab and cylindrically symmetric geometries, in which the essentially 1-D shape of the structure suppresses all convective derivatives. In view of this, we conclude that the gyrokinetic equation (3) can be used with caution also in plasmas with large ratios of thermodynamic and magnetic pressures, , to describe kinetic phenomena whose characteristic scales are somewhat bigger than the Larmor radius, .

ii.1 Integrals of motion (characteristics of the gyrokinetic equation)

We take that the unperturbed magnetic field is oriented along the -axis, viz. , and seek a localized, stationary, 2-D solution of Eq. (3) that is travelling with the velocity , where is an arbitrary phase velocity. This implies that the solution depends only on the variables , , , , and . Then, using , the gyrokinetic equation (3) can be rewritten as

(12)

where, for a stationary solution, we set and keeping only the terms of the orders and , we also have

(13)
(14)
(15)

The characteristics of the above stationary, 3-d gyrokinetic equation are determined from

(16)

from which we can calculate explicitly only two integrals of motion, the energy and the magnetic moment , viz.

(17)
(18)

Our expression (18) for the magnetic moment coincides with that derived by Davidson Davidson (1967) within the less restrictive gyro-drift scaling of Eq. (1), and also (in the appropriate limit) with the result of Jovanović and Shukla Jovanović and Shukla (2009) that permits also large perturbations of the compressional magnetic field. In a special case of a 2-D solution that is tilted relative to the axis by the small angle (where is the -component of the phase velocity), for which we have , we find one more conserved quantity, identified as the canonical momentum , viz.

(19)

Such tilted solution depends on four variables, , , , and , and the conserved quantities (17)-(19) constitute a complete set. Thus, an arbitrary travelling-tilted 2-D distribution function can be expressed as the function of three variables , , and . As the last one contains the explicit spatial variable , a distribution function can feature a -dependence only if it is spatially dependent in the unperturbed state. It should be noted also that the above integrals of motion have been calculated with the accuracy to first order in the small parameter , where in the expressions for the energy and the canonical momentum we neglected small terms of order , while the small variation of the magnetic moment is given by In a strictly 1-D case, this gives , and we expect that is of the same order also in 2-D and 3-D.

ii.2 Free and trapped particles

The stationary state under study has been established at a distant past and thus the solution of the 2-D stationary gyrokinetic equation (3) is constant along its characteristics. From the conservation laws (17) and (18) we can relate the particle velocities at the infinity, and , with those at the phase-space location . These "initial velocities" are the functions of integrals of motion and within the adopted accuracy take the form

(20)
(21)

where

(22)

One should keep in mind that cold and massless electrons efficiently short-circuit the parallel electric field and that, as a consequence, the term may become very small. As the latter appears to be the leading term within the scalings (6) and (7), it is necessary that we retain also the next-order term in Eqs. (20) and (22) although, at first sight, it appears to be a small quantity of higher order.

We take that that the electromagnetic field is localized, i.e. that that the potentials and , and the compressional field vanish at infinity, when , where . In such a case, there exist two fundamentally different shapes of the characteristics, i.e. of the particle trajectories in phase space , determined by :
i) Open characteristics, stretching to an infinitely distant point in real space . Particles following open characteristics are labeled as free.
ii) Characteristics that close on themselves and are confined to a limited domain in phase space. Particles on such trajectories are trapped.

On open characteristics, the distribution function is equal to its asymptotic value at , which we adopt to be a Maxwellian with anisotropic temperature, viz.

(23)

Here is the unperturbed ion density, and are the perpendicular and the parallel (to the magnetic field) ion temperatures, respectively, and and are the corresponding thermal velocities, . The "initial velocities" and are given in Eqs. (21) and (20). Clearly, the initial parallel velocity of free particles must be a real quantity, which is realized when . In the simple case , yielding , we find that inside a local minimum of the magnetic field, , the velocities of free particles belong to the loss cone in velocity space .

Conversely, for , i.e. for particles whose parallel velocities are in the region , the corresponding "initial velocity" is a complex quantity. Such result is unphysical and it implies that these particles have never been at, and will never come to, an asymptotic location . In other words, these particles are trapped on their characteristics which are closed curves in phase-space. As the particle trajectories do not cross, such closed characteristics occupy a region in phase-space that is inaccessible for free particles. This further implies that, in a distant past, the trapped particles have gone through some nonadiabatic process, during which time the potentials and the compressional magnetic field have been time-dependent, the term in the gyrokinetic equation (12) has been finite and their energy and magnetic moment have not been conserved. Over time, trapped particles perform a large number of bounces and we expect that the phase-averaging of the individual trajectories of trapped particles results in a shifted thermal distribution, with a parallel temperature , viz.

(24)

where such normalization has been adopted that the distribution functions and are continuous at the branch point , determined by Eq. (21). It should be noted that the trapped particles are isolated from those that are free and that the parallel temperature of trapped particles may be different than that of free particles and it can be even negative. This does not contradict the second law of thermodynamics, since trapped particles occupy only a limited phase-space volume within which their distribution function remains finite, irrespectively of the sign of the temperature. As a consequence, the relevant integrals of distribution function also remain finite.

Now we can calculate the necessary hydrodynamic quantities as the moments of the particle distribution function, performing the integration in velocity space with appropriate weight functions. It is instructive to separate nonresonant and resonant contributions in a specific moment , denoted by the superscripts "" and "", as follows

(25)
(26)

where denotes the principal value of an integral. In the above, the resonant distribution function is defined as and the weight function takes the values , , , and , in the expressions for the number density , the parallel hydrodynamic flow , the parallel pressure , and the perpendicular pressure , respectively. In the computation of nonresonant contributions, Eq. (25), we conveniently expand the free distribution function using the small quantity , which permits us to rewrite in the form

(27)

This enables a straightforward integration in Eq. (25), yielding

(28)
(29)
(30)
(31)

The parameter is the real part of the Fried-Conte plasma dispersion function (also called the -function):

(32)

and it has simple asymptotic values for and for . Although the finite Larmor radius terms in Eqs. (28)-(31) are small within our scaling, viz. , we may need them later since they provide the dispersion of MHD-like modes, such as the field swelling, electron-, and ion-mirror modes. As already mentioned, FLR terms are not calculated accurately from the gyrokinetic Frieman et al. (1966); Davidson (1967); Dippolito and Davidson (1975) equation (3). Making a comparison with the solutions of fluid equations (10) and (11) [see also Eq.s (16) and (17) in Ref. Jovanović et al. (2015)], we note that an appropriate description of the grad-, polarization, and weak FLR effects is obtained when in Eqs. (28)-(31) we implement the substitution , where the leading-order expression for is used, . We note also that the densities and parallel fluid velocities of nonresonant particles, Eqs. (28)-(31), have the same form as the linear solutions of the fluid equations (8)-(9). This implies that the adopted simple form (23) of the distribution function exist only when the sought-for coherent nonlinear structure possesses a geometry for which the nonlinearities due to convective derivatives vanish (e.g. 1-D slab or cylindrically symmetric geometries).

In Eq. (26), the integration is performed over the domain of trapped particles, viz. , inside which it is convenient to rewrite in the following way

(33)

which is further simplified setting

(34)

Using the above and comparing Eq.s (33) and (24), we note that in the domain of resonant parallel velocities, the trapped particles’ distribution and the even part of have identical forms, but with different parallel temperatures. Now we can easily write down the effective distribution in the resonant domain, viz.

(35)

where small terms of order have been neglected and we have conveniently separated even and odd parts.

Finally, performing the integrations in velocity space, we obtain the moments Eq. (26) in a closed form, as

(36)
(37)
(38)
(39)

where we used the notations

(40)
(41)

and erfc is the complementary error function . The function in Eqs. (36)-(39) behaves asymptotically as , with a full agreement for both and . We note that in Eq. (41) we can safely set , since in the regime the term represents a small FLR correction and it is negligible in the above setting. Conversely, for , the contribution of trapped particles can be neglected altogether, since from Eq. (40) it scales as .

ii.3 Field equations

Now we can easily write down the Poisson’s equation, and the components of the Ampere’s law that are parallel and perpendicular to the magnetic field, viz.

(42)