Whistler envelope solitons. I. Dynamics in inhomogeneous plasmas

C. Krafft and A. Volokitin

A self-consistent Hamiltonian model based on equations describing the coupled dynamics of whistler and lower frequency waves in inhomogeneous plasmas is built. On this basis, different aspects of whistler turbulence are studied, concerning mainly the development of modulational instabilities and the dynamics of envelope solitons in irregular plasmas. Numerical simulations based on the model show that modulational instability can lead to the generation of a beating of stable nonlinear whistlers propagating with a speed near the group velocity. The whistler envelope soliton is determined analytically and its propagation in plasmas presenting random density fluctuations and weakly irregular density structures of different scales and amplitudes is studied, showing that the envelope is very weakly affected by these inhomogeneities, whereas the wavelengths and the amplitudes of the phase oscillations strongly vary. Moreover, simulations show for the first time that two whistler solitons moving with different but close velocities and colliding one with the other remain unchanged after this collision, independently of their initial amplitudes and velocities. Finally we study the dynamics of sonic whistler envelope solitons and show that the propagation of their lower frequency perturbation is governed by a KdV-type equation.

I. INTRODUCTION

The emission of whistlers excited by electron beams in the presence of very low frequency acoustic-type waves and plasma inhomogeneities has been observed in several space experiments. For example, low frequency electromagnetic waves identified as whistlers were detected in the inhomogeneous solar wind plasma in association with Langmuir and ion acoustic waves 1,2 . The Wind spacecraft observed parallel propagating whistlers and Langmuir waves within a coronal mass ejection 3 , which were very good time-correlated with a loss-cone electron distribution within a magnetic cloud. Whistler waves propagating along magnetic loops during solar flares and generated by loss-cone distributions of relativistic electrons have been shown to be responsible for the generation of fiber bursts in the solar radio radiation [START_REF] Vocks | Whistler wave excitation by relativistic electrons in coronal loops during solar flares[END_REF] . Normal cyclotron mechanisms driven by temperature anisotropy have been identified [START_REF] Thorne | The generation mechanism for magnetosheath lion roars[END_REF] to be the cause of lion roars, which are intense and narrowband whistlers of low frequency, that are one of the characteristic features of the magnetosheath. Moreover, this mechanism is responsible for whistler emission upstream of the Earth's bowshock and at interplanetary shocks [START_REF] Wong | Electron beam excitation of upstream waves in the whistler mode frequency range[END_REF] .

Bursts of narrowband and short-living whistlers have been frequently observed by spacecraft [START_REF] Zhang | Bursts of whistler mode waves in the upstream of the bow shock: Geotail observations[END_REF] ; most of these waves propagate in the electron foreshock quasi-parallel to the ambient magnetic field, and are likely produced by electron fluxes through cyclotron resonances and temperature anisotropy; they are more rarely observed in the solar wind, where they are believed to propagate along the field lines and originate from distant sources (such as interplanetary shocks [START_REF] Coroniti | Whistler mode turbulence in the disturbed solar wind[END_REF] ) or halo electrons presenting temperature anisotropy and carrying heat flux [START_REF] Gary | Solar wind heat flux regulation by the whistler instability[END_REF] . The spacecraft Cluster observed in the solar wind narrow band coherent fluctuations propagating quasi-parallel to the magnetic field, interpreted as whistler waves [START_REF] Lacombe | Whistler mode waves and the electron heat flux in the solar wind: Cluster observations[END_REF] . The specific problem of interaction between narrow whistler waves packets and warm electron beams (i.e. with finite perpendicular and parallel temperatures) is particularly interesting. Indeed, as whistlers excited by such beams present in many cases positive linear growth rates only in narrow spectral bands 11 (see also below), one can suppose that, even in the quasilinear and nonlinear stages of the evolution, the waves with resonant velocities in the corresponding narrow velocity domain should play the most significant role. In this case a broadband packet is not necessary for producing an efficient interaction with electron fluxes and a subsequent significant radiation. Such whistler waves, for example, can be generated due to some natural physical processes, as evidenced by the space observations mentioned just above, or be emitted in the magnetosphere by ground transmitters 12 .

The coupling of large amplitude whistlers as those observed in space plasmas 13 with lower frequency waves as ion or electron acoustic waves can lead to various phenomena developing in the nonlinear stage of the evolution, in presence or absence of resonant particles, as ponderomotive effects, modulational instabilities, formation of stable structures as solitons or localized wave packets, etc. The satellite Ogo 5 observed isolated discrete whistler packets in the solar wind 14 . Observations by spacecraft revealed the existence of localized electric fields and density structures in the high and low frequency ranges. Indeed, the satellite Freja observed right-polarized wave packets accompanied by density cavities with depths of the order of 1% of the background plasma, identified by the authors as envelope whistler solitary waves 15 . The space mission Cluster detected whistler emissions near the plasmapause, correlated with density fluctuations aligned along the ambient magnetic field 16 , which are modulated structures associated with the density perturbations. Moreover, the satellite Fast observed envelope solitary electron-acoustic waves propagating along the magnetic field lines in auroral density cavities, in regions where two electrons populations, hot and cold, exist 17 . Finally, some authors report the observations of envelope soliton fine structures observed in solar radio metric-wave emissions occurring during solar radio bursts 18 .

In order to explain such observations, we present hereafter a self-consistent Hamiltonian model based on equations describing the coupled dynamics of electromagnetic whistlers and lower frequency waves 19,20 in randomly inhomogeneous plasmas. This model is used, owing to an associated numerical code, to study the characteristic features of nonlinear structures as solitons and physical effects as modulational instabilities, which can rise in space plasmas, as mentioned above. The present paper extends and generalizes the existing studies [21][22][23] , by enlarging the domain of parameters' validity (e.g. by considering the full range of whistler frequencies, and not only the limiting case of very low frequencies) and by providing detailed numerical simulations' results on different aspects of whistler turbulence in inhomogeneous plasmas. In particular and to our knowledge, the dynamics of the collision between whistler envelope solitons is presented for the first time. Concerning modulational instabilities, we also show how they can generate beatings between stable nonlinear structures. Moreover the whistler soliton propagation is studied in plasmas presenting external preexisting irregularities in the form of random fluctuations, which are particularly important as they can strongly influence on the nonlinear effects at work, as it was evidenced recently for the case of Langmuir turbulence in the solar wind [24][25][26][27] . Note that the model provides useful invariants which allow to control the numerical simulations; moreover it can be completed by adding populations of resonant particles interacting with the solitons, as discussed in the companion paper 32 .

II. INTERACTION OF WHISTLERS WITH LOWER FREQUENCY WAVES : THEORETICAL MODEL

In a magnetized plasma, whistlers can couple nonlinearly to lower frequency waves, due to the ponderomotive force and the interactions between their fields and the plasma fluid density's, velocity's and magnetic field's perturbations. In this frame, specific physical effects can rise and influence strongly on the dynamics of the whistlers.

Let us consider the case of whistler wave packets propagating parallel to the ambient magnetic field B 0 ; their magnetic field is assumed to be of the form

B = Re B (z, t) e -iω 0 t+ik 0 z (x+iy) , (1)
where B = B x -iB y is the slowly varying (with space and time) envelope of the right circularly polarized wave; x and y are the unitary vectors along the axes perpendicular to the direction z of the wave propagation and the ambient magnetic field B 0 ; B x and B y are the coordinates of B along x and y; ω 0 and k 0 are the central frequency and wavenumber of the whistler packet. The dispersion relation is given by

D (ω 0 , k 0 ) = k 2 0 c 2 -ω 2 0 (ε ⊥ -q) = k 2 0 c 2 -ω 2 0 + ω 2 p ω 0 ω 0 -ω c = 0 (2)
with the elements of the cold dielectric tensor

ε ⊥ = 1 - ω 2 p ω 2 0 -ω 2 c , q = ω c ω 2 p ω 0 (ω 2 0 -ω 2 c ) , (3)
where ω p and ω c are the electron plasma and cyclotron frequencies, respectively. Note that, contrary to previous studies 22 which considered only waves with ω 0 ω c , our model takes into account the full range of whistler frequencies, i.e. ω lh < ω 0 < ω c , where ω lh is the lower hybrid frequency. Combining the Maxwell equations we obtain that

1 c 2 ∂ 2 B ∂t 2 -∇ 2 B = 4π c ∇ × j, ( 4
)
where j is the current density involving the external currents j ext (which can be carried by an electron beam injected in the plasma, for example). Using the Ohm's law for the linear current density, and taking into account the slow space and time variations of the magnetic field envelope B, we obtain that D(ω, k)B 0 when j ext = 0 (for more details see the Appendix A). Performing a Taylor development of D(ω, k) around (ω 0 , k 0 ) up to order two (parabolic approximation) and keeping only the main terms, we can write

D(ω, k) i ∂D ∂ω 0 ∂ ∂t -i ∂D ∂k 0 ∂ ∂z - 1 2 ∂ 2 D ∂ω 2 0 ∂ 2 ∂t 2 -2 ∂ 2 D ∂k∂ω 0 ∂ 2 ∂z∂t + ∂ 2 D ∂k 2 0 ∂ 2 ∂z 2 +ρ ∂D ∂ρ 0 + δv e ∂D ∂δv e 0 + δB z B 0 ∂D ∂ (δB z /B 0 ) 0 , (5)
where ρ = δn e /n 0 and δv e are the perturbations of the slowly varying electron fluid density and velocity; δB z is the slowly varying magnetic field perturbation along z; n 0 is the unperturbed plasma density at equilibrium. Then the equation of evolution of the wave magnetic field envelope is obtained in the following form (see also the Appendix A)

i ∂B ∂t + iv g0 ∂B ∂z + v g0 2 ∂ 2 B ∂z 2 = - ω 0 ω 2 p D 0 (ω 0 -ω c ) ρ + ω c ω 0 -ω c k 0 ω 0 δv e + δB z B 0 B, (6)
with

D 0 = ∂D ∂ω 0 = -2ω 0 + ω 2 p ω c (ω c -ω 0 ) 2 . ( 7
)
The right hand side of (6) consists of three nonlinear terms representing the coupling between the whistler field and the lower frequency perturbations of density, velocity and ambient magnetic field. Note that D 0 = -ω 2 0 ε 0 , where ε 0 is the derivative of the dielectric constant at (ω 0 , k 0 ). The group velocity and its derivative are given by v g0 = (dω/dk) 0 = -(∂D/∂k) 0 / (∂D/∂ω) 0 = -2c 2 k 0 /D 0 and

v g0 = d 2 ω dk 2 0 = v g0 k 0 1 - v 2 g0 c 2 1 + ω 2 p ω c (ω c -ω 0 ) 3 . ( 8
)
Below the term with δB z in ( 6) will be omitted as the magnetic field B 0 is considered here to be constant. where δϕ is the low frequency potential; T e , m e and -e < 0 are the electron plasma temperature, mass and charge, respectively. The ponderomotive force F pz of whistlers acting on the particles is (see also the calculations of the non stationary part of the force in the Appendix B)

F pz = - ω 2 p ω 0 (ω 0 -ω c ) ∂ ∂z - ω c (ω 0 -ω c ) k 0 ω 0 ∂ ∂t |E| 2 16πn 0 , (9)
where E = (ω 0 /ck 0 ) B; the expression ( 9) is similar to that obtained in another way 28,29 . The ion dynamics is described by

∂ ∂t δn i + ∂ ∂z (n i δv i ) = 0, (10)
∂ ∂t δv i + δv i ∂ ∂z δv i = - e m i ∂ ∂z δϕ - T i n i m i ∂ ∂z δn i , (11)
where δn i (n i = n 0 + δn i ) and δv i are the proton density and velocity perturbations; T i and m i are the ion plasma temperature and mass. In the case of quasineutral low frequency oscillations (ρ = δn e /n 0 δn i /n 0 and consequently, due to charge conservation, V = δv e δv i ), one can get the low frequency equations in the form

∂ ∂t V - ω 2 p (ω 0 -ω c ) 2 ω c k 0 |B| 2 16πn 0 m i c 2 + ∂ ∂z c 2 s ρ + ω 2 p ω 0 k 2 0 (ω 0 -ω c ) |B| 2 16πn 0 m i c 2 0, ( 12
)
∂ρ ∂t + ∂V ∂z 0, ( 13
)
where the hydrodynamic nonlinear terms have been neglected and the ponderomotive terms have been expressed as a function of the magnetic field enevelope B; c s is the ion sound velocity. Defining |B| 2 = |B| 2 /16πn 0 m i c 2 and introducing the hydrodynamic flux Ψ in the form

∂Ψ ∂z = V - ω c ω 2 p k 0 (ω c -ω 0 ) 2 |B| 2 , (14)
the equations ( 12)-( 13) can be written as

∂Ψ ∂t + c 2 s ρ + ω 2 p ω 0 k 2 0 (ω 0 -ω c ) |B| 2 = 0, (15)
and

∂ 2 Ψ ∂z 2 + ∂ρ ∂t + ω 2 p ω c k 0 (ω 0 -ω c ) 2 ∂ |B| 2 ∂z = 0. ( 16
)
Adding the equation for the whistler envelope in the following form (δB z = 0)

i ∂B ∂t + iv g0 ∂B ∂z + v g0 2 ∂ 2 B ∂z 2 = - ω 2 p (ω 0 -ω c ) D 0 ω 0 ρB + k 0 ω c (ω 0 -ω c ) ∂Ψ ∂z B + ω 2 c ω 2 p |B| 2 B (ω 0 -ω c ) 3 , (17)
we get a system of three differential equations whose Hamiltonian can be expressed as (see also the Appendix A)

H = dz L H = dz L × iv g0 2 B ∂B * ∂z -B * ∂B ∂z + v g0 2 ∂B ∂z 2 + ω 2 p ω 0 ρ |B| 2 D 0 (ω c -ω 0 ) - k 2 0 2D 0 V 2 + c 2 s ρ 2 , ( 18
)
where L is the size of the system and V has been expressed using (14). The three first terms of (18) represent the energy of the whistler waves, the fourth one the energy of interaction between the whistlers and the lower frequency waves, and the two last the energy of the lower frequency waves. Another invariant, i.e. the momentum P of the system, can be deduced from ( 12)-( 13)-( 17)

P = i B ∂B * ∂z -B * ∂B ∂z dz L + 2k 2 0 D 0 ρV dz L - 2ω 2 p D 0 k 0 ω c (ω c -ω 0 ) 2 ρ |B| 2 dz L . ( 19
)
The last invariant is the number of quanta

H 0 = ω 0 |B| 2 dz L , (20)
which can also be added to (18). The two couples of canonical variables are (B,

B * ) = (B, B * )/ √ 16πn 0 m i c 2 and (Ψ, Π) = (Ψ, -k 2 0 ρ/D 0 )
. One can check that the corresponding Hamilton equations are given by

∂Π ∂t = δH δΨ , ∂Ψ ∂t = - δH δΠ , ∂B ∂t = -i δH δB * , ( 21
)
where δ is the functional derivative. This Hamiltonian system can used to study the interactions of whistlers with ion sound waves, for example. Note that, due to ( 5), the model is valid for the case of narrowband coherent whistlers propagating quasi-parallel to the magnetic field lines. Such whistlers can be generated naturally in space or artificially in laboratory experiments, using pump waves or particles' fluxes, as discussed in the companion paper of this article 32 . Our purpose is now to solve the system formed by ( 18) and ( 21) (or ( 6), ( 12) and ( 13)) in order to understand the dynamics of a whistler packet interacting with ion sound waves. A numerical code has been built which is based on the normalized model's equations, which are solved using pseudo-spectral methods, discretization schemes and Fast Fourier Transforms (see the Appendix C for more details). Packets of 1024 -4096 waves of wavenumbers k 0 ± k (0 < k k 0 ) and frequencies ω k 0 ±k are used, which present initially narrowband spectra peaked at the central wavenumber k 0 at frequency ω 0 . The one-dimensional simulation box extends typically over a normalized distance Lω c /c 10 3 -10 5 . External plasma inhomogeneities can be introduced as initial perturbations. The accuracy of the calculations is controlled owing to the model's invariants, i.e. the number of quanta (20), the system's energy (18) and momentum (19).

III. MODULATIONAL INSTABILITY

As shown in previous studies 21 , whistler waves propagating along magnetic field lines can be modulationally unstable with respect to perturbations of the lower frequency range. On the basis of the above presented Hamiltonian model we will first determine the growth rate of such instabilities and then perform numerical simulations.

Let us express the magnetic field envelope as

B = B k 0 + B k 0 +κ e iκz-iΩt + B k 0 -κ e -iκz+iΩt
, where the amplitude of the Fourier component B k 0 corresponding to the central wavenumber k 0 is large compared to the amplitudes of the other components B k 0 ∓κ . So the density and velocity perturbations satisfy ρ = ρ κ e iκz-iΩt +ρ -κ e -iκz+iΩt and V = V κ e iκz-iΩt +V -κ e -iκz+iΩt . From ( 13) one can deduce that Ωρ κ = κV κ . Moreover, in the Fourier space the equations ( 6) and ( 12) lead respectively to the relations

±Ω ∓ κv g0 - κ 2 v g0 2 B k 0 ±κ = - ω 0 ω 2 p D 0 (ω 0 -ω c ) κ Ω + ω c ω 0 -ω c k 0 ω 0 V ±κ B k 0 , (22)
and

Ω V κ - ω 2 p (ω 0 -ω c ) 2 ω c k 0 B * k 0 B k 0 +κ + B k 0 B * k 0 -κ 16πn 0 m i c 2 = κ c 2 s ρ κ + ω 2 p ω 0 k 2 0 (ω 0 -ω c ) B * k 0 B k 0 +κ + B k 0 B * k 0 -κ 16πn 0 m i c 2 ,
(23) where the asterisk indicates the complex conjugate. Combining the equations ( 22) and ( 23) we get

Ω 2 -κ 2 c 2 s Ω -κv g0 - κ 2 v g0 2 Ω -κv g0 + κ 2 v g0 2 = - ω 2 0 ω 4 p v g0 κ 2 D 0 (ω 0 -ω c ) 2 κ k 0 + ω c (ω 0 -ω c ) Ω ω 0 2 |B k 0 | 2 16πn 0 m i c 2 (24)
or also

Ω 2 -κ 2 c 2 s = Γ 3 κ 2 v g0 κ k 0 + ωc (ω 0 -ωc) Ω ω 0 2 (Ω -κv g0 ) 2 -κ 2 v g0 /2 2 , (25)
where

Γ 3 = ω 2 0 ω 2 p ck 0 ω 2 c (ω 0 -ω c ) 2 v g0 8c ω 2 p ω 2 c v A c 2 |B k 0 | 2 B 2 0 > 0, (26)
v A being the Alfven velocity; note that v g0 has the same sign as k 0 . Then, searching solutions of ( 25) for small κ that satisfy Ω κv g0 + iγ with κ 2 v g0 /2 |κv g0 | , we get

γ 2 Γ 3 v g0 c 2 s -v 2 g0 κ k 0 2 1 - ω c (ω c -ω 0 ) k 0 v g0 ω 0 2 , ( 27
)
which shows that modulational instability can exist for supersonic (v g0 > c s ) or subsonic (v g0 < c s ) whistlers only if v g0 < 0 or v g0 > 0, respectively. Note that γ is proportional to

|B k 0 /B 0 | through Γ 3 .
For the strongly supersonic case when Ω 2 c 2 s κ 2 (i.e. v g0 c s ) we get the growth rate

γ = Γ 3/2 v g0 1/2 v g0 κ k 0 1 - ω c (ω c -ω 0 ) k 0 v g0 ω 0 . (28)
If v g0 c s (sonic case), the growth rate given by ( 27) presents a singularity. However the relations (??) allow to write in this case that κ (c s -v g0 ) iγ so that we get from (27) that

γ i 1/3 κΓ v g0 2k 2 0 v g0 1/3 1 - ω c (ω c -ω 0 ) k 0 v g0 ω 0 2/3 , (29)
with i 1/3 = 0.866 -0.5i. Let us solve numerically the equation ( 25) and show in Fig. 1 the variations of the growth rates Im(Ω/ω 0 ) as a function of the normalized wavenumbers κ/k 0 for three examples of space plasma parameters, i.e. for a very low frequency subsonic whistler in the solar wind (Fig. 1, upper panel), a supersonic whistler and a sonic whistler of high frequency (ω 0 ω c ) in the Earth magnetosphere (Fig. 1, lower panels). As expected by the nature of the modulational instability, the calculated growth rates are rather weak, even if at large B/B 0 ∼ 0.3 they can reach a significative value around 10 -3 (see Fig. 1, upper panel) and can thus a priori compete with other nonlinear phenomena, as the three-wave resonant interaction process where a whistler decays into a backscattered whistler and an ion sound wave, for example. However, according to Ref. 28 where the authors considered the different classes of parametric instabilities for whistlers propagating along magnetic field lines in a dense plasma, these instabilities develop in different ranges of k. Moreover, in the case of narrowband whistlers, different effects should not compete one with another in narrow k ranges, for the same physical parameters. In Fig. 2 we present results of numerical simulations performed for the parameters of Fig. 1 (middle panel). It shows that not only large amplitude whistlers but also smaller ones (B/B 0 ∼ 0.01) may be subject to modulational instabilities; however, in the latter cases, the actual observation of such instabilities in space plasmas could be difficult if not impossible, due to the small value of Im(Ω/ω 0 ). In the left column of Fig. 2 one can see the profiles of the magnetic field envelope B/B 0 (together with its real and imaginary parts) for four moments of time, i.e. ω c t = 1.7 10 8 , 1.9 10 8 , 2.5 10 8 , and 3 10 8 . A modulation instability develops near ω c t 10 8 , as expected from the growth rate values of Fig. 1 (middle panel). Correspondingly, the right column of Fig. 2 shows for the same time moments the profiles of the lower frequency perturbations, i.e. the normalized plasma fluid velocity V /c s . The modulation instability manifests itself by the occurrence of nonlinear wave structures forming a set of several soliton-like peaks, which are shown in the frame moving with a velocity close to the group velocity v g0 where they are quasi-immobile. The distance between them (Λ 3500c/ω c ) corresponds to the wavelength of the most unstable perturbation in Fig. 1 (middle panel), i.e. to the wavenumber value κ m /k 0 = 0.00045 where the growth rate is maximum; indeed one easily verifies that 2π/κ m 3700c/ω c . This distance Λ is not very large in comparison with the normalized width of a soliton-like structure, which is of the order of 1000c/ω c . As a consequence, it is more reasonable to interpret the nonlinear structures as a periodic nonlinear wave, and not as a soliton or a train of solitons. Moreover the simulation demonstrates a more complicated picture: the growth of the modulational instability leads to the generation of a superposition (or beating) of nonlinear waves which are responsible for the large quasiperiodic oscillations of the ion sound energy density W IS during its saturation stage, which starts near ω c t 1.5 10 8 , as shown in Fig. 3. Note that Parameters are those of Fig. 2.

during these oscillations the whistler wave modulation tends to disappear (see e.g. the third left panel of Fig. 2 representing B/B 0 at ω c t = 2.5 10 8 ) as it is roughly 10 times smaller than its maximum value at ω c t 3 10 8 . An attempt to describe theoretically such interactions between nonlinear waves was presented in Ref. 29.

IV. WHISTLER ENVELOPE SOLITON

This Section is devoted to determine the analytical form of a whistler envelope soliton and then to study, owing to numerical simulations, its characteristics of propagation, i.e. mainly its dynamics in a inhomogeneous plasma and its collision with another soliton. Therefore let us find a stationary solution of the equations ( 6), ( 12) and ( 13). We suppose that the whistler envelope soliton is moving with the velocity U and define the variable ξ = z -U t, so that for a function f (z, t) we can write that ∂f /∂t = -U ∂f /∂z. Then ( 12) and ( 13) provide that ρ = V /U and

ρ(B) = ω 0 (ω 0 -ω c ) ω 2 p k 2 0 (U 2 -c 2 s ) 1 + ω c (ω 0 -ω c ) k 0 U ω 0 |B| 2 16πn 0 m i c 2 . ( 30
)
Moreover, searching stationary solutions of ( 6) in the form B = Φ (ξ) e -iλt+iµz , we get that

Φ + 2 v g0 λ -µv g0 -µ 2 v g0 2 Φ + bΦ 3 = 0, (31)
where

Φ = ∂ 2 Φ/∂ 2 ξ and b = - 2ω 4 p v g0 D 0 (ω 0 -ω c ) 2 ω 2 0 k 2 0 (c 2 s -U 2 ) 1 + ω c ω 0 -ω c k 0 U ω 0 2 1 16πn 0 m i c 2 . ( 32
)
One can find a family of solutions of (31) in the form Φ (ξ) = B s sec (ξ/l s ) with

µ = U -v g0 v g0 , λ = - v g0 2l 2 s + µv g0 + µ 2 v g0 2 , (33)
l 2 s = 16πn 0 m i c 2 B 2 s (ω 0 -ω c ) 2 ω 4 p 2k 0 v g0 v g0 c 2 k 2 0 ω 2 0 c 2 s -U 2 1 + ω c ω 0 -ω c k 0 U ω 0 -2 . ( 34
)
Then the lower frequency perturbations are given by ρ = ρ 0 sec 2 (ξ/l s ) and

V = V 0 sec 2 (ξ/l s ) = U ρ 0 sec 2 (ξ/l s )
, where ρ 0 = ρ(B s ) (30). Equation ( 34) shows that the whistler soliton exists only if v g0 (c 2 s -U 2 ) > 0. So the condition v g0 < 0 (resp. v g0 > 0) has to be fulfilled for a supersonic whistler soliton with U > c s (resp. a subsonic soliton with U < c s ). One can see that the width l s of the soliton is inversely proportional to its amplitude B s . For a supersonic soliton, ρ and V propagate in the form of density and velocity humps if

ω 0 ck 0 (ω c -ω 0 ) ω c < U c , (35)
and in the form of cavities for the opposite case. When the soliton velocity is equal to the group velocity, i.e. U = v g0 , one can show using ( 2) and ( 7) that the term 1 + ω c k 0 U/ω 0 / (ω 0 -ω c ) is always negative, so that the supersonic solitary whistler wave presents in this case always density and velocity humps and the subsonic one always density and velocity cavities. Note that the difference between U and the group velocity v g0 is formal, as it becomes clear if one notes that a soliton moving with velocity U = v g0 gets a correction µ to its wavenumber (33); thus the soliton always moves with the group velocity corresponding to its full wavenumber µ + k 0 . But it is suitable to distinguish U and v g0 when we consider the collision of two solitons (see below).

The questions arise whether a inhomogeneous plasma can support the propagation of stable whistler solitons and whether they can be accelerated or decelerated in such a medium, as a linear whistler wave does, and if they can keep their identities when passing through density gradients or randomly fluctuating plasma irregularities. So let us study the propagation of a whistler soliton in a depleted plasma with a density well of depth reaching 2% of the ambient plasma density and a scale significantly larger than the soliton width. Numerical simulations show that the solitary whistler propagating in this structure keeps its stability. Figure 4 presents the profiles of the whistler envelope soliton B/B 0 at three different times and positions when propagating through a small density depletion. The envelope is not deformed during its travel but its phase exhibits oscillations with varying wavelengths; those decrease more and more as the soliton approaches the bottom of the density well. The soliton moves with a quasi-constant velocity, it decelerates (resp. accelerates) when it crosses the negative (resp. positive) density gradient, in agreement with the behavior expected for a linear wave. When it reaches the simulation box at ω c t ∼ 10 5 , its velocity has eventually increased by less than 1%. These variations are tiny due to the small corresponding variations of the whistler frequency.

The dynamics of an envelope soliton crossing a density hump of comparable width is shown in Fig. 5 at four time moments, together with the corresponding evolutions of the density and the velocity external perturbations (the soliton's lower frequency perturbations are too small to be visible). When the soliton passes through the hump (see the second column), the external density hump splits in two parts of identical size but of half amplitude that are propagating further in opposite directions; moreover the corresponding velocity perturbation is amplified and appears as a hump accompanied by a depletion of similar scale and amplitude. Meanwhile the soliton envelope keeps its stability, even if its phase strongly varies.

Moreover, the propagation of a whistler envelope soliton in a plasma with random density fluctuations of comparable amplitudes and wavelengths has been studied, showing that the soliton is very weakly affected by these irregularities. Fig. 6 shows the soliton profile at five different time moments together with the density fluctuations' profile δn/n 0 . Note that such kind of plasma with random density inhomogeneities is typical of the solar wind. As discussed above, only the phase of the soliton is strongly varying when it crosses successive irregularities with negative or positive gradients, its envelope remaining very stable. The soliton is not accelerated nor decelerated during its travel through the whole box, i.e. during a time exceeding 1.5 10 6 ω -1 c . Finally, simulations show for the first time that two whistler solitons moving with different but close velocities and colliding one with the other are not destroyed during this process but propagate away with no changes after the collision is fulfilled, independently of their amplitudes and velocities. Figure 7 presents the magnetic field amplitudes B/B 0 of two envelope solitons moving along the simulation box with different velocities (both very close to the group velocity), at six different times during the collision. When the two solitons begin to overlap, their envelopes as well as the oscillations of their phases are strongly modified, interfering one with the other in a complex way. Note however that the wavelength of the real and imaginary parts of the fields are not changed; the phase oscillations with the smallest wavelength (belonging to the slowest soliton) modulate the phase oscillations with the longest wavelength (the fastest envelope). At the time ω c t 1.08 10 7 when the solitons coalesce, they exhibit a single structure with several peaks of wavelengths close to those of the slowest soliton. After the collision, when the fastest soliton (i.e. that with the smallest amplitude) has overtaken the slowest one (with the highest amplitude), both recover their initial shapes and continue their motion without noticeable variations of their amplitudes, forms and phases. The density and the velocity perturbations (not shown here) propagate in the form of sec 2 (ξ/l s ) functions as determined analytically, following the whistler soliton with the same speeds. However, similarly to the case of Langmuir solitons' collisions, emission of linear ion sound waves is detected after the collision is finished.

In a second example (see Fig. 8), the fastest soliton in its turn has the largest amplitude, but the conclusions are the same as mentioned previously. Moreover we present in Fig. 9 the time variations of the square maximum amplitude (B max /B 0 ) 2 of the overlapped structure formed by the two solitons and of its position ξ = z max -v g0 t in the frame moving with the group velocity, where z = z max for B = B max . This shows that the structure moves with a constant velocity and that its maximum amplitude oscillates weakly with time.

V. SONIC WHISTLER SOLITON

When the whistler group velocity v g0 is very close to the ion sound velocity c s , the width l s of the soliton (34) becomes infinitely small and its amplitude E s infinitely large. In order to suppress this singularity, we have to take into account small additional corrections in the low frequency equations (hydrodynamic nonlinearities and weak non quasineutrality). Let us consider that v g0 c s and combine the low frequency equations ( 10)-( 11) as follows

∂ 2 ρ ∂t 2 - T i m i ∂ 2 ∂z 2 ρ + eϕ T i = ∂ ∂z V ∂V ∂z - ∂ ∂t (ρV ) , (36)
where we used the same notations V and ρ = δn i /n 0 = (n i -n 0 ) /n 0 as in the previous Section; ϕ is the low frequency potential. The density n h of the electrons, which are a hot population compared to the ions, is

n h = ∞ -∞ f (0) + f (1) dv z = n (0) h + n (1) h , (37)
where n (0) h and n

(1)

h are the zero and first order electron densities, corresponding to the parallel velocity distributions f (0) and f (1) ; n (0) h is obtained by writing that the resulting force acting on the hot electrons (Coulomb, thermal and stationary ponderomotive forces) is vanishing, which leads to where T h is the temperature of the electron population and n h0 is the electron density at equilibrium. Considering the higher order terms provided by the thermal force and the non stationary part of the ponderomotive force, we get similarly the derivative of the first order electron density ∂n

n (0) h n h0 exp eϕ T h - ω 2 p ω 0 (ω 0 -ω c ) |E| 2 16πn 0 T h , (38)
(1) h ∂z n h0 ω 2 p ω c ω 0 (ω 0 -ω c ) 2 k 0 ω 0 ∂ ∂t |E| 2 16πn 0 T h . (39)
The Poisson equation can be written as

∂ 2 ϕ ∂z 2 = -4πe (n i -n h ) = -4πe ρn 0 -n (0) h + n (1) h -n 0 , (40)
which leads to

1 -λ 2 h ∂ 2 ∂z 2 eϕ T h = ρ + ω 2 p ω 0 (ω 0 -ω c ) |E| 2 16πn 0 T h + 1 2 eϕ T h - ω 2 p ω 0 (ω 0 -ω c ) |E| 2 16πn 0 T h 2 - n (1) h n 0 , (41)
where λ 2 h = T h /4πn 0 e 2 is the electron Debye length. Excluding ϕ in the right hand side of (41) we get

∂ ∂z eϕ T h 1 + λ 2 h ∂ 2 ∂z 2 ∂ρ ∂z + ω 2 p ω 0 (ω 0 -ω c ) ∂ ∂z - ω c ω 0 -ω c k 0 ω 0 ∂ ∂t |E| 2 16πn 0 T h + 1 2 ∂ρ 2 ∂z . (42)
Then, using (36) we obtain that

∂ 2 ρ ∂t 2 -c 2 s ∂ 2 ρ ∂z 2 -c 2 s ∂ 2 ∂z 2 λ 2 0 ∂ 2 ρ ∂z 2 + T h T h + T i ρ 2 2 - ω 2 p ω 0 (ω 0 -ω c ) ∂ ∂z ∂ ∂z - ω c (ω 0 -ω c ) k 0 ω 0 ∂ ∂t |E| 2 16πn 0 m i = ∂ ∂z V ∂V ∂z - ∂ ∂t (ρV ) , (43)
where

λ 2 0 = T h λ 2 h /(T h + T i ).
For waves propagating along one direction with the velocity U we can write that ρ ρ (τ , ξ) , where ξ = z -U t, and suppose the slow dependence of ρ on τ (i.e. ∂/∂t -U ∂/∂z) . In the first approximation we get that V ρU and, as we can write that

∂ ∂z V ∂V ∂z - ∂ ∂t (ρV ) 3 2 U 2 ∂ 2 ρ 2 ∂z 2 , ∂ 2 ρ ∂t 2 -2U ∂ ∂z ∂ρ ∂τ + U 2 ∂ 2 ρ ∂z 2 , (44)
we can derive that

∂ρ ∂τ - (U 2 -c 2 s ) 2U ∂ρ ∂z + c 2 s 2U ∂ ∂z λ 2 0 ∂ 2 ρ ∂z 2 + λ 2 0 λ 2 h ρ 2 2 + ω 2 p 2U ω 0 (ω 0 -ω c ) ∂ ∂z - ω c (ω 0 -ω c ) k 0 ω 0 ∂ ∂t |E| 2 16πn 0 m i + 3U 4
∂ρ 2 ∂z = 0. (45)
With the required accuracy we can write that U c s and ∂/∂t -c s ∂/∂z, so that we obtain finally

∂ρ ∂τ + c s 2 λ 2 0 ∂ 3 ρ ∂z 3 + c s 2 ∂ ∂z c 2 s -U 2 c 2 s ρ + λ 2 0 λ 2 h + 3 ρ 2 2 = - c s χ 2 ∂ ∂z |E| 2 16πn 0 m i c 2 s , (46)
which is a KdV equation with a right hand side term describing the ponderomotive effects. The high frequency equation ( 6) coupled with (46) can be written in the form

i ∂B ∂t + iv g0 ∂B ∂z + v g0 2 ∂ 2 B ∂z 2 = - ω 0 ω 2 p D 0 (ω 0 -ω c ) 1 + ω c ω 0 -ω c k 0 c s ω 0 ρB = αχ 2 ρB, (47)
where we note

χ = ω 2 p ω 0 (ω 0 -ω c ) 1 + ω c (ω 0 -ω c ) k 0 c s ω 0 , α = - 2ω 2 0 D 0 . ( 48
)
The system formed by the equations ( 46)-( 47) has an Hamiltonian H s which is the sum of three Hamiltonians, i.e. H w , H ρ and H int . Indeed, multiplying (47) by -∂B * /∂t, adding the complex conjugate and integrating on space, we obtain the first part H w of the total Hamiltonian H s

H w = dz L iv g0 2 B ∂B * ∂z -B * ∂B ∂z + v g0 2 ∂B ∂z 2 , (49)
which is the whistler wave energy. The KdV equation ( 46) provides the following Hamiltonian, which is related to the energy of the low frequency dynamics

H ρ = αγ dz L c 2 s -U 2 c 2 s ρ 2 2 - λ 2 0 2 ∂ρ ∂z 2 + λ 2 0 λ 2 h + 3 ρ 3 6 , (50)
where γ = 8π(c 2 k 2 0 /ω 2 0 )n 0 m i c 2 s , whereas the interaction of the plasma fluid and the whistler field is governed by the third Hamiltonian

H int = αχ 2 dz L ρ |B| 2 . ( 51
)
The total Hamiltonian H s = H w + H int + H ρ of the system is given by

H s = dz L iv g0 2 B ∂B * ∂z -B * ∂B ∂z + v g0 2 ∂B ∂z 2 + dz L αχ 2 ρ |B| 2 + αγ c 2 s -U 2 c 2 s ρ 2 2 - λ 2 0 2 ∂ρ ∂z 2 + T h T h + T i + 3 ρ 3 6 , (52)
where the canonical variables are the couples (B, B * ) and (ρ, ψ), with ψ = 2αγρ/c s , which satisfy the Hamilton equations

∂B ∂t = -i δH s δB * , ∂B * ∂t = i δH s δB , (53)
∂ρ ∂t = - ∂ ∂z δH s δψ , ∂ψ ∂t = - ∂ ∂z δH s δρ . (54)
In order to determine the form of the soliton which can propagate in the system, let us define the variables τ = t and ξ = z -v g0 t and write the equations ( 46)-( 47) in the frame moving with the group velocity as

∂ρ ∂τ + (c s -v g0 ) ∂ρ ∂ξ + c s 2 ∂ ∂ξ λ 2 0 ∂ 2 ρ ∂ξ 2 + 1 2 λ 2 0 λ 2 h + 3 ρ 2 + c s χ 2 ∂ ∂ξ |E| 2 16πn 0 m i c 2 s = 0, ( 55
) i ∂E ∂τ + v g0 2 ∂ 2 E ∂ξ 2 = αχ 2 ρE. (56)
Let us search in this frame stationary solutions in the form ρ = ρ(ξ) and E = E s (ξ) e -i t e iµ z . We get

2 (c s -v g0 ) c s ρ + λ 2 0 ∂ 2 ρ ∂ξ 2 + 1 2 λ 2 0 λ 2 h + 3 ρ 2 + χ |E s | 2 16πn 0 m i c 2 s = 0, (57)
and v g0 2

∂ 2 E s ∂ξ 2 + E s - αχ 2 ρE s = 0. ( 58
)
Note that the expected mutual compensation of dispersive and nonlinear effects is realized in the higher frequency equation ( 58) but unlikely in the lower frequency equation ( 57), due to the smallness of the dispersive term λ 2 0 ∂ 2 ρ/∂ξ 2 , which is typically 100 times smaller than (v g0 /2)∂ 2 E/∂ξ 2 . In this case the nonlinear term proportional to ρ 2 , which could lead to steepening effects and wave breaking, is compensated by the ponderomotive effects proportional to |E s | 2 . The equations ( 57)-( 58) have solutions in the form ρ (ξ) = ρ 0 sec 2 (ξ/l s ) and E s (ξ) = ±E s0 sec (ξ/l s ) tanh (ξ/l s ), with = -v g0 /2l 2 s and

ρ 0 = - 6v g0 αχl 2 s - (c s -v g0 ) c s 4 λ 2 0 /λ 2 h + 3 -2αχλ 2 0 /3v g0 , ( 59
)
where the term 2 αχ/v g0 λ 2 0 /3 1 in the denominator can be neglected compared to the other ones. With the same accuracy we get also

|E s | 2 16πn 0 m i c 2 s 8 (c s -v g0 ) 2 χc 2 s λ 2 0 /λ 2 h + 3 , c 2 ω 2 c l 2 s (c s -v g0 ) c s c 2 ω 2 c 2αχ 3v g0 λ 2 0 /λ 2 h + 3
.

Note that α is positive whatever the parameters are and that χ > 0 when v g0 c s . One can show that, similarly to ρ 0 , |E s | and 1/l 2 s are proportional to c s -v g0 : when v g0 → c s the soliton's amplitude decreases and its width increases. If v g0 > 0, the soliton exists whatever the parameters are, it is slightly subsonic (v g0 c s ) and propagates with a density dip ρ 0 < 0. On the other hand, if v g0 < 0, the soliton only exists if the following condition is fulfilled

T h T h + T i + 3 > 2αχλ 2 0 v g0 . ( 60
)
In this case, it is slightly supersonic (c s v g0 ) and its density perturbation propagates as a hump (ρ 0 > 0). In both cases we have v g0 (c s -v g0 ) > 0.

The variation of v g0 with ω 0 /ω c shows that the condition v g0 c s can be fulfilled in two frequency domains : ω 0 ω c or ω 0 ω c . In the latter case, we get the following approximate expressions for a plasma with ω p > ω c :

v g0 2c (ω c /ω p ) (1 -ω 0 /ω c ) 1/2 , v g0 -8c 2 ω c /ω 2 p / (1 -ω 0 /ω c ) and αχ 2ω p c s (1 -ω 0 /ω c ) -1/2 /c
, where we note that v g0 < 0; then the condition (60) can be written as

T h T h + T i 1 - v 2 T c 2 1 - ω 0 ω c + 3 > 0,
which is always true : solitons of the KdV equation exist for sonic whistlers with frequencies near ω c . Moreover their electric field amplitude is given by

|E s | 2 16πn 0 m i c 2 s 8 (c s -v g0 ) 2 c 2 s c c s ω 3 c ω 3 p (1 -ω 0 /ω c ) 5/2 λ 2 0 /λ 2 h + 3
, with the corresponding width and density dip

c 2 ω 2 c l 2 s - (c s -v g0 ) 6c ω 3 p ω 3 c (1 -ω 0 /ω c ) 1/2 λ 2 0 /λ 2 h + 3 , ρ 0 - 4 (c s -v g0 ) c s λ 2 0 /λ 2 h + 3
.

The soliton is slightly supersonic and the density ρ propagates as a hump. No subsonic soliton exists in this case.

VI. CONCLUSION

A self-consistent Hamiltonian model based on equations describing the coupled dynamics of electromagnetic whistlers and lower frequency waves in inhomogeneous plasmas is built. Based on this model, the paper extends and generalizes the existing studies by enlarging the domain of their parameters' validity and by providing detailed numerical simulations controlled by the model's invariants and involving various weakly inhomogeneous plasma density structures. Different aspects of whistler turbulence are studied, concerning mainly the development of modulational instabilities and the dynamics of envelope solitons in irregular plasmas.

Modulational instability can develop for supersonic or subsonic whistlers, if the derivative v g0 = dv g0 /dk of the group velocity v g0 is negative or positive, respectively. Simple analytical expressions of the growth rates are provided for the supersonic, subsonic as well as sonic cases. Moreover numerical simulations show that modulational instability can lead to the generation of a beating of stable nonlinear whistlers propagating with a speed near the group velocity.

The whistler envelope soliton has been determined analytically by using the developed model and its propagation in plasmas presenting random density fluctuations and irregular density structures of different scales and amplitudes has been studied, showing that the envelope is very weakly affected by these inhomogeneities, even if the wavelengths and the amplitudes of the phase oscillations inside the envelopes strongly vary. As a linear wave, the whistler soliton can accelerate or decelerate when crossing density gradients, these effects remaining however very weak. Moreover, simulations show for the first time that two whistler solitons colliding one with the other remain unchanged after this collision, independently of their initial amplitudes and velocities.

The propagation of sonic whistler solitons is actually different from that of subsonic and supersonic ones. Their lower frequency perturbation is governed by a KdV-type equation where dispersive effects are however weak. But they are compensated by ponderomotive effects due to the whistler fields which counterbalance the steepening and the wave breaking processes due to the nonlinearity.

VII. ACKNOWLEDGMENTS

This work was granted access to the HPC resources of IDRIS under the allocation 2017-A0010510106 made by GENCI. This work has been done within the LABEX Plas@par project, and received financial state aid managed by the Agence Nationale de la Recherche, 27 A. Voshchepynets, A. Volokitin, V. Krasnoselskikh, and C. Krafft, Statistics of electric fields' amplitudes in Langmuir turbulence: A numerical simulation study, J. Geophys. Res.: Space Phys. 122(4), 3915-3934 (2017). doi:10.1002/2017JA023898. 28 H. Washimi, Self-focusing of transverse waves in a magnetoplasma, J. Phys. Soc. Jpn. 34, 1373-1383 (1973)

VIII. APPENDIX A : MODEL'S EQUATIONS A. Equation for the whistler field envelope

Using (1) we can write that ∂B/∂t = (∂/∂t -iω 0 ) Be -iω 0 t+ik 0 z (x+iy) /2 + c.c. Taking into account that the envelopes are varying slowly in time and space, we obtain that ∂ 2 B/∂t 2 (-2iω 0 ∂/∂t -ω 2 0 ) Be -iω 0 t+ik 0 z (x+iy) /2 + c.c. and ∇ 2 B (2ik 0 ∂/∂z -k 2 0 ) B e -iω 0 t+ik 0 z (x+iy) /2 + c.c. So we get from (4) that

1 c 2 -2iω 0 ∂B ∂t -ω 2 0 B -2ik 0 ∂B ∂z -k 2 0 B = 4π c (∇ × j L + ∇ × j ext ) , (A1)
where j L is the linear current density corresponding to the plasma (nonresonant) particles, which can be calculated using the Ohm' law as j L = -(iω/4π) ε -I E; ε is the cold dielectric tensor with the non vanishing elements ε xx = ε yy = ε ⊥ , ε zz = 1-ω 2 p /ω 2 and ε xy = -ε yx = iq; I is the unity tensor. Then Maxwell equations lead to

∇ × j L iω 4πc ε -I ∂B ∂t ω 2 8πc c 2 k 2 ω 2 - D(ω, k) ω 2 -1 Be -iω 0 t+ik 0 z (x + iy) + c.c.
When no external current density is included (j ext = 0), we get from (A1) that D(ω, k) B 0, where D(ω, k) can be presented in the form of a differential operator (5). Note that, obviously, order two terms of D(ω, k) in the small nonlinear terms including ρ, δv e and δB z are neglected. Therefore we can write the equation for the magnetic field envelope as

i ∂D ∂ω 0 ∂B ∂t -i ∂D ∂k 0 ∂B ∂z - 1 2 ∂ 2 D ∂ω 2 0 ∂ 2 B ∂t 2 -2 ∂ 2 D ∂k∂ω 0 ∂ 2 B ∂z∂t + ∂ 2 D ∂k 2 0 ∂ 2 B ∂z 2 +ρ ∂D ∂ρ 0 B + δv e ∂D ∂δv e 0 B + δB z B 0 ∂D ∂ (δB z /B 0 ) 0 B 0.
Dividing this expression by (∂D/∂ω) 0 and noting that v g0 = (dω/dk) 0 = -(∂D/∂k) 0 / (∂D/∂ω) 0 as well as

v g0 = ∂v g0 ∂k + v g0 ∂v g0 ∂ω - ∂ 2 D ∂k 2 0 + 2v g0 ∂ 2 D ∂k∂ω + v 2 g0 ∂ 2 D ∂ω 2 0 / ∂D ∂ω 0 and ∂ 2 D ∂ω 2 0 ∂ 2 B ∂t 2 -2 ∂ 2 D ∂k∂ω 0 ∂ 2 B ∂z∂t + ∂ 2 D ∂k 2 0 ∂ 2 B ∂z 2 ∂ 2 D ∂k 2 0 + 2v g0 ∂ 2 D ∂k∂ω + v 2 g0 ∂ 2 D ∂ω 2 0 ∂ 2 B ∂z 2 ,
we get finally (6), observing that

D (ω, k) k 2 c 2 -ω 2 + ω 2 p (1 + ρ)ω ω -kδv e -ω c (1 + δB z /B 0 )
.

B. Hamiltonian of the system

In order to obtain the Hamiltonian (18) of the system of equations ( 12)-( 13)-( 17), let us multiply (17) by ∂B * /∂t, add the complex conjugate and integrate over the plasma size L. The first term is obviously vanishing. The second term can be expressed as follows, using integration by parts and the fact that the field and its derivatives are vanishing at infinity

iv g0 dz L ∂B ∂z ∂B * ∂t - ∂B * ∂z ∂B ∂t = - iv g0 2 ∂ ∂t dz L B ∂B * ∂z -B * ∂B ∂z . (A2)
For the same reasons, the third term can be written as

v g0 2 dz L ∂ 2 B ∂z 2 ∂B * ∂t + ∂ 2 B * ∂z 2 ∂B ∂t = - v g0 2 ∂ ∂t dz L ∂B ∂z 2 . ( A3
)
For the fourth term we use the following decomposition

dz L ρB ∂B * ∂t + ρB * ∂B ∂t = dz L ρ ∂ ∂t |B| 2 = ∂ ∂t dz L ρ |B| 2 - dz L |B| 2 ∂ρ ∂t . ( A4
)
In the fifth term we insert the low frequency equations ( 15)-( 16) so that we can write Let us define v + = v x + iv y , v -= v x -iv y , E + = E x + iE y and E -= E x -iE y , which provide the motion equations in the form

dv ± dt = - e m e E ± ± iω c v ± .
Searchig solutions proportional to e -iω 0 t we get the velocity perturbations as a function of the electric field perturbation so that δF n 0 = 1 8π

ω 2 p ω c ω 0 (ω 0 -ω c ) 2 k ω 0 ∂ ∂t |δE x | 2 .
Averaging on time, we get the non stationary part of the ponderomotive force as

F n 0 = ω 2 p ω c ω 0 (ω 0 -ω c ) 2 k ω 0 ∂ ∂t |E| 2 16π
.

X. APPENDIX C : NORMALIZED MODEL'S EQUATIONS AND NUMERICAL SCHEME

Figures are presented using dimensionless variables. The time t, the distance z and the whistler magnetic field B are normalized by the electron gyrofrequency ω c , by ω c /c and by the modulus B 0 of the ambient magnetic field, respectively. All velocities, as the group velocity v g0 , the ion acoustic velocity c s and the electron thermal velocity v T are normalized by the light velocity c, except of the slow velocity V of the electrons which is normalized by c s . The normalized forms of the equations ( 6), ( 12) and ( 13) are (the same name is given to the normalized variables used below, and to the non normalized ones used in the body of the text) where Ω κ = v g0 κ+v g0 κ 2 /2 and = τ 2 κ 2 c s /4; τ is the time step ; ρ n κ , V n κ , B n κ are the Fourier components at time n of the density, the fluid velocity and the magnetic field envelope.

i ∂B ∂t + iv g0 ∂B ∂z + v g0 2 ∂ 2 B ∂z 2 = - ω 0 ω 2 p ω c (ω 0 -ω c ) D 0 ρ + ω c ω 0 -ω c c s k 0 V ω 0 B, ∂ ∂t c s V - ω 3 c (ω 0 -ω c ) 2

FIG. 1 .

 1 FIG. 1. Growth rate Im(Ω/ω 0 ) of the modulational instability as a function of the normalized wavenumber κ/k 0 . (Upper panel) : Subsonic case (v g0 < c s ), for parameters typical of the solar wind : ω p /ω c = 110, ω 0 /ω c = 0.0013, ck 0 /ω c = 4, c s = 0.0008, B/B 0 = 0.28, v g0 = 0.00065, v g0 > 0. (Middle panel) : Supersonic case (v g0 > c s ), for parameters typical of the magnetosphere : ω p /ω c = 3, ω 0 /ω c = 0.59, ck 0 /ω c = 3.7, B/B 0 = 0.017, c s = 0.002, v g0 = 0.13, v g0 < 0. (Lower panel): Sonic case (v g0 c s ), for parameters typical of the magnetosphere : ω p /ω c = 2, ω 0 /ω c = 0.96, ck 0 /ω c = 10, B/B 0 = 0.01, c s = v g0 = 0.0073.

FIG. 2 .

 2 FIG. 2. (Color online) (Left column) Profiles of the magnetic field envelope B/B (thick black curve), its real and imaginary parts (thin black and gray lines -blue and gray lines onlinerespectively), for four moments of time, i.e. ω c t = 1.7 10 8 , 1.9 10 8 , 2.5 10 8 , and 3 10 8 . (Right column) Corresponding profiles of the lower frequency fluid velocity perturbation V /c s , at the same times. Parameters are those of Fig. 1, middle panel.

FIG. 3 .

 3 FIG.3. Time variation of the normalized ion sound energy density W IS (in logarithmic scale).

FIG. 4 .

 4 FIG. 4. Whistler envelope soliton (in thick lines) at three time moments ω c t = 20, 45000 and 90000 during its motion through a density depletion reaching δn/n 0 -0.02 (dashed lines); phase oscillations inside the envelope are presented in thin lines. Main parameters are : ω p /ω c = 17, ω 0 /ω c = 0.41, ck 0 /ω c = 14.3, B/B 0 = 0.015, c s = 0.00014, v g0 = 0.0034.

FIG. 5 .

 5 FIG. 5. (Upper row) Whistler envelope soliton (thick black lines) at four time moments ω c t = 500, 2.65 10 4 , 5.25 10 4 , 1.05 10 5 during its motion through a density hump with maximum δn/n 0 0.001 (bottom row, thick black lines); the phase oscillations inside the envelope are presented in thin black lines. (Bottom row) : Corresponding profiles of the external density (black lines) and velocity (gray lines) perturbations. In all places where black and gray lines do not appear separately, they are actually superposed. Main parameters are : ω p /ω c = 3, ω 0 /ω c = 0.73, ck 0 /ω c = 5.0, B/B 0 = 0.03, c s = 0.02, v g0 = 0.079.

FIG. 6 .

 6 FIG. 6. (Upper panel) : Whistler envelope soliton B/B 0 (thick lines) at five time moments ω c t = 800, 300800, 600800, 1000800, and 1400800 during its motion through an inhomogeneous plasma with random density fluctuations; phase oscillations inside the envelope are presented in thin lines. (Lower panel) : Profile of the external density fluctuations δn/n 0 . Main parameters are : ω p /ω c = 10, ω 0 /ω c = 0.28, ck 0 /ω c = 6.3, B/B 0 = 0.001, c s = 0.00015, v g0 = 0.065.

FIG. 7 .

 7 FIG. 7. (Color online) Time evolution of the collision between two whistler envelope solitons. Six moments of time are shown, as indicated on each panel : ω c t = 6 10 6 , 8.4 10 6 , 1.08 10 7 , 1.32 10 7 , 1.57 10 7 , and 1.8 10 7 . The first soliton, the fastest one, has the amplitude B/B 0 = 0.03 (upper left panel); the second soliton, the slowest one, has the amplitude B/B 0 = 0.05. The envelopes are shown in thick black lines, whereas the real and imaginary parts are presented in thin and dotted (red online) lines. Main parameters are : ω p /ω c = 3, ω 0 /ω c = 0.49, ck 0 /ω c = 3, c s = 0.001, v g0 = 0.17.

FIG. 8 .

 8 FIG. 8. (Color online) Time evolution of the collision between two whistler envelope solitons. Six moments of time are shown, as indicated on each panel : ω c t = 4 10 7 , 5.12 10 7 , 6.24 10 7 , 7.36 10 7 , 8.48 10 7 , and 9.6 10 7 . The first soliton, the fastest one, has the amplitude B/B 0 = 0.03 (upper left panel); the second soliton, the slowest one, has the amplitude B/B 0 = 0.01. The envelopes are shown in thick black lines, whereas the real and imaginary parts are presented in thin and dotted (red online) lines. Main parameters are : ω p /ω c = 3, , ω 0 /ω c = 0.63, ck 0 /ω c = 4, c s = 0.001, v g0 = 0.117.

FIG. 9 .

 9 FIG. 9. (Upper panel) : Time variation, during the collision, of the square maximum amplitude (B max /B 0 ) 2 of the structure formed by the two overlapped solitons in Fig. 8. (Lower panel) : Corresponding time variation of the position ξ = z max -v g0 t in the frame moving with the group velocity, where z = z max for B = B max . Parameters are those of Fig. 8.

  IX. APPENDIX B : NON STATIONARY PART OF THE PONDEROMOTIVE FORCELet us calculate the non stationary part of the ponderomotive force. The slowly varying parallel fluid motion can be written asm e ∂v z ∂t + m e v z ∂ ∂z v z + eE z = -e c (v ⊥ × B ⊥ ) j x B y -j y B x ,where E z is the low frequency part of the parallel electric field and F is the ponderomotive force per unit volume. At the first order, we can write the Newton equations for the electrons in the perpendicular plane in the form

  δv ± = -e m e (ω 0 ± ω c ) 2 ∂δE ± ∂t .Keeping only the field E -, as for whistlers we have E x = -iE y , i.e. E + = 0, we get the real values of the velocities(Re v x , Re v y ) = e 2m e 1 ω 0 -ω c Im E -, Re E -, (Re δv x , Re δv y ) = e 2m e (ω 0 -ω c ) 2 ∂ ∂t -Re δE -, Im δE -.For the fields we get similarly the first order perturbations Re δB x -Re δB y -Re v y Re δB x + Re δv x Re B y -Re δv y Re B x ]

  ω c = 1, ρ = δn i /n 0 and V = δv e /c s . The corresponding numerical scheme used for integration of the equations is given byB n+1 κ = (1 -iτ Ω κ /2) (1 + iτ Ω κ /2τ 2 κ 2 c s /4) k 0 c s (ω 0 -ω c )

  The evolution of the lower frequency oscillations of the electron density δn e and velocity δv e can be obtained by using the momentum conservation equation in the

	steady state
	∂ ∂t	δv e + δv e	∂ ∂z	δv e =	e m e	∂ ∂z	δϕ -	T e n e m e	∂ ∂z	δn e +	F pz m e	0,

  . doi:http://dx.doi.org/10.1143/JPSJ.34.1373. 29 H. Washimi and V. I. Karpman, The ponderomotive force of a high-frequency electromagnetic field in a dispersive medium, JETP 71,1010-1016 (1976). 30 D.W. Forslund, J.M. Kindel, and E.L. Lindman, Parametric excitation of electromagnetic waves, Phys. Rev. Lett. 29,249-252 (1972). 31 G.M. Zaslavskii and N.N. Filonenko, Instability of nonlinear periodic waves in a dispersive medium, JETP 29(3),572 (1969). 32 C. Krafft and A.Volokitin, Interaction of whistler envelope solitons with non-relativistic electron beams in plasmas with density inhomogeneities, Phys. Plasmas, in press, 2018.


de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Fabienne El-Khoury
email: fabienne.khoury@inserm.fr

Anne-Laure Sutter-Dallay

Lidia Panico

Marie-Aline Charles

Elie Azria

Judith Van Der Waerden

Maria Melchior

Fabienne El

Women's mental health in the perinatal period according to migrant status : the French representative ELFE birth cohort

Keywords: Migrant status, women, second generation migrants, postpartum depression, mental health

Background

Mental health problems in the perinatal period are common. We examined associations between different categories of migrant status and region of origin in relation to mental health during pregnancy and at two months postpartum.

Methods

We analysed data from the French nationally representative ELFE birth cohort (n=17,988). Migrant status was divided into five categories: 'majority population', 'descendants with one migrant parent', 'descendants with two migrant parents', 'naturalised migrant ' and 'non-naturalised migrant women'. Multivariate logistic regression models were implemented to examine associations between migrant status and mental health outcomes: persistent psychological difficulties during pregnancy as well as mother's depression and poor self-reported health at two months postpartum.

Results

After adjusting for covariates, migrant status was not associated with psychological difficulties during pregnancy. Descendants of migrants had comparable mental health to the majority population. Non-naturalised migrant women were more likely to experience depression (OR= 1.66, 95%CI:1.27,2.20) and poor self-reported health (OR=1.45, 95%CI:1.06,1.98) during the postpartum period. The region of origin was associated with postpartum health independently of migrant status, such that women from Africa and Turkey were most likely to have depression or poor selfrated health.

Conclusion

First, but not second, generation migrant women appear to have high levels of mental health difficulties during the postpartum period. Women from North Africa, Sub-Saharan Africa, and Turkey have higher levels of distress than those from other regions. In particular, non-naturalised migrant appear to be a vulnerable group; they may disproportionately face stressors that increase their risk for postpartum depressive symptoms.

Introduction

The perinatal period is fraught with socio-emotional, practical and hormonal changes and challenges that make women susceptible to mental health problems: approximately 20% experience depression in the antenatal period or in three months following childbirth. [START_REF] Apter | Perinatal Mental Health[END_REF][START_REF] Gaynes | Perinatal Depression: Prevalence, Screening Accuracy, and Screening Outcomes[END_REF][START_REF] Leung | Perinatal Depression: Prevalence, Risks, and the Nutrition Link-A Review of the Literature[END_REF] Maternal mental health in the perinatal period has important consequences for women and their children; therefore it is essential to understand the associated risk and protective factors.

In western countries, migrant women or those who descend from migrant parents are generally more prone to poor mental health than the majority population, [START_REF] Missinne | Depressive symptoms among immigrants and ethnic minorities: a population based study in 23 European countries[END_REF] or those who stay in the country of origin.(5) A recent meta-analysis reported that migrant women are two times more likely to experience depressive symptoms after giving birth than non-migrant women. [START_REF] Falah-Hassani | Prevalence of postpartum depression among immigrant women: A systematic review and meta-analysis[END_REF] However this review included little data from European countries and was limited by the lack of a clear and standardised definition of migrant status. Indeed, in most studies migrant status is conceptualised as a binary variable (foreign born vs. native born or White vs. ethnic minority), and neither second generation migrants nor geographic origins are accounted for. [START_REF] Onozawa | High EPDS scores in women from ethnic minorities living in London[END_REF][START_REF] Huang | Depressive Symptomatology and Mental Health Help-Seeking Patterns of U.S.-and Foreign-Born Mothers[END_REF] This is problematic as migrant communities in Europe are diverse. Across European countries, there is also heterogeneity in the policies and services available to migrants, as well as their level of exposure to racism and discrimination. [START_REF] Safi | Immigrants' Life Satisfaction in Europe: Between Assimilation and Discrimination[END_REF][START_REF] Ersanilli | Rewarding Integration? Citizenship Regulations and the Socio-Cultural Integration of Immigrants in the Netherlands, France and Germany[END_REF] In France, in 2014, 11% of the population had at least one migrant parent (born not in France and non-French), [START_REF] Bouvier | Démographie des descendants d'immigrés[END_REF] with 13.3% of new-borns having a migrant mother. [START_REF] Saurel-Cubizolles | Santé périnatale des femmes étrangères en France[END_REF] Also, approximately 9% of residents of France are migrant, among them about 37% were born in Europe and 44% in Sub-Saharan or North Africa. [START_REF] Brutel | Populations française, étrangère et immigrée en France depuis[END_REF] Mental health problems in the perinatal period among migrant women may be explained by socio-economic factors, [START_REF] Rechel | Migration and Health in the European Union[END_REF] but also an impaired adaptation to the surrounding environment. [START_REF] Jurado | World region of origin and common mental disorders among migrant women in Spain[END_REF][START_REF] Shoshani | School engagement, acculturation, and mental health among migrant adolescents in Israel[END_REF] However, this susceptibility is probably different between first and second generation migrants. [START_REF] Carta | Migration and mental health in Europe (the state of the mental health in Europe working group: appendix 1)[END_REF] Further, mental health among women of migrant descent (second generation) could differ depending on whether they have one or two migrant parents, [START_REF] Viruell-Fuentes | Beyond acculturation: Immigration, discrimination, and health research among Mexicans in the United States[END_REF] as well as their region of origin. The same applies to first generation migrants; there could be differences in coping with the challenges of the perinatal period as well as associated mental health problems between naturalised and non-naturalised women. To our knowledge, these differences have not been adequately examined.

In this study, we examine whether mental health during the perinatal period differs according to women's migrant status. Thus we studied the association between migrant status, region of origin and mental health indicators during pregnancy and at 2 months postpartum, adjusting for socio-demographic and economic characteristics, using data from the French nationally representative ELFE birth cohort study.

Methods

Study design

ELFE (Etude Longitudinale Française depuis l'Enfance) is an ongoing prospective, longitudinal nationally representative French study (www.elfe-france.fr).( 19)

Participants

A total of 17,988 mothers and their infants born in 2011 (n=18,275) were initially recruited in a representative random sample of 320 maternity wards throughout metropolitan France on a number of pre-selected days throughout the year (four waves: April; June-July; September-October; November-December). To be included, women had to be at least 18 years of age, plan on staying in metropolitan France for the following 3 years, and be capable of giving informed consent in one of the study languages (French, English, Arabic, and Turkish). Also, only single or twin births, born at or after 33 weeks gestation, were included.

Data sources

At birth, eligible mothers were interviewed by midwifes via face-to-face interviews at the maternity ward of delivery. Additionally, study participants completed a selfreported questionnaire (n=17,988). The second stage of data collection took place about two months after the child's birth, through telephone interviews with the mother with a response rate of 91% (n=16,280) and the father, if the mother participated (n=12,511).

Indicators of mental health

We considered three different outcome variables:

-at the time of the child's birth, women were asked whether they experienced persistent psychological difficulties during pregnancy (yes vs. no).

-at two months post-partum, depression was measured by the Edinburgh Postnatal Depression Scale (yes (EPDS≥12) vs. no), [START_REF] Guedeney | Validation study of the French version of the Edinburgh Postnatal Depression Scale (EPDS): new results about use and psychometric properties[END_REF] and women reported on their "general health" (average/bad vs. good/very good). Self-reported health has been known to capture not only physical, but also mental health and social well-being.(21)

Mother's migration status

At study enrolment, women were asked about their nationality and country of birth, as well as those of their partner. At two months post-partum, women reported their parents' country of birth. This information was used to identify women's 'migrant status' according to five categories: 'majority population' (includes foreign-born persons whose parents were French at birth), 'descendants (second generation) with one migrant parent', 'descendants with two migrant parents', 'naturalised migrant women ' (first generation; women born non-French outside of France who acquired French citizenship) and 'non-naturalised migrant women' (women non-French outside of France and who did not have French nationality at the time of data collection).

Region of origin

Participants' region of origin was defined as a) own region of birth for descendants of migrants, and b) parents' region of birth for migrants (North Africa and Turkey, Sub-Saharan Africa, non-EU Eastern Europe and Asia, other vs. France/ EU).

Other covariates

Covariates included known risk factors of women's mental health difficulties and characteristics related to migrant status:

-demographic characteristics: maternity unit of delivery, maternal age (≤ 30 years vs. > 30 years), educational level (< high school, high school vs. > high school degree), overall number of children (two or more vs. one), residential area (Paris Basin, North, East, West, South-West, Centre-East, others, vs. Paris region), recruitment wave (April, June-July, September-October vs November-December); -socio-economic characteristics: highest household occupational grade (none, low [e.g. clerk, manual worker], intermediate [e.g. mid-level manager, technician], vs.

high [e.g. manager]), maternal employment status at the time of pregnancy (unemployed, out of the job market vs. employed); -partner characteristics: employment status at the time of pregnancy (unemployed, out of the job market vs. employed), sufficient partner support during pregnancy (not living with a partner, no, vs. yes), nationality (naturalized, migrant vs. native French), and region of origin (North Africa and Turkey, Sub-Saharan Africa, Eastern Europe and Asia, other vs. France/ European Union); -pregnancy and child characteristics: pregnancy timing (anticipated vs. not anticipated), tobacco use during pregnancy (smoker, former smoker who quit during pregnancy, vs. non-smoker), unplanned caesarean section (yes, vs. no -including planned caesarean section, vaginal and instrumental extraction), child gestational age (<37 weeks vs ≥37 weeks), child sex (girl vs boy), and feeding method at 2 months (formula vs breast feeding (including exclusive and partial breastfeeding).-

Statistical Analysis

Missing data on covariates were imputed using multiple imputations by chained equations under fully conditional specification and assuming that data were missing at random. [START_REF] Buuren | Multiple imputation of discrete and continuous data by fully conditional specification[END_REF] The first imputation procedure included all the outcome variables and covariates measured at baseline as well as the maternity unit. The second imputation procedure included all outcome variables measured at 2 months as well as all covariates and the maternity unit. Ten imputed datasets were created for each imputation procedure. A variable that indicated missing values for each outcome was created before imputation, and served to select only complete outcome observations for each subsequent analysis on imputed datasets.

Logistic regression models were implemented to identify potential confounders among the list of previously described covariates. Variables associated with these outcomes in bivariate analyses (p<0.20) were included in the multivariate logistic regression models. These models examined associations between migrant status and the region of origin and each study outcome while adjusting for identified confounders, with a random intercept corresponding to the maternity unit of delivery. Interactions between migrant status and region of origin were examined separately for each study outcome. The majority group constituted the reference group in all analyses.

Analyses were adjusted for all the listed covariates, except in the analysis of psychological difficulties during pregnancy where covariates measured subsequently (ex. unplanned caesarean, child sex, gestational age and child feeding method) were not included.

All analyses were performed with SAS, version 9.4. Multiple imputations were implemented using the 'PROC MI' procedure with the FCS (fully conditional specification) statement. Bivariate and multivariate analyses were conducted separately for each imputed dataset and the results (OR, CI 95%) were pooled using the 'PROC MIANALYZE' procedure.

Results

Table 1 presents descriptive statistics of women included in the analysis.

Out of 17,988 mothers included in the study, 68.4% were in the "majority population" group, 5.8% were descendants with one migrant parent (n=1,046), 5.3% were descendants with two migrant parents (n=954), 3.6% were naturalised (n=654), and 8.5% were non-French citizens (n=1,525). Data on migrant status was missing for 8.4% (n=1,515) of the participants. More than half of descendants originated from another country in the European Union (53.4%) and 63.1% of migrants were from North Africa and Turkey.

In bivariate analyses, migrant status and the region of origin were associated with all study outcomes (supplementary table s2); however health outcomes among descendants with one migrant parent were comparable to those of members of the majority population. Additionally, we found a gradient of postpartum depression and self-reported health 2 months postpartum: non-naturalised migrants had worse health than naturalised migrants, who in turn had worse health than descendants with two migrant parents.

After adjusting for confounding factors (Table 2), particularly socio-economic variables and mother's age, migrant status and region of origin were no longer associated with persistent psychological difficulties during pregnancy.

Nevertheless, the association between migrant status and post-partum depression and self-rated health subsisted. In particular, non-naturalised migrant women were more likely to have post-partum depression (OR= 1.66, 95% CI 1.27-2.18) and more likely to report being in poor health (OR=1.45, 95% CI 1.06-1.98) 2 months after giving birth compared to the majority population. The region of origin was also associated with these 2 outcomes : women from North Africa and Turkey as well as 'other countries' were especially likely to be depressed; women from Sub-Saharan Africa and 'other countries' were especially likely to report poor health. There were no statistically significant interactions between migrant status and region of origin for the three mental health indicators.

Discussion

Key results

To our knowledge, this is the first European study that examines women's perinatal mental health in first and second-generation migrant women accounting for geographical region of origin. The present results show that after adjusting for multiple characteristics, only Non-French citizens display higher levels of postpartum depressive symptoms and poor perceived-health than the majority population. Additionally, the region of origin is also associated with postpartum depression (Turkey, North and Sub-Saharan Africa) and poor self-reported health (North Africa, Turkey, Other regions). Overall, women who descend from migrants have similar profiles to those who were born in France of French-citizen parents; however foreign women may require special attention from primary care and mental health professionals during pregnancy and the period that follows.

Interpretation

The finding that migrant women have worse self-reported health and higher levels of postpartum depression than native women is consistent with findings from other studies.( 6) Compared to the general population, migrant women disproportionately face stressors such as poor social support, low socioeconomic position, and/or difficulties in adaptation to the host culture that could increase their risk of postpartum depressive symptoms. [START_REF] Collins | Refugee, asylum seeker, immigrant women and postnatal depression: rates and risk factors[END_REF] In our study, neither naturalised nor secondgeneration migrant women were at increased risk of postpartum depression or poor perceived health, unlike non-French citizens, which could be due to differences in adaptation and social inclusion between these two groups. However, we did find an association between women's region of origin and mental health outcomes, independently of migrant status, which could reflect stressors not related to cultural and societal adaptation such as social adversity and discrimination. A recent review hypothesised that the reason migrants' life satisfaction in European countries is low is because of discrimination, which does not seem to improve across time or generation. [START_REF] Safi | Immigrants' Life Satisfaction in Europe: Between Assimilation and Discrimination[END_REF] In another study, perceived discrimination among pregnant lowincome women was associated with depressive symptoms independently of potential confounders, including ethnicity. [START_REF] Bennett | Perceived Discrimination and Depressive Symptoms, Smoking, and Recent Alcohol Use in Pregnancy[END_REF] In our study perceived persistent psychological difficulties during pregnancy were not associated with migrant status or the region of origin. It is possible that this binary indicator could not discriminate increased mental health problems persistently experienced by women with a migrant origin (4) from psychological difficulties directly linked to perinatal factors such as a mistimed pregnancy or the lack of partner support. In fact, these two factors had the strongest association with perceived persistent psychological difficulties in the antenatal period. Also, migrant women, especially those who were not French citizens, were more likely to be single mothers and to report a mistimed pregnancy compared to the majority population, therefore these factors possibly "mediate" the association between migrant status and psychological difficulties observed in our bivariate analysis.

Limitations

A limitation of our study is the lack of data on women's integration, acculturation, and perceived discrimination, even though, among migrants 'citizenship' (or naturalisation) can be considered as an adequate proxy for acculturation. [START_REF] Ersanilli | Rewarding Integration? Citizenship Regulations and the Socio-Cultural Integration of Immigrants in the Netherlands, France and Germany[END_REF] This is especially the case as naturalised women in our study lived in France an average of 8 years more than non-naturalised women (17.2 (sd=8) vs 9.3 (sd=8) and are therefore likely to have arrived at a much earlier age, which is related with the degree of acculturation. An alternative proxy would be the languages spoken at home. [START_REF] Lopez-Class | Conceptions of acculturation: A review and statement of critical issues[END_REF] Approximately 7% of women participating in the ELFE cohort reported using at least 2 languages in their household, but since most migrants in France and in the study originate from French-speaking countries, language may not be a precise proxy for acculturation in this setting. Another limitation is attrition at two months, which may have resulted in an underestimation of the association between migrant status and mental health problems, if most vulnerable women were not included in our followup. Nevertheless the associations we report were statistically significant even after adjusting for multiple socio-demographic, economic and partner characteristics.

Conclusion

Migrant women tend to have comparable mental health difficulties in the perinatal period as women who are not migrant; however, they also face some specific difficulties. Women of North African, Sub-Saharan or Turkish descent are more likely to report post-partum depression, as are women who do not hold French citizenship. Further research should compare women's perinatal health outcomes in different European countries, which might help distinguish the effects of migration itself, from those of socioeconomic circumstances and patterns of integration in the host country.

  Our study's main strengths are: a) the use of data from a large nationally representative cohort of women and children, b) measurement of multiple covariates, and c) the examination of mental health across different categories of migrant status and region of origin.

Table 1 :

 1 Descriptive characteristics of women of the ELFE cohort (n=17,988)

	Demographic characteristics (n. %*)		Health characteristics (n. %*)
		<30 years	7724 (42.9%)	Persistent	Yes	2222 (12.4%)
				psychological
	Age	≥30 years	10134 (56.3%)	difficulties	No	15524 (86.3%)
		< High school degree	3415 (19.0%)	Postpartum	No	13630 (75.8%)
	Educational level	High school	3760 (20.9%)	depression	Yes	1903 (10.6%)
		> High school degree	10757 (59.8%)	Self-reported health at	Good	14197 (78.9%)
		One	8327 (46.3%)	2 months postpartum	Bad	1345 (7.5%)
	Number of children
		≥Two	9661 (53.7%)	Pregnancy and child characteristics (n. %*)
		Paris	3184 (17.7%)		Yes	2156 (12.0%)
				Mistimed pregnancy
		Paris region	2699 (15.0%)		No	15555 (86.5%)
		North-Pas de Calais	1411 (7.8%)		Non-smokers	10220 (56.8%)
	Residential area	East	1592 (8.9%)	Smoking during	Former smoker	4020 (22.3%)
				pregnancy
		West	2282 (12.7%)		Smoker	3506 (19.5%)
		South-West	1300 (7.2%)	Unplanned caesarean	Yes	1869 (10.4%)

*Missing values are included in the percentage calculation

Table 2 : Associations between migrant status and region of origin with mental health in the perinatal period (multivariate logistic regression models, OR; 95% CI): the ELFE birth cohort study (n=17,988).

 2

	Psychological	Poor perceived	Post-partum
	difficulties during pregnancy	health, 2 months postpartum	depression, 2 months postpartum
	ORa (95%CI)	ORa (95%CI)	ORa (95%CI)

Acknowledgements

The ELFE study is a joint project between INED (Institut National d'Etudes Démographiques), INSERM (Institut National de la Santé et de la Recherche Médicale), EFS (Etablissement Français du Sang), InVS (Institut de Veille Sanitaire), INSEE (Institut National de la Statistique et des Etudes Economiques), the Ministry of Health (DGS, Direction Générale de la Santé), the Ministry of Environment (DGPR, Direction Générale de la Prévention des Risques), the Ministries of Health and Employment (DREES, Direction de la Recherche, des Etudes, de l'Evaluation et des Statistiques), and the CNAF (Caisse Nationale des Allocations Familiales), with the support the Ministry of Research and CCDSHS (Comité de Concertation pour les Données en Sciences Humaines et Sociales) and the Ministry of Culture (DEPS, Département des études, de la prospective et des statistiques). The ELFE study received approvals from the consultative committee for information treatment for health research (Comité Consultatif sur le Traitement des Informations pour la Recherche en Santé: CCTIRS) and the national data protection authority (Commission National Informatique et Libertés: CNIL).

This work was supported by state funding from the ANR (Agence Nationale de la Recherche/ National Agency for Research) within the framework of the "Future Investments" programme (reference: ANR-11-EQPX-0038), as part of the RECONAI platform.

Funding

This work was supported by funding from the ANR (Agence Nationale de la Recherche) within the framework of the "Future Investments" programme (reference: ANR-11-EQPX-0038), as part of the RECONAI platform. Additionally, this analysis was funded by the SOCIALRISK_MH project funded through the ANR 'Social determinants of health' program (2012).

Conflicts of interest

None declared

Keypoints

 Mental health problems are common in the perinatal period, especially among migrant women, as compared to those who are non-migrant.

 Descendants of migrants and naturalised migrants have comparable mental health to the majority population of France.

 Non-naturalised migrant women especially likely to have postpartum depression and poor self-reported health.

 Our findings highlight the need to screen for women's mental health difficulties the perinatal period and develop targeted psychological and social interventions for non-naturalised migrant women who need such support.