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[1] Recently, observations from laboratory experiments, which are relevant to space observations as well, have conclusively revealed the amplitude modulation of whistlers by low-frequency perturbations. Our objective here is to present theoretical and simulation studies of amplitude modulated whistler packets on account of their interaction with background low-frequency density perturbations that are reinforced by the whistler ponderomotive force. Specifically, we show that nonlinear interactions between whistlers and finite amplitude density perturbations are governed by a nonlinear Schrödinger equation for the modulated whistlers, and a set of equations for arbitrary large amplitude density perturbations in the presence of the whistler ponderomotive force. The governing equations are solved numerically to show the existence of large scale density perturbations that are self-consistently created by localized modulated whistler wavepackets. Our numerical results are found to be in good agreement with experimental results, as well as have relevance to observations from magnetized space plasmas.

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[2] In his classic paper, Stenzel [1976] experimentally demonstrated the creation of a magnetic field-aligned density trough by the ponderomotive force of localized electron whistlers. Observations from a recent laboratory experiment [Kostrov et al., 2003] exhibit a clear evidence of modulated whistler wavepackets due to nonlinear effects. Furthermore, instruments on board CLUSTER spacecraft have been observing broadband intense electromagnetic waves, correlated density fluctuations and solitary waves near the plasmapause as well as at the magnetopause and in the terrestrial foreshock [Moullard et al., 2002], revealing the signature of whistler turbulence in the presence of density depletions and enhancements. Observations from Freja satellite [Huang et al., 2004] also exhibit the formation of envelope whistler solitary waves accompanied by plasma density cavities.

[4] In this Letter, we investigate the dynamics of nonlinearly interacting electron whistlers and arbitrary large amplitude ion-acoustic perturbations, by using computer simulations, and find analytical expressions for whistler solitary waves in the low-amplitude limit.

2. The Mathematical Model

[5] Let us consider the propagation of nonlinearly coupled whistlers and ion-acoustic perturbations in a fully ionized electron-ion plasma in a uniform external magnetic field B_{0}, where is the unit vector along the z direction and B_{0} is the magnitude of the magnetic field strength. We consider the propagation of right-hand circularly polarized modulated whistlers of the form E = (1/2)E(z, t)( + i) exp [i(k_{0}Z − ω_{0}T)] + c.c, where E(z, t) is the slowly varying envelope of the whistler electric field, and and are the unit vectors along the x and y axes, respectively, and c.c. stands for the complex conjugate. The whistler frequency ω_{0} (≫), and the wavenumber k_{0} are related by the cold plasma dispersion relation ω_{0} = k_{0}^{2}c^{2}ω_{ce}/(ω_{pe,0}^{2} + k_{0}^{2}c^{2}), where c is the speed of light in vacuum, ω_{ce} = eB_{0}/m_{e}c (ω_{ci} = eB_{0}/m_{i}c) is the electron (ion) gyrofrequency, ω_{pe,0} = (4πn_{0}e^{2}/m_{e})^{1/2} is the electron plasma frequency, e is the amplitude of the electron charge, B_{0} is the background magnetic field, m_{e} (m_{i}) is the electron ion mass, and n_{0} is the unperturbed background electron number density.

[6] The dynamics of modulated whistler wavepacket in the presence of electron density perturbations associated with low-frequency ion-acoustic fluctuations and nonlinear frequency-shift caused by the magnetic field-aligned free streaming of electrons (with the flow speed v_{ez}) is governed by a nonlinear Schrödinger equation [Spatschek et al., 1979] i(∂_{t} + v_{g}∂_{z})E + (v′_{g}/2)∂_{zz}^{2}E + (ω_{0} − ω)E = 0, where ω = k_{0}^{2}c^{2}ω_{ce}/(ω_{pe}^{2} + k_{0}^{2}c^{2}) + k_{0}v_{ez}, and ω_{pe}^{2} = ω_{pe,0}^{2}n_{e}/n_{0} is the local plasma frequency including the electron density n_{e} of the plasma slow motion. The group velocity and the group dispersion of whistlers are v_{g} = ∂ω_{0}/∂k_{0} = 2(1 − ω_{0}/ω_{ce})ω_{0}/k_{0} and v′_{g} = ∂^{2}ω_{0}/∂k_{0}^{2} = 2(1 − ω_{0}/ω_{ce}) (1 − 4ω_{0}/ω_{ce})ω_{0}/k_{0}^{2}, respectively.

[7] The equations for the ion motion involved in the low-frequency (in comparison with the whistler wave frequency) ion-acoustic perturbations are ∂_{t}n_{i} + ∂_{z}(n_{i}v_{iz}) = 0 and ∂_{t}v_{iz} + (1/2)∂_{z}v_{iz}^{2} = −(e/m_{i})∂_{z}ϕ − (∂_{z}p_{i})/m_{i}n_{i}, where, for an adiabatic compression in one space dimension, the ion pressure is given by p_{i} = p_{i,0}(n_{i}/n_{0})^{3}. Here, the unperturbed ion pressure is denoted by p_{i,0} = n_{0}T_{i}, where T_{i} is the ion temperature.

[8] The electron dynamics in the plasma slow motion is governed by the continuity and momentum equations, viz. ∂_{t}n_{e} + ∂_{z}(n_{e}v_{ez}) = 0 and 0 = (e/T_{e})∂_{z}ϕ − ∂_{z}ln(n_{e}/n_{0})) + F, where T_{e} is the electron temperature, ϕ is the ambipolar potential, and the low-frequency ponderomotive force of electron whistlers is F = [ω_{pe,0}^{2}/ω_{0}(ω_{ce} − ω_{0})] (∂_{z} + ∂_{t})∣E∣^{2}/4πn_{0}T_{e}. The system of equations is closed by means of Poisson's equation ∂_{zz}^{2}ϕ = 4πe(n_{e} − n_{i}). However, simplification occurs if one assumes the quasi–neutrality n_{i} = n_{e} ≡ n, which is justified if ω_{0} < ω_{ce} is fulfilled with some margin. Then, the whistler dispersion relation gives k_{0}^{2}r_{D}^{2} = (V_{Te}^{2}/c^{2})/[(ω_{ce}/ω_{0}) − 1] ≪ 1 if V_{Te}^{2}/c^{2} ≪ 1, which is fulfilled for nonrelativistic plasmas, where V_{Te} = (T_{e}/m_{e})^{1/2} and r_{D} = V_{Te}/ω_{pe,0} are the electron thermal speed and the electron Debye radius, respectively. We have also assumed that scalelengths of the density and potential variations are larger or of the same order as the wavelength of the whistlers, viz. ∂_{z} ≤ k_{0}.

[9] The continuity equations for the electrons and ions give v_{iz} = v_{ez} ≡ v_{z}, so that ∂_{t}n + ∂_{z}(nv_{z}) = 0. Eliminating ∂_{z}ϕ from the governing equations for low- frequency density perturbations, we have ∂_{t}v_{z} + (1/2)∂_{z}v_{z}^{2} = −(T_{e}/m_{i}) [∂_{z}ln(n/n_{0}) − F] − (∂_{z}p_{i})/m_{i}n. The nonlinear Schrödinger equation for the whistler electric field together with the low-frequency equations form a closed set for our purposes.

2.1. The Scaled System of Equations

[10] In order to investigate numerically the interaction between whistlers and large amplitude ion-acoustic perturbations, it is convenient to normalize the governing equations into dimensionless units, so that relevant parameters can be chosen. We introduce the dimensionless variables ξ = ω_{pi,0}z/C_{s}, where the sound speed is C_{s} = [(T_{e} + 3T_{i})/m_{i}]^{1/2}, τ = ω_{pi,0}t, N = n/n_{0}, u = v_{z}/C_{s} and ℰ = E/; the only free dimensionless parameters of the system are Ω_{c} = ω_{ce}/ω_{pi,0}, κ = k_{0}c/ω_{pe,0}, η = T_{i}/T_{e} and V_{g} = v_{g}/C_{s}. The normalized system of equations are of the form

and

where the constants are α = (1 + κ^{2})^{2}ω_{pe,0}^{2}/ω_{ce}^{2}κ^{2} and P = (1 + κ^{2})(1 − 3κ^{2})V_{g}^{2}/4κ^{2}Ω_{c}. The sign of the coefficient P, multiplying the dispersive term in equation (3), depends on κ: When κ < 1/, P is positive and for κ < 1/ we see that P is negative.

3. Solitary Waves in the Small-Amplitude Limit

[11] In the small-amplitude limit, viz. N = 1 + N_{1}, u = u_{1}, where N_{1}, u_{1} ≪ 1, equations (1)–(3) yield

and

where the only nonlinearity kept is the ponderomotive force terms involving ∣ℰ∣^{2}. It is important to remember that our nonlinear Schrödinger equation for the whistler field is based on a Taylor expansion of the dispersion relation for whistler waves around a wavenumber k_{0}. Thus, this model is only accurate for wave envelopes moving with speeds close to the group speed V_{g}, and other speeds of the wave envelopes may give unphysical results. Here, we look for whistler envelope solitary waves moving with the group speed V_{g}, so that N_{1} and u_{1} depends only on χ = ξ − V_{g}τ, while the electric field envelope is assumed to be of the form ℰ = W(χ) exp (iΩτ − ikξ), where W is a real-valued function of one argument. Using the boundary conditions N_{1} = 0, u_{1} = 0 and W = 0 at ∣ξ∣ = ∞, we have k = 0, N_{1} = −W^{2}α/(1 − V_{g}^{2}) and u_{1} = V_{g}N_{1}. We here note that subsonic (V_{g} < 1) solitary waves are characterized by a density cavity while supersonic (V_{g} > 1) envelope solitary waves are characterized by a density hump. The system of equations (4)–(6) is then reduced to the cubic Schrödinger equation

where Q = αΩ_{c}κ^{2}/(1 + κ^{2}) (1 − V_{g}^{2}). Localized solutions of equation (7) only exist if the product PQ is positive. We note that P > 0 (P < 0) when the whistler frequency ω_{0} < ω_{ce}/4 (ω_{0} > ω_{ce}/4), and that Q < 0 (Q > 0) when ∣V_{g}∣ < 1 (∣V_{g}∣ > 1), so in the frequency band where ω_{0} < ω_{ce}/4, only subsonic solitary waves, characterized by a localized density cavity can exist, while in the frequency band ω_{0} > ω_{ce}/4, only supersonic solitary waves characterized by a localized density hump exist. Equation (7) has exact solitary wave solutions of the form W = (2Ω/Q)^{1/2}sech[(Ω/P)^{1/2}(ξ − V_{g}τ − ξ_{0})], where V_{g} and Ω and the displacement ξ_{0} are the three free parameters for a given set of physical plasma parameters. Finally, we recall that the dispersion relation for the electron whistlers used here is valid if ω_{0} > . For subsonic whistlers having the group speed v_{g} = C_{s}V_{g} (where V_{g} < 1), where v_{g} ≈ 2ω_{0}/k_{0} and ω_{0} ≈ k_{0}^{2}c^{2}ω_{ce}/ω_{pe,0}^{2}, this gives ck_{0}/ω_{pe,0} = (C_{s}/c)(ω_{pe,0}/ω_{ce})V_{g}/2 > (m_{e}/m_{i})^{1/4}.

4. Numerical Results

[12] We have investigated the properties of modulated whistlers wave packets by solving numerically equations (1)–(3). We have here chosen parameters from a recent experiment, where the formation of localized whistler envelopes have been observed [Kostrov et al., 2003]. In the experiment, one has n_{0} = 1.2 × 10^{12} cm^{−3} and B_{0} = 100 G, so that ω_{pe,0} = 6.7 × 10^{10} s^{−1} and ω_{ce} = 1.76 × 10^{9} s^{−1}, respectively. Hence, ω_{ce}/ω_{pe,0} = 0.026. The frequency of the whistler wave is ω_{0} = 2π × 160 × 10^{6} s^{−1} = 1.0 × 10^{9}s^{−1}, so that ω_{0}/ω_{ce} ≈ 0.57 > 0.25. Thus, the whistlers have negative group dispersion. From the dispersion relation of whistlers, we have κ ≈ 1.15, which gives k_{0} ≈ 257 m^{−1}. The latter corresponds to whistlers with a wavelength of 2.4 cm. Furthermore, the whistler group velocity is v_{g} = 3.36 × 10^{6} m/s. The argon ion-electron plasma (m_{i}/m_{e} = 73400) had the temperatures of T_{e} = 10 eV and T_{i} = 0.5 eV, giving the sound speed 5.25 × 10^{3} m/s, and the normalized group velocity V_{g} = v_{g}/C_{s} = 640. In Figure 1, we have illustrated the existence of localized whistler envelope solitons, in which the electric field envelope (left panels) is accompanied with a density hump (right panels). We notice that the density hump is relatively small, due to the large group velocity of the whistler waves. In Figure 2, we have presented the development of a large-amplitude whistler pulse, which was launched in a plasma perturbed by ion-acoustic waves, with a density modulation of one percent (see the caption of Figure 2). This simulates, to some extent, the experiment by Kostrov et al. [2003], where the density and magnetic field were perturbed by a low-frequency conical refraction wave, giving rise to a modulation of the electron whistlers. Here, as in the experiment, we observe that a modulated electron whistler pulse (middle panel of Figure 2) develops into isolated solitary electron whistlers (lower panel). We note that the wavelength of the whistlers is ≈2.5 cm, while the typical width of a solitary pulse is Δξ ≈ 3 × 10^{4} in the scaled length units, corresponding to ≈64 cm, so that each solitary wave train contains 25 wavelengths of the high-frequency whistlers. In one experiment, illustrated in the lower panel of Figure 4 in Kostrov et al. [2003], one finds that the width of the solitary whistler pulse in time is 0.2 μs, which with the group speed v_{g} = 3.36 × 10^{6} m/s gives the width ∼60 cm in space of the solitary wave packets, in good agreement with our numerical results. From the relation N_{1} = −W^{2}α/(1 − V_{g}^{2}) valid for solitary whistlers in the small-amplitude limit, and with the amplitude of W = ∣ℰ∣ approximately 0.3 seen in the lower panel of Figure 2, we can estimate the relative amplitude of the density hump associated with the solitary waves to be of the order 10^{−3}, i.e., much smaller than the modulation ∼10^{−2} due to the ion-acoustic waves excited in the initial condition.

[13] Next, we study the properties of subsonic whistler envelope solitary pulses which have the normalized group speed V_{g} = 0.5. Here, the restrictive condition ck_{0}/ω_{pe,0} = (C_{s}/c)(ω_{pe,0}/ω_{ce})V_{g}/2 > (m_{e}/m_{i})^{1/4} requires somewhat higher values of the plasma temperature and ω_{pe,0}/ω_{ce} for their existence. With m_{i}/m_{e} = 30000, we have (m_{e}/m_{e})^{1/4} ≈ 0.1. We take κ = ck_{0}/ω_{pe,0} = 0.2, C_{s} = 10^{5} m/s (corresponding to T_{e} ∼ 1400 eV) η = 0.1, and ω_{pe,0}/ω_{ce} = 2400. Thus, Ω_{c} = 0.072 and ω_{0}/ω_{ce} ≈ 0.039. For these values of the parameters, there exist solitary whistler pulse solutions, which we have displayed in Figure 3. Here, we have used the exact solution in the small-amplitude limit as an initial condition for the simulation of the full system of equations (1)–(3). The bell-shaped whistler electric field envelope is accompanied with a large-amplitude plasma density cavity.

5. Discussion

[14] In this paper, we have presented theoretical and simulation studies of nonlinearly interacting electron whistlers and arbitrary large amplitude ion-acoustic perturbations in a magnetized plasma. For this purpose, we have derived a set of equations which describe the spatio-temporal evolution of a modulated whistler packet. The ponderomotive force of the latter, in turn, modifies the local plasma density in a self-consistent manner. Numerical analyses of the governing nonlinear equations reveal that subsonic envelope whistler solitons are characterized by a bell- shaped whistler electric fields that are trapped in self-created density cavity. This happens when the whistler wave frequency is smaller than ω_{ce}/4, where the waves have positive group dispersion. When the whistler wave frequency is larger than ω_{ce}/4, one encounters negative group dispersive whistlers and the supersonic whistler envelope solitons are characterized by a bell-shaped whistler electric fields which create a density hump. Modulated whistler wavepackets have indeed been observed in a laboratory experiment [Kostrov et al., 2003] as well as near the plasmapause [Moullard et al., 2002] and in the auroral zone [Huang et al., 2004]. Our results are in excellent agreement with the experimental results [Kostrov et al., 2003], while we think that a multi-dimensional study, including channelling of whistler waves in density ducts, is required to interpret the observations by the Cluster and Freja satellites.

Acknowledgments

[15] This work was partially supported by the European Commission (Brussels, Belgium) through contract No. HPRN-CT-2001-00314, as well as by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 591.