Dynamics of the 3D FMS Soliton-like Beam Structures Propagating in Plasma of Ionosphere and Magnetosphere

Vasily Yu. Belashov*, Elena S. Belashova* and Sergey V. Vladimirov**
*Kazan State Power Engineering University, Kazan, Russia
**School of Physics, University of Sydney, Sydney, Australia
Abstract. The dynamics of the 3D fast magnetosonic soliton-like beam structures propagating in plasma of ionosphere
and magnetosphere near to cone of close to transverse angles between the wavevector and magnetic field is considered.
Keywords: Fast magnetosonic waves, Wave beam, Nonlinear waves, Plasma, Ionosphere, Magnetosphere, Numerical
simulation, generalized KP equation
The plasma is quasi-neutral in such movements, for ω<< ω0i where ω0i is the ion Lengmure frequency. Following
from (2), the dispersion is positive (phase velocity increases with k) except in the immediate neighbourhood of the
angles θ= 0 and θ= π / 2 . For near-to-transverse to field B propagation when π / 2 − θ < (β/ 4)1/ 2 the dispersion is
negative and is defined by the effects associated with the finiteness of the ion Larmor radius, ρ= vTi / ωB . As it is
known, to describe the small-amplitude FMS waves having a close angular distribution the KP equation in its
standard form is used [3]. With this equation, for the angles where the dispersion for small k is positive (for FMS
waves this requires a rather high ion temperature), the 3D wave packet of FMS waves with β> me / mi does not
form stable stationary solutions and scatters at π / 2 − θ ≤ (me / mi )1/ 2 or collapses outside this cone [1]. In the
latter case when a rather intense FMS wave beam is limited in the k⊥ direction one can observe the self-focusing
phenomenon. In ref. [3] the problem of the self-focusing of the FMS wave beam for the KP equation model is solved
by averaging the initial equations and subsequent numerical solution of the obtained ones∗). However, eq. (2) is
invalid for angles ( )1/ 2
θ < kc /ω0i where intense rebuilding of the oscillations dispersion mechanism takes place.
For β< me / mi the dispersion is defined for any angles θ from hydrodynamic equations and the FMS wave structure
depends on the sign of the dispersion coefficient

which is defined by the value of the angle θ, namely: for the near-to-transverse propagation the dispersion is
negative at π / 2 − θ ≤ (me / mi )1/ 2 , and it is positive outside this cone. In this case, the KP equation is also correct
[1] and for a rather intense FMS waves beam limited in the k⊥ direction one can also expect the self-focusing
phenomenon with propagation at angles θ where the dispersion is positive.
As an example, let us consider the results obtained in [1] for β< me / mi when for description of the small-amplitude
FMS waves having a close angular distribution the KP equation,

is valid, where h = B~ / B is the dimensionless FMS wave amplitude, B~ is the wave’s magnetic field; the
coefficient at nonlinear term is defined with help of the results obtained in [1].
As contrasted to usual GKP equation [1, 2], the damping is included into equation (3), at this the image of
operator νˆ is a logarithmic decrement ν(k ) of the monochromatic wave, stipulated by both collision and
collisionless mechanisms: ν = ν col + νL where ν L describes the collisionless damping associated in main with
For both cases, β> me / mi and β< me / mi , it is necessary to take into account that near the cone where the
dispersion changes its sign the dispersion coefficient γ1 →0 . That does not necessarily mean the disappearance of
dispersion in the medium and consequent invalidity of the KP equation in its standard form. Therefore, near the cone of
π / 2 − θ ≤ (β/ 4)1/ 2 , where γ1 →0 for β> me / mi , the results obtained in ref. [3] need a more exact definition;
relation (1) must be supplemented the next-order dispersion term which then plays a major role [1, 2]. A similar
situation occurs for β< me / mi near the cone of θ= arctan (mi / me )1/ 2 . At this, the next order dispersive term can
be obtained by keeping the next term in the Taylor’s expansion of the full dispersive relation as k, and it has the form
γ2kx , where in the latter case, β< me / mi , that will be covered below, according the results obtained in [1], the
coefficient is defined by expression

Thus, the character of the dispersion becomes more complicated and is defined by correlation of the signs of γ1 and
γ 2 (see fig. 1). Thus, for γ1 > 0, γ 2 < 0 negative dispersion takes place (region B in fig. 1), and for γ1 > 0 ,
γ 2 > 0 (region A) and γ1 < 0,γ2 < 0 (region C), there is “mixed” dispersion (when the dispersion sign is different
for small and big k). Then the propagation of small-amplitude FMS waves having a close angular distribution is
described by the KP equation generalization − the equation obtained by Belashov and Karpman (see [2]), which, for
the non-dissipative case, has the following form
∗) The angle θ= π / 2 (i.e. B|| k ⊥ ) is indicated wrongly in ref. [3] because in this case the

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