The application of the constants of motion to nonlinear stationary waves in complex plasmas: a unified fluid dynamic viewpoint

Abstract

Perturbation reductive procedures, as used to analyse various weakly nonlinear plasma waves (solitons and periodic waves), normally lead to the dynamical system being described by KdV, Burgers' or a nonlinear Schrödinger-type equation, with properties that can be deduced from an array of mathematical techniques. Here we develop a fully nonlinear theory of one-dimensional stationary plasma waves, which elucidates the common nature of various diverse wave phenomena. This is accomplished by adopting an essentially fluid dynamic viewpoint. In this unified treatment the constants of the motion (for mass, momentum and energy) lead naturally to the construction of the wave structure equations. It is shown, for example, that electrostatic, Hall–magnetohydrodynamic and ion–cyclotron–acoustic nonlinear waves all obey first-order differential equations of the same generic type for the longitudinal flow field of the wave. The equilibrium points, which define the soliton amplitude, are given by the compressive and/or rarefactive roots of a total plasma ‘energy’ or ‘momentum’ function characterizing the wave type. This energy function, which is an algebraic combination of the Bernoulli momentum and energy functions for the longitudinal flow field, is the fluid dynamic counterpart of the pseudo-potentials, which are characteristic of system structure equations formulated in other than fluid variables. Another general feature of the structure equation is the phenomenon of choked flow, which occurs when the flow speed becomes sonic. It is this trans-sonic property that limits the soliton amplitudes and defines the critical collective Mach numbers of the waves. These features are also obtained in multi-component plasmas where, for example, in a bi-ion plasma, momentum exchanges between protons and heavier ions are mediated by the Maxwell magnetic stresses. With a suitable generalization of the concept of a sonic point in a bi-ion system and the corresponding choked flow feature, the wave structures, although now more complicated, can also be understood within this overall fluid framework. Particularly useful tools in this context are the momentum hodograph (an algebraic relation between the bi-ion speeds and the electron speed, or magnetic field, which follows from the conservation of mass, momentum and charge-neutrality) and a generalized Bernoulli energy density for each species. Analysis shows that the bi-ion solitons are essentially compressive, but contain the remarkable feature of the presence of a proton rarefactive core. A new type of soliton, called an ‘oscilliton’ because embedded spatial oscillations are superimposed on the classical soliton, is also described and discussed. A necessary condition for the existence of this type of wave is that the linear phase velocity must exhibit an extremum where the phase speed matches the group speed. The remarkable properties of this wave are illustrated for the case of both whistler waves and bi-ion waves where, for the latter, the requisite condition is met near the cross-over frequencies. In the case of the whistler oscilliton, which propagates at speeds in excess of one half of the Alfvén speed (based on the electrons), an analytic solution has been constructed through a phase-portrait integral of the system in which the proton and electron dynamics must be placed on the same footing. The relevance of the different wave structures to diverse space environments is briefly discussed in relation to recently available high-time and spatial resolution data from satellite observations.

Heliosheath Magnetic Field and Plasma Observed by Voyager 2 During 2012 in the Rising Phase of Solar Cycle 24

“…Magnetic holes and magnetic humps can become unstable, producing oscillations that resemble those observed by Fitzenreiter & Burlaga (1978) at 1 au and by Burlaga et al (2007a) in the heliosheath. This model is an extension of the soliton models published by Baumgärtel (1999), McKenzie et al (2001McKenzie et al ( , 2004 and Avinash & Zank (2007). An example of a magnetic hump in the heliosheath observed by V2 on day 250, 2012 is shown in Figure 14.…”

Whistler-Langmuir oscillitons and their relation to auroral hiss

“…(8)- (14) admits the following constants of motion (momentum flux conservation), (e.g. McKenzie et al, 2004McKenzie et al, , 2005.…”

Study of nonlinear ion- and electron-acoustic waves in multi-component space plasmas

“…Panels a, b and c show the critical Mach numbers for the slow ion-acoustic, ion-acoustic and the electron-acoustic modes, respectively. Further, in a fluid dynamic formalism, when inertia for all species is retained (as done here), the respective Mach numbers are limited by sonic points, where the flow of one species is choked (McKenzie et al, 2004;Verheest et al, 2004). The critical Mach numbers correspond to this situation.…”