J. F. McKenzie

The ion-acoustic soliton: A gas-dynamic viewpoint

The properties of fully nonlinear ion-acoustic solitons are investigated by interpreting conservation of total momentum as the structure equation for the proton flow in the wave. In most studies momentum conservation is regarded as the first integral of the Poisson equation for the electric potential and is interpreted as being analogous to a particle moving in a pseudo-potential well. By adopting an essentially gas-dynamic viewpoint, which emphasizes momentum conservation and the properties of the Bernoulli-type energy equations, the crucial role played by the proton sonic point becomes apparent. The relationship (implied by energy conservation) between the electron and proton speeds in the transition yields a locus—the hodograph of the system–which shows that, in the first half of the soliton, the electrons initially lag behind the protons until the charge neutral point is reached, after which they run ahead of the protons. The system reaches an equilibrium point (the center of the soliton) before the p...

Nonlinear stationary whistler waves and whistler solitons (oscillitons): Exact solutions

A fully nonlinear theory for stationary whistler waves propagating parallel to the ambient magnetic field in a cold plasma has been developed. It is shown that in the wave frame proton dynamics must be included in a self-consistent manner. The complete system of nonlinear equations can be reduced to two coupled differential equations for the transverse electron or proton speed and its phase, and these possess a phase-portrait integral which provides the main features of the dynamics of the system. Exact analytical solutions are found in the approximation of ‘small’ (but nonlinear) amplitudes. A soliton-type solution with a core filled by smaller-scale oscillations (called ‘oscillitons’) is found. The dependence of the soliton amplitude on the Alfven Mach number, and the critical soliton strength above which smooth soliton solutions cannot be constructed is also found. Another interesting class of solutions consisting of a sequence of wave packets exists and is invoked to explain observations of coherent wave emissions (e.g. ‘lion roars’) in space plasmas. Oscillitons and periodic wave packets propagating obliquely to the magnetic field also exist although in this case the system becomes much more complicated, being described by four coupled differential equations for the amplitudes and phases of the transverse motion of the electrons and protons.

The properties of fast and slow oblique solitons in a magnetized plasma

This work builds on a recent treatment by McKenzie and Doyle [Phys. Plasmas8, 4367 (2001)], on oblique solitons in a cold magnetized plasma, to include the effects of plasma thermal pressure. Conservation of total momentum in the direction of wave propagation immediately shows that if the flow is supersonic, compressive (rarefactive) changes in the magnetic pressure induce decelerations (accelerations) in the flow speed, whereas if the flow is subsonic, compressive (rarefactive) changes in the magnetic pressure induce accelerations (decelerations) in the flow speed. Such behavior is characteristic of a Bernoulli-typeplasma momentum flux which exhibits a minimum at the plasma sonic point. The plasma energy flux (kinetic plus enthalpy) also shows similar Bernoulli-type behavior. This transonic effect is manifest in the spatial structure equation for the flow speed (in the direction of propagation) which shows that soliton structures may exist if the wave speed lies either (i) in the range between the fast and Alfven speeds or (ii) between the sound and slow mode speed. These conditions follow from the requirement that a defined, characteristic “soliton parameter” m exceeds unity. It is in this latter slow soliton regime that the effects of plasma pressure are most keenly felt. The equilibrium points of the structure equation define the center of the wave. The structure of both fast and slow solitons is elucidated through the properties of the energy integral function of the structure equation. In particular, the slow soliton, which owes its existence to plasma pressure, may have either a compressive or rarefactive nature, and exhibits a rich structure, which is revealed through the spatial structure of the longitudinal speed and its corresponding transverse velocity hodograph.

Compressive and rarefactive ion-acoustic solitons in bi-ion plasmas

Nonlinear propagation of ion-acoustic solitary structures in plasmas with an admixture of heavy ions is studied in the wave frame, where they are stationary, using a recently developed gas-dynamic approach, as an alternative to the conventional Sagdeev pseudopotential method. This viewpoint brings out the gas-dynamic aspects, which then allow a characterization of the solitary wave structures in terms of the species’ sonic points, the global charge neutral points, and critical collective Mach numbers. It is shown that the concepts of a critical density in the Korteweg–de Vries (KdV) treatment, and of a changeover from compressive to rarefactive soliton character, correspond to the formation of a second charge neutral point (outside equilibrium) in the rarefactive regime, at which the electric stresses maximize. It is possible therefore that in certain regions of parameter space compressive and rarefactive solitons can co-exist. The compressive solitons are not predicted by a weakly nonlinear KdV treatment...

Oblique solitons in a cold magnetized plasma

A fully nonlinear theory for stationary waves, propagating obliquely to the ambient magnetic field in a cold plasma, has been developed. Soliton solutions, representing both compressions and rarefactions in the magnetic field, exist for sub-fast flow conditions and in certain cones of magnetic obliquity. The soliton is explicitly characterized, in terms of the wave speed and its obliquity, by a parameter m (the “soliton number”). Compressive (“bright”) solitons are found to have a maximum attainable compression amplitude of three, corresponding to the condition m=1. Rarefactive (“dark”) solitons attain complete rarefaction when m=4. The properties of these stationary waves are described both in terms of magnetic hodographs, and of a spatial structure equation, whose equilibrium points yield the maximum compression and rarefaction at the center of the waves. An analytic solution, in terms of elementary transcendental functions, is also presented and highlights the role played by the soliton number m in det...

The fluid-dynamic paradigm of the dust-acoustic soliton

In most studies, the properties of dust-acoustic solitons are derived from the first integral of the Poisson equation, in which the shape of the pseudopotential determines both the conditions in which a soliton may exist and its amplitude. Here this first integral is interpreted as conservation of total momentum, which, along with the Bernoulli-like energy equations for each species, may be cast as the structure equation for the dust (or heavy-ion) speed in the wave. In this fluid-dynamic picture, the significance of the sonic points of each species becomes apparent. In the wave, the heavy-ion (or dust) flow speed is supersonic (relative to its sound speed), whereas the protons and electrons are subsonic (relative to their sound speeds), and the dust flow is driven towards its sonic point. It is this last feature that limits the strength (amplitude) of the wave, since the equilibrium point (the centre of the wave) must be reached before the dust speed becomes sonic. The wave is characterized by a compression in the heavies and a compression (rarefaction) in the electrons and a rarefaction (compression) in the protons if the heavies have positive (negative) charge, and the corresponding potential is a hump (dip). These features are elucidated by an exact analytical soliton, in a special case, which provides the fully nonlinear counterpoint to the weakly nonlinear sech 2 -type solitons associated with the Korteweg–de Vries equation, and indicates the parameter regimes in which solitons may exist.

Solitons and oscillitons in cold bi-ion plasmas: a parameter study

We investigate the structure of nonlinear stationary waves propagating obliquely to the magnetic field in a cold bi-ion plasma. By using the constants of motion that follow from the multi fluid equations, the system may be described by four coupled first-order differential equations. A new constant of motion characterizing a bi-ion flow (called the ‘energy difference integral’) is found. The combination of relations between the flow speeds derived from the conservation laws, which we call the ‘momentum–energy hodographs’, reveal some important features of stationary waves and solitons. Soliton solutions representing both compressions and rarefactions in the ion fluids exist in specific windows in the ‘Alfven Mach number–obliquity’ space. In other windows, solutions characterized by both oscillating and soliton properties (‘oscillitons’) exist. Critical Mach numbers and propagation angles narrow the size of the windows where smooth soliton solutions can be constructed.

Stationary waves in a bi-ion plasma transverse to the magnetic field

We investigate the nature of stationary structures streaming at subfast magnetosonic speeds perpendicular to the magnetic field in a bi-ion plasma consisting of protons and a heavy ion species in which the magnetic field is frozen into the electrons, whose inertia may be neglected. The study is based on the properties of the structure equation for the system, which is derived from the equations of motion and the Maxwell equations, and therefore reflects the coupling between the two ion fluids and the electrons through the Lorentz forces and charge neutrality. The basic features of the structure equation are elucidated by making use of conservation of total momentum and charge neutrality, which provide relations between the ion speeds in the unperturbed flow direction and the electron speed. This combination of relations, which we call the momentum hodograph of the system, reveals the structure of the flow and the magnetic field in a solitary-type pulse. In particular, we find that in the initial portion of a compressive soliton, heavy ions run ahead of the electrons and the protons lag between them until a point is reached where they all once more attain the same speed, after which the protons run ahead and are accelerated whereas the heavies now lag behind the continuously decelerating electrons. The second half of the wave is a mirror image of the first portion. The strength of the compression (the amplitude of the wave) is determined from the momentum hodograph, and depends upon the initial Mach number, abundance ratio of heavies to protons and the mass ratio. The analysis is relevant to subfast flows of mass-loaded plasmas and pile-up boundaries, which appear near comets and non-magnetic planets.

Nonlinear waves propagating transverse to the magnetic field

We generalize the classical work of Adlam and Allen [ Phil. Mag. 3 , 448 (1958)] on solitons in a cold plasma propagating perpendicular to the magnetic field to include the effects of plasma pressure. This is done by making extensive use of the properties of total momentum conservation (denoted by the term ‘momentum hodograph’, since it yields a locus in the plane of the electron and proton speeds in the direction of the wave) and the energy integral of the system as a whole. These relations elucidate the phase and integral curves of stationary flows, from which soliton solutions may be constructed. In general, only compressive solitons are permitted, and we have found an analytical expression for the critical fast Mach number as a function of the proton acoustic Mach number, which shows that it varies from its classical value of 2 (at large proton acoustic Mach numbers) to unity, where the incoming flow is proton-sonic. At the critical fast Mach number, two possible soliton-like solutions can be constructed. One is the classical compression, in which the magnetic field develops a cusp in the centre of the wave. The other is a compression in the magnetic field followed by a deep depression in the centre of the wave, which is completed by the mirror image of this signature of compression–rarefaction. This structure involves a smooth supersonic–subsonic transition in the proton flow. For Mach numbers in excess of the critical one, this kind of structure can also be constructed, but now the magnetic field is cusp-like at the points of maximum compression.

Hamiltonian Approach to Nonlinear Travelling Whistler Waves

A Hamiltonian formulation of nonlinear, parallel propagating, travelling whistler waves is discussed. The model is based on the equations of two‐fluid electron‐proton plasmas. In the cold gas limit, the complete system of equations reduces to two coupled differential equations for the transverse electron speed u and a phase variable φ = φp − φe representing the difference in the phases of the transverse complex velocities of the protons and the electrons. Two integrals of the equations are obtained. The Hamiltonian integral H, is used to classify the trajectories in the (φ, w) phase plane, where φ and w = u2 are the canonical coordinates. Periodic, oscilliton solitary wave and compacton solutions are obtained, depending on the value of the Hamiltonian integral H and the Alfven Mach number M of the travelling wave. The individual electron and proton phase variables φe and φp are determined in terms of φ and w. An alternative Hamiltonian formulation in which φ = φp + φe is the new independent variable repl...