Energy transformation between fast magnetosonic waves and Alfvén waves is considered. It is shown that the efficiency of this process can be significantly enhanced if the Alfvén wave propagation directions are appropriately chosen. The relevance of our investigation to the non-linear excitation of magnetohydrodynamic waves in space plasmas is discussed.

In 1989, it was proved that thunder is not caused by the thermal expansion of the lightning channel. The present paper suggests that it is the result of an air explosion driven by the liberation of chemical bond energy from the N2 and O2 molecules. This has been confirmed by laboratory experiments with air arcs of lightning strength. These arcs are short pieces of lightning strokes and their formation is accompanied by thunder-like sonic booms.

We investigate the conditions under which ‘high-frequency’ electron acoustic–Langmuir solitons can be constructed in a plasma consisting of protons and two electron populations: one ‘cold’ and the other ‘hot’. Conservation of total momentum can be cast as a structure equation either for the ‘cold’ or ‘hot’ electron flow speed in a stationary wave using the Bernoulli energy equations for each species. The linearized version of the governing equations gives the dispersion equation for the stationary waves of the system, from which follows the necessary – but not sufficient – conditions for the existence of soliton structures; namely that the wave speed must be less than the acoustic speed of the ‘hot’ electron component and greater than the low-frequency compound acoustic speed of the two electron populations. In this wave speed regime linear waves are ‘evanescent’, giving rise to the exponential growth or decay, which readily can give rise to non-linear effects that may balance dispersion and allow soliton formation. In general the ‘hot’ component must be more abundant than the ‘cold’ one and the wave is characterized by a compression of the ‘cold’ component and an expansion in the ‘hot’ component necessitating a potential dip. Both components are driven towards their sonic points; the ‘cold’ from above and the ‘hot’ from below. It is this transonic feature which limits the amplitude of the soliton. If the ‘hot’ component is not sufficiently abundant the window for soliton formation shrinks to a narrow speed regime which is quasi-transonic relative to the ‘hot’ electron acoustic speed, and it is shown that smooth solitons cannot be constructed. In the special case of a very cold electron population (i.e. ‘highly supersonic’) and the other population being very hot (i.e. ‘highly subsonic’) with adiabatic index 2, the structure equation simplifies and can be integrated in terms of elementary transcendental functions that provide the fully non-linear counterpart to the weakly non-linear sech$^{2}$-type solitons. In this case the limiting soliton is comprised of an infinite compression in the cold component, a weak rarefaction in the ‘hot’ electrons and a modest potential dip.

Lagrangian and Hamiltonian formalisms for ideal fluids and plasmas have, during the last few decades, developed much in theory and applications. The recent formulation of ideal fluid/plasma dynamics in terms of the Euler–Poincaré equations makes a self-contained, but mathematically elementary, form of Lagrange–Hamilton theory possible. The starting point is Hamilton's principle. The main goal is to present Lagrange–Hamilton theory in a way that simplifies its applications within usual fluid and plasma theory so that we can use standard vector analysis and standard Eulerian fluid variables. The formalisms of differential geometry, Lie group theory and dual spaces are avoided and so is the use of Lagrangian fluid variables or Clebsch potentials. In the formal ‘axiomatic’ setting of Lagrange–Hamilton theory the concepts of Lie algebra and Hilbert space are used, but only in an elementary way. The formalism is manifestly non-canonical, but the analogy with usual classical mechanics is striking. The Lie derivative is a most convenient tool when the abstract Lagrange–Hamilton formalism is applied to concrete fluid/plasma models. This directional/dynamical derivative is usually defined within differential geometry. However, following the goals of this paper, we choose to define Lie derivatives within standard vector analysis instead (in terms of the directional field and the div, grad and curl operators). Basic identities for the Lie derivatives, necessary for using them effectively in vector calculus and Lagrange–Hamilton theory, are included. Various dynamical invariants, valid for classes of fluid and plasma models (including both compressible and incompressible ideal magnetohydrodynamics), are given simple and straightforward derivations thanks to the Lie derivative calculus. We also consider non-canonical Poisson brackets and derive, in particular, an explicit result for incompressible and inhomogeneous flows.

The equations of ideal magnetohydrodynamics (MHD) form a non-strict hyperbolic system with a non-convex flux function and admit non-regular, so-called intermediate shocks. The presence of non-regular waves in the MHD system causes the Riemann problem to be not unique in some cases. This paper investigates the uniqueness of Riemann solutions of ideal MHD. To determine uniqueness conditions we discuss the correspondence of non-regular solutions and non-uniqueness. Additionally the structure of the Hugoniot curves and its non-regular behaviour are demonstrated. It follows that the degree of freedom for solving a Riemann problem is reduced in the case of a non-regular solution. From this, we can deduce uniqueness conditions depending on the initial conditions of an MHD Riemann problem. The results also allow one to construct non-unique solutions. We give an example for the case of non-planar initial conditions.