In this review we will focus on a topic of fundamental importance for both plasma physics and astrophysics, namely the occurrence of large-amplitude low-frequency fluctuations of the fields that describe the plasma state. This subject will be treated within the context of the expanding solar wind and the most meaningful advances in this research field will be reported emphasizing the results obtained in the past decade or so. As a matter of fact, Ulysses’ high latitude observations and new numerical approaches to the problem, based on the dynamics of complex systems, brought new important insights which helped to better understand how turbulent fluctuations behave in the solar wind. In particular, numerical simulations within the realm of magnetohydrodynamic (MHD) turbulence theory unraveled what kind of physical mechanisms are at the basis of turbulence generation and energy transfer across the spectral domain of the fluctuations. In other words, the advances reached in these past years in the investigation of solar wind turbulence now offer a rather complete picture of the phenomenological aspect of the problem to be tentatively presented in a rather organic way.

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The Solar Wind as a Turbulence Laboratory

A method of solution for the ’’derivative nonlinear Schrodinger equation’’ i q t =−q x x ±i (q*q 2) x is presented. The appropriate inverse scattering problem is solved, and the one‐soliton solution is obtained, as well as the infinity of conservation laws. Also, we note that this equation can also possess ’’algebraic solitons.’’

An exact solution for a derivative nonlinear Schrödinger equation

The investigation of nonlinear wave phenomena has been one of the main direc tions of research in optics for the last few decades. Soliton concepts applied to the description of intense electromagnetic beams and ultrashort pulse propagation in various media have contributed much to this field. The notion of solitons has proved to be very useful in describing wave processes in hydrodynamics, plasma physics and condensed matter physics. Moreover, it is also of great importance in optics for ultrafast information transmission and storage, radiation propagation in resonant media, etc. In 1973, Hasegawa and Tappert made a significant contribution to optical soliton physics when they predicted the existence of an envelope soliton in the regime of short pulses in optical fibres. In 1980, Mollenauer et al. conducted ex periments to elucidate this phenomenon. Since then the theory of optical solitons as well as their experimental investigation has progressed rapidly. The effects of inhomogeneities of the medium and energy pumping on optical solitons, the interaction between optical solitons and their generation in fibres, etc. have all been investigated and reported. Logical devices using optical solitons have been developed; new types of optical solitons in media with Kerr-type nonlinearity and in resonant media have been described.

Optical Solitons

I review the regular multiannual population fluctuations in voles and lemmings of northern latitudes. Periodically fluctuating small rodent populations all exhibit a clear two-dimensional density-dependent structure. This implies both a direct and a delayed annual density dependence, and suggests that either a predator-prey type of interaction or a specialised plant-herbivore type of interaction (but not both) may be the underlying cause of these multiannual density cycles. Clear-cut experimental testing relating to these propositions is, however, lacking. A two-dimensional annual density-dependent structure is typically non-linear in a way which may be modelled as a threshold type of non-linearity and interpreted as a phase dependence in the density-dependent structure (implying that the density-dependent structure is different in the increase and the decrease phase). Two clear geographic gradients in the annual density-dependent structure are reviewed: the Fennoscandian gradient and the Hokkaidian gradient. The seasonal nature of the annual density-dependent structure is furthermore reviewed: for voles in both Fennoscandia and Hokkaido the direct annual density dependence during the winter (measured per time unit) is concluded to be the strongest. I close with a survey of the main challenges within the field of small rodent cycles' - the greatest of which is suggested to be the integration of demography and population dynamics. I recommend to look at other populations for potentially applicable model systems. I conclude that multiannual population cycles seen in voles and lemmings will continue to be a strong source of conceptual and methodological developments within the field of population ecology.

Population cycles in voles and lemmings : density dependence and phase dependence in a stochastic world

By means of a numerical simulation, nonlinear evolution of large amplitude dispersive Alfven waves is studied. An energy transfer from the parent wave to two daughter Alfven-like waves and a soundlike wave is observed (a stimulated Brillouin scattering process). The observed growth rates and propagation characteristics of these daughter waves agree with the analytical results, which we obtain by extending the previous treatments by Goldstein, Derby, Sakai, and Sonnerup. Ions are first trapped by the electrostatic potential of the daughter soundlike waves. Along with the eventual decay (ion Landau damping) of the soundlike waves, ions are phase-mixed and left heated in the parallel direction. The increased parallel energy of ions is transferred to the perpendicular thermal energy through the nonresonant scattering process in the colliding Alfven waves (parent and daughter waves). We further observe that the daughter Alfven waves, which still have a large amplitude, are also unstable for further decay, and that the wave energy is continuously transferred to the longer wavelength regime (inverse cascading process).

Decay instability of finite-amplitude circularly polarized Alfven waves - A numerical simulation of stimulated Brillouin scattering

The stability of circularly polarized waves of finite amplitude propagating parallel to the magnetic field is studied. A set of equations for slowly varying waves of arbitrary amplitude is obtained. A discussion of the stability of the waces is based on this set of equations. Earlier results are confirmed; in addition we find that finite amplitude always promotes stability. An amplitude dependent stability condition for long waves, previously obtained by the author, is confirmed.

On the modulational instability of hydromagnetic waves parallel to the magnetic field

The transmission and reflexion coefficients are calculated for the coupling of an ordinary polarized wave to the Z mode when the angle of incidence is near the critical angle.

Coupling to Z mode near critical angle

L'equation de Schrodinger non lineaire derivee pour les ondes MHD non lineaires paralleles et dispersives est etendue aux valeurs de beta fini ainsi qu'aux trois dimensions spatiales, au moyen d'une methode de perturbation reductible. Les effets cinetiques sont inclus au moyen du modele de centre guide cinetique et fluide hybride de Grad (1961). L'equation resultante contient un terme non lineaire et non local representant l'effet des particules resonnantes

nonlinear Alfvén waves in a finite-beta plasma

The DNLS equation describes weakly dispersive MHD waves propagating parallel as well as at a small angle to the magnetic field. For oblique propagation, the magnetosonic mode is described by the KdV equation, whereas certain aspects of the oblique Alfven wave are described by the MKdV equation. The two latter equations possess one-parameter families of solitons. One-parameter families of oblique soliton solutions to the DNLS equation exist, which match to those of the KdV and MKdV in appropriate asymptotic limits. For parallel propagation, the DNLS equation possesses a well-known two-parameter family of soliton solutions. Such a family also exists in the oblique case (Kawata and Inoue, J. Phys. Soc. Japan 44, 1968 (1978)), but has so far been paid little attention. In the parallel case, soliton formation is known to result from modulational instability of circularly polarized wave trains. In the oblique case, a rich family of periodic wave train solutions exists which matches to the cnoidal solutions to MKdV and KdV in the appropriate limits. The modulational stability of these wave trains is studied by Whitman's method. A conjecture analogous to a result of Driscoll and O'Neil (Phys. Fluids 17, 1196 (1976)) is supported numerically.

https://iopscience.iop.org/article/10.1088/0031-8949/40/2/013/pdf

Nonlinear Alfven Waves and the DNLS Equation: Oblique Aspects

An amplitude dependent criterion for modulational stability of long Alfven waves parallel to the magnetic field is interpreted in terms of a recently obtained inverse scattering solution to the modified nonlinear Schrodinger equation. It is found that the solitons formed are of two types. In the strongly unstable case, normal solitons are formed. In the transition region of weakly unstable and stable cases, the anomalous type, which in a limiting case becomes the algebraic soliton, dominates. In the strongly stable case, no solitons are formed.

Einar Mjølhus

Papers

On the modulational instability of hydromagnetic waves parallel to the magnetic field

Coupling to Z mode near critical angle

nonlinear Alfvén waves in a finite-beta plasma

Nonlinear Alfven Waves and the DNLS Equation: Oblique Aspects

A note on the modulational instability of long Alfvén waves parallel to the magnetic field