We study the normal and leaky kink oscillations of a thin magnetic tube embedded in a surrounding magnetized plasma. We adopt the cold plasma (zero plasma $\beta$) approximation, appropriate for typical solar coronal conditions. Using the Laplace transform, we solve the initial value problem to determine the motions of the flux tube. The asymptotic description of this motion is determined by the properties of a dispersion function $D(\omega)$, an infinitely valued function of the complex frequency $\omega$. Two cases arise. When the plasma density $\rho_{\rm i}$ inside the tube exceeds the density $\rho_{\rm e}$ of the surrounding plasma, the asymptotic behaviour is given by the normal mode that is associated with two zeros of $D(\omega)$. These zeros are real and symmetric with respect to the imaginary axis, and are situated on the principal sheet of the Riemann surface of $D(\omega)$. The asymptotics given by the normal mode is valid for times much longer than the period of tube oscillation, and is homogeneous with respect to the radial distance $r$. In the second case, where $\rho_{\rm i} < \rho_{\rm e}$, there are no zeros of $D(\omega)$ on the principal sheet. The asymptotics of the solution of the initial value problem is then given by a leaky mode of the tube kink oscillations. This mode is associated with two zeros of $D(\omega)$ on two different, non-principal, sheets of the Riemann surface of $D(\omega)$ attached to the principal sheet. The asymptotics given by the leaky mode is intermediate, i.e. valid for times much longer than the period of oscillations but smaller than (or of the order of) the damping time due to energy leakage. The asymptotics given by the leaky mode is inhomogeneous with respect to $r$. We also consider other solutions associated with the zeros of $D(\omega)$ on non-principal sheets, and argue that in the coronal loop case where $\rho_{\rm i} > \rho_{\rm e}$ these solutions have no physical meaning. In the opposite case, where $\rho_{\rm i} < \rho_{\rm e}$, only the two solutions associated with the two zeros of $D(\omega)$ lying on two different non-principal sheets attached to the principal sheet are physically meaningful; they determine the leaky mode. All other solutions of this type are devoid of physical meaning. On the basis of our results, we conclude that there is no radial wave energy leakage from a dense coronal loop oscillating in the kink mode; other mechanisms must be sought to explain the observed oscillation damping.

Until recently, the calculation of field-line resonance (FLR) frequencies from magnetic field data, generated by magnetospheric models such as BATSRUS and Tsyganenko (T01), was restricted to orthogonal coordinate systems. With this restriction, only dipolar and axisymmetric configurations are admissible. The matter of addressing more general configurations such as non-axisymmetric stretched and twisted magnetic field topologies requires the use of a non-orthogonal coordinate system. This coordinate system can be constrained by defining the magnetic field as the product of Euler potentials, ${\bf B} = \nabla \alpha_i \times \nabla \alpha _j$, and imposing the condition, $\nabla \cdot {\bf B} = 0$, everywhere. As a consequence, the coordinates, $\alpha_i$, must satisfy the partial differential equation, ${\bf B}\cdot \nabla \alpha_i = 0$. In other words, $\alpha_i$ must be constant along magnetic field lines. Upon solving this differential equation implicitly using known magnetic field intensities, the metric tensor for the resulting basis can be computed. The elements of this tensor can be substituted directly into the eigenvalue problem for general coordinate systems written in covariant notation. The equation for FLR modes has been developed for arbitrary incompressible magnetospheric conditions and has been specialized to the case where spatial variations are constrained along the magnetic field. The result is a fourth-order system of ordinary differential equations, which can be evaluated numerically, provided that the variation of the metric coefficients as a function of the distance along field lines is sufficiently smooth. The eigenvalue problem is solved at several latitudes for a broad range of magnetospheric conditions.

In the parameter regime where the ion/electron–dust collisions dominate with respect to the plasma binary collisions, the presence of plasma fluxes to the dust surfaces and dust charge fluctuations can significantly modify the dusty plasma response to low-frequency perturbations, with respect to the multi-component response (no charging collisions, fixed dust charge). Here the response found from the kinetic theory which consistently takes into account collisions and charge fluctuations is examined, analytic expressions are given and the parameter regime where significant differences with respect to the multi-component approach arise is determined. The streaming instability is considered as a particular example and it is shown that both the real and imaginary parts of the frequency can be significantly changed with respect to the multi-component results.

An analysis of relativistic electron trajectories in a free-electron laser with a helical magnetic wiggler and an ion channel is presented. The wiggler field amplitude and the ion number density are taken to be uniform. Also included are the self-electric and self-magnetic fields of the electron beam, which is assumed to be of constant velocity and electron number density. The Hamiltonian, which is a constant of the motion, is first expressed in cartesian coordinates and momenta. A second constant of the motion is obtained by canonical transformation. The steady-state orbits, Poincaré maps, and Liapunov exponents are employed to investigate the chaotic motion in the presence of the ion channel. Numerical calculations reveal conditions under which chaotic and non-chaotic orbits exist.

The two-dimensional dynamics of nonlinear ion-acoustic waves in a weakly magnetized plasma comprising cold ions and trapped as well as free electrons is considered. It is shown that owing to departure from the Boltzmann electron distribution to a vortex-like one, the dynamics of small but finite-amplitude ion-acoustic waves is governed by a new nonlinear equation which is valid for both unmagnetized and magnetized plasmas. For exactly vanishing magnetic fields the modified Kadomtsev–Petviashvili (mKP) equation is recovered. For weak magnetic fields, however, the dynamics is mainly different from the mKP equation, depending on the amplitude. By increasing the magnetic field, the new equation becomes similar (but not identical) to the modified Zakharov–Kuznetsov (mZK) equation, which is fulfilled for very strong magnetic fields. The plane periodic and solitary wave solutions of this equation are obtained using the appropriate scalings.

We present a three-dimensional analysis covering an investigation of the behavior of the magnetosheath plasma velocity and the magnetic field during the transfer from the magnetosheath to the magnetosphere. Magnetic reconnection is assumed to occur at an arbitrary line parallel to the $y$-axis, located in a region along the magnetopause stretching from the sub-solar point to the north. We do not focus on the process itself, but on its implications for the plasma flow and the magnetic field outside the diffusion region. Viscosity and resistivity are included in the magnetohydrodynamic equation of motion as non-ideal effects. The governing equations are solved approximately using the method of asymptotic expansions, with expansion in orders of large Reynolds and Lundquist numbers. We present two classes of solutions describing the properties north and south of the reconnection line, respectively. Results are presented for two locations of the reconnection line at distances of two and seven Earth radii from the sub-solar point correspondingly. It is shown that the velocity increases faster north of the reconnection line, and that the effect intensifies as the location of the reconnection line moves further away from the sub-solar point. It is also shown for this case, that there is less variation of the magnetic field and an increase in the field strength.

The spatial structure of the magnetohydrodynamic shear Alfvén and cusp continuum modes in axisymmetric toroidal geometry is analyzed for an incompressible plasma. The normal component $\xi_{\psi}$ of the plasma displacement is found to have an oscillatory type of singularity, $\xi_{\psi} \sim (\psi - \psi_0)^{\delta} \sim \sin(|\delta| \ln|\psi-\psi_0| + \mbox{const})$, or, for example, in the case of up/down symmetry, a purely logarithmic behavior. This result was also obtained in the more general case of a compressible plasma (Salat, A. and Tataronis, J. A. 1999 Phys. Plasmas6, 3207). However, in that work the incompressible limit could not be taken because the continuum equations were written in terms of a variable that becomes ill-defined in the limit. The present ab initio derivation shows that, in general, the incompressible limit in the previous work, $\gamma \rightarrow\infty$, where $\gamma$ is the ratio of the specific heats, does give the correct spatial dependence. Furthermore, we prove that it is possible to recover the oscillatory type of singularity for the cylindrical screw pinch from the logarithmic singularity of an up/down symmetric torus.

The nonlinear coupling between the Alfvén–Rao (AR) and dust–Alfvén (DA) modes in a uniform magnetized dusty plasma is considered. For this purpose, multi-fluid equations (composed of the continuity and momentum equations), the laws of Faraday and Ampère and the quasi-neutrality condition are adopted to derive a set of equations, which show how the fields of the modes are nonlinearly coupled. The equations are then used to investigate decay and modulational instabilities in magnetized dusty plasmas. Stationary nonlinear solutions of the coupled AR and DA equations are presented. The relevance of the investigation to nonlinear phenomena (instabilities and localized structures) in interstellar molecular clouds is also discussed.

A rectangular periodically corrugated waveguide is considered in the exponential gain regime. In general, gain obeys a nonlinear cubic equation. In this paper it is solved numerically. Also the behavior of gain diagrams versus beam velocities for various parameters of the problem is surveyed. Furthermore, in the context of a slow-wave periodic structure oscillator, the threshold current which cause the system to be self-sustained, is obtained from a nonlinear cubic equation similar to the one obtained for gain, and the variation of its diagrams with other parameters is also considered.

In the interstellar medium (ISM), a broad range of turbulent plasmas can be observed by using radio waves. However, only density fluctuations are directly measurable with these radio astronomic tools. If one wants to deduce the underlying features of the turbulence, it is necessary to calculate the relation between density and magnetic fluctuations. Here a magnetohydrodynamic (MHD)-ansatz is taken into account for the ISM turbulence to calculate the fluctuations for the three low-frequency MHD waves.