Aims. We investigate the standing kink modes of a cylindrical model of coronal loops. The density is stratified along the loop axis and changes discontinuously at the surface of the cylinder. The periods and mode profiles are studies with their deviation from those of the unstratified loops. The aim is to extract information on the density scale heights prevailing in the solar corona. Methods. The problem is reduced to solving a single second-order partial differential equation for δB z (r, z), the longitudinal component of the Eulerian perturbations of the magnetic field. This equation, in turn, is separated into two second-order ordinary, differential equations in r and z that are, however, connected through a dispersion relation between the frequencies and the longitudinal wave numbers. In the thin tube approximation, the eigensolutions are obtained by a perturbation technique, where the perturbation parameter is the density stratification parameter. Otherwise the problem is solved numerically. Results. 1) On functional dependencies of the dispersion relation the radial wave number is independent of the longitudinal stratification. 2) We verify the earlier computational finding that the first overtone frequencies increase with increasing stratification and the observational finding (from analysis of TRACE data) that the ratio of the first to the fundamental overtone frequency decreases with increasing stratification. The method we use to arrive at these conclusions, however, is more analytical than computational, and yet our numerical results agree with the earlier results.3) The mode profiles depart from the sinusoidal mode profiles of the unstratified loops. This departure and its dependence on the scale height is obtained, and might serve to determine scale heights once high resolution data become available.

Allowing for virtual paths in phase space permits an extension of Hamilton’s principle of least action, of lagrangians and of hamiltonians to phase space. A subsequent canonical quantization, then, provides a framework for quantum statistical mechanics. The classical statistical mechanics and the conventional quantum mechanics emerge as special case of this formalism. Von Neumann’s density matrix may be inferred from it. Wigner’s functions and their evolution equation may also be obtained by a unitary transformation.

We carry out an exact analysis of the average frequency ν + αx i in the direction x i of positiveslope crossing of a given level α such that, h(x, t) −h = α, of growing surfaces in spatial dimension d.Here, h(x, t) is the surface height at time t, andh is its mean value. We analyze the problem when the surface growth dynamics is governed by the Kardar-Parisi-Zhang (KPZ) equation without surface tension, in the time regime prior to appearance of cusp singularities (sharp valleys), as well as in the random deposition (RD) model. The total number N + of such level-crossings with positive slope in all the directions is then shown to scale with time as t d/2 for both the KPZ equation and the RD model. PACS number(s): 52.75.Rx, 68.35.Ct.

Nonassociative structure of quantum mechanics in curved space-time Conventional approach to quantum mechanics in phase space, ͑q , p͒, is to take the operator based quantum mechanics of Schrödinger, or an equivalent, and assign a c-number function in phase space to it. We propose to begin with a higher level of abstraction, in which the independence and the symmetric role of q and p is maintained throughout, and at once arrive at phase space state functions. Upon reduction to the q-or p-space the proposed formalism gives the conventional quantum mechanics, however, with a definite rule for ordering of factors of noncommuting observables. Further conceptual and practical merits of the formalism are demonstrated throughout the text.

We study the resonant absorption of MHD waves in magnetized flux tubes with a radial density inhomogeneity. Within the approximation that resistive and viscous processes are operative in thin layers surrounding the singularities of the MHD equations, we give the full spectrum of the eigenfrequencies and damping rates of the MHD quasi modes of the tube. Both surface and body modes are analyzed.

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Abstract. Wave propagation in a zero-β magnetic flux tube with a discontinuous Alfvèn speed at its surface is considered. The problem is reduced to solving a wave equation for the projection of the magnetic perturbation along the axis of the cylinder. The mathematical formalism is identical with that for the propagation of electromagnetic waves in optical fibers with a varying index of refraction in the cross section of the fiber. The dispersion relation is solved in its full generality and three wave numbers are assigned to the normal modes of the cylinder. There is a lower cutoff for the longitudinal wave number along the cylinder axis and an upper cutoff for the radial wave number. Eigenfrequencies and eigenfields (i.e. the magnetic and velocity fields of modes) are calculated. Resistive and viscous dissipation rates have mathematically identical forms, differing only in their being inversely proportional to the Lundquist and Reynolds numbers, respectively. These rates as well as the energy densities are obtained for each mode and are commented on.

The standing quasi‐modes of the ideal magnetohydrodynamics (MHD) in a zero‐β cylindrical magnetic flux tube that undergoes a longitudinal density stratification and radial density structuring are considered. The radial structuring is assumed to be a linearly varying density profile. Using the relevant connection formulae of the resonant absorption, the dispersion relation for the fast MHD body waves is derived and solved numerically to obtain both the frequencies and damping rates of the fundamental and first‐overtone, k= 1, 2, modes of both the kink (m= 1) and fluting (m= 2) waves, where k and m are the longitudinal and azimuthal mode numbers, respectively.