Multigrid method for nearly singular and slightly indefinite problems

This paper deals with nearly singular, possibly indefinite problems for which the usual multigrid solvers converge very slowly or even diverge. The main difficulty is related to some badly approximated smooth functions which correspond to eigenfunctions with nearly zero eigenvalues. A correction to the usual coarse-grid equations is derived, both in the correction scheme and in the full approximation scheme. The performance of the new algorithm using this correction is essentially as that of usual multigrid for definite problems

Multigrid solutions to quasi-elliptic schemes

Quasi-elliptic schemes arise from central differencing or finite element discretization of elliptic systems with odd order derivatives on non-staggered grids. They are somewhat unstable and less accurate then corresponding staggered-grid schemes. When usual multigrid solvers are applied to them, the asymptotic algebraic convergence is necessarily slow. Nevertheless, it is shown by mode analyses and numerical experiments that the usual FMG algorithm is very efficient in solving quasi-elliptic equations to the level of truncation errors. Also, a new type of multigrid algorithm is presented, mode analyzed and tested, for which even the asymptotic algebraic convergence is fast. The essence of that algorithm is applicable to other kinds of problems, including highly indefinite ones

Nonstationary westward translation of nonlinear frontal warm-core eddies

For the first time, an analytical theory and a very high-resolution, frontal numerical model, both based on the unsteady, nonlinear, reduced-gravity shallow water equations on a beta plane, have been used to investigate aspects of the migration of homogeneous surface, frontal warm-core eddies on a beta plane. Under the assumption that, initially, such vortices are surface circular anticyclones of paraboloidal shape and having both radial and azimuthal velocities that are linearly dependent on the radial coordinate (i.e., circular pulsons of the first order), approximate analytical expressions are found that describe the nonstationary trajectories of their centers of mass for an initial stage as well as for a mature stage of their westward migration. In particular, near-inertial oscillations are evident in the initial migration stage, whose amplitude linearly increases with time, as a result of the unbalanced vortex initial state on a beta plane. Such an initial amplification of the vortex oscillations is actually found in the first stage of the evolution of warm-core frontal eddies simulated numerically by means of a frontal numerical model initialized using the shape and velocity fields of circular pulsons of the first order. In the numerical simulations, this stage is followed by an adjusted, complex nonstationary state characterized by a noticeable asymmetry in the meridional component of the vortex's horizontal pressure gradient, which develops to compensate for the variations of the Coriolis parameter with latitude. Accordingly, the location of the simulated vortex's maximum depth is always found poleward of the location of the simulated vortex's center of mass. Moreover, during the adjusted stage, near-inertial oscillations emerge that largely deviate from the exactly inertial ones characterizing analytical circular pulsons: a superinertial and a subinertial oscillation in fact appear, and their frequency difference is found to be an increasing function of latitude. A comparison between vortex westward drifts simulated numerically at different latitudes for different vortex radii and pulsation strengths and the corresponding drifts obtained using existing formulas shows that, initially, the simulated vortex drifts correspond to the fastest predicted ones in many realistic cases. As time elapses, however, the development of a beta-adjusted vortex structure, together with the effects of numerical dissipation, tend to slow down the simulated vortex drift

Nonlinear transverse oscillations of a geostrophic front

A planar problem of nonlinear transverse oscillations of the surface (warm) front of a finite width is considered within the framework of a reduced-gravity model of the ocean. The source of oscillations is the departure of the front from its geostrophic equilibrium. When the current velocity is linear in the horizontal coordinate and the front's depth is quadratic in this coordinate, the problem is reduced to a system of four ordinary differential equations in time. As a result, the solution is obtained in a weakly nonlinear approximation and strongly nonlinear oscillations of the front are studied by numerically solving this system of equations by the Runge-Kutta method. The front's oscillations are always superinertial. Nonlinearity can lead to a decrease or increase in the oscillation frequency in comparison with the linear case. The oscillations are most intense when the current velocity is disturbed in the direction of the front's axis. A weakly nonlinear solution of the second order describes the oscillations very accurately even for initial velocity disturbances reaching 50% of its geostrophic value. An increase in the background-current shear leads to the damping of oscillations of the front's boundary. The amplitude of oscillations of the current velocity increases as the intensity of disturbances increases, and it is relatively small if background-current shears are small or large

Coherent Description for Hitherto Unexplained Radioactivities by Super- and Hyperdeformed Isomeric States

Recently long-lived high spin super- and hyperdeformed isomeric states with unusual radioactive decay properties have been discovered. Based on these newly observed modes of radioactive decay, consistent interpretations are suggested for previously unexplained phenomena seen in nature. These are the Po halos, the low-energy enhanced 4.5 MeV alpha-particle group proposed to be due to an isotope of a superheavy element with Z = 108, and the giant halos.Comment: 8 pages, 2 figures, 1 table, to be published in Int. J. Mod. Phys.

Quantum statistical correlations in thermal field theories: boundary effective theory

We show that the one-loop effective action at finite temperature for a scalar field with quartic interaction has the same renormalized expression as at zero temperature if written in terms of a certain classical field $\phi_c$, and if we trade free propagators at zero temperature for their finite-temperature counterparts. The result follows if we write the partition function as an integral over field eigenstates (boundary fields) of the density matrix element in the functional Schr\"{o}dinger field-representation, and perform a semiclassical expansion in two steps: first, we integrate around the saddle-point for fixed boundary fields, which is the classical field $\phi_c$, a functional of the boundary fields; then, we perform a saddle-point integration over the boundary fields, whose correlations characterize the thermal properties of the system. This procedure provides a dimensionally-reduced effective theory for the thermal system. We calculate the two-point correlation as an example.Comment: 13 pages, 1 figur