International audienc

In the present work, the integral equation approach and the non-local elasticity theory are employed to investigate the free transverse vibrations of nano-to-micron scale beams. The frequency equation is analytically formulated into an eigenvalue problem of a matrix with an infinite order. The numerical calculation is implemented by truncating this matrix to a finite order one. It is found that the impact of the non-local effect on the natural frequencies and vibrating modes is negligible for the beams with micrometre scale length. But when the length of beams reaches the nanoscale, the non-local effect becomes important, especially for the high-order natural frequencies and vibrating modes.

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In a recent paper [1] it was shown that a simple mean-field dynamo model with a spherically symmetric helical turbulence parameter α can exhibit a number of features which are typical for Earth's magnetic field reversals. In particular, the model produces asymmetric reversals (with a slow decay of the dipole of one polarity and a fast recreation of the dipole with opposite polarity), a positive correlation of field strength and interval length, and a bimodal field distribution. All these features are attributable to the magnetic field dynamics in the vicinity of an exceptional point of the spectrum of the non-selfadjoint dynamo operator where two real eigenvalues coalesce and continue as a complex conjugated pair of eigenvalues. Usually, this exceptional point is associated with a nearby local maximum of the growth rate dependence on the magnetic Reynolds number. The negative slope of this curve between the local maximum and the exceptional point makes the system unstable and drives it to the exceptional point and beyond into the oscillatory branch where the sign change happens. A weakness of this reversal model is the apparent necessity to fine-tune the magnetic Reynolds number and/or the radial profile of α in order to adjust the operator spectrum in an appropriate way. In the present paper, it is shown that this fine-tuning is not necessary in the case of higher supercriticality of the dynamo. Numerical examples and physical arguments are compiled to show that, with increasing magnetic Reynolds number, there is strong tendency for the exceptional point and the associated local maximum to move close to the zero growth rate line where the indicated reversal scenario can be actualized. Although exemplified again by the spherically symmetric α 2 dynamo model, the main idea of this "self-tuning" mechanism of saturated dynamos into a reversal-prone state seems well transferable to other dynamos. As a consequence, reversing dynamos might be much more typical and may occur much more frequently in nature than what could be expected from a purely kinematic perspective.

With only a few exceptions, the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method. However, the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics. The paper is intended to facilitate the use of integral equation solvers in dynamo theory. In concrete, the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with a spherically symmetric, isotropic helical turbulence parameter α. Three examples of the function α(r) with steady and oscillatory solutions are considered. A convergence rate proportional to the inverse squared of the number of grid points is achieved. Based on this method, a convergence accelerating strategy is developed and the convergence rate is improved remarkably. Typically, quite accurate results can be obtained with a few tens of grid points.