第110回「非線形・統計力学とその周辺」セミナーのご案内
日時:平成22年4月13日(火)15時から
場所:京都大学工学部8号館第5講義室(吉田キャンパス)
講演者:
Oleg N. Kirillov (Technische Universitat Darmstadt, Germany)
講演題目:
On multiple eigenvalues and singularities in MHD: Oscillatory dynamo and helical magnetorotational instability
講演要旨:
Polarity reversals of the Earth's magnetic field have fascinated geophysicists since their discovery a century ago. One of the simplest reversal models relies basically on the existence of an exceptional point in the spectrum of the non-self-adjoint dynamo operator, where two real eigenvalues coalesce and continue as a complex conjugated pair of eigenvalues. Using a homotopic family of boundary eigenvalue problems for the mean-field alpha2-dynamo with helical turbulence parameter alpha(r) = alpha0 + gamma*phi(r) and homotopy parameter 0<=beta<=1, we show that the underlying network of diabolical points for Dirichlet (idealized, beta = 0) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for mixed (physically realistic, beta = 1) boundary conditions. In the (alpha0, beta, gamma)-space the Arnold tongues of oscillatory solutions at beta = 1 end up at the diabolical points for beta = 0. In the vicinity of the diabolical points the space orientation of the 3D tongues, which are cones in first-order approximation, is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones. The Krein space induced geometry of the resonance zones explains the subtleties in finding alpha-profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field. The magnetorotational instability (MRI) plays a crucial role for cosmic structure formation by enabling turbulence in Keplerian disks which would be otherwise hydrodynamically stable. With particular focus on MRI experiments with liquid metals, which have small magnetic Prandtl numbers, it has been shown that the helical version of this instability (HMRI) has a scaling behaviour that is quite different from that of the standard MRI (SMRI). We discuss the relation of HMRI to SMRI by exploring various parameter dependencies. We identify the mechanism of transfer of instability between modes through a spectral exceptional point that explains both the transition from a stationary instability (SMRI) to an unstable travelling wave (HMRI) and the excitation of HMRI in the inductionless limit. For certain parameter regions we find new islands of the HMRI.