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.