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Arnold tongues and spectral points with definite and mixed Krein signature in MHD alpha2 dynamo

Hold Date
2010-03-05 16:30〜2010-03-05 18:00
Ito Campus, Faculty of Mathematics building, seminar room 6
Object person
Oleg N. Kirillov (Technical University of Darmstadt, Germany and Institute of Mechanics, Moscow M.V. Lomonosov State University, Russia)

It is accepted that the magnetic fields of planets, stars and galaxies 
are maintained by dynamo effects in conducting fluids or plasmas. 
These dynamo effects are caused by a topologically nontrivial 
interplay of fluid (plasma) motions and a balanced self-amplification 
of the magnetic fields - and can be described within the framework of 
magnetohydrodynamics (MHD).

For physically realistic dynamos the coupled system of Maxwell and 
Navier-Stokes equations has, in general, to be solved numerically. For 
a qualitative understanding of the occurring effects, semi-
analytically solvable toy models play an important role. One of the 
simplest dynamo models is the so called alpha2-dynamo with a 
spherically symmetric helical turbulence parameter alpha, which can be 
assumed real-valued sufficiently smooth function of the radius. For 
such a dynamo the magnetic field can be decomposed into poloidal and 
toroidal components and expanded over spherical harmonics. As a 
result, one arrives at a set of mode decoupled matrix differential 
eigenvalue problems with boundary conditions (BCs) which have to be 
imposed in dependence on the concrete physical setup. The differential 
expression of this operator matrix has the fundamental (canonical) 
symmetry. In case of the idealized BCs (super-conducting surrounding) 
that are compatible with this fundamental symmetry the operator turns 
out self-adjoint in a Krein space, while in case of physically 
realistic BCs (surrounding vacuum) this property is lost.

We present some recent results on the spectral properties of 
spherically symmetric MHD alpha2-dynamos. In particular, the spectra of sphere-confined fluid or plasma configurations with physically realistic boundary conditions (surrounding vacuum) and with idealized BCs (super-conducting surrounding) and connection between these two boundary eigenvalue problems are discussed. A striking result is that the underlying network of eigenvalue crossings and the Krein signatures at the crossings for idealized BCs substantially determine geometry of the domains (Arnold tongues) of oscillatory dynamo for physically realistic BCs and, in particular, the loci of exceptional points corresponding to eigenvalues with the algebraic multiplicity two and geometric multiplicity one, which are important in modern theories that are aimed to explain the polarity reversals of the geomagnetic field."