Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics

We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. In the presence of the magnetic field the hydrodynamically stable flow can demonstrate non - axisymmetric azimuthal magnetorotational instability (AMRI) both in the diffusionless case and in the double-diffusive case with viscous and ohmic dissipation. Performing stability analysis of amplitude transport equations of short-wavelength approximation, we find that the threshold of the diffusionless AMRI via the Hamilton-Hopf bifurcation is a singular limit of the thresholds of the viscous and resistive AMRI corresponding to the dissipative Hopf bifurcation and manifests itself as the Whitney umbrella singular point. A smooth transition between the two types of instabilities is possible only if the magnetic Prandtl number is equal to unity, $\rm Pm=1$. At a fixed $\rm Pm\ne 1$ the threshold of the double-diffusive AMRI is displaced by finite distance in the parameter space with respect to the diffusionless case even in the zero dissipation limit. The complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities the double-diffusive system reduces to a marginally stable system which is either Hamiltonian or parity-time (PT) symmetric.Comment: 34 pages, 8 figures, typos corrected, refs adde

Destabilization of rotating flows with positive shear by azimuthal magnetic fields

According to Rayleigh's criterion, rotating flows are linearly stable when their specific angular momentum increases radially outward. The celebrated magnetorotational instability opens a way to destabilize those flows, as long as the angular velocity is decreasing outward. Using a short-wavelength approximation we demonstrate that even flows with very steep positive shear can be destabilized by azimuthal magnetic fields which are current-free within the fluid. We illustrate the transition of this instability to a rotationally enhanced kink-type instability in case of a homogeneous current in the fluid, and discuss the prospects for observing it in a magnetized Taylor-Couette flow.Comment: 4 pages, 4 figur

Standard and helical magnetorotational instability: How singularities create paradoxal phenomena in MHD

The magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable rotating shear flows, e.g., in accretion disks. What laws of differential rotation are susceptible to the destabilization by axial, azimuthal, or helical magnetic field? The answer to this question, which is vital for astrophysical and experimental applications, inevitably leads to the study of spectral and geometrical singularities on the instability threshold. The singularities provide a connection between seemingly discontinuous stability criteria and thus explain several paradoxes in the theory of MRI that were poorly understood since the 1950s.Comment: 25 pages, 10 figures. A tutorial paper. Invited talk at SPT 2011, Symmetry and Perturbation Theory, 5 - 12 June 2011, Otranto near Lecce (Italy

Extending the range of the inductionless magnetorotational instability

The magnetorotational instability (MRI) can destabilize hydrodynamically stable rotational flows, thereby allowing angular momentum transport in accretion disks. A notorious problem for MRI is its questionable applicability in regions with low magnetic Prandtl number, as they are typical for protoplanetary disks and the outer parts of accretion disks around black holes. Using the WKB method, we extend the range of applicability of MRI by showing that the inductionless versions of MRI, such as the helical MRI and the azimuthal MRI, can easily destabilize Keplerian profiles ~ 1/r^(3/2) if the radial profile of the azimuthal magnetic field is only slightly modified from the current-free profile ~ 1/r. This way we further show how the formerly known lower Liu limit of the critical Rossby number, Ro=-0.828, connects naturally with the upper Liu limit, Ro=+4.828.Comment: Growth rates added, references modified; submitted to Physical Review Letter

Perturbation of multiparameter non-self-adjoint boundary eigenvalue problems for operator matrices

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter and on the vector of real physical parameters. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of the parameters. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD alpha2-dynamo and circular string demonstrates the efficiency and applicability of the theory.Comment: 17 pages, 4 figures, presented at the International Conference "Modern Analysis and Applications - MAA 2007" dedicated to the centenary of Mark Krein. Odessa, Ukraine, April 9-14, 2007. Minor typos correcte

Instabilities of rotational flows in azimuthal magnetic fields of arbitrary radial dependence

Using the WKB approximation we perform a linear stability analysis for a rotational flow of a viscous and electrically conducting fluid in an external azimuthal magnetic field that has an arbitrary radial profile B_{phi}(R). In the inductionless approximation, we find the growth rate of the three-dimensional perturbation in a closed form and demonstrate in particular that it can be positive when the velocity profile is Keplerian and the magnetic field profile is slightly shallower than R^{-1}.Comment: 15 pages, 2 figures, slightly extended, the case of finite Rm treated, results were partially presented at the IUTAM Symposium on Vortex Dynamics, Fukuoka, Japan, March 10 201

Determining role of Krein signature for 3D Arnold tongues of oscillatory dynamos

Using a homotopic family of boundary eigenvalue problems for the mean-field $\alpha^2$-dynamo with helical turbulence parameter $\alpha(r)=\alpha_0+\gamma\Delta\alpha(r)$ and homotopy parameter $\beta \in [0,1]$, 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 Robin (physically realistic, $\beta=1$) boundary conditions. In the $(\alpha_0,\beta,\gamma)-$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.Comment: 4 pages, 3 figures, presented at the GAMM 2008, Bremen, Germany Introduction extended, refs adde

Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler's paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler's paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation and related bifurcations.Comment: 35 pages, 11 figure