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

Membrane flutter induced by radiation of surface gravity waves on a uniform flow

We consider stability of an elastic membrane being on the bottom of a uniform horizontal flow of an inviscid and incompressible fluid of finite depth with free surface. The membrane is simply supported at the leading and the trailing edges which attach it to the two parts of the horizontal rigid floor. The membrane has an infinite span in the direction perpendicular to the direction of the flow and a finite width in the direction of the flow. For the membrane of infinite width we derive a full dispersion relation that is valid for arbitrary depth of the fluid layer and find conditions for the flutter of the membrane due to emission of surface gravity waves. We describe this radiation-induced instability by means of the perturbation theory of the roots of the dispersion relation and the concept of negative energy waves and discuss its relation to the anomalous Doppler effect

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

Bifurcation of the roots of the characteristic polynomial and destabilization paradox in friction induced oscillations

Paradoxical effect of small dissipative and gyroscopic forces on the stability of a linear non-conservative system, which manifests itself through the unpredictable at first sight behavior of the critical non-conservative load, is studied. By means of the analysis of bifurcation of multiple roots of the characteristic polynomial of the non-conservative system, the analytical description of this phenomenon is obtained. As mechanical examples two systems possessing friction induced oscillations are considered: a mass sliding over a conveyor belt and a model of a disc brake describing the onset of squeal during the braking of a vehicle

Locating the sets of exceptional points in dissipative systems and the self-stability of bicycles

Sets in the parameter space corresponding to complex exceptional points have high codimension and by this reason they are difficult objects for numerical location. However, complex EPs play an important role in the problems of stability of dissipative systems where they are frequently considered as precursors to instability. We propose to locate the set of complex EPs using the fact that the global minimum of the spectral abscissa of a polynomial is attained at the EP of the highest possible order. Applying this approach to the problem of self-stabilization of a bicycle we find explicitly the EP sets that suggest scaling laws for the design of robust bikes that agree with the design of the known experimental machines

Sensitivity of Sub-critical Mode-coupling Instabilities in Non-conservative Rotating Continua to Stiffness and Damping Modifications

Mode-coupling instability is a widely accepted mechanism for the onset of friction-induced vibrations in car brakes, wheel sets, paper calendars, to name a few. In the presence of damping, gyroscopic, and non-conservative positional forces the merging of modes is imperfect, that is two modes may come close together in the complex plane without collision and then diverge so that one of the modes becomes unstable. In non-conservative rotating continua that respect axial symmetry this movement of eigenvalues is very sensitive to the variation of parameters of the system. Our study reveals some general rules that govern sub-critical mode-coupling instabilities in non-conservative rotating continua to stiffness and damping modifications and provide useful insight for optimisation of such systems and interpretation of experimental results