21 research outputs found
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, . At a fixed 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
From rotating fluid masses and Ziegler's paradox to Pontryagin- and Krein spaces and bifurcation theory
Three classical systems, the Kelvin gyrostat, the Maclaurin spheroids, and the Ziegler pendulum have directly inspired development of the theory of Pontryagin and Krein spaces with indefinite metric and singularity theory as independent mathematical topics, not to mention stability theory and nonlinear dynamics. From industrial applications in shipbuilding, turbomachinery, and artillery to fundamental problems of astrophysics, such as asteroseismology and gravitational radiation —- that is the range of phenomena involving the Krein collision of eigenvalues, dissipation-induced instabilities, and spectral and geometric singularities on the neutral stability surfaces, such as the famous Whitney's umbrella
Orbital Hypernormal Forms
In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can be carried out degree by degree using quasi-homogeneous expansions of the vector field of the system by means of reduced time-reparametrizations and near-identity transformations, achieving an important reduction in the computational effort. Moreover, although the orbital hypernormal form procedure is essentially nonlinear in nature, our results show that orbital hypernormal forms are characterized by means of linear operators. Some applications are considered: the case of planar vector fields, with emphasis on a case of the Takens–Bogdanov singularit
Natural doubly diffusive convection with vibration
We present a numerical and analytical study of doubly diffusive convection driven by horizontal thermal and solutal gradients in square and rectangular enclosures with no-slip walls subjected to high-frequency vibration. The two vertical walls of the enclosure are maintained at different but uniform temperatures and concentrations while the horizontal walls are assumed to be impermeable and insulating. The resulting system is described by time-averaged Boussinesq equations. These equations possess a doubly diffusive quasi-equilibrium solution provided the thermal and solutal buoyancy forces are equal and opposite. This solution is linearly stable up to a critical value of the stability parameter independently of the strength and orientation of the vibration. The solutions in the neighborhood of the bifurcation point are described analytically as a function of the strength and orientation of the vibration, and the larger amplitude states are computed numerically using a spectral collocation method. For vertical oscillation increasing the vibration amplitude decreases the subcriticality of the solutions and may even reverse it; the opposite occurs with horizontal vibration