Abstract
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, Pm=1. At a fixed Pm≠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-symmetric.
Highlights
While common sense tends to assign to dissipation the role of a vibration damper, as early as 1879 Kelvin and Tait predicted viscosity-driven instability of Maclaurin’s spheroids, presenting a class of Hamiltonian equilibria, which, stable in the absence of dissipation, become unstable due to the action of dissipative forces [4,5]
A classical example is given by secular instability of the Maclaurin spheroids due to both fluid viscosity and gravitational radiation reaction, where the critical eccentricity of the meridional section of the spheroid depends on the ratio of the two dissipative mechanisms and reaches its maximum, corresponding to the onset of dynamical instability in the ideal system, exactly when this ratio equals 1 [2,22]
We have studied azimuthal magnetorotational instability (AMRI) of a circular Couette–Taylor flow of an incompressible electrically conducting Newtonian fluid in the presence of an azimuthal magnetic field of arbitrary radial dependence
Summary
While common sense tends to assign to dissipation the role of a vibration damper, as early as 1879 Kelvin and Tait predicted viscosity-driven instability of Maclaurin’s spheroids (proved by Roberts & Stewartson in 1963 [1,2,3]), presenting a class of Hamiltonian equilibria, which, stable in the absence of dissipation, become unstable due to the action of dissipative forces [4,5]. To illustrate stability of the equipartition solution (1.9) with respect to non-axisymmetric perturbations, we substitute it into the following criterion of destabilization of a hydrodynamically stable rotating flow of an inviscid and perfectly conducting fluid by an azimuthal magnetic field: 4Ro m2. (2.1I)na1n9d56(2, .C2)hwanitdhraΩse=khBa0φr/([r4√4]ρoμb0s)earvneddPt=hactofnosrt.thine exact stationary solution the ideal case, i.e. when ν (2.3) of equations = 0 and η = 0, the kinetic and magnetic energies are in equipartition, ρ(Ωr)2/2 = (B0φ)2/(2μ0), and Ro = Rb = −1 The latter equality follows from the condition of constant total pressure and from the fact that, in the steady state, the centrifugal acceleration of the background flow is compensated by the pressure gradient, rΩ2 = (1/ρ)∂rp0 [43]. Substituting expansions (2.7) in (2.5) and collecting terms at −1 and 0, we find [43]
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