We perform a local stability analysis of rotational flows in the presence of a constant vertical magnetic field and an azimuthal magnetic field with a general radial dependence. Employing the short-wavelength approximation we develop a unified framework for the investigation of the standard, the helical, and the azimuthal version of the magnetorotational instability, as well as of current-driven kink-type instabilities. Considering the viscous and resistive setup, our main focus is on the case of small magnetic Prandtl numbers which applies, e.g., to liquid metal experiments but also to the colder parts of accretion disks. We show that the inductionless versions of MRI that were previously thought to be restricted to comparably steep rotation profiles extend well to the Keplerian case if only the azimuthal field slightly deviates from its current-free (in the fluid) profile. We find an explicit criterion separating the pure azimuthal inductionless magnetorotational instability from the regime where this instability is mixed with the Tayler instability. We further demonstrate that for particular parameter configurations the azimuthal MRI originates as a result of a dissipation-induced instability of the Chandrasekhar's equipartition solution of ideal magnetohydrodynamics.

Three-dimensional motion of a thin vortex filament with axial velocity, embedded in an inviscid incompressible fluid, is investigated. The deflections of the core centreline are not restricted to be small compared with the core radius. We first derive the equation of the vortex motion, correct to the second order in the ratio of the core radius to that of curvature, by a matching procedure, which recovers the results obtained by Moore & Saffman (1972). An asymptotic formula for the linear dispersion relation is obtained up to the second order. Under the assumption of localized induction, the equation governing the self-induced motion of the vortex is reduced to a nonlinear evolution equation generalizing the localized induction equation. This new equation is equivalent to the Hirota equation which is integrable, including both the nonlinear Schrödinger equation and the modified KdV equation in certain limits. Therefore the new equation is also integrable and the soliton surface approach gives the N-soliton solution, which is identical to that of the localized induction equation if the pertinent dispersion relation is used. Among other exact solutions are a circular helix and a plane curve of Euler's elastica. This local model predicts that, owing to the existence of the axial flow, a certain class of helicoidal vortices become neutrally stable to any small perturbations. The non-local influence of the entire perturbed filament on the linear stability of a helicoidal vortex is explored with the help of the cutoff method valid to the second order, which extends the first-order scheme developed by Widnall (1972). The axial velocity is found to discriminate between right- and left-handed helices and the long-wave instability mode is found to disappear in a certain parameter range when the successive turns of the helix are not too close together. Comparison of the cutoff model with the local model reveals that the non-local induction and the core structure are crucial in making quantitative predictions.

We develop a general matrix method to analyze from a far field the dynamics of an accelerated interface between incompressible ideal fluids of different densities with interfacial mass flux and with negligible density variations and stratification. We rigorously solve the linearized boundary value problem for the dynamics conserving mass, momentum, and energy in the bulk and at the interface. We find a new hydrodynamic instability that develops only when the acceleration magnitude exceeds a threshold. This critical threshold value depends on the magnitudes of the steady velocities of the fluids, the ratio of their densities, and the wavelength of the initial perturbation. The flow has potential velocity fields in the fluid bulk and is shear-free at the interface. The interface stability is set by the interplay of inertia and gravity. For weak acceleration, inertial effects dominate, and the flow fields experience stable oscillations. For strong acceleration, gravity effects dominate, and the dynamics is unstable. For strong accelerations, this new hydrodynamic instability grows faster than accelerated Landau-Darrieus and Rayleigh-Taylor instabilities. For given values of the fluids' densities and their steady bulk velocities, and for a given magnitude of acceleration, we find the critical and maximum values of the initial perturbation wavelength at which this new instability can be stabilized and at which its growth is the fastest. The quantitative, qualitative, and formal properties of the accelerated conservative dynamics depart from those of accelerated Landau-Darrieus and Rayleigh-Taylor dynamics. New diagnostic benchmarks are identified for experiments and simulations of unstable interfaces. Published by AIP Publishing.

A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched of matched asymptotic expansions is extended to a higher order in a small parameter ε = (v/Γ)1/2, where v is the kinematic viscosity of fluid and Γ is the circulation. Alternatively, ε is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics.We establish a general formula for the translation speed of the ring valid up to third order in ε. This is a natural extension of Fraenkel–Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and R˜s(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R˜p(t) as Rs ≈ R0 + [2 log(4R0/√vt) + 1.4743424] vt/R0, R˜s ≈ R0 + 2.5902739 vt/R0, and Rp ≈ R0 + 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet & Widmayer (1974) and Weigand & Gharib (1997).The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.

While looking from a far field at a discontinuous front separating incompressible ideal fluids of different densities, we identify two qualitatively different behaviors of the front-unstable and stable-depending upon whether the energy flux produced by the perturbed front is large or small compared to the flux of kinetic energy across the planar front. Landau's solution for the Landau-Darrieus instability is consistent with one of these cases.

The linear stability of unbounded strained vortices in a stably stratified fluid is investigated theoretically. The problem is reduced to a Floquet problem which is solved numerically. The three-dimensional elliptical instability of Pierrehumbert type [Phys. Rev. Lett. 57, 2157 (1986)] is shown to be suppressed by the stable stratification and it disappears when the Brunt–Väisälä frequency exceeds unity. On the other hand, two classes of new instability mode are found to occur. One appears only when the Brunt–Väisälä frequency is less than 2, whereas the other persists for all values of the Brunt–Väisälä frequency. The former mode is related to a parametric resonance of internal gravity waves, and the latter modes are related to superharmonic parametric instability.

A new instability mechanism is found for Kelvin's vortex ring, which may surpass the Widnall instability. The effect of ring curvature emerges at O() in the asymptotic solution of the Euler equations in powers , the ratio of core to ring radii. We show that the O() field causes a parametric resonance between a pair of Kelvin waves whose azimuthal wavenumbers are separated by 1. A closed-form solution enables us to calculate the maximum growth rate to be 165/256 and to make headway to nonlinear stability. MOTIVATION AND MAIN RESULT Vortex rings are invariably susceptible to wavy distortions, leading to violent wiggles and sometimes to disruption. It has been widely accepted that the Moore-Saffman-Tsai-Widnall instability (the MSTW instability) is responsible for genesis of unstable waves [1]-[5]. Remember that this is an instability for a straight vortex tube subjected to a straining field.

We revisit the Moore–Saffman–Tsai–Widnall instability, a parametric resonance between left- and right-handed bending waves of infinitesimal amplitude, on the Rankine vortex strained by a weak pure shear flow. The results of Tsai & Widnall (1976) and Eloy & Le Dizès (2001), as generalized to all pairs of Kelvin waves whose azimuthal wavenumbers $m$ are separated by 2, are simplified by providing an explicit solution of the linearized Euler equations for the disturbance flow field. Given the wavenumber $k_0$ and the frequency $\omega_0$ of an intersection point of dispersion curves, the growth rate is expressible solely in terms of the modified Bessel functions, and so is the unstable wavenumber range. Every intersection leads to instability. Most of the intersections correspond to weak instability that vanishes in the short-wave limit, and dominant modes are restricted to particular intersections. For helical waves $m=\pm 1$, the growth rate of non-rotating waves is far larger than that of rotating waves. The wavenumber width of stationary instability bands broadens linearly in $k_0$, while that of rotating instability bands is bounded. The growth rate of the stationary instability takes, in the long-wavelength limit, the value of $\varepsilon/2$ for the two-dimensional displacement instability, and, in the short-wavelength limit, the value of $9\varepsilon/16$ for the elliptical instability, being larger at large but finite values of $k_0$. Here $\varepsilon$ is the strength of shear near the core centre. For resonance between higher azimuthal wavenumbers $m$ and $m+2$, the same limiting value is approached as $k_0\,{\to}\,\infty$, along sequences of specific crossing points whose frequency rapidly converges to $m+1$, in two ways, from above for a fixed $m$ and from below for $m\,{\to}\,\infty$. The energy of the Kelvin waves is calculated by invoking Cairns' formula. The instability result is compatible with Krein's theory for Hamiltonian spectra.