Donnelly-Glaberson Instability Research Articles

Cross-Component Energy Transfer in Superfluid Helium-4

The reciprocal energy and enstrophy transfers between normal fluid and superfluid components dictate the overall dynamics of superfluid 4He including the generation, evolution and coupling of coherent structures, the distribution of energy among lengthscales, and the decay of turbulence. To better understand the essential ingredients of this interaction, we employ a numerical two-way model which self-consistently accounts for the back-reaction of the superfluid vortex lines onto the normal fluid. Here we focus on a prototypical laminar (non-turbulent) vortex configuration which is simple enough to clearly relate the geometry of the vortex line to energy injection and dissipation to/from the normal fluid: a Kelvin wave excitation on two vortex anti-vortex pairs evolving in (a) an initially quiescent normal fluid, and (b) an imposed counterflow. In (a), the superfluid injects energy and vorticity in the normal fluid. In (b), the superfluid gains energy from the normal fluid via the Donnelly–Glaberson instability.

Spherically symmetric formation of localized vortex tangle around a heat source in superfluid He4

We study the dynamical process of the vortex tangle development under a spherically symmetric thermal counterflow around a heat source submerged into a bulk superfluid 4He. We reveal a peculiar vortex dynamics that is unique to this geometry, which is greatly diverse from the vortex dynamics in a homogeneous counterflow. Two types of heater are considered here, namely, a spherical heater with a solid wall and a point-like heater. In both cases, a spherical vortex tangle is formed surrounding the heater. The mechanism of vortex tangle development in the vicinity of a solid wall is strongly governed by Donnelly-Glaberson instability; while, far away from the heater or around a point heater, the mechanism is governed by the dynamics of polarized vortex loops in radial counterflow. The decay process of such localized vortex tangles is also investigated and is compared with that of homogeneous vortex tangles.

Instability of a vortex ring due to toroidal normal fluid flow in superfluid 4He

Vortex rings self-propelling in superfluid 4He are shown to be driven unstable by a toroidal normal fluid flow. This instability has qualitative similarities with the Donnelly-Glaberson instability of Kelvin waves on a vortex filament driven by the normal fluid flow along the vortex filament.

Dynamics of quantized vortex filaments under a local induction approximation with second-order correction

We study a second-order local induction approximation (LIA) for the dynamics of a single open quantized vortex filament (such as those filaments arising in superfluid helium). While for a classical vortex filament, this second-order correction can be interpreted as a correction due to the inclusion of axial flow within a filament core, in the quantized filament case, this second order correction can be viewed as a correction due to variable condensate healing length. We compare the evolution of the decay rate, transverse velocity, and rotational velocity of Kelvin waves along vortex filaments under this model to that of the first order LIA of Schwarz for quantized vortex filaments, as well as to a corresponding nonlocal model involving Biot-Savart integrals for the self-induced motion of the vortex filament. For intermediate wavenumbers, the second-order model solutions show improved agreement with the nonlocal Biot-Savart model, due to an additional control parameter. We also consider the stability of Kelvin waves under the second-order corrections; these results allow us to understand the Donnelly-Glaberson instability in the context of the second-order model. The second-order corrections tend to stabilize the resulting solutions, in agreement with what was previously found from the nonlocal Biot-Savart formulation, yet still permit a local description of the vortex filament in terms of a partial differential equation (akin to the first-order LIA) rather than an integro-differential equation.

Hydrodynamic Stability Analysis of the Neutron Star Core

Abstract Hydrodynamic instabilities and turbulence in neutron stars have been suggested to be related to observable spin variations in pulsars, such as spin glitches, timing noise, and precession (nutation). Accounting for the stabilizing effects of the stellar magnetic field, we revisit the issue of whether the inertial modes of a neutron star can become unstable when the neutron and proton condensates flow with respect to one another. The neutron and proton condensates are coupled through the motion of imperfectly pinned vorticity (vortex slippage) and vortex-mediated scattering (mutual friction). Previously-identified two-stream instabilities that occur when the two condensates rotate with respect to one another in the outer core are stabilized by the toroidal component of the magnetic field. This stabilization occurs when the Alfvén speed of the toroidal component of the magnetic field becomes larger than the relative rotational velocity of the condensates, corresponding to toroidal field strengths in excess of ≃1010 G. In contrast with previous studies, we find that spin-down of a neutron star under a steady torque is stable. The Donnelly–Glaberson instability is not stabilized by the magnetic field and could play an important role if neutron stars undergo precession.

Localized nonlinear waves on quantized superfluid vortex filaments in the presence of mutual friction and a driving normal fluid flow.

We demonstrate the existence of localized structures along quantized vortex filaments in superfluid helium under the quantum form of the local induction approximation (LIA), which includes mutual friction and normal fluid effects. For small magnitude normal fluid velocities, the dynamics are dissipative under mutual friction. On the other hand, when normal fluid velocities are sufficiently large, we observe parametric amplification of the localized disturbances along quantized vortex filaments, akin to the Donnelly-Glaberson instability for regular Kelvin waves. As the waves amplify they will eventually cause breakdown of the LIA assumption (and perhaps the vortex filament itself), and we derive a characteristic time for which this breakdown occurs under our model. More complicated localized waves are shown to occur, and we study these asymptotically and through numerical simulations. Such solutions still exhibit parametric amplification for large enough normal fluid velocities, although this amplification may be less uniform than would be seen for more regular filaments such as those corresponding to helical curves. We find that large rotational velocities or large wave speeds of nonlinear waves along the filaments will result in more regular and stable structures, while small rotational velocities and wave speeds will permit far less regular dynamics.

Decay of helical Kelvin waves on a quantum vortex filament

We study the dynamics of helical Kelvin waves moving along a quantum vortex filament driven by a normal fluid flow. We employ the vector form of the quantum local induction approximation (LIA) due to Schwarz. For an isolated filament, this is an adequate approximation to the full Hall-Vinen-Bekarevich-Khalatnikov dynamics. The motion of such Kelvin waves is both translational (along the quantum vortex filament) and rotational (in the plane orthogonal to the reference axis). We first present an exact closed form solution for the motion of these Kelvin waves in the case of a constant amplitude helix. Such solutions exist for a critical wave number and correspond exactly to the Donnelly-Glaberson instability, so perturbations of such solutions either decay to line filaments or blow-up. This leads us to consider helical Kelvin waves which decay to line filaments. Unlike in the case of constant amplitude helical solutions, the dynamics are much more complicated for the decaying helical waves, owing to the fact that the rate of decay of the helical perturbations along the vortex filament is not constant in time. We give an analytical and numerical description of the motion of decaying helical Kelvin waves, from which we are able to ascertain the influence of the physical parameters on the decay, translational motion along the filament, and rotational motion, of these waves (all of which depend nonlinearly on time). One interesting finding is that the helical Kelvin waves do not decay uniformly. Rather, such waves decay slowly for small time scales, and more rapidly for large time scales. The rotational and translational velocity of the Kelvin waves depend strongly on this rate of decay, and we find that the speed of propagation of a helical Kelvin wave along a quantum filament is large for small time while the wave asymptotically slows as it decays. The rotational velocity of such Kelvin waves along the filament will increase over time, asymptotically reaching a finite value. These decaying Kelvin waves correspond to wave number below the critical value for the Donnelly-Glaberson instability, and hence our results on the Schwarz quantum LIA correspond exactly to what one would expect from prior work on the Donnelly-Glaberson instability.

Comment on “Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation” [Phys. Fluids 25, 085101 (2013)

We comment on the paper by Van Gorder [“Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation,” Phys. Fluids 25, 085101 (2013)]. We point out that the flow of the normal fluid component parallel to the vortex will often lead into the Donnelly–Glaberson instability, which will cause the amplification of the Kelvin wave. We explain why the comparison to local nonlinear equation is unreasonable, and remark that neglecting the motion in the x-direction is not reasonable for a Kelvin wave with an arbitrary wavelength and amplitude. The correct equations in the general case are also derived.

Response to “Comment on ‘Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation”’ [Phys. Fluids 26, 019101 (2014)

I agree with the authors regarding their comments on the Donnelly-Glaberson instability for such helical filaments as those obtained in my paper. I also find merit in their derivation of the quantum LIA (local induction approximation) in the manner of the LIA of Boffetta et al. However, I disagree with the primary criticisms of Hietala and Hänninen. In particular, though they suggest LIA and local nonlinear equation modes are not comparable since the former class of models contains superfluid friction parameters, note that since these parameters are small one may take them to zero and consider a qualitative comparison of the models (which is what was done in my paper). Second, while Hietala and Hänninen criticize certain assumptions made in my paper (and the paper of Shivamoggi where the model comes from) since the results break-down when Ak → ∞, note that in my paper I state that any deviations from the central axis along which the filament is aligned must be sufficiently bounded in variation. Therefore, it was already acknowledged that Ak(=|Φx|) should be sufficiently bounded, precluding the Ak → ∞ case. I also show that, despite what Hietala and Hänninen claim, the dispersion relation obtained in my paper is consistent with LIA, where applicable. Finally, while Hietala and Hänninen claim that the dispersion parameter should be complex valued, I show that their dispersion relation is wrong, since it was derived incorrectly (they assume the complex modulus of the potential function is constant, yet then use this to obtain a potential function with non-constant modulus).

Vortex motion in superfluidHe4: Reformulation in the extrinsic vortex-filament coordinate space

Vortex motion in superfluid $^{4}\mathrm{He}$ is considered by reformulating the Hall-Vinen equation (originally formulated in the intrinsic geometric parameter space) in the extrinsic vortex filament coordinate space which is shown to provide a useful alternative approach. It provides insights into the physical implications of one aspect of the friction term, associated with the friction coefficient $\ensuremath{\alpha}$, in the Hall-Vinen notation, as well as the previous neglect of the other aspect, associated with the friction coefficient ${\ensuremath{\alpha}}^{\ensuremath{'}}$. A nonlinear Schrodinger equation for the propagation of nonlinear Kelvin waves on a vortex filament in a superfluid is given. The localized vortex kink solution of this equation is shown to be driven unstable by the normal fluid flow along the vortex in qualitative similarity with the Donnelly-Glaberson instability of Kelvin waves on a vortex. This instability is demonstrated in a more clear way by considering the problem of a rotating planar vortex filament in a superfluid. Though the friction term associated with $\ensuremath{\alpha}$, for very small $\ensuremath{\alpha}$, has little capacity to make significant contribution to the vortex motion in a quantitative way, it appears to be able to influence the vortex kink characteristics in a qualitative way.

Spontaneous radiation and amplification of Kelvin waves on quantized vortices in Bose-Einstein condensates

We propose a different type of Landau instability in trapped Bose-Einstein condensates by a helically moving environment. In the presence of quantized vortices, the instability can cause spontaneous radiation and amplification of Kelvin waves. This study gives a microscopic understanding of the Donnelly-Glaberson instability which was known as the hydrodynamic instability in superfluid helium. The Donnelly-Glaberson instability can be a powerful tool for observing the dispersion relation of Kelvin waves, vortex reconnections, and quantum turbulence in atomic Bose-Einstein condensates.

Donnelly-Glaberson instability exciting kelvin waves in atomic Bose-Einstein condensates

The Donnelly-Glaberson (DG) instability causes the amplification of Kelvin waves on vortex lines in superfluid 4He and 3He-B at finite temperatures due to mutual friction between vortex lines and the normal fluid component. In this letter, we theoretically show that the DG instability is possible in atomic Bose-Einstein condensates even at T = 0 K without the normal component. The DG instability in atomic condensates can lead to observe the dispersion relation of Kelvin waves, vortex reconnections and quantum turbulence.