W. F. Vinen

Quantized vortex dynamics and superfluid turbulence

to Superfluid Vortices and Turbulence.- Turbulence Experiments.- An Introduction to Experiments on Superfluid Turbulence.- The Experimental Evidence for Vortex Nucleation in 4He.- Applications of Superfluid Helium in Large-Scale Superconducting Systems.- The Temperature Dependent Drag Crisis on a Sphere in Flowing Helium II.- Experiments on Quantized Turbulence at mK Temperatures.- Grid-Generated He II Turbulence in a Finite Channel - Experiment.- Intermittent Switching Between Turbulent and Potential Flow Around a Sphere in He II at mK Temperatures.- Vortex Dynamics.- Vortex Filament Methods for Superfluids.- to HVBK Dynamics.- Magnus Force, Aharonov-Bohm Effect, and Berry Phase in Superfluids.- Using the HVBK Model to Investigate the Couette Flow of Helium II.- Turbulence Theory.- An Introduction to the Theory of Superfluid Turbulence.- Numerical Methods for Coupled Normal-Fluid and Superfluid Flows in Helium II.- From Vortex Reconnections to Quantum Turbulence.- Vortices and Stability in Superfluid Boundary Layers.- Grid Generated He II Turbulence in a Finite Channel-Theoretical Interpretation.- Vortex Tangle Dynamics Without Mutual Friction in Superfluid 4He.- Applications of the Gaussian Model of the Vortex Tangle in the Superfluid Turbulent He II.- Stochastic Dynamics of a Vortex Loop. Thermal Equilibrium.- Stochastic Dynamics of a Vortex Loop. Large-Scale Stirring Force.- Nonequilibrium Vortex Dynamics in Superfluid Phase Transitions and Superfluid Turbulence.- The NLSE and Superfluidity.- The Nonlinear Schrodinger Equation as a Model of Superfluidity.- Vortex Nucleation and Limit Speed for a Flow Passing Nonlinearly Around a Disk in the Nonlinear Schrodinger Equation.- Vortices in Nonlocal Condensate Models of Superfluid Helium.- Ginzburg-Landau Description of Vortex Nucleation in a Rotating Superfluid.- Weak Turbulence Theory for the Gross-Pitaevskii Equation.- Dissipative Vortex Dynamics and Magnus Force.- Transition to Dissipation in Two- and Three-Dimensional Superflows.- Bose-Einstein Condensation.- Motion of Objects Through Dilute Bose-Einstein Condensates.- Stability of a Vortex in a Rotating Trapped Bose-Einstein Condensate*.- Kinetics of Strongly Non-equilibrium Bose-Einstein Condensation.- Quantum Nucleation of Phase Slips in Bose-Einstein Condensates.- Vortex Reconnections and Classical Aspects.- Vortex Reconnection in Normal and Superfluids.- Helicity in Hydro and MHD Reconnection.- Tropicity and Complexity Measures for Vortex Tangles.- The Geometry of Magnetic and Vortex Reconnection.- Current-Sheet Formation near a Hyperbolic Magnetic Neutral Line.- Nonlocality in Turbulence.- Helium 3 and Other Systems.- Quantized Vorticity in Superfluid 3He-A: Structure and Dynamics.- Vortices in Metastable 4He Films.- Quantum Hall E.ect Breakdown Steps and Possible Analogies with Classical and Super.uid Hydrodynamics.- Atomic Bose Condensate with a Spin Structure: The Use of Bloch State.- Quantum Dynamics of Vortex-Antivortex Pairs in a Circular Box.

Mutual friction in a heat current in liquid helium II III. Theory of the mutual friction

As was shown in part II, the Gorter-Mellink mutual friction force in a heat current is probably associated with turbulence in the superfluid. Following Feynman, it is suggested that this turbulence takes the form of a tangled mass of quantized vortex lines, so that the mutual friction probably arises from collisions between thermal excitations and these vortex lines. From the observed properties of the mutual friction it is deduced that the walls of the channel carrying the heat current play no essential role in the generation, maintenance or decay of the turbulence, but merely introduce a number of incidental complications; the present paper ignores these complications and deals therefore with the idealized case of a homogeneous heat current in an unbounded volume of helium. The turbulence in this idealized case must be homogeneous, and it is shown from experimental evidence that it is probably also isotropic. Values of the force exerted on unit length of a vortex line, which have been derived from the study of the attenuation of second sound in uniformly rotating helium, are used to calculate the Gorter-Mellink force per unit volume in terms of the length of line per unit volume; then by a simple dimensional argument it is shown that the force must depend on (vs — vn) in a manner agreeing with experiment. An attempt is made to produce a detailed theory of the generation and decay of superfluid turbulence: it is shown first that owing to the Magnus effect the turbulence can probably be built up by the action of the mutual friction force exerted on the individual lines, although the way in which turbulence can be initiated in undisturbed helium is not known, and secondly that the turbulence can probably decay in a manner closely analogous to the decay of homogeneous turbulence in an ordinary fluid. Equations for the rate of generation and decay of turbulence are obtained by dimensional arguments, and by analogy with formulae known to apply to turbulence in an ordinary fluid. Comparison of the equations with the experimental results described in parts I and II reveals good agreement, and makes it possible to deduce the form and magnitude of a term describing the effect of the unknown initiation process.

The rotation of liquid helium ii II. The theory of mutual friction in uniformly rotating helium ii

A discussion is given of models for the rotation of helium II involving regions of concentrated vorticity, and it is shown thermodynamically that an arrangement of vortex lines is energetically preferable to an arrangement of vortex sheets. It is suggested that such models exhibit the property of mutual friction, owing to the possibility of collisions between normal fluid excitations and the regions of concentrated superfluid vorticity; the observed anisotropy of this mutual friction (part I of this paper) is consistent only with a vortex-line model, so that the theoretical decision in favour of this model is confirmed by experiment. A detailed calculation of the magnitude and temperature-dependence of this mutual friction is given for the quantized vortex-line model of Onsager (1949) and Feynman (1955). The vortex lines are treated as classical vortex lines belonging entirely to the superfluid. The force of mutual friction arising from the collision of rotons with these lines is calculated in terms of the roton-line collision diameter σ̅, taking into account a tendency for the lines to drag the gas of excitations (i. e. the normal fluid) in their vicinity, and a transverse motion of the lines due to the Magnus effect. The calculated mutual friction contains two components: one parallel to, and one perpendicular to, (vs — vn). The magnitude of the former component agrees well with the experimental results if σ̅ is taken to be about 10 Å. The agreement between theory and experiment confirms that the normal fluid is dragged by the lines, and shows that the spacing of the lines must be close to the theoretical value given by Feynman; but it provides no evidence for or against a motion of the lines due to the Magnus effect. A rough value for σ̅ is calculated in an appendix, and shown to agree as well as can be expected with the value derived from experiment.

The rotation of liquid helium II I. Experiments on the propagation of second sound in uniformly rotating helium II

An experimental investigation of the propagation of second sound in uniformly rotating resonators filled with liquid helium II has been made. It is found that in the uniformly rotating liquid the velocity of the second sound is not changed by more than 0·1%, but there is an excess attenuation which is, except near the λ point, proportional to the angular velocity ω, independent of second-sound amplitude, and independent of frequency in the range 1·5 to 4·5 kc/s. These results are described phenomenologically by a mutual friction force B(ρsρn/ρ) ω(vs — vn) per unit volume in the two-fluid model. The constant B is of order unity for second sound propagated at right angles to the axis of rotation; it is smaller by a factor of at least 5 when the second sound is propagated parallel to the axis of rotation. It is suggested that the mutual friction in rotating helium may contain a component perpendicular to (vs — vn), and that there should be no mutual friction in an irrotational circulation. Experiments to verify these predictions are proposed.

Friction on quantized vortices in helium II. A review

We present an analysis of recent data on friction and drag on quantized vortices in helium II. From these data, we deduce values of the phenomenological and microscopic coefficients of friction on vortex lines and rings over a wide range of temperatures at the saturated vapor pressure. We demonstrate that the microscopic parameters are unusually sensitive to the input data. We include brief discussions of the vortex core parameter, and present the results of precision fits of a number of thermodynamic and transport properties of He II which are used in mutual friction calculations.

Mutual friction in a heat current in liquid helium ii I. Experiments on steady heat currents

Experiments have been carried out on the conduction of heat through helium II in channels of large rectangular cross-section (~ 2 × 6 mm) for small heat current densities. The observed relationship between temperature gradient and heat-current density can be interpreted phenomenologically in terms of the Gorter-Mellink (1949) mutual friction force, Fsn ≈ Aρsρn(vs-vn)3 per unit volume, in the two-fluid model, and observed values of A have been found to agree fairly well with those deduced from earlier measurements. Evidence is presented to show that the magnitude of the mutual friction is determined entirely by the value of (vs-vn), independently of the boundary conditions imposed on the flow. A study of the propagation of second sound across the heat currents has shown that, while the presence of the heat current leads to no observable change in the velocity of the second sound, it does lead to an attenuation; the attenuation is linear and approximately proportional to the square of the heatcurrent density. This behaviour can be described phenomenologically in terms of the twofluid model, if it is assumed that, in the presence of both a steady heat current and a second sound wave, the Gorter-Mellink mutual friction must be generalized to the form Fsn = AρsρnU2u, where u is the instantaneous relative velocity between the two fluids and U is the time-average of this relative velocity. This result shows that in the presence of a steady heat current one or both of the fluids must become modified in some way, and that an essentially linear mutual friction is associated with this modification. Observation of changes in the attenuation of second sound provides a more sensitive method of measuring mutual friction than does the observation of temperature gradients, and it has been shown by the former technique that in the channels used in the present work there is a critical heat current below which the mutual friction is either absent or very small.

The detection of single quanta of circulation in liquid helium ii

An apparatus is described for detecting single quanta of superfluid circulation round a fine wire in liquid helium II. The wire is stretched down the centre of a cylindrical vessel containing helium, and the circulation may be established by rotating the whole apparatus about the axis of the wire and cooling from above the λ-point. The wire can be set into transverse vibration, and the circulation round it can then be obtained from the rate of precession of the plane of vibration. The technique proves to be sufficiently sensitive for the measurement of circulations of order h/m with an accuracy of about 3%. The method in its present form measures only an average of the circulation along the length of the wire, and it is found that this average is not quantized. Apparent circulations equal to a fraction of a quantum are attributed to quantized vortices that are attached to only a fraction of the length of the wire, and this interpretation has been confirmed by showing that an apparent circulation of exactly h/m has much greater stability than any other value. In this way the quantization of superfluid circulation in units of h/m has been experimentally verified. Observations made in the course of this work show clearly that superfluid circulations (including free vortex lines) can persist indefinitely even when the rotation of the apparatus is stopped. Values have also been obtained for the circulation round the wire as a function of the angular velocity of rotation, and it is shown from these that the energy of a free vortex line in the helium surrounding the wire may perhaps be considerably smaller than has hitherto been supposed.

Mutual Friction in a Heat Current in Liquid Helium II. II. Experiments on Transient Effects

It was shown in part I that, when helium II is carrying a steady heat current on which is superimposed a second-sound wave, the mutual friction acting between the two fluids (the Gorter-Mellink force) is of the form G(vs - vn), where (vs - vn) is the instantaneous relative velocity between the fluids, and the factor G is proportional to the square of the time average of this relative velocity. The present paper describes some experimental studies that have been made of the manner in which G changes when the heat current in a wide (~ 2 mm) channel is suddenly changed from one steady value to another; the changes in G have been observed as changes in the attenuation of second sound, and, where possible, as changes in the temperature gradient in the helium. It has been found, for example, that, when a steady supercritical heat current is suddenly switched on in initially undisturbed helium, G rises to its equilibrium value only after a delay time which is of the order of 1s, and that, when the heat current is removed, a non-zero value of G persists for at least 30s. The results indicate that the Gorter-Mellink force is probably associated with turbulence in the superfluid. It is suggested that the force may therefore be due fundamentally to the presence in the superfluid of motions for which curl vs ≠ 0, and it is recalled that experimental evidence in favour of this view has been provided by the recent discovery (Hall & Vinen 1956a) that a mutual friction acts in helium that is simply in a state of uniform rotation.

Mutual Friction in a Heat Current in Liquid Helium II. IV. Critical Heat Currents in Wide Channels

Experiments are described in which a heat current in a wide channel is suddenly increased from a small value W1 to a large value W2; the time characterizing the build-up of the Gorter-Mellink mutual friction to its equilibrium value in the heat current W2 is studied as a function of W1. Interpretation of the results on the basis of the idea that the mutual friction is associated with turbulence (in the form of vortex lines) in the superfluid shows that some mutual friction exists in the heat current W1 even when the latter is less than the critical value described in parts I and II, and that, as the channel width is increased or the temperature raised, the magnitude of the subcritical mutual friction increases until the critical heat current ceases to exist. It is shown that these observations on mutual friction in small heat currents can be described semi-quantitatively if a single term is added to the expression obtained in part III for the length of vortex line per unit volume in a heat current in a channel of infinite width, and that this term can probably arise either from an annihilation of vortex lines at the walls of the channel or from interference by the walls with the mechanisms of growth and decay of superfluid turbulence discussed in part III. Finally, an explanation is suggested of some of the results described in part II on the decay of mutual friction in the presence of a subcritical heat current.

Kelvin-wave cascade on a vortex in superfluid 4He at a very low temperature.

A study by computer simulation is reported of the behavior of a quantized vortex line at a very low temperature when there is continuous excitation of low-frequency Kelvin waves. There is no dissipation except by phonon radiation at a very high frequency. It is shown that nonlinear coupling leads to a net flow of energy to higher wave numbers and to the development of a simple spectrum of Kelvin waves that is insensitive to the strength and frequency of the exciting drive. The results are likely to be relevant to the decay of turbulence in superfluid 4He at very low temperatures.