Mutual Friction in Bosonic Superfluids: A Review

Abstract

Mutual friction caused by scattering of thermal excitations by quantized vortices is a phenomenon of key importance for the understanding of vortex dynamics and quantum turbulence in finite-temperature superfluids. The article reviews theoretical, numerical, and experimental results obtained during the last 40 years in the study of mutual friction in bosonic superfluids such as $^4$He and Bose–Einstein condensates. Among several research topics reviewed in this article particular attention is paid to the development of theory and numerical analysis of roton-vortex interactions and mutual friction in superfluid helium; an application of the two-fluid model for the analysis of the flow in the vicinity of the vortex core and the calculation of the longitudinal and transverse forces exerted on a vortex; vortex diffusivity in $^4$He films; and theoretical, numerical, and experimental studies of thermal dissipation and mutual friction in finite-temperature Bose–Einstein condensates.

1 Introduction

This article reviews theoretical, numerical, and experimental studies of mutual friction in $^4$He and Bose–Einstein condensates published since 1983.

The concept of a mutual friction force between the normal and superfluid components in the finite-temperature¹ superfluid helium was introduced in 1949 by Gorter and Mellink [1]. In mid-1950s pioneering experimental and theoretical studies by Hall and Vinen of the mutual friction in rotating helium [2,3,4], and by Vinen in the thermal counterflow [5,6,7] resulted in understanding of basic physical mechanisms responsible for the emergence of the friction force exerted by thermal quasiparticle excitations (phonons and rotons, which make the normal component of $^4$He at finite temperatures) on quantized vortices in superfluid helium. Vinen was the first to recognize the importance of mutual friction for understanding quantum turbulence, the phenomenon to the study of which Vinen’s contribution cannot be overestimated. Since then mutual friction remained for Joe Vinen a topic of lifelong deep interest. The work of Hall and Vinen was quickly followed by further experimental and theoretical studies. A significant contribution to the theory of mutual friction was made by early works [8,9,10,11,12,13,14,15,16,17] and a little later by theoretical studies by Hillel, Hall, and Vinen [18,19,20]. In 1983, Barenghi, Donnelly, and Vinen published a comprehensive, in-depth review [21] that described the concurrent state-of-the-art in experimental and theoretical studies of the mutual friction in $^4$He. Below in our article we will review the progress made since that time.

Beside the review article by Barenghi et al. other excellent texts should be mentioned here that cover many aspects of theoretical and experimental developments in the study of mutual friction. Among these texts are the 1974 review article on superfluid mechanics by Roberts and Donnelly [22], the book (now classical) by Donnelly on quantized vortices in helium II [23], a review on hydrodynamics of rotating superfluids by Sonin [24] and his book [25] on quantized vortices in superfluids. In this paper we will try, where possible, to avoid reviewing in detail the topics that have already been discussed in depth in the texts cited above (among such topics will be, for example, the phonon scattering by quantized vortices and the detailed analysis of controversy related to the Iordanskii force).

A mutual friction in $^4$He remains a very active area of research, but already from mid-1990s the researchers’ interests began gradually shifting toward the study of friction forces exerted on quantized vortices in finite-temperature atomic Bose–Einstein condensates (BECs). This hardly amounts to a paradigm shift as basic mechanisms that determine the mutual friction remain the same as in the superfluid helium. A study of mutual friction in atomic condensates is often conceptually simpler than that in superfluid helium (and in some cases can be done from first principles); besides, the results of such a study are often qualitatively applicable to the analysis of mutual friction in $^4$He. To the present date there emerged a large body of theoretical, numerical, and, more recently, experimental studies on mutual friction in the BECs. These have never been reviewed before; hopefully our article will fill the gap.

We will not review here the works on mutual friction in fermionic systems such as superfluid $^3$He-B and $^3$He-A, ultracold Fermi gases, and superconductors. An inclusion of these topics would make a review excessively large. More importantly, there already exists an excellent review by Kopnin [26] on mutual friction in superconductors (many aspects are also discussed in Sonin’s book [25]). The interested reader should also be directed to the work by Bevan et al. [27] which discussed and reviewed many important aspects of mutual friction in $^3$He.

The plan of this paper is as follows. Two large Sects. 2 and 3 are dedicated, respectively, to the mutual friction in $^4$He and the BEC. Section 4 summarizes the review and contains some suggestions on possible avenues of the future research.

Section 2.1 of Sect. 2 summarizes definitions of the pairs of friction parameters; the content of this subsection is not new and can be found elsewhere, e.g., in Refs. [21, 23, 25]; Sect. 2.2 reviews the work on the frequency and velocity dependence of friction parameters in non-stationary flow conditions; Sect. 2.3 discusses the advances in the theory of mutual friction near the $\lambda$-point; Sect. 2.4 reviews the development of the theory of roton-vortex interactions and calculations of the longitudinal and transverse friction coefficients; Sect. 2.5 and the beginning of Sect. 2.6 are just short notes on phonon scattering by quantized vortices and the Iordanskii force (a comprehensive discussion on these topics can be found in Sonin’s book [25]). The remainder of Sect. 2.6 discusses the contribution of Thouless, Vinen, and others to the resolution of controversy related to the existence of the Iordanskii force. Section 2.7 reviews a modeling of mutual friction in the new self-consistent approach to the numerical simulation of quantum turbulence in $^4$He. In Sect. 2.8 a quantum-mechanical approach is reviewed to the calculation of friction force exerted by elementary excitations on a massive vortex in the two-dimensional Bose superfluid. In two-dimensional $^4$He films the primary transport parameter is the vortex diffusivity which can be expressed via the mutual friction parameters; the works on measurements of the vortex diffusivity in superfluid helium films are reviewed in Sect. 2.9.

Section 3.1 of Sect. 3 is a brief introduction to vortex dynamics in (quasi) two-dimensional Bose–Einstein condensates; Sect. 3.2, also briefly, describes the first theoretical calculations of the mutual friction parameters in the finite-temperature² BEC; in Sect. 3.3 the theoretical work is reviewed which investigates a link between thermal dissipation and mutual friction; Sect. 3.4 discusses the theoretical/numerical studies of mutual friction in the framework of the dissipative (“damped”) Gross–Pitaevskii equation (GPE); Sect. 3.5 briefly lists the mean-field, GPE-based models that are currently being used for the simulation of vortex dynamics and the analysis of mutual friction in the finite-temperature BEC; in Sect. 3.6 we review the applications of the truncated and stochastic projected GPEs to the analysis of finite-temperature effects and calculations of the longitudinal and transverse friction coefficients; Sect. 3.7 reviews an analysis of mutual friction in the framework of the Zaremba–Nikuni–Griffin kinetic theory which couples a generalized GPE with a Boltzmann equation for thermally excited, non-condensed atoms; in Sect. 3.8 we review experimental studies of mutual friction in Bose-Einstein condensates.

2 Mutual Friction in Superfluid $^4$He

2.1 Preliminaries: Pairs of Friction Coefficients

In superfluid $^4$He a core radius $a_0$ of the quantized vortex is typically several orders of magnitude smaller than macroscopic and mesoscopic length scales such as a size of the system, characteristic length of a coarse-grain averaged flow field, mean intervortex distance $\ell ={\mathcal {L}}^{-1/2}$ in rotating helium or quantum turbulence (here ${\mathcal {L}}$ is the vortex line density, that is a vortex length per unit volume), or typical inverse curvature of the vortex line (we will not consider here phenomena associated with reconnections of vortex line in quantum turbulence). It will suffice to reduce the problem of interaction between quantized vortices and $^4$He to an analysis of forces exerted on a rectilinear vortex line whose orientation is defined by the unit vector ${\textbf{s}}'$ in the direction of circulation ${\varvec{\kappa }}=\kappa {\textbf{s}}'$, where $\kappa$ is the quantum of circulation; the vortex line will be assumed to be moving, with velocity ${\textbf{v}}_L$, in the direction orthogonal to ${\textbf{s}}'$. Far from the vortex core the flow of helium is assumed uniform with ${\textbf{v}}_n$ and ${\textbf{v}}_s$ being the velocities of the normal and superfluid components, respectively.

The forces, per unit length, acting on a vortex are the Magnus force (see, e.g., [28, 29])

$$\begin{aligned} {\textbf{f}}_M={\rho _s}{\varvec{\kappa }}\times ({\textbf{v}}_L-{\textbf{v}}_s), \end{aligned}$$

(1)

and the friction force

$$\begin{aligned} {\textbf{f}}_D=\gamma _0({\textbf{v}}_n-{\textbf{v}}_L)+\gamma '_0{\textbf{s}}'\times ({\textbf{v}}_n-{\textbf{v}}_L), \end{aligned}$$

(2)

where $\gamma _0$ and $\gamma '_0$ are temperature-dependent mutual friction coefficients. It should be mentioned here that, based on the full hydrodynamic equations for $^4$He, the friction force in the general form given by Eq. (2) was obtained by Chandler and Baym [30] from the analysis of dissipation in rotating superfluids at finite temperatures. Assuming that the vortex core is massless, the friction force must balance the Magnus force, that is ${\textbf{f}}_D+{\textbf{f}}_M=\textbf{0}$.

The physical mechanism of mutual friction is the scattering by a vortex of quasiparticle thermal excitations (mainly rotons at temperatures above 0.6 K, and phonons at lower temperatures) which form the normal component of superfluid $^4$He. It seems, therefore, that a more fundamental form of the mutual friction force should be formulated via the average drift velocity ${\textbf{v}}_R$ of quasiparticles in the close vicinity of the vortex core:

$$\begin{aligned} {\textbf{f}}_D=D({\textbf{v}}_R-{\textbf{v}}_L)+{D_t}{\varvec{\kappa }}\times ({\textbf{v}}_R-{\textbf{v}}_L). \end{aligned}$$

(3)

The microscopic coefficients D and ${D_t}$ are related to the scattering length of excitations by the vortex and, therefore, are independent of the relative motion of the vortex and the normal fluid. The coefficient D, which determines the longitudinal component of the friction force, can be found via the equilibrium distribution function of rotons in the quiescent liquid,

$$\begin{aligned} n_0(p)=\left[ (2\pi \hbar )^3\left( e^{E_0(p)/kT}-1\right) \right] ^{-1}, \end{aligned}$$

(4)

(where ${\textbf{p}}$ is the roton’s momentum, $p=\vert {\textbf{p}}\vert$, E₀(p) is the quasiparticle’s energy in the fluid at rest, T is temperature, and k is the Boltzmann constant), the scattering cross section (or, in two dimensions, the scattering length) ${\sigma _{\parallel }}(p)$, and the excitation’s group velocity ${v_G}=dE_0(p)/dp$ as [17, 25]

$$\begin{aligned} D=-\frac{1}{3(2\pi \hbar )^3}\int \frac{dn_0}{dE_0}p^2{\sigma _{\parallel }}{v_G}\,d^3{\textbf{p}}. \end{aligned}$$

(5)

In the vicinity of the roton minimum $p=p_0\approx 2\times 10^{-19}$ g cm/s the energy spectrum for elementary excitations in quiescent $^4$He is given by the well-known Landau approximation

$$\begin{aligned} E_0(p)=\Delta +\frac{(p-p_0)^2}{2\mu ^{\star }}, \end{aligned}$$

(6)

where the energy gap $\Delta \approx 8.7$ K, and $\mu ^{\star }\approx 0.16m$, with m being the mass of helium atom, is the effective mass of the roton. Making use of ${\sigma _{\parallel }}$ calculated in Refs. [17, 19], Sonin [24, 25] obtained the approximate expression for the longitudinal friction coefficient in the form³

$$\begin{aligned} D\approx 2.6\frac{\sqrt{\mu ^\star kT}}{p_0}\kappa {\rho _n}. \end{aligned}$$

(7)

The issue of the friction parameter ${D_t}$ which determines the transverse force is more subtle and remained, for a long time, rather controversial, see Sect. 2.6. It was argued (see, e.g., Ref. [19]) that the coefficient ${D_t}$ should include both the term determined by the transverse scattering cross section (length) ${\sigma _{\perp }}$ as⁴

$$\begin{aligned} D'=\frac{1}{3(2\pi \hbar )^3}\int \frac{dn_0}{dE_0}p^2{\sigma _{\perp }}{v_G}\,d^3{\textbf{p}}, \end{aligned}$$

(8)

and the contribution from the transverse Iordanskii force

$$\begin{aligned} {\textbf{f}}_I=-{\rho _n}{\varvec{\kappa }}\times ({\textbf{v}}_R-{\textbf{v}}_L), \end{aligned}$$

(9)

so that

$$\begin{aligned} {D_t}=D'-{\rho _n}\kappa . \end{aligned}$$

(10)

For phonons scattered by the quantized vortex, the force given by Eq. (9) has been discovered by Iordanskii [12, 13]. For roton scattering the identical force (albeit with a wrong sign later corrected in Ref. [17]) was discovered earlier by Lifshitz and Pitaevskii [8].

The controversy arose when it was argued (see, e.g., Ref. [19] and the review by Barenghi et al. [21]) that the contribution $-{\rho _n}\kappa$ from the Iordanskii force should always be added to $D'$ determined by the scattering cross section so that the total transverse friction coefficient should be determined by Eq. (10). An ambiguity of such a proposal is threefold (see, e.g., the review [24] for a detailed discussion). Firstly, various models of the transverse cross section may or may not contain the contribution of the Iordanskii force. Secondly, while the effective transverse scattering cross section for phonons might be an ‘ambiguous concept’ [24], the calculation of ${\sigma _{\perp }}$ for rotons based on the theory developed by Lifshitz and Pitaevskii [8] is rather straightforward and yields $D'=-{\rho _n}\kappa$ in Eq. (8) so that an addition in Eq. (10) of the contribution of the Iordanskii force actually means accounting for this force twice. Thirdly, an existence of the Iordanskii force was later put in doubt by a number of authors. This led to rather prolonged debates that lasted from mid-1990s when Thouless and his co-workers [31] argued that the Iordanskii force does not exist (so that the only transverse force acting on a vortex is the Magnus force) until 5 years later it was recognized that the original theory ignored the nonzero contribution from circulation of the roton fluid around the vortex core [32,33,34,35]. These debates will be briefly reviewed below in Sect. 2.6.

We will now turn to relations between the pairs of friction parameters D, ${D_t}$ (or $D'$) and $\gamma _0$, $\gamma '_0$. It should be emphasized that the average drift velocity ${\textbf{v}}_R$ of excitations in the close vicinity of the vortex core differs from the normal fluid velocity ${\textbf{v}}_n$ far from the core. This effect is known as a viscous drag. Hall and Vinen’s calculation [3] of this effect was based on Sewell’s classical solution [36] of the Navier–Stokes equation for the problem of scattering of sound waves by cylindrical and spherical obstacles but included a correction accounting for a finite mean free path L of the rotons. Hall and Vinen analyzed the relation between ${\textbf{v}}_R$ and ${\textbf{v}}_n$ in the context of the second sound propagation in rotating superfluids. Having assumed that the characteristic length of the macroscopic flow (e.g., the second sound wave length) is much larger than the intervortex distance $\ell$, and that $\ell \gg \delta \gg L\gg a_0$, where $\delta =(\eta /{\rho _n}\sigma )^{1/2}$, with $\sigma$ being the wave’s angular frequency, is the penetration depth of the normal fluid, and the mean free path

$$\begin{aligned} L=3\eta /({\rho _n}{{\overline{v}}_G}), \end{aligned}$$

(11)

where $\eta$ is the viscosity of the normal fluid and

$$\begin{aligned} {{\overline{v}}_G}=\left( \frac{2kT}{\pi \mu ^\star }\right) ^{1/2} \end{aligned}$$

(12)

is the average group velocity of rotons, Hall and Vinen found that (see the original paper [3] and the review by Barenghi et al. [21] for detail)

$$\begin{aligned} {\textbf{v}}_R-{\textbf{v}}_n={\textbf{f}}_D/E. \end{aligned}$$

(13)

In the steady flow conditions [7] (see also Ref. [37])

$$\begin{aligned} E=-4\pi \eta /M, \end{aligned}$$

(14)

where

$$\begin{aligned} M=\ln \left( L/2\delta _{\text {eff}}\right) +1 \end{aligned}$$

(15)

and

$$\begin{aligned} \delta _{\text {eff}}=2\eta /({\rho _n}\left| {\textbf{v}}_n-{\textbf{v}}_L\right| )\gg L \end{aligned}$$

(16)

is the effective penetration depth (also known as the Oseen length, see, e.g., Refs. [38, 39]).

In their classical paper [3] Hall and Vinen analyzed a time-dependent problem of second sound attenuation by quantized vortices in rotating helium. In this case M becomes complex:

$$\begin{aligned} M=\ln (L/2\delta )+1+{\text {i}}\pi /4, \end{aligned}$$

(17)

where now $\delta$ is the frequency-dependent viscous penetration depth. In Eq. (14), the real part of E is responsible for attenuation, while the imaginary part only modifies the second sound velocity.

Making use of Eq. (13), the relation between the pairs of friction coefficients D, ${D_t}$ and $\gamma _0$, $\gamma '_0$ (see Eq. (2)) can be found as [21]

$$\begin{aligned} \gamma _0=\frac{E\left[ D(D+E)+{D_t}^2\right] }{(D+E)^2+{D_t}^2}, \quad \gamma '_0=\frac{{D_t}E^2}{(D+E)^2+{D_t}^2}. \end{aligned}$$

(18)

Hillel [40] and Sonin [25] suggested a concise form for Eqs. (18) as a complex relation

$$\begin{aligned} \frac{1}{\gamma _0+{\text {i}}\gamma '_0}=\frac{1}{E}+\frac{1}{D+{\text {i}}{D_t}}. \end{aligned}$$

(19)

Another pair of friction coefficients emerges in the mutual friction force ${\textbf{F}}_{ns}$, per unit volume, that appears in the equations for the coarse-grained superfluid and normal velocities in the uniformly rotating helium. In standard notation these equations are [4, 22]

$$\begin{aligned} {\rho _s}\frac{D{\textbf{v}}_s}{Dt}&= -\frac{{\rho _s}}{\rho }{\varvec{\nabla }}P+{\rho _s}S{\varvec{\nabla }}T-{\textbf{F}}_{ns}, \end{aligned}$$

(20)

$$\begin{aligned} {\rho _n}\frac{D{\textbf{v}}_n}{Dt}&= -\frac{{\rho _n}}{\rho }{\varvec{\nabla }}P-{\rho _s}S{\varvec{\nabla }}T+\eta \nabla ^2{\textbf{v}}_n+{\textbf{F}}_{ns}, \end{aligned}$$

(21)

where $D/Dt=\partial /\partial t+(\textbf{v}\cdot {\varvec{\nabla }})$, with $\textbf{v}={\textbf{v}}_s$ or $\textbf{v}={\textbf{v}}_n$, is the Lagrangian derivative, $\rho ={\rho _s}+{\rho _n}$, p is the pressure, and S is the entropy per unit volume. For helium uniformly rotating with angular velocity ${\varvec{\omega }}$ Hall and Vinen [2, 3] found that, provided the angular velocity of rotation is not too small so that the uniform array forms of quantized vortex lines parallel to ${\varvec{\omega }}$, the mutual friction force can be expressed as

$$\begin{aligned} {\textbf{F}}_{ns}=\frac{B{\rho _s}{\rho _n}}{\rho }{\hat{{\varvec{\omega }}}}\times \left( {\varvec{\omega }}\times {\textbf{v}}_{ns}\right) +\frac{B'{\rho _s}{\rho _n}}{\rho }{\hat{{\varvec{\omega }}}}\times {\textbf{v}}_{ns}, \end{aligned}$$

(22)

where ${\textbf{v}}_{ns}={\textbf{v}}_n-{\textbf{v}}_s$, ${\textbf{v}}_n$ and ${\textbf{v}}_s$ are the coarse-grained velocities (that is, space averaged over distances large compared with the intervortex spacing), ${\hat{{\varvec{\omega }}}}={\varvec{\omega }}/\omega$ is a unit vector, and B and $B'$ are the phenomenological (macroscopic) mutual friction parameters.

We will not discuss here a generalization of Eqs. (20)–(22) for the fully developed vortex tangle rather than a regular array of straight vortex lines. Such a generalization is known as the Hall–Vinen–Bekarevich–Khalatnikov (HVBK) model [2, 3, 11] (see also the book by Khalatnikov [41] for detail).

A relation between the pair of friction parameters B, $B'$ and the fundamental friction coefficients D, ${D_t}$ has been found in the form (see, e.g., Ref. [21])

$$\begin{aligned} B=\frac{2\rho }{{\rho _n}{\rho _s}\kappa }\,\frac{a}{a^2+b^2}, \quad B'=\frac{2\rho }{{\rho _n}{\rho _s}\kappa }\,\frac{b}{a^2+b^2}, \end{aligned}$$

(23)

where

$$\begin{aligned} a=\frac{D}{D^2+{D_t}^2}+\frac{1}{E}, \quad b=-\frac{{D_t}}{D^2+{D_t}^2}+\frac{1}{{\rho _s}\kappa }. \end{aligned}$$

(24)

For inversion of these formulae see Ref. [21]. A concise representation of Eqs. (23)–(24) in the form of a single complex relation is given in Refs. [25, 40].

Finally, a pair of friction coefficients $\alpha$, ${\alpha '}$ emerges in the study of quantized vortex dynamics. The friction force, per unit length, acting on the vortex line specified in the parametric form ${\textbf{s}}={\textbf{s}}(\xi ,\,t)$ can be represented as [42, 43]

$$\begin{aligned} \frac{{\textbf{f}}_D}{\kappa {\rho _s}}=\alpha \left\{ {\textbf{s}}'\times \left[ {\textbf{s}}'\times ({\textbf{v}}_n-{\textbf{v}}_{sl})\right] \right\} +{\alpha '}{\textbf{s}}'\times ({\textbf{v}}_n-{\textbf{v}}_{sl}), \end{aligned}$$

(25)

where ${\textbf{s}}'=d{\textbf{s}}/d\xi$ is the tangent unit vector of the curve, and ${\textbf{v}}_{sl}$ is the local superfluid velocity at a given point of the vortex line. Introduced by Schwarz [42, 43] the coefficients $\alpha$ and ${\alpha '}$ are directly linked with the phenomenological mutual friction parameters B and $B'$ by simple relations

$$\begin{aligned} \alpha =\frac{{\rho _n}}{2\rho }B, \quad {\alpha '}=\frac{{\rho _n}}{2\rho }B'. \end{aligned}$$

(26)

In Eq. (25), the local superfluid velocity should be interpreted as the sum of the self-induced velocity ${\textbf{v}}_i$ and the imposed superflow ${\textbf{v}}_s$. Equation (25) then yields the equation of evolution of the vortex line derived by Schwarz [42, 43] in the form

$$\begin{aligned} d{\textbf{s}}/dt={\textbf{v}}_s+{\textbf{v}}_i+\alpha {\textbf{s}}'\times ({\textbf{v}}_n-{\textbf{v}}_s-{\textbf{v}}_i)-{\alpha '}{\textbf{s}}'\times [{\textbf{s}}'\times ({\textbf{v}}_n-{\textbf{v}}_s-{\textbf{v}}_i)]. \end{aligned}$$

(27)

The tables of mutual friction parameters B, $B'$ and $\alpha$, ${\alpha '}$ for temperatures in the range from 1.3 K to $T_\lambda$ are provided by Donnelly and Barenghi in Ref. [44].

2.2 Frequency and Velocity Dependence of Mutual Friction Parameters

Throughout this review we will mostly consider the mutual friction parameters calculated for (quasi) stationary flow conditions. However, in experimental conditions, particularly in quantum turbulence experiments which might involve measurements of second sound attenuation by a vortex tangle, typical frequencies of the flow field may range from the values of order 1 Hz to those exceeding 20 kHz, see, e.g., Refs. [45, 46]. For example, it was found that, in the low-amplitude limit, at temperature 1.8 K the friction parameter B increases from 0.788 at frequency $f=1$ Hz to 1.203 at $f=10$ kHz. Generally, an increase of B becomes more pronounced with an increase of temperature. Likewise, experiments in the turbulent counterflow also showed a strong dependence of the mutual friction coefficients B and $B'$ on the relative vortex line – normal fluid velocity $v_{nL}=\vert {\textbf{v}}_n-{\textbf{v}}_L\vert$, especially in the range of $v_{nL}$ between 10 and 100 cm/s typical of high heat fluxes. This dependence also becomes stronger with the increase of temperature from 1 K toward the $\lambda$-point.

The frequency and velocity dependences of mutual friction parameters were predicted by Hall and Vinen [2] and Vinen [5, 6] and confirmed, for a frequency dependence, by experiments of Miller et al. [47] and Mathieu and Simon [48]. (For the velocity dependence, the experimental results of Yarmchuk and Glaberson [37] qualitatively agree with Vinen’s predictions.)

Having recognized an importance of data for the mutual friction parameters in the wide range of frequencies and velocities, Swanson et al. [49] developed an efficient algorithm and provided data for calculating the frequency- and velocity-dependent friction coefficients $\alpha$ and ${\alpha '}$ (and hence B and $B'$, see Eqs. (26)). Below we briefly outline the procedure proposed in Ref. [49].

At any frequency f or velocity $v_{nL}$ the values of the mutual friction parameters $\alpha$ and ${\alpha '}$ can be calculated from the values $\alpha _1$ and ${\alpha '}_1$ of these parameters known at some reference frequency $f_1$ making use of the formulae

$$\begin{aligned} \alpha =\frac{\alpha _1+\left( \alpha _1^2+{\alpha '}_1^2\right) {\tilde{\Delta }}}{A}, \quad {\alpha '}=\frac{{\alpha '}_1}{A}, \end{aligned}$$

(28)

where $A=1+2\alpha _1{\tilde{\Delta }}+\left( \alpha _1^2+{\alpha '}_1^2\right) {\tilde{\Delta }}^2$, and

$$\begin{aligned} {\tilde{\Delta }}=\frac{{\rho _s}\kappa }{8\pi \eta }\ln \frac{f_1}{f} \quad \text {or} \quad {\tilde{\Delta }}=\frac{{\rho _s}\kappa }{8\pi \eta }\ln \frac{8\pi \eta f_1}{{\rho _n}v_{nL}^2}. \end{aligned}$$

(29)

The values of $\alpha _1(T)-\alpha _\lambda (T)$ and ${\alpha '}_1(T)-{\alpha '}_\lambda (T)$ for a reference frequency 100 Hz can be determined from the tables in Ref. [49] making use of cubic splines described in detail in Refs. [44, 50]; here $\alpha _\lambda (T)$ and ${\alpha '}_\lambda (T)$ are provided by asymptotic formulae [49] valid as $T\rightarrow T_\lambda$. The values of other temperature-dependent properties (densities, viscosities, etc.) can be found in Ref. [44] (note also that an interactive website [51] can be used for online evaluation of the properties of liquid helium at the saturated vapor pressure for any given temperature $T<T_\lambda$).

2.3 Mutual Friction Near the Superfluid Transition

The first calculation of the mutual friction coefficients in the close vicinity of the superfluid transition has been done by Ginzburg and Sobyanin [52, 53]. Their approach was based on the so-called $\Psi$-model by the same authors which itself is a further development of Pitaevskii’s celebrated phenomenological theory [54] of superfluidity near the $\lambda$-point.

Later the key experimental study of mutual friction in the vicinity of $\lambda$-transition was done by Mathieu et al. [55] followed by a theoretical interpretation of their findings in Ref. [56]. Based on the Ginzburg–Sobyanin model, the theory has been further developed by Sonin [57]. Both the experiment by Mathieu et al. and the theoretical calculation by Sonin showed that for temperature T close to $T_\lambda$ the friction coefficients in Eq. (2) behave as

$$\begin{aligned} \gamma _0=a\epsilon ^{1/3} \quad \text {and} \quad \gamma '_0=1+b\epsilon ^{1/3}, \end{aligned}$$

(30)

where

$$\begin{aligned} \epsilon =(T_\lambda -T)/T_\lambda \ll 1. \end{aligned}$$

(31)

From their experimental results Mathieu et al. found $a\approx 2.8$ and $b\approx 2$, while Sonin’s calculation yielded $a\approx 1$, $b\approx 1.5$.

Based on Halperin and his co-authors’ approach [58, 59], which, in turn, is the further generalization of Ginzburg and Sobyanin’s theory, Onuki [60] developed a theory which resulted in the values of $\gamma _0$ and $\gamma '_0$ in a better agreement than those found by Sonin with the experimental results of Mathieu et al. Considering the importance of Onuki’s findings, his work is described below in more detail.

Onuki’s model is based on the generalization of the Ginzburg–Pitaevskii dissipative equation for the complex order parameter $\psi ({\textbf{r}},\,t)$ complemented by the entropy conservation equation which includes the term responsible for the thermal conductance of helium:

$$\begin{aligned} {\text {i}}\hbar \frac{\partial \psi }{\partial t}= & {} -\frac{\hbar ^2}{2m}\nabla ^2\psi +\left( \mu _n+\mu _s\right) m\psi \nonumber \\{} & {} \quad -{\text {i}}\Lambda \left\{ \frac{1}{2}\left( \frac{{\text {i}}\hbar }{m}{\varvec{\nabla }}+{\textbf{v}}_n\right) ^2+\mu _s\right\} m\psi +f, \end{aligned}$$

(32)

$$\begin{aligned} \frac{\partial {\tilde{s}}}{\partial t}+{\textbf{v}}_n\cdot {\varvec{\nabla }}{\tilde{s}}= & {} s_0{\varvec{\nabla }}\cdot \left( \textbf{j}_s-{\rho _s}{\textbf{v}}_n\right) -{\overline{\lambda }}_0\nabla ^2\mu _n+\zeta , \end{aligned}$$

(33)

where m is the mass of helium atom, $\mu _n$ and $\mu _s$ have the usual meaning of chemical potentials, ${\tilde{s}}=\rho _0S/(k\rho )$ with $\rho _0$ being the average density of helium and S the entropy density, $s_0$ is the average entropy per unit mass, and ${\overline{\lambda }}_0$ is the (scaled) thermal conductivity; the Pitaevskii’s relaxation parameter $\Lambda$ is assumed to be complex with $\text {Re}\,\Lambda >0$. An important feature of the Onuki model is the presence in Eqs. (32)–(33) of the Gaussian Markov source terms f and $\zeta$ characterized by

$$\begin{aligned}&\langle f({\textbf{r}},\,t)f^{\star }({\textbf{r}}',\,t')\rangle \sim \text {Re}\,\Lambda \delta ({\textbf{r}}-{\textbf{r}}')\delta (t-t'), \nonumber \\&\text {and} \nonumber \\&\langle \zeta ({\textbf{r}},\,t)\zeta ({\textbf{r}}',\,t')\rangle \sim -{\overline{\lambda }}_0\nabla ^2\delta ({\textbf{r}}-{\textbf{r}}')\delta (t-t') \end{aligned}$$

(34)

with $\langle ff\rangle =\langle f\zeta \rangle =\langle f^{\star }\zeta \rangle =0$ (for a detailed discussion of the thermal noise terms f and $\zeta$ see, e.g., Ref. [61]).

As was noticed by Sonin [25], making use of Ginzburg–Sobyanin’s real relaxation parameter $\Lambda \approx 0.3\epsilon ^{-1/3}$ [53] Onuki’s estimates yield the mutual friction coefficients

$$\begin{aligned} \gamma _0\approx 2.07\epsilon ^{1/3} \quad \text {and} \quad \gamma '_0\approx 1+1.93\epsilon ^{1/3} \end{aligned}$$

(35)

which are in a better agreement with the experimental results by Mathieu et al. [55] than those calculated by Sonin [57] (although it is not clear if this improvement is due to accounting for a thermal conductivity or simply to a more precise calculation of the superfluid density and entropy distributions).

The critical behavior (as $\epsilon \rightarrow 0$) of the microscopic friction coefficients was found by Onuki in the form $D\sim \epsilon ^{2/3}$ and $D'={\rho _s}\kappa +O\left( \epsilon ^{2/3}\right)$, in apparent contradiction with the results by Pitaevskii [10] who found $D\sim \epsilon$ and $D'={\rho _s}\kappa +O(\epsilon )$. As was pointed out by Hillel [40], this contradiction resulted from an algebraic error; the corrected derivations showed the critical behavior in full agreement with that predicted by Pitaevskii.

In general, Onuki’s theory explains rather well the experimentally observed [55] behavior of the mutual friction parameters in the close vicinity of the $\lambda$-transition, including among other results a negative sign of $B'$.

2.4 Roton-Vortex Interactions and Mutual Friction Parameters in $^4$He

In the range of temperatures between 0.6 K and $T_\lambda$ the mutual friction between the normal fluid and quantized vortices is determined mainly by the roton-vortex interactions and the contribution of phonons can be ignored. The idea that the analysis of mutual friction in $^4$He can be based on the quasiclassical treatment of roton trajectories was first suggested by Lifshitz and Pitaevskii [8] and later used by Goodman [16] and Sonin [17] who calculated the momentum transfer along straight line trajectories. This quasiclassical approach, which has been discussed in detail by Rayfield and Reif [14] and Roberts and Donnelly [22], is briefly outlined below.

In quiescent $^4$He the roton’s group velocity is ${v_G}=dE_0/dp$, where $E_0(p)$ is given by the Landau approximation (6). Rotons with $p>p_0$ are called $R^{+}$-rotons and have their momentum parallel to their group velocity. Rotons with $p<p_0$ are $R^{-}$-rotons which have their group velocity being antiparallel to their momentum.

In the presence of the vortex flow field ${\textbf{v}}_s({\textbf{r}})=(\kappa /2\pi r)\hat{{\varvec{\phi }}}$, where $\hat{{\varvec{\phi }}}$ is the unit vector in the asymuthal direction in the $(x,\,y)$-plane orthogonal to the rectilinear vortex line, and $r=\sqrt{x^2+y^2}$ is the distance from the vortex core, the dispersion curve (6) modifies as

$$\begin{aligned} E(p)=E_0(p)+{\textbf{p}}\cdot {\textbf{v}}_s. \end{aligned}$$

(36)

The quasiclassical trajectories ${\textbf{r}}={\textbf{r}}(t)$ and rotons’ momenta ${\textbf{p}}={\textbf{p}}(t)$ can then be calculated solving the Hamiltonian equations of motion

$$\begin{aligned} \frac{d{\textbf{r}}}{dt}=\frac{\partial E({\textbf{p}},\,{\textbf{r}})}{\partial {\textbf{p}}}, \quad \frac{d{\textbf{p}}}{dt}=-\frac{\partial E({\textbf{p}},\,{\textbf{r}})}{\partial {\textbf{r}}}-{\varvec{\nabla }}({\textbf{p}}\cdot {\textbf{v}}_s). \end{aligned}$$

(37)

The term $-{\varvec{\nabla }}({\textbf{p}}\cdot {\textbf{v}}_s)$ in the second equation should be interpreted as a force acting on a quasiparticle excitation.

We start with the contribution to the theory of interactions between rotons and quantized vortices made by Hillel in his 1981 work [19] which was later corrected by him and Vinen [20] and then developed further in his post-1983 paper [40]. Summing the force $-{\varvec{\nabla }}({\textbf{p}}\cdot {\textbf{v}}_s)$ acting on excitations over the entire roton fluid, Hillel [19] first calculated the force exerted on the roton fluid by a vortex without a core, and then, making use of the Goodman’s model [16], calculated the contribution to this force made by a finite-size vortex core which absorbs and re-emits the rotons. Hillel found that the coreless vortex exerts a transverse force so that an analysis of interactions between rotons and the core is essential to account for a substantial longitudinal component of the friction force observed in experiments. Hillel’s calculation, revised and corrected later by him and Vinen [20] yielded the microscopic friction parameters in the form

$$\begin{aligned}&D=1.8{\rho _n}{{\overline{v}}_G}a_0, \end{aligned}$$

(38)

$$\begin{aligned}&D'=\tfrac{1}{2}{\rho _n}\kappa , \end{aligned}$$

(39)

where $a_0$ is the core radius, and ${{\overline{v}}_G}$ is roton’s average group velocity given by Eq. (12).

The theory developed in Refs. [19, 20] has been already reviewed by Barenghi et al. [21] so we do not need to provide further details here.

An ‘unpleasant’ (to quote Hillel [40]) feature of the theory developed in Ref. [19] was that the calculation of the force ${\textbf{f}}_{\text {ex}}$ exerted on rotons by a vortex depended on the choice of shape of the domain over which the force $-{\varvec{\nabla }}({\textbf{p}}\cdot {\textbf{v}}_s)$ has been integrated. This feature was also present in calculations by Sonin [17] and other authors. In his later work [40] Hillel reconsidered his and Vinen’s earlier approach with the aim to get rid of this rather unfortunate feature of the theory. Hillel distinguished between two alternative macroscopic treatments of the finite-temperature $^4$He: he argued that the liquid may be treated either as the normal fluid plus superfluid, or as excitations plus a background fluid. The former served as a basis for theory developed in his and Vinen’s earlier work [19, 20]. In his later, 1987 work [40] Hillel claimed that this was precisely this treatment that caused the ‘unpleasant’ feature mentioned above. He argued that the second approach enables one to develop a theory free of the dependence of results on the shape of integration domain and also to avoid an explicit introduction of the Iordanskii force (for more on the controversy related to this force see Sect. 2.6).

The first treatment ($^4$He = normal fluid + superfluid) is based on the balance of forces ${\textbf{f}}_s+{\textbf{f}}_n=\textbf{0}$, where ${\textbf{f}}_s$ and ${\textbf{f}}_n$ are forces exerted by the vortex on the superfluid and the normal fluid, respectively. In the framework of Hillel’s second approach, the balance of forces was formulated as

$$\begin{aligned} {\textbf{f}}_{\text {ex}}+{\textbf{f}}_b=\textbf{0}, \end{aligned}$$

(40)

where ${\textbf{f}}_b$ is the force exerted by the vortex on the background fluid. Hillel claimed that the force ${\textbf{f}}_b$ has never been calculated directly. Instead, in Refs. [19, 20] the equation ${\textbf{f}}_s+{\textbf{f}}_n=\textbf{0}$ was invoked, and then Hillel claimed that ${\textbf{f}}_s$ is determined by the Magnus force and that the difference between ${\textbf{f}}_n$ and ${\textbf{f}}_{\text {ex}}$ is the Iordanskii force given by Eq. (9). He then argued that such an approach is fundamentally incorrect as it does not involve any assumption regarding the background fluid and ultimately leads to the dependence on the shape of the integration domain in calculation of the force exerted on excitations. Hillel then concluded that the deficiency of theory [19, 20] can be cured if the force ${\textbf{f}}_b$ were calculated directly. To do so he invoked the equations of the Landau two-fluid model to estimate the rate of change of momentum of the background fluid as it passes the vortex core. For the force exerted on the background fluid he found

$$\begin{aligned} {\textbf{f}}_b={\rho _s}{\varvec{\kappa }}\times ({\textbf{v}}_s-{\textbf{v}}_L)+{\rho _n}{\varvec{\kappa }}\times ({\textbf{v}}_R-{\textbf{v}}_L), \end{aligned}$$

(41)

which is, apparently, the same result as in Refs. [19, 20] but now derived without explicitly invoking either the Magnus or the Iordanskii force.

In Refs. [19, 20], the force exerted on excitations has been calculated as ${\textbf{f}}_{\text {ex}}=-{\textbf{f}}_D$, where ${\textbf{f}}_D$ is given by Eq. (3) with D and ${D_t}$ given by Eqs. (38)–(39) and (10). Hillel argued that the calculations of forces ${\textbf{f}}_{\text {ex}}$ and ${\textbf{f}}_b$ should be consistent so that the contribution to ${\textbf{f}}_{\text {ex}}$ of the force $-{\varvec{\nabla }}({\textbf{p}}\cdot {\textbf{v}}_s)$ arising from the (infinite) surface of the domain of integration should be discarded. This contribution, calculated in Ref. [19] as $\tfrac{1}{2}{\rho _n}{\varvec{\kappa }}\times ({\textbf{v}}_R-{\textbf{v}}_s)$, was precisely the one that caused the shape-dependence problem. Then, instead of ${\textbf{f}}_{\text {ex}}=-{\textbf{f}}_D$ with ${\textbf{f}}_D$ given by Eq. (3), Hillel obtained [40]

$$\begin{aligned} {\textbf{f}}_{\text {ex}}=-\left[ 1.8{\rho _n}{{\overline{v}}_G}a_0\left( {\textbf{v}}_R-{\textbf{v}}_L\right) +\tfrac{1}{4}{\rho _n}{\varvec{\kappa }}\times \left( {\textbf{v}}_R-{\textbf{v}}_L\right) +\tfrac{1}{2}{\rho _n}{\varvec{\kappa }}\times \left( {\textbf{v}}_R-{\textbf{v}}_s\right) \right] . \end{aligned}$$

(42)

Making use of Eqs. (40) and (41) Hillel found the microscopic friction parameters D and ${D_t}=D'-{\rho _n}\kappa$ in the revised form

$$\begin{aligned} D=\frac{1.8{\rho _n}{{\overline{v}}_G}a_0}{1+{\rho _n}/2{\rho _s}}, \quad {D_t}=-\frac{{\rho _n}\kappa }{4(1+{\rho _n}/2{\rho _s})}. \end{aligned}$$

(43)

Making use of Eqs. (23) and (24) (or their representation in the complex form, see Ref. [25, 40]), Hillel calculated the friction coefficients B and $B'$ and compared them with experimental data [55, 62] without compensating for frequency effects which are small compared to the scatter of the experimental data. This comparison, illustrated in Fig. 1, shows a reasonably good agreement for temperatures below 2 K (note also some improvement in the behavior of $B'$ compared to earlier prediction of Ref. [19]).

Fig. 1

Mutual friction parameters B (a) and $B'$ (b). Open circles—data of Lucas [62], black circles—data of Mathieu et al. [55]. Solid lines are calculated using the core size tabulated in Refs. [21, 44]; dashed line in panel (a) corresponds to $a_0$ three times larger. From Hillel [40]. Reprinted by permission, ©1987 Institute of Physics Publishing

An important contribution to the analysis of the roton-vortex interactions and hence to the calculation of the mutual friction parameters was made in the beginning of 1990s by Samuels and Donnelly [63] (cited below as SD) who, having simulated numerically quasiclassical trajectories of rotons in the flow field of a fixed, rectilinear quantized vortex line, calculated the longitudinal and the transverse momentum transfer and hence the fundamental friction coefficients D and $D'$. Their approach has been followed by the work of Ferrel and Kyung [64] who noticed that for roton’s momenta in the close vicinity of the roton minimum the SD approach allows an analytical treatment in the framework of the perturbation theory. It is important to note that unlike the earlier studies [8, 16, 17] the works of SD and Ferrel and Kyung are based on the analysis of actual roton trajectories found by solving the quasiclassical Eqs. (37) without making use of approximation of rotons’ trajectories by straight lines.⁵

Having solved numerically Eqs. (37), for a vortex with clockwise circulation located at $x=0$, $y=0$, Samuels and Donnelly illustrated, for different impact parameters $b=y(0)$, the trajectories ${\textbf{r}}(t)=(x(t),\,y(t))$ of a single $R^{+}$ roton of initial momentum $2.1\times 10^{-19}$ g cm/s in the positive x-direction, see Fig. 2.

Fig. 2

Trajectories of a roton incident in the positive x-direction upon a vortex line (indicated by a dot). The initial momentum of the roton is $p_0=2.1\times 10^{-19}$ g cm/s. The step of impact parameter b is 20 Å. The reversal of direction for $b<0$ corresponds to the species change $R^{+}\rightarrow R^{-}$. From Samuels and Donnelly [63]. Reprinted by permission, ©1990 American Physical Society

The main feature of the roton’s kinematics in the flow field of the vortex is the “species change” $R^{+}\rightarrow R^{-}$ for $b_\text {crit}<b<0$, where $b_\text {crit}$ is a critical impact parameter which divides reflected from non-reflected rotons. (In the context of fermionic $^3$He-B and superconductors, a similar mechanism of scattering of thermal quasiparticle excitations by a flow field is known as the Andreev reflection [69, 70]).

Based on their calculation of roton’s trajectories and the momentum exchange that occurs at the reversal of the roton’s direction, and assuming a macroscopic drift of rotons with the mean velocity in the positive x-direction, SD used the method developed by Rayfield and Rief [14] to calculate the longitudinal friction parameter as

$$\begin{aligned} D=-2\pi \int \limits _0^\infty dp\,p^4\frac{dn_0}{dE_0}{v_G}\int \limits _0^\pi d\theta \,{\sigma _{\parallel }}\sin ^3\theta , \end{aligned}$$

(44)

where ${\sigma _{\parallel }}$ is the cross section determined by SD from their numerical simulation. However, they noticed that for small impact parameters ($\vert b\vert <10$ Å) their numerical simulation could not yield physically reasonable values of $\sigma _\parallel$ because their model and simulation ignored several physical mechanisms which become important at close collisions, see Ref. [63] for detail. Samuels and Donnelly then suggested that the longitudinal friction coefficient should be represented in the form

$$\begin{aligned} D=D_1+D_2, \end{aligned}$$

(45)

where $D_1$ is given by Eq. (44), and $D_2$, following the arguments of Hillel and Vinen [20], accounts for the absorption and subsequent anisotropic re-emission of rotons:

$$\begin{aligned} D_2=4{\rho _n}{{\overline{v}}_G}a_0. \end{aligned}$$

(46)

For $D_1$, using the approximation for the cross section $\sigma$ based on their simulation, SD found

$$\begin{aligned} D_1\approx \frac{3\kappa }{32\pi ^2}\left( \frac{2\pi \mu ^{\star }}{kT}\right) ^{1/2}\frac{p_0^4e^{-\Delta /kT}}{\hbar ^3}. \end{aligned}$$

(47)

For the transverse microscopic coefficient SD quoted Hillel and Vinen’s result given by Eq. (39). Samuels and Donnelly further commented that their single-excitation model is not applicable at temperatures above 1.8 K. Having combined formulae (45)–(47) and (39) with results obtained earlier by Pitaevskii [10], Onuki [60], and Hillel [40] for the behavior of friction parameters near the $\lambda$-transition, SD suggested the following semi-empirical formulae applicable for all temperature and pressures:

$$\begin{aligned} D&= (D_1+D_2)\left[ 1-\exp \left( -90\epsilon ^2\right) \right] , \end{aligned}$$

(48)

$$\begin{aligned} D'&=\kappa {\rho _n}\left[ 1-\epsilon +20\epsilon ^{4/3}\exp \left( -200\epsilon ^3\right) \right] , \end{aligned}$$

(49)

where $\epsilon$ is given by Eq. (31).

The behavior of D and $D'$ with temperature is shown in Fig. 3a. The values of parameters B and $B'$ calculated by means of formulae (23)–(24) are compared in Fig. 3 with experimental data reviewed in Ref. [21].

Fig. 3

Calculated from semi-empirical formulae (48)–(49), the microscopic friction coefficients D and $D'$ (left) and parameters B and $B'$ (right) versus temperature; D and B—solid lines; $D'$ and $B'$—dashed lines. In the right panel small triangles correspond to experimental data taken from Ref. [21]. From Samuels and Donnelly [63]. Adapted by permission, ©1990 American Physical Society

An interesting result following from SD findings is that the friction parameter B is practically independent of pressure in the range from $P=0$ to 24 bars, in agreement with experimental data by Mathieu et al. [55].

The publication of Samuels and Donnelly [63] was followed by the comment of Sanders [71] who pointed out that making use of the angular momentum conservation in addition to energy conservation roton trajectories can be calculated exactly, without numerical approximations. In his comment Sanders improved the calculation of the critical impact parameter, but he also noticed that the corrected result for $b_\text {crit}$ will not significantly affect the SD’s calculation of the mutual friction parameters and their impressive agreement with experiment.

Another interesting development of the Samuels and Donnelly approach was proposed by Ferrell and Kyung [64] whose analysis was based on the perturbation treatment of roton-vortex interactions, with a small parameter defined as $\varepsilon =(p_i-p_0)/p_i$, where $p_i$ is the initial momentum of roton far from the vortex line.⁶ Ferrel and Kyung calculated a roton-vortex transverse momentum transfer to the first order in $\varepsilon$ by integrating along unperturbed roton’s trajectories. The result, for the distribution of the transverse momentum transfer, was represented in the closed analytical form by elliptic integrals. The fractional transverse momentum transfer, $\Delta p_\perp /p_i$ as a function of impact parameter b was found in the perfect agreement with numerical calculation by Samuels and Donnelly [63]; for $\varepsilon =0.033$ and the value of the characteristic impact parameter $b^{\star }=\kappa \mu ^{\star }p_i/\pi (p_i-p_0)^2=150$ Å the comparison of Ferrell and Kyung’s analytical calculation with the results of numerical analysis by Samuels and Donnelly is illustrated in Fig. 4.

Fig. 4

Fractional transverse momentum transfer, $\Delta p_\perp /p_i$ as a function of impact parameter. Solid lines—numerical calculation by Samuels and Donnelly [63], circles—perturbation theory by Ferrell and Kyung. From Ferrell and Kyung [64]. Reprinted by permission, ©1991 American Physical Society

2.5 A Short Note on Phonon Scattering by Quantized Vortices

While at temperatures above 0.6 K the mechanism of mutual friction in $^4$He is dominated by scattering of rotons by quantized vortices, a scattering of phonons is, nevertheless, of fundamental interest; besides, studies of phonon scattering shed some light on the nature of transverse force exerted on a vortex by incident quasiparticle excitations.

A theory of mutual friction caused by scattering of phonons by quantized vortices was started by pioneering works of Pitaevskii [9], Iordanskii [12, 13], and Fetter [15], and developed further in the works by Sonin [17, 34, 35, 72], Demircan, Ao, and Niu [73], Thouless and his co-workers [31, 74, 75], Fortin [76], Flaig and Fisher [77], and others (more citations relevant in the context of debates on the Iordanskii force can be found in the next Sect. 2.6). The current state of the theory of phonon scattering has been reviewed in detail by Sonin in his book [25] so we just refer the interested reader to this excellent text.

In the next Sect. 2.6 the debates on the existence and nature of the Iordanskii force will be given only brief description which, however, will be followed by a somewhat longer review of their resolution in the later papers by Thouless et al. and Sonin.

2.6 Iordanskii Force

Known as the Iordanskii force, the transverse force exerted by quasiparticle excitations on a quantized vortex was first discovered for rotons by Lifhsitz and Pitaevskii [8] (with a wrong sign later corrected by Sonin [17]) and then for phonons by Iordanskii [12, 13]; in both cases the force is given by the same expression (9). The debates on the nature and existence of the Iordanskii force began with the publication by Thouless, Ao, and Niu [31] in which, based on their earlier analysis [78] of the Berry phase [79], the authors came to a conclusion that the only transverse force exerted on a vortex is the Magnus force and that the transverse force exerted by quasiparticle excitations is precisely zero. The debates pro [72, 77, 80,81,82] and contra [75, 76, 83, 84] existence of the Iordanskii force followed, and about 2000 some consensus has been reached in publications by Thouless and Vinen with their co-authors [32, 33] and Sonin [34, 35, 85].

A comprehensive review of the theory and its current developments, including the derivation of Iordanskii and Lifshitz–Pitaevskii forces for phonon and roton scattering, respectively, a link between the transverse force arising from phonon scattering and the Aharonov–Bohm effect [86], a connection between the Berry phase and the Magnus force, and, importantly, a detailed review of the debates on the existence and nature of the Iordanskii force can be found in a recent monograph by Sonin [25] and will not be repeated here. However, it seems that the work by Thouless, Vinen and their co-workers [33] which aimed at resolving the controversy around the Iordanskii force deserves a somewhat more detailed description which is given below.

The authors of Ref. [33] recognized that the discrepancy between the work of Thouless, Ao, and Niu [31] (and subsequent publications by Wexler et al. cited above) in which no transverse force has been found and the Iordanskii and Lifshitz–Pitaevskii theories was caused by an assumption made in Ref. [31] that there is no circulation in the normal fluid. The analysis in Ref. [33] showed that this assumption was wrong and that the coupling between the normal and the superfluid components should necessarily lead to the circulation $\kappa _n\ne \kappa$ of the normal fluid around the vortex core. Consequently, instead of Eq. (9) the transverse component of the force exerted on a vortex by excitations should be

$$\begin{aligned} \textbf{f}_\perp =-{\rho _n}\kappa _n{\textbf{v}}_R \end{aligned}$$

(50)

(note that in the model considered in Ref. [33] ${\textbf{v}}_L=\textbf{0}$).

The analysis [33] of Thouless, Vinen and their co-workers was based on the two-fluid model.⁷ Assuming the steady state conditions, and ignoring the second viscosity of the normal component, the equations of the two-fluid model are [41]

$$\begin{aligned}&{\varvec{\nabla }}\cdot \left( {\rho _s}{\textbf{v}}_s+{\rho _n}{\textbf{v}}_n\right) =0, \end{aligned}$$

(51)

$$\begin{aligned}&\nabla _k\left( {\rho _s}v_{si}v_{sk}+{\rho _n}v_{ni}v_{nk}\right) +\nabla _i P \nonumber \\&\quad =\nabla _k\left[ \eta \left( \nabla _k v_{ni}+\nabla _i v_{nk}\right) \right] -\tfrac{2}{3}\nabla _i(\eta {\varvec{\nabla }}\cdot {\textbf{v}}_n), \end{aligned}$$

(52)

$$\begin{aligned}&\mu -\mu _0+\tfrac{1}{2}v_s^2=0, \end{aligned}$$

(53)

$$\begin{aligned}&{\varvec{\nabla }}\cdot (S{\textbf{v}}_n-\kappa _T{\varvec{\nabla }}T/T)=R/T. \end{aligned}$$

(54)

Here the usual convention is applied of summation over repeated indices. Equations (51)–(54) are, respectively, those of mass conservation, momentum conservation, superfluid dynamics, and thermal balance; they are complemented by the thermodynamic relation

$$\begin{aligned} dP=S\,dT+\rho \,d\mu +\tfrac{1}{2}{\rho _n}d({\textbf{v}}_n-{\textbf{v}}_s)^2. \end{aligned}$$

(55)

In Eq. (54), S is the entropy density, $\kappa _T$ is the thermal conductivity, and R is the dissipation rate

The authors of Ref. [33] developed a theory for perturbations $\delta {\textbf{v}}_s$, $\delta {\textbf{v}}_n={\textbf{v}}_n$, $\delta P$, $\delta T$, $\delta \mu$ about the stationary vortex solution

$$\begin{aligned} {\textbf{v}}_n=\textbf{0}, \quad {\textbf{v}}_s=\frac{\kappa }{2\pi r}\hat{{\varvec{\phi }}}, \quad \mu =\mu _0-\frac{\hbar ^2}{2m^2r^2}, \quad \frac{dP}{dr}={\rho _s}(r)\frac{\hbar ^2}{m^2r^3}, \quad T=T_0 \end{aligned}$$

(56)

(in Eq. (54) R is quadratic in $\delta {\textbf{v}}_n$ and $\delta T$ and hence has been neglected).

The analysis of superflow yielded the usual expression (1) for the Magnus force. The authors then analyzed the two-dimensional two-fluid flow in the case where the vortex is pinned by a circular wire of radius $r_0$, as in the Vinen’s experiment [89]; the boundary conditions imposed on the inner boundary (that is, on the surface $r=r_0$) are those of no-slip for the normal velocity, absence of the normal component of the superfluid velocity, and of the uniform temperature on the boundary. Far from the surface the normal flow is assumed to be uniform with velocity $\textbf{U}$ in the x-direction. The problem then reduces to that similar to the classical problem of the incompressible viscous flow past a cylinder of radius $r_0$ in the case where the normal fluid Reynolds number is small,

$$\begin{aligned} \text {Re}_n=\frac{{\rho _n}Ur_0}{2\eta }\lesssim O(1). \end{aligned}$$

(57)

It is well known [29] that for the problem of incompressible viscous flow past a cylinder a regular, spatially uniform perturbative solution in terms of the small parameter $\text {Re}_n$ can be obtained by means of the singular perturbation theory making use of the technique of matching asymptotic expansions, see, e.g., Refs. [38, 39] (the original approach to this problem has been developed by Oseen in 1910). This technique requires to introduce two asymptotic regions, the near-field region $r\lesssim O(r_0/\text {Re}_n)$, and the far-field region $r\gtrsim O(r_0/\text {Re}_n)$ ($r_0/\text {Re}_n$ is known as the Oseen length). The former region is dominated by the viscous momentum transfer so that in the momentum conservation equation the nonlinear convective term ${\rho _n}({\textbf{v}}_n\cdot {\varvec{\nabla }}){\textbf{v}}_n$ can be omitted; this yields an expansion in terms of $\text {Re}_n$ which, however, is not spatially uniform as it contains logarithmically growing terms and hence cannot satisfy boundary conditions as $r\rightarrow \infty$. In the far-field region, the nonlinear term ${\rho _n}({\textbf{v}}_n\cdot {\varvec{\nabla }}){\textbf{v}}_n$ should be replaced by ${\rho _n}(\textbf{U}\cdot {\varvec{\nabla }}){\textbf{v}}_n$, where $\textbf{U}$ is a constant velocity vector as $r\rightarrow \infty$. A general solution of the resulting momentum conservation equation for the far-field region should be asymptotically matched at a distance of the order of Oseen length with the solution in the near-field region and also should satisfy the condition of the uniform flow at large distance from the wire. The resulting solution, represented by two matched expansions in their respective asymptotic regions, is uniformly valid in the whole flow domain.

The analysis [33] indeed demonstrated that in the absence of circulation of the normal fluid the transverse component of the force exerted on a vortex is precisely zero. More importantly, Thouless, Vinen and their co-workers found that their solution yields a nonzero circulation in the normal fluid,

$$\begin{aligned} \kappa _n=\frac{3\kappa L_{\star }^2}{r_0^2\left[ \ln \left( 4\eta L_{\star }/{\rho _n}r_0U\right) -\gamma /2\right] ^2}, \end{aligned}$$

(58)

where

$$\begin{aligned} L_{\star }=\frac{2}{S}\left( \frac{\kappa \eta }{3T}\right) ^{1/2} \end{aligned}$$

(59)

is the parameter of the order of the mean free path of excitations L given by Eq. (11), and $\gamma$ is the Euler constant, and hence recovered Eq. (50) for the transverse force. For the longitudinal component of the force they found

$$\begin{aligned} f_\parallel =\frac{8\pi \eta U}{2\ln \left( 4\eta L_{\star }/{\rho _n}r_0U\right) -2\gamma +1+3L_{\star }/2r_0}. \end{aligned}$$

(60)

In the context of previous subsections, U should be replaced by $v_R$. Since $L_{\star }\gg r_0$, the results given by Eqs. (50), (58) and (60) show the behavior of the longitudinal and transverse components, respectively, as

$$\begin{aligned} f_\parallel \sim \frac{v_R}{\ln v_R} \quad \text {and} \quad f_\perp \sim \frac{v_R}{(\ln v_R)^2}. \end{aligned}$$

(61)

The results reported in Ref. [33] do not seem immediately applicable to the case of a free vortex core for the reason that, unlike in the case of the vortex pinned by the wire where the ‘inner’ boundary conditions could be easily formulated, there is no theory for the transition region between the collisionless domain of the size of mean free path and the hydrodynamic region where the equations of the two-fluid model are applicable. The authors suggested that, at least for the purpose of qualitative analysis, two regions can be joined abruptly. They argued that, just like in the case of the solid core (wire), the logarithmic growth with r of the normal velocity will lead to an increasing alignment of the direction of the normal velocity with the direction of the force until the Oseen radius or some other cutoff distance (e.g., of the order of intervortex spacing in rotating $^4$He or quantum turbulence) is reached. This was further supported by their argument that in the case where $r_0$ is comparable with the excitations’ mean free path their theory predicts a transverse force comparable with the Iordanskii force given by Eq. (9).

Appendix of the paper by Thouless, Vinen et al. contains some important generalizations of the Hall–Vinen theory of mutual friction outlined in Ref. [3]; in particular, it clarifies the connection between the components of the force and the microscopic friction parameters D and $D'$. We revert now to the notation of earlier subsections but still will be considering the vortex with ${\textbf{v}}_L=\textbf{0}$. The authors noted that, as ${\textbf{v}}_R\ne {\textbf{v}}_n$, the components of the force exerted on the vortex should be written as

$$\begin{aligned} f_\parallel =Dv_{R\parallel }-D'v_{R\perp }, \quad f_\perp =Dv_{R\perp }+D'v_{R\parallel }, \end{aligned}$$

(62)

where the subscripts $\parallel$ and $\perp$ refer, respectively, to directions parallel and orthogonal to ${\textbf{v}}_n$, and relation (13) can be re-written as

$$\begin{aligned} v_n-v_{R\parallel }=\frac{f_\parallel }{E}, \quad -v_{R\perp }=\frac{f_\perp }{E}. \end{aligned}$$

(63)

In the steady state conditions, provided $v_n$ is sufficiently small, the parameter E determined by Eqs. (14)–(16) can be approximated as

$$\begin{aligned} E\approx \frac{4\pi \eta }{\ln \left( {\rho _n}v_n L_{\star }/4\eta \right) }, \end{aligned}$$

(64)

and the relations between the components of the force and the microscopic friction parameters can be formulated as

$$\begin{aligned} \frac{f_\parallel }{v_n}=E-\frac{(D+E)E^2}{(D+E)^2+D'^2}, \quad \frac{f_\perp }{v_n}=E-\frac{E^2D'}{(D+E)^2+D'^2}. \end{aligned}$$

(65)

The authors noticed that in the case where $v_n$ is not too large $D\gg E$; this yield the behavior of $f_\parallel$ and $f_\perp$ in the form given by Eqs. (61).

A similar approach has been developed by Sonin [25, 34, 35], although there is some difference between his and the authors’ of Ref. [33] interpretation of the results.⁸ For the phonon scattering Sonin found the transverse force as

$$\begin{aligned} \textbf{f}_\perp =\frac{D'}{1+(D'/E)^2}{\hat{{\varvec{\kappa }}}}\times ({\textbf{v}}_n-{\textbf{v}}_L), \end{aligned}$$

(66)

where now

$$\begin{aligned} E=\frac{4\pi \eta }{\ln (r_m/L)} \end{aligned}$$

(67)

with $r_m$ being the Oseen length defined as $r_m=O(\eta /(\rho \vert {\textbf{v}}_R-{\textbf{v}}_L\vert )$, and L the excitations’ mean free path. Having postulated that $D'$ is determined by the Iordanskii force, i.e., $D'=-\kappa {\rho _n}$, Sonin calculated the normal circulation at large distances from the core as

$$\begin{aligned} \kappa _n=\oint {\textbf{v}}_n\cdot d\textbf{l}=\frac{\kappa }{1+(\kappa {\rho _n}/E)^2}. \end{aligned}$$

(68)

Later Thompson and Stamp [90, 91] gave a fully quantum-mechanical description to the motion of a massive vortex in a two-dimensional Bose superfluid at very low temperatures such that the normal fluid density ${\rho _n}(T)$ is small. Although their results critically depend on the characteristic frequency of the vortex motion, they, nevertheless, found that the Iordanskii force is given by its usual expression ${\textbf{f}}_I=-{\rho _n}(T){\varvec{\kappa }}\times ({\textbf{v}}_R-{\textbf{v}}_L)$ for both the classical (low frequency) and the quantum (high frequency) regimes of motion.

2.7 Mutual Friction in the New Self-Consistent Model of Quantum Turbulence in $^4{\textbf{He}}$

Recent visualization experiments (see, e.g., Refs. [92,93,94] and references therein) encourage theorists to develop self-consistent models of $^4$He quantum turbulence that would account for full coupling of the dynamics of the normal and superfluid components at nonzero temperatures. Then it becomes important how to treat mutual friction. In this subsection we review a recent approach proposed by Galantucci et al. [95] to the development of such a model and to the modeling of mutual friction. As will be seen later, some basic ideas of their approach bear certain similarity to those of the theory developed by Thouless, Vinen, and their co-workers reviewed in the previous subsection.

The work by Galantucci et al. revisits and further develops the modeling approach proposed earlier by Kivotides [96]. The authors of Ref. [95] noted that in the framework of the fully coupled, self-consistent model the motion of the normal fluid should no longer be prescribed a priori (as, so far, has been the case for the most of numerical studies of quantum turbulence in superfluid helium), and that the model should account for the effects of backreaction of quantized vortices on the normal fluid. Consequently, neither the coarse-grained HVBK model, represented by Eqs. (20)–(21), which ignores the dynamics of individual vortex lines nor the Schwarz’s Eq. (27) which requires the velocity of the normal fluid to be prescribed can serve as a basis for a fully coupled, self-consistent model.

In their approach, Galantucci et al. modeled each element of the vortex line by a cylinder of the vortex core radius $a_0$ and length $L_v\gg a_0$. Unlike Thouless, Vinen et al. [33], who made use of the two-fluid theory to calculate the force exerted on the cylinder which mimics the vortex core, Galantucci et al. adopted an approach based on the classical fluid dynamics: a flow of the roton gas of excitations which constitute the normal fluid was modeled by invoking the classical solution [97, 98] (see also Refs. [38, 39]), obtained by the singular perturbation technique, for the viscous, small Reynolds number flow around the cylinder (typically, the Reynolds number defined by the core size and the viscosity of the normal fluid is between $10^{-5}$ and $10^{-4}$). This solution yields the force exerted on the cylinder (the vortex core)

$$\begin{aligned} {\textbf{f}}_D=-{\mathscr {D}}{\textbf{s}}'\times [{\textbf{s}}'\times ({\textbf{v}}_n-{\dot{{\textbf{s}}}})], \end{aligned}$$

(69)

where⁹

$$\begin{aligned} {\mathscr {D}}=4\pi \eta \left[ \frac{1}{2}-\gamma -\ln \left( \frac{\vert {\textbf{v}}_{n\perp }-{\dot{{\textbf{s}}}}\vert {\rho _n}a_0}{4\eta }\right) \right] ^{-1}. \end{aligned}$$

(70)

Here ${\textbf{v}}_n$ is the normal velocity evaluated on the vortex line (in notation of Sect. 2.1${\textbf{v}}_n$ should be replaced by ${\textbf{v}}_R$ but here it seems more convenient to keep the notation of the original publication [95]), and the subscript $\perp$ indicates the component orthogonal to the vortex line.

While in the treatment of Ref. [33] the nonzero circulation of the normal fluid around the vortex core and hence the transverse force exerted on the vortex followed from the two-fluid model, in the approach of Galantucci et al. the transverse Iordanskii force given by Eq. (9) had to be added to ${\textbf{f}}_D$ to make the total force acting on the vortex

$$\begin{aligned} {\textbf{f}}_{sn}({\textbf{s}})={\textbf{f}}_D+{\textbf{f}}_I=-{\mathscr {D}}{\textbf{s}}'\times [{\textbf{s}}'\times ({\textbf{v}}_n-{\dot{{\textbf{s}}}})]-{\rho _n}\kappa {\textbf{s}}'\times ({\textbf{v}}_n-{\dot{{\textbf{s}}}}). \end{aligned}$$

(71)

From the balance of ${\textbf{f}}_{sn}$ and the Magnus force given by Eq. (1) Galantucci et al. derived the equation of the vortex motion [cf. Equation (27)]

$$\begin{aligned} d{\textbf{s}}/dt={\textbf{v}}_{s\perp }+\beta {\textbf{s}}'\times ({\textbf{v}}_n-{\textbf{v}}_s)+\beta '{\textbf{s}}'\times [{\textbf{s}}'\times ({\textbf{v}}_n-{\textbf{v}}_s)], \end{aligned}$$

(72)

where the coefficients $\beta >0$ and $\beta '<0$ are fully determined by the parameter ${\mathscr {D}}$ and the normal and superfluid densities.

The normal fluid momentum conservation equation that explicitly accounts for the backreaction of superfluid vortices on the normal fluid was written in Ref. [95] in the form

$$\begin{aligned} {\rho _n}\frac{D{\textbf{v}}_n}{Dt}=-{\varvec{\nabla }}P+\eta \nabla ^2{\textbf{v}}_n-\oint _{\mathscr {L}}\delta ({\textbf{r}}-{\textbf{s}}){\textbf{f}}_{sn}(\textbf{s})\,d{\textbf{s}}+\textbf{F}_\text {ext}, \end{aligned}$$

(73)

where $\textbf{F}_\text {ext}$ is the external force. In the third term of the right-hand-side, an integration is assumed over all vortex lines. The closed mathematical formulation of the model [95] is given by Eqs. (71)–(73) together with the mass conservation equation for the normal component and the standard Biot–Savart integral that determines the superfluid velocity by the configuration of quantized vortex lines.

This model is an interesting and important development in the ongoing numerical study of quantum turbulence in $^4$He. Discussed in detail in Ref. [95], the numerical implementation of the approach by Galantucci et al. as well as its applications to particular problems of quantum turbulence are beyond the scope of this review and will not be discussed here, but some comments below on the approach of Galantucci et al. to modeling of mutual friction seem to be relevant.

An advantage of the model of mutual friction proposed by Galantucci et al. is twofold: first, it explicitly yields the friction parameters fully determined by the local flow properties, and second, it is easily incorporated into the self-consistent approach that accounts for the backreaction of quantized vortices on the normal fluid. However, the model [95] still remains qualitative rather than quantitative. Strictly speaking, the equations of motion of the Newtonian viscous fluid are not applicable for the analysis of motion of the roton gas around the vortex core. Indeed, the core radius is $10^{-8}$ cm while, as noted in Ref. [95], the roton mean free path is between $13\times 10^{-8}$ cm at $T=1.5$ K and $2.2\times 10^{-8}$ cm at $T=2.15$ K. This difficulty has already been encountered and discussed in Ref. [33] by Thouless, Vinen and their co-workers.¹⁰

Closely related to difficulties discussed in Ref. [33] of modeling of the roton flow in the vicinity of the vortex core is also the issue of circulation $\kappa _n$ of the normal fluid around the vortex core. Based on the two-fluid model, the theory of Thouless, Vinen et al. predicted an existence of such a circulation (see Sect. 2.6). A fully classical treatment [95] of the interaction between the vortex core and the roton fluid rules out such a circulation (this is the reason why the Iordanskii force had to be added “by hand”).

Addressing the problems mentioned above presents substantial technical difficulties so that it is unlikely that a more quantitative self-consistent model of quantum turbulence can be developed very soon. Self-consistent approaches like the one developed by Galantucci et al. are important not only because they provide a better understanding of quantum turbulence but also because they highlight major fundamental modeling issues that may need to be addressed in the future.

2.8 Other Microscopic Models

Here we briefly review the quantum-mechanical approach to the problem of the friction force exerted by elementary excitations on a massive vortex in the two-dimensional Bose superfluid developed by the work of Cataldo and his co-workers [100,101,102,103,104,105] and later by Thompson and Stalp [90, 91].

In Refs. [100, 101] Cataldo et al. formulated a two-dimensional Hamiltonian model for the coupling between the vortex and elementary excitations in $^4$He. Making use of a standard approach of nonequilibrium statistical mechanics, the authors treated a vortex as a finite-mass quantum particle that experiences a quantal Brownian motion in a heat reservoir composed of quasiparticle excitations (in the absence of the vortex the normal and superfluid velocities are zero). Having formulated the generalized master equation for the irreversible time evolution of the density operator of the vortex, Cataldo et al. derived and analyzed solutions of the dissipative equations of the vortex motion. Crucially, the Hamiltonian and these equations include the parameter defined by the vortex mass, $M_v$ per unit length of the vortex line as

$$\begin{aligned} \Omega ={\rho _s}\kappa /M_v \end{aligned}$$

(74)

called by the authors, who invoked an analogy between the Magnus and Lorentz forces, a cyclotron frequency.

In the range of temperatures below 0.4 K such that the excitations are phonons the longitudinal and transverse forces have been calculated in Ref. [102]. For the considered two-dimensional motion of the vortex in the $(x,\,y)$-plane, the generalized master equation [100] reduces to the following equation for the mean value of the vortex position operator $R=x+{\text {i}}y$:

$$\begin{aligned} M_v\langle {\ddot{R}}\rangle ={\text {i}}(M_v\Omega +\gamma _M)\langle {\dot{R}}\rangle , \end{aligned}$$

(75)

where the coefficient $\gamma _M(\Omega )$ is determined by the Markovian time-correlation tensor ($-M_v/\gamma _M$ gives a time scale at which $\langle {\dot{R}}\rangle$ tends to zero). The authors then claimed that in the right-hand-side of Eq. (75) one can clearly identify the Magnus force and the drag force, the latter having two components

$$\begin{aligned} F_\parallel =-D\langle {\dot{R}}\rangle \quad \text {and} \quad F_\perp ={\text {i}}D'\langle {\dot{R}}\rangle . \end{aligned}$$

(76)

The calculation [102] of the friction parameters D and $D'$ made use of properties of the Markov process which, in turn, involved the distribution function of excitations. In order to select just a phonon part of the excitation spectrum the authors use the frequency cutoff $\omega _0=0.06\,\text {ps}^{-1}$.

The friction coefficients D and $D'$ calculated in Ref. [102] are strongly dependent, via the cyclotron frequency $\Omega$, on the vortex mass $M_v$. However, the issue of vortex inertia remains rather controversial (see, e.g., a discussion in Refs. [106, 107] and references therein).¹¹ As this issue is beyond the aim of this review we will not further discuss the results obtained in Ref. [102].

It seems that a somewhat more promising development of the model formulated in Refs. [100, 101] has been made in the work by Cataldo and Jezek [103] on calculation of the longitudinal component of the friction force. As in the earlier work by Cataldo et al., the formulation of the quantum-mechanical problem included the vortex Hamiltonian, the Hamiltonian of noninteracting quasiparticle excitations, the assumption that a gas of excitations is in thermal equilibrium, and the Markov approximation to second order in the parameter of coupling between the vortex and the heat reservoir. The main result of this work is the longitudinal friction coefficient (still dependent on the cyclotron frequency $\Omega$)

$$\begin{aligned} D_\Omega&=\frac{A^2}{(2\pi )^4\hbar \Omega }\int d^3{\textbf{k}}\int d^3{\textbf{q}}\,\delta (k_z-q_z)\vert \Lambda (q,\,k)\vert ^2({\textbf{k}}-{\textbf{q}})^2 \nonumber \\&\quad \times \left[ n(\omega _k)-n(\omega _q)\right] \delta (\omega _q-\omega _k-\Omega ), \end{aligned}$$

(77)

where $n(\omega )=\left[ \exp (\hbar \omega /kT)-1\right] ^{-1}$ is the Bose occupation number for quasiparticle excitations, A is the area of the system in the $(x,\,y)$-plane, $\hbar {\textbf{k}}$ is momentum, a subscript z indicates a direction orthogonal to the plane, and $\Lambda (q,\,k)$ is the coupling parameter yet to be determined.

Cataldo and Jezek argued that, although in Eq. (77) the limit $\Omega \rightarrow 0$ of vanishing cyclotron frequency is formally inadmissible, they can still regard such a limit as a good approximation provided a gap between Landau levels of the vortex Hamiltonian scales down with the vortex size. Modeling the coupling parameter as $\vert \Lambda (q,\,k)\vert ^2=\beta \left| \omega '_q\right| \,\left| \omega '_k\right|$, where $\omega '_k=\partial \omega _k/\partial k$ is the group velocity of excitations, and $\beta$ is a scaling parameter, assuming $\Omega \rightarrow 0$, and choosing $\beta$ such that in the phonon-dominated regime ($T\lesssim 0.4$ K) the limiting value of the longitudinal friction coefficient reproduces exactly Iordanskii’s result [13] $D\sim (kT)^5$, for the roton-dominated regime Cataldo and Jezek derived the expression

$$\begin{aligned} D=\frac{38}{105}\,\frac{\hbar ^3p_0^5}{m^2c_s^2}e^{-\Delta /kT} \end{aligned}$$

(78)

(here $c_s$ is the speed of sound) which practically coincides with the classical Rayfield and Rief’s formula [14] with only a slightly higher (by about 7%) coefficient of the exponent.

An interesting feature of the theory developed by Cataldo and Jezek [103] is that it seems to account for several mechanisms of interaction between quasiparticle excitations and the vortex such as pure phonon and roton scattering that conserves the magnitude of quasiparticles’ momentum, and phonon $\leftrightarrow R^\pm$ and $R^\pm \leftrightarrow R^\mp$ transitions (for a definition of $R^{+}$ and $R^{-}$ rotons see Sect. 2.4).

A behavior of the longitudinal friction coefficient with temperature is shown in Fig. 5 where it is compared with the values summarized in tables in the 1983 review by Barenghi. As can be seen from this figure, for temperatures below 1.5 K the data shown in Ref. [21] are reproduced rather well by the theoretical curve corresponding to $\Omega \rightarrow 0$. A discrepancy at higher temperatures are explained in Ref. [103] by interactions in the quasiparticle gas and by the viscous drag on the vortex by the normal fluid, both effects neglected in the Cataldo and Jezek’s approach.

Fig. 5

Longitudinal friction coefficient D versus temperature. Solid line corresponds to $\Omega \rightarrow 0$. Circles—values of microscopic friction coefficients, and black squares—values of phenomenological friction coefficients from Tables III and II, respectively, of Ref. [21]. From Cataldo and Jezek [103]. Reprinted by permission, ©2002 American Physical Society

Another useful development of his theory has been made by Ref. [105] in which Cataldo attempted to avoid the controversy associated with a vortex mass. In this version of his earlier approach he assumed a heat bath formed by a vortex lattice to be in thermal equilibrium and considered the interactions between the lattice and the flow of quasiparticle excitations. A crucial difference from his earlier approach is that in this version of the theory the main contribution to the heat capacity arises from the thermal excitation of Kelvin waves on individual vortices, whose frequency, $\omega _0$ is typically much higher than the cyclotron frequency $\Omega$, even for very heavy vortices. A generalization of approach developed in his earlier papers [100,101,102,103,104] enabled Cataldo to find an explicit expression for the longitudinal macroscopic friction parameter B that enters the coarse-grained HVBK equations and conclude that in the case where $\omega _0\gg \Omega$ the contribution of the cyclotron frequency is negligible. For the roton part of the spectrum, the resulting approximation independent of the vortex mass is

$$\begin{aligned} B=\frac{2079[1+n(\omega _0)]\hbar p_0^3\sqrt{kT}}{c_s^2{\rho _s}\lambda _\text {min}\sqrt{\mu ^{\star }}}, \end{aligned}$$

(79)

where λ_min ≈ 10³ Å is the “ultraviolet” cutoff which determines the total number of modes in the vortex heat bath. A typical frequency of thermally excited Kelvin waves behaves linearly with temperature, $\omega _0=aT$ (as noted by the author, the value $a=2\times 10^{10}\,\text {s}^{-1}\,\text {K}^{-1}$ reported in Ref. [105] is an overestimation).

Cataldo also derived an approximation for the parameter B in the phonon-dominated regime. As this regime is of less interest in the context of the HVBK equations (and also considering the absence of experimental data for B at temperatures below 1.3 K), the results of Ref. [105] for the phonon-dominated regime will not be reviewed here.

As was already mentioned in Sect. 2.6, a fully quantum-mechanical treatment to the problem of motion of a massive vortex and associated mutual friction effects in the two-dimensional Bose superfluid at very low temperatures such that the normal fluid density ${\rho _n}(T)$ is small has later been given by Thompson and Stamp [90, 91]. Their results crucially depend on a single nondimensional parameter ${\tilde{\Omega }}=\hbar \Omega /kT$, where $\Omega$ is the characteristic frequency of the vortex motion, and kT represents the thermal energy of quasiparticle excitations: ${\tilde{\Omega }}\ll 1$ corresponds to the classical regime of motion, while ${\tilde{\Omega }}\gg 1$ to the quantum regime which has not been explored before. Thompson and Stamp showed that in the classical regime the usual Hall–Vinen formulation of the equations of vortex motion can be recovered, while in the quantum regime the equations of motion become significantly retarded, with a value of the longitudinal friction coefficient reduced to a sixteenth of its value in the classical regime. Remarkably, the usual expression for the transverse Iordanskii force was found to be valid for the both regimes.

2.9 Vortex Diffusivity and Mutual Friction in Two-Dimensional $^4$He Films

An area of research closely related to the studies reviewed both in this and especially the next section of this review is that of vortex dynamics and diffusivity in superfluid helium films.

An interpretation of the dynamical (e.g., finite-frequency) experiments on the Kosterlitz–Thouless transition in $^4$He films is typically based on the AHNS (Ambegaokar–Halperin–Nelson–Siggia) theory [108, 109] in which the temperature-dependent vortex diffusivity ${\mathcal {D}}$ is the primary transport parameter¹² (see also Ref. [110] for the further development of the theory). In the framework of this theory, the motion of quantized vortices treated as vortex points in the $(x,\,y)$-plane was modeled using the approach of Hall and Vinen briefly described in Sect. 2.1. The forces acting on a vortex are the Magnus force given by Eq. (1), and the friction force caused by interactions between the vortex and thermal excitations (rotons and ripplons) in the two-dimensional film. The latter force is given by Eq. (25) in which, for the point vortex model, the density should be understood as that integrated across the film thickness. From the balance of these forces, Ambegaokar et al. [109] derived the equation of motion which, for the vortex point whose time-dependent position and velocity are ${\textbf{r}}(t)$ and ${\textbf{v}}_L=d{\textbf{r}}/dt$, respectively, coincides with the two-dimensional form of Schwarz’s Eq. (27) (cf. Equation (85) in the next Section):

$$\begin{aligned} {\textbf{v}}_L={\textbf{v}}_i+q\alpha \hat{\textbf{z}}\times ({\textbf{v}}_n-{\textbf{v}}_i)+q\alpha '({\textbf{v}}_n-{\textbf{v}}_i), \end{aligned}$$

(80)

where $q=\pm 1$ is the vortex charge, $\hat{\textbf{z}}$ is the unit vector orthogonal to the film, and ${\textbf{v}}_i$ is the superfluid velocity induced at the point ${\textbf{r}}(t)$ by all other point vortices in the system but excluding the divergent self-field of the vortex at that point. The friction coefficient $\alpha$ is linked with the vortex diffusivity defined [109, 111] as

$$\begin{aligned} {\mathcal {D}}=\frac{1}{2}\int \limits _0^\infty \langle {\textbf{v}}_L(t)\cdot {\textbf{v}}_L(0)\rangle \,dt \end{aligned}$$

(81)

by the relation

$$\begin{aligned} \alpha =\frac{2\pi \hbar \sigma _s}{mkT}{\mathcal {D}}, \end{aligned}$$

(82)

where $\sigma _s={\rho _s}h$, with h being the thickness of the film, is the areal (renormalized) superfluid density.¹³ As found in Refs. [109, 112], the diffusivity ${\mathcal {D}}$ must remain finite at the Kosterlitz–Thouless (KT) static transition temperature determined by the equation

$$\begin{aligned} T_\text {KT}=\frac{\pi \hbar ^2\sigma _s^0}{2km^2}, \end{aligned}$$

(83)

where $\sigma _s^0$ is the bare (unrenormalized) areal superfluid density. In a more advance treatment [109], the right-hand-side of Eq. (80) may also include the Gaussian fluctuating noise source $\eta _i^\alpha (t)$ with $\langle \eta _i^\alpha (t)\eta _i^\beta (t')\rangle =2{\mathcal {D}}\delta _{ij}\delta _{\alpha \beta }\delta (t-t')$.

The first direct measurements of vortex diffusivity were made by Kim and Glaberson [113] in rotating $^4$He films by observation of damping of the third-sound resonance associated with the motion of vortices induced by rotation and making use of the known relation between the diffusivity and contribution of vortices to the damping of third sound. At the KT transition, for $1.3~\text {K}<T_\text {KT}<1.5~\text {K}$, Kim and Glaberson obtained ${\mathcal {D}}\sim 0.4(\hbar /m)$ and found that the diffusivity increases rapidly as $T\rightarrow T_\text {KT}$. In the case where the temperature is not too low their results for diffusivity plotted versus $T/\sigma _s$ collapse on a universal curve

$$\begin{aligned} {\mathcal {D}}\approx 0.17(\hbar /m)^{-3}(kT/\sigma _s)^2. \end{aligned}$$

(84)

The universal behavior given by formula (84) appears to be valid for not too low temperatures (such that $(T/\sigma _s)^2> 10^{16}\,\text {K}^5\text {cm}^4\text {g}^{-2}$) both below and above the static KT transition; however, for lower temperatures ${\mathcal {D}}$ decreases with $T/\sigma _s$ more rapidly.

This work was continued by Adams and Glaberson [111, 114] who, based on the oscillating substrate method of Bishop and Reppy [115], performed in rotating $^4$He films more detailed measurements of the vortex diffusivity as a function of temperature through the superfluid transition. The authors noted that for interpretation of their experiments the Kosterlitz–Thouless static theory is not sufficient and applied a more comprehensive theory [108, 109, 116] that accounts for the dynamic response of vortices to an oscillating field. This, more general theory shows that oscillations shift the static transition temperature $T_\text {KT}$ to the dynamic transition temperature $T_c$ determined as a function of frequency in the cited work by Ambegaokar et al.

Typical results of diffusivity measurements for $T<T_c$ are illustrated in Fig. 6 (left). Adams and Glaberson found a strong divergence at the critical temperature of dynamic transition, $T_c$, shown in more detail in Fig. 6 (right) as ${\mathcal {D}}$ versus the reduced temperature $(T_c-T)/T_c$ for different film thicknesses. The authors commented that the observed behavior is linked to the dynamics of the KT transition itself although no theory existed yet at the time of publication of Ref. [111] to explain the details.

Fig. 6

Left: example of diffusivity measurements [111] for a 6.4 Å film rotating at angular velocity 8 rad/s. Right: Diffusivity plotted as a function of reduced temperature $(T_c-T)/T_c$ for films of various thicknesses h (Å). From Adams and Glaberson [111]. Reprinted by permission, ©1987 American Physical Society

While Adams and Glaberson measured diffusivity only below the dynamic transition temperature $T_c$, the later work by Finotello et al. [117] complemented these measurements by reporting the results obtained for $T>T_c$ which are illustrated in Fig. 7 where they are compared with measurements of Adams and Glaberson. Note that the both measurements show similar trends (however, one must be careful comparing the magnitudes as the measurements of Finotello et al. are only within a factor of the order unity). Finotello et al. also investigated the behavior with the dynamic transition temperature and the film thickness of the parameter ${\mathcal {D}}/a_0^2$, where $a_0$ is the vortex core radius which is roughly equal to the three-dimensional correlation length, and found that ${\mathcal {D}}/a_0^2$ is a decreasing function of the film thickness.

Fig. 7

The diffusivity ${\mathcal {D}}$ (in units of $\hbar /m$) versus temperature. Black symbols—measurements of Finotello et al. [117]; open circles—results of Adams and Glaberson [111]. From Finotello et al. [117]. Reprinted by permission, ©1990 American Physical Society

While the earlier experiments at temperatures above 1 K in films whose thickness was several atomic layers yielded the values of ${\mathcal {D}}/a_0^2$ of order $10^{11}\,\text {s}^{-1}$, the measurements of Agnolet et al. [118] at temperatures lower than 0.5 K in very thin films (about 1.5 of the atomic layer) showed much smaller values ${\mathcal {D}}/a_0^2\approx 5.6\times 10^7\,\text {s}^{-1}$ corresponding to the diffusivity of only $2\times 10^{-5}$ in units of $\hbar /m$.

Here we should also mention based on the torsional oscillation techniques experimental studies of Wada and his co-workers [119,120,121] on a dependence of ${\mathcal {D}}/a_0^2$ on the static ($T_\text {KT}$) and dynamic ($T_c$) critical temperatures, the latter itself depending on the frequency of oscillations.

More recently, making use of the third-sound method for vortex imaging, Sachkou et al. [122] observed coherent dynamics of two-dimensional vortices in a few nanometer-thick $^4$He films at temperature $T\approx 500$ mK which is far below that of the superfluid transition. Their measurements yielded a vortex diffusivity five orders of magnitude lower than in the earlier measurements [111, 113, 114, 117] done for temperatures closer to $T_\text {KT}$ or $T_c$.

Being of importance for the study of the Kosterlitz–Thouless transition, measurements of the vortex diffusivity (and hence of one of the mutual friction parameters) in helium films are also of interest for comparison with results of studies of mutual friction in other 2D or quasi-2D systems, first of all in atomic Bose–Einstein condensates. The latter studies will be reviewed in the next section. Here it is worth noting that the link between the diffusivity in $^4$He films and the friction parameters in (quasi) two-dimensional BECs was discussed very recently in Ref. [123]; in this work Wittmer et al. compared their calculation of the longitudinal friction coefficient in the BEC with the results of measurements of vortex diffusivity reviewed above in this subsection. This and other works on the mutual friction in atomic condensates will be reviewed in detail in the next section.

3 Mutual Friction in Atomic Bose–Einstein Condensates

During the last two decades a major effort in studying mutual friction shifted from superfluid helium to nonzero-temperature atomic Bose–Einstein condensates.

3.1 Vortex Dynamics and Mutual Friction

In the following subsections we will mostly be concerned with quasi-two-dimensional (“pancake-shape”) condensates. In such condensates, on the scales much larger than the vortex core size, which is of the order of coherence length, the vortex dynamics can be well described by the Helmholtz–Kirchhoff point vortex model (the system of many point vortices is known as the Onsager’s point vortex gas). For a point vortex whose time-dependent position is ${\textbf{r}}(t)$ a general formulation of the equation of motion follows from the Schwarz’s Eq. (27) as

$$\begin{aligned} {\dot{{\textbf{r}}}}={\textbf{v}}_s+{\textbf{v}}_i+q\alpha \hat{\textbf{z}}\times ({\textbf{v}}_n-{\textbf{v}}_s-{\textbf{v}}_i)-q\alpha '\hat{\textbf{z}}\left[ \times \hat{\textbf{z}}({\textbf{v}}_n-{\textbf{v}}_s-{\textbf{v}}_i)\right] , \end{aligned}$$

(85)

where $\hat{\textbf{z}}$ is the unit vector orthogonal to the $(x,\,y)$-plane, ${\textbf{v}}_s$ is any imposed superfluid velocity, ${\textbf{v}}_i$ is the velocity field generated by all other vortices in the system at the vortex point ${\textbf{r}}(t)$, and $q=\pm 1$ is the vortex charge; $\alpha$ should be interpreted as the longitudinal macroscopic friction coefficient. Note that, following the comment made by Vinen and Niemela [124] that the transverse friction coefficient $\alpha '$ is usually much smaller than $\alpha$, in some (but not all) of the numerical studies of vortex dynamics and mutual friction in the BECs the last term in Eq. (85) is omitted. In this case, for the point vortex in the reservoir with ${\textbf{v}}_n={\textbf{v}}_s=\textbf{0}$ Eq. (85) takes a simplified form

$$\begin{aligned} {\dot{{\textbf{r}}}}={\textbf{v}}_i-q\alpha \hat{\textbf{z}}\times {\textbf{v}}_i. \end{aligned}$$

(86)

Törnkvist and Schröder [125] were probably the first to rigorously derive the equation of vortex motion in dissipative systems described by the Ginzburg–Landau equation with complex coefficients (although some ideas have been formulated earlier [126, 127])

$$\begin{aligned} \frac{\partial \psi }{\partial t}=a\nabla ^2\psi +\left( \mu -b\vert \psi \vert ^2\right) \psi , \end{aligned}$$

(87)

where $\psi =\vert \psi \vert \exp ({\text {i}}S)$ is the complex order parameter, and the coefficients a, b, and $\mu$ are complex. The dissipative Gross–Pitaevskii equation that will be encountered in the following subsections is the particular case of Eq. (87). For two-dimensional, dilute systems of point vortices the equation derived by Törnkvist and Schröder takes the forms of Eqs. (85) or (86).

3.2 Early Theoretical Developments

One of the first theoretical calculations of the mutual friction parameters in the finite-temperature ($\mu \ll T\lesssim T_c$, where $\mu$ is the chemical potential, and $T_c$ is the superfluid transition temperature) Bose–Einstein condensate was done by Fedichev and Shlyapnikov [128] who analyzed dissipation caused by scattering on the mean-field potential of non-interacting, single-particle thermal excitations incident on a vortex in the direction orthogonal to the vortex filament. The derivations which followed the procedure outlined by Iordanskii [12, 13] yielded the force exerted on the vortex by thermal excitations in the form given by Eq. (3) (not surprisingly, with ${D_t}=-\kappa {\rho _n}$). The microscopic longitudinal friction coefficient was found in the form¹⁴

$$\begin{aligned} D=\kappa {\rho _n}(T)\left( \frac{n_0g}{T}\right) ^{1/2}, \end{aligned}$$

(88)

where $n_0$ is the number density of the condensate far from the vortex core, the coupling parameter

$$\begin{aligned} g=4\pi \hbar ^2a_s/m, \end{aligned}$$

(89)

with $a_s$ being the s-wave scattering length, characterizes the strength of interatomic interactions, and the density of the normal component

$$\begin{aligned} {\rho _n}\approx 0.1\frac{m^{5/2}T^{3/2}}{\hbar ^3}. \end{aligned}$$

(90)

Given by Eqs. (88)–(90), the results [128] for the microscopic friction parameter D can be conveniently re-formulated in terms of the friction coefficient $\alpha$ as

$$\begin{aligned} \alpha \approx \frac{n_\text {th}}{n_0}\sqrt{\frac{\mu }{kT}}=0.1\times \frac{4\pi a_s m^{1/2}}{\hbar \mu ^{1/2}}kT, \end{aligned}$$

(91)

where $n_\text {th}$ is the number density of the thermal cloud.

In Sect. 3.7 the theoretical prediction (91) will be compared with the values of $\alpha$ found from numerical simulations and experimental measurements.

3.3 A Link Between Thermal Dissipation and Mutual Friction

A study which shed much light on microscopic mechanisms of thermal dissipation and a link between the dissipation and the mutual friction in the dilute Bose–Einstein condensate was done by Kobayashi and Tsubota [129] who started with the many-body Hamiltonian (here and below, following the original publication [129] the formulation is given in the non-dimensional form)

$$\begin{aligned} {\hat{H}}=\int d{\textbf{r}}\,{\hat{\Psi }}^\dag \left[ -\nabla ^2-\mu +\frac{g}{2}\vert {\hat{\Psi }}\vert ^2\right] {\hat{\Psi }}, \end{aligned}$$

(92)

where ${\hat{\Psi }}({\textbf{r}},\,t)$ is the boson field operator, and the coupling parameter g is given by Eq. (89). In the framework of the first-order perturbation theory ${\hat{\Psi }}$ was represented in terms of the mean-field ansatz as ${\hat{\Psi }}=\Phi +{\hat{\chi }}$, where the second term corresponds to the first-order excitations. Then, making use of the Bogoliubov transformation,

$$\begin{aligned} {\hat{\chi }}=\frac{1}{\sqrt{V}}\sum _j\left( u_j{\hat{\alpha }}_j+v_j{\hat{\alpha }}_j^\dag \right) , \end{aligned}$$

(93)

where V is the volume of the system, $u_j$ and $v_j$ are the Bogoliubov coefficients, ${\hat{\alpha }}_j$ and ${\hat{\alpha }}_j^\dag$ are, respectively, the annihilation and creation operators of a Bogoliubov quasiparticle, Kobayashi and Tsubota derived the dissipative Gross–Pitaevskii equation

$$\begin{aligned} {\text {i}}\frac{\partial \Phi }{\partial t}=\left[ -\nabla ^2-\mu +g\left( \vert \Phi \vert ^2+2\langle {\hat{\chi }}^\dag {\hat{\chi }}\rangle \right) \right] \Phi +g\langle {\hat{\chi }}^2\rangle \Phi ^\star \end{aligned}$$

(94)

coupled with the Bogoliubov–de Gennes (BdG) equations for $u_j$ and $v_j$. Kobayashi and Tsubota further assumed that quasiparticles are coupled with a thermal bath of temperature T, and made use of the local equilibrium assumption

$$\begin{aligned} \langle {\hat{\alpha }}_j^\dag {\hat{\alpha }}_j\rangle =N_j=\frac{1}{\exp (E_j/T)-1}, \end{aligned}$$

(95)

where $E_j$ is the excitation spectrum of quasiparticles. Equation (95) explicitly controls the temperature; the rate of dissipation caused by interactions between condensed and non-condensed particles can be calculated as

$$\begin{aligned} \gamma _D=-g\,\text {Im}\left( \langle {\hat{\chi }}^2\rangle \Phi ^\star \Phi ^{-1}\right) . \end{aligned}$$

(96)

To analyze the dissipation, Kobayashi and Tsubota first considered spatially uniform excitations and a macroscopic wave function $\Phi ({\textbf{r}},\,t=0)$ which included several randomly placed vortices whose reconnections and evolution lead to a formation of the dynamic vortex tangle. For simulations at temperatures $T=0.01T_c$, $0.1T_c$, and $0.5T_c$, where $T_c$ is the critical temperature of Bose condensation, the authors calculated the Fourier-transformed dissipation rate, $\gamma _D(k,\,t)$ and found that at the lowest temperature dissipation is essential only at wave numbers greater than $2\pi /a_0$, where $a_0$ is the vortex core size. As the temperature increases, dissipation starts working at larger scales so that an effect of $\gamma _D$ could be expected similar to that of mutual friction.

To calculate the mutual friction parameters, Kobayashi and Tsubota considered a single straight vortex line, perpendicular to the $(x,\,y)$-plane, moving in the thermal reservoir under the uniform superfluid velocity field ${\textbf{v}}_s=v_s\textbf{i}$ (here and below $\textbf{i}$ and $\textbf{j}$ are the unit vectors in the x- and y-directions). For the vortex position ${\textbf{r}}(t)=x(t)\,\textbf{i}+y(t)\,\textbf{j}$, the solution of the Schwarz’s Eq. (85) yields

$$\begin{aligned} {\textbf{r}}(t)-{\textbf{r}}_0=\alpha 'v_st\,\textbf{i}+\alpha v_st\,\textbf{j}. \end{aligned}$$

(97)

To determine the values of the friction parameters $\alpha$ and $\alpha '$, the vortex trajectory determined by Eq. (97) has been compared with that found from the numerical solution of the system of equations consisting of the dissipative GPE coupled with two BdG equations for $u_j$ and $v_j$; under the velocity field ${\textbf{v}}_s$ these equations include, respectively, the terms ${\text {i}}({\textbf{v}}_s\cdot {\varvec{\nabla }})\Phi$, ${\text {i}}({\textbf{v}}_s\cdot {\varvec{\nabla }})u_j$, and ${\text {i}}({\textbf{v}}_s\cdot {\varvec{\nabla }})v_j$. The resulting temperature-dependent mutual friction coefficients are shown in Fig. 8.

Fig. 8

Friction coefficients versus temperature (lines show the fitting of numerical data). From Kobayashi and Tsubota [129]. Reprinted by permission, ©2006 American Physical Society

Note that the values of the longitudinal friction parameter $\alpha$ found by Kobayashi and Tsubota are nearly an order of magnitude higher than those estimated by more recent theoretical and numerical studies (such as, e.g., Ref. [130]) and measured later in the experiment [131], see Sects. 3.4, 3.6, 3.7 and 3.8.

3.4 Mutual Friction in Phenomenological Dissipative Models of the BEC

The simplest model of the finite-temperature Bose–Einstein condensate is represented by the dissipative (“damped”) Gross–Pitaevskii equation (DGPE)

$$\begin{aligned} \hbar ({\text {i}}-\gamma )\frac{\partial \psi }{\partial t}=\left[ -\frac{\hbar ^2}{2m}\nabla ^2+U({\textbf{r}})-\mu +g\vert \psi \vert ^2\right] \psi . \end{aligned}$$

(98)

Equation (98) was derived by Choi et al. [132] based on the method of phenomenological damping near the $\lambda$-point in superfluid $^4$He developed much earlier by Pitaevskii [54]. In Eq. (98), $U({\textbf{r}})$ is the external potential, $\mu$ is the chemical potential, and the coupling parameter g is given by Eq. (89). The temperature-dependent phenomenological damping parameter $\gamma$ modeling interactions between the condensate and the thermal cloud was found in Refs. [132, 133] in the form

$$\begin{aligned} \gamma =\frac{4mga_s^2kT}{\pi \hbar ^2}. \end{aligned}$$

(99)

Typically, $\gamma =O(10^{-2})$ (for instance, the experimental results [134] yield $\gamma \approx 0.03$).

Based on this model, a numerical study of vortex dynamics and mutual friction in the two-dimensional condensate trapped by an external potential with the angular frequency $\omega _\perp$,

$$\begin{aligned} U=\tfrac{1}{2}m\omega _\perp ^2(x^2+y^2) \end{aligned}$$

(100)

was undertaken by Madarassy and Barenghi [135]. The aim of their study was to find relations between the dissipation parameter $\gamma$ and the mutual friction coefficients $\alpha$ and $\alpha '$

The authors first analyzed a decay of a single 2D vortex placed initially at some off-center location $(x_0,\,y_0)$. They imprinted a vortex at this location taking for initial condition $\psi (x,\,y,\,t=0)$ the Thomas–Fermi approximation multiplied by a function proportional to $(x-x_0)+{\text {i}}(y-y_0)$. Their numerical solution of Eq. (98) then showed that in the absence of dissipation ($\gamma =0$) the vortex, under the action of the Magnus force, precesses around the trap along roughly circular orbit of constant energy. In the presence of dissipation the vortex loses energy and spirals outward, vanishing eventually at the edge of the condensate. Having calculated the trajectories of the vortex for several nonzero values of $\gamma$, Madarassy and Barenghi compared them with trajectories predicted by the solution of the Schwarz’s Eq. (85) for the stationary thermal cloud (${\textbf{v}}_n={\textbf{v}}_s=\textbf{0}$); this solution, in the cylindrical coordinates, is

$$\begin{aligned} r(t)=r(0)e^{-\alpha \omega _0t}, \quad \theta (t)=\theta (0)+\omega _0(1-\alpha ')t, \end{aligned}$$

(101)

where $\omega _0(x_0,\,y_0)$ is the angular frequency of the vortex precession in the absence of dissipation. The values of $\alpha$ and $\alpha '$ shown in Fig. 9 were then found by fitting the trajectories calculated, for several values of $\gamma$, from the numerical solution of Eq. (98) with those given by Eqs. (101) (as the condensate was not homogeneous the results depended slightly on the initial position of the vortex). Clearly, the longitudinal friction coefficient is proportional to $\gamma$. However, the value of the transverse coefficient $\alpha '$ appears somewhat more uncertain, mainly because $\alpha '$ is much smaller than $\alpha$ and hence is more difficult to estimate by the method described above. However, based on Fig. 9 it can be seen that $\alpha '$ is practically proportional to the dissipation rate at small $\gamma$ and perhaps saturates for $\gamma \gtrsim 0.015$.

Fig. 9

Friction coefficients $\alpha$ (left) and $\alpha '$ (right) versus $\gamma$. Circles and triangles correspond to trajectories with different initial positions of the vortex. From Madarassy and Barenghi [135]. Reprinted by permission, ©2008 Springer Nature

Madarassy and Barenghi also addressed an important question of the temperature dependence of the phenomenological dissipation parameter $\gamma$ by comparing the values of $\alpha$ and $\alpha '$ obtained in Ref. [135] with those obtained in the later work [130] (yet unpublished at the time of publication of Ref. [135]) by means of numerical simulation based on the Zaremba–Nikuni–Griffin finite-temperature theory [136, 137] which allows a direct control of the condensate’s temperature. Having compared the mutual friction coefficients obtained by two numerical approaches, Madarassy and Barenghi found that, for instance, $\gamma =0.044$ and 0.08 for $T/T_c=0.15$ and 0.27, respectively. They also found that the smallness of the transverse friction coefficient $\alpha '$ agrees with the results obtained earlier by Berloff and Youd [138] and later by Jackson et al. [130].

More recently the dissipative Gross–Pitaevskii equation was used to investigate the mutual friction parameters in the two-dimensional dilute Bose-gas by Wittmer et al. [123] (note that most of the essential details of this work are provided in the Supplemental Material to the cited paper). The main aim of this work was to determine the friction parameters of the holographic superfluid. To achieve this Wittmer and his co-workers compared the results obtained by analyzing a two-dimensional motion of the vortex-antivortex pair in the frameworks of three different models: a holographic model of a superfluid in two spatial dimensions [139, 140]; dissipative Gross–Pitaevskii equation; and the Hall–Vinen theory which for point vortices reduces to Schwarz’s equation in the form (85). As the holographic theory is beyond the scope of this review, below we review only the results obtained by Wittmer et al. by means of comparison of their numerical solution of the DGPE with the evolution of the vortex dipole predicted by Eq. (85).

Wittmer et al. considered, in the absence of trapping potential, the non-dimensionalized DGPE which they wrote in the form equivalent to Eq. (98) as

$$\begin{aligned} \frac{\partial \psi }{\partial t}= - ({\text {i}}+\Lambda )\left( -\frac{\hbar ^2}{2m}\nabla ^2-\mu +g\vert \psi \vert ^2\right) \psi . \end{aligned}$$

(102)

In the case where the temperature of the 2D Bose gas is much larger than the zero-point energy but much smaller than the Berezinskii–Kosterlitz–Thouless transition temperature $T_\text {KT}$, the dissipation parameter can be found following Ref. [141] as

$$\begin{aligned} \Lambda \approx 12ma_s^2kT/(\pi \hbar ^2). \end{aligned}$$

(103)

Assuming that the normal velocity and the imposed superfluid velocity can be neglected so that the dynamics of each vortex of the dipole is determined entirely by the superfluid velocity generated by the other vortex, for the distance d(t) between the vortex points of the dipole and the distance R(t) traveled by the center of coordinates of the pair the solution of Eq. (85) yields [123]

$$\begin{aligned} d(t)=\left( d^2(0)-8\alpha t\right) ^{1/2}, \quad R(t)=R(0)+q(1-\alpha ')(2\alpha) ^{-1}[d(0)-d(t)]. \end{aligned}$$

(104)

The trajectories determined by Eqs. (104) were then compared with those obtained in Ref. [123] by numerical solution of the DGPE (102). For the mutual friction coefficient $\alpha$ the results are summarized in Table 1.¹⁵

Table 1 The values of the dissipation parameter $\Lambda$ and the friction coefficient $\alpha$ obtained by comparing the numerical solution of the DGPE and the trajectories of vortex dipoles given by Eq. (104) (note also that the values of $\alpha$ are within 2% of those obtained in Ref. [123] by means of the holographic superfluid theory)

An interesting aspect of this work is the comparison of the friction parameter $\alpha$ with the results of measurements of the vortex diffusivity ${\mathcal {D}}$ in superfluid $^4$He films reviewed above in Sect. 2.9. The values of $\alpha$ found [123] in the range from 0.01 to 0.03 (see Table 1) are similar to the values calculated by means of Eq. (82) from the experimental results reported in Refs. [111, 113, 117, 118]. Also note that the values of $\alpha$ obtained by Wittmer et al.  agree with those found experimentally [122, 131] in atomic BECs.

3.5 A Short Note on the Mean-Field Modeling of Finite-Temperature Bose–Einstein Condensates

During the last decade and a half it became apparent that the damped Gross–Pitaevskii Eq. (98) might not be sufficient to account for detailed mechanisms of mutual friction in the atomic Bose–Einstein condensates so that further developments of the GPE-based theoretical and numerical models are necessary. We will briefly describe the main features of these, more elaborate approaches below and then review them in more detail in Sects. 3.6 and 3.7.

In the general case of nonlocal interatomic interactions, the Bose–Einstein condensate is described by the macroscopic, complex order parameter $\psi ({\textbf{r}},\,t)$ which obeys the generalized Schrödinger equation

$$\begin{aligned} {\text {i}}\hbar \frac{\partial \psi }{\partial t}=\left[ -\frac{\hbar ^2}{2m}\nabla ^2+\int \vert \psi ({\textbf{r}}')\vert ^2V(\vert {\textbf{r}}-{\textbf{r}}'\vert )\,d{\textbf{r}}'-\mu \right] \psi , \end{aligned}$$

(105)

where $V(\vert {\textbf{r}}-{\textbf{r}}'\vert )$ is an interparticle interaction potential. As found by Berloff and Roberts [144, 145], assuming V in the form of the Lennard–Jones potential yields, after a suitable choice of coefficients, the dispersion curve reproducing the Landau dispersion curve for $^4$He. However, such a model still remains only a qualitative one for superfluid helium (for a detailed discussion see, e.g., Ref. [146]).

For a dilute, weakly interacting Bose gas the nonlocality of interactions can be ignored and the interparticle interaction potential can be replaced by the $\delta$-function of strength g given by Eq. (89) so that Eq. (105) reduces to the classical Gross–Pitaevskii equation

$$\begin{aligned} {\text {i}}\hbar \frac{\partial \psi }{\partial t}=\left[ -\frac{\hbar ^2}{2m}\nabla ^2+g\vert \psi \vert ^2-\mu \right] \psi . \end{aligned}$$

(106)

The numerical studies reviewed in the following subsections are based on the local GPE (106) and its generalizations.¹⁶ It was shown by Putterman and Roberts [147] that in the framework of the classical GPE (106) and the hydrodynamic equations derived from it by means of the Madelung transformation it is possible to describe both the dynamics of the condensate and of thermal non-condensed excitations (and hence to derive the Landau-type two-fluid model), with lower Fourier modes corresponding to the former and higher modes to the latter. Detailed and comprehensive reviews of the classical GPE-based models and their numerical implementation for the finite-temperature condensate have been written by Proukakis and Jackson [148] and Berloff et al. [146]. Below we simply summarize the basic ideas of these models; a more detailed review of implementation of these models for calculation of the mutual friction parameters in the finite-temperature BECs will be given in the following two subsections.

The basic ideas on which these models and their numerical implementation are based are the following [146, 148]: a numerical truncation of the modes that separate the superfluid and the normal components; a method to determine the temperature corresponding to the separating modes; and the distribution of classical modes corresponding to the normal component. Among the versions of numerical implementation of these models are the projected GPE [149, 150], truncated GPE [138, 142, 143], and the stochastic (projected) GPE [151, 152]. An alternative approach has been provided by the Zaremba–Nikuni–Griffin (ZNG) model [136, 137] which couples the dissipative GPE with kinetic equations accounting for collisions between noncondensate atoms as well as for those between condensate and noncondensate atoms.

3.6 Truncated and Stochastic Projected GPEs: Mutual Friction and Finite-Temperature Effects

The truncated Gross–Pitaevskii equation (TGPE), first introduced for the study of finite-temperature BECs by Davis, Morgan, and Burnett [149], is obtained by truncating the higher modes in the Fourier space such that ${\mathcal {P}}_G[{\hat{\psi }}_{\textbf{k}}]=\theta \left( k_\text {max}-\vert {\textbf{k}}\vert \right) {\hat{\psi }}_{\textbf{k}}$, where ${\hat{\psi }}_{\textbf{k}}$ is the Fourier transform of the wave function and $\theta$ is the Heaviside function. The TGPE can be written as

$$\begin{aligned} {\text {i}}\hbar \frac{\partial \psi }{\partial t}={\mathcal {P}}_G\left\{ -\frac{\hbar ^2}{2m}\nabla ^2+g{\mathcal {P}}_G\left[ \vert \psi \vert ^2\right] \right\} \psi . \end{aligned}$$

(107)

The truncated GPE exactly conserves the energy and the particle number. In the framework of the TGPE, high wave number modes are partially thermalized. Davis et al. [149] found that the numerical solution of the TGPE evolves toward thermodynamic (microcanonical) equilibrium and hence concluded that the truncated GPE provides a description of finite-temperature effects in the BEC. In the framework of the TGPE, the condensation transition was also studied later by Connaughton et al. [153].

The first work in which the mutual friction was investigated in the framework of the classical approach based on a single, truncated Gross–Pitaevskii equation was that of Berloff and Youd [138]. The aim of their work was to determine a temperature-dependent universal scaling for the dissipation-driven decay of the vortex length in the range of temperatures from 0 to the critical temperature of condensation, $T_c$. Such a scaling then enabled them to estimate the value of the longitudinal friction coefficient $\alpha$ (although an accurate estimate of a smaller transverse coefficient $\alpha '$ remained somewhat more elusive).

The first, crucial step in the analysis was to achieve thermal equilibrium and hence determine the temperature and the condensate fraction. This was done based on the thermodynamic description of the condensation proposed earlier by Connaughton et al. [153]. The link between the temperature and the fraction of noncondensed particles is then given by the Rayleigh–Jeans distribution modified by interactions with the condensate (see, e.g., Refs. [154, 155]). It is important to note that in the numerical analysis by Berloff and Youd the ultraviolet cutoff for this distribution was provided by the spatial discretization which, in turn, played the role of truncation of the GPE. Based on the analysis of thermodynamic equilibrium, Berloff and Youd suggested a semi-empirical formula that links the non-dimensional numerical density $\rho _0$ of condensed particles with temperature and the non-dimensional total numerical density $\rho$:

$$\begin{aligned} \frac{T}{T_c}=1-\left( 1-\beta \sqrt{\rho }\right) \frac{\rho _0}{\rho }-\beta \sqrt{\rho }\left( \frac{\rho _0}{\rho }\right) ^2 \end{aligned}$$

(108)

with $\beta =0.227538$.

Having established the equilibrium state of the condensate, the authors then numerically imprinted the vortex ring of the non-dimensional (in units of the healing length $\xi$) initial radius $R_0$ and followed its decay due to interactions with noncondensed particles. They found that, for all temperatures between 0 and $T_c$, the evolution of the vortex line length ${\mathscr {L}}$ obeys the scaling law

$$\begin{aligned} d{\mathscr {L}}^2/dt=-\gamma _L(\rho ,\,T/T_c), \end{aligned}$$

(109)

where $\gamma _L$ is time-independent. This result agrees with theoretical [21] and experimental [156] studies of decay of vortex rings in the finite-temperature $^4$He. Berloff and Youd found that $\gamma _L$ scales linearly with the total number density $\rho$. The behavior of $\gamma _L/\rho$ with temperature is illustrated in Fig. 10.

Fig. 10

$\gamma _L/\rho$ versus temperature. Numerical results for various total number densities are shown by colored dots. Adapted from Berloff and Youd [138] by permission, ©2007 American Physical Society

The values of the friction coefficient $\alpha$ can be extracted from the solution of Schwarz’s Eq. (27) which, for a single ring whose time-dependent radius is R(t), yields, in the non-dimensional units,

$$\begin{aligned} {\dot{R}}=\alpha u_i, \end{aligned}$$

(110)

where the ring’s velocity is given by $u_i=[\ln (8R)-\delta +1]/R$ (for Gross–Pitaevskii vortices $\delta \approx 0.38$, see Ref. [138] and references therein). Having integrated Eq. (110), Berloff and Youd found the following relation between the longitudinal friction parameter $\alpha$ and the parameter $\gamma _L$:

$$\begin{aligned} \alpha =\frac{\gamma _L}{8\pi ^2[\ln (8{\hat{R}})-\delta +1]}, \end{aligned}$$

(111)

where ${\hat{R}}$ is the ring’s mean radius. For typical values of the non-dimensional total number density in the range from 0.25 to 0.75 and, for the authors’ simulation with the initial ring’s radius $R_0=10$, ${\hat{R}}$ close to 5, the values of the longitudinal friction coefficient, for, e.g., $T=0.5T_c$, appear to be in the range from 0.01 to 0.03. Such estimates are in agreement with the experimental results reported in Refs. [131, 157] as well as with the estimates obtained later from other GPE-based models [123, 130, 135, 158] reviewed here.

Berloff and Youd also suggested the method to estimate from the results of their simulation the value of the transverse friction coefficient $\alpha '$; they found that $\alpha '$ is small but did not provide an actual, more precise estimate of $\alpha '$. Jackson et al. [130] found that $\vert \alpha '\vert$ should be smaller than 0.02; a better estimate for $\alpha '$ as a function of temperature was found later by Krstulovic and Brachet [142, 143] who made use of their version of the TGPE-based numerical simulation.

We now turn to the work by Krstulovic and Brachet cited above. Having studied the dynamics and thermalization (full details are provided in Ref. [143]), they investigated numerically the thermodynamic equilibrium and concluded that in the framework of the TGPE-based approach the microcanonical and grand canonical descriptions are equivalent. The grand canonical equilibrium state that allows a direct control of temperature was obtained from a convergence of the stochastic process defined by the stochastic truncated Ginzburg–Landau equation

$$\begin{aligned} \hbar \frac{\partial \psi }{\partial t}= & {} {\mathcal {P}}_G\left\{ \frac{\hbar ^2}{2m}\nabla ^2\psi +\mu \psi -g{\mathcal {P}}_G\left[ \vert \psi \vert ^2\right] \psi -{\text {i}}\hbar {\textbf{v}}_n\cdot {\varvec{\nabla }}\psi \right\} \nonumber \\{} & {} +\sqrt{\frac{2\hbar }{V\beta }}{\mathcal {P}}_G[\zeta ({\textbf{r}},\,t)], \end{aligned}$$

(112)

where $\zeta$ is the white noise that satisfies $\langle \zeta ({\textbf{r}},\,t)\zeta ^\star ({\textbf{r}}',\,t')\rangle =\delta (t-t')\delta ({\textbf{r}}-{\textbf{r}}')$, and $\beta$ is the inverse temperature. In the considered model, the mean velocity of condensed particles, ${\textbf{v}}_s=\textbf{0}$, and ${\textbf{v}}_n$ represents the mean velocity of thermalized particles, that is, the normal velocity and hence describes an equilibrium state with the counterflow.

Having generated the equilibrium counterflow solution, $\psi _\text {eq}$ of Eq. (112), Krstulovic and Brachet imprinted the lattice of rectilinear vortices orthogonal to the counterflow thus setting the initial condition $\psi ({\textbf{r}},\,0)=\psi _\text {lattice}\psi _\text {eq}$ which was then evolved by solving numerically the TGPE (107). In the plane orthogonal to the vortex filaments, the motion of one of the vortices is then compared with that predicted by the Schwarz’s Eq. (85); this yielded the transverse friction coefficient as $\alpha '=v_\parallel /v_n$, where $v_\parallel$ is the counterflow-induced vortex velocity in the direction parallel to the counterflow. Having measured $v_\parallel$ for various counterflow velocities $v_n$ and temperatures, Krstulovic and Brachet calculated the transverse friction coefficient $\alpha '$ shown in Fig. 11 and found that their results are fitted with formula (26) with $B'=0.8334$ and the normal fluid density ${\rho _n}$ found from the numerical solution of Eq. (112), see Fig. 12.

Fig. 11

The transverse friction coefficient $\alpha '$ versus temperature for various counterflow velocities. Dashed line shows predictions based on Eq. (26) with $B'=0.8334$. Adapted from Krstulovic and Brachet [142] by permission, ©2011 American Physical Society

Fig. 12

Temperature dependence of the normal density. Points are predicted by the equilibrium solution of Eq. (112); solid line—exact low-temperature calculation [143]. Adapted from Krstulovic and Brachet [142] by permission, ©2011 American Physical Society

One important conclusion that can be made from this study is that although the friction parameter $\alpha '$ is small, it does not vanish, in agreement with arguments made by Thouless, Vinen, and their co-workers [33], and Sonin [34, 35] (and later by Thompson and Stamp [90, 91]) in their papers contributing to a long discussion on the existence of the Iordanskii force in $^4$He.

We should mention here a very recent work by Mehdi et al. [158] on the mutual friction and vortex diffusion in the two-dimensional point-vortex gas. Based on the stochastic projected Gross–Pitaevskii equation, this numerical study yielded the friction coefficient $\alpha$ close to that predicted by simulations by Jackson et al. [130] based on the kinetic ZNG theory. The work by Mehdi et al. appeared in Phys. Rev. Research when this review has already been accepted for publication. For this reason, given above is only a brief mention of this interesting work which otherwise would deserve a more detailed description.

3.7 Mutual Friction in the Framework of the ZNG Kinetic Theory

Based on the Zaremba–Nikuni–Griffin (ZNG) kinetic theory [136, 137] which couples a generalized, dissipative Gross–Pitaevskii equation for a condensate order parameter with a Boltzmann equation for a thermal cloud, a study of vortex dynamics in the atomic BEC by Jackson et al. [130] significantly clarified microscopic origins of the mutual friction and led to an ab initio estimates for values of friction coefficients as functions of the condensate temperature.

In the framework of the ZNG formalism, in the Hartree–Fock–Popov approximation the dynamics of the condensate are governed by the generalized GPE for the condensate wave function $\psi ({\textbf{r}},\,t)$,

$$\begin{aligned} {\text {i}}\hbar \frac{\partial \psi }{\partial t}=\left( -\frac{\hbar ^2}{2m}\nabla ^2+U+gn_c+2g{\tilde{n}}-{\text {i}}R\right) \psi \end{aligned}$$

(113)

coupled with the Boltzmann equation

$$\begin{aligned} \frac{\partial f}{\partial t}+\frac{{\textbf{p}}}{m}\cdot {\varvec{\nabla }}f-{\varvec{\nabla }}U_\text {T}\cdot {\varvec{\nabla }}_{\textbf{p}}f=C_{12}+C_{22} \end{aligned}$$

(114)

for the thermal cloud phase density $f({\textbf{p}},\,{\textbf{r}},\,t)$. In Eqs. (113) and (114)

$$\begin{aligned} n_c=\vert \psi \vert ^2 \quad \text {and} \quad {\tilde{n}}=\frac{1}{(2\pi \hbar )^3}\int \frac{d{\textbf{p}}}{f} \end{aligned}$$

(115)

are the condensate and the thermal cloud densities, respectively, $U({\textbf{r}})=m\left( \omega _\perp ^2r^2+\omega _z^2z^2\right) /2$ is the external trap potential, and $U_\text {T}({\textbf{r}})=U({\textbf{r}})+2g(n_c+{\tilde{n}})$ is the effective potential of thermal atoms; the collision integrals $C_{ij}$ in Eq. (114) are responsible for binary collisions between condensate and non-condensate atoms ($C_{12}$) and between atoms of the thermal cloud ($C_{22}$), see Refs. [136, 137] as well as [130] for details. The non-Hermitian dissipative term $-{\text {i}}R\psi$ in Eq. (113) is linked with the collision integral $C_{12}$ by the relation

$$\begin{aligned} R({\textbf{r}},\,t)=\frac{\hbar }{2n_c}\int \frac{d{\textbf{p}}}{(2\pi \hbar )^3}C_{12}. \end{aligned}$$

(116)

Following the method outlined in the earlier work by Jackson and Zaremba [159], the authors started their numerical simulation by generating a state of the condensate in equilibrium with the thermal cloud at temperature T (see the cited work for details of the iterative procedure which explicitly sets the temperature). The initial vortex state, with a rectilinear vortex imprinted at $(x_0,\,y_0)$, was then generated by multiplying the equilibrium wave function in the absence of vortex by the phase factor $\exp [{\text {i}}S({\textbf{r}})]$, where $S({\textbf{r}})=\arctan [(y-y_0)/(x-x_0)]$. Assuming trap frequencies such that $\omega _z=\sqrt{8}\omega _\perp$, the dynamics of this, single vortex was investigated numerically in a quasi-2D pancake-shaped condensate so that the vortex could be assumed rectilinear in the z-direction throughout its motion. The choice of physical parameters in Eqs. (113)–(116) corresponded to $N=10^4$ $^{87}$Rb atoms, the trap frequency $\omega _\perp =2\pi \times 129$ Hz, and the critical temperature of condensation $T_c=177$ nK.

An investigation of motion of the off-centered vortex (but placed initially relatively close to the center to avoid unwanted effects related to the depletion of the condensate density closer to its edges) was done along the same lines as those in the earlier work by Madarassy and Barenghi [135]. The resulting mutual friction coefficient $\alpha$ obtained by comparison of the solution (101) of Schwarz’s Eq. (85) with the trajectory of the vortex found by numerical solution of the ZNG Eqs. (113)–(116) is shown, as a function of temperature, in Fig. 13.

Fig. 13

Friction coefficient $\alpha$ versus temperature. See the text for the physical and trap parameters. From Jackson et al. [130]. Reprinted by permission, ©2009 American Physical Society

As was already mentioned above, the magnitudes of $\alpha$ agree with those found by numerical studies in Refs. [123, 135, 138, 158] and experimentally in Refs. [131, 157]. Like in earlier works by Berloff and Youd [138] and Madarassy and Barenghi [135], a precise value of the non-dissipative transverse friction coefficient $\alpha '$ could not be confidently extracted from numerical solutions, mainly because of errors in measuring a vortex precession, although the simulations by Jackson et al. [130] indicated that $\vert \alpha '\vert <0.02$ for $0<T/T_c<0.8$, in agreement with earlier results [135, 138]. Jackson et al. also investigated an effect of the total number of atoms on the friction parameters and found no clear dependence of $\alpha$ on N.

3.8 Experimental Studies of Mutual Friction in Bose–Einstein Condensates

Perhaps the most compelling experimental study so far of the mutual friction in Bose–Einstein condensates was that by Moon and his co-authors [131] who studied the dynamics of a vortex pair in the pancake-shaped BEC of $^{23}$Na atoms in the optical dipole trap to which a magnetic quadrupole field was applied. The experimental setup has been described in detail in some of the authors’ earlier publications [160, 161]. A highly oblate shape of the condensate and hence the effectively two-dimensional vortex dynamics were achieved by means of the high ratio, $\omega _z/(\omega _x^2+\omega _y^2)^{1/2}\approx 35$ of the trap frequencies. Images of vortices have been obtained by means of the time-of-flight absorption technique.

In their experiment, Moon et al., making use of the topological phase impurity method [162,163,164], created first, near the center of the condensate, a doubly charged vortex which then split into a co-rotating vortex pair whose center, ${\textbf{r}}_c$ performs an orbiting motion with the intervortex distance, ${\textbf{r}}_{12}$ decreasing with time due to friction. The evolution of ${\textbf{r}}_c(t)$ and ${\textbf{r}}_{12}(t)$ was then determined by the time-of-flight technique which showed, as expected, an increasing rate of pair separation $r_{12}$ with temperature (here and below $r_{12}=\vert {\textbf{r}}_{12}\vert$ and $r_c=\vert {\textbf{r}}_c\vert$). The observations showed that while the scatter of $r_c$ was rather large, the fluctuations of $r_{12}$ were small; this enabled Moon et al. to determine the mutual friction parameter $\alpha$ with high degree of confidence, see below.

To extract values of the friction coefficient as a function of temperature, Moon and his co-workers assumed a stationary thermal cloud and, neglecting the transverse mutual friction parameter $\alpha '$, compared the observed dynamics of the vortex pair with that obtained from the point vortex version of Schwarz’s Eq. (85) which can be re-formulated as a pair of coupled equations for ${\textbf{r}}_{12}(t)$ and ${\textbf{r}}_c(t)$. In the case where two vortices remain not very far from the center of the inhomogeneous condensate the precession frequency of the center of vortex pair can be approximated as

$$\begin{aligned} \Omega \approx \Omega _0=\frac{3\hbar }{2mR^2}\ln \left( \frac{R}{\xi }\right) , \end{aligned}$$

(117)

where R is the radial extent of the condensate, and the equations for ${\textbf{r}}_{12}(t)$ and ${\textbf{r}}_c(t)$ decouple as

$$\begin{aligned} {\dot{{\textbf{r}}}}_{12}=\omega _{p0}[\alpha +\hat{\textbf{z}}\times ]{\textbf{r}}_{12}, \quad {\dot{{\textbf{r}}}}_c=\Omega _0[\alpha +\hat{\textbf{z}}\times ]{\textbf{r}}_c, \end{aligned}$$

(118)

where $\omega _{p0}=\Omega _0+b\hbar /(mr_{12}^2)$. The parameter $b\approx 1.35$ accounts for a modification of interaction between two vortices in a harmonic trap [165]. It is clear that if vortices remain relatively close to the center of the trap the friction coefficient $\alpha$ can be estimated, making use of the first Eq. (118), from the observed behavior of $r_{12}(t)$ alone.

The results obtained by Moon and his co-workers are shown in Fig. 14. The authors found the values of $\alpha$ to be comparable with those obtained theoretically by Fedichev and Shlyapnikov [128] (see Eq. (91)) and to agree with the results of calculation [130] based on the ZNG kinetic theory.

Fig. 14

Friction coefficient $\alpha$ versus temperature. Solid line—theoretical prediction by Fedichev and Shlyapnikov [128]; dashed line—linear fit of data points [131] for low temperatures. Inset: $\alpha$ as a function of $T/T_c$; open circles—numerical results of Jackson et al. [130]. From Moon et al. [131]. Reprinted by permission, ©2015 American Physical Society

Later the experimental results by Moon et al. were directly applied by Kim et al. [166] to the analysis of correlation between the mutual friction and the vortex decay rate in the two-dimensional turbulent BEC. Based on the point-vortex model which incorporated only the longitudinal friction determined by the parameter $\alpha$, Kim et al. performed numerical simulations of the point vortex gas and found that the rate of decay driven by annihilation of vortex-antivortex pairs behaves almost linearly with the friction coefficient $\alpha$. This suggests yet another method of measurement of the friction parameter $\alpha$ in the BEC.

Of a considerable interest is a more recent experimental study by Kwon et al. [157] of dissipative vortex dynamics in ultracold Fermi gases. The work addresses two mechanisms of relaxation of the vortex energy: sound emission and mutual friction. Although an analysis of mutual friction in fermionic superfluids is outside the scope of our paper, one aspect of the work by Kwon and his co-authors is of direct relevance to this review: by tuning the s-wave scattering length in the superfluid gas of fermionic $^6$Li atoms, the authors studied, besides the regime of BCS superfluid and that of the unitary Fermi gas, the BEC regime of tightly bound molecules. The experimental study was performed in a thin, uniform superfluid of paired $^6$Li atoms trapped inside a circular box at temperature $T=0.3T_c$. Similarly to earlier numerical and experimental works the mutual friction coefficient $\alpha$ was determined by visualizing dynamics of the vortex-antivortex dipoles and then comparing the intervortex distances and dipole self-annihilation times with those predicted by the dissipative point-vortex model (like in Ref. [131], the transverse friction parameter $\alpha '$ has been neglected). In order to create a dipole at a desired position, Kwon et al. used the chopstick method [167] designed and implemented by Anderson and his co-authors. In the BEC regime vortices have been detected by the time-of-flight imaging. The obtained values of the friction coefficient $\alpha$ agree rather well with the earlier experiment by Moon et al. [131] and theoretical/numerical predictions based on the ZNG kinetic approach [130] as well as with the recent estimates obtained for 2D holographic superfluids in Ref. [123].

4 Conclusions and Possible Further Developments

This article reviewed the progress made since 1983 in understanding the mutual friction in finite-temperature bosonic superfluids. The year in the previous sentence is not coincidental: the first substantial review [21] of theoretical and experimental studies of the mutual friction in superfluid helium was published that year by Barenghi, Donnelly, and Vinen.

In our review the problem of mutual friction is addressed first in the context of superfluid $^4$He and then in the context of atomic Bose–Einstein condensates. The latter, second part of the review seems to be particularly timely with the emergence, since mid-1990s, of the large body of theoretical, numerical, and experimental work on mutual friction in cold atomic BECs. These studies have not been reviewed yet; more importantly, a comparison of results of these studies with those obtained for $^4$He can provide a valuable information essential for understanding of friction and dissipation mechanisms in the both systems.

In the author’s view, since 1983 the most important contributions to the study of mutual friction were the following:

$^4$He—the development of theory of the roton-vortex interactions; the application by Thouless, Vinen, and their co-workers of the two-fluid model to the study of vortex dynamics (this work significantly contributed to the resolution of controversy related to the existence of the Iordanskii force);

BEC—theoretical calculation of the friction parameters; investigations of a link between thermal dissipation and mutual friction parameters; applications of the mean-field models based on the damped, truncated, and stochastic projected Gross–Pitaevskii equations to the calculations of vortex dynamics and friction parameters; calculation of friction parameters in the framework of the Zaremba–Nikuni–Griffin kinetic theory; experimental measurements which yielded, for the longitudinal friction coefficient, the results in good agreement with numerical simulations.

Some suggestions on possible future developments in the study of mutual friction are formulated below. These suggestions are by no means comprehensive but simply reflect the author’s current interests.

A breakthrough in understanding of nature of the transverse force exerted by excitations on a quantized vortex was made by the work of Thouless, Vinen, and their co-workers [33]. Making use of the method of matching asymptotic expansions, they developed a theory for the two-fluid dynamics in the vicinity of the vortex core pinned by a cylinder (e.g., by a wire). This modeling approach enabled them to formulate a complete set of boundary conditions and then find the solution that clearly indicated the existence of circulation in the normal fluid around the vortex core and hence of the transverse force. However, the problem of the two-fluid flow around a free vortex is expected to be much more difficult. As the authors of Ref. [33] commented, an approach to this problem should involve an analysis of the transition region between the collisional region of the order of a mean free path (such an analysis should be based on a kinetic theory for elementary excitations) and the hydrodynamic region beyond a few mean free paths from the vortex core. A solution of this, undoubtedly very difficult problem would provide a better understanding of the nature of circulation of the normal fluid around the vortex core, and hence of the transverse force.

Another possible development may combine an approach by Samuels and Donnelly [63] to the theoretical and numerical analysis of quasiclassical roton trajectories incident on a single, stationary quantized vortex in $^4$He with that of the numerical studies [168, 169] of Andreev scattering of thermal quasiparticle excitations by the fully developed vortex tangle in $^3$He-B. Such a study would provide a tool for an analysis of roton trajectories and statistical properties of the longitudinal and transverse momentum transfer by quasiparticles on their interactions with vortices in quantum turbulence.

In the study of cold atomic Bose–Einstein condensates, a development could be useful of a model that would link closer an analysis of the mutual friction in the BECs with that in $^4$He. The models that have been employed so far were all based on generalizations of the classical Gross–Pitaevskii Eq. (106). The excitation spectrum determined by this equation is strictly monotonic; this means that the friction parameters obtained in these studies are in some sense analogous to the friction parameters obtained for $^4$He in the phonon part of the spectrum. However, the generalized, nonlocal Schrödinger Eq. (105), with a suitable choice of the interaction potential V (e.g., in the Lennard–Jones form) and its parameters, reproduces very closely the Landau $^4$He excitation spectrum with its phonon, maxon, and roton regions. Although, even with such a choice of nonlocal interaction potential, Eq. (105) cannot be regarded as a quantitative model for $^4$He (see Sect. 3.5 for references), a numerical analysis based on Eq. (105) and its generalizations may shed more light on the nature of mutual friction in both BECs and $^4$He, and also provide a better link between these two subareas of research. A drawback is that a numerical implementation of Eq. (105) is difficult and often leads to an appearance of numerical artefacts that might adversely affect an interpretation of excitations’ scattering in terms of mutual friction.

Also of interest might be a related study of mutual friction in dipolar bosonic quantum gases described by the nonlocal GPE (105) with an effective interaction potential (see, e.g., Refs. [170, 171])

$$\begin{aligned} V({\textbf{r}}-{\textbf{r}}')=g\delta ({\textbf{r}}-{\textbf{r}}')+\frac{C_\text {dd}}{4\pi }\,\frac{1-\cos ^2\theta }{\vert {\textbf{r}}-{\textbf{r}}'\vert ^3}, \end{aligned}$$

(119)

where the constant $C_\text {dd}$ characterizes the strength of dipole–dipole interactions, and $\theta$ is the angle between the vector joining the dipoles and the direction of polarization. The dispersion curve of Eq. (105) with the effective potential given by Eq. (119) possesses a roton-like minimum and so resembles the Landau excitation spectrum for $^4$He. A numerical treatment of the nonlocal GPE with the interaction potential in the form (119) is likely to be simpler than that of Eq. (105) with V in the Lennard–Jones form (this seems to be confirmed by a large number of numerical studies of the dipolar BECs), although a strong anisotropy of the dipolar quantum gas may make an interpretation of results in terms of the mutual friction rather challenging.