GENERALIZATION OF THE KUTTA-JOUKOWSKI THEOREM FOR THE HYDRODYNAMIC FORCES ACTING ON A QUANTIZED VORTEX

One of the most remarkable manifestations of superfluidity and superconductivity is the formation of topological defects. First directly observed[1, 2] in a lead-indium alloy, niobium and helium-4, individual quantized vortices have been routinely detected in many other superfluid and superconducting systems. Conversely, their presence in recently produced dilute atomic condensates has been interpreted as a proof of superfluidity (see, e.g., Refs. \refciteabo-shaeer2001,zwierlein2005,yao2016observation).

The dynamics of vortices plays a crucial role in various phenomena, such as the dissipation of electric currents in type-II superconductors or the development of quantum turbulence in superfluids.[6, 7] However, a complete understanding of the forces governing the motion of vortices is still lacking. The situation remains particularly unsatisfactory for the emblematic case of superfluid helium at finite temperatures. According to the two-fluid model[8], the superfluid with mass density $\rho_{\,\text{s}}$ and velocity $\boldsymbol{v_{\,\text{s}}}$ coexists with a “normal” viscous fluid with mass density $\rho_{\,\text{n}}$ and velocity $\boldsymbol{v_{\,\text{n}}}$. The existence of a relative superfluid flow round a vortex gives rise to a force (per unit length) of the Magnus type[9]

where $\bar{\rho}_{\,\text{s}}$ and $\boldsymbol{\bar{v}_{\,\text{s}}}$ denote the asymptotically uniform superfluid density and velocity respectively, $\kappa=h/m$ is the quantum of circulation ($h$ being the Planck constant and $m$ the mass of the superfluid particles), and $\boldsymbol{\hat{z}}$ is a unit vector in the direction of the vortex. This force has been experimentally confirmed by measurements of a vortex trapped on a wire.[10, 11, 12] The additional forces induced by the normal-fluid flow have been highly controversial (see, e.g., Refs. \refciteputterman74superfluid,thouless1996tranverse,kopnin2002vortex,sonin2010 and references therein). These forces have been most often empirically parametrized as

where $\boldsymbol{\bar{v}_{\,\text{n}}}$ denotes the asymptotically uniform velocity of the normal fluid. The normal forces have been alternatively described in terms of the relative velocity $\boldsymbol{\bar{v}_{\,\text{n}}}-\boldsymbol{\bar{v}_{\,\text{s}}}$ as (see, e.g., Ref. \refcitedonnelly2005)

The coefficients $\gamma_{0}$, $\gamma_{0}^{\prime}$ and $\alpha$, $\alpha^{\prime}$ have been experimentally found to have some dependence on the temperature as well as on the relative velocities.[6, 17, 18, 19] Second-sound experiments seem to suggest the existence of a small axial friction force directed along the vortex line.[20] Nevertheless, this force, whose origin remains poorly understood, is usually neglected (see, e.g., Section 3.6 in Ref. \refcitedonnelly2005). Different microscopic interpretations of the forces (2) or (3) have been proposed.[15, 21] However, comparisons with experiments have been hampered by the matching to macroscopic scales. In any case, the microscopic theory should reproduce the superfluid hydrodynamic equations, since the latter are based solely on fundamental conservation laws, thermodynamic principles, and symmetries, as pointed out in Ref. \refcitesonin2016 (see also Ref. \refcitethouless1996tranverse). The alternative approach is therefore to rely on the two-fluid model. The two-fluid hydrodynamic equations are generally linearized and solved in the vicinity of the vortex, where the normal-fluid flow is governed by viscous effects.[22, 23, 24] The flows are usually further assumed to be incompressible. However, such an analysis faces two serious problems: (i) the hydrodynamic description breaks down at short length scales (typically of the order of the mean-free path of the excitations), (ii) the inner boundary conditions to impose on the hydrodynamic equations remain uncertain since the vortex is not an external body immersed in the superfluid but is part of the flow, as stressed, e.g., in Ref. \refciteputterman74superfluid. The resulting expressions for the forces thus depend on somewhat arbitrary cutoffs. As a matter of fact, the analyses of Refs. \refcitemathieu1980hydro,thouless2001vortex,sonin2001LNP lead to different predictions.

To avoid the pitfalls that plague previous studies, we propose in this paper a new treatment, originally developed in the astrophysical context of compact stars, consisting in solving the hydrodynamic equations only in the region far from the vortex, where the perturbations of the asymptotically uniform flows are small. To illustrate the method, we have considered in Section 2 the two-fluid model of superfluid helium developed by Landau and Khalatnikov.[8] Making minimal and standard assumptions on the flows, we have thus generalized the classical Kutta-Joukowski formula[25, 26] to superfluid systems. Results are discussed and summarized in Section 3, where it is shown that the same approach can be followed to determine the forces acting on vortices in cold atomic condensates and superfluid mixtures.

We have presented a new approach, originally developed in the astrophysical context of compact stars, to determine the forces acting on a quantized vortex in a superfluid in the framework of Landau-Khalatnikov two-fluid model. Considering small perturbations of the asymptotically uniform flows, we need only consider the linearized hydrodynamic equations. The complete solution far from the vortex can thus be obtained by considering separately both the irrotational and solenoidal parts of the normal flow in the Helmholtz decomposition (10). The general force can then be calculated by summing the two contributions. The additivity of the different force terms is one of the main advantages of our approach compared to previous studies focused on the inner region where the hydrodynamic equations are nonlinear and even break down near and inside the vortex core. As a first step, we focused on the irrotational part of the flows. In this way, we have generalized the Kutta-Joukowski theorem[25, 26] from classical hydrodynamics and shown that the hydrodynamic force exerted by the superfluid and normal fluid can be expressed in terms of different well-defined contributions including Magnus-type forces, see Eq. (20). Calculations of the complete expression of the force is left for future work.

The same approach can be also followed to predict the hydrodynamic forces acting on a vortex in systems described by more complicated hydrodynamic equations, such as the A phase of superfluid helium-3 (see, e.g., Ref. \refcitevollhardt1990), starting from Eq. (6) and using the appropriate expression for the momentum-flux tensor. This approach could be also of interest for studying superfluid mixtures of the kind recently produced.[5, 46, 47, 48, 49] Depending on the interactions between the components, different vortex states have been suggested (see, e.g., Refs. \refcitemueller2002two,kasamatsu2005vortices,kuopanportti2012exotic and references therein). If the vortices from the different condensates are arranged into distinct lattices, the force acting on a vortex of component $X$ at zero temperature is easily shown to be given by

where $\kappa_{a}$ denotes the corresponding quantum of circulation and we have ignored interactions between vortices. If on the other hand, the vortices overlap, the force will be given by

These formulas remain valid if the superfluids are mutually coupled by Andreev-Bashkin effects.[53]

Other possible applications include superconductors and more exotic systems such as neutron stars, whose core is thought to contain a strongly coupled mixture of a neutron superfluid and a type-II proton superconductor.[54] The conditions prevailing in these astrophysical environments are so extreme that they cannot be reproduced in the laboratory. Studies of the superfluid dynamics in neutron stars have so far relied on bold extrapolations from superfluid helium (see, e.g., Ref. \refciteperalta2006). Existing theories developed for laboratory superfluids, such as that of Ref. \refcitekopnin1995spectral, are not directly applicable to neutron stars since they contain fitting parameters that remain unknown in this context. On the contrary, our approach provides a general theoretical framework for a more realistic description of the vortex dynamics in these compact stars.

The authors thank M. Gusakov and A. Sedrakian for valuable discussions. This work was supported by the Fonds de la Recherche Scientifique (Belgium) under grants no. CDR J.0115.18, PDR T.004320 and no. 1.B.410.18F.

Derivation of the generalized Kutta-Joukowski formula (20)

Following the analysis of Ref. \refcitecarter2005covariant within a fully 4D covariant framework, we start by rewriting the momentum-flux tensor in the region far from the vortex in the more convenient form

where $p_{j}=mv_{\,\text{s}\,j}$ denotes the superfluid momentum per particle and $\Theta_{j}$ (19) the normal momentum per unit entropy, $n^{i}=nv^{i}$ is the atomic current ($n=\rho/m$) and $s^{i}=sv_{\,\text{n}}^{i}$ is the entropy current³³3The currents are the fundamental objects from which the hydrodynamic equations are derived in the variational formalism developed by Carter and collaborators (see, e.g., Refs. \refciteprix2004variational,carter2005covariant,carter1989). When applied to the two-fluid model, Carter’s approach was shown to be fully equivalent to the more traditional Landau-Khalatnikov formulation[35, 36, 37], at least in the nondissipative limit considered in Section 2.2.. Let us recall that we consider here only the irrotational part of the normal velocity in the Helmholtz decomposition and since the normal flow is also incompressible, viscous effects can be ignored. Within our assumptions, $s$ and $\rho_{\,\text{n}}$ are uniform therefore the irrotationality condition (12) can be equivalently expressed as

The first-order perturbation in the momentum-flux tensor reads

where asymptotic uniform values are denoted with a bar. Likewise, the thermodynamic relation (18) can be rewritten as

where we have introduced the shorthand notations $p_{0}=-m\mu_{\,\text{s}}-mv_{\,\text{s}}^{2}/2$ and $\Theta_{0}=-T-\boldsymbol{\Theta}\cdot\boldsymbol{v_{\,\text{n}}}$, which in the 4D covariant formulation of Ref. \refcitecarter2005covariant correspond to the time-components of the superfluid and normal 4-momenta respectively.

Substituting Eq. (36) into Eq. (35), the hydrodynamic force (9) experienced by the vortex can thus be decomposed as follows:

where

and

These four force terms are evaluated in the following.

First, let us focus on $\mathcal{F}^{a}_{i}$. The Euler equation for the superfluid (14) implies that $\nabla_{i}\,\delta p_{0}=0$, which means that $p_{0}$ is uniform, i.e., $p_{0}=\bar{p}_{0}$. This evidently leads to $\delta p_{0}=0$. Given the assumptions adopted here, the same result actually applies to the normal fluid, i.e., $\delta\Theta_{0}=0$. To see this, let us note that

which follows from the very definition of $\Theta_{0}$. The first two terms appearing on the right-hand side vanish as a consequence of our assumptions that $\delta T=\delta s=\delta\rho_{\,\text{n}}=0$ (see Section 2.2). Besides, the linearized total momentum balance equation (16) ensures that the remaining terms (inside brackets) are also zero. Therefore, one simply has $\mathcal{F}^{a}_{i}=0$. The vanishing of $\delta p_{0}$ and $\delta\Theta_{0}$ arises naturally in the 4D covariant approach, as a consequence of the irrotationality conditions (11) and (34), together with the invariance of the fluids under the action of the Killing vector associated with the time translation symmetry.[41]

Introducing the coefficients $\mathcal{D}$ and $\mathcal{D}_{\,\text{n}}$ as

the force term $\mathcal{F}_{i}^{b}$ can be rewritten as

where we have used the fact that $\delta\mathcal{D}=\mathcal{D}-\bar{\mathcal{D}}=\mathcal{D}$ and $\delta\mathcal{D}_{\,\text{n}}=\mathcal{D}_{\,\text{n}}-\bar{\mathcal{D}}_{\,\text{n}}=\mathcal{D}_{\,\text{n}}$, since the uniform background values necessarily vanish. Using Stokes’ theorem, $\mathcal{D}$ and $\mathcal{D}_{\,\text{n}}$ can be equivalently expressed as

and

where the integrals are over the surface $\mathcal{S}\left(\mathcal{C}\right)$ delimited by the contour $\mathcal{C}$ and d$S$ is the corresponding surface element. In view of (13) and (15), these two integrals are independent of the chosen contour in the region of interest, such that the quantities $\mathcal{D}$ and $\mathcal{D}_{\,\text{n}}$ are well-defined. Furthermore, since the continuity equation (13) must hold everywhere (i.e., not only in the region far from the vortex core, but also in the innermost vortex regions), one has $\mathcal{D}=0$. The force term $\boldsymbol{\mathcal{F}^{b}}$ therefore reduces to

where $D_{\,\text{n}}=\mathcal{D}_{\,\text{n}}/\bar{s}$.

As shown in Ref. \refcitecarter2005covariant, the existence of another Killing vector associated with the longitudinal invariance along the vortex line and the irrotationality conditions (11) and (34) lead to $\hat{z}^{j}\delta p_{j}=0$ and $\hat{z}^{j}\delta\Theta_{j}=0$. This result can be alternatively derived by writing explicitly the irrotationality conditions (11) and (34) in cylindrical coordinates and considering that the momenta are independent of $z$. Defining $\perp^{i}_{j}$ as the operator of projection orthogonal to the vortex line, i.e., $\perp^{i}_{j}=\delta^{i}_{j}-\hat{z}^{i}\hat{z}_{j}$, one thus has

The force term $\mathcal{F}^{c}_{i}$ can therefore be recast as

Introducing the unit vector $\boldsymbol{\beta}$ along the contour such that $\nu_{i}=-\varepsilon_{ijk}\beta^{j}\hat{z}^{k}$ (see Fig. 2), where $\varepsilon_{ijk}$ is the Levi-Civita symbol, and making use of the identity

the third term in (37) can be equivalently written as

where the superfluid momentum integral $\mathcal{C}_{\,\text{s}}$ is given by

with $\boldsymbol{\text{d}\ell}=\text{d}\ell\,\boldsymbol{\beta}$. Since the background value $\bar{\mathcal{C}}_{\,\text{s}}$ vanishes and using the fact that the superfluid momentum circulation integral is quantized, i.e., one has

in the presence of a single vortex line, the force term $\boldsymbol{\mathcal{F}^{c}}$ finally reads

Similar arguments also apply to the normal fluid, with the only but important difference that the normal momentum circulation integral

is not quantized. The use of the Stokes’ theorem, i.e.,

in conjunction with the irrotationality condition (34) ensure that $\mathcal{C}_{\,\text{n}}$ is well-defined. The last force term $\boldsymbol{\mathcal{F}^{d}}$ thus reads

Collecting terms in Eq. (37) leads to the expression given in Eq. (20) for the Kutta-Joukowski force experienced by the vortex line.

Derivation of the Kutta-Joukowski formula for the force felt by a solid body immersed in an ordinary fluid

To check the validity of our approach, we rederive here the classical expression of the Kutta-Joukowski formula (28) for the force felt by a longitudinally-invariant solid body plunged into an ordinary fluid, using a procedure similar to that followed in Appendix A for the treatment of a superfluid at finite temperatures.

As in the superfluid case, we focus on the irrotational part of the flow far from the solid body (using a Helmholtz decomposition similar to that given in Eq. (10)), such that we have

We also neglect the first-order fluctuations of the mass density $\rho$ and entropy density $s$. These assumptions also imply that $\delta T=0$ since the entropy density of an incompressible fluid only depends on the temperature. The linearized continuity equation with $\delta\rho=0$ thus leads to $\boldsymbol{\nabla}\cdot\delta\boldsymbol{v}=0$. Viscous effects can thus be ignored in the region far from the body considered here (but note that we do not neglect viscous effects in the vicinity of the body). The momentum-flux tensor reduces to (see, e.g., Ref. \refciteprix2004variational)

where $n^{i}=nv^{i}$ is the atomic current ($n=\rho/m$ being the atom number density, $m$ the atomic mass, and $v^{i}$ the fluid velocity) and $p_{j}=mv_{\,j}$ is the fluid momentum per particle. By contrast with the superfluid case, the entropy does not contribute to the momentum-flux tensor although there exists an associated current $s^{i}$. The reason lies in the vanishing of the momentum per unit entropy $\Theta_{i}$, which is nonzero in the superfluid case only due to nondissipative couplings between the two fluids (see Secs. IV.A and IV.C of Ref. \refciteprix2004variational).

The first-order perturbation in the momentum-flux tensor reads

where asymptotic uniform values are denoted with a bar. The thermodynamic relation governing the system, i.e.,

where $\mu$ denotes the fluid chemical potential per unit mass, can be rewritten as

where we have introduced the shorthand notations $p_{0}=-m\mu-mv^{2}/2$ and $\Theta_{0}=-T$, which in the 4D covariant formulation of Ref. \refcitecarter2005covariant correspond to the time-components of the fluid and entropy 4-momenta, respectively.

Substituting Eq. (62) into Eq. (60), the hydrodynamic force (9) experienced by the solid body can thus be decomposed as follows:

where

Let us first concentrate on $\mathcal{F}^{a}_{i}$. The Euler equation governing the fluid motion (we recall here that viscous effects do not play any role far from the body given the assumptions followed here) implies that $\nabla_{i}\,\delta p_{0}=0$, which means that $p_{0}$ is uniform, i.e., $p_{0}=\bar{p}_{0}$. This evidently leads to $\delta p_{0}=0$. Given the assumptions adopted here, we also have $\delta\Theta_{0}=-\delta T=0$. Therefore, one simply has $\mathcal{F}^{a}_{i}=0$, as in the superfluid case.

Introducing

the force term $\mathcal{F}_{i}^{b}$ can be rewritten as

where we have used the fact that $\delta\mathcal{D}=\mathcal{D}-\bar{\mathcal{D}}=\mathcal{D}$. Using Stokes’ theorem, $\mathcal{D}$ can be equivalently expressed as

where the integral is over the surface $\mathcal{S}\left(\mathcal{C}\right)$ delimited by the contour $\mathcal{C}$ and d$S$ is the corresponding surface element. Since the continuity equation $\boldsymbol{\nabla}\cdot\left(n\,\boldsymbol{v}\right)=0$ must hold everywhere (i.e., not only in the region far from the solid body, but also in the innermost regions), one has $\mathcal{D}=0$. The force term $\boldsymbol{\mathcal{F}^{b}}$ therefore vanishes.

The existence of another Killing vector associated with the longitudinal invariance along the solid body and the irrotationality condition (58) lead to $\hat{z}^{j}\delta p_{j}=0$. Using a similar procedure as in Appendix A, the force term $\mathcal{F}^{c}_{i}$ can therefore be recast as

where the coefficient $\Gamma$ is given by the velocity circulation

Note that the irrotationality condition (58) ensures that $\Gamma$ is well-defined.

Collecting terms in Eq. (63) leads to the well-known expression (28) for the Kutta-Joukowski force experienced by a solid body immersed in an ordinary fluid. The presence of additional terms in the superfluid case arises from the existence of nondissipative couplings between the two fluids.