Modeling quantum fluid dynamics at SPECIAL FEATURE nonzero temperatures Natalia G. Berloffa,b,1, Marc Brachetc, and Nick P. Proukakisd aDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom; bCambridge-Skoltech Quantum Fluids Laboratory, Skolkovo Institute of Science and Technology ul. Novaya, Skolkovo 143025, Russian Federation; cCentre National de la Recherche Scientifique, Laboratoire de Physique Statistique, Université Pierre-et-Marie-Curie Paris 06, Université Paris Diderot, Ecole Normale Supérieure, 75231 Paris Cedex 05, France; and dJoint Quantum Centre (JQC) Durham–Newcastle, School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom Edited by Katepalli R. Sreenivasan, New York University, New York, NY, and approved December 13, 2013 (received for review July 15, 2013) The detailed understanding of the intricate dynamics of quantum similar [e.g., Kolmogorov spectrum (9–13)], features sensitive to fluids, in particular in the rapidly growing subfield of quantum the vortex core structure arising at lengthscales smaller than the turbulence which elucidates the evolution of a vortex tangle in average intervortex spacing [e.g., velocity statistics (20–22) and a superfluid, requires an in-depth understanding of the role of finite pressure (23)] show starkly different behavior––see, e.g., the temperature in such systems. The Landau two-fluid model is the most recent reviews (24, 25). In addition to liquid helium (2, 26), there successful hydrodynamical theory of superfluid helium, but by the has been some recent interest in the observation of turbulence in nature of the scale separations it cannot give an adequate descrip- smaller inhomogeneous confined weakly interacting gases (27, tion of the processes involving vortex dynamics and interactions. In 28); such systems offer unprecedented experimental control and our contribution we introduce a framework based on a nonlinear ab initio modeling (29), with their finite size, high compress- classical-field equation that is mathematically identical to the Landau ibility, and lower disparity between intervortex distances and model and provides a mechanism for severing and coalescence of core sizes (compared with superfluid helium) leading to the vortex lines, so that the questions related to the behavior of quan- probing of a very different turbulent regime (30). Despite the large body of research on superfluid turbulence, tized vortices can be addressed self-consistently. The correct equation numerous questions remain that can only be answered by proper PHYSICS of state as well as nonlocality of interactions that leads to the existence modeling of superfluid and normal fluid behavior and interactions of the roton minimum can also be introduced in such description. We including processes associated with quantized vortices. The aim of review and apply the ideas developed for finite-temperature de- this article is to introduce a framework for achieving this task scription of weakly interacting Bose gases as possible extensions and while making use of existing methods for modeling the thermal numerical refinements of the proposed method. We apply this method component and to apply one of these methods to an important to elucidate the behavior of the vortices during expansion and con- research question in liquid helium experiments (31, 32). We do traction following the change in applied pressure. We show that at low not thus attempt to give an overview of existing results in the temperatures, during the contraction of the vortex core as the negative context of superfluid turbulence––discussed in detail in recent pressure grows back to positive values, the vortex line density grows reviews (3–5, 22, 33, 34)––but instead highlight the key predictions through a mechanism of vortex multiplication. This mechanism is sup- of such methods in describing single-vortex dynamics, as a first pressed at high temperatures. step toward a “bottom-up” approach to superfluid turbulence. superfluidity | ZNG theory | (truncated) Gross–Pitaevskii equation | Brief Historical Overview stochastic Ginzburg–Landau equation | quantum Boltzmann equation The first successful theory of superfluid helium was proposed by Landau shortly after the discovery of superfluidity (35). Using n this article we propose a framework for modeling the Landau purely classical arguments and the principle of Galilean in- two-fluid dynamics in superfluids with the intention of laying variance, Landau modeled superfluid helium as an interpen- I etrating mixture of a superfluid and normal fluid components the groundwork for the study of the effects of interconnecting s n ’ mixture of superfluid and normal fluid components on vortices (denoted, respectively, by subscripts and ). Landau s formu- and quantum turbulence in superfluid helium. lation of the two-fluid model was in the form of conservation laws for the total density ρ, the specific entropy s, the total We are concerned here with systems that are described by an ðρ v + ρ v Þ v SI ψ momentum n n s s , and the superfluid velocity s (see order parameter , a classical matter field characterizing the col- Text lective behavior of the ensemble of particles. The quantum me- for an explicit formulation and ref. 29 for discussions). In chanical interpretation of the order parameter gives the this context, the normal fluid represents a sea of elementary n v excitations that ride on the superfluid. Below the transition number densitypffiffiffi and velocity via the Madelung trans- T formation ψ = n exp½iϕ ; v = Z=m∇ϕ. The single-valuedness of temperature λ, a superfluid state emerges; with further decrease ψ in temperature the density of the normal component decreases leads to a key aspect of a superfluid: quantization of circulation T = Γ ≡ ∮v · dl = jh=m,whereh is Planck’s constant, m is atomic mass, and the density of superfluid increases until at 0 the normal j is an integer, and the integral is taken for any closed contour component disappears. In the context of quantum turbulence, moving with the fluid. The presence of vortices can lead both to the normal fluid determines the properties of the turbulent state organized arrays known as vortex lattices (1), and to random for relatively high temperatures, with superfluid vorticity fol- arrays of tangles of vortices (2). Unlike classical systems, charac- lowing the normal fluid vorticity distribution. Close to zero terized by continuous vorticity, quantum systems exhibit quantized temperature the role of the components is reversed, with the circulation; here turbulence can manifest itself in a number of ways,summarized,e.g.,inref.3. Quantum turbulence studies are concerned with the relaxation Author contributions: N.G.B. designed research; N.G.B. performed research; M.B. and N.P.P. kinetics and dynamics of a vortex tangle, with some key issues in contributed new analytic tools; and N.G.B., M.B., and N.P.P. wrote the paper. this context relating to the universal features of steady-state The authors declare no conflict of interest. forced 3D superfluid turbulence (3–13), the emergence of an This article is a PNAS Direct Submission. inverse cascade in 2D systems (14–16), and the precise mecha- 1To whom correspondence should be addressed. E-mail: [email protected]. nisms and dynamics of the decay of turbulence (17–19). Whereas This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. large-scale features of classical and quantum turbulence are 1073/pnas.1312549111/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1312549111 PNAS | March 25, 2014 | vol. 111 | suppl. 1 | 4675–4682 Downloaded by guest on September 30, 2021 turbulence in the superfluid determining the turbulence of the dispersion becomes EðkÞ = Z2k2=2m,wherem is the particle normal component. mass. The chemical potential μ is connected to pressure P via The Landau two-fluid model predated the discovery of quantized dμðnÞ = ð1=nÞdPðnÞ and comes from the equation of state of the vorticity. The tension between vortices and mutual friction system. The small perturbations of the uniform number density between vortices and normal fluid were added phenomenologi- n0 = N=V, where N is the number of particles occupying the cally to the two-fluid model by Hall et al. (36) and Bekarevich volume V, satisfy the dispersion law and Khalatnikov (37). The resulting Hall–Vinen–Bekarevich– Khalatnikov (HVBK) theory scored some success in cases of Z2 ω2 = k4 + c2k2 + ðn =mÞV~ ðkÞk2; [4] high density of superfluid vortex lines of known orientation, for m2 0 0 instance, in explaining the instability of Taylor–Couette flow (38). 4 R HVBK cannot be applied when the dynamics and relaxation ~ where VðkÞ = VðrÞexp½−ik · rd r, and the speed of sound c0 is set of the vortex tangle are important, as it treats superfluid vorticity c2 = ð =mÞ∂P=∂n by 0 1 0. The pressure relates to the energy per par- as continuum and therefore excludes the processes leading to P = n2∂ðE=NÞ=∂n vortex reconnections that drive the vortex tangle relaxation. To ticle by 0 0. Neglecting the nonlocality of interac- model the evolution of the vortex tangle another classical ap- tions leads to the cubic or higher-order nonlinear Schrödinger proach was introduced (39–41), which is now the most widely equations (NLSEs). For the weakly interacting Bose gases the E=N = V n = accepted method in studying quantum turbulence. In this ap- equation of state is known to be 0 0 2, leading to the proach superfluid vortices are modeled as classical Eulerian cubic NLSE [in the context of dilute weakly interacting Bose vortices of zero cross-section with δ-function vorticity that move gases also commonly referred to as the Gross–Pitaevskii equa- according to the Biot–Savart law tion (GPE) (42)], Z ∂ψ Z2 κ s′ × ðx − sÞ dξ 2 2 vðxÞ = ; [1] iZ = − ∇ ψ + V0jψj ψ: [5] 4π jx − sj3 ∂t 2m The higher-order NLSE results for denser fluids such as superfluid where sðξÞ is the vortex filament parameterized by arc length ξ, 4He for which the equation of state is well approximated by the and s′ðξÞ = ds=dξ is the unit tangent vector.