Energy cascades and spectra in turbulent Bose-Einstein condensates
EEnergy cascades and spectra in turbulent Bose-Einsteincondensates
Davide Proment, ∗ Sergey Nazarenko, and Miguel Onorato Dipartimento di Fisica Generale, Universit`a di Torino,Via Pietro Giuria 1, 10125 Torino, Italy Mathematics Institute, The University of Warwick, Coventry, CV4-7AL, UK (Dated: November 7, 2018)
Abstract
We present a numerical study of turbulence in Bose-Einstein condensates within the 3D Gross-Pitaevskii equation. We concentrate on the direct energy cascade in forced-dissipated systems. Weshow that behavior of the system is very sensitive to the properties of the model at the scales greaterthan the forcing scale, and we identify three different universal regimes: (1) a non-stationary regimewith condensation and transition from a four-wave to a three-wave interaction process when thelargest scales are not dissipated, (2) a steady weak wave turbulence regime when largest scales aredissipated with a friction-type dissipation, (3) a state with a scale-by-scale balance of the linearand the nonlinear timescales when the large-scale dissipation is a hypo-viscosity.
PACS numbers: 03.75.Kk, 94.05.Lk, 05.45.-a, 94.05.Pt ∗ Electronic address: [email protected] a r X i v : . [ n li n . C D ] M a y xperimental discovery of Bose-Einstein condensates (BEC) in 1995 [1, 2], some sixtyyears after their theoretical prediction [3, 4], sparked a renewed interest in this subject.Besides the obvious importance of such systems for fundamental physics, BEC experimentsprovide an excellent opportunity to build and study new nonlinear dynamical systems fabri-cated with high degree of control and flexibility supplied by optical means. For the nonlinearscience and applied mathematics such an opportunity is extremely valuable, because it allowsto implement and test dynamical and statistical regimes previously predicted theoreticallyand to gain insights about new ones for which the theory is yet to be developed. Thisis because BEC can be described by one of the most important and universal PDE’s, thenonlinear Schr¨odinger equation, called in this case Gross-Pitaevskii equation (GPE) [5]: i ∂ψ∂t + ∇ ψ − | ψ | ψ = F + D , (1)where ψ is the order parameter indicating the condensate wave function, F and D representpossible external forcings and dissipation mechanisms. In general, when F = 0 and D = 0,GPE conserves total energy and particles H = (cid:90) |∇ ψ | d x + (cid:90) | ψ | d x = H LIN + H NL , (2a) N = (cid:90) | ψ | d x . (2b)As GPE model describes a Bose gas at very low temperature, it has been used to study theformation of a condensate in [6, 7, 8]. Moreover GPE can be mapped, using the Madelungtransformation, to the Euler equation for ideal fluid flows with the extra quantum pressureterm. This is why many concepts arising from the fluid dynamics have been discussed andstudied with GPE, for example vortices and their reconnection [9]. It was also suggestedthat this model allows statistical motions similar to classical fluid turbulence and a numberof papers was devoted to finding Kolmogorov spectrum in such GPE turbulence [10, 11, 12,13, 14].On the other hand, GPE solutions also include dispersive waves which may be involvedin nonlinear interactions, and an approach known as weak wave turbulence (WWT) can bemade for GPE. Generally, WWT describes statistics on large ensembles of weakly nonlinearwaves in different applications, i.e. water waves or waves in plasmas [15]. Such waves interactwith each other in a resonant way, e.g. in triads ar quartets, thereby transferring energy2or/and any other invariants) through the scale space forming turbulent cascades similar tothe classical Kolmogorov cascade in hydrodynamic turbulence. One remarkable property ofWWT is that, in contrast to hydrodynamic turbulence, power law spectra correspondingto such cascades, known as Kolmogorov-Zakharov (KZ) spectra, have been found as exactstationary solutions of the corresponding wave kinetic equation [15].WWT for GPE turbulence was developed in [6, 16]; the following wave kinetic equationwas derived ∂n ∂t = 4 π (cid:90) n n n n (cid:18) n + 1 n − n − n (cid:19) × (3) δ ( k + k − k − k ) δ ( ω + ω − ω − ω ) d k where n i = ( L/ π ) d (cid:104) ˆ ψ ( k i , t ) ˆ ψ ∗ ( k i , t ) (cid:105) is the wave-action spectrum averaged over many re-alisations (here L is the size of the bounding box and d is the dimension of the space), ω i = k i is the wave frequency, and k i = | k i | . Equation (3) conserves the total wave-action N = (cid:82) n d k and the total energy E = (cid:82) ω n d k which correspond to the GPE invari-ants (2a) and (2b) respectively. Besides thermodynamic solutions, equation (3) has twonon-equilibrium stationary isotropic solutions of the form of n ( k ) ∼ k − α corresponding toconstant fluxes of energy or wave-action (KZ spectra). In 3D the direct energy cascade spectrum has α = 3 and the inverse wave-action cascade has α = 7 /
3. We will present ourresults in terms of the 1D wave-action spectrum n D ( k ) = 4 πk n ( k ), i.e. after integrationover the solid angle. For such a spectrum the WWT prediction for the direct cascade is n D ( k ) ∼ k − . Note that in hydrodynamic turbulence the results are usually discussed interms of the 1D energy spectrum E D ( k ) (e.g. Kolmogorov E D ( k ) ∼ k − / ). For WWT wehave the relation E D ( k ) = ω ( k ) n D ( k ), which for GPE means E D ( k ) ∼ k − α +4 . Further, itwas predicted in [6] that in presence of a condensate (due to the inverse wave-action cascade)the four-wave resonant interaction will eventually be replaced by a three-wave process withan acoustic-type KZ spectrum.Previously, there has been a number of numerical simulations of tubulence in 2D GPEcase and comparisons with the WWT predictions [6], [8]. For the 3D case, we are aware of anumber of simulations of GPE in freely decaying case, [13], or for the unforced, undampedsimulation where the initial condition relaxes to the thermodynamic solution, [7]. As far a weknow no steady state (in the sense of cascade) has ever been reached in any simulation, andno direct comparison with the WWT predictions has ever been attempted. The purpose3f the present work is to revisit the problem of 3D GPE turbulence in the direct energycascade range using the numerical simulations. Our goal will be to find the spectrum andto see if (and when) it agrees with the WWT prediction. In cases when the numerics yielda spectrum which is different from WWT, we will aim to establish the physical mechanismsbehind the formation of this state.Our numerical domain is a cube with uniform mesh of 256 points and periodic boundaryconditions. We integrate equation (1) by a standard split step method. In order to observethe cascade, energy and wave-action are injected directly in Fourier space at wave numbers ∈ [9∆ k, k ] by a forcing term ˆ F = − if e iϕ ( k ) with ϕ randomly distributed in k -space andtime and f defining the forcing coefficient. To absorb energy at high wave numbers andprevent accumulation or thermalization, a dissipative hyper-viscous term D = iν h ( −∇ ) n ψ is included in (1), with ν h = 2 × − and n = 8.If GPE is integrated with a forcing at low wave numbers and a dissipation at only highwave numbers, it will never reach a steady solution. This is because it admits also the inversecascade of wave-action which will start feeding wave number k = 0 and its close vicinity,building a strong condensate c = | ˆ ψ (0 , t ) | that changes the type of interactions from fourto three-wave resonances [6]. This become clear by looking at figure 1 where we show thespectrum at two stages. In the first one, (a) , the spectrum exhibits at wave numbers largerthan forcing a power low close to k − , consistently with the WWT prediction. As thesimulation evolves, the condensate grows and the spectrum starts to deviate from the pure − (b) . These peaksare probably the result of a three-wave interaction similar to ones previously observed for athree-wave system in [17]. In the inset we present the condensate as a function of time: thedots (a) and (b) correspond to the instant of times at which the two spectra are computed.Regime where the condensate is prevalent was theoretically considered in [6, 18]. Inthis case, the wave field in (1) can be decomposed as ψ ( x , t ) = c ( t ) + φ ( x , t ), where φ represents small fluctuations, i.e. φ (cid:28) c . The condensate part evolves as c ( t ) = c e iρ t , with ρ = | ˆ ψ (0 , t ) | . Linearizing the system the new dispersion relation ω ( k ) = ρ ± k (cid:112) k + 2 ρ can be found, known in the literature as Bogoliubov dispersion. In figure 2 we presentthe numerical evaluation of the dispersion relation taken at the final stage of simulation,case (b) ; results show excellent agreement with the theoretical Bogoliubov curve. For very4 IG. 1: Spectrum n D ( k, t ) at t = 1 × (a) and at t = 6 × (b) . In this simulation thedissipation is only hyper-viscosity. Slopes k − and k − / are also plotted. Inset: c = | ˆ ψ (0 , t ) | asa function of time.FIG. 2: Dispersion relation in the simulation with only hyper-viscosity evaluated at final time, seefigure 1 case (b) . The Bogoliubov dispersion curve (only positive branch) is superposed with whitedashed line. Inset: zoom in the zone of low k ’s to observe the condensate (horizontal branch). strong condensate, ρ (cid:29) k , the Bogoliubov waves become acoustic and ω ( k ) = k √ ρ (in areference frame rotating with the condensate speed ρ ). Such acoustic WWT was consideredin [19] and the respective KZ spectrum, E D ( k ) ∼ k − / , is very close to Kolmogorov − / E D ∼ ρ n D [18]. By looking at the late time spectrum (b) in figure 1 we see that ourresults are consistent with the Zakharov-Sagdeev prediction k − / for the three-wave acousticturbulence, although the scaling range is tiny and not well developed because of the longtransient range with peaks [17].Note that the condensate keeps growing in this simulation. In order to avoid such growth,a dissipation can be included at wave numbers lower that the ones corresponding to forcing.5 IG. 3: Spectrum n D ( k, t ) at final stage of simulation in the presence of the friction term. The k − prediction of WWT is also shown. Inset: c = | ˆ ψ (0 , t ) | as a function of time. Different options are available. Firstly, we will use a friction -type dissipation which takes theform, in Fourier space, ˆ D = iµθ ( k ∗ − k ) ˆ ψ , where θ is the Heaviside step function, k ∗ = 9∆ k corresponds to lowest wavenumber forced and µ is a friction coefficient which has been setto µ = 1 × − . We present our stationary state solution in figure 3. The resulting spectralslope is consistent with the prediction of the WWT theory. The growth of the condensateis now stopped by friction, as shown in the inset, and transition from the four-wave to athree-wave regime is prevented.Another common way of damping the low wave numbers consists, in analogy to what isdone at high wave numbers, in including a hypo-viscosity term D = iν l ( ∇ − ) m ψ in equation(1) and suppressing the condensate in Fourier space (mode k = 0). In our simulations, wehave chosen ν l = 1 × − and m = 8. In figure 4 we show the stationary states achievedwith this new damping term for different forcing coefficient f . The observed spectrum isclearly much steeper than the WWT prediction and it is reasonably fitted by a power law k − for forcing f in a wide range (two orders of magnitude). It seems that the direct energycascade is strongly influenced by the accumulation of wave-action at wave numbers belowthe forcing. In other words, a sharp dissipative term at low wave numbers can cause an infrared bottleneck effect . Similar behavior (steeper spectrum) has been observed recently innumerical simulations for water waves [20].To understand these results we try to catch the level of nonlinearity in the system byconsidering the ratio η = H NL /H LIN . Note that integral quantities are not always relevantbecause we are interested at η in the inertial range and both energies may be strongly6 IG. 4: Wave-action n D ( k, t f ) spectrum at final stage of simulation with hypo-viscosity for dif-ferent forcing coefficient: f = 0 . (A) , f = 0 . (B) , f = 0 . (C) , f = 1 . (D) , f = 2 (E) , f = 3 (F) . A k − slope is also plotted.FIG. 5: Energy ratio η = H NL /H LIN evaluated at steady non-equilibrium state with the hypo-viscous dissipation for different forcing coefficient f (see figure 4 and its label). influenced by what happens, for example, in the forcing or in the low wave number region.In the case where WWT prediction are confirmed (figure 3), η ≈ .
06; apparently WWTcondition is not valid in this case but probably most of the nonlinear energy, in Fourierspace, is stacked at low wave-numbers and so, in the inertial range, the nonlinearity remainsweak. It is instructive to look now at η in simulations with hypo-viscosity that give the k − slope. As we can see in figure 5, even by increasing the forcing coefficient f by two orderof magnitude, the ratio η remains of order one. In those cases it is reasonable to think thatthe infrared bottleneck accumulation lead to the growth of the nonlinear terms until theybecome comparable to the linear ones in the inertial range.7his observations lead to a ”critical balance” (CB) conjecture that the systems saturatesin a state where the linear and the nonlinear timescales are balanced on a scale-by-scale basis. The name CB is borrowed from MHD turbulence were it was originally proposedin [21]. Even though not called by this name, CB-like ideas were put forward in the pastfor several other physical wave systems, notably the water surface gravity waves where theCB condition leads to the famous Phillips spectrum. Indeed, in the Phillips spectrum thewave steepness is saturated by the wave breaking process when a fluid particle cannot stayattached to the water surface because its downward acceleration becomes equal to the gravityconstant, which occurs when the nonlinear time scale becomes of order of the linear one.Now we propose a similar CB state may form in GPE turbulence with hypo-viscousdissipation (or another kind of low- k dissipation which is sharp enough to lead to the infraredbottleneck). Namely, when the low- k range is over-dissipated by strong hypo-viscosity, theinverse cascade tendency tends to accumulate the spectrum at low k ’s until the criticalbalance is reached and the spectrum is saturated. When the size of the nonlinear term,locally in Fourier space, becomes of the same order as the linear, which is the critical balancecondition, the inverse cascade is arrested and the further (infrared) bottleneck accumulationis halted. One could also qualitatively view this as a set of nonlinear coherent structures, inthis case solitons or/and vortices, whose amplitude is limited by the linear dispersion (i.e.stronger solitons would break into the weaker ones and incoherent waves).We now present an estimate the CB spectrum in the GPE model. Equating the linearand the nonlinear terms for equation (1) written in Fourier space, we have k | ˆ ψ k | ∼ | ˆ ψ k | k , (4)which gives for the 1D wave action spectrum n D ( k ) = 4 πk n ( k ) = 4 πk ( L/ π ) | ˆ ψ k | ∼ k − . (5)Note that in evaluating the nonlinear term in (4) we replaced each integration over k with k which implies that the modes with wave numbers k are correlated with other k -modeswith wavenumbers of the similar values ∼ k . This is consistent with our assumption thatthe nonlinearity is scale-by-scale of the same order as the linear term. The k − predictionin equation (5) is consistent with our numerical simulations shown in figure 4.Concluding, we have performed numerical simulations of the 3D GPE with forcing anddissipation. The direct energy cascade range is strongly influenced by the second conserved8uantity, the wave-action N , which has an inverse cascade tendency; results depend onhow the low k ’s are damped. We have observed three different types of universal behaviorroughly corresponding to situations where the largest scales are either non-dissipative, ordamped by an efficient (e.g. friction-type) dissipation, or or damped by an inefficient (e.g.hypo-viscosity) dissipation. In the first case turbulence is not steady: initial direct energycascade, with a spectrum in good agreement with predictions of the WWT theory, is followedby condensation at the largest scales and a transition from a four to a three-wave interactionswith a clearly Bogoliubov dispersion relation characteristic to this regime. In the secondcase, the wave-action cascade is effectively absorbed so that there is no condensation, and weobserve a steady state spectrum which is in good agreement with the WWT theory. In thethird case, the dissipation is not so efficient and an infrared bottleneck forms in the spectrum.In this regime we observe a robust steady state spectrum which could be explained by aphenomenological ”critical balance” proposition where the linear and nonlinear timescalesare balanced on the scale-by-scale basis. [1] M. Anderson, J. Ensher, M. Matthews, C. Wieman, and E. Cornell, Science , 198 (1995).[2] K. Davis, M. Mewes, M. Andrews, N. Van Druten, D. Durfee, D. Kurn, and W. Ketterle,Physical Review Letters , 3969 (1995).[3] S. Bose, Zeitschrift fur Physik , 178 (1924).[4] A. Einstein, Klasse, Sitzungsberichte (1925).[5] L. Pitaevskii and S. Stringari, Bose-Einstein condensation (Oxford University Press, USA,2003).[6] S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D: Nonlinear Phe-nomena , 96 (1992).[7] N. Berloff and B. Svistunov, Physical Review A , 13603 (2002).[8] S. Nazarenko and M. Onorato, Physica D: Nonlinear Phenomena , 1 (2006).[9] J. Koplik and H. Levine, Physical Review Letters , 1375 (1993).[10] C. Nore, M. Abid, and M. Brachet, Physics of Fluids , 2644 (1997).[11] M. Abid, C. Huepe, S. Metens, C. Nore, C. Pham, L. Tuckerman, and M. Brachet, FluidDynamics Research , 509 (2003).
12] N. Parker and C. Adams, Physical Review Letters , 145301 (2005).[13] M. Kobayashi and M. Tsubota, Physical Review Letters , 65302 (2005).[14] M. Kobayashi and M. Tsubota, Physical Review A , 045603 (2007).[15] V. Zakharov, V. L’vov, and Falkovich, Kolmogorov Spectra of Turbulence 1: Wave Turbulence (Springer-Verlag, 1992).[16] V. Zakharov, S. Musher, and A. Rubenchik, Physics Reports (1985).[17] G. Falkovich and A. Shafarenko, Soviet Physics - JETP , 1393 (1988).[18] V. Zakharov and S. Nazarenko, Physica D: Nonlinear Phenomena , 203 (2005).[19] V. Zakharov and R. Sagdeev, Soviet Physics - Doklady (1970).[20] A. Korotkevich, Physical Review Letters , 074504 (2008).[21] P. Goldreich and S. Sridhar, The Astrophysical Journal , 763 (1995)., 763 (1995).