模拟超流

id n i a rsit 4; p stu or hin nsis important st these experi This can be and the norm mutual fricti t�. The vor- e-connected d only by a vanishes so ex core size PRL 94, 065302 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005 conventional fluid [8]. Since the normal fluid is negligible at very low temperatures, an important question arises: even without the normal fluid, is ST still similar to CT or not? To address this question, we consider the statistical law of CT [7]. The steady state for fully developed turbulence is given by the healing length � � 1= g . Although quan- tized vortices at finite temperatures can decay through mutual friction with the normal fluid, at very low tempera- tures the vortices can decay only by two mechanisms; one is the sound emission through vortex reconnections [11], and the other is that vortices reduce through the Richardson 0031-9007= atistical laws [7] of fully developed CT, so ments show a similarity between ST and CT. understood using the idea that the superfluid al fluid are likely to be coupled together by the on between them and thus to behave like a superfluid velocity given by v�x; t� � 2r �x; ticity rotv�x; t� vanishes everywhere in a singl region of the fluid; any rotational flow is carrie quantized vortex in the core of which ��x; t� that the circulation is quantized by 4�. The vort������p Kolmogorov law. The Kolmogorov law is one of the most stant, �x; t� is the condensate density, and v�x; t� is the Kolmogorov Spectrum of Superflu of the Gross-Pitaevskii Equatio Michikazu Kobayash Department of Physics, Osaka City Unive (Received 30 November 200 The energy spectrum of superfluid turbulence is equation. We introduce the dissipation term which w to remove short wavelength excitations which may inertial range. The obtained energy spectrum is co DOI: 10.1103/PhysRevLett.94.065302 The physics of quantized vortices in liquid 4He is one of the most important topics in low temperature physics [1]. Liquid 4He enters the superfluid state at 2.17 K. Below this temperature, the hydrodynamics is usually described using the two-fluid model in which the system consists of invis- cid superfluid and viscous normal fluid. Early experimental works on the subject focused on thermal counterflow in which the normal fluid flowed in the opposite direction to the superfluid flow. This flow is driven by the injected heat current, and it was found that the superflow becomes dissipative when the relative velocity between two fluids exceeds a critical value [1]. Feynman proposed that this is a superfluid turbulent state consisting of a tangle of quan- tized vortices [2]. Vinen confirmed Feynman’s picture experimentally by showing that the dissipation comes from the mutual friction between vortices and the normal flow [3]. After that, many studies on superfluid turbulence (ST) have been devoted to thermal counterflow [4]. However, as thermal counterflow has no analogy with conventional fluid dynamics, we have not understood the relation between ST and classical turbulence (CT). With such a background, experiments of ST that do not include thermal counterflow have recently appeared. Such studies represent a new stage in studies of ST. Maurer et al. studied a ST that was driven by two counterrotating disks [5] in superfluid 4He at temperatures above 1.4 K, and they obtained the Kolmogorov law. Stalp et al. observed the decay of grid turbulence [6] in superfluid 4He at tempera- tures above 1 K, and their results were consistent with the 05=94(6)=065302(4)$23.00 06530 Turbulence: Numerical Analysis with a Small-Scale Dissipation nd Makoto Tsubota y, Sumiyoshi-Ku, Osaka 558-8585, Japan ublished 14 February 2005) died numerically by solving the Gross-Pitaevskii ks only in the scale smaller than the healing length der the cascade process of quantized vortices in the tent with the Kolmogorov law. PACS numbers: 67.40.Vs, 47.37.+q, 67.40.Hf of an incompressible classical fluid follows the Kolmogorov law for the energy spectrum. The energy is injected into the fluid at some large scales in the energy- containing range. This energy is transferred in the inertial range from large to small scales without being dissipated. The inertial range is believed to be sustained by the self- similar Richardson cascade in which large eddies are bro- ken up into smaller ones, having the Kolmogorov law E�k� � C�2=3k�5=3: (1) Here the energy spectrum E�k� is defined as E � RdkE�k�, where E is the kinetic energy per unit mass and k is the wave number from the Fourier transformation of the ve- locity field. The energy transferred to smaller scales in the energy-dissipative range is dissipated by the viscosity with the dissipation rate, which is identical with the energy flux � of Eq. (1) in the inertial range. The Kolmogorov constant C is a dimensionless parameter of order unity. In CT, the Richardson cascade is not completely under- stood, because it is impossible to definitely identify each eddy. In contrast, quantized vortices in superfluid are defi- nite and stable topological defects. A Bose-Einstein con- densed system yields a macroscopic wave function ��x; t� � ������������� �x; t�p ei �x;t�, whose dynamics is governed by the Gross-Pitaevskii (GP) equation [9,10] i @ @t ��x; t� � ��r2 ��� gj��x; t�j2 ��x; t�; (2) where � is the chemical potential, g is the coupling con- 2-1  2005 The American Physical Society cascade process and eventually change to elementary ex- we used a spatial resolution �x � 0:125 and V � 323, where the length scale is normalized by the healing length �. With this choice, �k � 2�=32. Numerical time evolu- tion was given by the Runge-Kutta-Verner method with the time resolution �t � 1� 10�4. To obtain a turbulent state, we start from an initial configuration in which the condensate density 0 is uni- form and the phase 0�x� has a random spatial distribution. Here the random phase 0�x� is made by placing random numbers between �� to � at every distance � � 4 and connecting them smoothly (Fig. 1). The initial velocity v�x; t � 0� � 2r 0�x� given by the initial random phase is random; hence, the initial wave function is dynamically unstable and soon produces homo- geneous and isotropic turbulence with many quantized vortex loops. To confirm the accuracy of our simulation, we calculate PRL 94, 065302 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005 30 citations at the healing length scale. In any case, dissipation occurs only at scales below the healing length. Therefore, ST at very low temperatures, consisting of such quantized vortices, can be a prototype to study the inertial range, the Kolmogorov law, and the Richardson cascade. There are two kinds of formulation to study the dynam- ics of quantized vortices: one is the vortex-filament model [12], and the other the GP model. By using the vortex- filament model with no normal fluid component, Araki et al. studied numerically a vortex tangle starting from a Taylor-Green flow, thus obtaining the energy spectrum consistent with the Kolmogorov law [13]. By eliminating the smallest vortices whose size is comparable to the numerical space resolution, they introduced the dissipation into the system. Nore et al. used the GP equation to numerically study the energy spectrum of ST [14]. The kinetic energy consists of a compressible part due to sound waves and an incompressible part coming from quantized vortices. Excitations of wavelength less than the healing length are created through vortex reconnections or through the disappearance of small vortex loops [11,15], so that the incompressible kinetic energy transforms into compress- ible kinetic energy while conserving the total energy. The spectrum of the incompressible kinetic energy is tempo- rarily consistent with the Kolmogorov law. However, the consistency becomes weak in the late stage when many sound waves created through those processes hinder the cascade process [16–18]. Our approach here is to introduce a dissipation term that works only on scales smaller than the healing length �. This introduction of the dissipation term supposes some possible dissipation like the emission of elementary exci- tations which can happen only in smaller scales than the healing length � even at T � 0 K [11,15]. The cascade process in larger scales must be independent of the detail of the dissipative mechanism at small scales. This dissipation removes not vortices but short wavelength excitations, thus preventing the excitation energy from transforming back to vortices. Compared to the usual GP model, this approach enables us to more clearly study the Kolmogorov law. To solve the GP equation numerically with high accu- racy, we use the Fourier spectral method in space with a periodic boundary condition in a box with spatial resolu- tion containing 2563 grid points. We solve the Fourier transformed GP equation i @ @t ~��k; t� � �k2 �� ~��k; t� � g V2 X k1;k2 ~��k1; t� ~���k2; t� � ~��k� k1 � k2; t�; (3) where V is the volume of the system and ~��k; t� is the spatial Fourier component of ��x; t� with the wave number k, numerically given by a fast Fourier transformation [19]. We consider the case of g � 1. For numerical parameters, 065 the total energy E, the interaction energy Eint, the quantum energy Eq, and the kinetic energy Ekin [14]: E� R dx����r2�g=2j�j2 �R dx ; Eint � g2 R dxj�j4R dx ; Eq � R dx�rj�j 2R dx ; Ekin � R dx�j�jr 2R dx ; (4) and compare them with those given by the dynamics with the different time resolutions �t � 2� 10�5 or different spatial resolutions 5123 grids. For the comparison, we estimate the relative errors F12�A� � j�hAi1 � hAi2�=hAi1j and F13�A� � j�hAi1 � hAi3�=hAi1j at t � 12 for A � E, Eint, Eq, and Ekin, where h i1, h i2, and h i3 are, respectively, the values given by three simulations with different reso- lutions: (i) �t�1�10�4 and 2563 grids, (ii) �t � 2� 10�5 and 2563 grids, and (iii) �t � 1� 10�4 and 5123 grids. Comparison is shown in Table I; the relative errors are extremely small, which allows us to use resolution (i). Furthermore, the total energy is conserved by the accu- racy of 10�10 in our simulation. -π π 0 32 φ0(x,y=0,z=0) random number x λ - 0 32 32 0 (x,y,z=0) x y φ π π (a) (b) FIG. 1 (color online). How to make the random phase 0�x� with � � 4 (a) and one example of an initial random phase in a cross section 0�x; y; 0� (b). 2-2 We introduced a dissipation term in Eq. (3) to remove excitations of a wavelength shorter than the healing length. The imaginary number i in the left-hand side of Eq. (3) was replaced by �i� ��k� , where ��k� � �0��k� 2�=�� with ��x� being the step function. The effect of this dis- sipation is shown in Fig. 2. We divide up to the kinetic energy Ekin into the compressible part Eckin �R dx��j�jr �c 2=R dx , due to sound waves, and the incompressible part Eikin � R dx��j�jr �i 2=Rdx , due c i This figure shows that our ST satisfies the Kolmogorov law Eikin�k� / k�5=3 for times 4 & t & 10, when the system may be almost homogeneous and isotropic turbulence. We also calculated the energy dissipation rate � � �@Eikin=@t and compared the results quantitatively with the Kolmogorov law. Figure 3(b) shows that � is almost con- stant in the period 4 & t & 10, which means that the Kolmogorov spectrum �2=3k�� is also constant in the period. Without the dissipation term ��k�, the ST could not satisfy the Kolmogorov law. This means that dissipat- ing short wavelength excitations are essential for satisfying the Kolmogorov law. Figure 3(c) shows the time develop- ment of the exponent � for �0 � 0. In this case, � is consistent with the Kolmogorov law only in the short period 4 & t & 7; active sound waves affect the dynamics of quantized vortices for t * 7. Such an effect of sound waves is also clearly shown in Fig. 3(d). Compared with Fig. 3(b), � is unsteady and smaller than that of �0 � 1 and even becomes negative at times. When � is negative, the energy flows backward from sound waves to vortices, which may prevent the energy spectrum from satisfying the Kolmogorov law for t * 7. According to Fig. 3(a), the agreement between the energy spectrum and the Kolmogorov law becomes weak at a later stage t * 10, which may be attributable to the following reasons. In the period 4 & t & 10, the energy spectrum agrees with the Kolmogorov law, which may support that the Richardson cascade process works in the system. The dissipation is TABLE I. Dependence of F12 and F13 on E, Eint, Eq, and Ekin. F12�A� F13�A� A � E 2:4� 10�12 6:3� 10�10 A � Eint 3:7� 10�12 8:8� 10�10 A � Eq 2:6� 10�12 6:9� 10�10 A � Ekin 5:1� 10�12 9:4� 10�10 PRL 94, 065302 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005 to vortices, where rot�j�jr � � 0 and div�j�jr � � 0 [14,15]. Figure 2 shows the time development of E, Ekin, Eckin, and Eikin in the cases of 2(a) �0 � 0 and 2(b) �0 � 1. Without dissipation, the compressible kinetic energy Eckin is increased in spite of conserving the total energy [Fig. 2(a)], which is consistent with the simulation by Nore et al. [14]. The dissipation suppresses Eckin and thus causes Eikin to be dominant. This dissipation term works as a viscosity only at the small scale. At other scales, which are equivalent to the inertial range �k < k < 2�=�, the energy does not dissipate. We calculated the spectrum of the incompressible ki- netic energy Eikin�k� defined as Eikin � R dkEikin�k�. Initially, the spectrum Eikin�k� significantly deviates from the Kolmogorov power law; however, the spectrum ap- proaches a power law as the turbulence develops. We assumed that the spectrum Eikin�k� is proportional to k�� in the inertial range �k < k < 2�=�, and determined the exponent � by making a straight line fit to a log-log plot of the spectrum Eikin�k� at each time. The time development of � is shown in Fig. 3(a). 5 10 15 E Ekin Ekin c Ekin i 5 10 15 E Ekin Ekin c Ekin i (a) (b) 00 1 2 3 4 5 6 t 00 1 2 3 4 5 6 t FIG. 2. Time development of E, Ekin, Eckin, and Eikin. (a) Case with �0 � 0; (b) case with �0 � 1. 06530 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 η 5/3 η t -2 -1 0 1 0 2 4 6 8 10 12 ε t 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0 2 4 6 8 10 12 η 5/3 η t -2 -1 0 1 0 2 4 6 8 10 12 ε t (a) (b) (c) (d) FIG. 3. Time dependence of the exponent of the spectrum � (a) and the dissipation rate �. (a) � for �0 � 1. (b) � for �0 � 1. (c) � for �0 � 0. (d) � for �0 � 0. In (a) and (c), the error bars are the standard deviations of the data from the fit. The line � � 5=3 is shown to compare the results with the Kolmogorov law. 2-3 the energy and broadening the inertial range of the energy spectrum. We will report on these results in near future. We acknowledge W. F. Vinen for useful discussions. M. T. and M. K. both acknowledge support by a Grant-in- Aid for Scientific Research (Grants No. 15340122 and No. 1505983, respectively) by the Japan Society for the Promotion of Science. [1] R. J. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991). [2] R. P. Feynman, in Progress in Low Temperature Physics, edited by C. J. Gorter (North-Holland, Amsterdam, 1955), 0 32 32 32 x y z -5 -4.5 -4 -3.5 -3 -2.5 -2 -0.5 0 1.1 Ekin c(k) Ekin i(k) Cε2/3k-5/3 (C=1) L og E (k ) Log k k=2 π/ξ (a) (b) FIG. 4. Vorticity rotv�x; t� and the energy spectrum at t � 6:0. The contour in (a) is 98% of maximum vorticity. The energy PRL 94, 065302 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005 caused mainly by removing short wavelength excitations emitted at vortex reconnections. However, the system at the late stage t * 10 has only small vortices after the Richardson cascade process, being no longer turbulent. The energy spectrum, therefore, disagrees with the Kolmogorov law of � � 5=3. And emissions of excitations through vortex reconnections hardly occur, which reduces greatly the energy dissipation rate �. Spatially, the system appears to develop full turbulence. Shown in Fig. 4(a) is the spatial distribution of vortices. The vorticity plot suggests fully developed turbulence. The energy spectrum in Fig. 4(b) agrees quantitatively with the Kolmogorov law. We thus conclude that the incompress- ible kinetic energy of ST without the effect of short wave- length excitations satisfies the Kolmogorov law. This inertial range 0:20 & k & 6:3 is larger than those in the simulations by Araki et al. [13] and Nore et al. [14]. The Kolmogorov constant is estimated to be C ’ 10�0:5 � 0:32, being just smaller than that in classical turbulence. In conclusion, we investigate the energy spectrum of ST by the numerical simulation of the GP equation. Suppressing the compressible sound waves which are cre- ated through vortex reconnections, we can make the quan- spectrum was obtained by making an ensemble average for 20 different initial states The solid line refers to the Kolmogorov law in Eq. (1). titative agreement between the spectrum of incompressible kinetic energy and Kolmogorov law. Our next subjects are making the steady turbulence by introducing injection of 06530 2-4 Vol. I, p. 17. [3] W. F. Vinen, Proc. R. Soc. London A 240, 114 (1957); 240, 128 (1957); 240, 493 (1957). [4] J. T. Tough, in Progress in Low Temperature Physics, edited by C. J. 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