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Kolmogorov spectrum of the Gross–Pitaevskii turbulence
Journal of Physics and Chemistry of Solids 66 (2005) 1498–1500 www.elsevier.com/locate/jpcs
Kolmogorov spectrum of the Gross–Pitaevskii turbulence Makoto Tsubota*, Michikazu Kobayashi Department of Physics, Faculty of Physics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan
Abstract The energy spectrum of quantum turbulence is studied numerically by solving the Gross–Pitaevskii equation. We introduce the dissipation term which works only in the scale smaller than the healing length, to remove short wavelength excitations which may hinder the cascade process of quantized vortices in the inertial range. The obtained energy spectrum is consistent with the Kolmogorov law. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Non-crystalline materials; D. Defeats
1. Introduction The physics of quantized vortices is one of the most important topics in low temperature physics [1]. Liquid 4He enters the superfluid state at 2.17 K, and its hydrodynamics is usually described using the two-fluid model in which the system consists of inviscid 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. Feynman proposed that this is a quantum turbulent state consisting of a tangle of quantized vortices [2], which was confirmed experimentally by Vinen [3]. After that, many studies on quantum turbulence (QT) have been devoted to thermal ounterflow. However, as thermal counterflow has no analogy with conventional fluid dynamics, we have not understood the relation between QT and classical turbulence (CT). Experiments of QT that do not include thermal counterflow have recently appeared [4,5], making the studies of QT enter a new stage [6]. One of the important interests is a similarity between QT and CT. To address this question, we consider the statistical law of CT. The steady state for fully developed turbulence of an incompressible classical fluid follows the Kolmogorov law * Corresponding author. E-mail address: [email protected] (M. Tsubota).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.05.042
for the energy spectrum. The 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 broken up to smaller ones, having the Kolmogorov law EðkÞZ ÐCe2=3 kK5=3 . Here, the energy spectrum E(k) is defined as EZ dkEðkÞ, where E is the kinetic energy per unit mass and k is the wave number from the Fourier transformation of the velocity 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 e in the inertial range. The Kolmogorov constant C is a dimensionless parameter of order unity. There are two kinds of formulation to study the dynamics of quantized vortices; one is the vortex-filament model [7] and the other the Gross–Pitaevskii (GP) model [8,9]. By using the vortex-filament model with no normal fluid component, Araki et at. [10] studied numerically a vortex tangle starting from a Taylor–Green flow, thus obtaining the energy spectrum consistent with the Kolmogorov law. By eliminating the smallest vortices whose size is comparable to the numerical space resolution, they introduced the dissipation into the system. Nore et al. [11] used the GP equation to numerically study the energy spectrum of QT. 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 [12,13], so that the incompressible kinetic energy transforms into compressible kinetic energy while conserving the total energy. The spectrum of the incompressible kinetic energy
M. Tsubota, M. Kobayashi / Journal of Physics and Chemistry of Solids 66 (2005) 1498–1500
is temporarily consistent with the Kolniogorov law. However, the consistency becomes weak in the late stage when many sound waves created through those processes hinder the cascade process. In this work, in order to preserve the cascade process, we introduce into the GP equation some dissipation that works only in the scales smaller than the healing length. Then we study numerically the GP equation for the macroscopic wave function in the wave number space and show the energy spectrum of QT takes the Kolmogorov law [14].
2. The Gross–Pitaevskii equation in the wave number space A Bose–Einstein condensed yields a macroscopic pffiffiffiffiffiffiffiffiffiffiffisystem ffi wave function Fðx; tÞZ rðx; tÞeifðx;tÞ , whose dynamics is governed by the Gross–Pitaevskii (GP) equation i
v Fðx; tÞ Z ½KV2 K m C gFðx; tÞj2 Fðx; tÞ; vt
(1)
where m is the chemical potential, g is the coupling constant, r(x, t) is the condensate density, and v(x, t) is the superfluid velocity given by v(x, t)Z2Pf(x, t). The vorticity rot v(x, t) vanishes every-where in a single-connected region of the fluid; any rotational flow is carried only by a quantized vortex in the core of which F(x, t) vanishes so that the circulation is quantized by 4p. The vortex core size is given pffiffiffiffiffi by the healing length xZ 1 gr. To solve the GP equation numerically with high accuracy, we use the Fourier spectral method in space with periodic boundary condition in a box with spatial resolution containing 2563 grid points. We solve the Fourier transformed GP equation i
v ~ g X ~ ~ Fðk; tÞ Z ½k2 K mFðk; Fðk1 ; tÞ tÞ C 2 vt V k ;k 1
2
~ ðk2 ; tÞ ! Fðk ~ K k1 C k2 ; tÞ; !F
(2)
~ where V is the volume of the system and Fðk; tÞ is the spatial Fourier component of F(x, t) with the wave number k. We consider the case of gK1. For numerical parameters, we used a spatial resolution DxZ0.125 and VZ323, where the length scale is normalized by the healing length x. With this choice, DkZ2p/32. Numerical time evolution was given by the Runge–Kutta–Verner method with the time resolution DtZ1!10K4. Our approach here is to introduce a dissipation term that works only on scales smaller than the healing length x. 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 [11], this approach enables us to more clearly study the Kolmogorov law. This dissipation term in. Eq. (2) is introduced by replacing the imaginary number i in
1499
the left-hand side by [iKg(k)], where gðkÞZ g0 qðkK 2p=xÞ with q(x) being the step function. 3. Numerical analysis of the GP equation and the energy spectrum of QT To obtain a turbulent state, we start from an initial configuration in which the condensate density r0 is uniform and the phase f0(x) has a random spatial distribution. Here, the random phase f0(x) is made by placing random numbers between Kp to p at every distance lZ4 and connecting them smoothly. The initial velocity v(x, tZ0)Z2Pf0(x) given by the initial random phase is random, hence the initial wave function is dynamically unstable and soon produces homogeneous and isotropic turbulence with many quantized vortex loops, as shown in Fig. 1(a)–(c). In order to study the Kolmogorov spectrum, we need take property the energy component of vortices. The total energy E of the GP model is given by the sum of the interaction energy Eint, the quantum energy Eq and the kinetic energy Ekin [11,13]: Ð dxF ½KV2 C g=2jFj2 F Ð ; EZ dxr Ð g dxjFj4 Ð Eint Z ; 2 2 dxr Ð Eq Z
dx½VjFj2 Ð ; dxr Ð
Ekin Z
dx½jFjVf2 Ð : dxr
The kinetic energy Ekin is furthermore divided into the compressible part Ð dx½ðjFjVfÞc 2 c Ð Ekin (3) Z dxr due to sound waves, and the incompressible part Ð dx½ðjFjVfÞi 2 i Ð Ekin Z dxr
(4)
due to vortices, where rot(jFjPf)cZ0 and div(jFjPf)iZ0. i It is the incompressible kinetic energy Ekin that can take the Kolmogorov law reflecting the Richradson cascade process. i We calculated the spectrum of Ekin . Initially, the i spectrum Ekin ðkÞ significantly deviates from the Kolmogorov power-law, however, the spectrum approaches a power-law as the turbulence develops. We assumed that the spectrum is proportional to kKh in the inertial range Dk! k!2p/x, and we determined the exponent h. It is found that h is almost 5/3 for times 4!t!10, when the system may be almost homogeneous and isotropic turbulence. We also i calculated the energy dissipation rate eZKvEkin =vt. We found that e is almost constant in the period 4!t!10,
1500
M. Tsubota, M. Kobayashi / Journal of Physics and Chemistry of Solids 66 (2005) 1498–1500
Fig. 1. Time development of the vorticity and the energy spectrum. The contours in (a)–(c) are 98% of the maximum vorticity. (d) The spectrum of the incompressible kinetic energy at tZ6. The energy spectrum was obtained by making am ensemble average for 20 initial Antes. The solid line refers to the Kolmogorov law.
which means that the Kolmogorov spectrum e2/3kKh is also constant in the period. The energy spectrum using this e agrees quantitatively with the Kolmogorov law obtained numerically, as shown in Fig. 1(d). Without the dissipation term g(k), the QT could not satisfy the Kolmogorov law. This means that dissipating short wavelength excitations are essential for satisfying the Kolmogorov law. The agreement between the energy spectrum and the Kolmogorov law becomes weak at a later stage tO10, 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 caused mainly by removing short wavelength excitations emitted at vortex reconnections. However, the system at the late stage tO10 has only small vortices after the Richardson cascade process, being no longer turbulent. The energy spectrum, therefore, disagrees with the Kolmogorov law of hZ5/3.
4. Conclusions We investigate the energy spectrum of QT by the numerical simulation of the GP equation. Suppressing the compressible sound waves which are created through vortex reconnections, we can make the quantitative agreement between the spectrum of the incompressible kinetic energy and the Kolmugorov law. Our next subjects are making the steady turbulence by introducing injection of the energy at
a large scale and broadening the inertial range of the energy spectrum. Acknowledgements We acknowledge W.F. Vineu for useful discussions. MT and MK both acknowledge support by a grant-in-aid for Scientific Research (Grant Nos 15340122 and 1505983, respectively) by the Japan Society for the Promotion of Science. References [1] R.J. Donnelly, Quantized Vortices in Helium II, Cambridge University Press, Cambridge, 1991. [2] R.P. Feymnan, in: C.J. Gorter (Eds), Progress in Low Temperature Physics, vol. I, North Holland Publishing Company, 1955, p.17. [3] W.F. Vinen, Proc. R. Soc. A240 (1957) 114; W.F. Vinen, Proc. R. Soc. 240 (1957) 128; W.F. Vinen, Proc. R. Soc. 240 (1957) 493. [4] J. Maurer, R. Tabeling, Europhys. Lett. 43 (1998) 29. [5] S.R. Stalp, L. Skrbek, R.J. Donnelly, Phys. Rev. Lett. 82 (1999) 4831. [6] W.F. Vinen, J.J. Niemela, J. Low Temp. Phys. 128 (2002) 167. [7] K.W. Schwarz, Phys. Rev. B31 (1985) 5782; K.W. Schwarz, Phys. Rev. 38 (1988) 2398. [8] E.P. Gross, J. Math. Phys. 4 (1963) 195. [9] L.P. Pitaevskii, Soviet Phys. JETP 13 (1963) 195. [10] T. Araki, M. Tsubota, S.K. Nemirovskii, Phys. Rev. Lett. 89 (2002) 145301. [11] C. Nore, M. Abid, M.E. Brachet, Phys. Fluids 9 (1997) 2644. [12] M. Leadbeater, T. Winiecki, D.C. Samuels, C.F. Barenghi, C.S. Adams, Phys. Rev. Lett. 86 (2001) 1410. [13] S. Ogawa, M. Tsubota, Y. Hattori, J. Phys. Soc. Jpn 71 (2002) 813. [14] M. Kobayashi, M. Tsubota, Phys. Rev. Lett. 94 (2005) 065302.
Kolmogorov spectrum of the Gross–Pitaevskii turbulence Makoto Tsubota*, Michikazu Kobayashi Department of Physics, Faculty of Physics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan
Abstract The energy spectrum of quantum turbulence is studied numerically by solving the Gross–Pitaevskii equation. We introduce the dissipation term which works only in the scale smaller than the healing length, to remove short wavelength excitations which may hinder the cascade process of quantized vortices in the inertial range. The obtained energy spectrum is consistent with the Kolmogorov law. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Non-crystalline materials; D. Defeats
1. Introduction The physics of quantized vortices is one of the most important topics in low temperature physics [1]. Liquid 4He enters the superfluid state at 2.17 K, and its hydrodynamics is usually described using the two-fluid model in which the system consists of inviscid 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. Feynman proposed that this is a quantum turbulent state consisting of a tangle of quantized vortices [2], which was confirmed experimentally by Vinen [3]. After that, many studies on quantum turbulence (QT) have been devoted to thermal ounterflow. However, as thermal counterflow has no analogy with conventional fluid dynamics, we have not understood the relation between QT and classical turbulence (CT). Experiments of QT that do not include thermal counterflow have recently appeared [4,5], making the studies of QT enter a new stage [6]. One of the important interests is a similarity between QT and CT. To address this question, we consider the statistical law of CT. The steady state for fully developed turbulence of an incompressible classical fluid follows the Kolmogorov law * Corresponding author. E-mail address: [email protected] (M. Tsubota).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.05.042
for the energy spectrum. The 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 broken up to smaller ones, having the Kolmogorov law EðkÞZ ÐCe2=3 kK5=3 . Here, the energy spectrum E(k) is defined as EZ dkEðkÞ, where E is the kinetic energy per unit mass and k is the wave number from the Fourier transformation of the velocity 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 e in the inertial range. The Kolmogorov constant C is a dimensionless parameter of order unity. There are two kinds of formulation to study the dynamics of quantized vortices; one is the vortex-filament model [7] and the other the Gross–Pitaevskii (GP) model [8,9]. By using the vortex-filament model with no normal fluid component, Araki et at. [10] studied numerically a vortex tangle starting from a Taylor–Green flow, thus obtaining the energy spectrum consistent with the Kolmogorov law. By eliminating the smallest vortices whose size is comparable to the numerical space resolution, they introduced the dissipation into the system. Nore et al. [11] used the GP equation to numerically study the energy spectrum of QT. 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 [12,13], so that the incompressible kinetic energy transforms into compressible kinetic energy while conserving the total energy. The spectrum of the incompressible kinetic energy
M. Tsubota, M. Kobayashi / Journal of Physics and Chemistry of Solids 66 (2005) 1498–1500
is temporarily consistent with the Kolniogorov law. However, the consistency becomes weak in the late stage when many sound waves created through those processes hinder the cascade process. In this work, in order to preserve the cascade process, we introduce into the GP equation some dissipation that works only in the scales smaller than the healing length. Then we study numerically the GP equation for the macroscopic wave function in the wave number space and show the energy spectrum of QT takes the Kolmogorov law [14].
2. The Gross–Pitaevskii equation in the wave number space A Bose–Einstein condensed yields a macroscopic pffiffiffiffiffiffiffiffiffiffiffisystem ffi wave function Fðx; tÞZ rðx; tÞeifðx;tÞ , whose dynamics is governed by the Gross–Pitaevskii (GP) equation i
v Fðx; tÞ Z ½KV2 K m C gFðx; tÞj2 Fðx; tÞ; vt
(1)
where m is the chemical potential, g is the coupling constant, r(x, t) is the condensate density, and v(x, t) is the superfluid velocity given by v(x, t)Z2Pf(x, t). The vorticity rot v(x, t) vanishes every-where in a single-connected region of the fluid; any rotational flow is carried only by a quantized vortex in the core of which F(x, t) vanishes so that the circulation is quantized by 4p. The vortex core size is given pffiffiffiffiffi by the healing length xZ 1 gr. To solve the GP equation numerically with high accuracy, we use the Fourier spectral method in space with periodic boundary condition in a box with spatial resolution containing 2563 grid points. We solve the Fourier transformed GP equation i
v ~ g X ~ ~ Fðk; tÞ Z ½k2 K mFðk; Fðk1 ; tÞ tÞ C 2 vt V k ;k 1
2
~ ðk2 ; tÞ ! Fðk ~ K k1 C k2 ; tÞ; !F
(2)
~ where V is the volume of the system and Fðk; tÞ is the spatial Fourier component of F(x, t) with the wave number k. We consider the case of gK1. For numerical parameters, we used a spatial resolution DxZ0.125 and VZ323, where the length scale is normalized by the healing length x. With this choice, DkZ2p/32. Numerical time evolution was given by the Runge–Kutta–Verner method with the time resolution DtZ1!10K4. Our approach here is to introduce a dissipation term that works only on scales smaller than the healing length x. 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 [11], this approach enables us to more clearly study the Kolmogorov law. This dissipation term in. Eq. (2) is introduced by replacing the imaginary number i in
1499
the left-hand side by [iKg(k)], where gðkÞZ g0 qðkK 2p=xÞ with q(x) being the step function. 3. Numerical analysis of the GP equation and the energy spectrum of QT To obtain a turbulent state, we start from an initial configuration in which the condensate density r0 is uniform and the phase f0(x) has a random spatial distribution. Here, the random phase f0(x) is made by placing random numbers between Kp to p at every distance lZ4 and connecting them smoothly. The initial velocity v(x, tZ0)Z2Pf0(x) given by the initial random phase is random, hence the initial wave function is dynamically unstable and soon produces homogeneous and isotropic turbulence with many quantized vortex loops, as shown in Fig. 1(a)–(c). In order to study the Kolmogorov spectrum, we need take property the energy component of vortices. The total energy E of the GP model is given by the sum of the interaction energy Eint, the quantum energy Eq and the kinetic energy Ekin [11,13]: Ð dxF ½KV2 C g=2jFj2 F Ð ; EZ dxr Ð g dxjFj4 Ð Eint Z ; 2 2 dxr Ð Eq Z
dx½VjFj2 Ð ; dxr Ð
Ekin Z
dx½jFjVf2 Ð : dxr
The kinetic energy Ekin is furthermore divided into the compressible part Ð dx½ðjFjVfÞc 2 c Ð Ekin (3) Z dxr due to sound waves, and the incompressible part Ð dx½ðjFjVfÞi 2 i Ð Ekin Z dxr
(4)
due to vortices, where rot(jFjPf)cZ0 and div(jFjPf)iZ0. i It is the incompressible kinetic energy Ekin that can take the Kolmogorov law reflecting the Richradson cascade process. i We calculated the spectrum of Ekin . Initially, the i spectrum Ekin ðkÞ significantly deviates from the Kolmogorov power-law, however, the spectrum approaches a power-law as the turbulence develops. We assumed that the spectrum is proportional to kKh in the inertial range Dk! k!2p/x, and we determined the exponent h. It is found that h is almost 5/3 for times 4!t!10, when the system may be almost homogeneous and isotropic turbulence. We also i calculated the energy dissipation rate eZKvEkin =vt. We found that e is almost constant in the period 4!t!10,
1500
M. Tsubota, M. Kobayashi / Journal of Physics and Chemistry of Solids 66 (2005) 1498–1500
Fig. 1. Time development of the vorticity and the energy spectrum. The contours in (a)–(c) are 98% of the maximum vorticity. (d) The spectrum of the incompressible kinetic energy at tZ6. The energy spectrum was obtained by making am ensemble average for 20 initial Antes. The solid line refers to the Kolmogorov law.
which means that the Kolmogorov spectrum e2/3kKh is also constant in the period. The energy spectrum using this e agrees quantitatively with the Kolmogorov law obtained numerically, as shown in Fig. 1(d). Without the dissipation term g(k), the QT could not satisfy the Kolmogorov law. This means that dissipating short wavelength excitations are essential for satisfying the Kolmogorov law. The agreement between the energy spectrum and the Kolmogorov law becomes weak at a later stage tO10, 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 caused mainly by removing short wavelength excitations emitted at vortex reconnections. However, the system at the late stage tO10 has only small vortices after the Richardson cascade process, being no longer turbulent. The energy spectrum, therefore, disagrees with the Kolmogorov law of hZ5/3.
4. Conclusions We investigate the energy spectrum of QT by the numerical simulation of the GP equation. Suppressing the compressible sound waves which are created through vortex reconnections, we can make the quantitative agreement between the spectrum of the incompressible kinetic energy and the Kolmugorov law. Our next subjects are making the steady turbulence by introducing injection of the energy at
a large scale and broadening the inertial range of the energy spectrum. Acknowledgements We acknowledge W.F. Vineu for useful discussions. MT and MK both acknowledge support by a grant-in-aid for Scientific Research (Grant Nos 15340122 and 1505983, respectively) by the Japan Society for the Promotion of Science. References [1] R.J. Donnelly, Quantized Vortices in Helium II, Cambridge University Press, Cambridge, 1991. [2] R.P. Feymnan, in: C.J. Gorter (Eds), Progress in Low Temperature Physics, vol. I, North Holland Publishing Company, 1955, p.17. [3] W.F. Vinen, Proc. R. Soc. A240 (1957) 114; W.F. Vinen, Proc. R. Soc. 240 (1957) 128; W.F. Vinen, Proc. R. Soc. 240 (1957) 493. [4] J. Maurer, R. Tabeling, Europhys. Lett. 43 (1998) 29. [5] S.R. Stalp, L. Skrbek, R.J. Donnelly, Phys. Rev. Lett. 82 (1999) 4831. [6] W.F. Vinen, J.J. Niemela, J. Low Temp. Phys. 128 (2002) 167. [7] K.W. Schwarz, Phys. Rev. B31 (1985) 5782; K.W. Schwarz, Phys. Rev. 38 (1988) 2398. [8] E.P. Gross, J. Math. Phys. 4 (1963) 195. [9] L.P. Pitaevskii, Soviet Phys. JETP 13 (1963) 195. [10] T. Araki, M. Tsubota, S.K. Nemirovskii, Phys. Rev. Lett. 89 (2002) 145301. [11] C. Nore, M. Abid, M.E. Brachet, Phys. Fluids 9 (1997) 2644. [12] M. Leadbeater, T. Winiecki, D.C. Samuels, C.F. Barenghi, C.S. Adams, Phys. Rev. Lett. 86 (2001) 1410. [13] S. Ogawa, M. Tsubota, Y. Hattori, J. Phys. Soc. Jpn 71 (2002) 813. [14] M. Kobayashi, M. Tsubota, Phys. Rev. Lett. 94 (2005) 065302.