The Study of Turbulent The Study of Turbulent State in Quantum Fluid State in Quantum Fluid Department of Physics, The University of Tokyo Department of Physics, The University of Tokyo Michikazu Kobayashi Michikazu Kobayashi International Workshop on Bio-soft Matter June 9 th , 2008
Contents Contents • I study the dynamics and statistics of turbulence in q uantum fluid such as superfluid 4 He in which all rota tional fluid is carried by quantized vortices. • By numerically solving the nonlinear Schrödinger eq uation, I obtain the dynamics of quantized vortices i n turbulence as tangled state and investigate the st atistics like energy spectrum, fractal structure, etc .
Quantized Vortices and Quantized Vortices and Quantum Turbulence Quantum Turbulence Ordinary fluid (air, water) Classical fluid (He-II) There is viscosity There is no viscosity Circulations of vortices are From large to small vortices quantized and turbulence is exist in turbulence realized as vortex tangle
Quantum Turbulence As an Quantum Turbulence As an Ideal System of Turbulence Ideal System of Turbulence Vortices in ordinary turbulence (Navi Vortices in quantum fluid turbulence er-Stokes simulation by S. Kida) All circulations are quantized and All circulations are quantized and All circulations around vortices All circulations around vortices vortices are definite vortices are definite have arbitrary value and vortices have arbitrary value and vortices Vortex skeletons in turbulence! → Vortex skeletons in turbulence! are indefinite → are indefinite
Experiment of Superfluid Experiment of Superfluid He-II 4 He-II 4 J. Maurer and P. Tabeling, Europh G. P. Bewley, et al, Natu ys. Lett. 43 (1), 29 (1998) re 441 , 588(2006). Visualization of Two-counter rotating disks quantized vortices in turbulence Quantum turbulence obeys the ordinary Kolmogorov law
Proposed Statistics of Vortices Proposed Statistics of Vortices and the Energy Spectrum and the Energy Spectrum 3 regions : classical, quantum, and dissipative with elementary excitations 3 regions : classical, quantum, and dissipative with elementary excitations ? Structure of quantized vortices and the energy Structure of quantized vortices and the energy spectrum are closely related with each other. spectrum are closely related with each other.
Model of Quantum Fluid Model of Quantum Fluid and Turbulence and Turbulence Nonlinear Schrödinger equation Vortex filament model Nonlinear Schrödinger equation Vortex filament model Quantized vortex Quantized vortex
Numerical Simulation of Nonline Numerical Simulation of Nonline ar Schrödinger Equation ar Schrödinger Equation Details of Simulation Details of Simulation
Turbulent State in the Turbulent State in the Simulation Simulation Periodic box with 256 3 grids (stereogram)
Energy Spectrum of Energy Spectrum of Turbulence Turbulence Energy spectrum 10 5 k < 2 / l : Quantum fluid 10 4 turbulence shows the Kolmogorov law : there is a similarity between 10 3 quantum and ordinary fluid. E kini ( k ) E kini ( k ) 10 2 ∝ k -5/3 ∝ k -17/5 k > 2 / l : There is Kelvin-wave 10 1 turbulence characteristic in 10 0 quantum fluid. 10 -1 2 π / l 2 π / ξ 0.1 1 10 k
Huge Scale Simulations Huge Scale Simulations In Japan Atomic Energy Agency Some bottleneck effect between Richardson Hump (Kolmogorov) structure and Kelvin-wave cascade?
Connection between Richardson Connection between Richardson and Kelvin-wave Cascade and Kelvin-wave Cascade Two analytical proposals V. S. L’vov et. al, PRB 76, 024520 (2007). E. Kozik and B. Svistunov, cond-mat/0703047 Bottleneck region as statistical Complicated vortex bundle structure. equipartition. One of the big mystery in quantum fluid turbulence
Summary & Outlook of Summary & Outlook of Quantum Fluid Turbulence Quantum Fluid Turbulence Quantum fluid turbulence consists of quantized vortices and shows the Kolmogorov law Quantum fluid turbulence can become a ideal prototype to study turbulence from the view of elementary structure of vortices and the relation between dynamics of vortices and statistics like the Kolmogorov law.
Future Subject Future Subject • Details in the region of Kelvin-wave turbulence. • Calculation of statistical and dynamical properties of vortices in real space, such as size-distribution of vortex loops, fractal dimension of vortex lines, vortex linking number etc. • Investigation of relation between statistics and dynamics in real space and wave-number space. MK and M. Tsubota, PRL 94 , 065302 (2005). MK and M. Tsubota, JPSJ 74 , 3248 (2005). M K and M. Tsubota, PRL 97 , 145301(2006). MK and M. Tsubota, PRA 76 045603 (2007).
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