×

Vortex equilibria in turbulence theory and quantum analogues. (English) Zbl 0736.76025

Summary: A family of statistical vortex equilibria and related quasi-equilibria in models of classical incompressible flow is analyzed; it is suggested that these equilibria constitute approximate models of the inertial scales in turbulence. Inertial exponents \(\gamma\) are calculated; the equilibrium that has maximum entropy has a spectrum close to the Kolmogorov spectrum. At maximum entropy, the vortex filaments have axes that are self-avoiding random walks. The velocity statistics are non-Gaussian even at equilibrium. The classical vortex system is contrasted with a quantum vortex system; it is shown that classical vortex folding is a manifestation of the Kosterlitz-Thouless transition mechanism. Universality properties of the inertial exponent are discussed, as is the effect of a strong cascade on vortex reconnection. A relation with some recent work on the \(\lambda\) transition in superfluidity is pointed out.

MSC:

76F99 Turbulence
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Batchelor, G., The Theory of Homogeneous Turbulence (1960), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0098.41001
[2] Bell, J.; Marcus, D., The evolution of a perturbed vortex tube, (report UCRL-JC-105029 (1990), Lawrence Livermore Laboratory)
[3] Chorin, A. J., The evolution of a turbulent vortex, Commun. Math. Phys., 83, 517-535 (1982) · Zbl 0494.76024
[4] Chorin, A. J., Turbulence and vortex stretching on a lattice, Commun. Pure Appl. Math., 39, S47-S65 (1986) · Zbl 0595.60098
[5] Chorin, A. J., Scaling laws in the lattice vortex model of turbulence, Commun. Math. Phys., 114, 167-176 (1988) · Zbl 0638.76072
[6] Chorin, A. J., Spectrum, dimension and polymer analogies in fluid turbulence, Phys. Rev. Lett., 60, 1947-1949 (1988)
[7] Chorin, A. J., Constrained random walks and vortex filaments in turbulence theory, Commun. Math. Phys., 132, 519-536 (1990) · Zbl 0708.76077
[8] Chorin, A. J., Hairpin removal in vortex interactions, J. Comp. Phys., 91, 1-21 (1990) · Zbl 0711.76047
[9] Chorin, A. J., Vortices, turbulence and statistical mechanics, (Gustafson, K.; Sethian, J., Vortex Methods (1991), SIAM: SIAM Philadelphia) · Zbl 0931.76031
[10] Chorin, A. J., Physical and numerical aspects of the transition to turbulence on a vortex ring, (report PAM-515 (1990), Mathematics Department, University of California: Mathematics Department, University of California Berkeley) · Zbl 0184.20103
[11] A.J. Chorin, Statistical mechanics and vortex motion, in: the Proceedings of the 1990 AMS-SIAM Summer School, eds. C. Anderson and C. Greengard, to appear.; A.J. Chorin, Statistical mechanics and vortex motion, in: the Proceedings of the 1990 AMS-SIAM Summer School, eds. C. Anderson and C. Greengard, to appear.
[12] Feynman, R.; Cohen, M., The character of the roton state in liquid helium, Prog. Theor. Phys., 14, 261-263 (1955)
[13] Freed, K., Renormalization Group for Macromolecules (1987), Wiley: Wiley New York
[14] Frostman, O., Potential d’équilibre et théorie des ensembles, thesis (1935), Lund · JFM 61.1262.02
[15] de Gennes, P. G., Scaling Concepts in Polymer Physics (1971), Cornell Univ. Press: Cornell Univ. Press Ithaca
[16] Kohring, G.; Schrock, R., Properties of generalized 3D O(2) model with suppression/enhancement of vortex strings, Nucl. Phys. B, 288, 397-418 (1987)
[17] Kosterlitz, J. M., The critical properties of the two-dimensional XY model, J. Phys. C, 7, 1046-1060 (1974)
[18] Kosterlitz, J. M.; Thouless, D. J., Order, metastability and phase transitions in two-dimensional systems, J. Phys. C, 6, 1181-1203 (1973)
[19] Lamb, H., Hydrodynamics (1932), Dover: Dover New York · JFM 26.0868.02
[20] Landau, L. D.; Lifshitz, E. M., (Statistical Physics, Vol. I (1980), Pergamon Press: Pergamon Press Oxford), 221
[21] Madras, N.; Sokal, A., The pivot algorithm: a highly efficient Monte-Carlo method for self-avoiding walks, J. Stat. Phys., 50, 109-186 (1988) · Zbl 1084.82503
[22] Miller, J., (Ph.D. thesis (1990), Physics Department, California Institute of Technology)
[23] Moffatt, H. K., On the degree of knottedness of tangled vortex lines, J. Fluid Mech., 35, 117-132 (1969) · Zbl 0159.57903
[24] Onsager, L., Statistical hydrodynamics, Nuovo Cimento, 6, 279-287 (1949), (Suppl.)
[25] Shenoy, S. R., Vortex loop scaling in three-dimensional XY ferromagnet, Phys. Rev. B, 40, 5056-5068 (1989)
[26] Wiegel, F. W., Introduction to Path Integral Methods in Physics and Polymer Science (1986), World Scientific: World Scientific Singapore · Zbl 0583.57002
[27] Williams, G., Vortex ring model of the superfield λ transition, Phys. Rev. Lett., 59, 1926-1929 (1987)
[28] G. Williams, Vortices and finite size scaling of superfluid phase transitions, Physica B.; G. Williams, Vortices and finite size scaling of superfluid phase transitions, Physica B.
[29] Williams, G., Vortices and the superfluid \(^4He\) transition in two and three dimensions, (Wyatt, A. F.G., Proceedings Exeter Conference on Excitations in Quantum Fluids (1990), NATO publication)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.