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Two-dimensional vortex motion and “negative temperatures”
Volume 39A, number 1
TWO-DIMENSIONAL
PHYSICS LETTERS
VORTEX MOTION AND "NEGATIVE
10 April 1972
TEMPERATURES"*
D. MONTGOMERY Department of Physics and Astronomy, Yhe University of Iowa, Iowa City, Iowa 52240, USA Received 23 February 1972 A recent numerical integration of the two-dimensional Navier-Stokes equations has tentatively identified an "ergodic boundary" in the space of initial conditions for the turbulent flow. An explanation is suggested in terms of negative temperatures, for a point vortex model.
Renewed interest in two-dimensional, incompressible, inviscid hydrodynamics has arisen lately, in part because of the identity of its mathematical description with that of a two-dimensional, electro-static guidingcenter plasma [ 1, 2]. A recent numerical integration [3] of the two-dimensional Navier-Stokes equations at high Reynolds number has suggested a novel dynamical phenomenon: an "ergodic boundary" in the space of initial conditions, on opposite sides of which the turbulent fluctuation spectrum (energy per unit wave number) approaches two rather different shapes for large times. Stated qualitatively, if the initial total energy is distributed over the longest wavelengths, the tendency of the turbulent flow is to evolve toward a pair of large counter-rotating vortices (or, in the plasma analogue, toward a pair of regions of net positive and negative electric charge). If the energy is distributed over shorter wavelengths, the spectrum does not appear to evolve in this manner, and the large pair of vortices may not be formed. An explanation of this behavior based on a theory [4] whose essential ingredient is a variant of the random phase approximation has been suggested by Cook and Taylor [5]. They argue for evolution toward a thermal equilibrium distribution (this had been previously calculated [2] by a less restrictive weak-coupling expansion of the equilibrium BBGKY hierarchy) which is characterized by a two-dimensional Debye-Hiickel pair distribution and by spatially-uniform one-body distributions for the vortices. * This work was supported by the National Aeronautics and Space administration under grant NGL-16-001-043.
The present note suggests a different way of viewing the phenomenon. Something similar to the "ergodic boundary" seems to have been predicted some time ago by Onsager [6], on general statistical mechanical grounds. We review his arguments. The Hamiltonian of N point vortices of strength -1N'x N k i in two dimensions is H = - ( 2 n ) /-Ji Lm, the temperature T will be negative. In our case, however, a macroscopic instability is to be expected, since the component parts of the system possess a finite interaction. We expect no macroscopic spatially-uniform thermal equilibrium state. In the limit of very large E 0, we observe that most of the volume in phase space which permits large values of H corresponds to configurations in which the
Volume 39A, number 1
PHYSICS LETTERS
vortices with ki ~>0 are crowded together on one side of the box, while those with ki
I0 April 1972
of vorticily are closer to start with. It is difficult not to speculate that Case 1 of Deem and Zabusky corresponds to tire "negative temperature" side of the Onsager boundary, as a function of interaction energy. Two non-trivial developments which would help to sharpen the conjecture would be: (1)semi-quantitative estimate (in terms of N, A and the values of the k2) of the value of Era; and (2) a numerical simulation of the Deem-Zabusky situation using point vortices instead of the Navier-Stokes equations. It is only to the point vortices that one can be certain that the Onsager prediction applies
The author acknowledges valuable discussions with Drs. F. Tappert and R. Hardin.
References [11 J.B. Taylor and B. McNamara, Phys. Fluids 14 (1971) 1492. 121 G. Vahala and D. Montgomery, J. Plasma Phys. 6 (1971) 425; D. Montgomery and I. Iappert, Phys. Rev, Lett. 27 (1971) 1419 and Phys. Fluids, to be published. 13] G.S. Deem and N.J. Zabusky, Phys. Rev, Lett. 27 (1971) 396. [4] J.B. Taylor and W.B. Thompson, Culham Lab. Report CLM-P286. [5] 1. Cook and J.B. Taylor, Phys. Rev. Lett. 28 (1972) 82. [61 L. Onsager, Nuovo Cimento Suppl. Series 9, 6 (1949) 279. [7] D. ter Haar, Elements of thermostatistics, 2nd. Ed. (Holt, Rinehart and Winston, New York, 1966) Ch.5. [8] L. Landau and L.D. Lifshitz, Statistical physics, 2nd Ed. (Addison-Wesley,Reading, Mass,, 1969) pp. 211-214. [9] A. Abragam and W.G. Proctor, Phys. Rev. 106 (1957) 160.
TWO-DIMENSIONAL
PHYSICS LETTERS
VORTEX MOTION AND "NEGATIVE
10 April 1972
TEMPERATURES"*
D. MONTGOMERY Department of Physics and Astronomy, Yhe University of Iowa, Iowa City, Iowa 52240, USA Received 23 February 1972 A recent numerical integration of the two-dimensional Navier-Stokes equations has tentatively identified an "ergodic boundary" in the space of initial conditions for the turbulent flow. An explanation is suggested in terms of negative temperatures, for a point vortex model.
Renewed interest in two-dimensional, incompressible, inviscid hydrodynamics has arisen lately, in part because of the identity of its mathematical description with that of a two-dimensional, electro-static guidingcenter plasma [ 1, 2]. A recent numerical integration [3] of the two-dimensional Navier-Stokes equations at high Reynolds number has suggested a novel dynamical phenomenon: an "ergodic boundary" in the space of initial conditions, on opposite sides of which the turbulent fluctuation spectrum (energy per unit wave number) approaches two rather different shapes for large times. Stated qualitatively, if the initial total energy is distributed over the longest wavelengths, the tendency of the turbulent flow is to evolve toward a pair of large counter-rotating vortices (or, in the plasma analogue, toward a pair of regions of net positive and negative electric charge). If the energy is distributed over shorter wavelengths, the spectrum does not appear to evolve in this manner, and the large pair of vortices may not be formed. An explanation of this behavior based on a theory [4] whose essential ingredient is a variant of the random phase approximation has been suggested by Cook and Taylor [5]. They argue for evolution toward a thermal equilibrium distribution (this had been previously calculated [2] by a less restrictive weak-coupling expansion of the equilibrium BBGKY hierarchy) which is characterized by a two-dimensional Debye-Hiickel pair distribution and by spatially-uniform one-body distributions for the vortices. * This work was supported by the National Aeronautics and Space administration under grant NGL-16-001-043.
The present note suggests a different way of viewing the phenomenon. Something similar to the "ergodic boundary" seems to have been predicted some time ago by Onsager [6], on general statistical mechanical grounds. We review his arguments. The Hamiltonian of N point vortices of strength -1N'x N k i in two dimensions is H = - ( 2 n ) /-Ji
Volume 39A, number 1
PHYSICS LETTERS
vortices with ki ~>0 are crowded together on one side of the box, while those with ki
I0 April 1972
of vorticily are closer to start with. It is difficult not to speculate that Case 1 of Deem and Zabusky corresponds to tire "negative temperature" side of the Onsager boundary, as a function of interaction energy. Two non-trivial developments which would help to sharpen the conjecture would be: (1)semi-quantitative estimate (in terms of N, A and the values of the k2) of the value of Era; and (2) a numerical simulation of the Deem-Zabusky situation using point vortices instead of the Navier-Stokes equations. It is only to the point vortices that one can be certain that the Onsager prediction applies
The author acknowledges valuable discussions with Drs. F. Tappert and R. Hardin.
References [11 J.B. Taylor and B. McNamara, Phys. Fluids 14 (1971) 1492. 121 G. Vahala and D. Montgomery, J. Plasma Phys. 6 (1971) 425; D. Montgomery and I. Iappert, Phys. Rev, Lett. 27 (1971) 1419 and Phys. Fluids, to be published. 13] G.S. Deem and N.J. Zabusky, Phys. Rev, Lett. 27 (1971) 396. [4] J.B. Taylor and W.B. Thompson, Culham Lab. Report CLM-P286. [5] 1. Cook and J.B. Taylor, Phys. Rev. Lett. 28 (1972) 82. [61 L. Onsager, Nuovo Cimento Suppl. Series 9, 6 (1949) 279. [7] D. ter Haar, Elements of thermostatistics, 2nd. Ed. (Holt, Rinehart and Winston, New York, 1966) Ch.5. [8] L. Landau and L.D. Lifshitz, Statistical physics, 2nd Ed. (Addison-Wesley,Reading, Mass,, 1969) pp. 211-214. [9] A. Abragam and W.G. Proctor, Phys. Rev. 106 (1957) 160.