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Simulation of the “negative temperature” instability for line vortices
Volume 39A, number 5
PHYSICS LETTERS
SIMULATION OF THE "NEGATIVE
5 June 1972
TEMPERATURE"
INSTABILITY FOR LINE VORTICES G. JOYCE* and D. MONTGOMERY** Department o f Physics and Astronomy, The University o f lowa, Iowa City, lowa 52240, USA Received 17 April 1972 Numerical simulation of the Onsager "negative temperature" instability for a large number of discrete line vortices is reported.
We have numerically simulated what we believe to be the "negative temperature" instability [ 1,2] for the two-dimensional motions of interacting line vortices in ideal hydrodynamics. A not dissimilar phenomenon appeared earlier [3] in a numerical solution to the c o n t i n u u m Navier-Stokes equations. Due to the uncertain partition between "self" and "interaction" energies in the continuum limit, it has seemed desirable to repeat the experiment for a discrete vortex model [2]. The theoretical advantages of working with a Euclidean phase space and simple Hamiltonian mechanics are considerable, since the best-established results of statistical mechanics deal with such systems. The dimensionless equations by which the locations of N vortices of strength K (and N more of strength - K ) are advanced in time are dxi/dt = --3~/3y and dYi/dt = +3c~/3x, for the ith vortex. The potential ~ obeys the two-dimensional Poisson equation, ~72~b= -2rrD [n_ - n+]. The exact number densities of the vortices are n+ = EN16(x - x i ) 6 ( y - Yi), where the summations run over the positive and negative vortices, respectively. The constant 2nD is T K / L 2, where the unit o f length L is ~ of the linear dimension of the perfectly-reflecting box in which the vortices are confined. The unit of time T is the time required for two unit vortices of like sign a unit length apart to circulate once around each other. 2nD is taken to be 0.5, fixing K. Perfect reflection is achieved by replacing the effect of the walls by an infinite doubly-periodic array of image vortices with a *Supported in part by NSF Grant GA-31676 and USAEC Grant AT(I 1-1)-2059. **Supported in part by NASA Grant NGL-16-001-043.
period twice the box size. Poisson's equation is solved by fast Fourier transform methods. The x i and Yi are advanced in time by a second-order predictor-corrector algorithm. The vortex density assignments for solution of Poisson's equation are accomplished by particle-in-cell (P.1.C.) techniques, employing area weighting. The field E-= - V ¢ is Fourier-analyzed as E = Z k E k exp (ik "x), with a basic wavelength twice the box size. The total energy of the system is W = ½~klEkl 2. W is the sum of the " s e l f ' and the "interaction" energies and is a constant of the motion. The "self" energy part is a constant depending only upon the total number of vortices. The "interaction" energy can be varied by varying W, which in turn is determined by the initial values of the x i a n d y i. These cannot be loaded randomly, for then PCwould quickly converge to the total self energy. Instead squares containing randomly chosen numbers of positive or negative vortices are loaded inside the basic box in a regular array. The vortex locations inside the squares are then chosen randomly. The procedure does not guarantee exact equality of the number of positive and negative vortices. Therefore a compensating number of vortices of the deficient sign is loaded randomly into the basic box to guarantee overall equality of the two signs. (This accounts for the few excess negatives in fig. 1, which do not lie inside the squares.) By decreasing or increasing the size of the squares, vortices of like sign are brought into more or less intimate contact with each other, thereby raising or lowering W. Typical results are shown in figs. 1 and 2, which are thought to lie above and below the energy thresh371
Volume 39A, number 5
NEGATIVE
POSITIVE
,%:
• ,.'.
u
•
5 June 1972
PHYSICS LETTERS
NEGATIVE
POSITIVE
'$ ;:;'::;!:
:
....
t=O
|=0
.
,
••i
•
t
i;ii/i'/i i i i .
=6o Fig. 1. Spatial locations of the vortices (N = 525, negatives and positives shown separately) at t = 0 and t = 60. The circular sectors contain half the vortices of the given sign. W = 2.08, of which 0.054 is self-energy. old for the negative temperature instability, respectively. Fig. 1 has lg = 2.08 and fig. 2 has W = 0.14. The figures show the spatial locations of the vortices within the boxes at t = 0 and at a later time. For ease in visualization the positive and negative vortices are shown in separate boxes, though in reality they coexist in the same spatial domain. Various numerical checks on the accuracy of the program (which has some things in c o m m o n with the VORTEX code of Christiansen [4] ) have been made, and energy is conserved in both runs to better than 5 percent. The initial pattern breaks up quickly (before t = 20). Case 2 remained disordered throughout the run. Case 1 showed a clear tendency, as predicted [ 1 , 2 ] , for the positives and negatives to collect in separate regions of the box as seen in fig. 1. The macroscopic
371
<
i
t =80 Fig. 2. Same as fig. 1, except W= 0.14, for t = 0 and 80. No separation appears in this case. flow pattern is again [3] two large counter-rotating vortices. This pattern persists, with the locations of the two vortices remaining visually discernible throughout the duration of the run. Though these vortices preserve their identities throughout a run, they are not stationary, but appear to chase each other slowly around the box. There is also observed a corresponding shift of W to the smaller values of k. A later and more detailed account o f these results is planned.
References [1] L. Onsager, Nuovo Cimento Suppl., Ser. 9, 6 (1949) 279. [2] D. Montgomery, Physics Letters 39A (1972) 7. [3] G.S. Deem and N.J. Zabusky, Phys. Rev. Letters 27 (197l) 396. [4] J.P. Crhistiansen, Culham Laboratory Report CLM-P282, submitted to J. Comp. Phys.
PHYSICS LETTERS
SIMULATION OF THE "NEGATIVE
5 June 1972
TEMPERATURE"
INSTABILITY FOR LINE VORTICES G. JOYCE* and D. MONTGOMERY** Department o f Physics and Astronomy, The University o f lowa, Iowa City, lowa 52240, USA Received 17 April 1972 Numerical simulation of the Onsager "negative temperature" instability for a large number of discrete line vortices is reported.
We have numerically simulated what we believe to be the "negative temperature" instability [ 1,2] for the two-dimensional motions of interacting line vortices in ideal hydrodynamics. A not dissimilar phenomenon appeared earlier [3] in a numerical solution to the c o n t i n u u m Navier-Stokes equations. Due to the uncertain partition between "self" and "interaction" energies in the continuum limit, it has seemed desirable to repeat the experiment for a discrete vortex model [2]. The theoretical advantages of working with a Euclidean phase space and simple Hamiltonian mechanics are considerable, since the best-established results of statistical mechanics deal with such systems. The dimensionless equations by which the locations of N vortices of strength K (and N more of strength - K ) are advanced in time are dxi/dt = --3~/3y and dYi/dt = +3c~/3x, for the ith vortex. The potential ~ obeys the two-dimensional Poisson equation, ~72~b= -2rrD [n_ - n+]. The exact number densities of the vortices are n+ = EN16(x - x i ) 6 ( y - Yi), where the summations run over the positive and negative vortices, respectively. The constant 2nD is T K / L 2, where the unit o f length L is ~ of the linear dimension of the perfectly-reflecting box in which the vortices are confined. The unit of time T is the time required for two unit vortices of like sign a unit length apart to circulate once around each other. 2nD is taken to be 0.5, fixing K. Perfect reflection is achieved by replacing the effect of the walls by an infinite doubly-periodic array of image vortices with a *Supported in part by NSF Grant GA-31676 and USAEC Grant AT(I 1-1)-2059. **Supported in part by NASA Grant NGL-16-001-043.
period twice the box size. Poisson's equation is solved by fast Fourier transform methods. The x i and Yi are advanced in time by a second-order predictor-corrector algorithm. The vortex density assignments for solution of Poisson's equation are accomplished by particle-in-cell (P.1.C.) techniques, employing area weighting. The field E-= - V ¢ is Fourier-analyzed as E = Z k E k exp (ik "x), with a basic wavelength twice the box size. The total energy of the system is W = ½~klEkl 2. W is the sum of the " s e l f ' and the "interaction" energies and is a constant of the motion. The "self" energy part is a constant depending only upon the total number of vortices. The "interaction" energy can be varied by varying W, which in turn is determined by the initial values of the x i a n d y i. These cannot be loaded randomly, for then PCwould quickly converge to the total self energy. Instead squares containing randomly chosen numbers of positive or negative vortices are loaded inside the basic box in a regular array. The vortex locations inside the squares are then chosen randomly. The procedure does not guarantee exact equality of the number of positive and negative vortices. Therefore a compensating number of vortices of the deficient sign is loaded randomly into the basic box to guarantee overall equality of the two signs. (This accounts for the few excess negatives in fig. 1, which do not lie inside the squares.) By decreasing or increasing the size of the squares, vortices of like sign are brought into more or less intimate contact with each other, thereby raising or lowering W. Typical results are shown in figs. 1 and 2, which are thought to lie above and below the energy thresh371
Volume 39A, number 5
NEGATIVE
POSITIVE
,%:
• ,.'.
u
•
5 June 1972
PHYSICS LETTERS
NEGATIVE
POSITIVE
'$ ;:;'::;!:
:
....
t=O
|=0
.
,
••i
•
t
i;ii/i'/i i i i .
=6o Fig. 1. Spatial locations of the vortices (N = 525, negatives and positives shown separately) at t = 0 and t = 60. The circular sectors contain half the vortices of the given sign. W = 2.08, of which 0.054 is self-energy. old for the negative temperature instability, respectively. Fig. 1 has lg = 2.08 and fig. 2 has W = 0.14. The figures show the spatial locations of the vortices within the boxes at t = 0 and at a later time. For ease in visualization the positive and negative vortices are shown in separate boxes, though in reality they coexist in the same spatial domain. Various numerical checks on the accuracy of the program (which has some things in c o m m o n with the VORTEX code of Christiansen [4] ) have been made, and energy is conserved in both runs to better than 5 percent. The initial pattern breaks up quickly (before t = 20). Case 2 remained disordered throughout the run. Case 1 showed a clear tendency, as predicted [ 1 , 2 ] , for the positives and negatives to collect in separate regions of the box as seen in fig. 1. The macroscopic
371
<
i
t =80 Fig. 2. Same as fig. 1, except W= 0.14, for t = 0 and 80. No separation appears in this case. flow pattern is again [3] two large counter-rotating vortices. This pattern persists, with the locations of the two vortices remaining visually discernible throughout the duration of the run. Though these vortices preserve their identities throughout a run, they are not stationary, but appear to chase each other slowly around the box. There is also observed a corresponding shift of W to the smaller values of k. A later and more detailed account o f these results is planned.
References [1] L. Onsager, Nuovo Cimento Suppl., Ser. 9, 6 (1949) 279. [2] D. Montgomery, Physics Letters 39A (1972) 7. [3] G.S. Deem and N.J. Zabusky, Phys. Rev. Letters 27 (197l) 396. [4] J.P. Crhistiansen, Culham Laboratory Report CLM-P282, submitted to J. Comp. Phys.