- Email: [email protected]
Vorticity gradient in 2D turbulence of ideal fluid
28 February 2000
Physics Letters A 266 Ž2000. 377–379 www.elsevier.nlrlocaterphysleta
Vorticity gradient in 2D turbulence of ideal fluid E.A. Novikov ) Institute for Nonlinear Science, UniÕersity of California, San Diego, La Jolla, CA 92093-0402, USA Received 16 October 1999; received in revised form 4 January 2000; accepted 12 January 2000 Communicated by C.R. Doering
Abstract The statistical description of vorticity gradient in 2D turbulence of ideal fluid is considered. Using the connection between Lagrangian and Eulerian ensemble of averages, inequalities are found that impose certain conditions on the statistical measure for singularities and zeros of vorticity gradient in homogeneous turbulence. An infinitely divisible distribution for the logarithm of vorticity gradient is suggested. q 2000 Published by Elsevier Science B.V. All rights reserved.
The dynamics and statistics of turbulent flows is better understood in terms of characteristics that are local in physical space and have a mechanism of nonlinear amplification w1–7x. For three-dimensional Ž3D. turbulence the primary local characteristic is the vorticity field and amplification is due to the effect of vortex stretching. For 2D turbulence corresponding local characteristic is the vorticity gradient Ž Õg . and amplification is due to the compression of fluid element in the direction of Õg. In the ideal fluid, the nonlinear amplification can potentially lead to the finite-time singularities of local characteristics Žin 3D such singularities are not ruled out even for viscous fluid w8x.. It seems important to describe turbulent singularities statistically and start from a more simple case of 2D turbulence. The existence and uniqueness of classical solutions for 2D ideal incompressible fluid was proved in the case of bounded domain w9–12x. Equilibrium distributions where constructed for bounded 2D flow, )
based on a maximum entropy principle and some approximations w13–15x. From dynamical theorems w9–12x we can conclude that if the initial statistical measure is supported on smooth functions, then the measure will retain this property in time. However, we are not aware of any such dynamical or statistical results for unbounded statistically homogeneous turbulent flow. The vorticity gradient in homogeneous turbulence will be the focus of this Letter. Consider the equation for the 2D vorticity field v s E Õ 2rE x 1 y E Õ 1rE x 2 in ideal incompressible fluid: dv dt
E
d
'
dt
Et
E q Õk
E xk
,
E Õk E xk
s 0.
Ž 1.
Here d dt is the Lagrangian time derivative, Õ k are the velocity components and we have summation over the repeated indexes from 1 to 2. The spatial differentiation of Ž1. gives equation for Õg si ' EvrE x i : dsi
Fax: q1-619-534-7664. E-mail address: [email protected] ŽE.A. Novikov..
s 0,
dt
sy
E Õk E xi
sk .
0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 0 6 0 - 8
Ž 2.
E.A. NoÕikoÕr Physics Letters A 266 (2000) 377–379
378
The right-hand side of Ž2. represents the effect of quadratic amplification, because Õi is a linear functional of Õg w2x. The balance of s s Ž si2 .1r2 is obtained by multiplying Ž2. by si : ds dt
s yb s,
b'
E Õ k sk si E xi s2
.
Ž 3.
Finally, substitution of Ž8. into Ž5. gives: 1r2
y² v 02 :
1r2
t F ²ln s Ž t . :L F ² v 02 :
Ž 9.
We assume that ² v 02 : is finite. Suppose, that s Ž t . becomes infinite at a finite time t s t ) . Let m t )Ž s . be a statistical measure corresponding to 0 F s Ž t ) . F s . A measure for infinite s Ž t ) . is defined by `
Here b is the eigenvalue of the deformation rates tensor in the direction of si . Solution of Ž3. can be written in the form:
t.
H dm s™ ` s
q` s lim
Ž s X . G 0.
t)
Ž 10 .
At t s t ) we have: `
t
ln s Ž t . s y
H0 b Ž t . dt ,
s Ž t . ' s Ž t . rs0 ,
Ž 4.
where the integral is taken along the trajectory of the fluid particle. The statistical measure for this solution is determined by the initial vorticity field v 0 . By averaging Ž4. and using inequality ² b :2L F ² b 2 :L , we get: t
H0 ² b
y
2
1r2 L dtF
Žt . :
²ln s Ž t . :L
t
t
H0 ² b Ž t . : dtFH0 ² b
sy
L
2
1r2 L dt .
Žt . :
Ž 5.
Here subscript L indicates the Lagrangian ensemble of averaging. From definition Ž3. we have:
E Õi
2
¦ž / ;
2
² b :L F
E xk
.
Ž 6.
L
For the statistically homogeneous turbulence of incompressible fluid the one-point Lagrangian and Eulerian probability distributions coincide w16x. From this fact and incompressibility we get:
E Õi
2
E Õi
2
¦ž / ; ¦ž / ; s
E xk
L
E xk
s ² v 2 :s ² v 2 :L .
Ž 7.
Here ²: Žwithout subscript. corresponds to the Eulerian ensemble of averaging and we used standard manipulating:
E Õ1
2
E Õ1 E Õ 2
E Õ1 E Õ 2
¦ž / ; ¦ ; ¦ ; E x1
sy
E x1 E x 2
sy
E x 2 E x1
.
From Ž6., Ž7. and Ž1. we get:
² b 2 :L F ² v 02 :.
Ž 8.
²ln s Ž t ) . :L s H
ln Ž s . dm t ) Ž s .
0
s J0 q J` G J0 q ln Ž s 1 . q` ,
Ž 11 .
where 1
J0 '
H0 ln Ž s . dm
J` '
H1 ln Ž s . dm
t)
Ž s . F 0,
` t)
Ž s . G0
and s 1 can be chosen arbitrary large. We see that if J0 is finite and q` ) 0, then the left-hand side of Ž11. is infinite for t s t ) . The finiteness of J0 is a condition on the asymptotic of m t )Ž s . when s 0 and it is ‘‘logarithmically stronger’’ than the condition of zero measure for s Ž t ) . s 0: q0 ' lim s ™ 0 H0sdm t )Ž s X . s 0. Inequality Ž9. shows that with finite J0 the statistical measure for infinite s Ž t ) . is zero: q` s 0. Similarly, from the left-hand side of inequality Ž9. it follows that if J` is finite than q0 s 0. We note that inequality Ž9., formally, may coexist with infinite ² s Ž t ) . :L , for example, if m t )Ž s . s u Ž s y 1.w1 y sya x, where u Ž x . is the unit step function Ž J0 s 0. and 0 - a F 1. However, realizability of such regime for a particular statistics of the initial vorticity field is an open question. On other hand, the integral representation Ž4. and derivation of the inequality Ž9. suggest that, with appropriate initial statistics, the measure may evolve to the log-normal type of distribution. For the lognormal distribution ² s Ž t . :L is finite if, in addition 2 to Ž9., ² ln s Ž t . :L is finite. Let us do an estimate. From Ž4. we get:
™
² ln s Ž t . 2 :L s HtdtHtdt X² b Ž t . b Ž t X . :L . 0
0
Ž 12 .
E.A. NoÕikoÕr Physics Letters A 266 (2000) 377–379
We will use inequality:
² b Ž t . b Ž t X . :L F ² b 2 Ž t . :L² b 2 Ž t X . :L
1r2
379
ing in future to find upper bounds for the Hausdorff dimensions for the space–time sets of possible singularities and zeros for the vorticity gradient.
.
Ž 13 . Eqs. Ž12., Ž13. and Ž8. give: `
² ln s Ž t . 2 :L s H
0
References
2 Ž ln s . dm t Ž s . F ² v 02 :t 2 .
Ž 14 . This inequality imposes additional limitations on measures for singularities and zeros of Õg w substitute Žln s . 2 instead of ln s into the conditions, obtained above from Ž9. and Ž11.x. We note that the log-normal asymptotic of distribution, generally, does not guarantee that moments Žincluding ² s Ž t . :L . will tend to the moments of the asymptotic distribution w17–20x. It will be interesting to perform careful numerical experiments, characterized by two major nondimensional parameters: t ' y1r2 1r2 1r2 L, where L is a and g ' ² s02 : ² v 02 : t² v 02 : size of the domain Žwhich can be chosen periodic, for simplicity.. By fixing sufficiently large t and increasing g , we can approach regime of homogeneous turbulence. Such experiments can, particularly, verify if log-normal or more general log-infinitely divisible distribution w17–21x is realizable in 2D turbulence of ideal fluid. It will be also interest-
w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x
E.A. Novikov, Fluid Dyn. Res. 6 Ž1990. 79. E.A. Novikov, Fluid. Dyn. Res. 12 Ž1993. 107. E.A. Novikov, Phys. Rev. Lett. 71 Ž1993. 2718. E.A. Novikov, Phys. Rev. E 49 Ž1994. 975. E.A. Novikov, Phys. Rev. E 52 Ž1995. 2574. E.A. Novikov, Phys. Lett. A 236 Ž1997. 65. E.A. Novikov, Phys. Rev. Lett. 82 Ž1999. 3992. L. Cafarelly, R. Kohn, L. Nirenberg, Commun. Pure Appl. Math. 35 Ž1982. 771. W. Wolibner, Math. Z 37 Ž1933. 698. V. Judovich, Z. Vych. Mat. 3 Ž1963. 1032. T. Kato, Arch. Rat. Mech. 27 Ž1968. 188. D.G. Ebin, J. Marsden, Ann. Math. 92 Ž1970. 102. J. Miller, Phys. Rev. Lett. 65 Ž1990. 2137. R. Robert, J. Statist. Phys. 65 Ž1991. 531. B. Turkington, Comm. Pure and Applied Math. 7 Ž1999. 781. E.A. Novikov, Appl. Math. Mech. 33 Ž1969. 862. E.A. Novikov, Prikl. Mat. Meckh. 35 Ž1971. 266. E.A. Novikov, Appl. Math. Mech. 35 Ž1971. 231. E.A. Novikov, Phys. Fluids. A 2 Ž1990. 819; Phys. Rev. E 50 Ž1994. R 3303. E.A. Novikov, D.G. Dommermuth, Phys. Rev. E 56 Ž1997. 5479. W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1991, vol. 2.
Physics Letters A 266 Ž2000. 377–379 www.elsevier.nlrlocaterphysleta
Vorticity gradient in 2D turbulence of ideal fluid E.A. Novikov ) Institute for Nonlinear Science, UniÕersity of California, San Diego, La Jolla, CA 92093-0402, USA Received 16 October 1999; received in revised form 4 January 2000; accepted 12 January 2000 Communicated by C.R. Doering
Abstract The statistical description of vorticity gradient in 2D turbulence of ideal fluid is considered. Using the connection between Lagrangian and Eulerian ensemble of averages, inequalities are found that impose certain conditions on the statistical measure for singularities and zeros of vorticity gradient in homogeneous turbulence. An infinitely divisible distribution for the logarithm of vorticity gradient is suggested. q 2000 Published by Elsevier Science B.V. All rights reserved.
The dynamics and statistics of turbulent flows is better understood in terms of characteristics that are local in physical space and have a mechanism of nonlinear amplification w1–7x. For three-dimensional Ž3D. turbulence the primary local characteristic is the vorticity field and amplification is due to the effect of vortex stretching. For 2D turbulence corresponding local characteristic is the vorticity gradient Ž Õg . and amplification is due to the compression of fluid element in the direction of Õg. In the ideal fluid, the nonlinear amplification can potentially lead to the finite-time singularities of local characteristics Žin 3D such singularities are not ruled out even for viscous fluid w8x.. It seems important to describe turbulent singularities statistically and start from a more simple case of 2D turbulence. The existence and uniqueness of classical solutions for 2D ideal incompressible fluid was proved in the case of bounded domain w9–12x. Equilibrium distributions where constructed for bounded 2D flow, )
based on a maximum entropy principle and some approximations w13–15x. From dynamical theorems w9–12x we can conclude that if the initial statistical measure is supported on smooth functions, then the measure will retain this property in time. However, we are not aware of any such dynamical or statistical results for unbounded statistically homogeneous turbulent flow. The vorticity gradient in homogeneous turbulence will be the focus of this Letter. Consider the equation for the 2D vorticity field v s E Õ 2rE x 1 y E Õ 1rE x 2 in ideal incompressible fluid: dv dt
E
d
'
dt
Et
E q Õk
E xk
,
E Õk E xk
s 0.
Ž 1.
Here d dt is the Lagrangian time derivative, Õ k are the velocity components and we have summation over the repeated indexes from 1 to 2. The spatial differentiation of Ž1. gives equation for Õg si ' EvrE x i : dsi
Fax: q1-619-534-7664. E-mail address: [email protected] ŽE.A. Novikov..
s 0,
dt
sy
E Õk E xi
sk .
0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 0 6 0 - 8
Ž 2.
E.A. NoÕikoÕr Physics Letters A 266 (2000) 377–379
378
The right-hand side of Ž2. represents the effect of quadratic amplification, because Õi is a linear functional of Õg w2x. The balance of s s Ž si2 .1r2 is obtained by multiplying Ž2. by si : ds dt
s yb s,
b'
E Õ k sk si E xi s2
.
Ž 3.
Finally, substitution of Ž8. into Ž5. gives: 1r2
y² v 02 :
1r2
t F ²ln s Ž t . :L F ² v 02 :
Ž 9.
We assume that ² v 02 : is finite. Suppose, that s Ž t . becomes infinite at a finite time t s t ) . Let m t )Ž s . be a statistical measure corresponding to 0 F s Ž t ) . F s . A measure for infinite s Ž t ) . is defined by `
Here b is the eigenvalue of the deformation rates tensor in the direction of si . Solution of Ž3. can be written in the form:
t.
H dm s™ ` s
q` s lim
Ž s X . G 0.
t)
Ž 10 .
At t s t ) we have: `
t
ln s Ž t . s y
H0 b Ž t . dt ,
s Ž t . ' s Ž t . rs0 ,
Ž 4.
where the integral is taken along the trajectory of the fluid particle. The statistical measure for this solution is determined by the initial vorticity field v 0 . By averaging Ž4. and using inequality ² b :2L F ² b 2 :L , we get: t
H0 ² b
y
2
1r2 L dtF
Žt . :
²ln s Ž t . :L
t
t
H0 ² b Ž t . : dtFH0 ² b
sy
L
2
1r2 L dt .
Žt . :
Ž 5.
Here subscript L indicates the Lagrangian ensemble of averaging. From definition Ž3. we have:
E Õi
2
¦ž / ;
2
² b :L F
E xk
.
Ž 6.
L
For the statistically homogeneous turbulence of incompressible fluid the one-point Lagrangian and Eulerian probability distributions coincide w16x. From this fact and incompressibility we get:
E Õi
2
E Õi
2
¦ž / ; ¦ž / ; s
E xk
L
E xk
s ² v 2 :s ² v 2 :L .
Ž 7.
Here ²: Žwithout subscript. corresponds to the Eulerian ensemble of averaging and we used standard manipulating:
E Õ1
2
E Õ1 E Õ 2
E Õ1 E Õ 2
¦ž / ; ¦ ; ¦ ; E x1
sy
E x1 E x 2
sy
E x 2 E x1
.
From Ž6., Ž7. and Ž1. we get:
² b 2 :L F ² v 02 :.
Ž 8.
²ln s Ž t ) . :L s H
ln Ž s . dm t ) Ž s .
0
s J0 q J` G J0 q ln Ž s 1 . q` ,
Ž 11 .
where 1
J0 '
H0 ln Ž s . dm
J` '
H1 ln Ž s . dm
t)
Ž s . F 0,
` t)
Ž s . G0
and s 1 can be chosen arbitrary large. We see that if J0 is finite and q` ) 0, then the left-hand side of Ž11. is infinite for t s t ) . The finiteness of J0 is a condition on the asymptotic of m t )Ž s . when s 0 and it is ‘‘logarithmically stronger’’ than the condition of zero measure for s Ž t ) . s 0: q0 ' lim s ™ 0 H0sdm t )Ž s X . s 0. Inequality Ž9. shows that with finite J0 the statistical measure for infinite s Ž t ) . is zero: q` s 0. Similarly, from the left-hand side of inequality Ž9. it follows that if J` is finite than q0 s 0. We note that inequality Ž9., formally, may coexist with infinite ² s Ž t ) . :L , for example, if m t )Ž s . s u Ž s y 1.w1 y sya x, where u Ž x . is the unit step function Ž J0 s 0. and 0 - a F 1. However, realizability of such regime for a particular statistics of the initial vorticity field is an open question. On other hand, the integral representation Ž4. and derivation of the inequality Ž9. suggest that, with appropriate initial statistics, the measure may evolve to the log-normal type of distribution. For the lognormal distribution ² s Ž t . :L is finite if, in addition 2 to Ž9., ² ln s Ž t . :L is finite. Let us do an estimate. From Ž4. we get:
™
² ln s Ž t . 2 :L s HtdtHtdt X² b Ž t . b Ž t X . :L . 0
0
Ž 12 .
E.A. NoÕikoÕr Physics Letters A 266 (2000) 377–379
We will use inequality:
² b Ž t . b Ž t X . :L F ² b 2 Ž t . :L² b 2 Ž t X . :L
1r2
379
ing in future to find upper bounds for the Hausdorff dimensions for the space–time sets of possible singularities and zeros for the vorticity gradient.
.
Ž 13 . Eqs. Ž12., Ž13. and Ž8. give: `
² ln s Ž t . 2 :L s H
0
References
2 Ž ln s . dm t Ž s . F ² v 02 :t 2 .
Ž 14 . This inequality imposes additional limitations on measures for singularities and zeros of Õg w substitute Žln s . 2 instead of ln s into the conditions, obtained above from Ž9. and Ž11.x. We note that the log-normal asymptotic of distribution, generally, does not guarantee that moments Žincluding ² s Ž t . :L . will tend to the moments of the asymptotic distribution w17–20x. It will be interesting to perform careful numerical experiments, characterized by two major nondimensional parameters: t ' y1r2 1r2 1r2 L, where L is a and g ' ² s02 : ² v 02 : t² v 02 : size of the domain Žwhich can be chosen periodic, for simplicity.. By fixing sufficiently large t and increasing g , we can approach regime of homogeneous turbulence. Such experiments can, particularly, verify if log-normal or more general log-infinitely divisible distribution w17–21x is realizable in 2D turbulence of ideal fluid. It will be also interest-
w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x
E.A. Novikov, Fluid Dyn. Res. 6 Ž1990. 79. E.A. Novikov, Fluid. Dyn. Res. 12 Ž1993. 107. E.A. Novikov, Phys. Rev. Lett. 71 Ž1993. 2718. E.A. Novikov, Phys. Rev. E 49 Ž1994. 975. E.A. Novikov, Phys. Rev. E 52 Ž1995. 2574. E.A. Novikov, Phys. Lett. A 236 Ž1997. 65. E.A. Novikov, Phys. Rev. Lett. 82 Ž1999. 3992. L. Cafarelly, R. Kohn, L. Nirenberg, Commun. Pure Appl. Math. 35 Ž1982. 771. W. Wolibner, Math. Z 37 Ž1933. 698. V. Judovich, Z. Vych. Mat. 3 Ž1963. 1032. T. Kato, Arch. Rat. Mech. 27 Ž1968. 188. D.G. Ebin, J. Marsden, Ann. Math. 92 Ž1970. 102. J. Miller, Phys. Rev. Lett. 65 Ž1990. 2137. R. Robert, J. Statist. Phys. 65 Ž1991. 531. B. Turkington, Comm. Pure and Applied Math. 7 Ž1999. 781. E.A. Novikov, Appl. Math. Mech. 33 Ž1969. 862. E.A. Novikov, Prikl. Mat. Meckh. 35 Ž1971. 266. E.A. Novikov, Appl. Math. Mech. 35 Ž1971. 231. E.A. Novikov, Phys. Fluids. A 2 Ž1990. 819; Phys. Rev. E 50 Ž1994. R 3303. E.A. Novikov, D.G. Dommermuth, Phys. Rev. E 56 Ž1997. 5479. W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1991, vol. 2.