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Chaotic vortex-body interaction
Volume 152, number 8
PHYSICS LETTERS A
4 February 1991
Chaotic vortex—body interaction E.A. Novikov Instil uteforNonlinear Science, 0402, University ofCalifornia, San Diego, La Jolla, CA 92093, USA
Received 30 July 1990; revised manuscript received 14 November 1990; accepted for publication 27 November 1990 Communicated by D.D. HoIm
It is shown, by using the Poincaré—Melnikov—Arnold method that the motion of a linear vortex in the flow past a cylindrical body is chaotic. In particular, a vortex can be captured by the body and then, after some complicated rotation near the body, can be lost. More general problems ofvortex—body interaction are discussed qualitatively. Possible applications of the theory to the problems of aviacatastrophicsand destruction ofbuildings by tornado are indicated.
The study of the chaotic interactions of linear vortices has a twelve year history with the participation of many authors (see refs. [1—16] and references therein). Generalization to the axisymmetric flows has been indicated in ref. [17]. The main conclusion from these studies is that two-dimensional and axisymmetrical flows of ideal incompressible fluid are generally non-integrable and with appropriate initial conditions exhibit chaos. This is in contrast with the opinion which prevailed earlier among the advocates of the solution approach to the hydrodynamics. The chaotic motion of vortices has various applications, in particular, to the problem of weather prediction [1,18] The main goal of this note is to indicate novel features of chaotic motion which arise in the presence of a moving body. Firstly, it is enough to have only one vortex in order to get chaotic motion. Secondly, the mechanism of generation of chaos is very transparent in the vortex—body system. Thirdly, we get new phenomena chaotic capture loss of vortex by a moving body. Vortex—body interactions are im—
—
portant in many situations. The most dramatic examples are the aviacatastrophics, caused by a vortex initiated by downdraft of cold air [19], and the destruction of buildings by tornado. We cannot expect in a visible future direct numerical simulations of these phenomena, based on Navier—Stokes equations, because the Reynolds numbers are extremely large. A natural way to deal with this kind of prob0375-9601/91/S 03.50 © 1991
—
lems is to develop a hierarchy of models with increasing complexity. We will start with an analytical description of the motion of a linear vortex in oscillating flow past a circular cylinder. Then we will make some qualitative remarks about more general problems. In the frame of reference moving with the cylinder, the velocity of a linear vortex in an ideal fluid is a Hamiltonian superposition of two parts of motion. The first part corresponds to the potential motion of fluid relative to the cylinder (see, for example, ref. [201), such as if the vortex has zero intensity. The second part of the motion is induced by the interaction of the vortex with the cylinder. In the case of a circular cylinder, this motion is induced by an image vortex, placed inside the cylinder at a distance from the center f=a2/r, where a is the radius ofthe cylinder and r corresponds to the position ofthe vortex. Both parts of motion have zero normal components of velocity at the surface of the cylinder. We will scale the distances by a and the time by a! u 0, where u0 is the characteristic fluid velocity at infinity. If the fluid velocity at infinity is constant, then the problem is characterized by only one nondimensional parameter a= k/2~tau0,where k is the vortex intensity. In the case of an oscillatingflow, which can model the turbulence of vibration ofthe cylinder, we have the relative fluid velocity at infinity u(1)=u0(l+ esinwt)
Elsevier Science Publishers B.V. (North-Holland)
,
(1) 393
Volume 152, number 8
PHYSICS LETTERS A
where e and w are the nondimensional amplitude and frequency of oscillation, In polar coordinates (p, 0) with origin in the center of the cylinder, the Hamiltonian system for the motion of the vortex has the form d I ~H / 1 \ = — = — sin 0 (~l —i)(1 + e sin WI), (2) ~ p dØ 1 äH = ~ —
—
=
—
~
0
(~ +
(1 + e sin WI)
p
H=H0 +eH1, p = na> 1, 2 1) + ~a ln (~2 H0 = p (p —
H
—
—
~
p
—
1
1)
(3)
(4)
2 1) sin wt. (5) p (p With e=0, the system (2)—(4) has a general analytical solution. The vortex trajectories (4) are presented in fig. 1. Without loss of generality, we as—
1
=
~
4 February 1991
The stationary trajectories ofvortex in the flow past a circular cylinder have been studied in ref. [211 without recognizing the Hamiltonian structure ofthe problem. The homoclinic trajectory has not been indicated ref. [211, probably because at that time the homoclinic trajectory was considered as something pathological and physically irrelevant to the problem. Now we know how important the homoclinic trajectories are for the generation of chaotic motion. For e small but non-zero, the system (2), (3) has no analytic integrals of motion. It possesses transversal stable and manifolds namely,intersecting the Poincaré maps, P(1unstable 0), which advance a —
solution by one period T= 2it/w starting at time I~, possess transversal homoclinic points. We will show this b~using the Poincaré—Melnikov—Arnold (PMA) method. This type ofbehavior of dynamical systems is called chaotic. According to the PMA method [22—24]we consider the Melnikov function I M(1
j
0)= sume that the motion of fluid around the vortex is clockwise (a< 0) and the direction of fluid velocity at infinity is from the top to the bottom on fig. 1. We see that there is a homoclinic vortex trajectory [Po(1), Ø~(I)I with the hyperbolic stationary point at 0=0, 3—p~).Thehomoclinictrajectoryseparates region where the vortex is captured by the .o=.o~’the (a=p; body.
/ —
2
-
—
—4
(
/
/7 //
/
//
-
\\
—
T
I 0
—
where
{
} denotes the Poisson brackets,
,
1 ‘ÔA {A, B}(p, 0
0.4 ~ —
)=
—
(
~°“
t9B (p, ~P0~t)
(p, 0~1)
p, I) OB(p, 0, 1)
_________________
The integral (6) is taken along the homoclinic trajectory [p 0(t—10), 0(1—I0)]. We will show that M(10) has simple zeros. From (4)—(6), by a change of variable, we get M(t0)= —a sin[00(I—10)J Sifl WI dt
=
—a
$
+cos wt
\\
5
Fig. I The vortex trajectories in the flow past a cylinder (p~= 3).
394
6
J
—
I-
{Ho~H1 }(po(tto), 00(I—t0); I) dt
(sin wt cos WI0 sin WI0) dt.
(7)
Since Ø~(i)-+O,Po(tY~P*exponentially near the hyperbolic stationary point (when 1—~±~), the integral (7) is convergent. It is convenient to choose the initial position Oo (0) = it. In this case we have
Volume 152, number 8
Oo(—t)
PHYSICS LETTERS A
0o(t), po(
M(10)=—acoswt0
1)po(t) and (7) reduces to
j sinOa(t) [00(1)] sin wi dl.
(8)
The integral in (8) is not identically zero, because it is the Fourier transform ofa function which is not identically zero. The function M( t~)clearly has simpie zeros. According to PMA theory, this proves that the system (2), (3) has no analytic integrals of motion and the vortex trajectory is chaotic. In particular, the vortex which is initially far from the body, can intersect the homoclinic loop and will be captured by the body. After several complicated revolutions near the body, the vortex will eventually escape In connection with the capture-loss phenomena, we have the following theorem. Let S be the set of positions of the vortex at time t~,for which the vortex for all t~i~will stay inside a circie C surrounding the body. It can be proven that the subset of S, for which the vortex was outside of C for some 1<10, has zero measure. The proof is the same as in Littlewood’s theorem [25 J for conservative (gravitational) systems. The only condition which matters is the preservation of volume in phase space. The nonautonomous Hamiltonian system clearly satislies this condition. The theorem remains true if instead of a circle C surrounding the body we choose any area in phase space. In the case of an arbitrary shape of the cylindrical body we still have Hamiltonian superposition of external and induced motion of the vortex in terms of corresponding Green functions for the Laplace operator. The external velocity is finite everywhere. The induced velocity is infinite near the body and zero at infinity. Having this in mind, we generally can expect the existence of a homoclinic separatrix with a hyperbolic stationary point, where two parts of yelocity are balanced (in the case ofstationary external velocity). Thus, the above described phenomena of chaotic vortex—body interaction seems to be generic. The local kinematic pressure, exerted on the sur2/ face of thedbody the vortex, is ofvortex the order k d2, where is thebydistance between andof body. ~.
~‘
This prediction has been recently confirmed in the series of numerical experiments for two-dimensional and axisymmetric flows, which were performed in collaboration with J. Kadtke and G. Pedrizzetti and will be published elsewhere.
4 February 1991
In the chaotic regime of motion, a vortex can come closer, it will spend more time near the body and is more likely to create a destructive impact on the body. The generalization to the case when we have additional circulation k0 around the cylinder is straightforward. In this case we have to add to (4) the term —00 Inp, where ao = k0/2itau0. By using the above described procedure, it is easy to show that in the time-periodic external flow, the motion of fluid particles becomes chaotic even without a vortex (0~...0)when 1001 >2. The condition aol >2 ensures the existence of a hyperbolic stagnation point when o= 0. Generally, when a 0 and C~~ 0, the unperturbed system has several stagnation points (hyperbolic and elliptic). With e ~ 0, the vortex can chaotically change the direction of rotation during the capture. This is physically ciear, but it has to be investigated in detail numerically. In practical problems, the vortex has a finite core, which leads to a system with an infinite number of degrees of freedom. If the size of the core is small in comparison with the size of the body, the multipole representation of the vortex can be used. In the simplest representation we have two closely located concentrated vortices, which rotate around each other. It this case we get chaos even without oscillation (e = 0). The proofis lengthy and will not be presented here, but the idea is simple the reduction of the Hamiltonian system [24]. The slow variables are the coordinates of the center ofvorticity, the angle of mutual rotation of vortices plays the role of time and the distance between vortices is a small parameter. In the case of bigger distances between two vortices, one of the vortices can be captured forever and the other will escape. The permanent capture of one of the vortices does not contradict Llttlewood’s theorem. This kind of partial capture of vorticity in a more complex situation happens when a tornado hits a building. The problem of three-dimensional chaotic vortex— body interactions with effects of vortex stretching and reconnection is the more difficult and profound. In this case we plan to use the appropriate modification of the method of three-dimensional solenoidal vortex singularities (vortons), which includes a —
mechanism of inviscid dissipation of energy [19,27]. Generally, the topic of chaotic vortex—body inter395
Volume 152, number 8
PHYSICS LETTERS A
action, connected with important applications, can serve as a link between the theory ofturbulence and vortex dynamics. In particular, it seems natural to connect the Markovian description of intermittent turbulence [27] with the symbolic dynamics of vortex systems, exhibiting the Markov property [1,2,6].
4 February 1991
[12] H. Aref, J. Fluid Mech. 143 (1984) 1. [13] J.C. Hardin and J.P. Mason, Phys. Fluids 27 (1984) 1583. [14] Y. Kimura and H. Hasimoto, Phys. Soc. Japan 55 (1986) [15] H. Aref, J.B. Kadtke, I. Zawadzki, L.J. Campbell and B. Eckhardt, Fluid Dyn. Res. 3 (1988) 63.
References
[16] V. Rom-Kedar, A. Leonard and S. Wiggins, J. Fluid Mech. 214 (1990) 347. [17] E.A. Novikov, Phys. Fluids 28 (1985) 2921. [18] E.A.Novikov and S.G. Chefranov,Phys. Atmos. 13 (1977) 414. [19] E.A. Novikov, Boundary-Layer Meteorol. 38 (1987) 305. [20] L.D. Landau and E.M. Lifshitz, Fluid mechanics (Pergamon, Oxford, 1959) p. 26. [21] E.T.S. Walton, Proc. R. Irish Academy 38A (1928) 29. [22] V.K. Melnikov, Trans. Moscow Math. Soc. 12 (1963) 1.
[1] E.A. Novikov and Y.B. Sedov, Soy. Phys. JETP 48 (1978) 440. [2] E.A. Novikov and Y.B. Sedov, JETP Lett. 29 (1979) 20. [3] N.A. Inogamov and S.V. Manakov, prepnnt (1979). [4] S.L. Ziglin, Soy. Math. DokI. 21(1980) 296. [5]H.ArefandN.Pomprey,Phys.Lett.A78 (1980) 279. [6] E.A. Novikov, Ann. N.Y. Acad. Sci. 357 (1981) 47. [7] H. Aref, Annu. Rev. Fluid Mech. 15(1983) 345. [8] J. Marsden and A. Weinstein, Physica D 7 (1983) 305. [9] S. Manakov and L. Schur, JETP Leti. 37 (1983) 54. [10] H. Hasimoto, K. Ishii, Y. Kimura and M. Sakiyama, in: Proc. IUTAM Symp. on Turbulence and chaotic phenomena in fluids, Kyoto, 1983 (North-Holland, Amsterdam, 1984) p.231. [ll]J.KoillerandS.P.Carvalho,preprint(l983).
[23]V. Arnold, Doki. Akad. Nauk SSSR 156 (1964) 9. [24] J. Guckenheimer and P. Holmes, Nonlinear oscillation, dynamical systems and bifurcations of vector fields (Springer, Berlin, 1983). [25] B. Bollobas, ed., Littlewood’s miscellany (Cambridge Univ. Press, Cambridge, 1986) p. 186. [26lE.A. Novikov, Zh. Eksp. Teor. Fiz. 84 (1983) 975 [Soy. Phys. JETP 57 (1983) 566]; Phys. Lett. A 112 (1985) 327; in: Proc. 6th Symp. on Energy engineering sciences, Argonne National Laboratory, Argonne, IL, 1988, p. 59; MI. Aksman, E.A. Novikov and S.A. Orszag, Phys. Rev. Lett. 54 (1985) 2410; M.I. Aksman and E.A. Novikov, Fluid Dyn. Res. 3 (1988) 239. [27] E.A. Novikov, Phys. Fluids 29 (1986) 3907; Phys. Fluids Al (1989)326;2(1990)814.
This work is supported by the U.S. Department of Energy under grant DE-FGO3O-87ER13801. I wish to thank M. Mikulska for help in preparing fig. 1.
396
PHYSICS LETTERS A
4 February 1991
Chaotic vortex—body interaction E.A. Novikov Instil uteforNonlinear Science, 0402, University ofCalifornia, San Diego, La Jolla, CA 92093, USA
Received 30 July 1990; revised manuscript received 14 November 1990; accepted for publication 27 November 1990 Communicated by D.D. HoIm
It is shown, by using the Poincaré—Melnikov—Arnold method that the motion of a linear vortex in the flow past a cylindrical body is chaotic. In particular, a vortex can be captured by the body and then, after some complicated rotation near the body, can be lost. More general problems ofvortex—body interaction are discussed qualitatively. Possible applications of the theory to the problems of aviacatastrophicsand destruction ofbuildings by tornado are indicated.
The study of the chaotic interactions of linear vortices has a twelve year history with the participation of many authors (see refs. [1—16] and references therein). Generalization to the axisymmetric flows has been indicated in ref. [17]. The main conclusion from these studies is that two-dimensional and axisymmetrical flows of ideal incompressible fluid are generally non-integrable and with appropriate initial conditions exhibit chaos. This is in contrast with the opinion which prevailed earlier among the advocates of the solution approach to the hydrodynamics. The chaotic motion of vortices has various applications, in particular, to the problem of weather prediction [1,18] The main goal of this note is to indicate novel features of chaotic motion which arise in the presence of a moving body. Firstly, it is enough to have only one vortex in order to get chaotic motion. Secondly, the mechanism of generation of chaos is very transparent in the vortex—body system. Thirdly, we get new phenomena chaotic capture loss of vortex by a moving body. Vortex—body interactions are im—
—
portant in many situations. The most dramatic examples are the aviacatastrophics, caused by a vortex initiated by downdraft of cold air [19], and the destruction of buildings by tornado. We cannot expect in a visible future direct numerical simulations of these phenomena, based on Navier—Stokes equations, because the Reynolds numbers are extremely large. A natural way to deal with this kind of prob0375-9601/91/S 03.50 © 1991
—
lems is to develop a hierarchy of models with increasing complexity. We will start with an analytical description of the motion of a linear vortex in oscillating flow past a circular cylinder. Then we will make some qualitative remarks about more general problems. In the frame of reference moving with the cylinder, the velocity of a linear vortex in an ideal fluid is a Hamiltonian superposition of two parts of motion. The first part corresponds to the potential motion of fluid relative to the cylinder (see, for example, ref. [201), such as if the vortex has zero intensity. The second part of the motion is induced by the interaction of the vortex with the cylinder. In the case of a circular cylinder, this motion is induced by an image vortex, placed inside the cylinder at a distance from the center f=a2/r, where a is the radius ofthe cylinder and r corresponds to the position ofthe vortex. Both parts of motion have zero normal components of velocity at the surface of the cylinder. We will scale the distances by a and the time by a! u 0, where u0 is the characteristic fluid velocity at infinity. If the fluid velocity at infinity is constant, then the problem is characterized by only one nondimensional parameter a= k/2~tau0,where k is the vortex intensity. In the case of an oscillatingflow, which can model the turbulence of vibration ofthe cylinder, we have the relative fluid velocity at infinity u(1)=u0(l+ esinwt)
Elsevier Science Publishers B.V. (North-Holland)
,
(1) 393
Volume 152, number 8
PHYSICS LETTERS A
where e and w are the nondimensional amplitude and frequency of oscillation, In polar coordinates (p, 0) with origin in the center of the cylinder, the Hamiltonian system for the motion of the vortex has the form d I ~H / 1 \ = — = — sin 0 (~l —i)(1 + e sin WI), (2) ~ p dØ 1 äH = ~ —
—
=
—
~
0
(~ +
(1 + e sin WI)
p
H=H0 +eH1, p = na> 1, 2 1) + ~a ln (~2 H0 = p (p —
H
—
—
~
p
—
1
1)
(3)
(4)
2 1) sin wt. (5) p (p With e=0, the system (2)—(4) has a general analytical solution. The vortex trajectories (4) are presented in fig. 1. Without loss of generality, we as—
1
=
~
4 February 1991
The stationary trajectories ofvortex in the flow past a circular cylinder have been studied in ref. [211 without recognizing the Hamiltonian structure ofthe problem. The homoclinic trajectory has not been indicated ref. [211, probably because at that time the homoclinic trajectory was considered as something pathological and physically irrelevant to the problem. Now we know how important the homoclinic trajectories are for the generation of chaotic motion. For e small but non-zero, the system (2), (3) has no analytic integrals of motion. It possesses transversal stable and manifolds namely,intersecting the Poincaré maps, P(1unstable 0), which advance a —
solution by one period T= 2it/w starting at time I~, possess transversal homoclinic points. We will show this b~using the Poincaré—Melnikov—Arnold (PMA) method. This type ofbehavior of dynamical systems is called chaotic. According to the PMA method [22—24]we consider the Melnikov function I M(1
j
0)= sume that the motion of fluid around the vortex is clockwise (a< 0) and the direction of fluid velocity at infinity is from the top to the bottom on fig. 1. We see that there is a homoclinic vortex trajectory [Po(1), Ø~(I)I with the hyperbolic stationary point at 0=0, 3—p~).Thehomoclinictrajectoryseparates region where the vortex is captured by the .o=.o~’the (a=p; body.
/ —
2
-
—
—4
(
/
/7 //
/
//
-
\\
—
T
I 0
—
where
{
} denotes the Poisson brackets,
,
1 ‘ÔA {A, B}(p, 0
0.4 ~ —
)=
—
(
~°“
t9B (p, ~P0~t)
(p, 0~1)
p, I) OB(p, 0, 1)
_________________
The integral (6) is taken along the homoclinic trajectory [p 0(t—10), 0(1—I0)]. We will show that M(10) has simple zeros. From (4)—(6), by a change of variable, we get M(t0)= —a sin[00(I—10)J Sifl WI dt
=
—a
$
+cos wt
\\
5
Fig. I The vortex trajectories in the flow past a cylinder (p~= 3).
394
6
J
—
I-
{Ho~H1 }(po(tto), 00(I—t0); I) dt
(sin wt cos WI0 sin WI0) dt.
(7)
Since Ø~(i)-+O,Po(tY~P*exponentially near the hyperbolic stationary point (when 1—~±~), the integral (7) is convergent. It is convenient to choose the initial position Oo (0) = it. In this case we have
Volume 152, number 8
Oo(—t)
PHYSICS LETTERS A
0o(t), po(
M(10)=—acoswt0
1)po(t) and (7) reduces to
j sinOa(t) [00(1)] sin wi dl.
(8)
The integral in (8) is not identically zero, because it is the Fourier transform ofa function which is not identically zero. The function M( t~)clearly has simpie zeros. According to PMA theory, this proves that the system (2), (3) has no analytic integrals of motion and the vortex trajectory is chaotic. In particular, the vortex which is initially far from the body, can intersect the homoclinic loop and will be captured by the body. After several complicated revolutions near the body, the vortex will eventually escape In connection with the capture-loss phenomena, we have the following theorem. Let S be the set of positions of the vortex at time t~,for which the vortex for all t~i~will stay inside a circie C surrounding the body. It can be proven that the subset of S, for which the vortex was outside of C for some 1<10, has zero measure. The proof is the same as in Littlewood’s theorem [25 J for conservative (gravitational) systems. The only condition which matters is the preservation of volume in phase space. The nonautonomous Hamiltonian system clearly satislies this condition. The theorem remains true if instead of a circle C surrounding the body we choose any area in phase space. In the case of an arbitrary shape of the cylindrical body we still have Hamiltonian superposition of external and induced motion of the vortex in terms of corresponding Green functions for the Laplace operator. The external velocity is finite everywhere. The induced velocity is infinite near the body and zero at infinity. Having this in mind, we generally can expect the existence of a homoclinic separatrix with a hyperbolic stationary point, where two parts of yelocity are balanced (in the case ofstationary external velocity). Thus, the above described phenomena of chaotic vortex—body interaction seems to be generic. The local kinematic pressure, exerted on the sur2/ face of thedbody the vortex, is ofvortex the order k d2, where is thebydistance between andof body. ~.
~‘
This prediction has been recently confirmed in the series of numerical experiments for two-dimensional and axisymmetric flows, which were performed in collaboration with J. Kadtke and G. Pedrizzetti and will be published elsewhere.
4 February 1991
In the chaotic regime of motion, a vortex can come closer, it will spend more time near the body and is more likely to create a destructive impact on the body. The generalization to the case when we have additional circulation k0 around the cylinder is straightforward. In this case we have to add to (4) the term —00 Inp, where ao = k0/2itau0. By using the above described procedure, it is easy to show that in the time-periodic external flow, the motion of fluid particles becomes chaotic even without a vortex (0~...0)when 1001 >2. The condition aol >2 ensures the existence of a hyperbolic stagnation point when o= 0. Generally, when a 0 and C~~ 0, the unperturbed system has several stagnation points (hyperbolic and elliptic). With e ~ 0, the vortex can chaotically change the direction of rotation during the capture. This is physically ciear, but it has to be investigated in detail numerically. In practical problems, the vortex has a finite core, which leads to a system with an infinite number of degrees of freedom. If the size of the core is small in comparison with the size of the body, the multipole representation of the vortex can be used. In the simplest representation we have two closely located concentrated vortices, which rotate around each other. It this case we get chaos even without oscillation (e = 0). The proofis lengthy and will not be presented here, but the idea is simple the reduction of the Hamiltonian system [24]. The slow variables are the coordinates of the center ofvorticity, the angle of mutual rotation of vortices plays the role of time and the distance between vortices is a small parameter. In the case of bigger distances between two vortices, one of the vortices can be captured forever and the other will escape. The permanent capture of one of the vortices does not contradict Llttlewood’s theorem. This kind of partial capture of vorticity in a more complex situation happens when a tornado hits a building. The problem of three-dimensional chaotic vortex— body interactions with effects of vortex stretching and reconnection is the more difficult and profound. In this case we plan to use the appropriate modification of the method of three-dimensional solenoidal vortex singularities (vortons), which includes a —
mechanism of inviscid dissipation of energy [19,27]. Generally, the topic of chaotic vortex—body inter395
Volume 152, number 8
PHYSICS LETTERS A
action, connected with important applications, can serve as a link between the theory ofturbulence and vortex dynamics. In particular, it seems natural to connect the Markovian description of intermittent turbulence [27] with the symbolic dynamics of vortex systems, exhibiting the Markov property [1,2,6].
4 February 1991
[12] H. Aref, J. Fluid Mech. 143 (1984) 1. [13] J.C. Hardin and J.P. Mason, Phys. Fluids 27 (1984) 1583. [14] Y. Kimura and H. Hasimoto, Phys. Soc. Japan 55 (1986) [15] H. Aref, J.B. Kadtke, I. Zawadzki, L.J. Campbell and B. Eckhardt, Fluid Dyn. Res. 3 (1988) 63.
References
[16] V. Rom-Kedar, A. Leonard and S. Wiggins, J. Fluid Mech. 214 (1990) 347. [17] E.A. Novikov, Phys. Fluids 28 (1985) 2921. [18] E.A.Novikov and S.G. Chefranov,Phys. Atmos. 13 (1977) 414. [19] E.A. Novikov, Boundary-Layer Meteorol. 38 (1987) 305. [20] L.D. Landau and E.M. Lifshitz, Fluid mechanics (Pergamon, Oxford, 1959) p. 26. [21] E.T.S. Walton, Proc. R. Irish Academy 38A (1928) 29. [22] V.K. Melnikov, Trans. Moscow Math. Soc. 12 (1963) 1.
[1] E.A. Novikov and Y.B. Sedov, Soy. Phys. JETP 48 (1978) 440. [2] E.A. Novikov and Y.B. Sedov, JETP Lett. 29 (1979) 20. [3] N.A. Inogamov and S.V. Manakov, prepnnt (1979). [4] S.L. Ziglin, Soy. Math. DokI. 21(1980) 296. [5]H.ArefandN.Pomprey,Phys.Lett.A78 (1980) 279. [6] E.A. Novikov, Ann. N.Y. Acad. Sci. 357 (1981) 47. [7] H. Aref, Annu. Rev. Fluid Mech. 15(1983) 345. [8] J. Marsden and A. Weinstein, Physica D 7 (1983) 305. [9] S. Manakov and L. Schur, JETP Leti. 37 (1983) 54. [10] H. Hasimoto, K. Ishii, Y. Kimura and M. Sakiyama, in: Proc. IUTAM Symp. on Turbulence and chaotic phenomena in fluids, Kyoto, 1983 (North-Holland, Amsterdam, 1984) p.231. [ll]J.KoillerandS.P.Carvalho,preprint(l983).
[23]V. Arnold, Doki. Akad. Nauk SSSR 156 (1964) 9. [24] J. Guckenheimer and P. Holmes, Nonlinear oscillation, dynamical systems and bifurcations of vector fields (Springer, Berlin, 1983). [25] B. Bollobas, ed., Littlewood’s miscellany (Cambridge Univ. Press, Cambridge, 1986) p. 186. [26lE.A. Novikov, Zh. Eksp. Teor. Fiz. 84 (1983) 975 [Soy. Phys. JETP 57 (1983) 566]; Phys. Lett. A 112 (1985) 327; in: Proc. 6th Symp. on Energy engineering sciences, Argonne National Laboratory, Argonne, IL, 1988, p. 59; MI. Aksman, E.A. Novikov and S.A. Orszag, Phys. Rev. Lett. 54 (1985) 2410; M.I. Aksman and E.A. Novikov, Fluid Dyn. Res. 3 (1988) 239. [27] E.A. Novikov, Phys. Fluids 29 (1986) 3907; Phys. Fluids Al (1989)326;2(1990)814.
This work is supported by the U.S. Department of Energy under grant DE-FGO3O-87ER13801. I wish to thank M. Mikulska for help in preparing fig. 1.
396