Turbulent 4-wave interaction of two type of waves

17 January 2000

Physics Letters A 265 Ž2000. 83–90 www.elsevier.nlrlocaterphysleta

Turbulent 4-wave interaction of two type of waves Maxim Lyutikov

)

Canadian Institute for Theoretical Astrophysics, St George Street, Toronto, Ontario, M5S 3H8, Canada Received 29 October 1999; received in revised form 30 November 1999; accepted 30 November 1999 Communicated by C.R. Doering

Abstract We consider turbulent 4-wave interaction of two types of waves: acoustic waves Ždispersion v s k . and electromagnetic-type waves Ždispersion V 2 s m 2 q p 2 .. For large wave vectors Ž k 4 m., when the dispersion of EM-type waves becomes quasiacoustic, the stationary spectra are obtained following a standard Zakharov approach. In the small wave number region Ž k, p < m. we derive nonlinear differential Kompaneets-type kinetic equation for arbitrary distributions of the interacting fields and find when this equation has Kolmogorov-type power law solutions. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 05.20.Dd; 05.30.Jp; 05.90.q m

1. Introduction In this Letter we consider 4-wave interaction of two types of waves: one with acoustic dispersion v s k, called massless field, and another with electromagnetic-type dispersion V 2 s m 2 q p 2 , called massive field Ž v , V are frequencies, k, p – wave vectors of the waves, the phase speed of the acoustic waves was set to unity.. This is a fairly common case of wave interaction. It includes, for example, interaction of electrons with photons and electromagnetic fields with Langmuir waves. We assume that the interacting fields are scalar fields obeying quantum Bose–Einstein statistics. Then the collisional

)

E-mail: [email protected]

integral becomes a nonlinear functional in occupation numbers of both types of waves. We are interested in finding stationary solutions of such collision integral. In particular we will be looking for power law type solutions Ž n A k a , N A p b . which can exist in the inertial ranges k, p 4 m or k, p < m. A standard procedure to find power law solutions of the kinetic equation is through Zakharov conformal transformations w1x. This method provides a simple rule for finding stationary power law solutions in the inertial region for differential scattering Žwhen the change in energy or momentum in each scattering is small.. For coupled kinetic equations the Zakharov transformation in the general cannot be done. Tractable interactions include Ži. 3-wave interaction of waves with similar dispersion relations Žusing conformal transformations., Žii. 3-wave interaction of high frequency and low frequencies waves w1x Žusing a diffusion approximation..

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 9 9 . 0 0 8 6 5 - 8

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M. LyutikoÕr Physics Letters A 265 (2000) 83–90

In this Letter we consider coupled kinetic equations for 4-wave interaction of waves. In the ‘ultrarelativistic’ regime, when the wave vectors of both particles is much larger than the mass k , p 4 m, both types of waves have similar acoustic-type dispersion and thus can be viewed as selfinteraction between quasiacoustic waves. The stationary spectra then can be found using conformal transformations. In the oposite limit of small wave vectors, k, p < m, different dispersion laws of interacting waves break the homogeneity of the kernel of the kinetic equation and do not allow exact Kolmogorov-type solutions. In this case we expand the collision integral in small changes of energy in each collision. Then the collision integral may be simply written as a nonlinear diffusion equation. This method was first applied by Kompaneets w5x who derived a nonlinear equation for the interaction of the low frequency boson particles Žphotons. with nonrelativistic classical electrons which were assumed to be in thermodynamic equilibrium due to quick relaxation by the long range Coulomb forces. The ‘ultrarelativistic’ regime, when both interacting waves have quasiacoustic dispersion requires a careful consideration – for exactly acoustic dispersion the weak turbulence theory may not be applicable Žsee, e.g., Refs. w2,4x.. Balk w2x and Newell and Aucoin w3x argued that in the multidimensional case the angular dispersion can justify the applicability of the weak turbulence theory to acoustic waves, while Zakharov and Sagdeev w4x resorted to small corrections to the acoustic dispersion. We follow the latter approach: in the limit p 4 m the correction to the dispersion of EM-type waves will provide the dispersion required for the final time of interaction of acoustic waves and will ensure that kinetic approach is applicable. The primary purpose of this Letter is to derive a low frequency diffusion approximation for the 4wave interaction for arbitrary distribution functions of both interacting waves and to find Kolmogorovtype solutions in the inertial ranges. In case of quantum fields, the Kolmogorov-type power law solution may be possible only in the inertial ranges, where energies of the interacting particles are not close to the only quantity with dimension of energy – m and the occupation numbers satisfy n , N 4 1 or n , N < 1.

We start with a Hamiltonian of the system of two interacting waves in terms of wave amplitudes a k and A p H s v k a k a )k d 3 k q V p A p A)p d 3 p

H

H

q 14 Tk p k X pX a )k A)p a k X A pX

H

=d Ž k q p y kX y pX . d 3 kd 3 pd 3 kX d 3 pX .

Ž 1.

We assume the simplest form of the relativistically invariant matrix element Žsee, e.g., Ref. w6x. Tk p k p s

g2

X X

v k V p v k X V pX

.

Ž 2.

Then the kinetic equations for occupation numbers n k , Np are

ž

E n krE t E NprE t

/

s pr2 Tk2p k X pX F w n, N x d Ž v k q V p y v k X y V pX .

H

=d Ž k q p y kX y pX .

d3p

ž / 3

d 3 kX d 3 pX ,

d k

Ž 3.

where F w n, N x s NpX n k X Ž n k q Np q 1 . y n k Np Ž NpX q n k X . .

Ž 4.

2. Applicability of the kinetic equation For the applicability of the weak turbulence theory Žbased on random phase approximation. we need that the typical interaction time t int f 1rŽTNtot ., Žwhere Ntot s Hd 3 pNp and T is a matrix element for amplitudes. be much larger than chaotization time for the phases of waves t collision s 1rD kÕg Ž Õg is the group velocity and the wave packet is centered on k 0 " D k . and diffusion time tdiff s 1rD k 2 ÕXg . For a power law v ; k a we have ÕXg ; a Ž a y 1. k ay2 , so t collision ; krŽ D k .1rŽ av . which is can be very short – of the order 1rv . On the other hand the diffusion time tdiff s Ž D krk .y2 rŽ a Ž a y 1. v .. Which obviously unapplicable to acoustic turbu-

M. LyutikoÕr Physics Letters A 265 (2000) 83–90

lence. Still, if a / 1, then the diffusion time tdiff also can be very short. This case Žnonacoustic turbulence. is applicable for small wave vectors: k < m. tdiff t int

2

s TNrkc;

g nk m

,

Ž 5.

where we estimated T Ž k < m. s g 2rŽ km., N s nk 3. Thus the applicability of the kinetic equation in this case requires m n< 2 , Ž 6. g k which for k F m reducesto n < 1rg 2 . On the other hand for semiacoustic waves with k 4 m we find v s k cŽ1 q m 2rk 2 . we have v XX s 2 m2rk 3, tdiff s Ž kr Ž D k . . krm2 f krm2 , then the applicability criterion is tdiff t int

g 2 nk 2

TNk s

m2

s

m2

< 1,

Ž 7.

Žthe matrix element in this limit is T s g 2rk 2 .. This will eventually break at m k max s . Ž 8. g'n Thus for applicability of a kinetic equation in a range m < k < k max the same condition should be satisfied, n < 1rg 2 . We can also estimate when the higher terms will be unimportant in the kinetic equation. The ratio of the second order terms of expansion in g 2 to the first order is typically 4

3 2

w 2 x r w 1 x s g k n rv 1 v 2 v .

Ž 9.

For acoustic turbulence k 4 m the frequencies are v i ; k, this gives

w 2 x r w 1 x a s g 4 n2 < 1 if n < 1rg 2 ,

Ž 10 .

while for small wave vectors v 1 s m v 2 s k

w 2 x r w 1 x s g 4 n2 krm < 1 if n < 1rg 2 .

Ž 11 .

Thus we conclude that for existence of inertial range where weak turbulence theory is applicable the occupation numbers should satisfy Np ,n k <

1 g2

Ž 12 .

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3. Interaction of particles at large wave vectors In the limit k, p 4 m Ž‘relativistic particles’. the dispersion law for the EM-type waves become almost acoustic

ž

Vfk 1q

m2 2k2

/

.

Ž 13 .

The applicability of the weak turbulence theory to the interaction of acoustic wave in more than one dimension is a long standing question. In 1-D weak turbulence theory is not applicable since the interaction time between two waves is infinite, so that mutual influence of waves becomes large. In larger dimensions there were claims Žw3,2x. that angular dispersion may limit the interaction time sufficiently to allow weak turbulence approximation. Another approach is to allow for small corrections to the dispersion which will limit the interaction time of wave packets and allows us to implement methods of the weak turbulence theory regardless of angular dispersion w4x. Assuming the weak turbulence theory is applicable Eq. Ž12. we can find stationary solutions to the kinetic equation using Zakharov conformal transformations. In the limit k, p 4 m the dispersion laws v A k a and V A p a with a s 1, and the interaction coefficient in this limit is a homogeneous function of the order y2: Tk p k X pX Ž l k , l p, l kX , l pX . s l p Tk123 Ž k , p, kX , pX . ,

p s y2.

Ž 14 .

Zakharov transformations then give stationary solutions n P , NP A vy5 r3 ,

n Q , NQ A vy4 r3 .

Ž 15 .

Here, P and Q symbolize Kolmogorov spectra with constant flux of energy and constant flux of action. It is possible to check that collision integral converges for both spectra, so that the interaction is local. The sign of fluxes signw P x ) 0 and signw Q x - 0 imply that two possible stationary spectra include the constant energy flux flowing to large k or constant flux of particles to small k. Realization of a particular spectra Eq. Ž15. depends on the boundary condition at the limits of the inertial range, e.i., by the

M. LyutikoÕr Physics Letters A 265 (2000) 83–90

86

location of the sinks and sources of energy and action. If the source of energy and of action are not at k s 0 and k s ` correspondingly, then, depending on values of p and a , the integrals of energy or action converge or diverge. Using criteria for the stability of Kolmogorov spectra w7x, we conclude that for sources located at k 0 , p 0 4 m it is KolmogoroÕin-action case that is realized Žthe action integral diverges at large k .. In this case the turbulence is Kolmogorov in action – particles are flowing to small k Žand may eventually be stored in Bose condensate.. For k, p - k 0 , p 0 a stationary Kolmogorov-in-action spectrum will form, while for for k, p ) k 0 , p 0 there will be a selfsimilar ‘thermal wave’ type solution carrying energy to large k and p: nk s

1 t 3r7

ž / t 5r7

F w n, N x

En

ž

s D N Ž N q 1.

D2

ž

q 2

E 2n

N Ž N q 1.

EN En y2

Ev

y n Ž n q 1.

Ev 2

EN Ev

/

q n Ž n q 1.

E 2N Ev 2

/

Ž n q N q 1. .

Ev Ev

Ž 19 .

In the following we concentrate on the kinetic equation for particles n. In the kinetic equations we next separate angular and momentum integrations Icoll Ž k .

v n0

Expansion of F w n, N x to the second order in D gives

,

k ) k0 ,

for

Ž 16 .

`

s 4ps 0

where it is eventually dissipated w7x. Thus, in the ‘thermal wave’ the mean frequency increases with time A t 5r7. This situation Žflow of particles to small wave numbers, where stationary spectrum is formed, and flow of energy to large wave numbers. is somewhat similar to interaction of Langmuir waves where in the low frequency region the dominant nonlinear process for Langmuir waves is the exchange of virtual ion sound waves which leads accumulation of particles with small k – Langmuir collapse.

4. Interaction of particles at small wave vectors

=

EN

ž

EV

1 q 2

½H

E 2n 2

Ev

d V p d V nX 4p

n Ž n q 1. y d V p d V nX 4p

ž

=

½H

p 2 dp

4p

= n Ž n q 1.

y2

E 2N

En Ev

4p

D Ž Ž k , p,nX .

N Ž N q 1.

D Ž Ž k , p,nX .

2

2

5

/

5

q N Ž N q 1.

EV 2

EN En EV Ev

Ž N q n q 1.

/

Ž 20 .

where

s0 s

4.1. Kompaneets approximation

H0

1

Mf i

16p

m2

2

.

Ž 21 .

Denoting the angle averages In the limit k, p < m the change in energy in each collision is small, so we can expand the collision integral in small changes of the energies of particles:

I1 Ž p,k . s s 0 s

v kX s v k q D ,

V pX s V p y D ,

v Ž v Ž 1 y n P nX . q p P Ž n y nX . . X

X

mqv Ž1ynPn . ypPn

.

s

4p

4p

D Ž Ž k , p,nX .

3 m2

I2 Ž p,k . s s 0

Ž 18 .

d V p d V nX

k Ž4 k 2 y 3 k m q p 2 .

Ž 17 .

where D to the first order in vrm is given by

Dsy

H

H

d V p d V nX 4p

4p

2 k 2 Ž2 k 2 q p 2 . 3 m2

s0 ,

D Ž Ž k , p,nX . s0 .

2

Ž 22 .

M. LyutikoÕr Physics Letters A 265 (2000) 83–90

We can rewrite collision integral in the form `

Icoll Ž k . s 4ps 0

H0

En y

p 2 dp I1 Ž p,k .

/

Ev

E 2N

ž

= n Ž n q 1.

EV

2

EV

n Ž n q 1.

E

I2 Ž p,k .

2

Icoll Ž k . s 2ps 0 p 2 dp

H

E 2n

q N Ž N q 1.

Ev

3

coll

Ž N q n q 1.

EV Ev

/

.

Ž 23 .

Ž k . s 0.

Ž 24 .

Partially integrating and collecting terms with different powers of distribution function we find

E

ž

Hd v d V Nn 1 q 2

ž

E2

y

Ev

E2

E 2V

Icoll Ž p . s 2ps 0 k 2 dk

H

/

i2

E 2v

yN X n Ž n q 1 .

H

EVEv 1

Ev

yi1 q

E q d v d V n2 N

H

EV

i1 y

1

/

i2

H

E

ž

2 Ev

E y

E

ž

EV

/

i2

v 02 s

E

2 Ev

y

EV

/

i 2 s 0,

I1 ,

i2 s k 2

Ek

p2

Ep

1 E i2 2

ž

EV

y

E i2 Ev

.

,

Hk dk 4 m n Ž n q 1. , Hk dkk n Ž n q 1. 2

I2 .

Ž 26 .

2

X

2

H p dpN p

2

B1 s y

, 2

/

2

k4

2

V0 s Ep

2

H p dpN Ž N q 1. p 2 H p dpN Ž N q 1. 1

2

Ev EV Ev EV Each term in Eq. Ž25. should be equal to zero. This gives a relation between the coefficients i 1 and i 2 :

i1 s

Ž 28 .

A 2 s 4p k 2 dkk 2 n Ž n q 1 . ,

where p2

..

H p dpN Ž N q 1. p

Ž 27 .

This is an important relation that is one of the main results of the Letter. It relates the coefficients in the diffusion expansion of the collisional integral independently of the particular form of the interaction.

nX N Ž N q 1 .

For massless particles n and massive particles N correspondingly. Eqs. Ž28. are the diffusion-type equations for the low frequency interaction of two types of quantum waves – ‘massless’ acoustic waves ‘n’ and ‘massive’ EM waves N. To reduce then to the more familiar Fokker–Planck form we introduce coefficients

Ž 25 . Ek

Ž 'V I2

H

E q dv dV N 2 n

i1 s k 2

EV

A1 s 4p p 2 dpN Ž N q 1 . ,

E2 y2

nX N Ž N q 1 .

.,

E

EV

q

Ž v 2 I2

Ev

yN X n Ž n q 1 .

2

There is a also a constraint condition on the Eq. Ž23. – it sould conserve total number of particles, which is an integral of motion of the Hamiltonian Eq. Ž1.:

Hd kI

Using relation Ž27. we can simplify considerably the collision integral equation Ž3.. Partially integrating we find that collision integrals take a particularly simple diffusion type form:

E

EN En y2

1

N Ž N q 1. q

EN

ž

87

2

Hk dkk

2 X

n

B2 s y 2

Hk dkk

2

n Ž n q 1.

2

M. LyutikoÕr Physics Letters A 265 (2000) 83–90

88 X

2

2

1

X

2

k

2

Under assumption C1,2 f 0 the Kompaneets-type equations for the occupation numbers n and N become

2

H p dpN p H p dpN Ž N q 1. C s H p dpN H p dpN Ž N q 1. p 2

y 1, 2

En

4 X

2

Et

2

Hk dk 4 m n Hk dkk n Ž n q 1. C s y 1. Hk dkk n Hk dk k n Ž n q 1. 4m 2

2 X

2

=

2

2

ž

Icoll Ž V . s

s0 2

ž

A1 v 4 Ž v 2 q v 02 .

= 1q

A2

v 02 C1 v 2 q v 02

//

ž

q B1 n Ž n q 1 .

Ev

Et

sy

En Ev

s0 2

=

ž

A1

EV

E

2

v Ev

v 4 Ž v 2 q v 02 .

q B1 n Ž n q 1 .

A2

EN

1

1

E

'V

EV

/

,

'V Ž V q V 0 .

q B2 N Ž N q 1 .

/

,

Ž 31 .

and the the stationary equations E nrE t s E NrE t s 0 should satisfy

v 4 Ž v 2 q v 02 . Ž nX q B1 n Ž n q 1 . . s P1 ,

'V Ž V q V 0 . Ž N X q B2 N Ž N q 1. . s P2 .

The physical meaning of the constants P1 and P2 is the total particle flux in the wave vector space. Eqs. Ž32. are nonlinear Riccatti-type differential equations. Using a conventional substitution N X Ž x . U XrU they can be reduce to linear equations of the form U XX q U X y G Ž x . U s 0.

ž

EN EV

V 0 C2 VqV0

Ž 32 .

™

,

'V Ž V q V 0 .

qB2 N Ž N q 1 . 1 q

En

ž

EN

Note, that in the thermodynamic equilibrium N X s yN Ž N q 1.rT, so that B1 , B2 s 1rT ŽT is equilibrium temperature. and C1 ,C2 s 0. Coefficients A1,2 and v , V are complicated even for thermodynamic equilibrium. In the classical limit they give A1,2 s Ntot , n tot - total number of particles, v 02 s 3mT -average - k 2 ) and V 0 s 5Tr4. Using notations 29 the collision integrals may be written as Icoll Ž v . s

2

4

Ž 29 .

s0

s0

sy

//

. Ž 30 .

It is not generally integrable in quadratures, but in some limits it may have a solution which is a sum of power laws w8x. 1 The best way to search for possible power law solutions is to use the substitution UŽ x .

4.2. Stationary solutions To find stationary power law solutions to the nonlinear Eqs. Ž30. we assume that C1,2 f 0 Žin this case the resulting solution are sums of power laws Žsee Eqs. Ž35. and Ž36... In case of C1 the dominant contribution to the number density comes from the short wave vector limit of the thermodynamic equilibrium Žsee Eq. Ž36.., for which we can assume C1 f 0. The coefficient C2 on the other hand is generally nonzero, but if the maximal energy of particle n Žcut-off of the power law. is much less than the mass m then the term with C2 is approximately pmaxrm – much smaller than unity.

Ž 33 .

™ Wexp ½ ž H(G Ž x . /

y1

dx

5

which takes a correct account of the leading power law term. Power law solutions of the Eq. Ž32. are possible in the inertial regions – where there is no quantity with a dimension of length: these are regions v much less or much larger than v 0 , V much less or much larger than V 0 and N,n much less or much larger than 1.

1 For the equilibrium distribution of heavy particle N various approximations for n in the limit v 4 v 0 and N <1 are summarized in Ref. w9x.

M. LyutikoÕr Physics Letters A 265 (2000) 83–90

89

In the linear regime N,n < 1 the stationary solutions are nA

H v Žv

P1

3

2

q v 02

dv ,

.

NA

P2'V

H Ž Vqv

dV . 0

. Ž 34 .

In the strongly nonlinear Ž N,n 4 1. limit the leading orders in power-laws are

° nv

P1

v A~

0v

1 2

q

P1

Bv 3

¢v q 2 B v °1 y P NV

A~

1

,

2

located at k, p < m ŽFig. 1Žb.., then the low energy tail of the distribution of massless particles is n A vy3 .

for V < V 0

V 01r2V 1r4

.

P2

¢ BV y V

Ž 35 .

for v 4 v 0

3

BV

for v < v 0

Ž 36 .

for V 4 V 0

3r4

5. Conclusion

Given the solutions 36, we can also calculate the coefficients V 0 and v 0 :

°k ~

V0 s

2 max

8m 2 k max

¢12 m v 0 s pmaxr'3 .

for

n A vy2

for

y3

Fig. 1. Stationary power law solutions of the system Ž12.. Ža. source of particles is located at k, p4 m, Žb. source of particles is located at k, p< m.

, nAv

Ž 37 .

where k max and pmax are the upper cut-offs limits for the corresponding power laws. We can verify that the integral in the expressions for the coefficients ŽEq. Ž29. converge at k, p s 0. Note also, that for massive particles N, the leading term in the small wave vector limit corresponds to the low frequency asymptotics of the thermodynamic equilibrium spectrum. Eqs. Ž35. – Ž37. give the selfconsistent Kolmogorov-type solutions to turbulent interaction of two types of waves. They are qualitatively shown in Fig. 1. In all cases the particles are flowing to small k, p with a constant flux in phase space. If the sources of particles are located at k, p 4 m ŽFig. 1Ža.., so that v 0 f m, then the low energy tail of the distribution of massless particles Ž v < v 0 . is n A vy2 . Alternatively, if the sources of particles are

We have considered turbulent 4-wave interaction of two types of waves with different dispersion laws: acoustic and electromagnetic-type. Both types of waves were assumed to obey a quantum boson statistics. We found possible stationary Kolmogorov-type spectra in the inertial ranges k, p 4 m and k, p < m. To reach the stationary solutions the particles should be generated at finite momenta k and p and flow to small wave numbers. In the ‘ultrarelativistic limit’ k, p 4 m, the turbulent interaction of two types of waves resembles self interaction of semiacoustic waves with the dispersive correction ensuring the applicability of the weak turbulence limit. In the small wave vector limit k, p < m the collisional integral may be reduced to diffusion-type differential equation. In this limit, the small wave number asymptotic of massive particles is dominated by the thermal equilibrium tail N A Vy1 , while the small wave number asymptotic of massless particles is steeper: n A vy2 % vy3 . Arguably, the massless particles n will eventually reach k s 0 to be absorbed in the Bose condensate. 2

2

Limitations of the kinetic approach to Bose condensation were discussed by Kogan et al. Ž1992.

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M. LyutikoÕr Physics Letters A 265 (2000) 83–90

Acknowledgements I would like to thank Alexander Balk for useful comments. References w1x V.E. Zakharov, V.S. L’vov, G. Falkovich, Kolmogorov spectra of turbulence, Springer, BerlinrNew York, 1992. w2x A. Balk, Phys. Lett. A 187 Ž1994. 302.

w3x A.C. Newell, P.J. Aucoin, J. Fluid Mech. 49 Ž1971. 593. w4x V.E. Zakharov, R.Z. Sagdeev, Sov. Phys. Dokl. 15 Ž1970. 439. w5x A.S. Kompaneets, Sov. Phys. JETP 31 Ž1956. 876. w6x L. Landau, Lifshits, Quantum Mechanics, Pergamon Press, OxfordrNew York, 1977. w7x V.E. Zakharov, in: A.A. Galeev, R.N. Sudan ŽEds.., Basic plasma physics, North-Holland, New York, NY, 1983. w8x E. Kamke, Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig, 1969. w9x R.A. Sunyaev, L.G. Titarchuk, Astron. Astrophys. 86 Ž1980. 121.