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Nonlocal angular instability of a Kolmogorov-like wave turbulence spectrum
Physics Letters A 168 ( 1992 ) 127-132 North-Holland
PHYSICS LETTERS A
Nonlocal angular instability of a Kolmogorov-like wave turbulence spectrum G.E. F a l k o v i c h Department of Nuclear Physics, Weizmann Institute of Science, Rehovot 76100, Israel and Institute of Automation, Siberian Division of the USSR Academy of Sciences, Novosibirsk, Russian Federation
and M . D . Spector Department of Fluid Mechanics and Heat Transfer, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel Received 19 May 1992; revised manuscript received 4 June 1992; accepted for publication 5 June 1992 Communicated by D.D. Holm
A new type of instability of Kolmogorov-like wave turbulence spectra is found. Such an instability is due to an interaction nonlocal in k-space and it strongly modifies the angular structure of the turbulence spectrum. However, the spectrum dependence on the modulus k is still a Kolmogorov-like one corresponding to energy transfer local in k-space. The specific case of capillary waves on shallow water is considered in detail. It is shown that the energy transfer is local while that of m o m e n t u m is nonlocal in k-space.
1. Introduction The picture of cascade turbulence was suggested by Richardson, Kolmogorov and Obukhov for incompressible fluids [ 1-4 ] and it was later extended to wave turbulence in other media [ 5 ]. It is based on the concept of interaction locality. That means that those modes (vortices or waves) effectively interact which are of comparable scales only. The question naturally arises: is the locality property satisfied on the steady Kolmogorovlike spectrum only or on slightly differing distributions as well? Proceeding from continuity-like speculations, one might suppose that in the general case interaction locality for the Kolmogorov distribution leads to that for close ones. Such a supposition is, however, incorrect since the Kolmogorov spectrum usually possesses a higher degree of symmetry (for example, being isotropic) than arbitrary yet close distributions. A stationary locality does not mean thus an evolutionary locality as it was stated in refs. [5,6]. What can we say about the properties of a system with a local Kolmogorov-like spectrum and with perturbations interacting nonlocally? Our paper deals with that matter. We demonstrate here that nonlocality can lead to either the growth or the decreasing of perturbation, depending on the number of the angular harmonic. The growth of perturbation means the instability of the stationary isotropic spectrum with respect to that harmonic. Nevertheless, we demonstrate the possibility of a Kolmogorov-like spectrum to survive on average which means that, despite the steady spectrum being anisotropic, the angle averaging spectrum decreases with the modulus k by the Kolmogorov law. The energy is thus transferred locally. As far as momentum is concerned, it can be transferred nonlocally in k-space. We give our consideration using weakly nonlinear wave turbulence as an example which allows analytical treatment and is of physical interest itself. Section 1 deals with the kinetic equation which describes weak wave 0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
127
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turbulence. Here we demonstrate the difference between stationary and evolutionary locality. In section 2 we consider a specific example, viz. capillary waves on shallow water and demonstrate that there exist stationary locality and evolutionary nonlocality of the Kolmogorov-like spectrum in this case. We prove that there is a local steady isotropic spectrum stable with respect to local even angular harmonics of perturbation. Odd harmonics are nonlocal: an arbitrary perturbation strongly interacts with both small and large k's. The angular harmonic with the number l= 2j+ 1 decreases for o d d j and increases for evenj. The Kolmogorov-like spectrum is thus unstable with respect to the angular harmonics with the numbers l = 4 i + l, i=0, l, .... So the steady spectrum should be anisotropic. Nevertheless, we prove the k-dependence of the spectrum to be Kolmogorovlike, so the energy is transferred locally (by a cascade process) while the momentum is transferred nonlocally. Our consideration is restricted to weak turbulence (which is the set of weakly interacting waves) and cannot be applied directly for describing turbulence with strong interaction, for instance, in an incompressible fluid. This latter case seems to correspond to both stationary and evolutionary locality according to ref. [7].
2. Stationary and evolutionary locality If the level of wave excitation is small enough to provide the weakness of nonlinear effects compared with those of dispersion, the turbulence is regarded as weak and can be described with the help of a kinetic equation. Such an equation is formulated in terms of pair correlators, called also wave occupation numbers, in k-space, ( a (k) a* (k') ) = n (k) ~ ( k - k' ), where a (k) are the wave amplitudes. With a small level of nonlinearity the three-wave processes are sufficient to restrict ourselves to, which is allowed if the dispersion law o~(k) admits the decay equality o ~ ( k + k ' ) = ~o(k)+ to(k'). In this case, the kinetic equation reads as follows,
On(k,
Ot
=I(k, t)= f Uk,2(n, n2--nkn,--nkn2) dk, dk2-2 f U~k2(nkn2--nknl--n, n2) dk~ dk2.
(1)
Here Uk~2= V2~2~( k - k~ - k2) ~( 0 4 - ~0~- 0)2) and Vk12is the matrix element of the three-wave interaction. The g-functions in ( 1 ) express the conservation laws of the values of the energy E=fcokn(k) dk and of the momentum H = fkn (k) dk in every elementary act of interaction. Consequently, the kinetic equation (1) can be written down, e.g., in the form of the continuity equation for the wave energy. For isotropic distributions, one can introduce the quantity e (k, t ) = (2k)d-lo)ktl (k, t), representing the density of the wave energy in k-space, so the kinetic equation can be represented as follows,
O~(k, t) OP(k, t) + _o. Ot Ok
(2)
The value of k
P(k, t) = ~ (2k)d-ltOkI(k, t) dk
(3)
0
thus represents the density of the energy flux (again in the spherical normalization), d being the dimension of k-space. The stationary turbulent distribution n (k), as is readily seen from (2), corresponds to a flux constancy. In the case of scale invariance of the wave system implying that ~o(/*k)=/z"to(k), V(/.tk, #k~, l~k2) = I*mV(k, kl, k2), a stationary solution of eqs. (1), (2) turns out to be the power n (k) =Ak-". The index v corresponds to the energy flux constancy and can be found from (3): v = m + d. The constant A, defined by the condition P ( k ) = P , is proportional to pw2. Distributions of the type of
n( k )ocpw2k -m-d
(4)
are called Kolmogorov-like spectra of weak wave turbulence in direct analogy with those for turbulence of an 128
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incompressible fluid. Such distributions are stationary solutions of the kinetic equation provided that the integrals involved converge. Let the matrix element have the following asymptotics, V2(k, k~, kz) ozk'~ k 2m-'m for k~ << k. Then it is easy to show that the collision integral in ( 1 ) defined on the power distributions n(k)ock -s converge under the condition ml + d - 1 +2c~=s~ > s > s 2 = 2 m - m l
+ d + 1-20¢.
To obtain that expression, it is necessary to account the mutual annihilation of the divergences in ( 1 ), which add the factors k? at kt-~oo and ki -2~ at k l ~ 0 , k. Therefore, for the existence of a Kolmogorov-like power spectrum one should demand s~ > s2 which means ml-m+2c~-
1> 0 .
(5)
Under such a condition there exists on interval (s~, s2) of indices which provide interaction locality for isotropic spectra. It is curious to note that the Kolmogorov index lies quite in the middle of the locality interval: u = ½(s~ +s2). Such a fitting seems to be closely connected with the fact that the contribution of all scales (from small to large ones) to the interaction are strictly balanced on the Kolmogorov solution (this is also proven for fluid turbulence [ 7 ] and for the general case of weak turbulence [ 9 ] ). However, the main contribution to the collision integral is provided by the region k~ ~ k which just allows one to speak about the interaction locality. Let us turn to the problem of the interaction locality for small perturbations of a Kolmogorov-like spectrum. To do that, it is necessary to study the convergence of the collision integral linearized against the background of the Kolmogorov solution. Supposing n(k, t) = n ( k ) + fin(k, t), tSn(k, t) << n(k), we obtain the linearized equation 1 O~n(k, t)
A
Ot
- f { Uk,2[ (kF ~ - k -~) ~n(k, ) + (k~-~ - k -~) ~n(kz) - (k7 v +k~ ~) 8n(k) ]
× U~k2[ ( k - ~ - k i -v) ~n (k2) + (k~ v + k -v) ~Sn(k~ ) + (k~ v - k ~ v) 6n(k) ]} elk, dk2.
(6)
Unlike the isotropic case, the cancellation of divergences arises in powers of k rather than mk for general (not even) perturbations. For even perturbations cancellation in powers of k 2 occurs. Therefore, since three-wave processes are allowed for the frequency index a larger than unity, the locality interval for arbitrary perturbation is shortened by the value 2 ( c e - 1 ). Indeed, for ~3n(k)=k-~f(k/k), we find the convergence condition in the form
m~ +d+o/.-s3 >S>$4 = 2 m - - m l + d - c ~ .
(7)
The following inequality is thus necessary for the existence of the locality interval, ml-m+ot>0,
(8)
which is more restrictive than (5) since ot > 1. I f condition (8) is satisfied, then the interaction is local for perturbations which decrease faster than k -s' at k ~ and increase slower than k -s~ at k ~ 0 . If condition (8) is invalid for a given system, then the interaction with largest and smallest motions affects the behavior of quite arbitrary small perturbations. Such a property is referred to as evolutionary nonlocality [ 5,6 ]. Let us turn to the consideration of the specific system for which inequality (5) is satisfied while (8) is not.
3. T h e turbulence of capillary waves on s h a l l o w water
The well-known expression for the dispersion law of surface waves, c02(k) = t a n h kh (gk+ak3/p) 129
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for the intermediate wave numbers
17 August 1992
x/pg/a<< k<< h - ~ can be written as follows,
0)2= ah k4 " P
(9)
Here, co, k are the frequency and wave number, p, a, h are the density, the surface tension and the depth of the fluid. The dispersion law (9) allows three-wave processes: co(kl + k2) = co(kl) + 0) (k2). The coefficient of triple interaction was obtained in ref. [ 8] and has the extremely simple form k4 ~/o"
VZ(k, kl, k 2 ) - 64n2
4ph"
(10)
So in this case, c¢=2, m = 2 , rnl = 0 and d = 2 . The following Kolmogorov-like stationary solution (4) has been found in ref. [81,
n(k) = 8pl/2k -4(phg2/ff)1/4
•
(
1 1)
According to (5), that solution is local. Let us consider the stability problem for the Kolmogorov-like spectrum ( 11 ). Supposing perturbations to be small, we arrive at the linearized kinetic equation (6). That equation can be considered separately for each angular harmonic. Expanding 8n(k, t) = ~ nt(k, t)e it° we get
l Onl(k,t) 2
O~
n / ( k ) ( [ k4ki-4 \~ ~
-
k5[(k2--k2)--2--k14]
dkl +
kx/-~l-k 2
dk,
k k
+ J cos(larccos(k,/k)]n,(kl )[ ( k 2 - k 2 ) - 2 - k -4]
k 4 dkl
0
dkl + i cos[larccos(k/kl)]nt(k~)[(k~-kZ)-2+k -4] k~k2_k2 k oo
+cos(½/n)
f n l ( x / ~ - k 2 ) ( k v 4 - k -4) kx/-~_k k~ dkl 2 .
(12)
k
One can directly verify that for even harmonics the divergences in (12) disappear for nl(k)ock-" with 3 < Re s < 5, which agrees with (5). Due to the presence of the locality interval, for solving the stability problem one can use the stability theory developed in refs. [5,6,9 ]. For this aim, it is necessary to represent nt with the help of the Mellin transform,
nt(k, t) = ~ at(s) exp[ W(s)t]k -s ds. Here the contour y goes from - i ~ to + ioo in the strip 3 < Re s < 5. The temporal behavior of the perturbations is governed by the function W(s) which is the MeUin image of the operator of the lineafized kinetic equation. It can be computed by direct integration. For example, f o r / = 0 it is
Wo(s)=B(½(1-s),-3)-B(½(I-s),½)+B(½(s-5),½)+B(½(s-1), 3 ) . Here B(x, y) is the Euler beta function. As one can see, Wo(4) = 0 which corresponds to the neutral stability of the Kolmogorov-like spectrum ( 11 ) with respect to the variation of the flux value. The derivative d Wo ( s ) / ds at s = 4 yields the group velocity of the perturbations [9 ] which is positive in this case. So the isotropic perturbations drift to the large k's (damping region), i.e. in the same direction as an energy flux carried by 130
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spectrum ( 11 ). According to ref. [ 5 ], solution ( 11 ) is thus stable with respect to isotropic perturbations. By using the well-known representation cos(larccosx) = ( - (2l1 )tlx/i--Z~-x2 dl l ( 1 - x 2 ) l - l / 2 ' 1)!! dx one can obtain a similar expression in terms of B( ½( 1 - s - l), 1+ ½) etc. for Wt with even I and prove the stability. (One can directly verify that Re W/(s)<0 for Re s = 4 which guarantees the absence of both absolute and convective instabilities [ 5,8 ]. ) Let us turn to odd harmonics. The absence of the locality strip for index s can be seen yet from (8) since m t - m + a = 0 in this case. Indeed, considering the main divergences in ( 12 ), we get for nt (k, t) ozk -s and for l = 2 j + 1: oo
0--[ oc ( - 1 ylk-4
k3~-s dk~ +
k 3-s dk~
.
(13)
0
Whatever Re s, the linearized collision integral diverges for the perturbations of power type. On the face of it, it might seem that only mathematical difficulty rejected the use of the Mellin (or Fourier) transform, while for any finite ~n(k, t) (such that ~n(k, t)/n(k)--,O as k ~ 0 , ~ ) the time derivative is finite at any k. Nevertheless, the time derivatives of the perturbation moments oo
Mr(s, t ) = ~ kS-~nt(k, t) dk 0
go to infinity. Strictly speaking, from the physical viewpoint integrals should be cut off on the left, ko (wave source), and right, km (sink), boundaries of the so-called inertial interval where the Kolmogorov-like spectrum can occur. So it is more meaningful to speak about large time derivatives than about infinite ones. Anyway, for the time derivative of the moment we obtain km
Ot
-(-1)JlM~(4't)
(f..
dk+2
f
ko
kS-Sdk
)
.
(14)
One can see the twice odd moments (with l = 2 j + 1 a n d j = 2 i + 1 ) to be quickly diminishing with time, so the spectrum ( 11 ) is stable with respect to those perturbations. On the contrary, moments with l---4i+ 1 quickly grow which corresponds to the instability of the isotropic spectrum ( 11 ). That some moment goes to infinity (or to a large value defined by cut off scales) means the arising of power asymptotics at k--*0, ~ (or at k~ko, kin). Since every moment is proportional to M ( 4 ) it is easy to see that the asymptotic of the perturbation arises in the form nt(k, t)--*k -4 at k--,0, oo. Indeed, to have the asymptotics k - 4 is the only way to satisfy the moment relations
M(s, t ) / M ( 4 , t)ock~-4,
s>4,
ock~-4,
s<4,
following from (14). Therefore, strongly changing the angular shape of the spectrum, instability preserves the Kolmogorov-like k-dependence corresponding to the cascade transfer of energy. The nonlocal character of the interaction of the anisotropic perturbations means that momentum is nonlocally transferred from source to sink.
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References [ 1 ] A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30 ( 1941 ) 299. [2] A.M. Obukhov, Izv. Akad. Nauk SSSR 4 ( 1941 ) 453. [ 3 ] A.S. Monin and A.M. Yaglom, Statistical fluid mechanics, Vol. 2 (MIT Press, Cambridge, MA, 1975 ). [ 4 ] M. Lesieur, Turbulence in fluids (Kluwer, Dordrecht, 1990). [ 5 ] V.E. Zakharov, V.S. Lvov and G.E. Falkovich, Kolmogorov spectra of developed turbulence, Vol. 1 (Springer, Berlin, 1991 ). [6] A.M. Balk and V.E. Zakharov, in: Integrability and kinetic equations for solitons (Naukova Dumka, Kiev, 1990) [in Russian]. [ 7 ] V. Lvov and G. Falkovich, Phys. Rev, to be published. [8] A.V. Kats and V.M. Kontorovich, Zh. PriN. Mekh. Tekh. Fiz. 6 (1974) 97. [9] G.E. Falkovich, Soy. Phys. JETP 66 (1987) 97.
132
PHYSICS LETTERS A
Nonlocal angular instability of a Kolmogorov-like wave turbulence spectrum G.E. F a l k o v i c h Department of Nuclear Physics, Weizmann Institute of Science, Rehovot 76100, Israel and Institute of Automation, Siberian Division of the USSR Academy of Sciences, Novosibirsk, Russian Federation
and M . D . Spector Department of Fluid Mechanics and Heat Transfer, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel Received 19 May 1992; revised manuscript received 4 June 1992; accepted for publication 5 June 1992 Communicated by D.D. Holm
A new type of instability of Kolmogorov-like wave turbulence spectra is found. Such an instability is due to an interaction nonlocal in k-space and it strongly modifies the angular structure of the turbulence spectrum. However, the spectrum dependence on the modulus k is still a Kolmogorov-like one corresponding to energy transfer local in k-space. The specific case of capillary waves on shallow water is considered in detail. It is shown that the energy transfer is local while that of m o m e n t u m is nonlocal in k-space.
1. Introduction The picture of cascade turbulence was suggested by Richardson, Kolmogorov and Obukhov for incompressible fluids [ 1-4 ] and it was later extended to wave turbulence in other media [ 5 ]. It is based on the concept of interaction locality. That means that those modes (vortices or waves) effectively interact which are of comparable scales only. The question naturally arises: is the locality property satisfied on the steady Kolmogorovlike spectrum only or on slightly differing distributions as well? Proceeding from continuity-like speculations, one might suppose that in the general case interaction locality for the Kolmogorov distribution leads to that for close ones. Such a supposition is, however, incorrect since the Kolmogorov spectrum usually possesses a higher degree of symmetry (for example, being isotropic) than arbitrary yet close distributions. A stationary locality does not mean thus an evolutionary locality as it was stated in refs. [5,6]. What can we say about the properties of a system with a local Kolmogorov-like spectrum and with perturbations interacting nonlocally? Our paper deals with that matter. We demonstrate here that nonlocality can lead to either the growth or the decreasing of perturbation, depending on the number of the angular harmonic. The growth of perturbation means the instability of the stationary isotropic spectrum with respect to that harmonic. Nevertheless, we demonstrate the possibility of a Kolmogorov-like spectrum to survive on average which means that, despite the steady spectrum being anisotropic, the angle averaging spectrum decreases with the modulus k by the Kolmogorov law. The energy is thus transferred locally. As far as momentum is concerned, it can be transferred nonlocally in k-space. We give our consideration using weakly nonlinear wave turbulence as an example which allows analytical treatment and is of physical interest itself. Section 1 deals with the kinetic equation which describes weak wave 0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
127
Volume 168, number 2
PHYSICS LETTERSA
17 August 1992
turbulence. Here we demonstrate the difference between stationary and evolutionary locality. In section 2 we consider a specific example, viz. capillary waves on shallow water and demonstrate that there exist stationary locality and evolutionary nonlocality of the Kolmogorov-like spectrum in this case. We prove that there is a local steady isotropic spectrum stable with respect to local even angular harmonics of perturbation. Odd harmonics are nonlocal: an arbitrary perturbation strongly interacts with both small and large k's. The angular harmonic with the number l= 2j+ 1 decreases for o d d j and increases for evenj. The Kolmogorov-like spectrum is thus unstable with respect to the angular harmonics with the numbers l = 4 i + l, i=0, l, .... So the steady spectrum should be anisotropic. Nevertheless, we prove the k-dependence of the spectrum to be Kolmogorovlike, so the energy is transferred locally (by a cascade process) while the momentum is transferred nonlocally. Our consideration is restricted to weak turbulence (which is the set of weakly interacting waves) and cannot be applied directly for describing turbulence with strong interaction, for instance, in an incompressible fluid. This latter case seems to correspond to both stationary and evolutionary locality according to ref. [7].
2. Stationary and evolutionary locality If the level of wave excitation is small enough to provide the weakness of nonlinear effects compared with those of dispersion, the turbulence is regarded as weak and can be described with the help of a kinetic equation. Such an equation is formulated in terms of pair correlators, called also wave occupation numbers, in k-space, ( a (k) a* (k') ) = n (k) ~ ( k - k' ), where a (k) are the wave amplitudes. With a small level of nonlinearity the three-wave processes are sufficient to restrict ourselves to, which is allowed if the dispersion law o~(k) admits the decay equality o ~ ( k + k ' ) = ~o(k)+ to(k'). In this case, the kinetic equation reads as follows,
On(k,
Ot
=I(k, t)= f Uk,2(n, n2--nkn,--nkn2) dk, dk2-2 f U~k2(nkn2--nknl--n, n2) dk~ dk2.
(1)
Here Uk~2= V2~2~( k - k~ - k2) ~( 0 4 - ~0~- 0)2) and Vk12is the matrix element of the three-wave interaction. The g-functions in ( 1 ) express the conservation laws of the values of the energy E=fcokn(k) dk and of the momentum H = fkn (k) dk in every elementary act of interaction. Consequently, the kinetic equation (1) can be written down, e.g., in the form of the continuity equation for the wave energy. For isotropic distributions, one can introduce the quantity e (k, t ) = (2k)d-lo)ktl (k, t), representing the density of the wave energy in k-space, so the kinetic equation can be represented as follows,
O~(k, t) OP(k, t) + _o. Ot Ok
(2)
The value of k
P(k, t) = ~ (2k)d-ltOkI(k, t) dk
(3)
0
thus represents the density of the energy flux (again in the spherical normalization), d being the dimension of k-space. The stationary turbulent distribution n (k), as is readily seen from (2), corresponds to a flux constancy. In the case of scale invariance of the wave system implying that ~o(/*k)=/z"to(k), V(/.tk, #k~, l~k2) = I*mV(k, kl, k2), a stationary solution of eqs. (1), (2) turns out to be the power n (k) =Ak-". The index v corresponds to the energy flux constancy and can be found from (3): v = m + d. The constant A, defined by the condition P ( k ) = P , is proportional to pw2. Distributions of the type of
n( k )ocpw2k -m-d
(4)
are called Kolmogorov-like spectra of weak wave turbulence in direct analogy with those for turbulence of an 128
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17 August 1992
incompressible fluid. Such distributions are stationary solutions of the kinetic equation provided that the integrals involved converge. Let the matrix element have the following asymptotics, V2(k, k~, kz) ozk'~ k 2m-'m for k~ << k. Then it is easy to show that the collision integral in ( 1 ) defined on the power distributions n(k)ock -s converge under the condition ml + d - 1 +2c~=s~ > s > s 2 = 2 m - m l
+ d + 1-20¢.
To obtain that expression, it is necessary to account the mutual annihilation of the divergences in ( 1 ), which add the factors k? at kt-~oo and ki -2~ at k l ~ 0 , k. Therefore, for the existence of a Kolmogorov-like power spectrum one should demand s~ > s2 which means ml-m+2c~-
1> 0 .
(5)
Under such a condition there exists on interval (s~, s2) of indices which provide interaction locality for isotropic spectra. It is curious to note that the Kolmogorov index lies quite in the middle of the locality interval: u = ½(s~ +s2). Such a fitting seems to be closely connected with the fact that the contribution of all scales (from small to large ones) to the interaction are strictly balanced on the Kolmogorov solution (this is also proven for fluid turbulence [ 7 ] and for the general case of weak turbulence [ 9 ] ). However, the main contribution to the collision integral is provided by the region k~ ~ k which just allows one to speak about the interaction locality. Let us turn to the problem of the interaction locality for small perturbations of a Kolmogorov-like spectrum. To do that, it is necessary to study the convergence of the collision integral linearized against the background of the Kolmogorov solution. Supposing n(k, t) = n ( k ) + fin(k, t), tSn(k, t) << n(k), we obtain the linearized equation 1 O~n(k, t)
A
Ot
- f { Uk,2[ (kF ~ - k -~) ~n(k, ) + (k~-~ - k -~) ~n(kz) - (k7 v +k~ ~) 8n(k) ]
× U~k2[ ( k - ~ - k i -v) ~n (k2) + (k~ v + k -v) ~Sn(k~ ) + (k~ v - k ~ v) 6n(k) ]} elk, dk2.
(6)
Unlike the isotropic case, the cancellation of divergences arises in powers of k rather than mk for general (not even) perturbations. For even perturbations cancellation in powers of k 2 occurs. Therefore, since three-wave processes are allowed for the frequency index a larger than unity, the locality interval for arbitrary perturbation is shortened by the value 2 ( c e - 1 ). Indeed, for ~3n(k)=k-~f(k/k), we find the convergence condition in the form
m~ +d+o/.-s3 >S>$4 = 2 m - - m l + d - c ~ .
(7)
The following inequality is thus necessary for the existence of the locality interval, ml-m+ot>0,
(8)
which is more restrictive than (5) since ot > 1. I f condition (8) is satisfied, then the interaction is local for perturbations which decrease faster than k -s' at k ~ and increase slower than k -s~ at k ~ 0 . If condition (8) is invalid for a given system, then the interaction with largest and smallest motions affects the behavior of quite arbitrary small perturbations. Such a property is referred to as evolutionary nonlocality [ 5,6 ]. Let us turn to the consideration of the specific system for which inequality (5) is satisfied while (8) is not.
3. T h e turbulence of capillary waves on s h a l l o w water
The well-known expression for the dispersion law of surface waves, c02(k) = t a n h kh (gk+ak3/p) 129
Volume 168, number 2
PHYSICS LETTERSA
for the intermediate wave numbers
17 August 1992
x/pg/a<< k<< h - ~ can be written as follows,
0)2= ah k4 " P
(9)
Here, co, k are the frequency and wave number, p, a, h are the density, the surface tension and the depth of the fluid. The dispersion law (9) allows three-wave processes: co(kl + k2) = co(kl) + 0) (k2). The coefficient of triple interaction was obtained in ref. [ 8] and has the extremely simple form k4 ~/o"
VZ(k, kl, k 2 ) - 64n2
4ph"
(10)
So in this case, c¢=2, m = 2 , rnl = 0 and d = 2 . The following Kolmogorov-like stationary solution (4) has been found in ref. [81,
n(k) = 8pl/2k -4(phg2/ff)1/4
•
(
1 1)
According to (5), that solution is local. Let us consider the stability problem for the Kolmogorov-like spectrum ( 11 ). Supposing perturbations to be small, we arrive at the linearized kinetic equation (6). That equation can be considered separately for each angular harmonic. Expanding 8n(k, t) = ~ nt(k, t)e it° we get
l Onl(k,t) 2
O~
n / ( k ) ( [ k4ki-4 \~ ~
-
k5[(k2--k2)--2--k14]
dkl +
kx/-~l-k 2
dk,
k k
+ J cos(larccos(k,/k)]n,(kl )[ ( k 2 - k 2 ) - 2 - k -4]
k 4 dkl
0
dkl + i cos[larccos(k/kl)]nt(k~)[(k~-kZ)-2+k -4] k~k2_k2 k oo
+cos(½/n)
f n l ( x / ~ - k 2 ) ( k v 4 - k -4) kx/-~_k k~ dkl 2 .
(12)
k
One can directly verify that for even harmonics the divergences in (12) disappear for nl(k)ock-" with 3 < Re s < 5, which agrees with (5). Due to the presence of the locality interval, for solving the stability problem one can use the stability theory developed in refs. [5,6,9 ]. For this aim, it is necessary to represent nt with the help of the Mellin transform,
nt(k, t) = ~ at(s) exp[ W(s)t]k -s ds. Here the contour y goes from - i ~ to + ioo in the strip 3 < Re s < 5. The temporal behavior of the perturbations is governed by the function W(s) which is the MeUin image of the operator of the lineafized kinetic equation. It can be computed by direct integration. For example, f o r / = 0 it is
Wo(s)=B(½(1-s),-3)-B(½(I-s),½)+B(½(s-5),½)+B(½(s-1), 3 ) . Here B(x, y) is the Euler beta function. As one can see, Wo(4) = 0 which corresponds to the neutral stability of the Kolmogorov-like spectrum ( 11 ) with respect to the variation of the flux value. The derivative d Wo ( s ) / ds at s = 4 yields the group velocity of the perturbations [9 ] which is positive in this case. So the isotropic perturbations drift to the large k's (damping region), i.e. in the same direction as an energy flux carried by 130
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spectrum ( 11 ). According to ref. [ 5 ], solution ( 11 ) is thus stable with respect to isotropic perturbations. By using the well-known representation cos(larccosx) = ( - (2l1 )tlx/i--Z~-x2 dl l ( 1 - x 2 ) l - l / 2 ' 1)!! dx one can obtain a similar expression in terms of B( ½( 1 - s - l), 1+ ½) etc. for Wt with even I and prove the stability. (One can directly verify that Re W/(s)<0 for Re s = 4 which guarantees the absence of both absolute and convective instabilities [ 5,8 ]. ) Let us turn to odd harmonics. The absence of the locality strip for index s can be seen yet from (8) since m t - m + a = 0 in this case. Indeed, considering the main divergences in ( 12 ), we get for nt (k, t) ozk -s and for l = 2 j + 1: oo
0--[ oc ( - 1 ylk-4
k3~-s dk~ +
k 3-s dk~
.
(13)
0
Whatever Re s, the linearized collision integral diverges for the perturbations of power type. On the face of it, it might seem that only mathematical difficulty rejected the use of the Mellin (or Fourier) transform, while for any finite ~n(k, t) (such that ~n(k, t)/n(k)--,O as k ~ 0 , ~ ) the time derivative is finite at any k. Nevertheless, the time derivatives of the perturbation moments oo
Mr(s, t ) = ~ kS-~nt(k, t) dk 0
go to infinity. Strictly speaking, from the physical viewpoint integrals should be cut off on the left, ko (wave source), and right, km (sink), boundaries of the so-called inertial interval where the Kolmogorov-like spectrum can occur. So it is more meaningful to speak about large time derivatives than about infinite ones. Anyway, for the time derivative of the moment we obtain km
Ot
-(-1)JlM~(4't)
(f..
dk+2
f
ko
kS-Sdk
)
.
(14)
One can see the twice odd moments (with l = 2 j + 1 a n d j = 2 i + 1 ) to be quickly diminishing with time, so the spectrum ( 11 ) is stable with respect to those perturbations. On the contrary, moments with l---4i+ 1 quickly grow which corresponds to the instability of the isotropic spectrum ( 11 ). That some moment goes to infinity (or to a large value defined by cut off scales) means the arising of power asymptotics at k--*0, ~ (or at k~ko, kin). Since every moment is proportional to M ( 4 ) it is easy to see that the asymptotic of the perturbation arises in the form nt(k, t)--*k -4 at k--,0, oo. Indeed, to have the asymptotics k - 4 is the only way to satisfy the moment relations
M(s, t ) / M ( 4 , t)ock~-4,
s>4,
ock~-4,
s<4,
following from (14). Therefore, strongly changing the angular shape of the spectrum, instability preserves the Kolmogorov-like k-dependence corresponding to the cascade transfer of energy. The nonlocal character of the interaction of the anisotropic perturbations means that momentum is nonlocally transferred from source to sink.
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References [ 1 ] A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30 ( 1941 ) 299. [2] A.M. Obukhov, Izv. Akad. Nauk SSSR 4 ( 1941 ) 453. [ 3 ] A.S. Monin and A.M. Yaglom, Statistical fluid mechanics, Vol. 2 (MIT Press, Cambridge, MA, 1975 ). [ 4 ] M. Lesieur, Turbulence in fluids (Kluwer, Dordrecht, 1990). [ 5 ] V.E. Zakharov, V.S. Lvov and G.E. Falkovich, Kolmogorov spectra of developed turbulence, Vol. 1 (Springer, Berlin, 1991 ). [6] A.M. Balk and V.E. Zakharov, in: Integrability and kinetic equations for solitons (Naukova Dumka, Kiev, 1990) [in Russian]. [ 7 ] V. Lvov and G. Falkovich, Phys. Rev, to be published. [8] A.V. Kats and V.M. Kontorovich, Zh. PriN. Mekh. Tekh. Fiz. 6 (1974) 97. [9] G.E. Falkovich, Soy. Phys. JETP 66 (1987) 97.
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