Finite-dimensional turbulence of planetary waves

Finite-dimensional turbulence of planetary waves

Citation for published version (APA):

L'vov, V. S., Pomyalov, A., Procaccia, I., & Rudenko, O. (2009). Finite-dimensional turbulence of planetary waves. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 80(6), 066319-1/25. [066319]. https://doi.org/10.1103/PhysRevE.80.066319

DOI:

10.1103/PhysRevE.80.066319

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Finite-dimensional turbulence of planetary waves

Victor S. L’vov,

*

Anna Pomyalov, Itamar Procaccia, and Oleksii Rudenko Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel

共Received 12 June 2009; published 30 December 2009兲

Finite-dimensional wave turbulence refers to the chaotic dynamics of interacting wave “clusters” consisting

of finite number of connected wave triads with exact three-wave resonances. We examine this phenomenon using the example of atmospheric planetary共Rossby兲 waves. It is shown that the dynamics of the clusters is determined by the types of connections between neighboring triads within a cluster; these correspond to substantially different scenarios of energy flux between different triads. All the possible cases of the energy cascade termination are classified. Free and forced chaotic dynamics in the clusters are investigated: due to the huge fluctuations of the energy exchange between resonant triads these two types of evolution have a lot in common. It is confirmed that finite-dimensional wave turbulence in finite wave systems is fundamentally different from kinetic wave turbulence in infinite systems; the latter is described by wave-kinetic equations that account for interactions with overlapping quasiresonances of finite amplitude waves. The present results are directly applicable to finite-dimensional wave turbulence in any wave system in finite domains with three-mode interactions as encountered in hydrodynamics, astronomy, plasma physics, chemistry, medicine, etc.

DOI:10.1103/PhysRevE.80.066319 PACS number共s兲: 47.27.ed, 92.60.hk, 47.32.⫺y

I. INTRODUCTION

A. Weak-wave turbulence in finite-size systems “Wave turbulence” refers to the chaotic dynamics of non-linearly coupled oscillatory modes 关1–5兴. The phenomenon

appears in a variety of physical context from surface water waves, through atmospheric planetary waves, plasma waves, acoustic waves in solids and fluids etc. Depending on the strength of nonlinear interaction one distinguishes weak-wave turbulence from strong-weak-wave turbulence. Weak-weak-wave turbulence is characterized by a smallness parameter␨which is roughly the root-mean square of the ratio of the nonlinear to the linear term in the equation of motion. For surface waves␨is about the ratio of the wave amplitude to the wave-length␭, for sound in continues media this is the ratio of the density variations to the mean density, etc.

The theory of weak-wave turbulence is particularly well developed in the limit of infinite systems where the ratio of the system size L to the characteristic wavelength ␭ is very large, L/␭→⬁. In that limit the observed energy spectrum 共energy distribution between modes兲 is well described by the so called “wave-kinetic equations” that received considerable attention in the last half century, see, e.g.,关1–9兴. We will call

this regime of weak wave turbulence “kinetic wave turbu-lence” to distinguish it from other regimes, “finite-dimensional wave turbulence” and “mesoscopic wave turbu-lence” that will be introduced later.

Notice that when the parameter L/␭ is of the order of unity the dynamics of waves can be very well described by low-dimensional chaotic models of the type studied inten-sively in recent decades, see e.g.,关10–13兴. In this paper we

explore the nonlinear dynamics of weakly interacting waves when the parameter L/␭ is neither of order unity nor very large. This regime of parameters cannot be described either by kinetic equations or as low-dimensional chaos; it calls for

new approaches and novel concepts, as partially demon-strated in this paper.

To clarify the possible new regimes of weak-wave turbu-lence in various finite-domain systems we consider the geral mathematical framework that takes the form of an en-ergy conserving partial differential nonlinear equation for a field ⌿共r,t兲. In a finite domain S one expands ⌿共r,t兲 in a complete set of eigenfunctions ⌽j共r兲 of the linearized

dy-namics that satisfy the boundary condition on the boundary

⳵S,

⌿共r,t兲 =

兺

j

Aj共t兲⌽j共r兲, 共1兲

where in general j can be a multiple index and the ampli-tudes Aj共t兲 are functions of time but not of space.

Accord-ingly the dynamics can be represented by a set of ordinary differential equations for the vector of amplitudes A共t兲 =兵Aj共t兲其, of the form

dAj共t兲

dt = i␻jAj共t兲 + NLj共A兲, 共2兲 where ␻j is the 共real兲 eigenfrequency of the jth mode; The

symbolic term NL stands for the nonlinear contributions in this equation. According to the Poincaré and Poincaré-Dulac theorems关14兴 the nonlinear contributions can be brought to a

normal form by a nonlinear change of variables. The nonlin-ear monomials that survive the change of variable are the resonant ones. The n-tuple共␻1, . . .␻n兲 of eigenfrequencies is

said to be resonant if there exists a relation of the form

␻j= m1␻1+ m2␻2+ . . . mn␻n. 共3兲

The order of the resonance is兺kmk. The resonant monomials

are of the form A₁m1_{. . . A} n

mn_.

For the case of weak nonlinearities we invoke the small-ness parameter ␨Ⰶ1 to discard all the higher order reso-nances, keeping only the lowest available order. In the atmo-sphere the condition ␨Ⰶ1 is applicable when the pressure *victor.lvov@weizmann.ac.il

variation due to Rossby waves is much smaller than the mean pressure. When this condition is applicable, and there exist solutions to the equation

␻j=␻m+␻n, 共4a兲

we keep only the resulting quadratic monomials which are also known as “three-wave interactions,” which satisfy the conservation law共4a兲. For example, in space-homogeneous,

scale-invariant, isotropic, infinite wave systems, in which the dependence of the wave frequency␻共k兲 on the wave vector k⬅兩k兩,␻共k兲⬀k␣, the three wave resonances

␻共k1兲 +␻共k2兲 =␻共兩k1+ k2兩兲, 共4b兲 are allowed if␣ⱖ1 关5兴. For␣⬍1 Eq. 共4b兲 has no solutions

and one needs to account for higher order resonances. In this paper we focus on problems for which Eq.共4a兲 has

solutions, determining the leading nonlinearity. We note that as the ratio L/␭ increases, there may be more and more eigenfrequencies that satisfy Eq. 共4a兲. In particular, while at

small values of L/␭ we can expect only isolated resonant triads of waves, for larger values of L/␭ triads can share a common mode and the number of coupled resonant triads increases considerably, finally forming infinite clusters of connected triads. Analysis of the ensuing dynamics under the influence of such growing clusters is the main subject of the present paper. We will focus here on the case of small enough nonlinearity parameter ␨ to ensure that only waves with exact resonances are important. In this case one can consider only clusters of connected resonant triads of acting waves. We will refer to the chaotic dynamics of inter-acting waves in this regime as “finite-dimensional wave tur-bulence,” to stress the importance of the finite number of interacting modes with exact wave resonances. With increas-ing of wave amplitudes one has to account also for qua-siresonances. This type of wave turbulence was called “dis-crete wave turbulence”关15兴. In contrast, in infinite systems,

the resonance conditions 关e.g., Eq. 共4b兲兴 has infinitely many

solutions; then usually the共kinetic兲 wave turbulence can be described by wave-kinetic equations. A more detailed analy-sis关16兴 shows that in the plane 共L/␭,␨兲 there exists a region

of parameters where there exists weak-wave turbulence whose properties are intermediate between finite-dimensional and kinetic regimes. Some features of this type of turbulence, called “mesoscopic wave turbulence,” were observed, for example, in关8,17兴.

To study finite-dimensional wave turbulence we focus here for concreteness on the example of the barotropic vor-ticity equation on a sphere; this is an idealized model for atmospheric planetary 共Rossby兲 waves 关11,18兴, shortly

de-scribed in Sec. II. Planetary-scale motions in the ocean and atmosphere are due to the shape and rotation of the Earth, and play a crucial role in weather and climate predictability 关18兴. Oceanic planetary waves influence the general

large-scale ocean circulation, can intensify the currents such as the Gulf Stream, as well as push them off their usual course. For example, a planetary wave can push the Kuroshio Current northwards and affect the weather in North America 关19兴.

Atmospheric planetary waves detach the masses of cold or warm air that become cyclones and anticyclones and are

re-sponsible for day-to-day weather patterns at midlatitudes 关20兴.

Recently a new model 关12兴 was developed for the

in-traseasonal oscillations in the Earth atmosphere, in terms of triads of planetary waves whose eigenfrequencies solve Eq. 共4a兲. The study of the complete cluster structure in various

spectral domains shows that both for atmospheric 关21兴 and

oceanic关22兴 planetary waves indeed the size of the clusters

increases with the growth of the spectral domain. In large clusters one finds both large and small wavenumbers, mean-ing that the energy flux between very different scales be-comes possible, bringing with it the hallmark of turbulence. Nevertheless, both numerical simulations 关17,23兴 and

labo-ratory experiments关24兴 indicate that the dynamics of wave

systems with intermediate value of L/␭ do not obey the sta-tistical description provided by wave-kinetic equations. This domain calls for a specialized investigation which is initiated in this paper.

B. Structure of this paper

SectionIIreviews the properties of atmospheric planetary waves which are important for our analysis: in Sec. II Awe consider barotropic vorticity Eq.共5兲 on a sphere and its

dy-namical invariants关Eq. 共6兲兴; in Sec.II Bwe project Eqs.共5兲

and 共6兲 on the spherical basis; and in Sec.II C we analyze the properties of the resulting interaction coefficients.

In Sections III and IVwe study the topology and other properties of finite size clusters of resonant triads of plan-etary waves that influence the dynamics of finite-dimensional wave turbulence. In Sec.IIIwe begin with small clusters of resonant triads. In Sec. III A we overview equation of mo-tion共14兲 for isolated resonant triad and its dynamical

invari-ants 关Eq. 共15兲兴, present them in the Hamiltonian form 共20兲,

and use the notion of “active” and “passive” modes 关13兴

denoted as A and P modes; these notions are crucial for our theory. In Sec.III Bwe move on to double-triad clusters, PP, PA, and AA butterflies, their Hamiltonian equation of motion 共25兲, Hamiltonian 共26兲 and Manley-Rowe dynamical

invari-ants 关Eq. 共27兲兴. Similar discussion for the triple-triad

clus-ters, stars, chains, and triangles, is given in Sec.III C. Spe-cific for atmospheric planetary waves, six-triad cluster, caterpillar, is presented in Sec. III D as an example of a more complicated cluster structure.

In Sec. IV we study clusters of atmospheric planetary waves in a large spectral domainᐉ,兩m兩ⱕ1000, presenting in Sec.IV Athe total number of clusters consisting of one, two, three, etc. triads with different topologies. The histogram of all cluster distributions with respect to the triad number in clusters, ranging from 1 to 3691 is also presented. In Sec.

IV Bwe use a notion of PP-irreducible clusters关13兴

共impor-tant for the discussion of the energy flux in finite-dimensional wave turbulence兲 and show the histogram of their distribution in size, ranging from 1 to 130 triads in the PP-irreducible clusters.

Sections V and VI are devoted to numerical simulation and preliminary analytical studies of finite-dimensional wave turbulence in clusters typical for numerous physical systems including the planetary waves described in Secs.IIIandIV.

In Sec. V we begin with the analysis of free evolution in small clusters: butterflies in Sec.V Band triple-triad clusters 共stars and triple chains兲 in Sec.V C. The main questions, that we discuss in this section are:

共i兲 reasonable choice of initial conditions, interaction co-efficients and data representation that allows to shed light on the typical features of finite-dimensional wave turbulence, that depend on many parameters;

共ii兲 how the energy flux between triads depends on the type of connections, on the interaction coefficients, on the initial conditions and on the cluster topology.

In Sec. VI we study finite-dimensional wave turbulence with a constant energy flux in the long-chain clusters, con-sisting of a large number of triads共NⰇ1兲. For this goal we introduce pumping of energy into the leading共first兲 triad and damping in the driven 共last兲 triad. We discuss in Sec. VI A

how to mimic the energy pumping and energy damping in our particular problem and what are the necessary conditions of stationarity, Sec. VI B. In Sec. VI C we show that the distribution of mode amplitudes in long chains is universal in the following sense: it is asymptotically independent of num-ber of triads in the chain in the limit of large N and of interaction coefficients共in a wide region of their definition兲. Moreover, the distribution for the forced case practically co-incides with that for free evolution from initial conditions, corresponding to the forced stationary case.

In Sec.VII A we briefly summarize the main features of finite-dimensional wave turbulence discovered in this paper and formulate in Sec.VII Bsome important questions in this field that remain unstudied. Our feeling is that the present paper presents many more questions than answers, and all these questions 共and many other related ones兲 belong to a new field of study of weak-wave turbulence: finite-dimensional and mesoscopic wave turbulence in finite-size physical systems.

II. ATMOSPHERIC PLANETARY WAVES A. Barotropic vorticity equation on a sphere

Planetary waves in the atmosphere pose a rich and com-plicated problem which is influenced by the earth topogra-phy, the vertical temperature profiles共varying between land and ocean兲, global winds etc. We do not attempt here to take into account all this richness. The essence of the interesting dynamics can be gleaned from simplified models. A very simplified model of atmospheric planetary waves共discussed in pioneering works by Silberman 关4,11兴 and by Reznik,

Piterbarg, and Kartashova关11兴兲 is provided by the barotropic

vorticity equation on a rotating sphere for the dimensionless stream function ␺共␪,␸, t兲. The variables t,␪, and␸ are the time, the latitude 共−␲/2ⱕ␪ⱕ␲/2兲 and longitude 共0ⱕ␸ ⱕ2␲兲 on the sphere. The equation reads 关4,11兴

⌬⳵␺ ⳵t = 1 sin␪

冉

⳵␺ ⳵␸ ⳵ ⳵␪− ⳵␺ ⳵␪ ⳵ ⳵␸

冊

共2⍀ cos␪+⌬␺兲, ⌬ = 1 sin␪

冋

⳵ ⳵␪

冉

sin␪ ⳵ ⳵␪

冊

+ 1 sin␪ ⳵2 ⳵␸2

册

, 共5兲

where⌬ is the angular part of the spherical Laplacian opera-tor. The stream function gives rise to the velocity v =⍀R关z

⫻⵱␺兴, where ⍀ and R are the angular velocity and radius of the Earth and z is the vertical unit vector.

Equation共5兲 conserves the energy E and the enstrophy H,

which are defined by关11兴:

E =1 2

冕

0 2␲ d␸

冕

0 ␲ 兩ⵜ␺兩2_{sin␪d␪, 共6a兲} H =1 2

冕

₀ 2␲ d␸

冕

0 ␲ 兩⌬␺兩2_{sin␪d␪. 共6b兲} B. Projection on the spherical basis

The eigenfunctions of the linear part of Eq. 共5兲 are

⌿j⬅ ⌿_ᐉ j

m_j共␪

,␸,t兲 = Yj共␪,␸兲exp共i␻jt兲, 共7a兲

where the frequencies of planetary waves are

␻j⬅␻共ᐉj,mj兲 = − 2mj⍀/ᐉj共ᐉj+ 1兲. 共7b兲

From now on we use the shorthand notation j =共ᐉj, mj兲 for

the eigennumbers ᐉj, mjof the spherical harmonic Y_ᐉ j mj共␪_,␸兲 Yj⬅ Y_ᐉ j m_j共␪ ,␸兲 = P_ᐉ j m_{j共cos␪兲exp共im} j␸兲, 共7c兲

with the associated Legendre polynomials Pj⬅ P_ᐉ j mj共cos␪兲, normalized as follows:

冕

0 ␲ P_ᐉmP_ᐉm_⬘sin␪d␪=␦共ᐉ,ᐉ

⬘

兲, P_ᐉ−m= P_ᐉm. 共7d兲 Here␦共ᐉ,ᐉ

⬘

兲 is the Kronecker symbol 共1 for ᐉ=ᐉ

⬘

and zero otherwise兲. The integer indices m and 共ᐉ−m兲 are the longi-tudinal and latilongi-tudinal wave numbers of theᐉ,m mode; they count the number of zeros of the spherical function along the longitudinal and the latitudinal directions. Below we refer to the range of m and ᐉ as the “spectral domain.” For the ap-proximation of two-dimensional atmosphere to hold, the wavelength is supposed to be much smaller than the atmo-sphere’s height. If we estimate the wavelength as the distance between the appropriate zeroes of spherical function, the length of the equator at about 40 000 km and the height of the atmosphere at about 40 km, we understand that the ap-proximation of a two-dimensional atmosphere holds up to ᐉⱗ1000.

Expanding the function␺共␪,␸, t兲 in the basis 共7a兲 we get

␺共␪,␸,t兲 =

兺

j

Aj⌿j. 共7e兲

Substituting in Eq. 共5兲 one obtains the governing

equa-tions for the “slow” amplitudes Aj⬅A_ᐉj

m_{j共t兲 of the planetary} waves, dAj dt = i 2Nj

兺

r,s Nr,sZj_兩r,sArAs⫻ exp关i共␻r+␻s−␻j兲t兴␦共mj,mr + ms兲, 共8a兲 Nj⬅ ᐉj共ᐉj+ 1兲, Nr,s⬅ Nr− Ns, 共8b兲

Zj兩r,s⬅ Z_ᐉj m_j兩 ᐉr,ᐉs m_r,ms₌

冕

0 ␲ Zj兩r,s共␪兲d␪, 共9a兲

with the “interaction integrand”

Zj兩r,s共␪兲 = P_ᐉ j m_j

冋

mrP_ᐉ r m_rdPᐉs m_s d␪ − msPᐉs m_sdPᐉr m_r d␪

册

. 共9b兲 We note that Eq.共8兲 is an exact consequence of the

barotro-pic vorticity equation, without any assumption about the ex-istence of a small parameter. Among the interactions appear-ing in this equations there are many nonresonant ones, in which the exponent exp关i共␻r+␻s−␻j兲t兴 is not unity. All

these interactions can be removed by a change of variables, which however will result in new nonlinear terms 共higher than quadratic兲, see, e.g., Sec. 1.1.4 in 关5兴. Assuming that the

wave amplitudes are small enough, all these can be disre-garded, bringing the final equations back to the same form as in Eq. 共8兲, but including only resonant triads for which

exp关i共␻r+␻s−␻j兲t兴=1.

C. Necessary conditions for nonvanishing interaction The first necessary condition that guarantees finite inter-action amplitudes follows from the axial symmetry of the problem and is reflected in the Kronecker symbol in Eq.共8a兲,

mj= mr+ ms. 共10a兲

Second, the spherical symmetry of the nonlinear term in Eq. 共5兲 leads to the conservation of the square of the total angular

momentum of the system. This translates to the triangle in-equality for vectors ᐉj=ᐉr+ᐉsin each triangle with nonzero

interaction amplitude,

兩ᐉr−ᐉs兩 ⬍ ᐉj⬍ ᐉr+ᐉs. 共10b兲

Third, the explicit form of Eq.共9a兲 requires that

ᐉj+ᐉr+ᐉs is odd. 共10c兲

Otherwise integrand共9b兲 is odd function of cos␪ and inte-gral共9a兲 is zero.

It can be shown by integrating Eq.共9兲 by parts that

when-ever Eq. 共10a兲 is fulfilled the three interaction coefficients

satisfy Z_ᐉ j m_j兩 ᐉr,ᐉs m_r,ms_{= Z} ᐉs m_s兩 ᐉr,ᐉj −mr,mj_{= Z} ᐉr m_r兩 ᐉj,ᐉs m_j,−ms_. 共11兲

From this follows that the interaction coefficients Z_{. . .}satisfy two Jacoby identities

Zj_兩r,s+ Zr_兩s,j+ Zs_兩j,r= 0,

NjZj兩r,s+ NrZr兩s,j+ NsZs兩j,r= 0. 共12兲

As a result, Eqs. 共8兲 have two integrals of motion

E =1 2

兺

j Nj兩Aj兩2, 共13a兲 H =1 2

兺

j N2j兩Aj兩2, 共13b兲

which are nothing else but the energy关Eq. 共6a兲兴 and

enstro-phy 关Eq. 共6b兲兴, presented in the basis 共7兲.

III. SMALL CLUSTERS OF RESONANT TRIADS In this section we formulate equations of motion and mo-tion invariants of small clusters of resonant triads, study their topology and other properties, that affect on the dynamics of finite-dimensional wave turbulence. Brief analysis of these questions was given in Ref.关13兴.

A. Active and passive modes in a resonant triad In this paper we refer to a “resonant triad” whenever we have three modes共j,r,s兲 whose frequencies satisfy the triad resonance condition ␻j=␻r+␻s. Accordingly, the equations

for the slow amplitudes of modes in resonant triads are writ-ten,共after relabeling according to r→1, s→2 and j→3兲 as follows: N₁dA1 dt = iN3,2ZA2 ⴱ_A 3, Z⬅ Z3兩1,2, N2 dA2 dt = iN1,3ZA1 ⴱ_A 3, N3 dA₃ⴱ dt = iN2,1ZA1 ⴱ_A 2 ⴱ_{. 共14兲}

In this case the conservation laws共13兲 take the form

E =1 2共N1兩A1兩

2_{+ N2兩A2兩}2_{+ N3兩A3兩}2_{兲, 共15a兲}

H =1 2共N1 2_兩A1兩2_{+ N} 2 2_兩A2兩2_{+ N} 3 2_兩A3兩2_{兲. 共15b兲} Taking for concreteness an example for which

ᐉ1⬎ ᐉ3⬎ ᐉ2, 共16兲

we make in Eqs. 共14兲 a linear change of variables Bi=␣iAi

such that

␣1= − i

冑

N1,2N1,3/

冑

N2N3,

␣2= i

冑

N1,2N3,2/

冑

N1N3,

␣3= i

冑

N1,3N3,2/

冑

N1N2. 共17兲

This results in equations with real coefficients that involve only one interaction amplitude Z,

dB1 dt = ZB2 ⴱ_B 3, dB2 dt = ZB1 ⴱ_B 3, dB₃ dt = − ZB1B2. 共18兲

This is a dynamical system corresponding to the simplest possible resonant cluster. Equations共18兲 are symmetric with

respect to replacing two low-frequency modes 1⇔2. The mode with highest frequency 共which in this paper will be always denoted by subscript “ 3“兲 is special. When Eq. 共16兲 does not hold one can find a similar change of variable for any relations between the magnitudes of the three indicesᐉj.

The system 共18兲 has two independent conservation laws

共known as Manley-Rowe integrals兲

冦

I23=兩B2兩2+兩B3兩2=共EN1− H兲N23/N1N2N3, I13=兩B1兩2+兩B3兩2=共EN2− H兲N13/N1N2N3, I12= I13− I23=兩B1兩2−兩B2兩2,

冧

共19兲 which are linear combinations of the energy E and enstrophy H defined by Eqs.共15兲.

Obviously, Eq. 共18兲 can be written in the Hamiltonian

form idBj dt = dHint dBⴱj , 共20a兲

with the interaction Hamiltonian

Hint= iZ共B1B2B3ⴱ− B1ⴱB2ⴱB3兲, 共20b兲 which is an additional integral of motion. In terms of old variables Aj

H_int= iZN1,2N1,3N3,2 N1N2N3

共A1A₂A₃ⴱ− A₁ⴱA₂ⴱA3兲. 共20c兲 By direct calculation it is easy to check that Eq. 共20c兲 is an

integral of motion of the dynamical system Eq.共14兲.

On the face of it Eqs.共18兲 involves six dynamical

vari-ables: i.e., B’s and their complex conjugates. In fact, using the standard representation of the complex amplitudes Bj in

terms of real amplitudes Cjand phases ␪j,

Bj= Cjexp共i␪j兲, 共21a兲

one recognizes that the right-hand side 共RHS兲 of Eq. 共18兲

depends only on a single combination of phases in the triad which affects the dynamics. We refer to this combination as the triad phase,

␸⬅␪1+␪2−␪3. 共21b兲

The triad phase appears in Eq. 共20b兲 as follows:

Hint= − 2Z兩B1B2B3兩sin␸. 共22兲 Thus we have a four-dimensional phase space with three integrals of motion, resulting in a simple periodic trajectory

for almost all conditions. For more details see 关12兴.

Never-theless even this simple dynamics offers the first opportunity to discuss the energy flow within a cluster of interacting modes.

To this aim we discuss the evolution of the triad of am-plitudes with special initial conditions, when only one mode is appreciably excited at zero time. If B1共t=0兲ⰇB2共t=0兲 and B1共t=0兲ⰇB3共t=0兲, then I23共t=0兲ⰆI13共t=0兲. The integrals of motion are independent of time, therefore I13ⰇI23at all later times, and hence 兩B1共t兲兩2_{Ⰷ兩B2共t兲兩}2_{. Moreover, 兩B1共t兲兩}2 Ⰷ兩B3共t兲兩2 _{at all times. Indeed, the assumption 兩B1共t兲兩}2 ⱗ兩B3共t兲兩2 _{yields I}

13⯝I23, which is not tenable. This means that the ␻1 mode, being the only essentially exited one at t = 0 cannot redistribute its energy to the other two modes in the triad. The same is true for the ␻₂ mode. For this reason we refer to the lower frequency modes with frequencies ␻₁ ⬍␻3and␻2⬍␻3 “passive modes,” or P modes.

On the other hand, the conservation laws 共19兲 cannot

re-strict the growing of P-modes from initial conditions when only␻₃-mode is appreciably excited. In this case the P-mode amplitudes will grow exponentially 关25兴: 兩B1共t兲兩, 兩B2共t兲兩

⬀exp关兩ZB3共t=0兲兩t兴 until all the modes will have comparable magnitudes of their amplitudes. Therefore we refer to the

␻3-mode as an “active mode,” or A mode. An A mode, being initially excited, is capable of shearing its energy with two P modes within the triad.

B. Double-triad clusters: Butterflies

An arbitrary cluster in our wave system is a set of con-nected triads. Examples of the simplest clusters, consisting of two triads connected via one common mode are shown in Fig. 2. They will be referred to as butterflies. Note that in principle, for another dispersion law, one could have two triads connected by two modes. Such a structure does not exist for the dispersion law of Rossby waves and will not be discussed here.

The dynamics of a cluster depends on the type of the mode which is common for the neighboring triads. Corre-spondingly we can distinguish three types of butterflies: PP, AP and AA butterflies, In this section we consider the equa-tions of motion, the invariants and the restricequa-tions on

dy-9,13 4,12 5,14

P

_P

A

1 2 3

= +

1 2

FIG. 1. 共Color online兲 Resonant triad ⌬1, see Table I below.

Wave numbers of modes m,ᐉ are shown inside ovals. Red arrows are coming from an active mode A 共with frequency ␻₃=␻₁+␻₂兲 and show directions of the energy flux to the passive P-modes共with frequencies␻₁and␻₂兲.

namical behavior, that follow from the existence of invari-ants for relatively small clusters consisting of two triads. In the following sections we consider these questions for clus-ters consisting of three and six triads.

Butterflies, as shown in Fig.2, consist of two triads a and b, with wave amplitudes Bj_兩a, Bj_兩b, j = 1 , 2 , 3, connected via

one common mode. For PP butterfly the common mode is passive in both triads, say

B_1兩a= B_1兩b PP-butterfly; 共23a兲 for an PA-butterfly the common mode is passive in a triad and active in the second, b triad,

B_1兩a= B_3兩b PA-butterfly; 共23b兲

while for an AA butterfly the common mode is active in both triads,

B_3兩a= B_3兩b AA-butterfly. 共23c兲

The equations of motion for these systems follow from the Eqs. 共8兲 under the condition of small nonlinearity and

from the resonance conditions in both triads,

␻1兩a+␻2兩a=␻3兩a, ␻1兩b+␻2兩b=␻3兩b, 共24兲 with the obvious requirement that the frequencies of the common modes are the same. After a change of variables similar to Eqs. 共17兲 and elimination of one common mode,

the resulting equations for PP butterfly共B1_兩a= B1兩b兲 are

冦

B˙_1兩a= ZaB_2兩aⴱ B3兩a+ ZbB_2兩bⴱ B3兩b, ,

B˙_2兩a= ZaB_1兩aⴱ B3兩a, B˙2兩b= ZbB_1兩aⴱ B3兩b,

B˙3兩a= − ZaB1兩aB2兩a, B˙3兩b= − ZbB1兩aB2兩b.

冧

共25a兲

For PA butterfly 共with B1兩a= B_3兩b兲 they are

冦

B˙_1兩b= ZbB2ⴱ兩bB3兩b, B˙3兩a= − ZaB3兩bB2兩a,

B˙_2兩b= ZbB_1兩bⴱ B3兩b, B˙2兩a= ZaB_3兩bⴱ B3兩a,

B˙3兩b= − ZbB1兩bB2兩b+ ZaB2ⴱ兩aB3兩a,

冧

共25b兲

and for AA butterfly共with B3兩a= B_3兩b兲,

冦

B˙_1兩a= ZaB_2兩aⴱ B3兩a, B˙1兩b= + ZbB_2兩bⴱ B3兩a,

B˙_2兩a= ZaB_1兩aⴱ B3兩a, B˙2兩b= ZbB_1兩bⴱ B3兩a,

B˙3兩a= − ZaB1兩aB2兩a− ZbB1兩bB2兩b.

冧

共25c兲

All these equations can be obtained from the canonical equations of motion 共20a兲 using the Hamiltonian

Hint= 2 Im兵ZaB_1兩aⴱ B_2兩aⴱ B3兩a+ ZbB_1兩bⴱ B_2兩bⴱ B3兩b其, 共26兲

in which the conditions 共23兲 have to be fulfilled for each

particular butterfly.

In addition to the Hamiltonian 共26兲 we have three more

invariants of the Manley-Rowe type for each butterfly. For the PP, PA, and AA butterflies they, respectively, are

I2,3兩a=兩B2兩a兩2+兩B3兩a兩2, I2,3兩b=兩B2兩b兩2+兩B3兩b兩2, 共27a兲 I_兩a,b=兩B1兩a兩2+兩B3兩a兩2+兩B3兩b兩2, PP;

I1,2兩b=兩B1兩b兩2−兩B2兩b兩2, I2,3兩a=兩B2兩a兩2+兩B3兩a兩2, 共27b兲 I_兩a,b=兩B1兩b兩2+兩B3兩b兩2+兩B3兩a兩2, PA;

I1,2兩a=兩B1兩a兩2−兩B2兩a兩2, I1,2兩b=兩B1兩b兩2−兩B2兩b兩2, 共27c兲 I_兩a,b=兩B1兩a兩2+兩B1兩b兩2+兩B3兩a兩2, AA.

The first two invariants for the PP butterfly, I2,3兩a and I2,3兩b, do not involve the common mode B1兩a= B1兩b, and are similar to the invariant I₂₃, Eq. 共19兲, for an isolated triad. We can

make the following observation: if at t = 0 the amplitudes in one triad exceed substantially the two remaining amplitudes of the butterfly, that is 兩B1_兩a兩,兩B2_兩a兩,兩B3_兩a兩Ⰷ兩B2_兩b兩,兩B3_兩b兩, this relation persists. In other words, in PP butterfly when any of the two triads, a or b, has initially very small amplitudes, it is unable to absorb the energy from the second triad during the nonlinear evolution.

The invariants I1,2兩band I2,3兩a for the PA butterfly do not involve the common mode B3兩b= B1兩a; they are similar to the corresponding integrals I₁₂and I₂₃, Eqs.共19兲, for an isolated

triad. If at t = 0 the b triad is excited much more than the a triad 共and thus I1,2_兩bⰇI2,3_兩a兲 the smallness of the positively 2,6 5,7 3,8 6,9 4,14 P P a b

B

0

B

0

~

13,19 6,18 7,20 2,15 5,24 A P a b

B

0

B

0

~

29,376 85,187 114,208 9,116 105,231 A A a b

B

0

B0

~

(b) (a) (c)

FIG. 2. 共Color online兲 Examples of isolated butterflies. All the notations are as in Fig.1. In particular: red arrows are coming from the active modes A and show directions of the energy flux to the passive P-modes. The letters “a” and “b” in square boxes denote triads and will be used as subscripts in the corresponding evolution equations for amplitudes and for integrals of motion. In studies of free evolution in Sec. Vthe initial energy is concentrated in the “leading” a triad in “individual” modes with amplitudes B0and B˜0and then goes to the “driven” b triad.

defined invariant I_2,3兩a prevents the a-triad from absorbing energy from b-triad during the time evolution. The situation is different when the a triad is initially excited and I2,3兩a ⰇI1,2_兩b. In this case the initial energy of a triad can be easily shared with b triad. The smallness of I1,2兩bonly requires that during evolution 兩B1_兩b兩⬇兩B2_兩b兩. Under this type of the initial conditions we will call the a triad “leading” triad, while the b triad will be referred to as “driven” triad.

Finally, the invariants for the AA butterfly, I1,2兩aand I1,2兩b, do not involve the common mode B3兩a= B1兩b and are similar to I12, Eqs. 共19兲, for isolated triad. Simple analysis of these integrals of motion shows that energy, initially held in one of the triads will be shared between both triads dynamically.

The conclusion that we can draw from these examples is general: any triad which is connected to any given cluster of any size whatsoever where the connection occurs via its pas-sive mode cannot absorb the energy from the cluster, if ini-tially the triad is not excited. In contrast, a triad connected to a cluster of any given size via an active mode can freely adsorb energy from the cluster during the nonlinear evolu-tion.

C. Triple-triad clusters: stars, chains, and triangles Triple triad clusters consist of three triads, denoted as a, b, and c triads with the mode amplitudes denoted as Bj_兩a, Bj_兩b,

Bj兩c, j = 1 , 2 , 3. There are three topologically different types

of triple-triad clusters: with one common mode—stars, shown in Fig.3; with two common modes—chains, and with three common modes—triangles, these clusters are shown in Fig.4. Having in mind different types of common modes one distinguishes 13 types of triple-triad clusters, including four stars 共AAA, AAP, APP, PPP stars, Fig. 3兲, seven types of

three-chain, and two types of triangle clusters, in Fig.4. All motion equations can be written in the canonical form共20a兲

with the Hamiltonian

Hint= 2 Im兵ZaB1ⴱ兩aB2ⴱ兩aB3兩a+ ZbBⴱ1兩bB2ⴱ兩bB3兩b+ ZcB1ⴱ兩cB2ⴱ兩cB3兩c其, 共28兲 in which one has to equate amplitudes of common modes.

共a兲 Stars have one common mode in three triads. For ex-ample, taking B_3兩a= B_3兩b= B_3兩c in Eq.共28兲 one gets from Eq.

共20兲 equations of motion for AAA-stars, 435,464 18,186 417,527 345,527 90,340 225,399 210,588

A

A

A

a

b

c

271,812 125,720 146,927 541,811 270,810 126,783 145,840

P

A

A

a

c

b

231,902 385,737 154,602 226,903 5,860 22,615 253,860

P

P

A

a

c

b

303,909 238,713 541,805 546,714 261,594 311,819 8,287

P

P

P

a

c

b

(b) (a) (c) (d)

FIG. 3.共Color online兲 Examples of isolated triple stars in the spectral domain mⱕᐉⱕ1000. The letters “a,” “b,” and “c” in square boxes denote triads and will be used as subscripts in the corresponding evolution equations for amplitudes and for integrals of motion. All other notations as in Fig.1and2.

冦

B˙_1兩a= ZaB2兩aB3兩a, B˙2兩a= ZaB1兩aB3兩a,

B˙_1兩b= ZbB2兩bB3兩a, B˙2兩b= ZaB1兩bB3兩a,

B˙_1兩c= ZcB2兩cB3兩a, B˙2兩c= ZcB1兩cB3兩a,

B˙3兩a= − ZaB1兩aB2兩a− ZbB1兩bB2兩b− ZcB1兩cB2兩c.

冧

共29a兲 Taking B_3兩a= B_3兩b= B_1兩cone gets for AAP stars,

冦

B˙_1兩a= ZaB2兩aB3兩a, B˙2兩a= ZaB1兩aB3兩a,

B˙_1兩b= ZbB2兩bB3兩a, B˙2兩b= ZaB1兩bB3兩a,

B˙_3兩c= − ZcB3兩aB2兩c, B˙2兩c= ZcB3兩aB3兩c,

B˙_3兩a= − ZaB1兩aB2兩a− ZbB1兩bB2兩b+ ZcB2兩cB3兩c.

冧

共29b兲 Similarly one gets motion equations for

PPA-star: B_1兩a= B_1兩b= B_3兩c,

冦

B˙_3兩a= − ZaB1兩aB2兩a, B˙2兩a= ZaB1兩aB3兩a,

B˙_3兩b= − ZbB1兩aB2兩b, B˙2兩b= ZaB1兩aB3兩b,

B˙_1兩c= ZcB_2兩cⴱ B1兩a, B˙2兩c= ZcB1兩aB1兩c,

B˙1兩a= ZaB2ⴱ兩aB3兩a+ ZbBⴱ2兩bB3兩b− ZcB1兩cB2兩c.

冧

共29c兲

PPP-star: B_1兩a= B_1兩b= B_1兩c,

冦

B˙_3兩a= − ZaB1兩aB2兩a, B˙2兩a= ZaB1兩aB3兩a,

B˙_3兩b= − ZbB1兩aB2兩b, B˙2兩b= ZaB1兩aB3兩b,

B˙_3兩c= − ZcB1兩aB2兩c, B˙2兩c= ZcB1兩aB3兩c,

B˙1兩a= ZaB2ⴱ兩aB3兩a+ ZbBⴱ2兩bB3兩b+ ZcB2兩cB3兩c.

冧

共29d兲

In addition to Hamiltonian, all triple-star clusters have four invariants of the Manley-Rowe type, three of them does not involve the common mode. For example, for

冦

PPA-star: I_1,2兩a=兩B1兩a兩2−兩B2兩a兩2, I_2,3兩b=兩B2兩b兩2_{+兩B3兩b兩}2_{, I}

2,3兩c=兩B2兩c兩2+兩B3兩c兩2, I_兩a,b,c=兩B1兩a兩2_{+兩B3兩a兩}2_{+兩B3兩b兩}2_{+兩B3兩c兩}2_.

冧

共30兲

Integrals I2,3兩band I2,3兩cprevent b and c triads共connected via the P mode兲 from adopting energy of initially excited a triad. In cases when a b and/or a c triad are initially exited, the a triad can freely share their energy via the connecting A mode.

共b兲 Triple-chains have two common modes in two triads. As we mentioned, there are seven types of triple chains, that differ in type of connections, see Fig. 4. Similarly to the triple-star clusters, one gets equation of motion for triple chains from the canonical Eq. 共20a兲 with the Hamiltonian

共28兲, in which one has to equate two pairs of amplitudes of

common modes. For example, for PA-PA chain,

PA-PA-chain: B1兩a= B3兩b, B1兩b= B3兩c, 共31a兲

A A P 212,795 159,597 493,713 371,689 27,650 95,874 P _122,805 a b c P A A P A 171,360 403,455 315,636 18,224 83,741 232,608 153,399 391,759A P A 1,94 392,704 372,836 19,360 17,399 2,224 P A A_434,459A P 280,620 154,340 320,575 114,323 38,228 76,458 P A A_82,90 A P 69,90 12,65 3,35 66,104 70,98 16,63 P A P A 356,800 37,434 161,575 3,279 34,464 517,704 322,899 P P A P A P P A 118,341 228,539 204,636 14,154 80,741 104,154 124,588 340,779 495,615 171,455 155,455 16,455 324,819 P a c b P P A A A 417,779 88,615 646,779 558,819 _141,987 229,779 a c b P A P P P P (b) (a) (c) (d) (e) (f) (h) (i) (g)

FIG. 4.共Color online兲 In the spectral domain mⱕᐉⱕ1000 there are 66 isolated triple-chain clusters of seven types 关examples are shown in panels A–F兲 and four triangle clusters with 共AA-PA-PP兲 and 共PA-PP-PP兲 connections 共examples are shown in panels H and I兲兴. Common PP modes are split, denoting difficulty in the energy exchange between corresponding triads. Dashed lines separate PP-irreducible subclus-ters, discussed in Sec.IV B.

冦

B˙1兩a= ZaB_2兩aⴱ B3兩a− ZbB1兩bB2兩b,

B˙_2兩a= ZaB1ⴱ兩aB3兩a, B˙3兩a= − ZaB1兩aB2兩a,

B˙_2兩b= ZbB_1兩bⴱ B3兩b,

B˙1兩b= ZbB2ⴱ兩bB3兩b− ZaB1兩cB2兩c,

B˙_1兩c= ZcB_2兩cⴱ B1兩b, B˙2兩c= ZcB_1兩cⴱ B1兩b.

冧

共31b兲

Again, besides Hamiltonian, all triple-chain clusters have four invariants of the Manley-Rowe type, two of them do not involve the common mode. In particular, PA-PA chain, gov-erned by Eq.共31a兲 has the following invariants:

冦

PA-PA chain: I_兩a,b=兩B1兩a兩2−兩B2兩a兩2+兩B2兩b兩2, I_兩b,c=兩B1兩b兩2−兩B2兩b兩2+兩B2兩c兩2,

I_2,3兩a=兩B2兩a兩2+兩B3兩a兩2, I_1,2兩c=兩B1兩c兩2−兩B2兩c兩2.

冧

共31c兲 共c兲 Triple-triangles have three common modes in three triads, see Figs.4共H兲and4共I兲. Correspondingly, equations of motion for AA-PA-PP and PA-PP-PP triangles one gets from the canonical Eq.共20a兲 with the Hamiltonian 共28兲, in which

one has to equate three pairs of amplitudes of common modes. In particular, for AA-PA-PP triangle one has

B1兩a= B1兩b, B3兩a= B3兩c, B3兩b= B1兩c, 共32a兲

冦

B˙_1兩a= ZaB2ⴱ兩aB3兩a+ ZbB2ⴱ兩bB3兩b, B˙_2兩a= ZaB_1兩aⴱ B3兩a,

B˙3兩a= − ZaB1兩aB2兩a− ZcB2兩cB3兩b,

B˙_2兩b= ZbB_1兩aⴱ B3兩b,

B˙_3兩b= − ZbB1兩aB2兩b+ ZcB_2兩cⴱ B3兩a,

B˙2兩c= ZcB3ⴱ兩bB3兩a.

冧

共32b兲

Triangle clusters have three Manley-Rowe invariants. In par-ticular, AA-PA-PP triangle, governed by Eq. 共32b兲 has the

following invariants:

冦

AA-PA-PP triangle: I_兩a,b=兩B2兩b兩2−兩B1兩a兩2, I_兩a,c=兩B2兩a兩2+兩B3兩a兩2+兩B2兩c兩2,

I_兩b,c=兩B2兩b兩2−兩B3兩b兩2−兩B2兩c兩2.

冧

共32c兲 Notice, that triple-stars and triple-chains have ten real variables共seven amplitudes and three triad phases兲 and five invariants 共Hamiltonian and four Manley-Rowes’兲, while triple-triangles have only nine real variables共six amplitudes and three triad phases兲 and four invariants 共Hamiltonian and three Manley-Rowes’兲. Therefore, all triple-triad clusters have five-dimensional effective phase space. Recall, that but-terflies have three-dimensional phase space. Moreover, one can prove that any n-triad cluster has 共2n−1兲-dimensional effective phase space.

D. Caterpillar: six-triad cluster

The largest cluster found in the domainᐉ⬍21 consists of six resonant triads⌬11. . .⌬16with three PP, one AP, and one AA connection, see Fig. 5. The equation of motion for this cluster, 共called “caterpillar”兲 can be obtained from Hamil-tonian, similar to Eq.共28兲, but consisting of six terms

H_cat= 2 Im

兺

n=11

16

ZnB_1兩nⴱ B_2兩nⴱ B3兩n, 共33a兲

in which we have to equate amplitudes of common modes:

再

B1兩11= B1兩12, B3兩12= B1兩13, B2兩12= B1兩14,

B_3兩14= B_3兩15, B_1兩14= B_1兩16.

冎

共33b兲

Due to these five connections one has only 3⫻6−5=13 complex equations for remaining amplitudes,

冦

B˙1兩11= Z11B2ⴱ兩11B3兩11+ Z12B2ⴱ兩12B3兩12, B˙_2兩11= Z11B1兩11ⴱ B3兩11, B˙3兩11= − Z11B1兩11B2兩11, B˙2兩12= Z12B_1兩11ⴱ B3兩12+ Z14B_1兩14ⴱ B3兩14, B˙_3兩12= − Z12B1兩12B2兩12+ Z13B2ⴱ兩13B3兩13 B˙_1兩14= Z₁₄B_2兩12ⴱ B_3兩14+ Z₁₆B_2兩16ⴱ B_3兩16, B˙3兩14= Z14B2ⴱ兩12B1兩14+ Z16B2ⴱ兩16B3兩16, B˙_1兩15= Z15B2兩15ⴱ B3兩14, B˙2兩15= Z15B1兩15ⴱ B3兩14, B˙_2兩16= Z₁₆B_1兩14ⴱ B_3兩16, B˙_3兩16= − Z₁₆B_1兩14B_2兩16.

冧

共33c兲 Caterpillar has six Manley-Rowe invariants:

P P P P P P P P P P A A A A A 11.4 14 16 15 A A A A A P P _P P P _P P P P P 11.3 11.2 11.1 13 12 11

FIG. 5. 共Color online兲 Triads belonging to caterpillar are drawn by bold lines. New, connected to them triads, appearing in spectral domain m ,ᐉⱕ1000 are drawn by thin lines. The PP reduction of the extended caterpillar to three triads共⌬₁₁,⌬₁₆and⌬_11.4, accord-ing to TableI兲, PA butterfly q12,13, AA butterflyq14,15, and AAA

star is shown by dashed lines. The rest of notations as in Figs.1and 4.

冦

I_2,3兩11=兩B2兩11兩2+兩B3兩11兩2, I_2,3兩13=兩B2兩13兩2_{+兩B3兩13兩}2_, I_1,2兩15=兩B1兩15兩2−兩B2兩15兩2, I2,3兩16=兩B2兩16兩2+兩B3兩16兩2, I_兩11,12,13=兩B1_兩11兩2+兩B3_兩11兩2+兩B3_兩12兩2+兩B3_兩13兩2, I_兩14,15,16=兩B1_兩14兩2+兩B1_兩15兩2+兩B3_兩15兩2+兩B3_兩16兩2.

冧

共33d兲 IV. HOW CLUSTERS ARE ORGANIZED

This section is devoted to the analysis of the structure of clusters of resonant triads of atmospheric planetary waves 共based on the data set of the exact solutions of resonance conditions ␻j=␻r+␻s with restrictions 关Eq. 共10兲兴, provided

by Kartashova关26兴兲. This analysis is important for the study

of finite-dimensional wave turbulence throughout the present paper. A preliminary study of this issue can be found in Ref. 关13兴.

A. “Meteorologically significant” clusters and their extension in large spectral domain

Dealing with atmospheric waves one learns that the “me-teorologically significant” wave numbers are believed to be limited toᐉ⬍21. Nevertheless, as explained above, the spec-tral domain for the approximate two-dimensional atmosphere extends toᐉⱗ1000. Counting explicitly how many clusters we have in this spectral domain we find that there exists altogether 1965 isolated triads and 424 clusters consisting from 2 to 3691 connected triads. Among them there are 234 butterflies, 95 triple-triad clusters, etc.共cf. the histogram in Fig.6共A兲兲. For clarity of presentation we did not display in this histogram the largest 3691 cluster, which we refer to as the monster.

It can be seen that about 82.2% of all clusters are pre-sented by isolated triads and their dynamics has been inves-tigated in 关12兴 in all details; the main findings are that the

energy oscillates between the three modes in the triads, with a period of oscillation that is much larger then the wave period. This period is inversely proportional to the root-mean-square of the wave amplitude.

234 clusters in the spectral domain共⯝10.5%兲 are the but-terflies discussed above; these are further analyzed below. Among them there are 131 PP, 69 AP, and 35 AA butterflies. The 95 triple-triad clusters include 25 “triple-star” clusters with one triple connection, see Fig. 3. This set includes 3 AAA, 5 AAP, 6 APP, and 11 PPP stars. There are also 66 chain clusters with two pair connections and seven combina-tions of the connection types, shown in Fig.4. We found also four 共two pairs兲 triple-triad clusters with three pair connec-tions, belonging to two different types, see Fig.4.

Similar classification can be performed for all the other clusters. For example, the monster includes one mode 共218 545兲, participating in ten triads, three modes, participat-ing in nine triads, five modes—in eight triads, 23—in seven, 50—in six, 90 in five, 236—in four, 550—in three and 1428

modes—in two triads 共butterflies兲. The analysis of their dy-namical behavior depends crucially on the connection type as shown above and detailed below.

B. PP reduction of larger clusters

Large clusters can be divided into “almost separated” sub-clusters connected by PP-connections 共e.g., Figs. 5, 7, and

8兲. If such a subcluster cannot be divided further into smaller

clusters connected by PP connection we refer to it as a PP-irreducible cluster. For example, the triangle cluster in Fig.

4共I兲 can be PP reduced to a triad and a PA-butterfly, while clusters in Figs.8共A兲and8共B兲can be PP reduced to two and

(b) (a)

FIG. 6. 共Color online兲 Horizontal axes denote the number of triads in the cluster while vertical axes show the number of corre-sponding clusters 共panel A兲 and PP irreducible subclusters 共panel B兲. P P P P P P P P P P A A A A A 8 7 7.1 7.2 7.3

FIG. 7. 共Color online兲 Triads ⌬7 and ⌬8, that belong to PP

butterfly _q7,8 are drawn by bold lines. New, connected to them

triads ⌬_7.1, ⌬_7.2, and ⌬_7.3, appearing in spectral domain m ,ᐉ ⱕ1000 共for numeration of triads see Table I兲, are drawn by thin lines. The PP reduction of this cluster to three triads ⌬₇, ⌬₈, and ⌬7.1共according to the notation in TableI兲 and one AP butterfly is

three individual triads, respectively. The cluster in Fig.7 is PP reduced into three triads and PA butterfly, while extended caterpillar in Fig.5 is PP reduced into three triads, PA, AA butterfly, and AAA star. One can see that the 14-triad cluster, shown in Fig. 9共A兲can be PP-reduced into eight triads and six-triad cluster. The 16-triad cluster 共Fig. 9共B兲兲 can be PP

reduced into five triads, AA and PA butterfly, AAP star, and four-triad cluster, consisting of an AAA star with one AP-connected triad. Note that the concept of PP-reducible cluster is different from a disconnected cluster. While a PP connec-tion is reluctant to transfer energy from a highly excited sub-cluster to a lowly excited one, it can still redistribute energy between similarly excited subclusters. A disconnected cluster is clearly unable to do that.

In the spectral domain ᐉ,兩m兩ⱕ1000 the largest PP irre-ducible subcluster belonging to the monster consists of 130 triads and is shown in Fig. 10. The statistics of PP-irreducible subclusters are presented in Fig.6共B兲.

We learn from this analysis that many clusters cannot carry energy flux through PP connections; their dynamics is naturally reduced to the dynamics of PP-irreducible subclus-ters. Therefore it is sufficient to study carefully the dynamics of these PP-irreducible clusters to understand the properties of any cluster.

V. NUMERICAL ANALYSIS OF FREE EVOLUTION OF TYPICAL SUB-CLUSTERS

Our goal is to describe the energy flux through resonant triads in the regime of finite-dimensional wave turbulence. In the first Sec.V Bwe consider free evolution of the smallest clusters—butterflies—from asymmetrical initial conditions, in which only one triad is excited to high amplitudes, ex-ceeding by orders of magnitudes the initial amplitudes in the other triad. The questions are how the energy flux from the energetic “leading” triad depends on the type of connection, on the ratios of the interaction coefficients etc. Sec. V C is devoted to the free evolution of the triple-triad clusters: stars and chains from initial conditions in which only the leading a triad is substantially excited, the levels of excitation of the two other triads are much smaller. All these examples, and PA-PA-..PA chains, studied in the next section, can serve as building blocks of bigger clusters and the knowledge about the energy flux through them allows to qualitatively predict efficiency of the energy transfer through bigger clusters.

A. Methodology and numerical procedure

The equations of motion for all clusters were prepared for numerical analysis using a specially designed algorithm that

2,6 5,7 3,8 6,9 4,14

P

P

238,713 541,805 546,714 261,594 311,819 8,287

P

P

P

303,909

a

c

b

(b) (a)

FIG. 8. 共Color online兲 Examples of PP reduction of pair connection, panel A and of triple connection, panel B. Common 共PP- and PPP-兲 modes are split stressing the difficulty of energy exchange between the corresponding triads. Dashed lines separate PP irreducible subclus-ters, discussed in Sec.IV B. The rest of the notations are as in Fig.1.

(b) (a)

FIG. 9.共Color online兲 Large PP-reducible clusters. Common PP and PPP modes are shown by empty circles. The rest of modes are denoted by full共blue兲 circles. As in previous figures, outgoing 共red兲 arrows indicate A modes.

FIG. 10. 共Color online兲 Largest PP-irreducible cluster in the spectral domainᐉ,兩m兩ⱕ1000.

allowed an automatic implementation for any cluster, given the number of triads and the connectivity table. This served to avoid human errors in implementing large sets of equa-tions.

The equations of motion for the free evolution of butter-flies and triple clusters are stiff and were integrated using a multistep adaptive method based on numerical differentiation formulas关27兴.

For each system of equations the accuracy of integration was controlled by testing the conservation of the relevant integrals of motion. The integration parameters were ad-justed to keep the standard deviation ␴共I兲 below a given threshold for the duration of the numerical runs. Since the main parameter that affects the accuracy in terms of the con-servation of the integrals of motion is the maximal allowed time step, the preliminary calculations were carried out with a requirement ␴共I兲ⱕ10−5; actually in most cases ␴共I兲 ⱕ10−8 _{was achieved.}

The evolution starting from several hundred initial condi-tions was analyzed and several representative condicondi-tions were chosen for the study of energy transfer in the clusters. All the conclusions regarding the discovered dependencies were verified by control calculations with stricter accuracy requirements.

The equation of motion for forced chain clusters, studied in the next Sec. VI, were integrated by both adaptive meth-ods关27兴 and by fourth-order constant time-step Runge-Kutta

for better control of accuracy. By construction of the model, these equations required accurate description of the last triad in the cluster to ensure proper energy dissipation. The inte-gration parameters were adjusted to reproduce this fastest evolution, and therefore were automatically suitable for all other triads. We verified the convergence of the resulting statistics with respect to all relevant parameters.

B. Free evolution in butterflies

The simplest topology that allows consideration of the energy flux between resonant triads is the double-triad clusters—butterflies.

1. Initial conditions, choice of the interaction coefficients and data representation

In this Subsection we show that details of the time evolu-tion in butterflies are very sensitive to the initial condievolu-tions, which define the values of the dynamical invariants. There-fore a reasonable choice of initial conditions, allowing to shed light on a “typical” time evolution in a relatively com-pact form is not obvious. At initial time t = 0 we assign most of the energy to two individual共not common兲 modes of one 共leading兲 triad, denoted below for concreteness as a triad. The initial amplitudes of these two modes we denote as B0 and B˜0共Fig.2兲. To study the influence of the energy distri-bution between these modes we will use two types of initial conditions,

Type I: B0= 3.9 + 0.50i, B˜0= 3.7 + 0.93i, 共34a兲

Type II: B0= 5.3 + 0.50i, B˜0= 0.9 + 0.93i. 共34b兲 Both distributions共34兲 are complex and normalized such that

兩B0兩2_+兩B˜0兩2_{⬇30. The difference between Eqs. 共34a兲 and} 共34b兲 is that in Eq. 共34a兲 both amplitudes are similar, while

in Eq.共34b兲 they are quite different.

For different butterflies we choose in the leading triad, PP-butterfly with B_1兩a共0兲 = B1兩b共0兲: 共35a兲

B_2兩a共0兲 = B˜0, B_3兩a共0兲 = B0;

AA-butterfly with B3兩a= B3兩b: 共35b兲 B_1兩a共0兲 = B0, B_2兩a共0兲 = B˜0;

PA-butterfly with B_1兩a= B_3兩b: 共35c兲 B2兩a共0兲 = B˜0, B3兩a共0兲 = B0,

as it is shown in Fig.2.

The initial conditions in the driven b triad we choose the same for all types of butterflies. They have much smaller initial amplitudes, for example

冦

B_1兩b共0兲 = B1,0, B1,0⬅ C共0.05 + 0.02i兲, B_2兩b共0兲 = B2,0, B2,0⬅ C共0.02 + 0.05i兲, B_3兩b共0兲 = B3,0, B3,0⬅ C共0.10 − 0.02i兲.

冧

共36兲 To study the dependence of the energy flow between triads on the initial level of excitation of the driven triad we vary the energy content in the driven triad changing the coeffi-cient C in Eq. 共36兲, taking in addition to C=1 also C=0.1

and C = 0.01. In all the further simulations C = 1 if else is not mentioned.

In this way the initial conditions for all butterflies are as similar as possible and we can study the difference in time evolutions, caused by different types of connections.

Last but not least are the interaction coefficients. For con-creteness we chose interaction coefficients corresponding to ⌬14 and ⌬16 共Z14⬇75, and Z16⬇15兲 as prototypes and use either 兵Za= 75, Zb= 15其 or vise versa and sometimes 兵Za

= Zb= 15其. Since the change in the interaction coefficient

renormalizes the corresponding time scale, only their ratio is important for the dynamics. In our case these ratios are =5, 1/5 or 1; this allows us to study the butterfly dynamics with very different values of the interaction amplitudes, which is typically the case. Having in mind that special choices of the ratios of interaction coefficients may lead to integrability of clusters of resonant triads关28兴 共and see also 关29兴兲 we verified

that small variations of these ratios does not changed our conclusions concerning energy transfer in clusters.

Recall that any butterfly has three quadratic integrals of motion, that involve only squares of five amplitudes of modes and therefore only two combinations of them are in-dependent. For the presentation we chose such combinations that are orthogonal to the corresponding invariants. Namely, for PP butterflies, connected via B_1兩a= B_1兩bmodes,

J2,3兩a⬅ 兩B2兩a兩2−兩B3兩a兩2, J2,3兩b⬅ 兩B2兩b兩2−兩B3兩b兩2; 共37a兲 for AA butterflies, connected via B_3兩a= B_3兩bmodes,

J1,2兩a⬅ 兩B1兩a兩2+兩B2兩a兩2, J1,2兩b⬅ 兩B1兩a兩2+兩B2兩b兩2, 共37b兲 and for AP-butterflies, connected via B_1兩a= B_3兩bmodes, J_2,3兩a and J_1,2兩b.

Time evolutions for these three types of butterflies with various initial conditions and choices of the ratio Za/Zb共5 or

1/5兲 are shown in Figs. 11–15. Following Sec. V B 3is de-voted to discussion of these numerical results.

2. Effect of the type of connections and of the ratio ZaÕ Zb

共a兲 PP butterfly has the most trivial time evolution, see Fig. 11 for Za= 75, Zb= 15,共panel A兲 and Za= 15, Zb= 75,

共panel B兲. As expected, there is practically no energy ex-change between triads: amplitudes J_2,3兩a, J_2,3兩b, defined by

0 1 2 3 4 5

−30 −28 −26

J 3,2|a

PP butterfly, leading triad, Z

a= 75 0 1 2 3 4 5 6 7.5 8 8.5 9x 10 −3 time J 3,2|b

PP butterfly, driven triad, Z_b= 15

0 1 2 3 4 5

−30 −28 −26

J 3,2|a

PP butterfly, leading triad, Z

a= 15 0 1 2 3 4 5 6 −0.02 0 0.02 time J3,2|b

PP butterfly, driven triad, Z

b= 75 0 1 2 3 4 5 6 −40 −20 0 20 J3,2|a

PA butterfly, leading triad, Z= 75

0 1 2 3 4 5 6 4 6 8 10x 10 −3 time J 1,2|b

PA butterfly, driven triad, Z= 15

0 1 2 3 4 5 6 −40 −20 0 20 J 3,2|a

PA butterfly, leading triad, Z= 15

0 1 2 3 4 5 6 0 10 20 30 time J 1,2|b

PA butterfly, driven triad, Z= 75

0 1 2 3 4 5 6

0 20 40

J 1,2|a

AA butterfly, leading triad, Z= 75

0 1 2 3 4 5 6 0 0.005 0.01 0.015 time J1,2|b

AA butterfly, driven triad, Z= 15

(b) (a)

(c) (d)

(e)

FIG. 11. 共Color online兲 Time evolution of PP, PA, and AA butterflies with Za= 75, Zb= 15, left panels and Za= 15, Zb= 75, right panels.