Discrete of rotational capillary water

Adrian Constantin†, Elena Kartashova], Erik Wahlén‡ † Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria ] RISC, J. Kepler University, Linz 4040, Austria ‡ Centre for Mathematical Sciences, Lund University, PO Box 118, 221 00 Lund, Sweden; FR 6.1 - Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany∗

We study the discrete wave turbulent regime of capillary water waves with constant non-zero vorticity. The explicit Hamiltonian formulation and the corresponding coupling coefficient are ob- tained. We also present the construction and investigation of clustering. Some physical implications of the obtained results are discussed.

I. INTRODUCTION (Fig.2, [25]). Non-resonant modes with comparatively big broadening do not change their at the appropriate time scales (Fig.3, [25]). Capillary water waves are water waves, the range of being λ < 17mm. In spite of A similar effect for irrotational capillary waves has their small amplitude, capillary waves play an important been discovered by Pushkarev and Zakharov [38, 39] who role in the dynamics of the surface. gen- investigated how the discreteness of the spectrum of ir- erated capillary waves provide roughness on the ocean rotational capillary water waves influences the statistical surface which then yields subsequently the appearance of properties of a weakly nonlinear wave field described by capillary-gravity and gravity waves [34, 37, 43]. On the the kinetic equation. It was shown in a series of numer- other hand, capillary waves can be pumped by gravity ical simulations with the dynamical equations of motion waves [36, 54] and therefore they also transport energy (with energy pumping into small ) that the 2 from bigger to smaller scales. wave spectrum consists of ∼ 10 excited harmonics which do not generate an toward high wavenum- The role of capillary water waves in the general theory bers — frozen or fluxless modes. "There is virtually of wave turbulence is invaluable. One can say that most no energy absorption associated with high-wavenumbers new groundbreaking results and effects in (weak) wave damping in this case" ([39], p.107). The quasi-resonance turbulent systems have been first discovered for capillary channels of the energy flux have also been observed and water waves. Seminal work of Zakharov and Filonenko it was shown that the size of (quasi)-resonance clusters [58] laid in 1967 the very foundation for what later be- grows with increasing of the resonance broadening ([39], came known as statistical (or kinetic) wave turbulence p.113). Two mechanisms for increasing the number of theory: the first power energy spectrum has been ob- flux-generating modes have been identified: 1) growing tained for capillary water waves, as an exact solution of the modes‘ amplitudes, and 2) enlarging the number of the wave kinetic equation. The wave kinetic equation modes. describes the wave field evolution at the finite inertial in- The difference between the results of [25] and [38, 39] terval 0 < (k1, k2) < ∞ in k-space. At this interval the energy of a wavesystem is supposed to be constant while is as follows. Irrotational capillary waves do not pos- sess exact [26] which yields the appearance of forcing (usually in the small scales 0 < kfor < k1) and dissipation (in the scales k < k ) are balanced [59]. only fluxless modes; on the other hand, resonances do 2 dis exist among atmospheric planetary waves. Accordingly, Wave kinetic equations are deduced for the continu- in [25] two types of modes had been singled out — frozen ous k-spectrum in Fourier space while in bounded or and flux-generating. The specifics of the discrete wave periodical systems the wave spectrum is discrete and turbulent regime is due to the fact that its time evolu- described by resonance clustering [29]. An important tion is governed by the dynamical equations, not by the fact first shown by Kartashova in [25] is that indepen- kinetic equation [29]; accordingly, phases of single modes dent resonance clusters might exist in the infinite spectral are coherent and random phase approximation (necessary space and are nonlocal in k-space [24, 25]. Moreover, it for deducing the wave kinetic equation) does not hold. was shown numerically (for atmospheric planetary waves) The pioneering result of Zakharov et al. [61] was that increasing the wave amplitudes till some limiting the discovery of the mesoscopic wave turbulence regime magnitude yields no energy spreading from a cluster to which is characterized by the coexistence of discrete and all the modes of the spectral domain but rather goes kinetic regimes in numerical simulations modeling tur- along some "channels" formed by quasi-resonances, that bulence of gravity waves on the surface of a deep ideal is, resonances with small enough frequency broadening incompressible fluid. The mathematical justification of this co-existence was given in [27] in the frame of the model of laminated wave turbulence. ∗Electronic address: [email protected],lena@ A peculiar property of irrotational capillary waves is risc.uni-linz.ac.at,[email protected] that they do not have exact resonances in periodic boxes 2 and therefore their discrete regime is exhibited in the The discrete layer is formed by exact resonances and form of frozen turbulence, with distinct modes just keep- quasi-resonances governed by (2) and (3), while the con- ing their initial energy. As it was shown quite recently, tinuous layer is formed by approximate interactions given rotational capillary waves do have exact resonances, for by (4). Four main distinctions between the time evolu- specific positive magnitudes of the vorticity [8], which tion of these two layers have to be accounted for: means that their discrete regime will have a different dy- 1. dependence or independence on the initial condi- namics. The simplest physical system where non-zero tions, vorticity is important is that of tidal flows, which can be 2. locality or non-locality of interactions, realistically modeled as two-dimensional flows with con- 3. existence or nonexistence of the inertial interval, stant non-zero vorticity, with the sign of the vorticity 4. random or coherent phases. distinguishing between ebb/ [16]. The main goal of our paper is to study the discrete The continuous layer wave turbulent regime of rotational capillary water waves with constant nonzero vorticity. In Section II we give a brief overview of different turbulent regimes for three- This layer is described by the wave kinetic equations and wave resonance systems and we present known results its stationary solutions in the form of power energy spec- on irrotational capillary waves (with zero vorticity). De- tra. The main necessary assumptions in kinetic wave tailed presentation of discrete wave turbulent regime is turbulence theory are random phase approximation, lo- given in Section III. In Section IV we provide some back- cality of interactions and existence of a finite inertial in- ground material on water waves with vorticity. Reso- terval. In the kinetic regime the time evolution of the nance clustering is constructed and studied in Section V. wavefield does not depend on the details of the initial A brief discussion concludes the paper. conditions and not even on the magnitude of the initial energy. Only the fact that the complete energy of the wave system is conserved in the inertial interval is im- II. WAVE TURBULENT REGIMES portant. Interactions are local, that is, only wave vectors with lengths of order k are allowed to interact. More- To introduce the three generic wave turbulent regimes over, interactions take place only at some finite interval, we need to distinguish between the notions of quasi- and no information is given about time evolution of the resonance and an approximate (non-resonant) interac- modes with wavenumbers lying outside this interval. tion. Let us call a parameter δω, The discrete layer |ω1 ± ω2 ± ... ± ωs| = δ(ω) ≥ 0, (1) resonance width. Regarding the function ωj This layer is governed by the set of dynamical systems as a function of integer variables which are indices of describing resonance clusters; its time evolution depends Fourier harmonics (corresponding to a finite-size box), crucially on the initial conditions. Modes’ phases are it was shown by [28] that the resonance width can not coherent; the linear combination of the phases corre- be arbitrary small. This means that if δ(ω) 6= 0, then sponding to the choice of resonance conditions (called δ(ω) ≥ R(ω) where R(ω) is some positive number. Ac- dynamical phase of a resonant cluster) describes phases’ cordingly, exact resonances, quasi-resonances and ap- time evolution (see Section III B). Resonant interactions proximate interactions can be introduced as follows: are not local, that is, waves with substantially different (1) exact resonances are solutions of (1) with k-scales may form resonances. For instance, gravity- capillary waves with wave lengths of order of k and of δ(ω) = 0; (2) the order of k3 can form a resonance; more examples are (2) quasi-resonances are solutions of (1) with given in [30]. Also the existence or absence of inertial in- terval is not important: resonances can occur all over the 0 < δ(ω) < R(ω); (3) k-spectrum. Moreover, in some wave systems waves with (3) approximate interactions are solutions of (1) with arbitrary wavelengths can form a resonance, for instance rotational capillary waves [8]. We will discuss this aspect R(ω) ≤ δ(ω) < Rmax(ω), (4) in detail in Section V. where Rmax(ω) defines the distance to the nearest res- onance. The number R(ω) can easily be found numeri- Main aspects cally, for a given dispersion function, as in [46]. The main features of the model of laminated turbu- lence can be summarized as follows. A. The model of laminated wave turbulence • Independent clusters of resonantly interacting modes can exist all over k-spectrum. According to (2)–(4), discrete and continuous layers in • The time evolution of the modes belonging to a clus- a wave turbulent system are defined as follows. ter are not described by KZ-spectra, thus leaving the 3

"gaps" in the spectra; their energetic behavior is de- scribed by finite (sometimes quite big) dynamical sys- tems. Depending on their form and/or the initial con- ditions, the energy flux within a cluster can be regular (integrable) or chaotic. • The places of the "gaps" in the spectral space are defined by the integer solutions of resonance conditions (e.g. see (34) below). They are completely determined by the geometry of the wave system. • The size of the "gaps" depends on a) the form of dis- persion function, b) the number of modes forming mini- mal possible resonance, and c) sometimes — though not always — also on the size of the spectral domain. • The discrete layer can be observable all over k-space and is characterized by the energy exchange among a finite number of resonant modes. The continuous layer can be observable within the inertial interval (k1, k2) and is characterized by the energy transport over the k-scales. Accordingly, three distinct wave turbulent regimes can FIG. 1: Color online. Schematical representation of three be identified with substantially different time evolution, wave turbulent regimes. In the upper panel two co-existing as shown schematically in Fig.1. Energetic behavior of layers of turbulence are shown; (black) solid arrows are di- the modes in the discrete regime is governed by a set of rected to the wave turbulent regimes in which a layer can be dynamical systems (see next section). observable (shown in the middle panel). (Red) dashed arrows going from the lower panel to the middle one show the evo- Energy transport in the kinetic regime is described by lutionary characteristics of the wavefield which manifest the the kinetic energy cascade, corresponding regime.

−ν Ekin ∼ k , (5) B. Parameter distinguishing among wave turbulent where ν > 0 is a positive number depending on the dis- regimes persion function and the number of interacting waves, [59]. In order to observe the various wave turbulent regimes Much less is known about the mesoscopic regime where in laboratory experiments and in numerical simulations both types of the wavefield time evolution can be de- one has to carefully make a few choices. tected simultaneously: energy exchange within resonance First of all, it is necessary to define the range of al- clusters and energy cascades. The existence of energy- lowed amplitude magnitude so that the waves belong to cascading clusters has been first shown in [30]. The sim- the "corridor" of weak nonlinearity. This means that plest dynamic cascade (without ) has the form the amplitudes should be big enough to leave the linear regime but should not be too big (to prevent strong tur- −n Edyn ∼ p , (6) bulence). For water waves the wave steepness ε, which is the characteristic ratio of the wave amplitude to the where p is a constant, 0 < p < 1, n = 1, 2, ..., N and wavelength, is taken as a suitable small parameter, the N is the number of cascading modes within a cluster. usually choice being ε = 0.1. Dissipation can also be included, i.e. dynamic energy Secondly, the choice of the spectral domain, i.e. the cascade is not affected by the existence or non-existence number of Fourier modes, is of the utmost importance. of the inertial interval. It is usually regarded as the main parameter allowing Whether an energy cascade in the mesoscopic regime to distinguish between different wave turbulent regimes. has kinetic or dynamic origin can easily be checked in Namely, the ratio of the characteristic λ to experimental data. the size of the experimental tank L is, as a rule, cho- In various physical systems either continuous or dis- sen as the appropriate parameter for estimating whether crete or both layer(s) can be observable. For instance, in or not the waves "notice" the boundaries. For capillary the laboratory experiments reported in [17] only discrete water waves, if L/λ > 50, the wave system can be re- dynamics has been identified. Coexistence of both types garded as infinite and the dispersion function, both in of the wavefield evolution has been detected in laboratory circular and in rectangular tanks, can be taken in the experiments, [57], and in numerical simulations, [61]. In- usual form as k3/2 (M. Shats, private communication). clusion of additional physical parameters could yield the The different choices explain why various laboratory ex- transition from the kinetic to the discrete or mesoscopic periments with capillary water waves demonstrate dif- regime, [8]. ferent turbulent regimes. For instance, in [44] discrete 4 modes have been identified while in [20] the development of full broadband spectra is observed.

Characteristic wavelengths

It is tempting to use the parameter

L/λ (7) as a characteristic allowing to distinguish between dif- ferent wave turbulent regimes: taking a sufficiently large number of modes one will get the classical kinetic regime. However, the situation is more complicated. As it was shown in [57], Fig.4, low-frequency excitation (beginning FIG. 2: Color online. The energy power spectrum Ek is de- with 25 Hz) generates distinct peaks of wave frequen- picted as the (grey) solid curve. of resonant modes cies while increasing the excitation to 300 Hz yields an are shown by (red) vertical solid T-shaped lines having a isotropic turbulent regime with a power law distribution string-like part. The energy of fluxless modes is shown by of energies. Further increasing the excitation frequency (blue) vertical solid T-shaped lines. The empty circles in corresponding places of the energy power spectrum represent generates an unexplained peak at 400 Hz. Moreover, cap- "gaps", indicating that for these modes the energy is given not illary waves in Helium do notice the geometry of the ex- by the solid curve but by the T-shaped lines. The horizontal perimental facilities for L/λ ∼ 30 − 300, the correspond- and vertical axes show wavenumbers and energy correspond- ing dispersion in circular tanks being described by Bessel ingly, while k1 and k2 denote the beginning and the end of functions (G. Kolmakov, private communication). In this the inertial interval. case there exist nonlinear resonances and their manifes- tation has been recently established in laboratory exper- iments with capillary waves on the surface of superfluid in superfluid Helium where rogue waves have been de- Helium, [1]. The local maximum of the wave amplitudes tected and the fact that the energy balance is nonlocal in is detected at of the order of the viscous cut- k-space was established [19]. off frequency in the case when the surface is driven by a periodical low-frequency force. This means that the en- Characteristic resonance broadening ergy is concentrated in a few discrete modes at the very The usual way to characterize the physical meaning of end of the inertial interval (k , k ), in the beginning of 1 2 the resonance width δ is to regard it as a shift in the the dissipative range. ω resonant wave frequencies, cf. [40, 47, 61], yielding reso- These novel results are paramount both from theoret- nance broadening in a wide wave spectrum with a large ical and practical point of view. This is the first exper- number of modes. imental confirmation of the predictions of the model of To characterize the broadening in a three-wave system laminated wave turbulence that resonant modes can ex- governed by (12) one can choose as an appropriate pa- ist within and outside the inertial interval in the large rameter the inverse nonlinear oscillation time of resonant k-scales. modes The origin of these peaks (similar to rogue waves in the ) in the wave spectrum can be seen from the −1 τk ≈ |ZBk|. (8) Fig.2: whether or not a is generated depends both on the driving frequency ωdrive and on the reso- In discrete regimes the phases of individual modes are nant frequencies in a wave system. Indeed, suppose the coherent; however if the width |∆ω,B3 | of the high- modes M1,M2,M3 have frequencies ω1, ω2, ω3 and form frequency mode in a resonant triad becomes substantially −1 a resonance ω1 + ω2 = ω3. If ωdrive ≈ ω3 (the mode with larger that τk , the coherence is lost — this being a nec- high frequency which is unstable), then the appearance of essary condition for the occurrence of the kinetic regime. peaks with frequencies ω1 and ω2 is to be expected. If the One concludes then that modes M2,M3,M4 form a resonance with ω2 = ω3 + ω4, • The discrete regime corresponds to then an excitation with ω ≈ ω yields peaks at ω drive 2 3 −1 and ω4. The importance of the high-frequency mode is |∆ω,B3 | ¿ τk , (9) discussed in the next section. • The kinetic regime corresponds to Notice that rogue waves can appear within and out- side the inertial interval while discrete wave turbulent −1 |∆ω,B | À τ , (10) regimes are not connected with the existence of the in- 3 k ertial interval. The situation is quite general and does • The mesoscopic regime corresponds to not depend on the type of waves under consideration. −1 This is in agreement with the experimental observations |∆ω,B3 | ' τk . (11) 5

In the case when exact resonances are absent and only The high-frequency modes B3|a and B3|b are approximate interactions have to be accounted for, the called active, or A-modes, and the other modes estimates (9)-(11) can be made general (cumulative) by (B1|a,B2|a,B1|b,B2|b) are called passive, or P -modes [32], the substitution |∆k| instead of |∆ω,B3 |, where |∆k| is because due to the Hasselman’s criterion of instability defined as a broadening of a mode with wave vector k. [22] A-modes are unstable under infinitesimal excitation The fact that the generation of the kinetic regime occurs and P -modes are neutrally stable. The cluster dynamics via spectral broadening of discrete harmonics has been is then defined by three possible connection types within established in [40] for irrotational capillary waves. the cluster, namely AA−, AP − and PP −connections. In There are three main reasons for the absence of ex- the system (13) the connection B1|a = B1|b is of PP -type act resonances in a wave turbulent system with decay and B3|a = B2|b is of AP -type. type of dispersion function: a. the resonance conditions The graphical representation of an arbitrary resonance do not have any solutions (ω ∼ k3/2); b. the resonance cluster in the form of a NR-diagram (NR for nonlinear conditions do not have solutions of a specific form (for ex- resonance), first introduced in [29], allows us to keep this ample, for ω ∼ k−1 solutions exist in a square but not in dynamical information. The vertices of a NR-diagram a majority of rectangular domains); c. all resonances are are triangles corresponding to resonant triads and the formed in such a way that at least one mode in each pri- half-edges drawn as bold and dashed lines denote A- and mary cluster lies outside the inertial interval. In this case P -modes correspondingly. Examples of NR-diagrams are other modes will become "frozen" ([38, 39]) and no reso- shown in Fig. 3. nance occurs in the inertial range of the wavenumbers. In Each mode of a cluster generates a gap in the KZ- the case when exact resonances are not absent, one has spectrum. The maximal possible number of gaps can to be very careful while introducing any cumulative esti- also be seen from the form of the NR-diagram, though mate: for approximate interactions the upper boundary the actual number can be smaller, in the case when two Rmax(ω), which depends on k, has to be included. The different wavevectors k1 and k2 have the same length. problem of introducing a suitable cumulative parameter for distinguishing among various wave turbulent regimes is contingent on future extensive research.

III. DISCRETE REGIME

The discrete wave turbulent regime is characterized by resonance clusters which can be formed by a big number FIG. 3: Color online. NR-diagrams for clusters generating of connected triads. The dynamical system correspond- maximally 3, 4, 5, 6 and 7 gaps (from left to right and from ing to a cluster formed by N triads can be written out up to down). explicitly by coupling the N systems for a triad An illustrative example can be found in [4] where B˙ = Z B∗ B , B˙ = Z B∗ B , 1|j j 2|j 3|j 2|j j 1|j 3|j laboratory experiments with two-dimensional gravity- ˙ B3|j = −ZjB1|jB2|j, (12) capillary waves are described. By analyzing the mode frequencies in the measured data, only five different fre- with j = a, b, c, ... and equaling the appropriate Bi|j. For quencies were identified: 10, 15, 25, 35 and 60 Hz. How- instance, the dynamical system of the first two-triad clus- ever, theoretical consideration allowed to conclude that ter shown in Fig.3 reads in fact seven distinct modes take part in the nonlinear  ˙ ∗ ˙ ∗ resonant interactions, and the corresponding resonance B = ZaB B , B = ZaB B ,  1|a 2|a 3|a 2|a 1|a 3|a cluster has the form of a chain formed by three connected  ˙ ˙ ∗ B3|a = −ZaB1|aB2|a, B1|b = ZbB2|bB3|b, triads. The amplitudes and frequencies were identified as B˙ = Z B∗ B , B˙ = −Z B B , ⇒  2|b b 1|b 3|b 3|b b 1|b 2|b  A1 ↔ 60Hz,A2 ↔ 35Hz,A3 ↔ 25Hz, B1|a = B1|b ,B3|a = B2|b  A4 ↔ 25Hz,A5 ↔ 10Hz,A6 ↔ 25Hz,A7 ↔ 15Hz, ˙ ∗ ∗ B = ZaB B + ZbB B ,  1|a 2|a 3|a 3|a 3|b B˙ = Z B∗ B , and resonance conditions for frequencies as 2|a a 1|a 3|a (13) B˙ = −Z B B + Z B∗ B ,  3|a a 1|a 2|a b 1|a 3|b ω1 = ω2 + ω3, ω2 = ω4 + ω5, ω6 = ω5 + ω7 (14)  ˙ B3|b = −ZbB1|aB3|a, "The waves A3,A4 and A6 all have the same frequency The system (13) is unique, up to the change of indices (25 Hz), but they must have different wavevectors, in or- 1 ↔ 2. On the other hand, the form of the system der to satisfy the kinematic resonance conditions. We changes depending on the fact whether or not the con- assume that the mechanical means to generate a test necting mode is a high-frequency mode in one or both or wave at 25 Hz also generates perturbative waves in other in no triads. directions at 25 Hz" ([4], p.70). The NR-diagram of this 6

amplitudes Bj, j = 1, 2, 3) and only 3 conservation laws (15), (16) which, generally speaking, are not enough for FIG. 4: Color online. NR-diagram of the resonance cluster ap- integrability. However, if we rewrite it in the standard pearing in the laboratory experiments with gravity-capillary amplitude-phase representation Bj = Cj exp(iθj) as water waves reported in [4].  C˙ = ZC C cos ϕ,  1 2 3 C˙ = ZC C cos ϕ, 2 1 3 (17) cluster is shown in Fig.4: the cluster is a 3-chain with one C˙ = −ZC C cos ϕ,  3 1 2 PA- and one PP-connections. The dynamical system cor-  −2 −2 −2 ϕ˙ = −ZHT (C + C − C ), responding to the cluster has been solved numerically: 1 2 3 for all calculations the dynamical phase of the initially it becomes clear immediately that in fact we only have 4 excited triad was set to π, while during the experiments independent variables: three real amplitudes Cj and one the phases of the modes have not been measured (D. dynamical phase Henderson, private communication). This yielded quali- tative agreement of the results of numerical simulations ϕ = θ1 + θ2 − θ3 (18) with measured data but higher magnitudes of observed amplitudes. according to the choice of resonance conditions. In this The evolution of the amplitudes and phases in the dis- section we will concentrate on the wave amplitudes while crete turbulent regime is discussed in Section III A and the importance of dynamical phases will be discussed in in Section III B correspondingly. the Section III B. Making use of the addition theorems sn2 +cn2 = 1 and dn+µ2sn = 1 (here µ is a known func- tion of conservation laws, µ = µ(H,I23,I13) and of the A. Evolution of the amplitudes amplitude-phase representation Bj = Cj exp(iθj), one can easily get explicit expressions for the squared ampli- In Fig.2 we show the difference in dynamics of discrete tudes: ³ ´ modes of capillary waves with and without vorticity. The 2 2 energies of fluxless modes in the frozen turbulence regime Cj (t) = α1j + α2j · dn α3j · t · Z, α4j − α5j (19) for irrotational capillary waves are shown as (blue) bold vertical lines and the energies of resonant modes appear- with the coefficients α1j, ..., α5j being explicit functions of ing due to the non-zero constant vorticity are depicted by I13,I23 and HT , for j = 1, 2, 3, [31]. The energy of each (red) bold lines with a spring part inside. The "springs" mode Ej is proportional to the square of its amplitude, 2 2 show symbolically that the magnitudes of energies can Ej(t) ∼ Cj (t) ∼ dn , i.e. its changing range is finite and change in time, similarly to the process of compressing defined by the initial conditions as 0 < dn2(t) ≤ 1. or pulling an elastic string. Notice that this situation There is no analytical expression for the amplitudes is quite general and does not depend on the wave type (energies) of a cluster formed by two or more triads. On (see [25] for numerical simulations of atmospheric plan- the contrary, as it was shown in [2, 49, 50], the corre- etary waves — in Fig.1 the periodical time evolution of sponding dynamical systems are integrable only in a few resonant modes is shown and in Fig.3 the fluxless modes exceptional cases. For instance, clusters formed by N demonstrate no time changes in the magnitudes of am- triads all having one common mode and with all con- plitudes). nections of AA- or PP -type are integrable for arbitrary The analogy with a string is quite helpful for the un- initial conditions if Zi/Zj = 1/2, 1 or 2. Some clusters derstanding of the dynamics of resonant modes. A string are known to be integrable for arbitrary coupling coeffi- can be compressed or pulled (without destruction) only cients but only for some specific initial conditions. On in a finite range of the applied forces; similarly, the range the other hand, already the smallest possible cluster of of amplitudes’ changing is also finite. Indeed, the sim- two connected triads can present chaotic behavior [30], plest way to show it is just to look at the smallest possible though the modes energies are still bounded as the cor- resonant cluster — a triad. The dynamical system for a responding dynamical systems are energy conserving. resonant triad (12) has the Hamiltonian In the case of a generic cluster, the time evolution of

∗ the amplitudes has to be studied numerically and de- HT = Im(B1B2B3 ) , (15) pends crucially on the initial conditions. In order to de- crease the number of degrees of freedom one has to use and two Manley-Rowe constants of motion [35] in the the fact that each triad has 3 integrals of motion given form by (15)(16), i.e. N isolated triads have 2N conservation 2 2 2 2 I23 = |B2| + |B3| ,I13 = |B1| + |B3| , (16) laws and N Hamiltonians. A cluster formed by N triads with n connections has (2N − n) conservation laws and providing the integrability of (12) in terms of the Jaco- one Hamiltonian. The integrability of a generic cluster bian elliptic functions sn, cn and dn [55]. Notice that depends on the connection types within a cluster and on (12) has 6 real variables (real and imaginary parts of the the ratios of coupling coefficients Zj. Some results of 7 numerical simulations with two-triad clusters (fixed con- Π nection type, various coupling coefficients) can be found in [2].

Π €€€€€€ 2 B. Evolution of the dynamical phases

As one can see from (17), the time evolution of a res- onant triad does not depend on the individual phases of 0 the resonant modes but on the dynamical phase ϕ cor- 0 20 40 60 80 100 120 responding to the resonance conditions. In this case, the 2.5 closed expression for the evolution of dynamical phase, 2 first found in [31], reads ³ ³ ´´ 1.5 ϕ(t) = sign(ϕ0) arccot k1 F k2 (t − t0), µ , (20) 1 where 0.5 µ ¶ µ 2K(µ) 3 K(µ) k = − , k = 2 0 1 2 0 20 40 60 80 100 120 |HT | Z τ τ and FIG. 6: Upper panel. Color online. Plots of the dynamical phases as functions of time, for each frame: ϕa(t) is (red) F(x, µ) ≡ sn(x, µ) cn(x, µ) dn(x, µ), solid and ϕb(t) is (black) dashed, with ϕa,in = ϕb,in = π/2. Lower panel. Plots of the squared amplitudes, as functions while ϕ0 is the initial dynamical phase and the constants of time. The connecting mode C1 is blue (the upper one), C and C are green (in the middle), while C and C µ, K(µ), τ are known explicitly as functions of I13,I23 2|a 3|a 2|b 3|b are purple (at the bottom). To facilitate the view, C2 ,C2 and HT . 2|b 3|b 2 The time evolution of a generic cluster again does not and C1 are shifted upwards. The horizontal axis stands for depend on the individual phases but on the dynamical non-dimensional time and the vertical axis for the phase in phases; the corresponding equations for the dynamical the upper panel and for the squared amplitudes in the lower phases have a form similar to the last equation of (17). panel. For instance, in case of two triads a and b connected via

then this range is substantially diminished: depending on the initial energy distribution within a triad it can de- crease 10 times and more, being minimal for ϕin = π/2. FIG. 5: NR-diagram for a two-triad cluster with one PP- For the cluster shown in Fig.5 the maximal range of the connection. amplitudes is observed for ϕa,in = ϕb,in = 0. An interesting phenomenon has been uncovered in the one PP-connection (the NR-diagram is shown in Fig.5), case when at least one of the phases ϕa and ϕb is initially the equations for the dynamical phases non-zero: a new time-scale is clearly observable where the amplitudes of the resonant modes are modulated by the ϕa = θ1|a + θ2|a − θ3|a, ϕb = θ1|b + θ2|b − θ3|b (21) evolution of the dynamical phases. The characteristic form of evolution is shown in Fig.6 for ϕ = ϕ = π/2, read a b [3]. −2 −2 −2 This is a manifestation of one more crucial distinction ϕ˙ a = −H(C1 + C2|a − C3|a), (22) −2 −2 −2 between discrete and statistical wave turbulent regimes: ϕ˙ b = −H(C1 + C2|b − C3|b ). (23) while in the first regime phases are locked, in the latter they are supposed to be random. The effect of phase where the joint mode is denoted by C1 and H is the randomization among three-wave interactions of capillary Hamiltonian waves has been studied recently experimentally in [40]. ¡ ¢ The transition from a coherent-phase to a random- H = C1 ZaC2|aC3|a sin ϕa + ZbC2|bC3|b sin ϕb . (24) phase system occurs above some excitation threshold The effect of the dynamical phases on the evolution of when coherent wave harmonics broaden spectrally. The the amplitudes has been studied numerically in [3]. It waves were excited parametrically in the range of the turned out that if initially ϕin = 0 for a triad, then it shaker frequencies 40 ÷ 3500 Hz and the modulation in- stays zero and the changing range of amplitudes is max- stability of capillary waves, which can be approximated 2 imal. If ϕin is very small but non-zero, say, ϕin = 0.01, by the squared secant function sech , has been estab- 8 lished. Experiments show that the gradual development irrotational flows) can be established by introducing a of the wave continuum occurs due to the spectral broad- generalized velocity potential. In particular there is a ening. Similar effects have been previously observed in Hamiltonian formulation in terms of two scalar variables, liquid Helium [41] and in spin waves in ferrites [33]. one of which is the free surface elevation [51]. In contrast However, the form of the resulting KZ-spectrum does to the three-dimensional waves, it is convenient to regard not allow to conclude automatically whether the energy the y-axis as the vertical axis, with the waves propagating cascade is due to three- or four-wave interactions, and in the x-direction. The problem can be written in terms both explanations can be found in the literature. A di- of a generalized velocity potential, satisfying rect way to clarify the matter would be to estimate the ( periods of the excited modes for the case of three- and ∆φ = 0, −∞ < y < η, (25) four-wave interactions at the corresponding time scale. |∇φ| → 0, y → −∞, The experimental confirmation can then be obtained by measuring the corresponding wave fields [42]. with φx = u + Ωy and φy = v, [51]. In this coordinate system the free surface oscillates about the mean water level y = 0. IV. FLOWS WITH CONSTANT VORTICITY Introducing the new variable ζ, with

Ω −1 A rotational water wave is a wave in which the un- ζ = ξ − ∂x η, (26) derlying fluid flow exhibits non-zero vorticity. Physi- 2 cally rotational waves describe wave-current interactions: where ξ = φ|y=η is the restriction of the generalized ve- a uniform current is described by zero vorticity (irrota- locity potential to the free surface, allows to obtain a tional flow) [5, 7, 15], while a linearly sheared current has canonical Hamiltonian system in the form [51]: constant non-zero vorticity [18]. The irrotational flow δH δH setting is appropriate for waves generated by a distant η˙ = , ξ˙ = − , (27) storm and entering a region of water in uniform flow δξ δη [34], while tidal flows are modelled by constant vortic- ity [16]. The existence of periodic trains where the notation δF/δu is used for the variational with small and large amplitudes [12, 14] was established, derivative of a functional F with respect to the variable and qualitative properties of such waves were studied u. [6, 10, 11, 13, 14, 21, 23, 48]. In the case of capillary- Passing to Fourier variables can be seen as a change of or gravity-capillary waves the rigorous theory is limited variables and this transforms Hamilton’s equations into to the small-amplitude regime [52], although some par- δH ˙ δH tial results pertaining to the existence of large-amplitude η˙k = , ξk = − , (28) δξ∗ δη∗ waves have recently been obtained [53]. Note that al- k k ready the small-amplitude theory is more involved in the for k ∈ Z∗. presence of surface tension, including such phenomena as The change of variables Wilton ripples [56] in which two harmonics with the ratio s s 1 : 2 are in resonance. |k| ω˜(k) η = (a + a∗ ), ζ = −i (a − a∗ ) An important new result on rotational capillary waves k 2˜ω(k) k −k k 2|k| k −k is the theorem on the dimension of flows with constant vorticity [8]: with r Theorem. Capillary wave trains can propagate at the Ω2 free surface of a layer of water with a flat bed in a flow ω˜(k) = σ|k|3 + , (29) of constant non-zero vorticity only if the flow is two- 4 dimensional. linearizes the quadratic Hamiltonian and transforms Hamilton’s equations into This theorem allows us to consider for flows of constant vorticity only two-dimensional flows propagating in the δH a + i = 0. (30) x-direction. For irrotational flows the existence of a ve- k δa∗ locity potential enables the transformation of the govern- k ing equations for capillary water waves to a Hamiltonian In the new variables we have that system expressed solely in terms of the free surface and X 2 of the restriction of the velocity potential to the free sur- H2 = ω(k)|ak| , (31) face [60]. The absence of a velocity potential for non-zero k∈Z∗ vorticities complicates the analysis considerably and one where can not expect results of this type. r However, for flows of constant vorticity the remarkable Ω Ω2 ω(k) = − sgn k + σ|k|3 + , (32) preservation of the main features of the above theory (for 2 4 9 is the dispersion relation for two-dimensional 2π-periodic of vorticity. In other words, any vorticity computed by capillary waves with constant vorticity [52]. (37) will generate an isolated triad; its dynamics has been The non resonant terms in the Hamiltonian can be discussed in detail in Section III. eliminated up to any desired order by the normal form However, in laboratory experiments the magnitude of transformation, i.e. we can replace the Hamiltonian func- vorticity can only be controlled within some non-zero ˆ res ˆ tion H = H2 + H3 + R4 by H = H2 + H3 + R4. Here ε error corresponding to the available accuracy of the res −2 H3 is the resonant part of H3, experiments. For capillary waves ε ∼ 10 is easy to achieve and more refitments allow to reach ε ∼ 10−4 in X some cases. In Section V we construct the resonance res 1 ∗ H = S(k1, k2, k3)(ak ak a + c.c.), clustering of rotational capillary waves for various ap- 3 2 1 2 k3 k3=k1+k2 proximate magnitudes of the vorticity. ω(k3)=ω(k1)+ω(k2) (33) with S(k1, k2, k3) being known functions of wavenumbers V. RESONANCE CLUSTERING kj first found in [9]. Let us now restrict our attention to a resonant triad To see how the resonance clustering depends on the with resonance conditions taken in the form available accuracy ε, let us define the ε-vicinity of vortic- ity Ω as ω(k3) = ω(k1) + ω(k2), k3 = k1 + k2. (34) ¯ ¯ ¯Ωmax − Ωmin ¯ If we approximate the Hamiltonian using the quadratic ε = ¯ ¯. (39) Ω and cubic terms, Hamiltonian equations take the form max  and let us construct examples of clustering for a few dif- ∗ a˙ 1 + iω1a1 = −iS(k1, k2, k3)a2a3, ferent choices of ε. ∗ √ a˙ 2 + iω2a2 = −iS(k1, k2, k3)a1a3, Notice that since the vorticity depends linearly on σ,  the previously defined ε does not depend on σ and conse- a˙ 3 + iω3a3 = −iS(k1, k2, k3)a1a2. quently the resonance clustering constructed below will

−iωj t be the same for capillary waves in an arbitrary liquid. Setting aj = iBje , these equations transform into the standard form However, even for the same liquid, say water, the magni- tude of σ depends on the temperature and atmospheric ˙ ∗ ˙ ∗ ˙ B1 = ZB2 B3, B2 = ZB1 B3, B3 = −ZB1B2, (35) pressure while performing experiments. All numerical simulations discussed below have been performed in the where the coupling coefficient reads, [9], spectral domain k1, k2 ≤ 100. −8 −5 r µ ¶ • 10 ≤ ε ≤ 10 . −8 k1k2k3 ω˜1ω˜2 Ω We began our numerical computations with ε = 10 Z = − + ω˜3 . (36) πω˜1ω˜2ω˜3 2 4 which does not generate any other clusters but one iso- lated triad for each magnitude of the approximate vor- −8 −5 Note that in this formula the variables kj, ωj, ω˜j and ticity |Ω ± ε|. Increasing ε from 10 to 10 still leaves Ω are not independent, since they are related by the res- us with isolated triads. onance conditions (34). All possible magnitudes of the • 10−5 ≤ ε ≤ 10−4. constant non-zero vorticity generating resonances, i.e. so- For this values of ε, four new clusters appear, each lutions of (34), can be directly computed, [8], as formed by two connected triads:  Ω(k1, k2) = (20, 94, 114), (24, 70, 94); r  σ k k (9k2 + 9k2 + 14k k )(k + k )−1/2 (17, 71, 88), (15, 88, 103); · p 1 2 1 2 1 2 1 2 (37) (40) 4 3 2 2 3 4 6 (6k + 15k k2 + 22k k + 15k1k + 6k ) (11, 83, 94), (12, 71, 83); 1 1 1 2 2 2  (10, 47, 57), (12, 35, 47). for arbitrary k1, k2, k3 satisfying (34). It follows from (37) the magnitude of a positive vortic- Each line in (40) consists of two resonance triads and the ity triggering a resonance can not be too small. The min- exact magnitudes of the generating vorticities are given imal magnitude of the constant non-zero vorticity which by (37), say Ω(20, 94) and Ω(24, 70). These magnitudes generates resonances is do not coincide, Ω(20, 94) 6= Ω(24, 70), however r σ Ω(20, 94) − Ω(24, 70) Ωmin = 2 . (38) | | < ε. (41) 3 Ω(20, 94) The main conclusion one can deduce from (37) is that All four two-triad clusters have the same NR-diagram any two rotational capillary waves with arbitrary wave- shown in Fig.7. No results on the integrability of cor- lengths can form a resonance for a suitable magnitude responding dynamical systems are presently known, and 10

namical system reads  ˙ ∗ ˙ ∗ −4 B = ZaB B , B = ZaB B , FIG. 7: NR-diagram for clusters appearing for ε ≤ 10 .  1|a 2|a 3|a 2|a 1|a 3|a B˙ = Z B∗ B , B˙ = Z B∗ B , 1|b b 2|b 3|a 2|b b 1|b 3|a (45) B˙ = Z B∗ B , B˙ = Z B∗ B ,  1|c c 2|c 3|a 2|c c 1|c 3|a we expect chaotic energy exchange among the modes of  ˙ B3|a = −ZaB1|aB2|a − ZbB1|bB2|b − ZcB1|cB2|c such a cluster. • 10−4 ≤ ε ≤ 10−3. where the indices a, b and c are taken for three triads; The structure of resonance clustering is substantially all connections within this cluster are of AA-type, i.e. richer in this case. For instance, 83 different vortici- B3|a = B3|b = B3|c. ties can generate two-cluster clusters, among those most The cluster of four triads clusters have AP-connections but also clusters with AA- connections appear, for instance (48, 48, 96), (47, 49, 96), (28, 96, 124), (46, 50, 96) (46)  has three AA-connections and one AP-connection. The (50, 50, 100), (49, 51, 100);  NR-diagram (see Fig.9) and the dynamical system can (47, 48, 95), (46, 49, 95); (42) be easily constructed but no results on the integrability  (44, 44, 88), (43, 45, 88) of the corresponding dynamical system are known. • 10−3 ≤ ε ≤ 10−2. and others. The size of the clusters grows exponentially with the growth of ε so that for ε = 10−2 some vorticities generate clusters formed by a few hundreds to few thousands of connected triads. The maximal cluster in the studied spectral domain consists of about 4000 triads with more FIG. 8: NR-diagram for clusters described by (42). than 33.000 connections among them. It would be plausible to assume that in this situation The corresponding NR-diagram is shown in Fig.8. The no regular patterns will be observable for generic initial explicit form of the dynamical system for a cluster con- conditions. On the other hand, a special choice of initial sisting of a triad a and a triad b connected via two high- conditions — excitation of a P-mode — might produce a fluxless (frozen) regime. However, one has to construct frequency modes, B3|a = B3|b, reads the resonance clustering and to check whether or not  the "parasite" frequency of electronic equipment used B˙ = Z B∗ B , B˙ = Z B∗ B ,  1|a a 2|a 3|a 1|b b 2|b 3|a during experiments generates a resonance (see [4] for B˙ = Z B∗ B , B˙ = Z B∗ B , (43)  2|a a 1|a 3|a 2|b b 1|b 3|a more detail).  ˙ B3|a = −ZaB1|aB2|a − ZbB1|bB2|b. Last but not least. Let us rewrite (37) as The integrability of (43) has been studied in [2, 49] with 1/2 ˜ the following results. The system (43) is integrable for Ω(k1, k2) = σ · Ω(k1, k2) (47) arbitrary initial conditions, if Z /Z is equal to 1, 2 or a b where 1/2, [49]. In the case of arbitrary Za/Zb, the integrability can be proven for some specific initial conditions [31]. 2 2 −1/2 ˜ k1k2 (9k1 + 9k2 + 14k1k2)(k1 + k2) Beside two-triad clusters we also observe the appear- Ω(k1, k2) = p 6(6k4 + 15k3k + 22k2k2 + 15k k3 + 6k4) ance of various structures formed by three, four, five and 1 1 2 1 2 1 2 2 more connected triads; some examples are given below. does not depend on the properties of fluid: these are The cluster of three triads absorbed in the coefficient σ. The expression for ω˜(k) can also be rewritten as (79, 80, 159), (78, 81, 159), (77, 82, 159) (44) r s Ω2 Ω˜ 2 ω˜(k) = σ|k|3 + = σ1/2 |k|3 + , (48) has the NR-diagram shown in Fig.9; accordingly, its dy- 4 4

with (|k|3 + Ω˜ 2/4) not depending on σ. Correspondingly, the interaction coefficient Z has the form

Z = σ1/4 · Z˜ (49)

FIG. 9: NR-diagram for the clusters given by (44) (on the with Z˜ depending only on the wave numbers k1, k2, k3. left) and by (46) (on the right). This means that the cluster integrability defined by the 11 ratios of the corresponding coupling coefficients [49, 50] from of (36) it follows that in fact this expression gives also does not depend on the properties of fluid σ while the correct coupling coefficient for rotational capillary waves, not only in water but in arbitrary liquid. 0 ˜ ˜ 0 Zj1 /Zj = Zj/Zj (50) • Last but not least. The discrete turbulent dynamics shown in Fig.2 occurs for very general types of flow mo- ˜ ˜ 0 where Zj and Zj0 do not depend on σ and the indexes j, j tions and is also valid for wave systems possessing 4-wave correspond to the j-th and the j0-th triad in a cluster. resonances though the corresponding theoretical study is However, the magnitude of the vorticity Ω generating more involved. Due to Hasselman’s criterion for 4-wave the corresponding cluster will depend, of course, on the systems [22], there is no analog to the A-mode in this properties of a fluid as a function of σ. case, while the excitation of any one mode in a generic Indeed, σ is ratio of the coefficient of surface tension resonant quartet does not yield an energy flow within a σ˜ to the liquid density ρ, i.e. σ =σ/ρ ˜ , and is different quartet. At least two resonant modes have to be excited. even for the same liquid if experiments are performed The only exception is given by the quartets of the form under different conditions (such as a change of the ambi- ω1 + ω2 = 2ω3, which can be regarded essentially as a ent temperature). For instance, for water with ρ = 998 3-wave resonance and all the results presented above can kg/m3 and standard pressure, σ˜ = 72.14 mN/m if the be applied directly, with the A-mode having frequency ◦ temperature of water is 25 C and σ˜ = 75.09 mN/m if 2ω3. The dynamical system for a generic quartet with ◦ the temperature is 5 C. Accordingly, σ = 7.23·10−5 and frequency resonance condition σ = 7.52 · 10−5 for these two cases. ω(k1) + ω(k2) = ω(k3) + ω(k4), (51)

VI. DISCUSSION is also integrable in terms of Jacobian elliptic functions [45]. The time evolution of a quartet is defined by the Our main conclusions can be formulated as follows: fact whether or not two initially excited modes belong to • Among the rotational capillary waves with constant the same side of the (51) or to the different sides. Corre- non-zero vorticity resonances can occur only if 1) the flow spondingly, the pairs (ω1, ω2) and (ω3, ω4) are called 1- is two-dimensional, and 2) the magnitude of vorticity is pairs, while the pairs (ω1, ω3), (ω1, ω4), (ω2, ω3), (ω2, ω4) are called 2-pairs. The excitation of a 2-pair again does larger than Ωmin given by (38). • Two arbitrary rotational one-dimensional capillary not generate an energy exchange among the resonant waves can form a resonance only for appropriate magni- modes, while the energy pumping into a 1-pair will pro- tudes of the constant non-zero vorticity, with the exact duce it under some conditions (see [30], Ch. 4, for more magnitude given by (37). Thus a chosen magnitude of details). The NR-diagrams keep track of this dynami- the vorticity generates one isolated resonance triad with cal information in the form of the edges [29]. Applying a periodic energy exchange among the modes of the triad these results for gravity water waves (work in progress) and the magnitudes of the resulting amplitudes depend we can see that in this case the generation of rogue waves on the initial dynamical phase (20). In a laboratory ex- demands more conditions to be fulfilled than in a 3-wave periment where initially the A-mode of the triad is ex- system. cited we expect the appearance of some regular patterns on the surface of the liquid. • If some non-zero experimental accuracy is taken Acknowledgments into account, clusters of more complicated structure oc- cur. Their time evolution can be regular (integrable) or We acknowledge L.V. Abdurakhimov, D. Henderson, chaotic, depending on the ratios of coupling coefficients G.V. Kolmakov, S. Nazarenko and M. Shats for fruitful and sometimes also on the initial conditions. For an ar- discussions. A.C. was supported by the Vienna Science bitrary cluster its dynamical system can be written out and Technology Fund (WWTF). E.K. acknowledges the explicitly and solved numerically, thus predicting the re- support of the Austrian Science Foundation (FWF) un- sults of a laboratory experiment beforehand. der the project P20164-N18 “Discrete resonances in non- • The coupling coefficient for the corresponding dy- linear wave systems". E.W. was supported by an Alexan- namical system is given explicitly by (36). Note that der von Humboldt Research Fellowship.

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