form of frozen turbulence, with distinct modes just keep- ing their initial energy. As it was shown quite recently, rotational capillary waves do have exact resonances, for specific positive magnitudes of the vorticity [8], which means that their discrete regime will have a different dy- namics. The simplest physical system where non-zero vorticity is important is that of tidal flows, which can be realistically modeled as two-dimensional flows with con- stant non-zero vorticity, with the sign of the vorticity distinguishing between ebb/tide [16]. The discrete layer is formed by exact resonances and quasi-resonances governed by (2) and (3), while the con- tinuous layer is formed by approximate interactions given by (4). Four main distinctions between the time evolu- tion of these two layers have to be accounted for: This layer is described by the wave kinetic equations and its stationary solutions in the form of power energy spec- tra. The main necessary assumptions in kinetic wave turbulence theory are random phase approximation, lo- cality of interactions and existence of a finite inertial in- terval . In the kinetic regime the time evolution of the wavefield does not depend on the details of the initial 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- portant. Interactions are local , that is, only wave vectors with lengths of order k are allowed to interact. More- over, interactions take place only at some finite interval , and no information is given about time evolution of the modes with wavenumbers lying outside this interval. 1. dependence or independence on the initial condi- tions, 2. locality or non-locality of interactions, 3. existence or nonexistence of the inertial interval, 4. random or coherent phases. To introduce the three generic wave turbulent regimes we need to distinguish between the notions of quasi- resonance and an approximate (non-resonant) interac- tion. Let us call a parameter δ, resonance width . Regarding the dispersion function j as a function of integer variables which are indices of Fourier harmonics (corresponding to a finite-size box), it was shown by [28] that the resonance width can not be ar- bitrary small. This means that if δ () = 0 , then δ () ≥ R () where R () is some positive number. Accordingly, ex- act resonances, quasi-resonances and approximate inter- actions can be introduced as follows: where R $_{max}$() defines the distance to the nearest reso- nance. The number R () can easily be found numerically, for a given dispersion function, as in [46]. This layer is governed by the set of dynamical systems describing resonance clusters; its time evolution depends crucially on the initial conditions . Modes’ phases are coherent; the linear combination of the phases corre- sponding to the choice of resonance conditions (called dynamical phase of a resonant cluster) describes phases’ time evolution (see Section III B). Resonant interactions are not local , that is, waves with substantially different k -scales may form resonances. For instance, gravity- capillary waves with wave lengths of order of k and of the order of k 3 can form a resonance; more examples are given in [30]. Also the existence or absence of inertial in- terval is not important: resonances can occur all over the k -spectrum. Moreover, in some wave systems waves with arbitrary wavelengths can form a resonance, for instance rotational capillary waves [8]. We will discuss this aspect in detail in Section V. According to (2)–(4), discrete and continuous layers in a wave turbulent system are defined as follows. The continuous layer The discrete layer II. WAVE TURBULENT REGIMES The main goal of our paper is to study the discrete 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- wave resonance systems and we present known results on irrotational capillary waves (with zero vorticity). De- tailed presentation of discrete wave turbulent regime is given in Section III. In Section IV we provide some back- ground material on water waves with vorticity. Reso- nance clustering is constructed and studied in Section V. A brief discussion concludes the paper. 2 | 1 ± 2 ± ... ± s | = δ () ≥ 0 , (1) (1) exact resonances are solutions of (1) with (2) quasi-resonances are solutions of (1) with (3) approximate interactions are solutions of (1) with The main features of the model of laminated turbu- lence can be summarized as follows. δ () = 0; (2) 0 < δ () < R (); (3) R () ≤ δ () < R $_{max}$() , (4) A. The model of laminated wave turbulence Main aspects • Independent clusters of resonantly interacting modes can exist all over k -spectrum . • The time evolution of the modes belonging to a clus- ter are not described by KZ-spectra, thus leaving the