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Next: Three-wave case Up: Wave Turbulence "Recent developments Previous: Introduction

Setting the stage I: Dynamical Equations of motion

Wave turbulence formulation deals with a many-wave system with dispersion and weak nonlinearity. For systematic derivations one needs to start from Hamiltonian equation of motion. Here we consider a system of weakly interacting waves in a periodic box [1],
$\displaystyle i\dot c_l$ $\displaystyle =$ $\displaystyle \frac{\partial {\cal H}}{\partial \bar c_l},$ (1)

where $ c_l$ is often called the field variable. It represents the amplitude of the interacting plane wave. The Hamiltonian is represented as an expansion in powers of small amplitude,

$\displaystyle {\cal H} = {\cal H}_2 + {\cal H}_3+ {\cal H}_4 + {\cal H}_5 + \dots,$ (2)

where $ H_j$ is a term proportional to product of $ j$ amplitudes $ c_l$,

$\displaystyle {\cal H}_j = \sum\limits_{q_1,q_2,q_3, \dots q_n, p_1,p_2,\dots p...
...ts \bar c_{q_m} c_{p_1} c_{p_2}\dots c_{p_m}+{\rm c.c}.\right), \ \ \ n+m=j \\ $    

where $ q_1,q_2,q_3, \dots q_n$ and $ p_1,p_2,\dots p_m$ are wavevectors on a $ d$-dimensional Fourier space lattice. Such general $ j$-wave Hamiltonian describe the wave-wave interactions where $ n$ waves collide to create $ m$ waves. Here $ T^{q_1 q_2 \dots
q_n}_{p1,p2 \dots p_m}$ represents the amplitude of the $ n\to m$ process. In this paper we are going to consider expansions of Hamiltonians up to forth order in wave amplitude.

Under rather general conditions the quadratic part of a Hamiltonian, which correspond to a linear equation of motion, can be diagonalised to the form

$\displaystyle {\cal H}_2 = \sum_{n=1}^\infty \omega_n\vert c_n\vert^2.$ (3)

This form of Hamiltonian correspond to noninteracting (linear) waves. First correction to the quadratic Hamiltonian is a cubic Hamiltonian, which describes the processes of decaying of single wave into two waves or confluence of two waves into a single one. Such a Hamiltonian has the form
$\displaystyle {\cal H}_3=
\sum_{l,m,n=1}^\infty V^l_{mn} \bar c_{l} c_m c_n\delta^l_{m+n}+c.c.,$      

where $ \epsilon \ll 1$ is a formal parameter corresponding to small nonlinearity ( $ \epsilon$ is proportional to the small amplitude whereas $ c_n$ is normalised so that $ c_n \sim 1$.) Most general form of three-wave Hamiltonian would also have terms describing the confluence of three waves or spontaneous appearance of three waves out of vacuum. Such a terms would have a form

$\displaystyle \sum_{l,m,n=1}^\infty \ U^{lmn} c_{l} c_m c_n\delta_{l+m+n}+c.c.$

It can be shown however that for systems that are dominated by three-wave resonances such terms do not contribute to long term dynamics of systems. We therefore choose to omit those terms.

The most general four-wave Hamiltonian will have $ 1\to 3$, $ 3\to 1$, $ 2\to
2$, $ 4\to 0 $ and $ 0\to 4$ terms. Nevertheless $ 1\to 3$, $ 3\to 1$, $ 4\to 0 $ and $ 0\to 4$ terms can be excluded from Hamiltonian by appropriate canonical transformations, so that we limit our consideration to only $ 2\to
2$ terms of $ {\cal H}_4$, namely

$\displaystyle {\cal H}_4=
W^{lm}_{\mu\nu} \bar c_{l} \bar c_m c_\mu c_\nu.$      

It turns out that generically most of the weakly nonlinear systems can be separated into two major classes: the ones dominated by three-wave interactions, so that $ {\cal H}_3$ describes all the relevant dynamics and $ {\cal H}_4$ can be neglected, and the systems where the three-wave resonance conditions cannot be satisfied, so that the $ {\cal H}_3$ can be eliminated from a Hamiltonian by an appropriate near-identical canonical transformation [25]. Consequently, for the purpose of this paper we are going to neglect either $ {\cal H}_3$ or $ {\cal H}_4$, and study the case of resonant three-wave or four-wave interactions.

Examples of three-wave system include the water surface capillary waves, internal waves in the ocean and Rossby waves. The most common examples of the four-wave systems are the surface gravity waves and waves in the NLS model of nonlinear optical systems and Bose-Einstein condensates. For reference we will give expressions for the frequencies and the interaction coefficients corresponding to these examples.

For the capillary waves we have [1,5],

$\displaystyle \omega_j = \sqrt{\sigma k^3},$ (4)


$\displaystyle V^l_{mn} = {1 \over 8 \pi \sqrt{2 \sigma}} (\omega_{l} \omega_m \...
...r ( k_l k_m)^{1/2} k_n } - {L_{k_l, -k_n} \over ( k_l k_n)^{1/2} k_m } \right],$ (5)


$\displaystyle L_{k_m, k_n} = ({\bf k}_m \cdot {\bf k}_n) + k_m k_n$ (6)

and $ \sigma$ is the surface tension coefficient.

For the Rossby waves [13,14],

$\displaystyle \omega_j = {\beta k_{jx} \over 1 + \rho^2 k_j^2},$ (7)


$\displaystyle V^l_{mn} = -{i \beta \over 4 \pi} \vert k_{lx} k_{mx} k_{nx} \ver...
... - { k_{my} \over 1 + \rho^2 k_m^2} - { k_{ny} \over 1 + \rho^2 k_n^2} \right),$ (8)

where $ \beta$ is the gradient of the Coriolis parameter and $ \rho$ is the Rossby deformation radius.

The simplest expressions correspond to the NLS waves [15,7],

$\displaystyle \omega_j=\vert k_j\vert^2, \hspace{1cm} W^{lm}_{\mu \nu} = 1.$ (9)

The surface gravity waves are on the other extreme. The frequency is $ \omega = \sqrt{gk}$ but the matrix element is given by notoriously long expressions which can be found in [1,17].

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Next: Three-wave case Up: Wave Turbulence "Recent developments Previous: Introduction
Dr Yuri V Lvov 2007-01-23