Scaling laws and intermittency in phase turbulence

2014 We investigate numerically the statistical properties of the Kuramoto-SivashinskyTsuzuki model [1-3], which describes the phase fluctuations of extended systems near the transition to turbulence. This model has some similarities with hydrodynamic turbulence [4]. The time fluctuations exhibit an unexpected scaling corresponding to anomalous diffusion of the phase. Other results also point to an intermittent behaviour. J. Physique Lett. 46 (1985) L-793 L-797 ler SEPTEMBRE 1985, Classification Physics Abstracts 05.40 47.25 1. Phase turbulence as hydrodynamics with many chaotic degrees of freedom. Many pattern-forming instabilities can be modelled by the equation : where subscripts stand for derivatives. It is shown to describe the temporal phase of coupled chemical oscillators [1]. It also describes the behaviour of other extended pattern forming systems, such as flame fronts [2] and fluid interfaces [3]. The derivative of the phase satisfies an equation of hydrodynamical type : where v = ~x. In this form the model is reminiscent of the Burgers equation which was also proposed as a model for turbulence. But in contrast with the Burgers equation, (1) displays a sustained chaotic behaviour [4, 5]. This occurs for large enough values of L and for the boundary conditions Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019850046017079300 L-794 JOURNAL DE PHYSIQUE LETTRES The chaos then involves a number of degrees of freedom proportional to L. Rigorous bounds on this number were given for even periodic solutions [6]. The model (2) somehow mimics hydrodynamical turbulence where a chaotic behaviour with a large number of degrees of freedom is also expected [7]. It would hence be interesting to know whether statistical theories of turbulence can be applied in this case. A detailed check of such theories is made possible by modem computing facilities. The analogy with hydrodynamics is relevant because, as pointed out by several authors, basic ideas about turbulence, like the Kolmogorov theory of energy cascade [8], depend very little on the accurate form of the Navier-Stokes equations. Actually, the Kolmogorov theory mainly relies on the existence of a quantity approximately conserved in the equations, the kinetic energy. In equation (2), the quantity V2 might play a similar role. However, it has been shown that no inertial range cascade existed for (2) [4, 9]. There seems to be, however, another scaling in the space spectrum : a range of equipartition for the « energy » v2 has been found numerically [4] and using the D.I.A. perturbation method [10-11]. Further pursuing the above analogy, we note that v is exactly conserved, just as the actual momentum in hydrodynamics. « Exact conservation » means that there exists a current J(x, t) such that