Heuristic model for the energy spectrum of phase turbulence

Abstract

We present a heuristic model for the energy spectrum of the one-dimensional phase turbulence in the steady state of the Kuramoto-Sivashinsky equation. Our model contains an energy transfer mechanism from low-to high-wave-vector modes. The energy transfer is written as the sum of local and nonlocal interactions. Our analytical results show good agreement with numerical simulations, particularly for the hump in the energy spectrum, which is mainly due to the local interactions. DOI: 10.1103/PhysRevE.64.057301 PACS number͑s͒: 47.27.Eq, 05.45.Ϫa, 47.52.ϩj, 47.54.ϩr A satisfactory understanding of spatially extended systems, although fascinating, is a difficult task. The evolution of these systems is generally described by nonlinear partial differential equations where analytical results are rather scarce ͓1͔. Phase turbulence is the irregular behavior of an extended system described by the paradigmatic KuramotoSivashinsky equation ͑KSE͒, one of the simplest partial differential equations exhibiting chaotic behavior ͓2͔. It appears in a variety of physical systems driven away from equilibrium such as reaction-diffusion chemical systems ͓3͔ or flame front propagation ͓4͔. In one dimension it readswith , Ͼ0. Defining uϭ x as the velocity field, one gets the alternative equationThis equation can also be written in the standard scaled form u t ϩuu x ϩu xx ϩu xxxx ϭ0. The periodic boundary conditions normally used are u(xϩL)ϭu(x), u x (xϩL)ϭu x (x), etc. L is the length of the system. In the thermodynamic limit (L →ϱ) there is no free control parameter. The unstable growth of fluctuations given by the term u xx acts as an energy source in the large-wavelength region. In the shortwavelength region the fluctuations are attenuated by the term u xxxx which acts as a stabilizing energy sink. We can say that and play the roles of an ''antiviscosity'' and a ''hyperviscosity,'' respectively. Several papers have addressed the study of the longwavelength behavior-the hydrodynamic limit-of the KSE in one dimension. It was conjectured by Yakhot ͓5͔, using a perturbative renormalization group approach, that the statistical behavior of the KSE, written as in Eq. ͑2͒, is equivalent to the stochastic Burgers equation ͓6͔ u t ϩuu x Ϫ u xx ϩ u ϭ0, where Ͼ0 is a renormalized viscosity and u is a Gaussian white noise forcing. Alternatively, the KSE written as in the Eq. ͑1͒, is equivalent to the Kardar-Parisi-Zhang ͑KPZ͒ equation ͓7͔ t ϩ x 2 /2Ϫ xx ϩ ϭ0. The numerical work of Sneppen et al. ͓8͔ strongly supports Yakhot's conjecture. An analytical demonstration of the connection between KS and KPZ equations was given by Chow and Hwa ͓9͔ by explicitly coarse-graining the KSE. Generically, the system forms cells of a preferred size. These cells are locally compressed or stretched giving rise to cell creation or annihilation. This mechanism provides a positive renormalized viscosity and the cell interactions are sufficiently uncorrelated to give rise to the random forcing .One of the quantities of primary interest when the KSE is numerically...

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