dorsal/arxiv
View SchemaThe decay of multiscale signals - deterministic model of the Burgers turbulence
| Authors | S. N. Gurbatov, A. V. Troussov |
|---|---|
| Categories | |
| ArXiv ID | physics/0002043 |
| URL | https://arxiv.org/abs/physics/0002043 |
| DOI | 10.1016/S0167-2789(00)00090-7 |
Abstract
This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers' equation in the limit of vanishing viscosity. A deterministic model of turbulence-like evolution is considered. We con- struct the initial perturbation as a piecewise linear analog of the Weierstrass function. The wavenumbers of this function form a "Weierstrass spectrum", which accumulates at the origin in geometric progression."Reverse" sawtooth functions with negative initial slope are used in this series as basic functions, while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wavenumbers is the same as for the continuous spectrum in Burgers turbulence. Combining these two ideas allows us to obtain an exact analytical solution for the velocity field. We also notice that such multiscale waves may be constructed for multidimensional Burgers' equation. This solution has scaling exponent h=-(1+n)/2 and its evolution in time is self-similar with logarithmic periodicity and with the same average law L(t) as for Burgers turbulence. Shocklines form self-similar regular tree-like struc- tures. This model also describes important properties of the Burgers turbulence such as the self-preservation of the evolution of large scale structures in the presence of small scales perturbations.
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"abstract": "This work is devoted to the study of the decay of multiscale deterministic\nsolutions of the unforced Burgers\u0027 equation in the limit of vanishing\nviscosity.\n A deterministic model of turbulence-like evolution is considered. We con-\nstruct the initial perturbation as a piecewise linear analog of the Weierstrass\nfunction. The wavenumbers of this function form a \"Weierstrass spectrum\", which\naccumulates at the origin in geometric progression.\"Reverse\" sawtooth functions\nwith negative initial slope are used in this series as basic functions, while\ntheir amplitudes are chosen by the condition that the distribution of energy\nover exponential intervals of wavenumbers is the same as for the continuous\nspectrum in Burgers turbulence. Combining these two ideas allows us to obtain\nan exact analytical solution for the velocity field. We also notice that such\nmultiscale waves may be constructed for multidimensional Burgers\u0027 equation.\n This solution has scaling exponent h=-(1+n)/2 and its evolution in time is\nself-similar with logarithmic periodicity and with the same average law L(t) as\nfor Burgers turbulence. Shocklines form self-similar regular tree-like struc-\ntures. This model also describes important properties of the Burgers turbulence\nsuch as the self-preservation of the evolution of large scale structures in the\npresence of small scales perturbations.",
"arxiv_id": "physics/0002043",
"authors": [
"S. N. Gurbatov",
"A. V. Troussov"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1016/S0167-2789(00)00090-7",
"title": "The decay of multiscale signals - deterministic model of the Burgers turbulence",
"url": "https://arxiv.org/abs/physics/0002043"
},
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