dorsal/arxiv
View SchemaOn the decay of Burgers turbulence
| Authors | S. N. Gurbatov, S. I. Simdyankin, E. Aurell, U. Frisch, G. Tóth |
|---|---|
| Categories | |
| ArXiv ID | physics/9709002 |
| URL | https://arxiv.org/abs/physics/9709002 |
| DOI | 10.1017/S0022112097006241 |
| Journal | J. Fluid Mech. 344 (1997) 339--374 |
Abstract
This work is devoted to the decay ofrandom solutions of the unforced Burgers equation in one dimension in the limit of vanishing viscosity. The initial velocity is homogeneous and Gaussian with a spectrum proportional to $k^n$ at small wavenumbers $k$ and falling off quickly at large wavenumbers. In physical space, at sufficiently large distances, there is an ``outer region'', where the velocity correlation function preserves exactly its initial form (a power law) when $n$ is not an even integer. When $1<n<2$ the spectrum, at long times, has three scaling regions : first, a $|k|^n$ region at very small $k$\ms1 with a time-independent constant, stemming from this outer region, in which the initial conditions are essentially frozen; second, a $k^2$ region at intermediate wavenumbers, related to a self-similarly evolving ``inner region'' in physical space and, finally, the usual $k^{-2}$ region, associated to the shocks. The switching from the $|k|^n$ to the $k^2$ region occurs around a wave number $k_s(t) \propto t^{-1/[2(2-n)]}$, while the switching from $k^2$ to $k^{-2}$ occurs around $k_L(t)\propto t^{-1/2}$ (ignoring logarithmic corrections in both instances). The key element in the derivation of the results is an extension of the Kida (1979) log-corrected $1/t$ law for the energy decay when $n=2$ to the case of arbitrary integer or non-integer $n>1$. A systematic derivation is given in which both the leading term and estimates of higher order corrections can be obtained. High-resolution numerical simulations are presented which support our findings.
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"abstract": "This work is devoted to the decay ofrandom solutions of the unforced Burgers\nequation in one dimension in the limit of vanishing viscosity. The initial\nvelocity is homogeneous and Gaussian with a spectrum proportional to $k^n$ at\nsmall wavenumbers $k$ and falling off quickly at large wavenumbers. In physical\nspace, at sufficiently large distances, there is an ``outer region\u0027\u0027, where the\nvelocity correlation function preserves exactly its initial form (a power law)\nwhen $n$ is not an even integer. When $1\u003cn\u003c2$ the spectrum, at long times, has\nthree scaling regions : first, a $|k|^n$ region at very small $k$\\ms1 with a\ntime-independent constant, stemming from this outer region, in which the\ninitial conditions are essentially frozen; second, a $k^2$ region at\nintermediate wavenumbers, related to a self-similarly evolving ``inner region\u0027\u0027\nin physical space and, finally, the usual $k^{-2}$ region, associated to the\nshocks. The switching from the $|k|^n$ to the $k^2$ region occurs around a wave\nnumber $k_s(t) \\propto t^{-1/[2(2-n)]}$, while the switching from $k^2$ to\n$k^{-2}$ occurs around $k_L(t)\\propto t^{-1/2}$ (ignoring logarithmic\ncorrections in both instances). The key element in the derivation of the\nresults is an extension of the Kida (1979) log-corrected $1/t$ law for the\nenergy decay when $n=2$ to the case of arbitrary integer or non-integer $n\u003e1$.\nA systematic derivation is given in which both the leading term and estimates\nof higher order corrections can be obtained. High-resolution numerical\nsimulations are presented which support our findings.",
"arxiv_id": "physics/9709002",
"authors": [
"S. N. Gurbatov",
"S. I. Simdyankin",
"E. Aurell",
"U. Frisch",
"G. T\u00f3th"
],
"categories": [
"physics.flu-dyn",
"chao-dyn",
"cond-mat.stat-mech",
"nlin.CD"
],
"doi": "10.1017/S0022112097006241",
"journal_ref": "J. Fluid Mech. 344 (1997) 339--374",
"title": "On the decay of Burgers turbulence",
"url": "https://arxiv.org/abs/physics/9709002"
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