{
  "id": "0904.3028",
  "version": "v1",
  "published": "2009-04-20T13:39:21.000Z",
  "updated": "2009-04-20T13:39:21.000Z",
  "title": "Number of degrees of freedom of two-dimensional turbulence",
  "authors": [
    "Chuong V. Tran",
    "Luke Blackbourn"
  ],
  "comment": "7 journal pages, to appear in PRE",
  "categories": [
    "physics.flu-dyn",
    "gr-qc"
  ],
  "abstract": "We derive upper bounds for the number of degrees of freedom of two-dimensional Navier--Stokes turbulence freely decaying from a smooth initial vorticity field $\\omega(x,y,0)=\\omega_0$. This number, denoted by $N$, is defined as the minimum dimension such that for $n\\ge N$, arbitrary $n$-dimensional balls in phase space centred on the solution trajectory $\\omega(x,y,t)$, for $t>0$, contract under the dynamics of the system linearized about $\\omega(x,y,t)$. In other words, $N$ is the minimum number of greatest Lyapunov exponents whose sum becomes negative. It is found that $N\\le C_1R_e$ when the phase space is endowed with the energy norm, and $N\\le C_2R_e(1+\\ln R_e)^{1/3}$ when the phase space is endowed with the enstrophy norm. Here $C_1$ and $C_2$ are constant and $R_e$ is the Reynolds number defined in terms of $\\omega_0$, the system length scale, and the viscosity $\\nu$. The linear (or nearly linear) dependence of $N$ on $R_e$ is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number. This result is in a sharp contrast to the forced case, for which well-known estimates for the Hausdorff dimension $D_H$ of the global attractor scale highly superlinearly with $\\nu^{-1}$. We argue that the \"extra\" dependence of $D_H$ on $\\nu^{-1}$ is not an intrinsic property of the turbulent dynamics. Rather, it is a \"removable artifact,\" brought about by the use of a time-independent forcing as a model for energy and enstrophy injection that drives the turbulence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-20T13:39:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47.27.-i",
      "05.45.-a"
    ],
    "keywords": [
      "two-dimensional turbulence",
      "phase space",
      "attractor scale",
      "navier-stokes turbulence freely decaying",
      "smooth initial vorticity field"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1103/PhysRevE.79.056308",
      "journal": "Physical Review E",
      "year": 2009,
      "month": "May",
      "volume": 79,
      "number": 5,
      "pages": "056308"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009PhRvE..79e6308T",
      "inspire": 1369976
    }
  }
}