{
  "id": "1804.05784",
  "version": "v1",
  "published": "2018-04-16T16:54:23.000Z",
  "updated": "2018-04-16T16:54:23.000Z",
  "title": "On passive scalars advected by two-, three- and $d$-dimensional incompressible flows",
  "authors": [
    "Jian-Zhou Zhu"
  ],
  "categories": [
    "physics.flu-dyn",
    "hep-th",
    "nlin.CD"
  ],
  "abstract": "When a passive scalar in the incompressible two-dimensional (2D) flow is so smart as to track the vorticity field to have a rugged correlation, it may form large scale patterns as the vortex with inverse transfer of the scalar energy, together with the 2D kinetic energy. For the $1$-form velocity $\\verb\"U\"$ solving the 4D ideal Euler equation, L. Tartar's relevant invariant $\\mathscr{N}=\\int_{\\mathcal{D}} d\\verb\"U\"\\wedge d\\verb\"U\"$ vanishes in the case of no boundary contribution from the integration by parts (typically for $\\mathcal{D} = \\mathbb{T}^4$), becoming a null correlation constraint for the corresponding 3D passive scalar $\\theta$ from the cylinder condition, so the absolute-equilibrium energy of $\\theta$ is expected to be equipartitioned and transferred forwardly to smaller scales in turbulence. General situation of dimension $d$, odd or even, is also remarked.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-16T16:54:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "passive scalar",
      "dimensional incompressible flows",
      "4d ideal euler equation",
      "form large scale patterns",
      "null correlation constraint"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}