{
  "id": "2309.01602",
  "version": "v1",
  "published": "2023-09-04T13:38:12.000Z",
  "updated": "2023-09-04T13:38:12.000Z",
  "title": "Fingering convection in a spherical shell",
  "authors": [
    "T. Tassin",
    "T. Gastine",
    "A. Fournier"
  ],
  "comment": "43 pages, 22 figures, 3 tables, submitted to JFM",
  "categories": [
    "physics.flu-dyn",
    "astro-ph.EP",
    "astro-ph.SR",
    "physics.geo-ph"
  ],
  "abstract": "We use 120 three dimensional direct numerical simulations to study fingering convection in non-rotating spherical shells. We investigate the scaling behaviour of the flow lengthscale, mean velocity and transport of chemical composition over the fingering convection instability domain defined by $1 \\leq R_\\rho \\leq Le$, $R_\\rho$ being the ratio of density perturbations of thermal and compositional origins. We show that the horizontal size of the fingers is accurately described by a scaling law of the form $\\mathcal{L}_h/d \\sim |Ra_T|^{-1/4} (1-\\gamma)^{-1/4}/\\gamma^{-1/4}$, where $d$ is the shell depth, $Ra_T$ the thermal Rayleigh number and $\\gamma$ the flux ratio. Scaling laws for mean velocity and chemical transport are derived in two asymptotic regimes close to the two edges of the instability domain, namely $R_\\rho \\lesssim Le$ and $R_\\rho \\gtrsim 1$. For the former, we show that the transport follows power laws of a small parameter $\\epsilon^\\star$ measuring the distance to onset. For the latter, we find that the Sherwood number $Sh$, which quantities the chemical transport, gradually approaches a scaling $Sh\\sim Ra_\\xi^{1/3}$ when $Ra_\\xi \\gg 1$; and that the P\\'eclet number accordingly follows $Pe \\sim Ra_\\xi^{2/3} |Ra_T|^{-1/4}$, $Ra_\\xi$ being the chemical Rayleigh number. When the Reynolds number exceeds a few tens, a secondary instability may occur taking the form of large-scale toroidal jets. Jets distort the fingers resulting in Reynolds stress correlations, which in turn feed the jet growth until saturation. This nonlinear phenomenon can yield relaxation oscillation cycles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-04T13:38:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spherical shell",
      "mean velocity",
      "yield relaxation oscillation cycles",
      "fingering convection instability domain",
      "large-scale toroidal jets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}