{
  "id": "1306.2495",
  "version": "v2",
  "published": "2013-06-11T12:00:52.000Z",
  "updated": "2014-09-23T11:26:39.000Z",
  "title": "Large-scale dynamo action due to $α$ fluctuations in a linear shear flow",
  "authors": [
    "S. Sridhar",
    "Nishant K. Singh"
  ],
  "comment": "33 pages, 6 figures, accepted for publication in Monthly Notices of the Royal Astronomical Society",
  "categories": [
    "astro-ph.GA",
    "physics.flu-dyn",
    "physics.plasm-ph"
  ],
  "abstract": "We present a model of large-scale dynamo action in a shear flow that has stochastic, zero-mean fluctuations of the $\\alpha$ parameter. This is based on a minimal extension of the Kraichnan-Moffatt model, to include a background linear shear and Galilean-invariant $\\alpha$-statistics. Using the first order smoothing approximation we derive a linear integro-differential equation for the large-scale magnetic field, which is non perturbative in the shearing rate $S\\,$, and the $\\alpha$-correlation time $\\tau_\\alpha\\,$. The white-noise case, $\\tau_\\alpha = 0\\,$, is solved exactly, and it is concluded that the necessary condition for dynamo action is identical to the Kraichnan-Moffatt model without shear; this is because white-noise does not allow for memory effects, whereas shear needs time to act. To explore memory effects we reduce the integro-differential equation to a partial differential equation, valid for slowly varying fields when $\\tau_\\alpha$ is small but non zero. Seeking exponential modal solutions, we solve the modal dispersion relation and obtain an explicit expression for the growth rate as a function of the six independent parameters of the problem. A non zero $\\tau_\\alpha$ gives rise to new physical scales, and dynamo action is completely different from the white-noise case; e.g. even weak $\\alpha$ fluctuations can give rise to a dynamo. We argue that, at any wavenumber, both Moffatt drift and Shear always contribute to increasing the growth rate. Two examples are presented: (a) a Moffatt drift dynamo in the absence of shear; (b) a Shear dynamo in the absence of Moffatt drift.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-11T12:00:52.000Z",
      "title": "Dynamo action due to alpha fluctuations in a shear flow: mean--field theory",
      "abstract": "We present an analytical theory of the growth of a large-scale mean magnetic field in a linear shear flow with fluctuations in time of the alpha parameter (equivalently, kinetic helicity). Using shearing coordinates and Fourier variables we derive a set of coupled integro-differential equations, governing the dynamics of the mean magnetic field, that are non perturbative in the rate of shear. When the alpha fluctuations are of white-noise form, the mean electromotive force (EMF) is identical to the negative diffusive form derived by Kraichnan for the case of no shear; the physical reason is that shear takes time to act, and white-noise fluctuations have zero correlation time. We demonstrate that the white-noise case does not allow for large-scale dynamo action. We then allow for a small but non zero correlation time and show that, for a slowly varying mean magnetic field, the mean EMF has additional terms that depend on a combination of shear and alpha fluctuations; the mean-field equations now reduce to a set of coupled partial differential equations. A dispersion relation for modes is derived and studied in detail for growing solutions. Our salient results are: (i) a necessary condition for dynamo action giving the minimum value of shear required; (ii) two types of dynamos depending on the different forms taken by the growth rate as a function of wavenumber; (iii) explicit expressions for the growth rate and wavenumber of the fastest growing mode; these are not only consistent with the scalings with shear seen in numerical simulations, but also provide an estimate of the strength of alpha fluctuations.",
      "comment": "19 pages, 3 figures, submitted to Monthly Notices of the Royal Astronomical Society",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-23T11:26:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large-scale dynamo action",
      "linear shear flow",
      "fluctuations",
      "moffatt drift",
      "white-noise case"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1093/mnras/stu1981",
      "journal": "Monthly Notices of the Royal Astronomical Society",
      "year": 2014,
      "month": "Dec",
      "volume": 445,
      "number": 4,
      "pages": 3770
    },
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1238078,
      "adsabs": "2014MNRAS.445.3770S"
    }
  }
}