{
  "id": "1310.5914",
  "version": "v2",
  "published": "2013-10-22T13:36:46.000Z",
  "updated": "2014-11-24T10:19:25.000Z",
  "title": "Hierarchical Bayesian analysis of the velocity power spectrum in supersonic turbulence",
  "authors": [
    "Lukas Konstandin",
    "Rahul Shetty",
    "Philipp Girichidis",
    "Ralf S. Klessen"
  ],
  "comment": "10 pages, 6 figures",
  "categories": [
    "astro-ph.SR"
  ],
  "abstract": "Turbulence is a dominant feature operating in gaseous flows across nearly all scales in astrophysical environments. Accordingly, accurately estimating the statistical properties of such flows is necessary for developing a comprehensive understanding of turbulence. We develop and employ a hierarchical Bayesian fitting method to estimate the parameters describing the scaling relationships of the velocity power spectra of supersonic turbulence. We demonstrate the accuracy and other advantages of this technique compared with ordinary linear regression methods. Using synthetic power spectra, we show that the Bayesian method provides accurate parameter and error estimates. Commonly used normal linear regression methods can provide estimates that fail to recover the underlying slopes, up to 70% of the instances, even when considering the 2std.dev. uncertainties. Additionally, we apply the Bayesian methods to analyse the statistical properties of compressible turbulence in 3D numerical simulations. We model driven, isothermal, turbulence with rms Mach numbers in the highly supersonic regime M~15. We study the influence of purely solenoidal (divergence-free) and purely compressive (curl-free) forcing on the scaling exponent of the power spectrum. In simulations with solenoidal forcing and 1024^3 resolution, our results indicate that there is no extended inertial range with a constant scaling exponent. The bottleneck effect results in a curved power spectrum at all wave numbers and is more pronounced in the transversal modes compared with the longitudinal modes. Therefore, this effect is stronger in stationary turbulent flows driven by solenoidal forcing compared to the compressive one. The longitudinal spectrum driven with compressive forcing is the only spectrum with constant scaling exponent z=-1.94 +- 0.01, corresponding to slightly shallower slopes than the Burger's prediction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-22T13:36:46.000Z",
      "title": "Hierarchical Bayesian analysis of the velocity power spectrum in subsonic and supersonic turbulence",
      "abstract": "We analyse the statistical properties of turbulence using a suite of three-dimensional numerical simulations. We model driven, compressible, isothermal, turbulence with r.m.s. Mach numbers ranging from the subsonic to the highly supersonic regime. We focus on the extreme cases of purely solenoidal (divergence-free) and purely compressive (curl-free) forcing. By employing a hierarchical Bayesian fitting method, we estimate the parameters describing the scaling relationships of the velocity power spectra. The method explicitly treats uncertainties and time-dependent fluctuations through Markov Chain Monte Carlo sampling. We find that the scaling exponents of the decomposed spectra strongly depend on the forcing mechanism, due to the energy transfer between the transverse and longitudinal components. Accordingly, we derive a phenomenological model describing this behaviour. The scaling exponents are in agreement with a Kolmogorov -5/3 spectrum in a tiny range of k [12:15] in the case of the transverse velocity spectrum driven with solenoidal forcing. With compressive forcing and with high Mach number, both longitudinal and transverse spectra show an almost universal behaviour and resemble the Burgers case with a slope of -2. We also analyse the spectra in the bottleneck regime and show that the bottleneck bump decreases with increasing r.m.s. Mach number. It is also more evident on smaller scales in the transverse spectrum in comparison with the longitudinal spectrum.",
      "comment": "25 pages, 8 figures",
      "journal": null,
      "doi": null,
      "authors": [
        "Lukas Konstandin",
        "Rahul Shetty",
        "Christoph Federrath",
        "Philipp Girichidis",
        "Ralf S. Klessen"
      ]
    },
    {
      "version": "v2",
      "updated": "2014-11-24T10:19:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "velocity power spectrum",
      "hierarchical bayesian analysis",
      "supersonic turbulence",
      "chain monte carlo sampling",
      "mach number"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1093/mnras/stu2154",
      "journal": "Monthly Notices of the Royal Astronomical Society",
      "year": 2015,
      "month": "Jan",
      "volume": 446,
      "number": 2,
      "pages": 1775
    },
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015MNRAS.446.1775K"
    }
  }
}