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Physics of Fluids / Volume 24 / Issue 2 / ARTICLES / Turbulent Flows
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Phys. Fluids 24, 025102 (2012); http://dx.doi.org/10.1063/1.3683556 (14 pages)

Wavelet decomposition of forced turbulence: Applicability of the iterative Donoho-Johnstone threshold

Jesse W. Lord1, Mark P. Rast1, Christopher Mckinlay2, John Clyne2, and Pablo D. Mininni3

1Laboratory for Atmospheric and Space Physics, Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, Colorado 80309-0391, USA
2Computational and Information Systems Laboratory, National Center for Atmospheric Research (NCAR), Boulder, Colorado 80307-3000, USA
3Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina

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(Received 9 May 2011; accepted 18 January 2012; published online 14 February 2012)

We examine the decomposition of forced Taylor-Green and Arn'old-Beltrami-Childress (ABC) flows into coherent and incoherent components using an orthonormal wavelet decomposition. We ask whether wavelet coefficient thresholding based on the Donoho-Johnstone criterion can extract a coherent vortex signal while leaving behind Gaussian random noise. We find that no threshold yields a strictly Gaussian incoherent component, and that the most Gaussian incoherent flow is found for data compression lower than that achieved with the fully iterated Donoho-Johnstone threshold. Moreover, even at such low compression, the incoherent component shows clear signs of large-scale spatial correlations that are signatures of the forcings used to drive the flows.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. FORCED TURBULENCE SIMULATIONS
  3. COHERENT VORTEX EXTRACTION
  4. CVE ANALYSIS OF TAYLOR-GREEN AND ABC FLOWS
  5. A CLOSER LOOK AT THE ITERATIVE DONOHO-JOHNSTONE THRESHOLD
    1. Is the incoherent vorticity Gaussianly distributed?
    2. Does the incoherent flow helicity indicate random vector orientations?
    3. Does the incoherent kinetic energy scale as k 2 ?
  6. SPATIAL AND TEMPORAL CORRELATIONS
  7. CONCLUSION

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KEYWORDS and PACS

Keywords

Gaussian noise, iterative methods, turbulence, vortices

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3683556

PUBLICATION DATA

ISSN

1070-6631 (print)  
1089-7666 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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Figures (5) Tables (1)

Figures (click on thumbnails to view enlargements)

FIG.1
Logarithm of the probability density of the velocity P(ux) in (a) and vorticity P(ωx) in (b). The coherent (dashed) and incoherent (dotted-dashed) components of the original (solid) Taylor-Green flow were computed using the Coifman12 wavelet in the CVE analysis (iterative Donoho-Johnstone threshold applied to vorticity magnitude). Dotted curves indicate Gaussian probability density functions (PDFs) with the same mean and variance. Insets show incoherent flow velocity and vorticity PDFs and the Gaussian fits over a narrower range of values. The flow is quite isotropic with nearly identical PDFs for the y and z velocity and vorticity, though the z velocity is more Gaussian with somewhat smaller variance.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
Relative kinetic helicity distribution in the (a) Taylor-Green and (b) ABC flows. Black curves plot values for the original flow (solid) and the coherent (dashed) and incoherent (dotted-dashed) components, the later found using the Coifman12 wavelet in the CVE analysis (iterative Donoho-Johnstone threshold applied to the vorticity vector magnitude). Red and blue curves show the same components obtained using the most Gaussian vorticity and 60% compression thresholds, respectively. Green curves plot values obtained when applying the iterative Donoho-Johnstone threshold to individual vorticity component amplitudes. Insets display incoherent results over a limited range, and dotted curves in inset (a) indicate helicity distribution of randomized vector components (Sec. 5B).

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
Kinetic energy spectra of the (a) Taylor-Green and (b) ABC flows. Black curves for the original flow (solid), and coherent (dashed), and incoherent (dotted-dashed) components, the later determined using the Coifman12 wavelet in the CVE analysis (iterative Donoho-Johnstone threshold applied to the vorticity vector magnitude). Red and blue curves show the same components obtained using the most Gaussian vorticity and 60% compression thresholds, respectively. Green curves for values obtained by applying the iterative Donoho-Johnstone threshold to individual vorticity component amplitudes. Note that the wavelet decomposition introduces power above the Orszag frequency kmax into both the coherent and incoherent components (Sec. 3), but that these sum to zero in amplitude. Fiducial k−5/3 and k2 lines shown for reference.

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
In (a), the Anderson-Darling test statistic A2 measuring the Gaussianity of the incoherent component of the flow as a function of compression, and in (b) A2 as a function of threshold value ADJ. Filled and empty symbols indicate analysis on the Taylor-Green and ABC solutions, respectively. The incoherent component was extracted using the Haar (circles), Coifman12 (squares), and Coifman30 (triangles) wavelets. Black curves highlight Coifman12 and Haar analysis of the Taylor-Green flow. The incoherent A2 values obtained for thresholds (applied to vorticity vector magnitude of the Coifman12 wavelet decomposition) yielding the most Gaussian incoherent component are shown in red and those yielding 60% compression in blue (60% compression symbols for Taylor-Green and ABC flows directly overlap). The incoherent A2 values obtained using the iterative Donoho-Johnstone threshold applied to the individual vorticity amplitudes are shown by the six disconnected points with highest compression on the right of (a) without crosses. Symbols with superimposed crosses indicate results for incoherent flows determined by application of the single iteration threshold to the vorticity vector magnitude, as done in previous studies.

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.5
In (a), spatial autocorrelation ρz of the Taylor-Green enstrophy as a function of the spatial lag Δz, and in (b), the temporal correlation ρτ of Taylor-Green vorticity ωx as a function of temporal lag τ (vorticity chosen here only to reduce the computational cost of time series analysis). Black curves for the original flow (solid) and coherent (dashed) and incoherent (dotted-dashed) components, the later determined using the Coifman12 wavelet in the CVE analysis (iterative Donoho-Johnstone threshold applied to the vorticity vector magnitude). Red and blue curves show results for the same components obtained using the most Gaussian vorticity and 60% compression thresholds, respectively, and in green, those for components identified using the iterative Donoho-Johnstone threshold applied to the individual vorticity amplitudes. The coherent components (dashed lines) in (a) underlie the solid curve and are thus hidden from view. The double curve for each of the incoherent components (dotted-dashed lines) in (a) reflects grid oscillation of the incoherent component at the Nyquist frequency, not seen in the original flow, due to enhanced power above the Orszag frequency kmax as a result of the wavelet filtering (Sec. 3).

FIG.5 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

Tables

Table I. Compression achieved and kinetic energy and enstrophy retained in the coherent component of the flow after employing the iterative Donoho-Johnstone filter threshold on two-flows with three wavelets. Results differ when application of the filter is based on the vorticity magnitude |ω| vs. component amplitudes |ωx, y, z|.

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