Research ArticleMATHEMATICAL MODELING

A variational approach to probing extreme events in turbulent dynamical systems

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Science Advances  22 Sep 2017:
Vol. 3, no. 9, e1701533
DOI: 10.1126/sciadv.1701533
  • Fig. 1 State space geometry of extreme events.

    (A) A depiction of intermittent bursts of an observable. The highlighted regions mark an approximation of the growth phase of the extreme events. (B) In the state space, extreme events are viewed as fast excursions away from the background attractor (blue ball) owing to small regions of stability.

  • Fig. 2 Extreme energy dissipation in the Kolmogorov flow.

    (A) The time series of energy dissipation rate D at Reynolds number R = 40. (B) A close-up of the energy input I (solid red curves) and the energy dissipation D (dashed black curves) at Re = 40. The bursts in the energy dissipation are slightly preceded with a burst in the energy input. A similar behavior is observed for all bursts and at higher Reynolds numbers. (C) The vorticity field ∇ × u(x, t) = ω(x, t)e3 at time t = 433 over the domain x ∈ [0, 2π] × [0, 2π].

  • Fig. 3 The solutions of Eqs. 11A to 11C, with c0 = 1, as a function of the Reynolds number.

    The solid (dashed) black line corresponds to the exact solution u (u+). The red solid line (circles) corresponds to the global maximizer. The outset shows the scalar vorticity ∇ × u(x, y) = ω(x, y)e3 and the Fourier spectrum |a(kx, ky)| of the global maximizer at select Reynolds numbers.

  • Fig. 4 Internal energy transfers lead to extreme events.

    (A) Time series of the energy input I and the modulus of the Fourier mode a(1, 0) at Re = 40. The eddy turnover time at this Reynolds number is te = 0.46. (B) The joint probability density of the energy input I and the real and imaginary parts of the mode a(1, 0), approximated from 100,000 samples. The density decreases from dark green to light blue. The cone-shaped density indicates the strong correlation between the large values of the energy input rate I and small values of |a(1, 0)|. The axisymmetric nature of the probability density is a consequence of the translation invariance of the Kolmogorov flow.

  • Fig. 5 Prediction of extreme events.

    (A) The probability density associated with the conditional probability (Eq. 12). The vertical dashed line marks the extreme event threshold Embedded Image ≃ 0.194. The horizontal dashed line marks λ = 0.4. The quadrants correspond to the following: I, correct rejections; II, false positives; III, hits; and IV, false negatives. (B) The probability of extreme events Pee corresponding to the extreme event threshold De = 0.194.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/9/e1701533/DC1

    section S1. Derivation of the Euler-Lagrange equation

    section S2. The Navier-Stokes equation

    section S3. Newton iterations

    section S4. Sensitivity to parameters

    section S5. Computing the probability of extreme events

    section S6. Supporting computational results

    fig. S1. Evolution of the energy input versus mean flow.

    fig. S2. Triad interactions.

    fig. S3. Sensitivity of the optimal solutions.

    fig. S4. Joint PDFs for higher Reynolds numbers.

    fig. S5. Prediction of intermittent bursts at higher Reynolds numbers.

    table S1. Simulation parameters.

    movie S1. The prediction of an extreme event in the Kolmogorov flow.

    References (3942)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Derivation of the Euler-Lagrange equation
    • section S2. The Navier-Stokes equation
    • section S3. Newton iterations
    • section S4. Sensitivity to parameters
    • section S5. Computing the probability of extreme events
    • section S6. Supporting computational results
    • fig. S1. Evolution of the energy input versus mean flow.
    • fig. S2. Triad interactions.
    • fig. S3. Sensitivity of the optimal solutions.
    • fig. S4. Joint PDFs for higher Reynolds numbers.
    • fig. S5. Prediction of intermittent bursts at higher Reynolds numbers.
    • table S1. Simulation parameters.
    • Legend for movie S1
    • References (39–42)

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    Other Supplementary Material for this manuscript includes the following:

    • movie S1 (.mp4 format). The prediction of an extreme event in the Kolmogorov flow.

    Files in this Data Supplement:

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