Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Sandpile modeling of pellet pacing in fusion plasmas

C. A. Bowie and M. J. Hole
Phys. Rev. E 101, 013207 – Published 29 January 2020
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • INTERMITTENT EXTRA SAND: CORE FUELLING
  • HIGH DRIVING: CONSTANT FUELLING
  • LIMITS OF MODELS AT VERY HIGH DRIVING
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Sandpile models have been used to provide simple phenomenological models without incorporating the detailed features of a fully featured model. The Chapman sandpile model [Chapman et al., Phys. Rev. Lett. 86, 2814 (2001)] has been used as an analog for the behavior of a plasma edge, with mass loss events being used as analogs for edge-localized modes (ELMs). In this work we modify the Chapman sandpile model by providing for both increased and intermittent driving. We show that the behavior of the sandpile, when continuously fuelled at very high driving, can be determined analytically by a simple algorithm. We observe that the size of the largest avalanches is better reduced by increasing constant driving than by the intermittent introduction of “pellets” of sand. Using the sandpile model as a reduced model of ELMing behavior, we conject that ELM control in a fusion plasma may similarly prove more effective with increased total fuelling than with pellet addition.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    4 More
    • Received 7 July 2019

    DOI:https://doi.org/10.1103/PhysRevE.101.013207

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Magnetic confinement fusionNuclear fusion
    Plasma Physics

    Authors & Affiliations

    C. A. Bowie* and M. J. Hole

    • Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia

    • *craig.bowie@anu.edu.au

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 101, Iss. 1 — January 2020

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      Sandpile schematic, showing the key features of the sandpile model discussed in this paper. The schematic is the same for both the classic and running models, which differ only in respect to whether further sand is added only after an avalanche has concluded (classic model) or during an avalanche (running model).

      Reuse & Permissions
    • Figure 2
      Figure 2

      Ep, max MLE, and max ΔtA for varying pellet sizes, with ΔtP=70000 (top) and ΔtP=100000 (bottom). In all cases, fuelling occurs at the core.

      Reuse & Permissions
    • Figure 3
      Figure 3

      Ep versus dx up to dx=30, for classic and running models. It is notable that Ep is effectively constant for all values of dx shown here in the classic model, while Ep gradually increases in the running model.

      Reuse & Permissions
    • Figure 4
      Figure 4

      Potential energy/maximum potential energy vs. dx/Zc (all unitless). The potential energy measured is the average potential energy (given by the sum of the squares of the cells) after the system has evolved from a nil sandpile to a “steady state,” which typically takes several hundred thousand iterations. The maximum Ep is calculated on the basis that actual gradient is equal to Zc, i.e., that the sandpile is in a maximally critical state. The three curves which largely coincide represent data for different values of Zc but common values of Lf. The other curve represents data for a changed value of Lf.

      Reuse & Permissions
    • Figure 5
      Figure 5

      Ep (blue dotted line) and max MLE (red solid line) versus dx for the running model. Max MLE size decreases with increasing dx, while Ep peaks at about dx=35 (i.e., dx/Zc=0.3). Both Ep and max MLE size show elements of fine structure, although changes in Ep are more pronounced.

      Reuse & Permissions
    • Figure 6
      Figure 6

      Combined pdfs of waiting times between MLEs (classic model) dx/Zc=0.01 for dx=0.12 to 1.20 in increments of 0.12 and Zc=12 to 120 in increments of 12. The pdfs overlap entirely, suggesting that waiting times are identical for identical values of dx/Zc, regardless of the specific values of dx or Zc.

      Reuse & Permissions
    • Figure 7
      Figure 7

      Schematic sandpile at iteration prior to MLE for (a) dx=32,Zc=30. Iteration n+1 will trigger an avalanche, as one iteration is enough for gradient to exceed Zc. Thirty-two grains to be distributed. (b) dx=29,Zc=30. Iteration n+2 will trigger an avalanche, as two iterations required for gradient to exceed Zc. Fifty-eight grains to be distributed.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Actual gradient (Z) of the resulting sandpile in steady state, as a function of dx/Zc, for driving up to dx/Zc≈4.1. Elements of fine structure are observed up to dx/Zc≈3.1 after which Z increases linearly with dx/Zc. The actual gradient of the sandpile is closely related to Ep, as the total size of the sandpile is determined by its gradient. It is also related to Ep/Epmax. We have shown actual gradient, rather than Ep/Epmax in order to show the straight line relationship between dx/Zc and actual gradient for values of dx/Zc>3.1.

      Reuse & Permissions
    • Figure 9
      Figure 9

      dx/Z as a function of dx: Lf=5 and Lf=6.

      Reuse & Permissions
    • Figure 10
      Figure 10

      dx necessary to trigger systemwide avalanche in steady state for Lf=4.

      Reuse & Permissions
    • Figure 11
      Figure 11

      Classic model: Ep as a function of dx, up to dx=4100, for Lf=5. Ep remains constant up to dx≈480, after which elements of fine structure appear.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review E

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×