A model for coupled edge-core dynamics
R. Ball and W. Horton†
Mathematical Sciences Institute, The Australian National University, Canberra 0200 Australia.
† Institute for Fusion Studies, The University of Texas at Austin, Austin, TX 78712 USA.
Abstract. An exploratory study is made of a low-dimensional model for the edge dynamics of magnetic fusion plasmas, in
which entrainment by sawtooth relaxation oscillations in the core is simulated by a periodic forcing term. The bifurcation
structure of the constantly forced model is reviewed, to identify oscillatory behavior that could be beneficially modified by
mode-locking to the sawtooth period and amplitude. Some novel effects of entrainment are shown.
Keywords: Two-dimensional flows, Bifurcations, Edge-localized modes, L–H transitions, sawtooth oscillations
PACS: 52.25.Xz, 05.45.-a, 05.65.+b, 47.20.Ky, 47.27.-i, 47.27.De, 47.27.ed, 47.27.Rc
Magnetic fusion plasmas are strongly driven dissipative flows in which the kinetic energy of small-scale turbulence
can drive the formation of large-scale coherent structures such as sheared mass flows. This tendency to self-organise
is endemic to flows where Lagrangian fluid elements see globally prevalent or soliton-like two-dimensional velocity
fields, and is due to a net inverse energy cascade .
Quasi two-dimensional fluid motion is also behind natural phenomena such as zonal structuring of planetary flows
and the segregation of pebbles washed by the sea on a beach (Fig. 1). The ability of such flows to organise advected
FIGURE 1. Left: The sea has segregated stones on Pebbly Beach, NSW, according to size. Right: Secondary instabilities grow
from the edge of this streaming cloud formation, a fast-moving zonal flow. (Photos: R. Ball.)
or diffusively transported particles or heat is a fair prospect for achieving directed management of turbulent transport
in technologies, but it is only in magnetic fusion plasmas that this potential has been exploited.
In toroidal fusion devices two-dimensional fluid motion structures the flow poloidally and gives rise to confinement
or L–H transitions, characterized by dramatic enhancement of sheared zonal flows and suppression of high wavenumber
turbulence at the edge that degrades confinement , and associated ege-localized modes (ELMs). Two major
strands have emerged in the literature on the physics of confinement transitions: (1) They are an internal phenomenon
that occurs spontaneously when upscale energy transfer from turbulence to shear flows exceeds nonlinear dissipation
[3, 4]; (2) They are due to edge ion orbit losses or induced biasing, the resulting electric field providing a torque which
drives shear flows nonlinearly [5, 6, 7].
In this contribution we describe a low-dimensional dynamical model in which these two different strands are
effectively unified. We review and explain the bifurcation structure of the model, then examine the the oscillatory
dynamics using a periodic power input to the edge potential energy reservoir as a simple prototype for entrainment of
edge-localized modes (ELMs) by sawtooth relaxation oscillations in the core.
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Acknowledgment: This work is supported by the Australian Research Council. Part of this work was carried out while RB was a
visitor at Politecnico di Torino, partially supported by a Lagrange Fellowship.
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