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Energy transport and confinement in tokamak fusion plasmas is usuallydetermined by the coupled nonlinear interactions of small-scale driftturbulence and larger scale coherent nonlinear structures, such as zonal flows,together with free energy sources such as temperature gradients.Zero-dimensional models, designed to embody plausible physical narratives forthese interactions, can help identify the origin of enhanced energy confinementand of transitions between confinement regimes. A prime zero-dimensionalparadigm is predator-prey or Lotka-Volterra. Here we extend a successfulthree-variable (temperature gradient; microturbulence level; one class ofcoherent structure) model in this genre [M A Malkov and P H Diamond, Phys.Plasmas 16, 012504 (2009)], by adding a fourth variable representing a secondclass of coherent structure. This requires a fourth coupled nonlinear ordinarydifferential equation. We investigate the degree of invariance of thephenomenology generated by the model of Malkov and Diamond, given thisadditional physics. We study and compare the long-time behaviour of thethree-equation and four-equation systems, their evolution towards the finalstate, and their attractive fixed points and limit cycles. We explore thesensitivity of paths to attractors. It is found that, for example, anattractive fixed point of the three-equation system can become a limit cycle ofthe four-equation system. Addressing these questions - which we together referto as "robustness" for convenience, is particularly important for models which,as here, generate sharp transitions in the values of system variables which mayreplicate some key features of confinement transitions. Our results helpestablish the robustness of the zero-dimensional model approach to capturingobserved confinement phenomenology in tokamak fusion plasmas.

Robustness of predator-prey models for confinement regime transitions in fusion plasmas