Energy transport and confinement in tokamak fusion plasmas is usually determined by the coupled nonlinear interactions of small-scale drift turbulence and larger scale coherent nonlinear structures, such as zonal flows, together with free energy sources such as temperature gradients. Zerodimensional models, designed to embody plausible physical narratives for these interactions, can help identify the origin of enhanced energy confinement and of transitions between confinement regimes. A prime zero-dimensional paradigm is predator-prey or Lotka-Volterra. Here we extend a successful three-variable (temperature gradient; microturbulence level; one class of coherent structure) model in this genre [M A Malkov and P H Diamond, Phys. Plasmas 16, 012504 (2009)], by adding a fourth variable representing a second class of coherent structure. This requires a fourth coupled nonlinear ordinary differential equation. We investigate the degree of invariance of the phenomenology generated by the model of Malkov and Diamond, given this additional physics. We study and compare the long-time behaviour of the three-equation and four-equation systems, their evolution towards the final state, and their attractive fixed points and limit cycles. We explore the sensitivity of paths to attractors. It is found that, for example, an attractive fixed point of the three-equation system can become a limit cycle of the four-equation system. Addressing these questions which we together refer to as robustness for convenience, is particularly important for models which, as here, generate sharp transitions in the values of system variables which may replicate some key features of confinement transitions. Our results help establish the robustness of the zero-dimensional model approach to capturing observed confinement phenomenology in tokamak fusion plasmas.

No abstract

In this paper, the effect of an environmental temperature change on multilayer diffractive optical elements (MLDOEs) is studied in terms of the diffraction efficiency and the polychromatic integral diffraction efficiency (PIDE). The relation between the diffraction efficiency of MLDOEs and environmental temperature is analyzed theoretically, and examples of different MLDOEs are discussed in the visible and infrared wavebands. The result shows that the diffraction efficiency reduction is no more than 5% and the PIDE reduction is less than 1.5% for optical plastic MLDOEs in the visible waveband when the environmental temperature ranges from −62 to 71 • C, and that the decrease of both the diffraction efficiency and PIDE for MLDOEs is more significant in the mid-wave infrared than in the long-wave infrared. The analysis result can be considered during optical engineering design with MLDOEs.

It is shown that rapid substantial changes in heating rate can induce transitions to improved energy confinement regimes in zero-dimensional models for tokamak plasma phenomenology. We examine for the first time the effect of step changes in heating rate in the models of E-

No abstract