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

Abstract

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.

Sparse identification of a predator-prey system from simulation data of a convection model

“…Sugama and Horton 4 , and Ball, Dewar, and Sugama 5 define state variables for their L-H transition models which are equivalent to the definitions in (9). The time-derivatives of the energies (9) for the system (4) can be written aṡ…”

“…The solution is initialized at t = 0 and run with output time steps of ∆t = 0.05 until t = 400. At each output time of the simulation the three energies (9) and their time-derivatives are computed and saved. …”

“…The creation of an edge transport barrier formed by a sheared zonal flow is closely related to the L-H transition 3 . Ordinary differential equation (ODE) models for the L-H transition are based on the predator-prey relationship between zonal flow and turbulent flow, and incorporate a potential energy related to the pressure profile as an additional state variable [4][5][6][7][8][9][10][11][12] . Miki et al 13 and Wu et al 14 have both suggested 1D partial differential equation (PDE) models for the L-H transition based on this predator-prey relationship.…”

Study of improved confinement by a stepwise increase of the input heating power for tokamak plasmas

“…The bifurcation occurs when zonal flow causes a transport barrier (steep temperature gradient) [2][3][4][5][6], to be produced at the edge of the plasma, improving plasma confinement. Zero-dimensional models [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] [7], called MD model, which has one fewer variable for which oscillatory heating was studied in [21]. This distinctive phenomenology may offer a path to future experimental testing of the assumptions of zero-dimensional models, and perhaps distinguishing between them.…”

Detailed mathematical and numerical analysis of a dynamo model

“…This is particulary the case in view of the limitations inherent in any model (e.g., in parameterizations of transport coefficients in MHD simulations). We note that a simple parameterized model has attracted a lot of attention for fusion plasmas as a useful model for understanding regulation and bifurcation (L-H transition), which is crucial for improving plasmas (Zhu et al 2013;Malkov et al 2009;Kim & Diamond 2003).…”