Bridging between weakly and strongly driven quasilinear regimes with a convolutional resonance function

Quasilinear simulations can be an effective reduced tool for predicting and interpreting fast ion transport in fusion devices. At the core of these transport models is the form of their diffusion coefficients (or resonance functions), which have a leading role in determining the resonant wave saturation levels. While conventional quasilinear theory requires a random phase approximation to be satisfied via resonance overlap, it has been shown that near-threshold instabilities are naturally in the quasilinear regime even in the absence of any overlap, provided that the effective scattering rate felt by resonance particles well exceeds the instability net growth rate. This further motivates the deployment of resonance broadened quasilinear models to predict the dynamics of discrete Alfvénic eigenmodes upon their interaction with energetic ions, as they may alternate between the isolated and overlapping regimes.

This work describes how a generalization of the resonance function can be constructed to automatically enforce quasilinear simulations to replicate saturation levels obtained from nonlinear simulations across any level of instability excitation marginality. The resonance function is intuitively proposed in a convolutional form in phase space to encode the following properties: (i) its characteristic broadening is the sum of the broadenings of the two individual components due to the effective scattering frequency and due to wave amplitude, (ii) it integrates to the unity, as physically expected for a broadened function replacing a delta function, and (iii) it exactly recovers either of its constituents in the limits of marginal or strongly driven instability.

It is found that the broadening width is the key parameter that determines the mode evolution properties, with the resonance function shape playing a sub-dominant role. A discrepancy is found for the location of the parameter-space boundary between pulsating and quasi-steady saturation scenarios in quasilinear vs in nonlinear simulations in scenarios of low stochasticity. Such lack of agreement is verified to be largely a consequence of the diffusive resonant dynamics assumption embedded in quasilinear theory, rather than a lack of accuracy in the prescription of the resonance function.