Perturbative model for the saturation of energetic-particle-driven modes limited by self-generated zonal modes

TL;DR

This work introduces a time-local, energy-conserving reduced model that couples beat-driven zonal modes (ZM) to the saturation of energetic-particle–driven Alfvén modes. By deriving coupled amplitude equations for the pump and ZM energies and enforcing energy balance, the authors investigate both collisionless and collisional regimes, validating the framework with analytic reductions and BOT nonlinear simulations. In the collisionless limit, ZM generation reduces the pump saturation via energy diverted to ZMs, matching qualitative gyrokinetic trends; with sources and sinks, the ZM mainly influences saturation through turbulence suppression, slowing growth but leaving the steady-state set by γ_{d,p} and ν_eff largely intact. The model provides a simple, scalable tool to incorporate wave–wave nonlinearities into reduced kinetic codes, improving predictive capability for strongly driven instabilities and EP transport in fusion devices.

Abstract

We present a simplified approach enforcing energy conservation to incorporate the effects of zonal modes alongside wave-particle nonlinearities in the determination of the saturation amplitude. The model assumes that the zonal perturbations grow at a rate twice that of the original (pump) wave, consistent with a beat-driven (or force-driven) generation mechanism. The evolution and saturation of the mode amplitude are investigated both analytically and numerically within our reduced model assumptions, in both the collisionless and scattering-dominated regimes. These studies underscore the crucial role of sources and sinks in accurately capturing the impact and the role of beat-driven zonal perturbations on mode evolution. In the realistic case of saturation set by sources and sinks, we discuss the role of a finite amplitude zonal mode in reducing microturbulent particle scattering, thus limiting the energy source for the resonant mode. We then discuss comparisons between the model predictions and simulation results. The model reproduces key features observed in gyrokinetic simulations as the reduction in saturated mode amplitude and the onset of wave-wave nonlinear effects as functions of mode growth rate and amplitude. Thanks to its simplicity, it can be readily implemented into codes based on reduced models, thereby improving their predictive capability for strongly driven instabilities.

Figures (6)

• Figure 1: Value of $\delta B_r/B$ at saturation for different values of the model parameter $K_{hh}$. The normalized model result are compared with the data published in Chen 2018 ChenY_2018 coming from nonlinear collisionless gyrokinetic simulations using the code GEM. The non-zonal curve from ChenY_2018 follows $\delta B_r/B = 0.233(\gamma_L/\omega_p)^2$ and is reproduced as expected by the $K_{hh}=0$ case. Good agreement between our model and the GEM simulations is found for $K_{hh}=22$.
• Figure 2: Reduction in the saturated amplitude due to ZM suppression of $\nu_{\text{eff}}$ as a function of the expected saturation in the absence of ZM for different values of $K_z$. The other parameters values are set to $\gamma_c = r_s = B_0 = (\int d \mathcal{V} \, |\alpha_z(r)|^2)^{1/2} = 1$. The threshold between weak and strong shearing regimes corresponds to $\omega_{b,sat}/\omega_{b}|_{nz} = 2^{-2/3}$.
• Figure 3: In part (a), the normalized amplitude evolution of pump and associated beat-driven ZM $A_z(t) = K_{z}A_p^2(t)$ are plotted for $K_{z} =22$. The analytic solution of Eq. (\ref{['quintic2']}) for the expected saturation is showed as dashed black line. Part (b) shows the evolution of pump wave amplitudes for different values of the parameter $K_{z}$. The corresponding ZM amplitude, following the quadratic relation for each $K_{z}$ value, is not shown. The expected saturation for the non-zonal case with $\omega_b/\gamma_L \approx 3.2$, is showed as dashed line. The initial condition for the pump wave amplitude is $\omega_b^2/\gamma_L^2=10^{-6}$.
• Figure 4: Evolution of pump wave amplitude close to threshold for different values of $K_{z}$ compared with the saturation predicted by analytic theory in absence of ZM generation. Sources and sinks parameters used are: $\gamma_{d,p}/\gamma_L = 0.95$ and $\nu_{\text{eff}}/(\gamma_L-\gamma_{d,p}) = 30$. The initial condition for the pump wave amplitude is $\omega_b^2/\gamma_L^2=10^{-6}$.
• Figure 5: Evolution of pump wave amplitude far from threshold with (red curve) and without (blue curve) ZM generation shown in part (a). Zoom around the early exponential growth is shown in part (b). Sources and sinks parameters used are: $\gamma_{d,p}/\gamma_L = 0.01$ and $\nu_{\text{eff}}/(\gamma_L-\gamma_{d,p}) = 0.55$. The initial condition for the pump wave amplitude is $\omega_b^2/\gamma_L^2=10^{-6}$.