A simplified quasilinear critical gradient model (QLCGM) for Alfven eigenmode driven energetic particle transport is used to treat the intermittency (burstiness) of the transport flow time traces. The extreme burstiness of Alfven eigenmode (AE) driven energetic particle (EP) losses increasing with transport flow has been well demonstrated in recent DIII-D experiments. Research to assess the impact of intermittent energetic particle losses on wall damage has only begun. The goal of the present research is to demonstrate the close connection between strongly intermittent AE-EP transport and stiff critical gradient model (CGM) transport. Here intermittency is defined as the ratio of the root mean square deviation from mean flow to the mean flow. The model posits that the intermittency results from the micro-turbulent noise induced in the thermal plasma damping rate for the Alfven eigenmodes. The model is embedded in a time dependent energetic particle density transport code to generate the transport flow time traces from typical critical gradient and slowing down density profiles. The QLCGM appears to be in reasonable agreement with the level and characteristics of the intermittency observed in the DIII-D fast ion lost detector (FILD) flow time traces: high kurtosis O(40) of the burstiness with the intermittency O(1-2) increasing roughly as the square root of the mean flow.
Application of the time dependent quasilinear approximation in the context of a local critical gradient model requires some review. The CGM for time average transport was originally motivated by 2010 local nonlinear gyrokinetic simulations with low-n energetic particle (EP) driven Alfven eigenmodes (AEs) embedded in high-n micro-turbulence. The simulations demonstrated unbounded transport when the local EP density gradient exceeded a critical value at the EP driven AE linear instability threshold. A CGM embedded in an EP density transport code (ALPHA) has been used to predict the fusion alpha AE time average losses in ITER[3,4]. The CGM and ALPHA transport code with has been validated against DIII-D experiments. The radial profile of the transported EPs is invariant to large variations in the peakedness of the NBI source: strong evidence for “stiff” critical gradient transport. Long time scale nonlinear simulations with the MEGA code demonstrate strong “profile resiliency” and AE-EP transport intermittency increasing with transport flow .
The formulation of the QLCGM is straightforward. The EP density flux is modeled with an AE diffusivity added to a small background diffusivity from the micro-turbulence. As in Eq. (1) of Ref. , the time dependent EP density transport equation has a fixed EP source proportional to the slowing down density divided the slowing down time and a sink given by the time dependent EP density divided by the slowing down time. The AE time dependent quasilinear diffusivity is given by the product of a fixed quasilinear diffusivity weight and the time dependent AE mode energy. The local mode energy is exponentially advanced in time at twice the AE net linear growth rate as per the usual quasilinear model formulation. The net AE growth rate is given by a driving rated proportional to the EP density gradient relative to the critical density gradient less the AE damping rate on the thermal plasma. (It is important to note that the AE damping rate is independent of the presence of energetic particles.) If the radial maximum in the slowing down density gradient profile exceeds the minimum of the critical gradient profile (both typically near the mid-core radius), time traces of the local mode energy and transport flows increase and decrease as the EP density profile oscillates slightly above and below the critical gradient profile. The EP transport flows and density profile evolution time traces are completely independent of the size and profile of the quasilinear diffusivity weight. These limit cycle flows decay after a few slowing down times to the stationary critical gradient transport flow profiles (e.g. Fig. 3(d) of Ref. .) These initial value dependent decaying limit cycle flow “burst” are regular and have nothing to do with the stochastic and highly intermittent burst on very short time scales seen in the DIII-D experiments . The key postulate to account for the experimentally observed intermittent flows, is to assume the AE damping rate on the thermal plasma in the formulated QLCGM acquires a small x% random fluctuation from the microturbence in the thermal plasma.
For strong driving with the maximum slowing down EP density relative to the minimum critical EP density gradient of 5, with x%=4%(2%,1%) random fluctuations in the AE damping, and with a strong ITER like AE damping rate 10,000 times the EP slowing down rate, the edge EP transport flow intermittency is O[2.0(1.4,1.0)] respectively. For DIII-D like parameters with x%=15% edge (and 1% core) fluctuations, and with an AE damping rate 1000 times the EP slowing down rate, the QLCGM finds the intermittency O(1-2) with very high kurtosis in agreement with experiment. In general the intermittency scales like the square root of the AE damping rate, square root of the x% fluctuation level in the damping rate, and square root of the transport flow level.
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This work is supported by U.S. Department of Energy under Grants DE-FG02-95ER54309 (theory), DESC0018108 (SciDAC-ISEP project), and DE-FC02-04ER54698 (DIII-D).
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