Critical Gradient Model Research Articles

Isotope impact on Alfvén eigenmodes and fast ion transport in DIII-D

Measurements of beam driven Alfvén Eigenmode (AE) activity in matched deuterium (D) and hydrogen (H) DIII-D plasmas show a dramatic difference in unstable mode activity and fast ion transport for a given injected beam power. The dependence of the unstable AE spectrum in reversed magnetic shear plasmas on beam and thermal species is investigated in the current ramp by varying beam power in a sequence of discharges for fixed thermal and beam species at fixed density. In general, a spectrum of Reversed Shear Alfvén Eigenmodes (RSAEs) and Toroidal Alfvén Eigenmodes (TAEs) are driven unstable with sub-Alfvénic D beam injection while primarily only RSAEs are driven unstable for the H beam cases investigated. Further, for a given beam power, the driven AE amplitude is always reduced with H beams relative to D and for H thermal plasma relative to pure D or mixed D/H plasmas. Estimates of the fast ion stored energy combined with modeling using the hybrid kinetic-MHD code MEGA indicate that the dominant mechanism contributing to the difference between H and D beam drive is the faster classical slowing down of H beam ions relative to D and the resultant lower beam ion pressure. Calculations of the AE induced stored energy deficits using the reduced critical gradient model TGLFEP show quantitative agreement with the observed dependencies on injected power, isotope and minimum safety factor.

Energetic particle marginal stability profile for HL-2M integrated simulation based on neural network module

A critical gradient model is employed to develop a module of energetic particle (EP) marginal stability profiles in OMFIT integrated simulations for studying EP transport. Currently, each iteration of transport evolution is approximately 10 min in the integrated simulation, whereas, the EP marginal stability profile, which serves as an input in the integrated simulation could take much longer; the reason being a combination of the TGLFEP and EPtran codes is employed in our previous investigation. To reduce the simulation time, the critical gradient is predicted by a neural network instead of the TGLFEP code, and the EPtran code is revised with parallel computing, so that the running time of this module can be controlled to within 5 min. The predictions are in good agreement with previous approaches. The integrated simulation of HL-2M with Alfven eigenmodes transported by neutral beam EP profiles indicates that EP transport reduces the total pressure and current as expected, but could also under some conditions raise the safety factor in the core, which is favorable for reversed magnetic shear and high-performance plasmas.

PFPO plasma scenarios for exploration of long pulse operation in ITER

Long Pulse Scenarios (LPS) in ITER foreseen during the Pre-Fusion Power Operation (PFPO) phase of the ITER Research Plan (IRP) are assessed using 1.5D transport simulations within the ASTRA framework. Such assessment is required to predict the operational space for LPS operation in PFPO, as well as to evaluate which physics processes for LPS operation during Fusion Power Operation (FPO) could be studied during PFPO. An important aspect in the development of LPSs in PFPO is to minimize lifetime consumption of the Central Solenoid (CS) for these scenarios. The maximum pulse length achievable for LPSs in PFPO with no consumption of CS lifetime (currents in CS coils ⩽30 kA per turn) has been assessed for a range of heating schemes and heating mixes, confinement regimes (L-mode and H-mode) and for helium and hydrogen plasmas. The operational space of LPS and pulse length has been explored through density scans with the Heating and Current Drive mix required for the FPO Q ⩾ 5 steady-state plasma scenario (namely Neutral Beam Injection and Electron Cyclotron Heating) including acceptable shine through losses on the first wall for both helium and hydrogen plasmas. Fast particle physics aspects that are common between FPO plasmas and LPS PFPO H-mode plasmas at low densities are studied including MHD stability analysis with the KINX code and non-perturbative critical gradient model based on high-n Toroidal Alfven Eigenmodes (TAE) stability kinetic ballooning code HINST calculations.

Numerical Study of the Influence of the Thermal Gas Expansion on the Boundary Layer Flame Flashback in Channels with Different Wall Thermal Conditions

In recent years, boundary layer flame flashback (BLF) has re-emerged as a technological and operational issue due to the more widespread use of alternative fuels as a part of a global effort to promote carbon neutrality. While much understanding has been achieved in experiments and simulations of BLF in the past decades, the theoretical modeling of BLF still largely relies on the progress made as early as the 1940s, when the critical gradient model (CGM) for the laminar flame flashback was proposed by Lewis and von Elbe. The CGM does not account for the modification of the upstream flow by the flame, which has been recently shown to play a role in BLF. The aim of the present work is to gain additional insight into the effects of thermal gas expansion and confinement on the flame-flow interaction in laminar BLF. Two-dimensional simulations of the confined laminar BLF in a channel are performed in this work. The parametric study focuses on the channel width, the thermal gas expansion coefficient, and the heat losses to the wall. This study evaluates the influence of these factors on the critical condition for the flame flashback. By varying the channel width, it is demonstrated that at the critical condition, the incoming flow in narrow channels is modified globally by the thermal gas expansion, while in wider channels, the flow modification by the flame tends to be more local. In narrow channels, a non-monotonic dependence of the critical-condition centerline velocity on the channel width has been identified. The variation of the heat loss to the wall confirms that the wall’s thermal conditions can significantly alter the flashback limit, with the flashback propensity being larger when the thermal resistance of the wall is high. To assess the general applicability of the CGM, the flame consumption speed and the flow velocity near the wall are quantified. The results confirm that the assumption of flame having no influence on the upstream flow, employed in the CGM, is not fulfilled under confinement for a realistic thermal gas expansion. This results in a general disagreement between the simulations and the CGM, which implies that the thermal expansion effects should be accounted for when considering the confined boundary layer flashback limits. It is shown that the critical velocity gradient increases with the gas expansion coefficient for the given channel width and wall thermal condition.

Improvement of critical gradient model and establishment of an energetic particle module for integrated simulation

Based on the critical gradient model , the combination of the TGLFEP code and EPtran code is employed to predict energetic particle (EP) transport induced by Alfvén eigenmodes (AEs). To be consistent with the experimental results, the model was improved recently by taking into consideration the threshold evolution and orbit loss mechanism. The threshold is modified to be the normalized critical gradient ((d<i>n</i>/d<i>r</i>)/(<i>n</i>/<i>a</i>)) instead of the critical gradient (d<i>n</i>/d<i>r</i>), and the new threshold is defined as a function inversely proportional to the EP density as obtained by the TGLFEP code. Additionally, the EP loss cone calculated by ORBIT is added into the EPtran code, which provides an important additional core loss channel for EPs due to finite orbits. With these two improvements, the EP redistribution profiles are found to very well reproduce the experimental profiles of two DIII-D validation cases (#142111 and #153071) with multiple unstable AEs and large-scale EP transport. In addition, a neural network is established to replace TGLFEP for critical gradient calculation, and EPtran code is rewritten with parallel computing. Based on this, a module of EP is established and it is added into the integrated simulation of OMFIT framework. The integrated simulation of HL-3 with AE transported neutral beam EP profile indicates that EP transport reduces the total pressure and current as expected, but under some condition it could also raise the safety factor in the core.

The Mathematical Model and Research Progress of the Boundary Layer Flashback in Premixed Combustion

Flashback is a key problem influencing the normal operation of power equipment such as gas turbines. As one of the main mechanisms that cause flashback, the boundary layer flashback has an important effect on the design and operation of gas turbine combustors and other combustion devices. Since the critical gradient model for the boundary layer flashback was put forward by Lewis et al. in 1945, the theoretical models for the boundary layer flashback, such as the Peclet number model, the Damköhler number model and the flame angle theory, were developed one after another. However, these theoretical models still need improvements. Until now, the theoretical models for the boundary layer flashback are still in continuous development and modification. The history of the boundary layer flashback was reviewed, and the background, pertinence and shortcomings of the theoretical models were elucidated in the order of the model establishment time. In addition, the development status and research progress of the theoretical models for the boundary layer flashback in recent years were summarized, especially the progress made with new methods such as numerical simulation and statistical analysis. Further, the theoretical research direction and breakthrough points of the combustion boundary layer flashback at present and in the future were put forward.

Model and Experimental Studies of the Seepage Failure of Damaged Geotextile at the Joint between Tubes in a Geotextile Tube Dam

This paper focuses on the impact of geotextile tube damage at the joints between tubes on dam structures subjected to seepage. First, a seepage-induced failure critical gradient model for damaged geotextile tubes was developed. Tests were conducted using geotextile specimens with precut O-shaped holes to simulate the seepage erosion process. Various overburden loads (0, 5, 10, 20, and 30 kPa) and hole radii (0.5, 1.0, 1.5, and 2 cm) were examined. Based on the test phenomena and the changes in pore-water pressure and seepage flow, four progression stages (seepage stability, sand particle wash-out, preferential flow formation and development, and complete failure) were identified. The experimental critical gradients obtained under different conditions agreed well with the model results. The critical gradient is positively correlated with the overburden load and negatively correlated with the hole radius. Critical gradient growth gradually slows with increasing overburden load. The critical gradient difference caused by the hole size decreases rapidly. When the overburden load increases to 20 kPa, this difference is essentially unchanged. These findings can provide a better understanding of the performance of damaged geotextile tubes.

Prediction of the energetic particle redistribution by an improved critical gradient model and analysis of the transport threshold

Based on the theory of critical gradient model (CGM) and following the simulation method proposed by Waltz et al. [Nucl. Fusion 55, 123012 (2015)], a combination of TGLFEP and EPtran code is employed to predict the energetic particle (EP) transport induced by Alfvén eigenmodes (AEs). To be consistent with the experiment, recent improvements to the simulation method include consideration of threshold evolution and orbit loss due to finite orbit width. The revised CGM is applied to simulate two DIII-D experimental discharges (#142111 and #153071). It well reproduces the experimental profiles with multiple unstable AEs and large-scale EP transport. Discharge #142111 had previously been simulated using a nonlinear MHD-kinetic code MEGA [Todo et al., Nucl. Fusion 55, 073020 (2015)] with a transport mechanism based on stochasticity induced by overlapping AE. By comparing the simulated EP profiles, we find that the AE transport threshold is approximated by both the MEGA nonlinear stability threshold and the proposed CGM threshold (error <5% for single n and <17% for multiple n simulation). Both of them are larger than the linear stability threshold of the most unstable AE mode by a quantity of the order of the flux needed to sustain EP transport by the background turbulence. We have also applied the improved CGM to simulate the α particle redistribution for a China Fusion Engineering Test Reactor steady state scenario. Because of the clear separation between the AE unstable region and the loss cone, only a moderate α particle loss of ∼9.6% is predicted.

Quasilinear critical gradient model for Alfven eigenmode driven energetic particle transport with intermittency

A simplified quasilinear critical gradient model (QLCGM) for Alfven eigenmode (AE) driven energetic particle (EP) transport is used to treat the intermittency (burstiness) of the 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 AE. The model is embedded in a time-dependent EP 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 particle loss detector flow time traces: high kurtosis [O(40)] of the burstiness with the intermittency O(1–2) increasing with the level of mean flow.

Nonlinear verification of a linear critical gradient model for energetic particle transport by Alfvén eigenmodes

A “stiff transport” critical gradient model of energetic particle (EP) transport by EP-driven Alfvén eigenmodes (AEs) is verified against local nonlinear gyrokinetic simulations of a well-studied beam-heated DIII-D discharge 146102. A greatly simplifying linear “recipe” for the limiting EP-density gradient (critical gradient) is considered here. In this recipe, the critical gradient occurs when the local AE linear growth rate, driven mainly by the EP gradient, exceeds the ion temperature gradient (ITG) or the trapped electron mode (TEM) growth rate, driven by the thermal plasma gradient, at the same toroidal mode number (n) as the AE peak growth, well below the ITG/TEM peak n. This linear recipe for the critical gradient is validated against the critical gradient determined from far more expensive local nonlinear simulations in the gyrokinetic code GYRO, as identified by the point of transport runaway when all driving gradients are held fixed. The reduced linear model is extended to include the stabilization from local equilibrium E × B velocity shear. The nonlinear verification unambiguously endorses one of two alternative recipes proposed in the study by Waltz et al. [Nucl. Fusion 55, 123012 (2015)]: the EP-driven AE growth rate should be determined with rather than without an added thermal plasma drive.

First steps towards modeling of ion-driven turbulence in Wendelstein 7-X

Due to foreseen improvement of neoclassical confinement in optimised stellarators—like the newly commissioned Wendelstein 7-X (W7-X) experiment in Greifswald, Germany—it is expected that turbulence will significantly contribute to the heat and particle transport, thus posing a limit to the performance of such devices. In order to develop discharge scenarios, it is thus necessary to develop a model which could reliably capture the basic characteristics of turbulence and try to predict the levels thereof. The outcome will not only be affordable, using only a fraction of the computational cost which is normally required for repetitive direct turbulence simulations, but would also highlight important physics. In this model, we seek to describe the ion heat flux caused by ion temperature gradient (ITG) micro-turbulence, which, in certain heating scenarios, can be a strong source of free energy. With the aid of a relatively small number of state-of-the-art nonlinear gyrokinetic simulations, an initial critical gradient model (CGM) is devised, with the aim to replace an empirical model, stemming from observations in prior stellarator experiments. The novel CGM, in its present form, encapsulates all available knowledge about ion-driven 3D turbulence to date, also allowing for further important extensions, towards an accurate interpretation and prediction of the ‘anomalous’ transport. The CGM depends on the stiffness of the ITG turbulence scaling in W7-X, and implicitly includes the nonlinear zonal flow response. It is shown that the CGM is suitable for a 1D framework turbulence modeling.

Alfvén eigenmode stability and critical gradient energetic particle transport using the Trapped-Gyro-Landau-Fluid model

The Trapped-Gyro-Landau-Fluid (TGLF) transport model is a physically realistic and comprehensive theory based on a local quasilinear transport model fitted to linear and nonlinear GYRO gyrokinetic simulations [Staebler et al., Phys. Plasmas 14, 55909 (2007)]. This work presents the first use of the TGLF model to treat low-n Alfvén eigenmode (AE) stability and energetic particle (EP) transport. TGLF accurately recovers the local GYRO toroidicity-induced AE (TAE) and energetic particle mode (EPM) linear growth and frequency rates for a fusion alpha case. With a very high grid resolution, TGLF can quickly find the critical EP pressure gradient profile for stiff EP transport based on an AE linear threshold given the background thermal plasma profiles in DIII-D. The TGLF critical gradient profile using the recipe γAE = 0, that is the linear AE growth rate without additional driving rates from the background plasma gradients, matches the more expensive linear GYRO results with a single worst toroidal mode number n. TGLF can easily find the minimum critical gradient profile with testing multiple ns. From a database of runs using a newly developed TGLFEP code, a rough but insightful parametric “power law” scaling for critical EP beta is demonstrated. An important toroidal stabilization condition on the EP pressure gradient pEP/LpEP drive is isolated: R/LpEP>CR ∼ 3, where LpEP is the EP pressure gradient length and R is the tokamak major radius. This paper also demonstrates that relaxation of the fixed slowing down EP profile shape approximation often used to find the critical EP density profile has little effect on the resulting EP transport. The single EP species critical gradient model is generalized to handle two EP species.

Phase-space dependent critical gradient behavior of fast-ion transport due to Alfvén eigenmodes

Experiments in the DIII-D tokamak show that many overlapping small-amplitude Alfvén eigenmodes (AEs) cause fast-ion transport to sharply increase above a critical threshold in beam power, leading to fast-ion density profile resilience and reduced fusion performance. The threshold is above the AE linear stability limit and varies between diagnostics that are sensitive to different parts of fast-ion phase-space. Comparison with theoretical analysis using the nova and orbit codes shows that, for the neutral particle diagnostic, the threshold corresponds to the onset of stochastic particle orbits due to wave-particle resonances with AEs in the measured region of phase space. The bulk fast-ion distribution and instability behavior was manipulated through variations in beam deposition geometry, and no significant differences in the onset threshold outside of measurement uncertainties were found, in agreement with the theoretical stochastic threshold analysis. Simulations using the ‘kick model’ produce beam ion density gradients consistent with the empirically measured radial critical gradient and highlight the importance of including the energy and pitch dependence of the fast-ion distribution function in critical gradient models. The addition of electron cyclotron heating changes the types of AEs present in the experiment, comparatively increasing the measured fast-ion density and radial gradient. These studies provide the basis for understanding how to avoid AE transport that can undesirably redistribute current and cause fast-ion losses, and the measurements are being used to validate AE-induced transport models that use the critical gradient paradigm, giving greater confidence when applied to ITER.

Analysis of edge stability for models of heat flux width

Detailed measurements of the ne, Te, and Ti profiles in the vicinity of the separatrix of ELMing H-mode discharges have been used to examine plasma stability at the extreme edge of the plasma and assess stability dependent models of the heat flux width. The results are strongly contrary to the critical gradient model, which posits that a ballooning instability determines a gradient scale length related to the heat flux width. The results of this analysis are not sensitive to the choice of location to evaluate stability. Significantly, it is also found that the results are completely consistent with the heuristic drift model for the heat flux width. Here the edge pressure gradient scales with plasma density and is proportional to the pressure gradient inferred from the equilibrium in accordance with the predictions of that theory.

Improving fast-ion confinement in high-performance discharges by suppressing Alfvén eigenmodes

We show that the degradation of fast-ion confinement in steady-state DIII-D discharges is quantitatively consistent with predictions based on the effects of multiple unstable Alfvén eigenmodes on beam-ion transport. Simulation and experiment show that increasing the radius where the magnetic safety factor has its minimum is effective in minimizing beam-ion transport. This is favorable for achieving high performance steady-state operation in DIII-D and future reactors. A comparison between the experiments and a critical gradient model, in which only equilibrium profiles were used to predict the most unstable modes, show that in a number of cases this model reproduces the measured neutron rate well.

Validating predictive models for fast ion profile relaxation in burning plasmas

The redistribution and potential loss of energetic particles due to MHD modes can limit the performance of fusion plasmas by reducing the plasma heating rate. In this work, we present validation studies of the 1.5D critical gradient model (CGM) for Alfvén eigenmode (AE) induced EP transport in NSTX and DIII-D neutral beam heated plasmas. In previous comparisons with a single DIII-D L-mode case, the CGM model was found to be responsible for 75% of measured AE induced neutron deficit []. A fully kinetic HINST is used to compute mode stability for the non-perturbative version of CGM (or nCGM). We have found that AEs show strong local instability drive up to violating assumptions of perturbative approaches used in NOVA-K code. We demonstrate that both models agree with each other and both underestimate the neutron deficit measured in DIII-D shot by approximately a factor of 2.On the other hand in NSTX the application of CGM shows good agreement for the measured flux deficit predictions. We attempt to understand these results with the help of the so-called kick model which is based on the guiding center code ORBIT. The kick model comparison gives important insight into the underlying velocity space dependence of the AE induced EP transport as well as it allows the estimate of the neutron deficit in the presence of the low frequency Alfvénic modes. Within the limitations of used models we infer that there are missing modes in the analysis which could improve the agreement with the experiments.

Investigation of transport models in ASDEX Upgrade current ramps

Understanding the physics of transport during current ramps in tokamaks is essential in order to prepare adequate ramp-up and ramp-down scenarios for ITER, where constraints on the transformer flux and the internal inductance will be more stringent than in present-day devices. In this paper, five transport models (Coppi–Tang, Neo-Alcator, Bohm/gyro-Bohm, critical gradient model and H98/2 scaling-based model) are used to reproduce the experimental data during the current ramps of ASDEX Upgrade with its metallic wall. The calculated temperature profiles are compared with the experimental temperature profiles under different ramp-up conditions. Our study reveals important differences between boronized and non-boronized walls which are reproduced with variable success by the models. We also investigated ramp-up phases heated centrally by electron cyclotron heating which creates experimental conditions with peaked electron temperature profiles, which are very different from the usual ohmic heated cases. Our study reveals that the models react very differently to this additional heating.

Initial results of the high resolution edge Thomson scattering upgrade at DIII-D

Validation of models of pedestal structure is an important part of predicting pedestal height and performance in future tokamaks. The Thomson scattering diagnostic at DIII-D has been upgraded in support of validating these models. Spatial and temporal resolution, as well as signal to noise ratio, have all been specifically enhanced in the pedestal region. This region is now diagnosed by 20 view-chords with a spacing of 6 mm and a scattering length of just under 5 mm sampled at a nominal rate of 250 Hz. When mapped to the outboard midplane, this corresponds to ~3 mm spacing. These measurements are being used to test critical gradient models, in which pedestal gradients increase in time until a threshold is reached. This paper will describe the specifications of the upgrade and present initial results of the system.

Analysis of electron heat transport with off-axis modulated ECRH in Tore Supra

Experiments to study inward heat transport phenomena have been performed in the Tore Supra tokamak with off-axis electron cyclotron resonance heating (ECRH). Both power balance and perturbation transport analysis have been done for low-frequency (1 Hz) ECRH modulation experiments. Heat diffusivity and heat pinch have been separately determined by fitting the experimental data of the amplitude and phase of the Fourier transform of the modulated temperature with a linear transport model including convection term. Comparison with the critical gradient model has shown that the heat pinch previously obtained could include a pseudo-pinch due to the non-linearity of the diffusivity and an additional non-diffusive heat pinch. The pinch effect is reduced for higher densities.

Front propagation and critical gradient transport models

This paper analyzes the properties of a two-field critical gradient model that couples a heat equation to an evolution equation for the turbulence intensity. It is shown that the dynamics of a perturbation is ballistic or diffusive depending on the shape of the pulse and also on the distance of the temperature gradient to the instability threshold. This dual character appears in the linear response of this model for a wave packet. It is recovered when investigating the nonlinear solutions of this system. Both self-similar diffusive fronts and ballistic fronts are shown to exist. When the propagation is ballistic, it is found that the front velocity is the geometric mean between the turbulent diffusion coefficient and a microinstability growth rate.