# Comparison of Saturation Rules Used for Gyrokinetic Quasilinear Transport Modeling

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Tokamak Plasma Parameters and Assumptions

## 3. Linear Analysis

## 4. Quasilinear Theory

#### 4.1. Quasilinear Expression of Fluxes Using Linear Gyrokinetic Simulation

#### 4.2. Saturation Rules

#### 4.3. Saturation Levels and Quasilinear Fluxes

^{2}], we take the values presented in Figure 4 and multiply by ${\left(\frac{{\rho}_{i}}{R}\frac{{T}_{i}}{e}\right)}^{2}$. The saturation rules in Equations (4), (6) and (7) are labeled “Lapillonne(2011)”, “Bourdelle(2007)”, and “Kumar(2021)”, respectively, for the convenience of the reader. TGLF SAT1 is labeled “SAT1”. No indication of the relative validity of the various models should be made here, as we are only using the various saturation rules for comparison. Additionally, the theory-based transport models calculate the fluxes differently than we do here, as we use the linear GENE results. The overall level of each saturation rule, i.e., the value of ${A}_{0}$, has little meaning, as quasilinear transport models calibrate the saturation rule using a constant coefficient.

## 5. Comparison with Nonlinear Gyrokinetic Simulations

## 6. Discussion and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Profiles for DIII-D 162940 ELMy H-mode just prior to ELM onset: (

**a**) electron and main ion density profiles, (

**b**) impurity density profiles, (

**c**) electron temperature profile, and (

**d**) main ion temperature profile.

**Figure 2.**Growth rate and real frequency at the radial locations $\rho =$ 0.8, 0.85, and 0.9 in Miller geometry from the local GENE linear initial value simulation.

**Figure 4.**The three saturation rules obtained from linear GENE along with TGLF SAT1 in GENE gyroBohm units, as described in the text.

**Figure 5.**Quasilinear fluxes from GENE versus ${k}_{y}{\rho}_{i}$ at $\rho =0.85$ for the three saturation rules in GENE gyroBohm units: (

**a**) deuterium heat flux, (

**b**) electron heat flux, (

**c**) carbon heat flux, (

**d**) deuterium particle flux, (

**e**) electron particle flux, and (

**f**) carbon particle flux. Fluxes are normalized such that the total flux matches SAT1.

**Figure 6.**Lapillonne QL flux model, NL from GENE, and GEM results versus ${k}_{y}{\rho}_{i}$ at $\rho =0.85$ in GENE gyroBohm units. GEM fluxes scaled by $3.49$. (

**a**) Deuterium heat flux, (

**b**) electron heat flux, (

**c**) carbon heat flux, (

**d**) deuterium particle flux, (

**e**) electron particle flux, (

**f**) carbon particle flux.

$\mathit{\rho}$ | $\frac{\mathit{R}}{{\mathit{L}}_{{\mathit{T}}_{\mathit{i}}}}$ | $\frac{\mathit{R}}{{\mathit{L}}_{{\mathit{T}}_{\mathit{e}}}}$ | $\frac{{\mathit{T}}_{\mathit{e}}}{{\mathit{T}}_{\mathit{i}}}$ | $\frac{\mathit{R}}{{\mathit{L}}_{\mathbf{ne}}}$ | $\frac{\mathit{R}}{{\mathit{L}}_{\mathbf{ni}}}$ | $\frac{\mathit{R}}{{\mathit{L}}_{\mathbf{nC}}}$ | $\frac{{\mathit{n}}_{\mathit{C}}}{{\mathit{n}}_{\mathit{e}}}\mathbf{[}\mathbf{\%}\mathbf{]}$ | $\mathit{q}$ | $\hat{\mathit{s}}$ | ${\mathit{\beta}}_{\mathit{e}}\mathbf{[}\mathbf{\%}\mathbf{]}$ | $\mathit{\kappa}$ | $\mathit{\delta}$ | $\mathit{\zeta}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.8 | 6.71 | 7.49 | 0.87 | 1.44 | 2.40 | −0.71 | 5.16 | 2.28 | 1.75 | 1.00 | 1.47 | 0.21 | −0.03 |

0.85 | 9.16 | 8.94 | 0.87 | 1.84 | 3.06 | −0.75 | 5.37 | 2.56 | 2.17 | 0.85 | 1.51 | 0.24 | −0.04 |

0.9 | 12.49 | 11.17 | 0.88 | 2.36 | 3.98 | −0.79 | 5.65 | 2.97 | 2.94 | 0.67 | 1.55 | 0.28 | −0.05 |

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**MDPI and ACS Style**

Parker, S.E.; Haubrich, C.S.; Tirkas, S.; Cai, Q.; Chen, Y.
Comparison of Saturation Rules Used for Gyrokinetic Quasilinear Transport Modeling. *Plasma* **2023**, *6*, 611-622.
https://doi.org/10.3390/plasma6040042

**AMA Style**

Parker SE, Haubrich CS, Tirkas S, Cai Q, Chen Y.
Comparison of Saturation Rules Used for Gyrokinetic Quasilinear Transport Modeling. *Plasma*. 2023; 6(4):611-622.
https://doi.org/10.3390/plasma6040042

**Chicago/Turabian Style**

Parker, Scott E., Calder S. Haubrich, Stefan Tirkas, Qiheng Cai, and Yang Chen.
2023. "Comparison of Saturation Rules Used for Gyrokinetic Quasilinear Transport Modeling" *Plasma* 6, no. 4: 611-622.
https://doi.org/10.3390/plasma6040042