# Nonlinear Problems of Equilibrium Charge State Transport in Hot Plasmas

## Abstract

**:**

## 1. Introduction

## 2. Charge State Equilibrium and Transport

#### 2.1. The Discrete Transport and Equilibrium Conditions

_{i}and x

_{j}are changed almost continuously. The function $g\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right)$ must satisfy the Smoluchowski Equation with integration over mD phase volume $\prod d{x}_{i}$. The derivation of Equation (3) can be found elsewhere [4,5].

^{−1}) could be denoted as ${w}_{kn}^{p}$ if the CS motion by p-transport is directed from n to n + 1 and as ${u}_{kn+1}^{p}$ if the motion is directed from n + 1 to n. Note that these introduced transport coefficients are just a modified form of the conventional D and V and present the same, albeit unidirectional, particle fluxes. So, for rough numerical estimates, D and V could be obtained from these 1D matrix coefficients simply as $D\propto \left({w}_{kn}^{p}+{u}_{kn+1}^{p}\right)/2$ and $V\propto \left({u}_{kn+1}^{p}-{w}_{kn}^{p}\right)$ for the most abundant k states (see in detail in [28]).

#### 2.2. Reduced Equilibrium Cell

#### 2.3. Reduction Schemes

## 3. Equilibrium Constant

#### 3.1. Time Scale of Transport Rates

^{t}is the transposed orthogonal matrix (see, e.g., in [20]).

#### 3.2. Impurity Equilibrium and Transport Coefficients

_{1}and A

_{2}with central accumulation correspond to a decrease in D in the plasma core, which is in qualitative agreement with the experiments [26] and modelling [17].

#### 3.3. Impurity Equilibrium and Confinement Time

#### 3.4. Types of Impurity CS Equilibrium

^{19}m

^{−3}. The selected profiles of temperatures, electron density, and relative density of hydrogen neutrals ${\xi}_{n}={n}_{n}/{n}_{e}$ remain the same as previously used for modeling Ar transport in JET.

_{f}, calculated for almost flat in the core and peaked at the plasma edge, also related to R-equilibrium (with He/Li—like ions). The calculations shown in Figure 7b show that a change of the levels He/Li (in R-equilibrium) to He/H (in S-equilibrium) leads to a significant decrease in D. This is obviously due to the low ionization and recombination rates of He-like and H-like ions.

#### 3.5. The Radial Scale of the CS Equilibrium

## 4. The Recovery of CS Equilibrium on the Density Profile

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1D, 2D, mD | one-dimensional, two-dimensional, multi-dimensional |

CS, CSs | charge state, charge states |

FPKE | Fokker-Planck-Kolmogorov Equation |

DCM | diffusive-convective model |

GM | grid model |

CEC | central equilibrium cell |

CE | coronal equilibrium, the term refers to the 1D balance of ionization/recombination processes, that is, to the charge state distribution of impurity particles |

DCE | double coronal equilibrium is the 2D balance of impurity charge states |

EC | equilibrium cell, the 2D reduction scheme shown in Figure 4 |

REC | reduced equilibrium cell, the 2D reduction scheme shown in Figure 2 |

CCs | coupled cells, the 2D reduction scheme shown in Figure 4 |

HL | horizontal line, the 1D reduction scheme shown in Figure 4 |

VL | vertical line, the 1D reduction scheme shown in Figure 4 |

SP | separate pair of charge states, the 1D reduction scheme shown in Figure 4 |

A_{1}, A_{2}, F, H | the types of the total density profiles in Figure 5. |

TICS | Transport of impurity charge states, numerical transport code described in [17] |

S-equilibrium is of a 2D one given by the values ${\alpha}_{1},{\alpha}_{2},s$ described by Equation (16) | |

R- equilibrium is of a 2D one given by the values ${\alpha}_{1}^{-1},{\alpha}_{2}^{-1},\tilde{r}$ described by Equation (17) |

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**Figure 3.**Dependence of the limit ratio of densities in the boundary EC under the condition of the defined ionization rate gradient in this cell.

**Figure 4.**The main schemes of GM reduction by the pseudo-state technique: GM—the initial grid model of impurity distributions, CCs—coupled cells, HL and VL denote horizontal and vertical CS lines, respectively, EC—an equilibrium cell, SP—a separate pair of states.

**Figure 5.**(

**a**) Typical total impurity density profiles (A

_{1}, A

_{2}denote accumulation, F—almost flat, H—hollow) obtained in experiments on the DIII-D [29] (A

_{1}for C), JT-60 [30] (A

_{2}for C) and JET [31,32,37] (A1, A2, F, H for Ar) tokamaks; (

**b**) reduced velocity profiles ${\tilde{w}}_{n}$ (dashed) and ${\tilde{u}}_{n}$ (dotted) calculated using Formulas (33).

**Figure 6.**Dependences of the smallest (in absolute value) non-zero eigenvalue of the matrices $\tilde{N}$, corresponding to the profiles ${n}_{Z}\left(\rho \right)$ in Figure 5a, on the number of m cells.

**Figure 7.**The equilibrium of Ar calculated with a recombination (R, R

_{f}curves) and ionization (S curves) equilibrium base: (

**a**) profiles with and without central accumulation; (

**b**) profiles of the diffusivity calculated by Formula (37); (

**c**) S- and R-equilibrium regions determined by the relative profiles of hydrogen neutrals ${\xi}_{n}={n}_{n}/{n}_{e}$ separated by dash-dotted curves showing the limits of the transition region; (

**d**) abrupt change in the equilibrium constant depending on the average impurity charge with a gradual transition between the R and S regions due to changes in the profiles ${\xi}_{n}\left(\rho \right)$. The dash-dotted curves on (

**d**) show the same limits of the transition region.

**Figure 8.**Model calculations of equilibrium constant (

**a**) and confinement time (

**b**) calculated for Ar by Formula (42) with m = 12, various $j/j+1$ (as denoted), the equilibrium base (for S-equilibrium—full symbols and for R– equilibrium—open symbols), ${T}_{e}\left(0\right)$ and ${T}_{i}\left(0\right)$ varying within 1.5–8 keV, corresponding ${\xi}_{n}\left(\rho \right)$, ${n}_{e}=(4-9)\times {10}^{19}{\mathrm{m}}^{-3}$. The points denoted by JET correspond to the modelling data from JET [17].

**Figure 9.**The equilibrium constant of Ar in dependence on (

**a**) the number of cells m; (

**b**) on the central plasma electron temperature ${T}_{e}\left(0\right)$.

**Figure 10.**A comparison of the calculations using the exact Equation (43) and the approximate Equation (44) in the various equilibrium cases of C and Ar. Points shown with open symbols are related to He/Li equilibrium of Ar, while full symbols are of the He/H equilibrium type. Carbon data ($j/j+1$ are H-like ions and nucleous) are shown in black triangles.

**Figure 11.**The calculated quasi-stationary equilibrium of C in L-mode at ${t}_{1}=1.84$ s along with modelled central accumulation (curves A) and in H-mode at ${t}_{2}=2.04$ s, ${t}_{3}=2.24$ s, ${t}_{4}=2.54$ s: (

**a**) input total density profiles measured in the experiment [43]; (

**b**) the profiles of diffusion coefficient, calculated according to Formula (37); (

**c**) the equilibrium functions; (

**d**) calculated profiles of the relative density of hydrogen neutrals.

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Shurygin, V.A.
Nonlinear Problems of Equilibrium Charge State Transport in Hot Plasmas. *Symmetry* **2021**, *13*, 324.
https://doi.org/10.3390/sym13020324

**AMA Style**

Shurygin VA.
Nonlinear Problems of Equilibrium Charge State Transport in Hot Plasmas. *Symmetry*. 2021; 13(2):324.
https://doi.org/10.3390/sym13020324

**Chicago/Turabian Style**

Shurygin, Vladimir A.
2021. "Nonlinear Problems of Equilibrium Charge State Transport in Hot Plasmas" *Symmetry* 13, no. 2: 324.
https://doi.org/10.3390/sym13020324