# Closures for Two-fluid Five-moment Equations for Magnetized Plasmas Grant Recipient: Utah State University Final Scientific Report

## Abstract

In the proposed work, we will develop a complete set of closures for two-fluid five-moment equations for magnetized plasmas. Specifically, we propose to obtain explicit formulas for electrons and ions to express heat flow, viscosity, frictional force, and collisional heating in terms of density, and temperature, flow velocity. For this purpose, we solve a set of general moment equations for the relevant closure moments. An infinite hierarchy of general moment equations is completely equivalent to the original Landau-Fokker-Planck kinetic equation. In the small gyro-period expansion, the parallel components of the general moment equations agree with the moments of the drift-kinetic equation (DKE). The advantages of solving the moment equations include (i) an exact treatment of the collision operator, (ii) closure and transport relations with no flux-surface average, and (iii) a unified form of closures for general collisionality. Practically we solve a truncated set of moment equations and hence must verify the convergence of the solutions by increasing number of moments. The parallel integral (nonlocal) closures obtained by solving a set of linearized parallel moment equations are useful only when the nonlinear coupling of moments to the temperature and magnetic field gradient terms is negligible. When the nonlinear coupling terms aremore »

- Authors:

- Publication Date:

- Research Org.:
- Jeong-Young Ji/Utah State University

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24)

- OSTI Identifier:
- 1481730

- Report Number(s):
- Final report: DOE-USU-SC0014033

- DOE Contract Number:
- SC0014033

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Ji, Jeong-Young.
```*Closures for Two-fluid Five-moment Equations for Magnetized Plasmas Grant Recipient: Utah State University Final Scientific Report*. United States: N. p., 2018.
Web. doi:10.2172/1481730.

```
Ji, Jeong-Young.
```*Closures for Two-fluid Five-moment Equations for Magnetized Plasmas Grant Recipient: Utah State University Final Scientific Report*. United States. doi:10.2172/1481730.

```
Ji, Jeong-Young. Mon .
"Closures for Two-fluid Five-moment Equations for Magnetized Plasmas Grant Recipient: Utah State University Final Scientific Report". United States. doi:10.2172/1481730. https://www.osti.gov/servlets/purl/1481730.
```

```
@article{osti_1481730,
```

title = {Closures for Two-fluid Five-moment Equations for Magnetized Plasmas Grant Recipient: Utah State University Final Scientific Report},

author = {Ji, Jeong-Young},

abstractNote = {In the proposed work, we will develop a complete set of closures for two-fluid five-moment equations for magnetized plasmas. Specifically, we propose to obtain explicit formulas for electrons and ions to express heat flow, viscosity, frictional force, and collisional heating in terms of density, and temperature, flow velocity. For this purpose, we solve a set of general moment equations for the relevant closure moments. An infinite hierarchy of general moment equations is completely equivalent to the original Landau-Fokker-Planck kinetic equation. In the small gyro-period expansion, the parallel components of the general moment equations agree with the moments of the drift-kinetic equation (DKE). The advantages of solving the moment equations include (i) an exact treatment of the collision operator, (ii) closure and transport relations with no flux-surface average, and (iii) a unified form of closures for general collisionality. Practically we solve a truncated set of moment equations and hence must verify the convergence of the solutions by increasing number of moments. The parallel integral (nonlocal) closures obtained by solving a set of linearized parallel moment equations are useful only when the nonlinear coupling of moments to the temperature and magnetic field gradient terms is negligible. When the nonlinear coupling terms are substantial, we propose a Fourier transform method to solve the parallel moment equations. By Fourier-expanding the magnetic field, we can convert the differential equations of parallel moments to linear algebraic equations for Fourier components of the moments. The convergence of the solutions can be confirmed by increasing the number of Fourier modes. Our preliminary study in axisymmetric circular magnetic geometry with a large aspect ratio shows that (i) the solutions for Fourier modes n=0 and n=1 converge when modes up to n=2 are included in the expansion, and (ii) lower collisionality requires more moments for convergent closures, as in the case of the integral closures. The proposed work will start from this simple magnetic geometry to find general parallel closures and then be generalized to explore a non-circular magnetic field with finite aspect-ratio. With parallel moments known, the perpendicular component of moment closures can be obtained by inverting the generalized cross product operator in the general moment equations. These efforts will be performed in conjunction with several multi-institutional projects, NIMROD, the Center for Extended MHD Modeling (CEMM), and the Plasma Science and Innovation (PSI) Center.},

doi = {10.2172/1481730},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2018},

month = {11}

}