# Exact collisional moments for plasma fluid theories

## Abstract

The velocity-space moments of the often troublesome nonlinear Landau collision operator are expressed exactly in terms of multi-index Hermite-polynomial moments of distribution functions. The collisional moments are shown to be generated by derivatives of two well-known functions, namely, the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The resulting formula has a nonlinear dependency on the relative mean flow of the colliding species normalised to the root-mean-square of the corresponding thermal velocities and a bilinear dependency on densities and higher-order velocity moments of the distribution functions, with no restriction on temperature, flow, or mass ratio of the species. The result can be applied to both the classic transport theory of plasmas that relies on the Chapman-Enskog method, as well as to derive collisional fluid equations that follow Grad's moment approach. As an illustrative example, we provide the collisional ten-moment equations with exact conservation laws for momentum-and energy-transfer rates.

- Authors:

- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
- Princeton Univ., NJ (United States); Harvard Univ., Cambridge, MA (United States); Harvard-Smithsonian Center for Astrophysics, Cambridge, MA (United States)

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

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

- Contributing Org.:
- Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA; Harvard-Smithsonian Center for Astrophysics, The Institute for Theory and Computation, Cambridge, Massachusetts 02138, USA

- OSTI Identifier:
- 1358661

- Alternate Identifier(s):
- OSTI ID: 1361817

- Grant/Contract Number:
- AC02-09CH11466; AC02-09-CH11466

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 4; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; Transport; Equation

### Citation Formats

```
Pfefferlé, D., Hirvijoki, E., and Lingam, M.
```*Exact collisional moments for plasma fluid theories*. United States: N. p., 2017.
Web. doi:10.1063/1.4979992.

```
Pfefferlé, D., Hirvijoki, E., & Lingam, M.
```*Exact collisional moments for plasma fluid theories*. United States. doi:10.1063/1.4979992.

```
Pfefferlé, D., Hirvijoki, E., and Lingam, M. Sat .
"Exact collisional moments for plasma fluid theories". United States.
doi:10.1063/1.4979992. https://www.osti.gov/servlets/purl/1358661.
```

```
@article{osti_1358661,
```

title = {Exact collisional moments for plasma fluid theories},

author = {Pfefferlé, D. and Hirvijoki, E. and Lingam, M.},

abstractNote = {The velocity-space moments of the often troublesome nonlinear Landau collision operator are expressed exactly in terms of multi-index Hermite-polynomial moments of distribution functions. The collisional moments are shown to be generated by derivatives of two well-known functions, namely, the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The resulting formula has a nonlinear dependency on the relative mean flow of the colliding species normalised to the root-mean-square of the corresponding thermal velocities and a bilinear dependency on densities and higher-order velocity moments of the distribution functions, with no restriction on temperature, flow, or mass ratio of the species. The result can be applied to both the classic transport theory of plasmas that relies on the Chapman-Enskog method, as well as to derive collisional fluid equations that follow Grad's moment approach. As an illustrative example, we provide the collisional ten-moment equations with exact conservation laws for momentum-and energy-transfer rates.},

doi = {10.1063/1.4979992},

journal = {Physics of Plasmas},

number = 4,

volume = 24,

place = {United States},

year = {Sat Apr 01 00:00:00 EDT 2017},

month = {Sat Apr 01 00:00:00 EDT 2017}

}

*Citation information provided by*

Web of Science

Web of Science