# An integral transform technique for kinetic systems with collisions

## Abstract

The linearized Vlasov-Poisson system can be exactly solved using the G-transform, an integral transform introduced in Morrison and Pfirsch [Phys. Fluids B 4, 3038–3057 (1992)] and Morrison [Phys. Plasmas 1, 1447 (1994); Transp. Theory Stat. Phys. 29, 397 (2000)] that removes the electric field term, leaving a simple advection equation. We investigate how this integral transform interacts with the Fokker-Planck collision operator. The commutator of this collision operator with the G-transform (the “shielding term”) is shown to be negligible. We exactly solve the advection-diffusion equation without the shielding term. This solution determines when collisions dominate and when advection (i.e., Landau damping) dominates. This integral transform can also be used to simplify gyro-/drift-kinetic equations. We present new gyrofluid equations formed by taking moments of the G-transformed equation. Since many gyro-/drift-kinetic codes use Hermite polynomials as base elements, we include an explicit calculation of their G-transform

- Authors:

- Univ. of Texas, Austin, TX (United States). Dept. of Physics and Institute for Fusion Studies
- Univ. of Texas, Austin, TX (United States). Dept. of Physics and Inst. for Fusion Studies

- Publication Date:

- Research Org.:
- Univ. of Texas, Austin, TX (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1515033

- Alternate Identifier(s):
- OSTI ID: 1465620

- Grant/Contract Number:
- [FG05-80ET53088; FG05-80ET-53088]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- [ Journal Volume: 25; Journal Issue: 8]; 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

### Citation Formats

```
Heninger, J. M., and Morrison, P. J. An integral transform technique for kinetic systems with collisions. United States: N. p., 2018.
Web. doi:10.1063/1.5046194.
```

```
Heninger, J. M., & Morrison, P. J. An integral transform technique for kinetic systems with collisions. United States. doi:10.1063/1.5046194.
```

```
Heninger, J. M., and Morrison, P. J. Tue .
"An integral transform technique for kinetic systems with collisions". United States. doi:10.1063/1.5046194. https://www.osti.gov/servlets/purl/1515033.
```

```
@article{osti_1515033,
```

title = {An integral transform technique for kinetic systems with collisions},

author = {Heninger, J. M. and Morrison, P. J.},

abstractNote = {The linearized Vlasov-Poisson system can be exactly solved using the G-transform, an integral transform introduced in Morrison and Pfirsch [Phys. Fluids B 4, 3038–3057 (1992)] and Morrison [Phys. Plasmas 1, 1447 (1994); Transp. Theory Stat. Phys. 29, 397 (2000)] that removes the electric field term, leaving a simple advection equation. We investigate how this integral transform interacts with the Fokker-Planck collision operator. The commutator of this collision operator with the G-transform (the “shielding term”) is shown to be negligible. We exactly solve the advection-diffusion equation without the shielding term. This solution determines when collisions dominate and when advection (i.e., Landau damping) dominates. This integral transform can also be used to simplify gyro-/drift-kinetic equations. We present new gyrofluid equations formed by taking moments of the G-transformed equation. Since many gyro-/drift-kinetic codes use Hermite polynomials as base elements, we include an explicit calculation of their G-transform},

doi = {10.1063/1.5046194},

journal = {Physics of Plasmas},

number = [8],

volume = [25],

place = {United States},

year = {2018},

month = {8}

}

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