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# Linear response of a Hall magnetic drift wave for verification of Hall MHD algorithms

## Abstract

Numerical implementations of Hall magnetohydrodynamics (Hall MHD) can be challenging due to the nonlinear multidimensional nature of the Hall term. Here, a model problem is presented that couples the hydrodynamic motion of the plasma to Hall MHD evolution of the magnetic field. The Hall MHD equations are linearized about unperturbed solutions in both cylindrical and Cartesian coordinates in two dimensions. The magnetic field is assumed to lie in the ignorable direction, and the linear response about the unperturbed solution is considered. The resulting ordinary differential equation is used to numerically compute the eigenfunctions and eigenfrequencies of the mode. The resulting eigenfunctions do not make the local wave approximation but are instead global solutions that depend on the spatial dependence of the unperturbed Alfvén speed. Hall MHD simulations are then performed in the Ares multiphysics code and shown to agree with the predicted phase velocities of the wave, and the simulated solutions are shown to numerically converge to the semianalytic modes. By varying the background density of the plasma (and correspondingly, the ion inertial length), the importance of Hall physics can be varied. This allows the test problem to transition from the classical MHD limit to the extreme Hall MHD limit.more »

- Authors:

- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1561457

- Alternate Identifier(s):
- OSTI ID: 1567932

- Report Number(s):
- LLNL-JRNL-768218

Journal ID: ISSN 1070-664X; 958627

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 26; Journal Issue: 7; 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

```
Farmer, W. A., Ellison, C. L., and Hammer, J. H. Linear response of a Hall magnetic drift wave for verification of Hall MHD algorithms. United States: N. p., 2019.
Web. doi:10.1063/1.5094349.
```

```
Farmer, W. A., Ellison, C. L., & Hammer, J. H. Linear response of a Hall magnetic drift wave for verification of Hall MHD algorithms. United States. doi:10.1063/1.5094349.
```

```
Farmer, W. A., Ellison, C. L., and Hammer, J. H. Tue .
"Linear response of a Hall magnetic drift wave for verification of Hall MHD algorithms". United States. doi:10.1063/1.5094349.
```

```
@article{osti_1561457,
```

title = {Linear response of a Hall magnetic drift wave for verification of Hall MHD algorithms},

author = {Farmer, W. A. and Ellison, C. L. and Hammer, J. H.},

abstractNote = {Numerical implementations of Hall magnetohydrodynamics (Hall MHD) can be challenging due to the nonlinear multidimensional nature of the Hall term. Here, a model problem is presented that couples the hydrodynamic motion of the plasma to Hall MHD evolution of the magnetic field. The Hall MHD equations are linearized about unperturbed solutions in both cylindrical and Cartesian coordinates in two dimensions. The magnetic field is assumed to lie in the ignorable direction, and the linear response about the unperturbed solution is considered. The resulting ordinary differential equation is used to numerically compute the eigenfunctions and eigenfrequencies of the mode. The resulting eigenfunctions do not make the local wave approximation but are instead global solutions that depend on the spatial dependence of the unperturbed Alfvén speed. Hall MHD simulations are then performed in the Ares multiphysics code and shown to agree with the predicted phase velocities of the wave, and the simulated solutions are shown to numerically converge to the semianalytic modes. By varying the background density of the plasma (and correspondingly, the ion inertial length), the importance of Hall physics can be varied. This allows the test problem to transition from the classical MHD limit to the extreme Hall MHD limit. Furthermore, this problem is a useful tool for the verification of Hall MHD routines implemented in various codes, and the robustness of a routine can be tested in regimes in which Hall physics is dominant.},

doi = {10.1063/1.5094349},

journal = {Physics of Plasmas},

number = 7,

volume = 26,

place = {United States},

year = {2019},

month = {7}

}

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