# Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria

## Abstract

The Hamiltonian structure of ideal translationally symmetric extended MHD (XMHD) is obtained by employing a method of Hamiltonian reduction on the three-dimensional noncanonical Poisson bracket of XMHD. The existence of the continuous spatial translation symmetry allows the introduction of Clebsch-like forms for the magnetic and velocity fields. Upon employing the chain rule for functional derivatives, the 3D Poisson bracket is reduced to its symmetric counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir invariants are identified, and used to obtain energy-Casimir variational principles for generalized XMHD equilibrium equations with arbitrary macroscopic flows. The obtained set of generalized equations is cast into Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria, where the electron inertia is neglected. The barotropic Hall MHD equilibrium equations are derived as a limiting case of the XMHD GSB system, and a numerically computed equilibrium configuration is presented that shows the separation of ion-flow from electro-magnetic surfaces.

- Authors:

- Univ. of Ioannina, Ioannina (Greece). Dept. of Physics
- 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:
- 1535314

- Alternate Identifier(s):
- OSTI ID: 1374779

- Grant/Contract Number:
- FG05-80ET-53088

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 9; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- physics

### Citation Formats

```
Kaltsas, D. A., Throumoulopoulos, G. N., and Morrison, P. J. Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria. United States: N. p., 2017.
Web. doi:10.1063/1.4986013.
```

```
Kaltsas, D. A., Throumoulopoulos, G. N., & Morrison, P. J. Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria. United States. doi:10.1063/1.4986013.
```

```
Kaltsas, D. A., Throumoulopoulos, G. N., and Morrison, P. J. Fri .
"Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria". United States. doi:10.1063/1.4986013. https://www.osti.gov/servlets/purl/1535314.
```

```
@article{osti_1535314,
```

title = {Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria},

author = {Kaltsas, D. A. and Throumoulopoulos, G. N. and Morrison, P. J.},

abstractNote = {The Hamiltonian structure of ideal translationally symmetric extended MHD (XMHD) is obtained by employing a method of Hamiltonian reduction on the three-dimensional noncanonical Poisson bracket of XMHD. The existence of the continuous spatial translation symmetry allows the introduction of Clebsch-like forms for the magnetic and velocity fields. Upon employing the chain rule for functional derivatives, the 3D Poisson bracket is reduced to its symmetric counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir invariants are identified, and used to obtain energy-Casimir variational principles for generalized XMHD equilibrium equations with arbitrary macroscopic flows. The obtained set of generalized equations is cast into Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria, where the electron inertia is neglected. The barotropic Hall MHD equilibrium equations are derived as a limiting case of the XMHD GSB system, and a numerically computed equilibrium configuration is presented that shows the separation of ion-flow from electro-magnetic surfaces.},

doi = {10.1063/1.4986013},

journal = {Physics of Plasmas},

number = 9,

volume = 24,

place = {United States},

year = {2017},

month = {8}

}

*Citation information provided by*

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