# Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

## Abstract

Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect helical symmetry by reducing the Poisson bracket for the three-dimensional dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD (Kaltsaset al.,Phys. Plasmas, vol. 24, 2017, 092504). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy–Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad–Shafranov–Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helically symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a magnetic configuration having non-planar helical magnetic axis.

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1538937

- DOE Contract Number:
- FG05-80ET53088

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Plasma Physics

- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 3; Journal ID: ISSN 0022-3778

- Publisher:
- Cambridge University Press

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Physics

### Citation Formats

```
Kaltsas, D. A., Throumoulopoulos, G. N., and Morrison, P. J.
```*Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles*. United States: N. p., 2018.
Web. doi:10.1017/s0022377818000338.

```
Kaltsas, D. A., Throumoulopoulos, G. N., & Morrison, P. J.
```*Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles*. United States. doi:10.1017/s0022377818000338.

```
Kaltsas, D. A., Throumoulopoulos, G. N., and Morrison, P. J. Thu .
"Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles". United States. doi:10.1017/s0022377818000338.
```

```
@article{osti_1538937,
```

title = {Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles},

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

abstractNote = {Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect helical symmetry by reducing the Poisson bracket for the three-dimensional dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD (Kaltsaset al.,Phys. Plasmas, vol. 24, 2017, 092504). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy–Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad–Shafranov–Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helically symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a magnetic configuration having non-planar helical magnetic axis.},

doi = {10.1017/s0022377818000338},

journal = {Journal of Plasma Physics},

issn = {0022-3778},

number = 3,

volume = 84,

place = {United States},

year = {2018},

month = {5}

}