# Eigenmode analysis of the sheared-flow Z-pinch

## Abstract

Experiments have demonstrated that a Z-pinch can persist for thousands of times longer than the growth time of global magnetohydrodynamic (MHD) instabilities such as the m=0 sausage and m=1 kink modes. These modes have growth times on the order of ta=a/vi, where vi is the ion thermal speed and a is the pinch radius. Axial flows with duz/dr ≲ vi/a have been measured during the stable period, and the commonly accepted theory is that this amount of shear is sufficient to stabilize these modes as predicted by numerical studies using the ideal MHD equations. However, these studies only consider specific equilibrium profiles that typically have a modest magnitude for the logarithmic pressure gradient, qP≡d ln P/d ln r, and may not represent experimental conditions. Linear stability of the sheared-flow Z-pinch is studied here via a direct eigen-decomposition of the matrix operator obtained from the linear ideal MHD equations. Several equilibrium profiles with a large variation of qP are examined. Considering a practical range of k, 1/3 ≲ ka ≲ 10, it is shown that the shear required to stabilize m=0 modes can be expressed as duz/dr≥Cγ0/(ka)α. Here, γ0=γ0(ka) is the profile-specific growth rate in the absence of shear, which scales approximately with |qP|. Both C and α are profile-specific constants, butmore »

- Authors:

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1737847

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Publisher's Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Name: Physics of Plasmas Journal Volume: 27 Journal Issue: 12; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics

- Country of Publication:
- United States

- Language:
- English

## Citation Formats

```
Angus, J. R., Van De Wetering, J. J., Dorf, M., and Geyko, V. I.. Eigenmode analysis of the sheared-flow Z-pinch. United States: N. p., 2020.
Web. doi:10.1063/5.0029716.
```

```
Angus, J. R., Van De Wetering, J. J., Dorf, M., & Geyko, V. I.. Eigenmode analysis of the sheared-flow Z-pinch. United States. https://doi.org/10.1063/5.0029716
```

```
Angus, J. R., Van De Wetering, J. J., Dorf, M., and Geyko, V. I.. Thu .
"Eigenmode analysis of the sheared-flow Z-pinch". United States. https://doi.org/10.1063/5.0029716.
```

```
@article{osti_1737847,
```

title = {Eigenmode analysis of the sheared-flow Z-pinch},

author = {Angus, J. R. and Van De Wetering, J. J. and Dorf, M. and Geyko, V. I.},

abstractNote = {Experiments have demonstrated that a Z-pinch can persist for thousands of times longer than the growth time of global magnetohydrodynamic (MHD) instabilities such as the m=0 sausage and m=1 kink modes. These modes have growth times on the order of ta=a/vi, where vi is the ion thermal speed and a is the pinch radius. Axial flows with duz/dr ≲ vi/a have been measured during the stable period, and the commonly accepted theory is that this amount of shear is sufficient to stabilize these modes as predicted by numerical studies using the ideal MHD equations. However, these studies only consider specific equilibrium profiles that typically have a modest magnitude for the logarithmic pressure gradient, qP≡d ln P/d ln r, and may not represent experimental conditions. Linear stability of the sheared-flow Z-pinch is studied here via a direct eigen-decomposition of the matrix operator obtained from the linear ideal MHD equations. Several equilibrium profiles with a large variation of qP are examined. Considering a practical range of k, 1/3 ≲ ka ≲ 10, it is shown that the shear required to stabilize m=0 modes can be expressed as duz/dr≥Cγ0/(ka)α. Here, γ0=γ0(ka) is the profile-specific growth rate in the absence of shear, which scales approximately with |qP|. Both C and α are profile-specific constants, but C is order unity and α≈1. It is further demonstrated that even a large value of shear, duz/dr=3vi/a, is not sufficient to provide linear stabilization of the m=1 kink mode for all profiles considered. This result is in contrast to the currently accepted theory predicting stabilization at much lower shear, duz/dr=0.1vi/a, and suggests that the experimentally observed stability cannot be explained within the linear ideal-MHD model.},

doi = {10.1063/5.0029716},

journal = {Physics of Plasmas},

number = 12,

volume = 27,

place = {United States},

year = {2020},

month = {12}

}

https://doi.org/10.1063/5.0029716

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