# A non-modal analytical method to predict turbulent properties applied to the Hasegawa-Wakatani model

## Abstract

Linear eigenmode analysis often fails to describe turbulence in model systems that have non-normal linear operators and thus nonorthogonal eigenmodes, which can cause fluctuations to transiently grow faster than expected from eigenmode analysis. When combined with energetically conservative nonlinear mode mixing, transient growth can lead to sustained turbulence even in the absence of eigenmode instability. Since linear operators ultimately provide the turbulent fluctuations with energy, it is useful to define a growth rate that takes into account non-modal effects, allowing for prediction of energy injection, transport levels, and possibly even turbulent onset in the subcritical regime. Here, we define such a non-modal growth rate using a relatively simple model of the statistical effect that the nonlinearities have on cross-phases and amplitude ratios of the system state variables. In particular, we model the nonlinearities as delta-function-like, periodic forces that randomize the state variables once every eddy turnover time. Furthermore, we estimate the eddy turnover time to be the inverse of the least stable eigenmode frequency or growth rate, which allows for prediction without nonlinear numerical simulation. Also, we test this procedure on the 2D and 3D Hasegawa-Wakatani model [A. Hasegawa and M. Wakatani, Phys. Rev. Lett. 50, 682 (1983)] and findmore »

- Authors:

- Univ. of California, Berkeley, CA (United States). Dept. of Physics and Astronomy; Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Univ. of California, Berkeley, CA (United States). Dept. of Physics and Astronomy

- Publication Date:

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

- Sponsoring Org.:
- USDOE; National Science Foundation (NSF)

- OSTI Identifier:
- 1395522

- Alternate Identifier(s):
- OSTI ID: 1421181

- Report Number(s):
- LLNL-JRNL-733803

Journal ID: ISSN 1070-664X

- Grant/Contract Number:
- AC52-07NA27344; PHY-1202007

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 22; Journal Issue: 1; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION; DRIFT-WAVE TURBULENCE; PLASMA TURBULENCE; EDGE TURBULENCE; STABILITY; INSTABILITY; FLOW; TRANSITION

### Citation Formats

```
Friedman, B., and Carter, T. A..
```*A non-modal analytical method to predict turbulent properties applied to the Hasegawa-Wakatani model*. United States: N. p., 2015.
Web. doi:10.1063/1.4905863.

```
Friedman, B., & Carter, T. A..
```*A non-modal analytical method to predict turbulent properties applied to the Hasegawa-Wakatani model*. United States. doi:10.1063/1.4905863.

```
Friedman, B., and Carter, T. A.. Thu .
"A non-modal analytical method to predict turbulent properties applied to the Hasegawa-Wakatani model". United States.
doi:10.1063/1.4905863. https://www.osti.gov/servlets/purl/1395522.
```

```
@article{osti_1395522,
```

title = {A non-modal analytical method to predict turbulent properties applied to the Hasegawa-Wakatani model},

author = {Friedman, B. and Carter, T. A.},

abstractNote = {Linear eigenmode analysis often fails to describe turbulence in model systems that have non-normal linear operators and thus nonorthogonal eigenmodes, which can cause fluctuations to transiently grow faster than expected from eigenmode analysis. When combined with energetically conservative nonlinear mode mixing, transient growth can lead to sustained turbulence even in the absence of eigenmode instability. Since linear operators ultimately provide the turbulent fluctuations with energy, it is useful to define a growth rate that takes into account non-modal effects, allowing for prediction of energy injection, transport levels, and possibly even turbulent onset in the subcritical regime. Here, we define such a non-modal growth rate using a relatively simple model of the statistical effect that the nonlinearities have on cross-phases and amplitude ratios of the system state variables. In particular, we model the nonlinearities as delta-function-like, periodic forces that randomize the state variables once every eddy turnover time. Furthermore, we estimate the eddy turnover time to be the inverse of the least stable eigenmode frequency or growth rate, which allows for prediction without nonlinear numerical simulation. Also, we test this procedure on the 2D and 3D Hasegawa-Wakatani model [A. Hasegawa and M. Wakatani, Phys. Rev. Lett. 50, 682 (1983)] and find that the non-modal growth rate is a good predictor of energy injection rates, especially in the strongly non-normal, fully developed turbulence regime.},

doi = {10.1063/1.4905863},

journal = {Physics of Plasmas},

number = 1,

volume = 22,

place = {United States},

year = {Thu Jan 15 00:00:00 EST 2015},

month = {Thu Jan 15 00:00:00 EST 2015}

}

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