^{1,a)}and Gary P. Zank

^{1,b)}

### Abstract

Turbulence simulations of a hydrodynamic fluid are performed to explore various nonlinear aspects of a nearly incompressible (NI) fluid in a regime where temperature fluctuations dominate. The NI model was developed primarily to understand weak compressive effects in interplanetary and interstellar media. Nonlinear structures generated by turbulent relaxation are shown to exist in our two-dimensional fluid simulations. Dynamically weak compressive effects are associated with passively convected thermal fluctuations which enhance the rate of selective decay in decaying NI turbulence.Turbulent relaxation leads to self-organization in thermally dominated NI velocity fluctuations and predicts the formation of large-scale steady-state coherent structures via an inverse cascade mechanism. In agreement with theoretical predictions, density fluctuations are slaved to the incompressible velocity fluctuations and exhibit a Kolmogorov-type power law. Thermal and density fluctuations are found to be anticorrelated in an adiabatic fluid. This suggests that a large fraction of the high plasma- fluid departs from a thermal equilibrium. Furthermore, compressional effects in nearly incompressible turbulence enhance decay rates significantly and lead to the formation of coherent vortices on much faster time scales when compared with incompressible turbulence.

D.S. and G.P.Z. have been supported in part by NASA Grants No. NAG5-11621 and No. NAG5-10932 and NSF Grant No. ATM0296113.

I. INTRODUCTION

II. THE HFD EQUATIONS

III. NONLINEAR SIMULATIONS

IV. SPECTRAL FEATURES

V. DENSITY TEMPERATURE CORRELATIONS

VI. CONCLUSIONS

### Key Topics

- Turbulent flows
- 69.0
- Rotating flows
- 30.0
- Interstellar medium
- 25.0
- Hydrodynamics
- 23.0
- Fluid equations
- 17.0

## Figures

(Color) Decaying turbulence in a NI fluid coupled to an IN fluid through nonlinear interactions. (a) Initial condition specified on the component of NI velocity field shows random fluctuations in a two-dimensional box. (b), (c), and (d) show fields at , respectively. Box size is . , . Other constants are , , , .

(Color) Decaying turbulence in a NI fluid coupled to an IN fluid through nonlinear interactions. (a) Initial condition specified on the component of NI velocity field shows random fluctuations in a two-dimensional box. (b), (c), and (d) show fields at , respectively. Box size is . , . Other constants are , , , .

(Color) Coherent structure formation corresponding to the component of the NI fluid velocity.

(Color) Coherent structure formation corresponding to the component of the NI fluid velocity.

The ratio of kinetic energies of incompressible and weakly compressible fluid, i.e., . The ratio shows a finite value at long times.

The ratio of kinetic energies of incompressible and weakly compressible fluid, i.e., . The ratio shows a finite value at long times.

Evolution of mean Fourier mode associated with selective decay rates. Dashed and solid curves are and rates for IN and NI turbulence, respectively. Decay rates are stronger in NI turbulence than in IN hydrodynamics.

Evolution of mean Fourier mode associated with selective decay rates. Dashed and solid curves are and rates for IN and NI turbulence, respectively. Decay rates are stronger in NI turbulence than in IN hydrodynamics.

(Color online) Comparison of analytic and numerical results of the vortex amplitude distribution in space. Shown is a 1D cut along the vorticity distribution. Clearly, the analytic and numerical solutions show excellent agreement.

(Color online) Comparison of analytic and numerical results of the vortex amplitude distribution in space. Shown is a 1D cut along the vorticity distribution. Clearly, the analytic and numerical solutions show excellent agreement.

Density power spectrum is plotted as a function of (along the horizontal axis) from a decaying NI hydrodynamics simulation. The incompressible velocity fluctuations follow a Kolmogorov spectrum close to (with an error ) in a forward (or enstrophy) cascade regime of decaying turbulence. The density fluctuations are passively convected by the incompressible velocity fluctuations and exhibit nearly the same spectrum. The intermediate curve represents the temperature spectrum.

Density power spectrum is plotted as a function of (along the horizontal axis) from a decaying NI hydrodynamics simulation. The incompressible velocity fluctuations follow a Kolmogorov spectrum close to (with an error ) in a forward (or enstrophy) cascade regime of decaying turbulence. The density fluctuations are passively convected by the incompressible velocity fluctuations and exhibit nearly the same spectrum. The intermediate curve represents the temperature spectrum.

A driven turbulence NI hydrodynamic simulation yields a Kolmogorov-type spectrum close to (with an error ) in the inverse (or energy) cascade regime for incompressible velocity fluctuations (left). Turbulence is driven by a random forcing in space and time. The compressible density fluctuations (right) follow the incompressible velocity spectrum closely in the inertial regime of turbulence. In the right panel, the temperature spectrum is shown below the density spectrum. The horizontal axis represents the modes .

A driven turbulence NI hydrodynamic simulation yields a Kolmogorov-type spectrum close to (with an error ) in the inverse (or energy) cascade regime for incompressible velocity fluctuations (left). Turbulence is driven by a random forcing in space and time. The compressible density fluctuations (right) follow the incompressible velocity spectrum closely in the inertial regime of turbulence. In the right panel, the temperature spectrum is shown below the density spectrum. The horizontal axis represents the modes .

Energy associated with temperature and density fluctuations in NI turbulence are shown, respectively, in (a) and (b). Shown here are solid , dashed , and dashed-dot curves. The decay rates depends critically upon the Prandtl number Pr. The density and the temperature fluctuations are clearly anticorrelated in agreement with the prediction of Ref. 8.

Energy associated with temperature and density fluctuations in NI turbulence are shown, respectively, in (a) and (b). Shown here are solid , dashed , and dashed-dot curves. The decay rates depends critically upon the Prandtl number Pr. The density and the temperature fluctuations are clearly anticorrelated in agreement with the prediction of Ref. 8.

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