^{1}

### Abstract

We show that some statistical properties of forced two-dimensional turbulence have an important sensitivity to the form of large-scale dissipation, which is required to damp the inverse cascade. We consider three models of large-scale dissipation: linear “Ekman” drag, nonlinear quadratic drag, and scale-selective hypo-drag that damps only low-wavenumber modes. In all cases, the statistically steady vorticity field is dominated by almost axisymmetric vortices, and the probability density function of vorticity is non-Gaussian. However, in the case of linear and quadratic drag, we find that the velocity statistics is close to Gaussian, with non-negligible contribution coming from the background turbulent flow. On the other hand, with hypo-drag, the probability density function of velocity is non-Gaussian and is predominantly determined by the properties of the vortices. With hypo-drag, the relative positions of the vortices and the exponential distribution of the vortex extremum are important factors responsible for the non-Gaussian velocity statistics.

I acknowledge useful discussions and correspondence with Antonello Provenzale and Jost von Hardenberg. I thank Bill Young for comments and discussions that improved the scientific content and presentation of the paper. This work was supported by the National Science Foundation under Grant No. OCE07-26320.

I. INTRODUCTION

II. A MODEL OF FORCED TWO-DIMENSIONAL TURBULENCE

III. STATISTICS OF THE SIMULATIONS

IV. ROLE OF COHERENT VORTICES IN DETERMINING THE VELOCITY PDF

A. Vortex census and Gaussian vortex reconstruction

B. Vortical contribution to velocity PDF with hypo-drag

C. Vortical contribution to velocity PDF with linear drag

D. Transition from hypo-drag to linear drag

V. CONCLUSION

### Key Topics

- Rotating flows
- 81.0
- Vortex dynamics
- 38.0
- Turbulent flows
- 26.0
- Statistical properties
- 8.0
- Probability density functions
- 6.0

## Figures

Snapshots of the vorticity from simulations using different models of . Upper panel is hypo-drag with ; middle panel is linear drag with ; and lower panel is quadratic drag with . Strong vortices are indicated by the larger scale used on the -axis in the top panel.

Snapshots of the vorticity from simulations using different models of . Upper panel is hypo-drag with ; middle panel is linear drag with ; and lower panel is quadratic drag with . Strong vortices are indicated by the larger scale used on the -axis in the top panel.

Lagrangian trajectories of selected particles from the simulation shown in the middle panel of Fig. 1 (linear drag ). Three of the particles were released close to a vorticity extremum and exhibit “looping” trajectories, characteristic of trapping within a vortex. The other two particles were released in the filamentary sea between the vortices.

Lagrangian trajectories of selected particles from the simulation shown in the middle panel of Fig. 1 (linear drag ). Three of the particles were released close to a vorticity extremum and exhibit “looping” trajectories, characteristic of trapping within a vortex. The other two particles were released in the filamentary sea between the vortices.

Energy spectra for the three simulations shown in Fig. 1.

Energy spectra for the three simulations shown in Fig. 1.

Top panel: vorticity PDFs corresponding to the three simulations in Fig. 1. The abscissa is , and the dotted parabola is a standard Gaussian. Lower panel: velocity PDFs and the standard Gaussian (the dotted parabola). The abscissa is normalized with and . PDFs of and almost coincide and are close to the standard Gaussian for linear drag and quadratic drag.

Top panel: vorticity PDFs corresponding to the three simulations in Fig. 1. The abscissa is , and the dotted parabola is a standard Gaussian. Lower panel: velocity PDFs and the standard Gaussian (the dotted parabola). The abscissa is normalized with and . PDFs of and almost coincide and are close to the standard Gaussian for linear drag and quadratic drag.

Vorticity (upper panel) and velocity (lower panel) PDFs from a sequence of simulations using linear drag, with decreasing from 0.1 to 0.002. The abscissa uses standardized variables etc. The velocity PDF remains close to the standard Gaussian (the smooth parabola) at all values of , while the vorticity PDF becomes increasingly non-Gaussian as is decreased.

Vorticity (upper panel) and velocity (lower panel) PDFs from a sequence of simulations using linear drag, with decreasing from 0.1 to 0.002. The abscissa uses standardized variables etc. The velocity PDF remains close to the standard Gaussian (the smooth parabola) at all values of , while the vorticity PDF becomes increasingly non-Gaussian as is decreased.

Snapshots of the velocity for hypo-drag (upper) and linear drag (lower), regions in which are inside the black solid bubbles.

Snapshots of the velocity for hypo-drag (upper) and linear drag (lower), regions in which are inside the black solid bubbles.

Distribution of vorticity extremum for hypo-drag , linear drag , and quadratic drag simulations. The solid straight line is a least-square-fit to the exponential.

Distribution of vorticity extremum for hypo-drag , linear drag , and quadratic drag simulations. The solid straight line is a least-square-fit to the exponential.

Insensitivity of velocity PDF obtained by the Gaussian vortex reconstruction Eq. (9) to the threshold for hypo-drag . Results are similar for linear and quadratic drag.

Insensitivity of velocity PDF obtained by the Gaussian vortex reconstruction Eq. (9) to the threshold for hypo-drag . Results are similar for linear and quadratic drag.

Comparison of velocity PDF from simulations (circles) to that obtained by the Gaussian vortex reconstruction Eq. (9) (solid) for hypo-drag (upper) and linear drag (lower). Velocity PDFs obtained by GVR but with randomized vortex positions (dashed) and the exponential distribution in vorticity extremum replaced by constant (dot-dashed) are also plotted in the hypo-drag case.

Comparison of velocity PDF from simulations (circles) to that obtained by the Gaussian vortex reconstruction Eq. (9) (solid) for hypo-drag (upper) and linear drag (lower). Velocity PDFs obtained by GVR but with randomized vortex positions (dashed) and the exponential distribution in vorticity extremum replaced by constant (dot-dashed) are also plotted in the hypo-drag case.

Probability distribution of vortex pair separation for hypo-drag and linear drag . Opposite-signed and like-signed vortex pairs are considered separately. The main figure focuses on small , and the inset shows for the full range of .

Probability distribution of vortex pair separation for hypo-drag and linear drag . Opposite-signed and like-signed vortex pairs are considered separately. The main figure focuses on small , and the inset shows for the full range of .

The evolution of standardized vorticity PDF , standardized velocity PDF , energy spectrum , and velocity kurtosis when the value of in hypo-drag simulations is varied. corresponds to the simulation with linear drag . The dotted curve in the lower left panel is the standard Gaussian. In the lower right panel, the dashed line indicates the value of for linear drag ; the solid curve is the fit in Eq. (10).

The evolution of standardized vorticity PDF , standardized velocity PDF , energy spectrum , and velocity kurtosis when the value of in hypo-drag simulations is varied. corresponds to the simulation with linear drag . The dotted curve in the lower left panel is the standard Gaussian. In the lower right panel, the dashed line indicates the value of for linear drag ; the solid curve is the fit in Eq. (10).

## Tables

A summary of the statistical properties of nine runs. All quantities are nondimensionalized using to scale length and to scale time. In the third column, is the rate of energy dissipation by large-scale drag (as opposed to hyperviscosity), and in the fourth column, .

A summary of the statistical properties of nine runs. All quantities are nondimensionalized using to scale length and to scale time. In the third column, is the rate of energy dissipation by large-scale drag (as opposed to hyperviscosity), and in the fourth column, .

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