Unifying Scaling Theory for Vortex Dynamics in Two-Dimensional Turbulence

We present a scaling theory for unforced inviscid two-dimensional turbulence. Our model unifies existing spatial and temporal scaling theories. The theory is based on a self-similar distribution of vortices of different sizes $A$. Our model uniquely determines the spatial and temporal scaling of the associated vortex number density which allows the determination of the energy spectra and the vortex distributions. We find that the vortex number density scales as $n(A,t)\sim t^{-2/3}/A$, which implies an energy spectrum $\mathcal{E}\sim k^{-5}$, significantly steeper than the classical Batchelor-Kraichnan scaling. High-resolution numerical simulations corroborate the model.

Figure 1

Vorticity field at $t=4$ (left) and $t=160$ (right) in one representative simulation. A linear gray scale is used, with the highest positive vorticity being white. Only $1/16$ of the domain is shown.

Figure 2

Full vorticity field $\omega (\mathbf{x},t=16)$ (left) and its corresponding coherent part (right). Only $1/16$ of the domain is shown.

Figure 3

Ensemble-averaged normalized number density (left) and mean vorticity ${\omega }_v$ (right) versus area, in logarithmic scales; $t=0$ is shown by a dashed line with diamonds, $20\le t\le 30$ (averaged) is shown by a thin line with squares, and $100\le t\le 160$ (averaged) is shown by a bold line with triangles.

Figure 4

Ensemble- and time-averaged ($100\le t\le 160$) enstrophy spectrum (thin line) and its coherent counterpart (bold line) versus wave number $k$, in logarithmic scales. Various slopes are indicated.

Figure 5

Temporal decay of total (resolved) enstrophy $Q(t)$ (top curve) and coherent vortex enstrophy $Q_v(t)$ (bottom curve), in logarithmic scales.