Numerical results

To directly check the prediction for the steepening of the enstrophy spectrum and for the statistics of small scale fluctuations of vorticity obtained in the previous section using the analogy with the passive scalar problem, we have performed numerical integration of Navier-Stokes equation for the vorticity field (Eq. (2.6))

The numerical integration is performed by a fully de-aliased pseudo-spectral code with a second-order Runge-Kutta scheme, on a doubly periodic square domain of size $L = 2\pi$ at different resolutions: $N^2 = 512^2, 1024^2, 2048^2$ grid points. Simulations have been partially performed at CINECA on IBM SP3 and SP4 parallel computers.

The vorticity fluctuations are injected by a large-scale forcing $f_\omega$ which is Gaussian, $\delta$-correlated in time, and limited to a shell of wave-numbers around $k_f=2\pi/L$. Forcing amplitude is chosen to provide an enstrophy injection rate $F=0.16$. At variance with other choices for $f_\omega$ commonly used (e.g. large-scale shear) this kind of forcing ensures the statistical isotropy and homogeneity of the vorticity field.

Figure 2.3: Snapshots of the vorticity field obtained from numerical simulations with two different values of friction: $\alpha = 0.15$ (left) and $\alpha = 0.23$ (right).

Different values of the friction $\alpha = 0.15, 0.23, 0.30$ has been tested, and a small viscosity (see Table (2.7)) is used to remove the remnant enstrophy flux at small scales.