Computation of the dimension of two-dimensional turbulence

The Lyapunov dimension of an incompressible, homogeneous, two-dimensional periodic shear flow with moderate resolution (from ${16}^{\mathrm{*}}$16 up to ${128}^{\mathrm{*}}$128 points) is calculated. The results show that the dimension of the chaotic attractor is much lower than the total number of degrees of freedom represented in the numerical system, and depends essentially on the scale at which energy is injected in the system, and not (or very weakly) on the Reynolds number, suggesting that only a fraction of the modes at a scale larger than that of the forced mode is turbulent.