The number of degrees of freedom of three-dimensional Navier--Stokes turbulence

In Kolmogorov's phenomenological theory of turbulence, the energy spectrum in the inertial range scales with the wave number $k$ as $k^{-5/3}$ and extends up to a dissipation wave number $k_\nu$, which is given in terms of the energy dissipation rate $\epsilon$ and viscosity $\nu$ by $k_\nu\propto(\epsilon/\nu^3)^{1/4}$. This result leads to Landau's heuristic estimate for the number of degrees of freedom that scales as $\Re^{9/4}$, where $\Re$ is the Reynolds number. Here we consider the possibility of establishing a quantitative basis for these results from first principles. In particular, we examine the extent to which they can be derived from the three-dimensional Navier--Stokes system, making use of Kolmogorov's hypothesis of finite and viscosity-independent energy dissipation only. It is found that the Taylor microscale wave number $k_T$ (a close cousin of $k_\nu$) can be expressed in the form $k_T \le CU/\nu = (CU/\norm{\u})^{1/2}(\epsilon/\nu^3)^{1/4}$. Here $U$ and $\norm{\u}$ are, respectively, a ``microscale'' velocity and the root mean square velocity, and $C\le1$ is a dynamical parameter. This result can be seen to be in line with Kolmogorov's prediction for $k_\nu$. Furthermore, it is shown that the minimum number of greatest Lyapunov exponents whose sum becomes negative does not exceed $\Re^{9/4}$, where $\Re$ is defined in terms of an average energy dissipation rate, the system length scale, and $\nu$. This result is in a remarkable agreement with the Landau estimate, up to a presumably slight discrepancy between the conventional and the present energy dissipation rates used in the definition of $\Re$.