  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 $\Re$.
