  Transition to condensate state in the degenerate gas of bosons is studied
using the Gross-Pitaevskii equation. It is shown that adiabatic invariance of
an enstrophy (mean squared vorticity) based on a generalized vorticity $curl
(|\psi|^2 {\bf v})/|\psi|^2$, controls dynamics of the transitional turbulence
just before formation of the condensate with its tangles of quantized vortices
($|\psi|^2 {\bf v}$ is the weighted velocity field defined on the macroscopic
wave function $\psi$). Scaling of the angle-averaged occupation number
spectrum, $N_k \sim k^{-1}$, has been obtained for the free developing
transitional turbulence for the weak nonlinear (and completely disordered)
initial conditions. Results of the three-dimensional numerical simulations have
been used to support the theoretical consideration.
