  We formalise a systematic method of constructing forward self-similar
solutions to the Navier-Stokes equations in order to characterise the late
stage of decaying process of turbulent flows. (i) In view of critical
scale-invariance of type 2 we exploit the vorticity curl as the dependent
variable to derive and analyse the dynamically-scaled Navier-Stokes equations.
This formalism offers the viewpoint from which the problem takes the simplest
possible form. (ii) Rewriting the scaled Navier-Stokes equations by Duhamel
principle as integral equations, we regard the nonlinear term as a perturbation
using the Fokker-Planck evolution semigroup. Systematic successive
approximations are introduced and the leading-order solution is worked out
explicitly as the Gaussian function with a solenoidal projection. (iii) By
iterations the second-order approximation is estimated explicitly up to
solenoidal projection and is evaluated numerically. (iv) A new characterisation
of nonlinear term is introduced on this basis to estimate its strength $N$
quantitatively. We find that $N=O(10^{-2})$ for the 3D Navier-Stokes equations.
This should be contrasted with $N=O(10^{-1})$ for the Burgers equations and $N
\equiv 0$ for the 2D Navier-Stokes equations. (v) As an illustration we
explicitly determine source-type solutions to the multi-dimensional the Burgers
equations. Implications and applications of the current results are given.
