  We demonstrate that numerical solutions of Burgers' equation can be obtained
by a scale-totality algorithm for fluids of small viscosity (down to one
billionth). Two sets of initial data, modelling simple shears and wall boundary
layers, are chosen for our computations. Most of the solutions are carried out
well into the fully turbulent regime over finely-resolved scales in space and
in time. It is found that an abrupt spatio-temporal concentration in shear
constitutes an essential part during the flow evolution. The vorticity surge
has been instigated by the non-linearity complying with instantaneous enstrophy
production, while ad hoc disturbances play no role in the process. In
particular, the present method predicts the precipitous vorticity
re-distribution and accumulation, predominantly over localised regions of
minute dimension. The growth rate depends on viscosity and is a strong function
of initial data. Nevertheless, the long-time energy decay is
history-independent and is inversely proportional to time. Our results provide
direct evidence of the vorticity proliferation embedded in the equations of
motion. The non-linear intensification is a robust feature, and is ultimately
responsible for the drastic succession in boundary layer profiles over the
intrinsic laminar-turbulent transition (Schubauer and Klebanoff 1955). The
dynamical inception of turbulence can be decrypted by solving the full
time-dependent Navier-Stokes equations which ascribe no instability stages.
