  In the present work fournontrivial stages of electrokinetic instability are
identified by direct numerical simulation (DNS) of the full
Nernst-Planck-Poisson-Stokes (NPPS) system: i) The stage of the influence of
the initial conditions (milliseconds); ii) 1D self-similar evolution
(milliseconds-seconds); iii) The primary instability of the self-similar
solution (seconds); iv) The nonlinear stage with secondary instabilities. The
self-similar character of evolution at intermediately large times is confirmed.
Rubinstein and Zaltzman instability and noise-driven nonlinear evolution to
over-limiting regimes in ion-exchange membranes are numerically simulated and
compared with theoretical and experimental predictions. The primary instability
which happens during this stage is found to arrest self-similar growth of the
diffusion layer and specifies its characteristic length as was first
experimentally predicted by Yossifon and Chang (PRL 101, 254501 (2008)). A
novel principle for the characteristic wave number selection from the
broadbanded initial noise is established.
