  The structure and dynamics of the M87 jet from sub-milli-arcsec to arcsecond
scales are continuously examined. We analysed the VLBA archival data taken at
43 and 86 GHz to measure the size of VLBI cores. Millimeter/sub-mm VLBI cores
are considered as innermost jet emissions, which has been originally suggested
by Blandford & K\"onigl. Those components fairly follow an extrapolated
parabolic streamline in our previous study so that the jet has a single
power-law structure with nearly five orders of magnitude in the distance
starting from the vicinity of the supermassive black hole (SMBH), less than 10
Schwarzschild radius ($r_{\rm s}$). We further inspect the jet parabolic
structure as a counterpart of the magnetohydrodynamic (MHD) nozzle in order to
identify the property of a bulk acceleration. We interpret that the parabolic
jet consists of Poynting-flux dominated flows, powered by large amplitude,
nonlinear torsional Alfv\'en waves. We examine the non-relativistic MHD nozzle
equation in a parabolic shape. The nature of trans-fast magnetosonic flow is
similar to the one of transonic solution of Parker's hydrodynamic solar wind;
the jet becomes super-escape as well as super-fast magnetosonic at around $\sim
10^{3}\,r_{\rm s}$, while the upstream trans-Alfv\'enic flow speed increases
linearly as a function of the distance at $\sim 10^{2}$ - $10^{3}\, r_{\rm s}$.
We here point out that this is the first evidence to identify these features in
astrophysical jets. We propose that the M87 jet is magnetically accelerated,
but thermally confined by the stratified ISM inside the sphere of gravitational
influence of the SMBH potential, which may be a norm in AGN jets.
