  We introduce the concept of bond spin current, which describes the spin
transport between two sites of the lattice model of a multiterminal spin-orbit
(SO) coupled semiconductor nanostructure, and express it in terms of the
spin-dependent nonequilibrium (Landauer-Keldysh) Green functions of the device.
This formalism is applied to obtain the spatial distribution of microscopic
spin currents in {\em clean} phase-coherent two-dimensional electron gas with
the Rashba-type of SO coupling attached to four external leads. Together with
the corresponding profiles of the stationary spin density, such visualization
of the phase-coherent spin flow allow us to resolve several key issues for the
understanding of mechanisms which generate pure spin Hall currents in the
transverse leads of ballistic devices due to the flow of unpolarized charge
current through their longitudinal leads. The local spin current profiles
crucially depend on whether the sample is smaller or greater than the spin
precession length, thereby demonstrating its essential role as the
characteristic mesoscale for the spin Hall effect in ballistic multiterminal
semiconductor nanostructures. Although static spin-independent disorder reduces
the magnitude of the total spin current in the leads, the bond spin currents
continue to flow through the whole diffusive 2DEG sample, without being
localized as edge spin currents around any of its boundaries.
