  Vortex structure of pure $d_{x^2-y^2}$-wave superconductors is
microscopically analyzed in the framework of the quasi-classical Eilenberger
equations. Selfconsistent solution for the $d$-wave pair potential is obtained
for the first time in the case of an isolated vortex. The vortex core
structure, i.e., the pair potential, the supercurrent and the magnetic field,
is found to be fourfold symmetric even in the case that the mixing of $s$-wave
component is absent. The detailed temperature dependences of these quantities
are calculated. The fourfold symmetry becomes clear when temperature is
decreased. The local density of states is calculated for the selfconsistently
obtained pair potential. From the results, we discuss the flow trajectory of
the quasiparticles around a vortex, which is characteristic in the
$d_{x^2-y^2}$-wave superconductors. The experimental relevance of our results
to high temperature superconductors is also given.
