  We present a comprehensive study of vortex structures in $d$-wave
superconductors from large-scale renormalized mean-field theory of the
square-lattice $t$-$t'$-$J$ model, which has been shown to provide a
quantitative modeling for high-$T_c$ cuprate superconductors. With an efficient
implementation of the kernel polynomial method for solving electronic
structures, self-consistent calculations involving up to $10^5$ variational
parameters are performed to investigate the vortex solutions on lattices of up
to $10^4$ sites. By taking into account the strong correlation of the model,
our calculations shed new lights on two puzzling results that have emerged from
recent scanning tunneling microscopy (STM) experiments. The first concerns the
issue of the zero-biased-conductance peak (ZBCP) at the vortex core for a
uniform $d$-wave superconducting state. Despite its theoretical prediction, the
ZBCP was not observed in most doping range of cuprates except in heavily
over-doped samples at low magnetic field. The second issue is the nature of the
checkerboard charge density waves (CDWs) with a period of about 8 unit cells in
the vortex halo at optimal doping. Although it has been suggested that such
bipartite structure arises from low-energy quasiparticle interference, another
intriguing scenario posits that the checkerboard CDWs originate from an
underlying bidirectional pair-density wave (PDW) ordering with the same period.
We present a coherent interpretation of these experimental results based on
systematic studies of the doping and magnetic field effects on vortex solutions
with and without a checkerboard structure. The mechanism of the emergent
intertwined orders within the vortex halo is also discussed.
