  In this paper, we are concerned with the tridimensional anisotropic
Boussinesq equations which can be described by {equation*}
  {{array}{ll}
  (\partial_{t}+u\cdot\nabla)u-\kappa\Delta_{h} u+\nabla \Pi=\rho
e_{3},\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3},
  (\partial_{t}+u\cdot\nabla)\rho=0,
  \text{div}u=0.
  {array}. {equation*} Under the assumption that the support of the
axisymmetric initial data $\rho_{0}(r,z)$ does not intersect the axis $(Oz)$,
we prove the global well-posedness for this system with axisymmetric initial
data. We first show the growth of the quantity $\frac\rho r$ for large time by
taking advantage of characteristic of transport equation. This growing property
together with the horizontal smoothing effect enables us to establish
$H^1$-estimate of the velocity via the $L^2$-energy estimate of velocity and
the Maximum principle of density. Based on this, we further establish the
estimate for the quantity $\|\omega(t)\|_{\sqrt{\mathbb{L}}}:=\sup_{2\leq
p<\infty}\frac{\norm{\omega(t)}_{L^p(\mathbb{R}^3)}}{\sqrt{p}}<\infty$ which
implies $\|\nabla u(t)\|_{\mathbb{L}^{3/2}}:=\sup_{2\leq
p<\infty}\frac{\norm{\nabla u(t)}_{L^p(\mathbb{R}^3)}}{p\sqrt{p}}<\infty$.
However, this regularity for the flow admits forbidden singularity since $
\mathbb{L}$ (see \eqref{eq-kl} for the definition) seems be the minimum space
for the gradient vector field $u(x,t)$ ensuring uniqueness of flow. To bridge
this gap, we exploit the space-time estimate about $ \sup_{2\leq
p<\infty}\int_0^t\frac{\|\nabla
u(\tau)\|_{L^p(\mathbb{R}^3)}}{\sqrt{p}}\mathrm{d}\tau<\infty$ by making good
use of the horizontal smoothing effect and micro-local techniques. The global
well-posedness for the large initial data is achieved by establishing a new
type space-time logarithmic inequality.
