  In this paper, we obtain a refined temporal asymptotic upper bound of the
global axially symmetric solution to the Boussinesq system with no thermal
diffusivity. We show the spacial $W^{1,p}$-Sobolev ($2\leq p<\infty$) norm of
the velocity can only grow at most algebraically as $t\to+\infty$. Under a
signed potential condition imposed on the initial data, we further derive that
the aforementioned norm is uniformly bounded at all times. Higher order
estimates are also given: We find the $H^1$ norm of the temperature fluctuation
grows sub-exponentially as $t\to+\infty$. Meanwhile, for any $m\geq 1$, we
deduce that the $H^m$-temporal growth of the solution is slower than a double
exponential function. As a result, these improve the results in
\cite{HR:2010AIHP} where the authors only provided rough temporal asymptotic
upper bounds while proving the global well-posedness.
