  In this paper we address the temperature patch problem of the 2D viscous
Boussinesq system without heat diffusion term. The temperature satisfies the
transport equation and the initial data of temperature is given in the form of
non-constant patch, usually called the temperature front initial data.
Introducing a good unknown and applying the method of striated estimates, we
prove that our partially viscous Boussinesq system admits a unique global
regular solution and the initial $C^{k,\gamma}$ and $W^{2,\infty}$ regularity
of the temperature front boundary with $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ and
$\gamma\in (0,1)$ will be preserved for all the time. In particular, this
naturally extends the previous work by Danchin $\&$ Zhang (2017) and Gancedo
$\&$ Garc\'ia-Ju\'arez (2017). In the proof of the persistence result of higher
boundary regularity, we introduce the striated type Besov space
$\mathcal{B}^{s,\ell}_{p,r,W}(\mathbb{R}^d)$ and establish a series of refined
striated estimates in such a function space, which may have its own interest.
