On the attractor of the semi-dissipative Boussinesq equations revised

In this paper, we continue investigate the structure of the weak $\sigma$-attractor $\mathcal{A}$ for the 2D Boussinesq equations with viscosity and without heat diffusion. Specifically, we show that the projection of the attractor $\mathcal{A}$ onto the temperature component is surjective, i.e., $P_{\theta }\mathcal{A}=L^2$; and, for any $r>0$, the projection of ${\mathcal{A}}_r$ onto the velocity component contains a infinite-dimensional Hilbert ellipsoid, which implies $P_u{\mathcal{A}}_r$ is infinite dimensional immediately.