Global Well-Posedness and Energy Decay for Hypodissipative 3D Tropical Climate Models with damping

In this paper we consider a three-dimensional Tropical Climate Model with fractional diffusion $\Lambda^{2\alpha} u$ and, in particular, nonlinear damping $|u|^{\beta-1}u$ in the equation for the barotropic mode of the velocity $u$. Assuming $\alpha\in (9/10, 1]$ and exploiting the regularizing effect due to the damping term, we establish the global existence and uniqueness of strong solutions, for $\beta$ greater than a suitable threshold depending on $\alpha$. In the given analysis, we place particular emphasis on the case of hypoviscosity. As a further result, we prove the $L^2$-energy decay of the global solution $(u,v,\theta)$ over the long-time period.