This paper studies the global well-posedness problem on a tropical climate model with fractional dissipation. This system allows us to simultaneously examine a family of equations characterized by the fractional dissipative terms $ (-Δ)^{\mathit{\alpha }}u$ in the equation of the barotropic mode $ u$ and $ (-Δ)^β v$ in the equation of the first baroclinic mode $ v$. We establish the global existence and regularity of the solutions when the total fractional power is 2, namely $ {\mathit{\alpha }}+ β = 2$.

In this paper, we establish the global regularity for the two-dimensional incompressible magneto-micropolar equations with almost Laplacian magnetic diffusion and Laplacian micro-rotational diffusion, but without kinematic dissipation. The key arguments are based on the maximal regularity property of the generalized heat operators and a combined quantity. For the two-dimensional incompressible magneto-micropolar equations with only Laplacian magnetic diffusion and Laplacian micro-rotational diffusion, we derive an improved regularity criterion which is less restrictive than the classical Beale-Kato-Majda regularity criterion. Consequently, our results improve and generalize the previous works.