Global existence for a 3D Tropical Climate Model with damping and small initial data in ${\dot{H}}^{1\mathrm{/}2}({\mathbb{R}}^3)$

We consider a 3D Tropical Climate Model with damping terms in the equation of the barotropic mode $u$ and in the equation of the first baroclinic mode $v$ of the velocity. The equation for the temperature $\theta$ is free from dampings. We prove global existence in time for this system assuming the initial data $(u_0,v_0,{\theta }_0)$ small, in terms of the homogeneous space ${\dot{H}}^{1\mathrm{/}2}({\mathbb{R}}^3)$.