  In this paper we prove the local well-posedness and global well-posedness
with small initial data of the strong solution to the reduced $3D$ primitive
geostrophic adjustment model with weak dissipation. The term reduced model
stems from the fact that the relevant physical quantities depends only on two
spatial variables. The additional weak dissipation helps us overcome the
ill-posedness of original model. We also prove the global well-posedness of the
strong solution to the Voigt $\alpha$-regularization of this model, and
establish the convergence of the strong solution of the Voigt
$\alpha$-regularized model to the corresponding solution of original model.
Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$
primitive geostrophic adjustment model with weak dissipation based on Voigt
$\alpha$-regularization.
