What is radiation divergence?
The way I understand it, the net longwave radiation $L^\star:=L^\downarrow-L^\uparrow$ (maybe with different sign), i.e. the difference between downwelling and upwelling radiation, can be viewed as a vector field (with vanishing horizontal components under the assumption of homogeneous horizontal radiation distribution). This vector field has a divergence $\nabla\cdot L^\star=\frac{\partial L^\star}{\partial z}$. 

This term occurs in equation (1) of the paper you referenced as part of the heating rate. Since non-vanishing divergence is in a vector field is associated with the field having a source or sink, this corresponds to heating/cooling of the atmosphere.