In a former post , “Early earth simple model”, I considered the toroidal poloidal decomposition of the velocity solenoidal field. Here below, in the embedded pdf file, I expose some conditions for such a decomposition to exist and how it can be determined. The conditions are about the smoothness of the velocity field and the boundedness of the expansions in spherical harmonics which must be satisfied by the radial components of the velocity field and vorticity field.