The aim of this work is to describe some operations in the category of regular holonomic $\cD$-modules with support a normal crossing and variation zero introduced in [J.Alvarez Montaner, R.Garcia Lopez and S.Zarzuela, "Local cohomology, arrangements of subspaces and monomial ideals ", Adv. in Math. 174 (2003), 35--56]. These operations will allow us to compute the characteristic cycle of the local cohomology supported on homogeneous prime ideals of these modules. In particular, we will be able to describe their Bass and dual Bass numbers.