Let G be a planar graph with maximum degree $Delta$. The linear 2-arboricity $la_2$(G) of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. In this paper, we prove that (1) $la_2(G){leq}{lceil}frac{Delta}{2} ceil+8$ if G has no adjacent 3-cycles; (2) $la_2(G){leq}{lceil}frac{Delta}{2} ceil+10$ if G has no adjacent 4-cycles; (3) $la_2(G){leq}{lceil}frac{Delta}{2} ceil+6$ if any 3-cycle is not adjacent to a 4-cycle of G.