  We study the evolution of orbital patterns in ferromagnetic layered
ruthenates due to the competition of Coulomb interactions, compressive c axis
and orthorhombic distortions in the presence of a polarizing orbital field
coupled to the angular momentum. By means of the exact diagonalization on a 2x2
cluster and a cluster embedded analysis where inter-plaquette interaction is
treated on mean field level, we determine the ground-state phase diagram.
Specifically, we demonstrate that, via the activation of two or three of t_2g
local orbital configurations, an external field applied along different
symmetry directions can lead to inequivalent orbital correlated states.
Starting from an antiferro-orbital pattern, for the easy axis case an orbital
ordered phase is induced, having strong next nearest neighbors ferro-orbital
correlations. Otherwise, a field applied along the hard axis leads a reduction
of local orbital moment in a way to suppress the orbital order.
