Floating Offshore Wind Turbines (FOWTs) are projected to undergo substantial expansion in the coming decades. However, the motion of their supporting platforms due to mooring lines and wave interaction makes it difficult to predict their wake dynamics. The vortex ring structure produced during surge motion has been subject of study for nearly a decade now but there are still many features to bring to light. As most of the studies have been under idealized, uniform flow there is little knowledge on how this structure behaves under Atmospheric Boundary Layer (ABL) flow. In this work, the authors propose to study this structure under three different inflow conditions: laminar and turbulent uniform flows, and ABL flow. Large Eddy Simulations are carried out in combination with an Actuator Disk (AD) as a wind turbine model, with a focus on surge motion and a Strouhal number ranging between 0 and 0.47. In order to quantify the energy of the vortex ring structure, Proper Orthogonal Decomposition (POD) is applied to a vertical plane parallel to the AD axis. A good visualization of the structure is achieved for all cases, and the energy of the vortex ring structure exhibits a local maximum for turbulent and ABL flows, with the highest energy at Strouhal number 0.30 and 0.32, respectively. However, the local maximum is narrower in the ABL case than in the turbulent one. Also, due to the presence of strong turbulent structures in ABL flow, the energy present in the vortex ring structure is considerably lower in this case than under uniform turbulent flow. For the laminar case, the POD method allocates almost 99.9 % of the energy to modes related to the vortex ring structure, as no other strong structures arise in the wake. A meandering phenomenon is detected under low-turbulence and ABL flows. In the first scenario, meandering is initiated by inflow conditions, while in the second, it is the consequence of the interplay between shear flow and surge motion. The study is replicated in a horizontal plane at hub height, thereby demonstrating that for the low-turbulence flow, meandering occurs with equal intensity in all directions. Conversely, in ABL conditions, lateral meandering is unrelated to the vortex ring structure. Finally, phase average is carried out to evaluate how the structure propagates in each case. The results obtained indicate a faster decay of the structure for the non-laminar inflow cases, although the impact on the growth rate is contingent on the energy content of the vortex structure for each surge frequency. Further analysis indicates that the wake is modulated by the surge motion, manifesting as expansions and contractions, for the laminar and low-turbulence cases. In contrast, an inclination of the structures towards the flow direction is identified in the ABL conditions, attributable to the shear flow.