Mechanical coupling of the atmosphere to the ocean surface in general circulation models is represented using bulk wind stress formulations. The stress is often based on either absolute wind velocity, τa, or the more correct wind velocity relative to the ocean surface currents, τr. Here, we use coarse-graining to disentangle wind work by these formulations at different length-scales. We show that both can be reasonably accurate in forcing the ocean at length-scales larger than the mesoscales, with τa overestimating wind work by 10%. However, τa and τr show stark and opposing systematic biases in how they drive the mesoscales; τa does negligible (albeit positive) work on the mesoscales, while τr yields eddy-killing (negative work) that is artificially exaggerated by a factor of ≈4. We derive an analytical criterion for eddy-killing to occur, which shows that exaggerated eddy killing is due to resolution mismatch between the atmosphere and ocean. Our criterion highlights the disproportionate effect small-scale winds Ο(100)km can have on the dynamics of mesoscale ocean eddies, despite the dominant atmospheric motions being at length-scales larger than Ο(103) km. The eddy-killing criterion shows that large-scale winds do not necessarily cause eddy-killing but are merely an amplification factor for wind work on the mesoscales, which can be either positive or negative depending on the local alignment of small-scale winds with the ocean eddies. We propose a simple reformulation of τr, without introducing tuning parameters, to remove spurious eddy-killing from air-sea resolution mismatch that is often present in climate models.