The influence of different wind stress bulk formulae on the response of the Southern Ocean circulation to wind stress changes is investigated using an idealised channel model. Surface/mixed layer properties are found to be sensitive to the use of the relative wind stress formulation, where the wind stress depends on the difference between the ocean and atmosphere velocities. Previous work has highlighted the surface eddy damping effect of this formulation, which we find leads to increased circumpolar transport. Nevertheless the transport due to thermal wind shear does lose sensitivity to wind stress changes at sufficiently high wind stress. In contrast, the sensitivity of the meridional overturning circulation is broadly the same regardless of the bulk formula used due to the adiabatic nature of the relative wind stress damping. This is a consequence of the steepening of isopycnals offsetting the reduction in eddy diffusivity in their contribution to the eddy bolus overturning, as predicted using a residual mean framework.