Abstract Wind stress imposed on the upper ocean by a hurricane can limit the hurricane’s intensity primarily through shear-induced mixing of the upper ocean and subsequent cooling of the sea surface. Since shear-induced mixing is a one-dimensional process, some recent studies suggest that coupling a one-dimensional ocean model to a hurricane model may be sufficient for capturing the storm-induced sea surface temperature cooling in the region providing heat energy to the hurricane. Using both a one-dimensional and a three-dimensional version of the same ocean model, it is shown here that the neglect of upwelling, which can only be captured by a three-dimensional ocean model, underestimates the storm-core sea surface cooling for hurricanes translating at <∼5 m s−1. For hurricanes translating at <2 m s−1, more than half of the storm-core sea surface cooling is neglected by the one-dimensional ocean model. Since the majority of hurricanes in the western tropical North Atlantic Ocean translate at <5 m s−1, the idealized experiments presented here suggest that one-dimensional ocean models may be inadequate for coupled hurricane–ocean model forecasting.