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Open-ocean polynyas in the Weddell Sea of Antarctica are the product of deep convection, which transports Warm Deep Water (WDW) to the surface and melts sea ice or prevents its formation. These polynyas occur only rarely in the observational record but are a near-permanent feature of many climate and ocean simulations. A question not previously considered is the degree to which the Weddell polynya affects the nearby Filchner-Ronne Ice Shelf (FRIS) cavity. Here we assess these effects using regional ocean model simulations of the Weddell Sea and FRIS, where deep convection is imposed with varying area, location, and duration. In these simulations, the idealized Weddell polynyas consistently cause an increase in WDW transport onto the continental shelf as a result of density changes above the shelf break. This leads to saltier, denser source waters for the FRIS cavity, which then experiences stronger circulation and increased ice shelf basal melting. It takes approximately 14 years for melt rates to return to normal after the deep convection ceases. Weddell polynyas similar to those seen in observations have a modest impact on FRIS melt rates, which is within the range of simulated interannual variability. However, polynyas that are larger or closer to the shelf break, such as those seen in many ocean models, trigger a stronger response. These results suggest that ocean models with excessive Weddell Sea convection may not be suitable boundary conditions for regional models of the Antarctic continental shelf and ice shelf cavities.