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Recent work demonstrates that high‐resolution global models forced simultaneously by atmospheric fields and the astronomical tidal potential contain a partial internal (gravity) wave (IW) spectral continuum. Regional simulations of the MITgcm forced at the horizontal boundaries by a global run that carries a partial IW continuum spectrum are performed at the same grid spacing as the global run and at finer grid spacings in an attempt to fill out more of the IW spectral continuum. Decreasing only the horizontal grid spacing from 2 to 0.25 km greatly improves the frequency spectra and slightly improves the vertical wavenumber spectra of the horizontal velocity. Decreasing only the vertical grid spacing by a factor of 3 does not yield any significant improvements. Decreasing both horizontal and vertical grid spacings yields the greatest degree of improvement, filling the frequency spectrum out to 72 cpd. Our results suggest that improved IW spectra in regional models are possible if they are run at finer grid spacings and are forced at their lateral boundaries by remotely generated IWs. Additionally, consistency relations demonstrate that improvements in the spectra are indeed due to the existence of IWs at higher frequencies and vertical wavenumbers when remote IW forcing is included and model grid spacings decrease. By being able to simulate an IW spectral continuum to 0.25 km scales, these simulations demonstrate that one may be able to track the energy pathways of IWs from generation to dissipation and improve the understanding of processes such as IW‐driven mixing.