We compare high-resolution optical and radio imaging of 12 luminous submm galaxies at z=2.2+/-0.2 observed with HST and the MERLIN and VLA at comparable spatial resolution, 0.3 (2kpc). The radio emission traces the likely far-infrared morphology of these dusty, luminous galaxies. In ~30% of the sample the radio appears unresolved, suggesting that the emission is compact: either an obscured AGN or nuclear starburst. However, in the majority, ~70% (8/12), the radio emission is resolved by MERLIN/VLA on scales of ~1 (10 kpc). For these galaxies the radio morphologies are broadly similar to their restframe UV emission seen by HST. We discuss the probable mechanisms for the extended emission and conclude that their luminous radio and submm emission arises from a large, spatially-extended starburst. The median SFRs are 1700Mo/yr occuring within a ~40kpc^2 region, giving a star formation density of 45Mo/yr/kpc^2. Such vigorous and extended starbursts appear to be uniquely associated with the submm population. A more detailed comparison of the distribution of UV and radio emission shows that the broad similarities on large scales are not carried through to smaller scales, where there is rarely a one-to-one correspondance. We interpret this as resulting from highly structured internal obscuration, suggesting that the vigorous activity is producing wind-blown channels through the obscuration in these galaxies. If correct this underlines the difficulty of using UV morphologies to understand structural properties of this population and also may explain the surprising frequency of Ly-alpha emission in their spectra. [Abridged]
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