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We analyse the multi-frequency radio spectral properties of $41$6GHz-detected ALMA-identified, submillimetre galaxies (SMGs), observed at610MHz, 1.4GHz, 6GHz with GMRT and the VLA. Combining high-resolution($\sim0.5''$) 6GHz radio and ALMA $870\,\mu$m imaging (tracing rest-frame$\sim20$GHz, and $\sim250\,\mu$m dust continuum), we study thefar-infrared/radio correlation via the logarithmic flux ratio $q_{\rm IR}$,measuring $\langle q_{\rm IR}\rangle=2.20\pm 0.06$ for our sample. We show thatthe high-frequency radio sizes of SMGs are $\sim1.9\pm 0.4\times$($\sim2$-$3$kpc) larger than those of the cool dust emission, and find evidencefor a subset of our sources being extended on $\sim 10$kpc scales at 1.4GHz. Bycombining radio flux densities measured at three frequencies, we can movebeyond simple linear fits to the radio spectra of high-redshift star-forminggalaxies, and search for spectral curvature, which has been observed in localstarburst galaxies. At least a quarter (10/41) of our sample show evidence of aspectral break, with a median $\langle\alpha^{1.4\,{\rm GHz}}_{610\,{\rmGHz}}\rangle=-0.60\pm 0.06$, but $\langle\alpha^{6\,{\rm GHz}}_{1.4\,{\rmGHz}}\rangle=-1.06\pm 0.04$ -- a high-frequency flux deficit relative to simpleextrapolations from the low-frequency data. We explore this result within thissubset of sources in the context of age-related synchrotron losses, showingthat a combination of weak magnetic fields ($B\sim35\,\mu$G) and young ages($t_{\rm SB}\sim40$--$80\,$Myr) for the central starburst can reproduce theobserved spectral break. Assuming these represent evolved (but ongoing)starbursts and we are observing these systems roughly half-way through theircurrent episode of star formation, this implies starburst durations of$\lesssim100$Myr, in reasonable agreement with estimates derived via gasdepletion timescales.

Radio spectra and sizes of ALMA-identified submillimetre galaxies: evidence of age-related spectral curvature and cosmic ray diffusion?