Limiter H-modes with centrally peaked density profiles have been obtained in TFTR using a highly conditioned graphite limiter. The transition to these centrally peaked H-modes takes place from the supershot to the H-mode rather than the usual L- to H-mode transition observed on other tokamaks. Bi-directional beam heating is required to induce the transition. Density peaking factors, ne(0)/〈ne〉, greater than 2.3 are obtained and at the same time the H-mode characteristics are similar to those of limiter H-modes on other tokamaks, while the global confinement, τE, can be > 2.5 times L-mode scaling. The transport analysis of the data shows that transport in these H-modes is similar to that of supershots within the inner 0.6 m core of the plasma, but the stored electron energy (calculated using measured values of Te and ne) is higher for the H-mode at the plasma edge. Microwave scattering data for the edge plasma shows broad spectra at k = 5.5 cm-1 which begin at the drop in Dα radiation and are strongly shifted in the electron diamagnetic drift direction. At the same time, beam emission spectroscopy (BES) shows a coherent mode near the boundary which propagates in the ion direction with m = 15-20 at 20-30 kHz. During an ELM event these apparent rotations cease and Mirnov fluctuations in the frequency range of 50-500 kHz increase in intensity.