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. Bidirectional beam heating is required to induce the transition. Density peaking factors, n{sub e}(0)/, 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, {tau}{sub E}, 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 T{sub e} and n{sub e}) 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{sup {minus}1} which begin at the drop in D{sub {alpha}} 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 ionmore » direction with m = 15--20 at 20--30 kHz. During the ELM event these apparent rotations cease and Mirnov fluctuations in the frequency range of 50--500 kHz increase in intensity. 16 refs., 8 figs.« less