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Multicarrier modulation in the form of orthogonal-frequency-division-multiplexing (OFDM) has been intensively pursued for underwater acoustic (UWA) communications recently due to its ability to handle long dispersive channels. Fast variation of UWA channels destroys the orthogonality of the sub-carriers and leads to inter-carrier interference (ICI), which degrades the system performance significantly. In this paper, we propose a progressive receiver dealing with time-varying UWA channels. The progressive receiver is in nature an iterative receiver, based on the turbo principle. However, it distinguishes itself from existing iterative receivers in that the system model for channel estimation and data detection is itself continually updated during the iterations. When the decoding in the current iteration is not successful, the receiver increases the span of the ICI in the system model and utilizes the currently available soft information from the decoder to assist the next iteration which deals with a channel with larger Doppler spread. Numerical simulation and experimental data collected from the 2008 Surface Processes and Acoustic Communications Experiment (SPACE08) show that the proposed receiver can self adapt to channel variations, enjoying low complexity in good channel conditions while maintaining excellent performance in adverse channel conditions.