We consider the underwater acoustic channel which is time-variant and doubly-spread in this work. Since conventional channel estimation and decision feedback equalizer (DFE) can not work well for this type of channel, a path-specific underwater acoustic channel tracking is proposed. It is based on the framework of Kalman filter. We provide a simplified sound propagation model as the state transition model. A multipath tracker is proposed which is tolerant of the model-mismatch. Then we can obtain the time-variant path number and path-specific parameters such as delay and Doppler scaling factor. We also consider the application of the proposed path-specific underwater acoustic channel tracking. We propose two types of passive time reversal mirror (PTRM) with our path-specific parameters for time-variant and doubly-spread underwater acoustic channel. With the path-specific parameters obtained by the proposed channel tracking, the proposed PTRM can not only match the time dispersion as conventional PTRM, but also the doubly-spread channel, since the path-specific delay and Doppler scaler factor can help to match the channel in both time and frequency domain. For extensive doubly-spread channel, we can further apply the path-specific compensation to the PTRM. Both simulations and experimental results by data from 2016 Qiandao Lake experiment show the efficiency of proposed path-specific channel tracking and proposed PTRMs with path-specific parameters.