Due to dynamic changes and instability of oceanic eddies, continuous tracking and sampling using mobile platforms is a challenging field. Aiming at the requirements of accurate observation of mesoscale eddies, this paper studies the problem of searching and tracking the eddy center in a mapless environment with an underactuated autonomous surface underwater vehicle (ASUV) and proposes a path planning method based on the deep reinforcement learning (DRL). Firstly, the existing observation methods are summarized, and the dynamic tracking framework of mesoscale eddies is established. Then, the DRL and long short-term memory (LSTM) are combined to train end-to-end and real-time planning strategies in an eddy environment. Finally, a high-fidelity simulation platform in the eddy environment is built, and actual data from the Kuroshio Extension region is used to verify the effectiveness and feasibility of the strategies. The experimental results show that the ASUV with the proposed strategies can realize the autonomous search and tracking of the eddy center, have good stability and real-time performance, and the tracking error is always within an acceptable range.