We propose real-time channel tracking for underwater acoustic communications under dynamic sea conditions. The key idea is to employ sophisticated sparse sensing techniques that are cognizant of stable or slowly time-varying channel components against a transient background. Shallow water acoustic channel is generally challenging to track under moderate to rough sea conditions. This is primarily due to non-stationary highly transient elements within the channel delay spread resulting from rapidly fluctuating multipath arrivals from unpredictable surface wave reflections. The proposed channel estimation method exploits two channel characteristics: (i) Inherent sparsity of the time-varying channel in the two-dimensional dual (Fourier) domain; and (ii) Relative dominance of the direct arrival and slowly varying multipath arrivals against the otherwise non-stationary channel impulse response. Specifically, we utilize this apriori information to compressed sensing (CS) framework and thus, achieve channel sensing cognizant of time-frequency localization across significant channel taps. Numerical evidence based on data-driven channel ground truths are presented.