The underwater acoustic channel is challenging to estimate and track due to channel dynamism, the ill-conditioned nature of the estimation problem, as well as the need to optimize over a complex field. An extension of previous work is presented where the sparse channel coefficients in the delay and delay-Doppler spread domains are estimated using mixed norm convex optimization. The estimation algorithm is extended to track the time-varying channel using a gradient descent approach, trading off optimality for tracking efficiency. A computationally inexpensive approximation of the dynamic gradient is used rather than re-computing the precise gradient every time. Results presented are based on field data collected over 15-m depth and a range of wave heights and distances using the normalized prediction error as the performance metric. Comparison of performance between tracking the delay-only and delay-Doppler coefficients is presented for different weightings on the mixed norm between L1 and L2 measures and over different lags. Performance and computational efficiency of the proposed tracking algorithm are compared to conventional sparse estimation techniques. [Work supported in part by ONR Grants N00014-05-10085 and N00014-07-10184 and in part by Postdoctoral Scholar Program at the Woods Hole Oceanographic Institution, with funding provided by the Doherty Foundation.]