An improved least mean square/fourth direct adaptive equalizer (LMS/F-DAE) is proposed in this paper for underwater acoustic communication in the Arctic. It is able to process complex-valued baseband signals and has better equalization performance than LMS. Considering the sparsity feature of equalizer tap coefficients, an adaptive norm (AN) is incorporated into the cost function which is utilized as a sparse regularization. The norm constraint changes adaptively according to the amplitude of each coefficient. For small-scale coefficients, the sparse constraint exists to accelerate the convergence speed. For large-scale coefficients, it disappears to ensure smaller equalization error. The performance of the proposed AN-LMS/F-DAE is verified by the experimental data from the 9th Chinese National Arctic Research Expedition. The results show that compared with the standard LMS/F-DAE, AN-LMS/F-DAE can promote the sparse level of the equalizer and achieve better performance.