This paper addresses the time-optimal trajectory planning for autonomous underwater vehicles. A detailed mixed integer nonlinear programming (MINLP) model is presented, explicitly taking into account vehicle kinematic constraints, obstacle avoidance, and a nonlinear flow field to represent the ocean current. MINLP problems pose great challenges because of the combinatorial complexity and nonconvexities introduced by the nature of the flow field. A novel solution approach in an optimization framework is developed to address associated difficulties. The main benefit of the proposed methodology is the ability to find multiple local minima. The contribution of the paper is fourfold: 1) a novel approach to integrate the flow field into the MINLP model; 2) a diversified initialization strategy using multiple waypoints, different solvers and approximated models, namely, a mixed integer linear programming model and the MINLP model with and without the flow field; 3) an algorithm that forces the solver to seek improved solutions; and 4) a parallel computing approach capitalizing on diversified initialization. The performance of the resulting methodology is illustrated on idealized case studies, and the results are used to gain insight into trajectory planning in the presence of flow fields.