This paper considers the problem of monitoring and adaptively estimating an environmental field, such as temperature or salinity, using an autonomous underwater vehicle (AUV). The AUV moves in the field and persistently measures environmental scalars and its position in its local coordinate frame. The environmental scalars are approximately linearly distributed over the region of interest, and an adaptive estimator is designed to estimate the gradient. By orthogonal decomposition of the velocity of the AUV, a linear time-varying system is equivalently constructed, and the sufficient conditions on the motion of the AUV are established, under which the global exponential stability of the estimation error system is rigorously proved. Furthermore, an estimate of the exponential convergence rate is given, and a reference trajectory that maximizes the estimate of the convergence rate is obtained for the AUV to track. Numerical examples verify the stability and efficiency of the system.