Numerical modeling of conserved atmospheric fields is critical to weather prediction and climate simulation. Atmospheric dynamics often distort these fields into long filamentary structures that are difficult to model, because the extreme aspect ratios of the filaments present a wide range of spatial scales that need to be resolved. A method of dealing with these differing spatial scales is to use a dynamic adaptive grid (DAG) technique, which continuously moves grid points in response to changes in the tracer field to give higher resolution where small spatial scales are prominent;The Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) of Smolarkiewicz is used to solve the advection equation on a moving grid. As the number of grid points increases, instabilities in MPDATA with an adaptive grid occur due to violation of the CFL condition. Analysis of the problem leads to a method of limiting grid point movement to maintain stability. This analysis shows the ideal grid movement is along Lagrangian trajectories;One- and two-dimensional models using grid point movement schemes based on equidistribution and variational techniques respectively demonstrate the effect of varying the number of grid points and applying several advanced MPDATA options. Using experience gained from these two models a three-dimensional code was designed to be driven by the Fifth-Generation Penn State/NCAR Mesoscale Model (MM5). Three test cases were performed with the MM5-DAG combination, These used: (1) idealized wind and tracer fields; (2) idealized tracer fields and realistic wind fields; and (3) realistic wind and tracer fields. These tests show that the MPDATA scheme is less diffusive than the standard MM5 centrally differenced leap-frog scheme and that the dynamic MPDATA has less diffusion and resolves the tracer fields better than does MPDATA on a static grid. Dynamic MPDATA was able to yield similar results with a quarter the grid points as MPDATA with a static grid.