This article describes a novel algorithmic development extending the contour advective semi-Lagrangian model to include nonconservative effects. The Lagrangian contour representation of finescale tracer fields, such as potential vorticity, allows for conservative, nondiffusive treatment of sharp gradients allowing very high numerical Reynolds numbers. It has been widely employed in accurate geostrophic turbulence and tracer advection simulations. In the present, diabatic version of the model the constraint of conservative dynamics is overcome by including a parallel Eulerian field that absorbs the nonconservative ( diabatic) tendencies. The diabatic buildup in this Eulerian field is limited through regular, controlled transfers of this field to the contour representation. This transfer is done with a fast newly developed contouring algorithm. This model has been implemented for several idealized geometries. In this paper a single-layer doubly periodic geometry is used to demonstrate the validity of the model. The present model converges faster than the analogous semi-Lagrangian models at increased resolutions. At the same nominal spatial resolution the new model is 40 times faster than the analogous semi-Lagrangian model. Results of an orographically forced idealized storm track show nontrivial dependency of storm-track statistics on resolution and on the numerical model employed. If this result is more generally applicable, this may have important consequences for future high-resolution climate modeling.