The numerical solution of the one-dimensional advection-diffusion equationin layered coordinates

The numerical solution of the vertical advection-diffusion equation in laye red coordinates is revisited. The objectives of this work are to propose a generalization of the discontinuous layered representation of the ocean tra cer held to higher-order, smoother representations (while retaining the qua si-Lagrangian character of the coordinate) and compare the solutions genera ted by several approaches in order to illustrate their respective advantage s and disadvantages. The one-dimensional advection-diffusion equation is ch osen as a test bed for layered coordinates because ocean simulation for cli matic purposes requires the inclusion of dianeutral diffusive processes. The layered approach is generalized by replacing the traditional stack of w ell-mixed layers by stacks of piecewise smooth profiles. All the well-known properties of quasi-Lagrangian coordinates are retained. Comparisons of th e quasi-Lagrangian solutions with coarse- and fine-resolution fixed grid so lutions illustrates the efficiency of the adaptive, quasi-Lagrangian coordi nate.