Multiscale finite elements through advection-induced coordinates for transient advection-diffusion equations

Long simulation times in climate sciences typically require coarse grids due to computational constraints. Nonetheless, unresolved subscale information significantly influences the prognostic variables and can not be neglected for reliable long term simulations. This is typically done via parametrizations but their coupling to the coarse grid variables often involves simple heuristics. We explore a novel up-scaling approach inspired by multi-scale finite element methods. These methods are well established in porous media applications, where mostly stationary or quasi stationary situations prevail. In advection-dominated problems arising in climate simulations the approach needs to be adjusted. We do so by performing coordinate transforms that make the effect of transport milder in the vicinity of coarse element boundaries. The idea of our method is quite general and we demonstrate it as a proof-of-concept on a one-dimensional passive advection-diffusion equation with oscillatory background velocity and diffusion.