Some insights on the spurious numerical mixing of the time-stepping of advection schemes

Spurious numerical mixing is still considered a problem in ocean models, especially in long simulations where errors can accumulate over time and severely bias the output. In order to be able to obtain relevant insights into the physics from numerical models, it is crucial to reduce this artificial mixing to an acceptable level.

The discretisation of the spatial component of the advection equation is the largest source of such errors, especially in models with fixed vertical coordinates, but the picture cannot be complete without considering the effect of the time-stepping procedure.

We first present some considerations on the damping effect of the coupled space-time procedure for linear advection schemes.
Recognising that commonly used time-stepping schemes can run stable with *a priori* unstable spatial components, we introduce a way to counteract the damping effect of the time integration by isolating, reformulating and introducing a so-called "backscattering" coefficient in the diffusive component of the advection scheme. Several formulations for this coefficient as well as the order of the scheme are then discussed, and finally an application of the method in a realistic simulation is carried out.

We conclude with some thoughts on the comparison of spurious numerical mixing with physical mixing and the implications for ocean models.