Hans Burchard and Hannes Rennau Baltic Sea Research Institute Warnemünde Comparative quantification of physically and numerically induced mixing in a coastal model application

What is mixing ? Non-averaged salinity equation: Reynolds decomposition: Mean salinity equation (with down-gradient turbulent flux)

Micro-structure salinity variance equation: Mean salinity variance equation: Mixing is dissipation of tracer variance micro-structure or ensemble-averaged

GETM is a 3D numerical model for estuarine, coastal and shelf sea hydrodynamics with Coupling to GOTM Turbulence Module Generalised vertical coordinates High-resolution TVD advection schemes Parallel execution …

GETM Western Baltic Sea hindcast

Model derived monthly mean vertically integrated salinity variance decay in Western Baltic Sea How does this relate to numerically induced mixing ?

Principle of numerical mixing diagnostics: First-order upstream (FOU) for s: FOU for s is equivalent to FOU for s² with variance decay :

Principle of numerical mixing diagnostics: Reformulation of FOU of s² into operator notation: We define for any advection scheme the numerical variance decay as: Reformulation of FOU of s into numerical operator: Advected tracer square minus square of advected tracer

Loss of salinity 2 due to numerical mixing (TVD scheme) Without physical mixingWith physical mixing Total salinity squared Loss of salinity squared

Physical and numerical diffusion for the DOME testcase

Model derived monthly mean vertically integrated physically and numerically induced salinity variance decay Physical mixing Numerical mixing

Conclusions In buoyancy driven currents, the numerically induced mixing may be of the same order or even much larger than the physical mixing due to entrainment (at least in GETM). The amount of tracer variance is limited. Physics & numerics compete for decaying it. Less numerical mixing allows for more physical mixing. Improved mixing formulations may not result in improved estimates of effective mixing. In order to assess the generality of this problem (for other models than GETM), the analysis of numerically and physically induced tracer squared decay needs to be carried out for many other applications, advection schemes and models. Numerical mixing is not our friend !