## Special Colloquium: Problems with the closure problem

Due to their limited resolution, numerical ocean models need to be interpreted as representing filtered or averaged equations. Due to the nonlinearity of the equations, this averaging introduces “eddy” contributions that cannot be expressed in terms of the averaged quantities that the model is solving for. To solve the equations we therefore need to introduce “parameterizations" (or “closures”) for these eddy terms. How to interpret models in terms of formally averaged equations, however, is not always clear, particularly in the case of hybrid or generalized vertical coordinate models, which is problematic as the eddy terms that need to be parameterized depend on the choice of average.

In this talk I will start by introducing the closure problem in general, and discuss an important caveat that affects attempts to infer parameterizations based on averaged data from high-resolution models or observations. I will then introduce the averaged hydrostatic Boussinesq equations in generalized vertical coordinates and discuss the extent to which the averaged equations are consistent with existing model formulations under different assumptions about the average. As previously discussed, the momentum equations in existing depth-coordinate models are best interpreted as representing Eulerian averages (i.e., averages taken at fixed depth), while the tracer equations can be interpreted as either Eulerian or thickness-weighted isopycnal averages (i.e. averages taken along potential density surfaces), which implies somewhat different interpretations of the terms represented by the parameterizations. Instead we find that no averaging is fully consistent with existing formulations of the parameterizations in widely used generalized vertical coordinate ocean models. Perhaps the most natural interpretation of generalized vertical coordinate models is to assume that the average follows the model's coordinate surfaces. However, the existing model formulations are generally not consistent with coordinate-following averages, which would require "coordinate-aware" parameterizations that can account for the changing nature of the eddy terms as the coordinate changes. Alternatively, the model variables can be interpreted as representing either Eulerian or (thickness-weighted) isopycnal averages, independent of the model coordinate that is being used for the numerical discretization. Existing parameterizations in generalized vertical coordinate models, however, are again not fully consistent with either of these interpretations. I will discuss what changes are needed to achieve consistency.

### Date

10.07.2024

### Time

13:30–15:00 h

### Place

- Bundesstr. 53, room 022/023
- Seminar Room 022/023, Ground Floor, Bundesstrasse 53, 20146 Hamburg, Hamburg