# M5: Reducing spurious diapycnal mixing in ocean models

Principal investigators: Prof. Armin Iske (Universität Hamburg), Prof. Hans Burchard (Institute for Baltic Research Rostock), Dr. Sergey Danilov (Alfred Wegener Institute of Polar and Marine Research Bremerhaven)

## Objectives

The proposed project aims to further develop, assess and analyse numerical algorithms leading to reduction in spurious diapycnal mixing in ocean circulation models. This goal will be achieved by (i) the design and implementation of vertical mesh motion algorithms that reduce spurious mixing; (ii) use of advective schemes with isopycnal diffusion and special design of limiters; (iii) development and analysis of high-order advection algorithms relying on high-order flux evaluation.

## Invited Guests

## Reports

## Solving equations faster and more accurately

Designing high-order numerical methods needs a good understanding of the mathematical equations we want to solve.

Hi, I’m Claus and I am a PostDoc in Project M5. The focus of my research in this project is the development and analysis of high-order advection methods and high-oder flux evaluation techniques for ocean models. Our goal is to reduce spurious diapycnal mixing in ocean models and I’m working on the intersection between mathematical analysis and numerical methods to help with that.

High-order numerical methods for flow computations are becoming increasingly more popular in computational engineering, but may be not as widespread in climate and ocean science. So what’s the deal with high and low order?

Classical finite differences and finite volume methods for the discretization of partial differential equations use one piece of information in each cell (say, a function value at the cell-center or an integral average over the cell) and put this information into a discrete version of the PDE. This allows for fast and robust algorithms, but in order to resolve small scale features, we often need very fine grids. On the other hand, the high-order methods we are interested in, use higher degree polynomials or other nonlinear functions in each cell. This extra information allow us resolve more features of the solution, even on coarser girds. In many computational fluid dynamics applications this leads to a smaller overall computational time and we would like to show that this is also true for problems in ocean science.

However, carelessly throwing high-order polynomials at your problem is a sure recipe for failure. If we want to include more features of the analytical solution in our numerical solution, we need a good understanding of the analytical properties and how they can be translated into our numerical scheme. For our particular application, e.g., we want tight control over diffusion properties and need to tune our methods to avoid artificial diffusion without losing stability properties that numerical diffusion brings.

In short, designing high-order numerical methods needs a good understanding of the mathematical equations we want to solve. I’m happy to be involved with learning more about the equations in ocean science, so that we can solve them faster and more accurately.

## Minimising spurious mixing in numerical ocean models

The possibility for direct application of newly developed model techniques within leading national climate model systems is very stimulating.

Hi. Last month I started working as a postdoctoral researcher in subproject M5. After being involved already in the project's proposal and review process, I am very happy to finally participate in this exciting TRR. As a co-developer of the coastal ocean model GETM, I am strongly interested in the development of energy-consistent modelling techniques. In ocean models the advective transport of water masses is prone to energetic inconsistency. On the discrete model level this transport is associated with truncation errors causing spurious diapycnal mixing, which artificially increases potential energy without any physical sources.

Recently, I developed (together with PI Hans Burchard) a new analysis method that can quantify spurious mixing locally in every single grid cell. In M5 this method will be applied now to assess the new adaptive grid techniques and advanced advection schemes, that will be developed at UHH (PI Armin Iske), AWI (PI Sergey Danilov) and IOW (PI Hans Burchard) in order to reduce spurious mixing. I will be responsible for the development of algorithms that during runtime adapt the discrete model layers to the fluid flow (in order to minimise vertical transports across the layer interfaces), to isopycnals (in order to minimise diapycnal transports along the model layers) as well as to regions where high vertical resolution is needed (in order to minimise truncation errors) in an optimal way.

Furthermore, I will successively implement all schemes and algorithms developed in M5 into GETM to identify promising combinations that should finally be included into FESOM (developed by Sergey Danilov), which is the ocean component of the state-of-the-art climate model system ECHAM6/FESOM. With Sergey Danilov and Armin Iske being also PIs in the synthesis project S2, this possibility for direct application of newly developed model techniques within leading national climate model systems is very stimulating.

In the frame of the TRR I am looking forward for the close collaboration with experienced oceanographers, meteorologists and mathematicians, which offers an optimal academic environment for me as a young scientist. To foster the internal collaboration within M5, I will be first employed at UHH for 1.5 years and afterwards at IOW.

## Publications

Stähler, S. C., Panning, M. P., Hadziioannou, C., Lorenz, R. D., Vance, S.,

**Klingbeil, K.**, & Kedar, S. (2019). Seismic signal from waves on Titan's seas.*Earth and Planetary Science Letters*,*520*, 250-259.Lorenz, M.,

**K. Klingbeil**, P. MacCready, and**H. Burchard**(2019). Numerical issues of the Total Exchange Flow (TEF) analysis framework for quantifying estuarine circulation,*Ocean Science*, 15, 601-614.Gräwe, U.,

**K. Klingbeil**, J. Kelln, and S. Dangendorf (2019). Decomposing mean sea level rise in a semi-enclosed basin, the Baltic Sea.*Journal of Climate*.**Klingbeil, K.**, J. Becherer, E. Schulz, H. E. de Swart, H. M. Schuttelaars, A. Valle-Levinson and**H. Burchard**(2019). Thickness-weighted averaging in tidal estuaries and the vertical distribution of the Eulerian residual transport.*Journal of Physical Oceanography*.**Burchard, H.**, X. Lange,**K. Klingbeil**, and P. MacCready (2019) Mixing estimates for estuaries,*J. Phys. Oceanogr.*, 49, 631-648.**Iske, A.**(2019). Approximation Theory and Algorithms for Data Analysis.*Texts in Applied Mathematics*, 68, Springer.**Klingbeil, K., H. Burchard, S. Danilov, C. Goetz, and A. Iske**(2019). Reducing spurious diapycnal mixing in ocean models. In*Energy Transfers in Atmosphere and Ocean*(pp. 245-286). Springer, Cham.Rybicki, M., C. Moldaenke, K. Rinke, H. Dahlhaus,

**K. Klingbeil**, P. L. Holtermann, ... and J. Zhu (2019). WP-C: A Step Towards Secured Drinking Water: Development of an Early Warning System for Lakes. In*Chinese Water Systems*(pp. 159-205). Springer, Cham.**Burchard, H.**, Bolding, K., Feistel, R., Gräwe, U., MacCready, P.,**Klingbeil, K.**, Mohrholz, V.,**Umlauf, L.**, and van der Lee, E. M. , (2018)**.**The Knudsen theorem and the Total Exchange Flow analysis framework applied to the Baltic Sea,*Progress in Oceanography*, 165, 268-286 , https://doi.org/10.1016/j.pocean.2018.04.004 .Slavik, K., Lemmen, C., Zhang, W., Kerimoglu, O.,

**Klingbeil, K.**, and Wirtz, K. W. (2018). The large-scale impact of offshore wind farm structures on pelagic primary productivity in the southern North Sea.*Hydrobiologia*, 1-19.