Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde How to make a three-dimensional numerical model that works for lakes and estuaries?

Essential problem in ocean models: To move a continuous property through a fixed grid without changing its form. Example: solid body rotation: Result of a high-order linear scheme: new artificial maxima and minima.

Two strategies for solutions: Non-linear limiter schemes. Vertically adaptive coordinates. + a numerical mixing analysis to quantify the numerical mixing.

Principle of non-linear schemes: combine the advantages of two linear schemes in a non-linear way. Use monotonicity from the diffusive first-order upstream and the 2nd order of accuracy from the non-diffusive Lax-Wendroff scheme.

What is mixing ? Salinity equation (no horizontal mixing): Salinity variance equation: ? Mixing is dissipation of tracer variance.

Principle of numerical mixing diagnostics: First-order upstream (FOU) for s: FOU for s is equivalent to FOU for s² with variance decay : numerical diffusivity Salinity gradient squared See Maqueda Morales and Holloway (2006) 1D advection equation for S: 1D advection equation for s 2 :

Transport pathways in the Baltic Sea Reissmann et al. (2009)

Pressure gradient problem of sigma coordinates Sigma coordinate problem Inflows

Inflow approximation problem of geopotential coordinates Geopotential coordinate problem Inflows Additionally, both coordinate types share the problem of numerical mixing.

Adaptive vertical grids in GETM hor. filtering of layer heights Vertical zooming of layer interfaces towards: a) Stratification b) Shear c) surface/ bottom z bottom Vertical direction Horizontal direction hor. filtering of vertical position Lagrangian tendency isopycnal tendency Solution of a vertical diffusion equation for the coordinate position Burchard & Beckers (2004); Hofmeister, Burchard & Beckers (2010)

The philosophy behind GETM GETM is a coastal and shelf sea (and lake?) hydrodynamic model. GETM is a Public Domain Community Model. GETM is released under the Gnu Public Licence. GETM is Open Source. GETM has a modular structure (open for extentions). GETM has an international developer and user community. GETM started in 1997 and has been steadily developed since then.

The physics & numerics of the GETM core GETM  is based on the 3D shallow water equations  has been extended to represent non-hydrostatic pressure  is using GOTM as turbulence closure model  is using bulk formulae to calculate surface fluxes  is a finite volume (i.e., conservative) model  uses Cartesian, spherical or curvilinear horizontal coordinates  uses general vertical coordinates (including adaptive)  uses high-resolution TVD advection schemes  is based on explicit mode splitting  is fully paralellised using MPI and domain decomposition  works with netCDF input and output

Baltic slice with adaptive vertical coordinates Fixed coordinatesAdaptive coordinates Hofmeister, Burchard & Beckers (2010) Numerical mixing Physical mixing

Adaptive vertical coordinates along transect in 600 m Western Baltic Sea model Gräwe et al. (in prep.)

Adaptive coordinates in Bornholm Sea

1 nm Baltic Sea model with adaptive coordinates - refinement partially towards isopycnal coordinates - reduced numerical mixing - reduced pressure gradient errors - still allowing flow along the bottom salinity temperature km Hofmeister, Beckers & Burchard (2011) Feistel et al., 2004 Observations November 2003

Channelled gravity current in Bornholm Channel sigma-coordinates adaptive coordinates - stronger stratification with adaptive coordinates - larger core of g.c. - salinity transport increased by 25% - interface jet along the coordinates Hofmeister, Beckers & Burchard (2011)

Gotland Sea time series 3d baroclinic simulation 50 adaptive layers vs. 50 sigma layers num. : turb. mixing 80% : 20% num. : turb. mixing 50% : 50% Hofmeister, Beckers & Burchard (2011)

Multi-scale applications Structured models Unstructured models Gräwe et al. (in prep.) de Bauwere et al. (2009)

Does this model also work for lakes? Yes, but you have to resolve the slopes which are generally steeper in lakes than in the shelf sea. Example: seiches in Lake Alpnach:

Study of boundary mixing (Lake Alpnach, Switzerland) Becherer & Umlauf (2011)

Simulation Lake Alpnacher (Switzerland) Becherer & Umlauf (2011)

Take home: It is a demanding numerical task to obtain efficient and accurate discretisations for the advection terms. Mass conservation can be obtained by finite-volume schemes, but variance conservation would only work in Lagrangean (particle tracking) models. Vertically adaptive coordinates are one efficient method to reduce numerical variance decay due to advection schemes. Question: What is the 3D type of model for the future? Will in 20 years still structured and unstructured models co-exist, or will one or the other or a third method rule out the others?