1. Modeling global ocean circulation on unstructured meshes: current status and perspectives S. Danilov, Q. Wang, D. Sidorenko, J. Schröter Alfred Wegener Institute, Bremerhaven, Germany

2. Goal: Modelling large-scale ocean circulation Main motivation: Representation of coastlines, straits, or need to resolve local processes will benefit from using variable resolution Existing models: ADCIRC, TELEMAC, FVCOM, UnTRIM,... main area – coastal oceanography FEOM/FESOM (AWI) – large-scale circulation SLIM (University Louvain la Neuve), ICOM (Imperial) As a rule the complex shape of ocean basin requires to invest degrees of freedom in geometry => low-order elements are used

3. What we have already tried in FEOM P1-P1 version (3D primitive equations) Current versions: prismatic and tetrahedral P1-P1 Prismatic: more symmetric vertical stencil quadratures on generalized meshes (slow) Tetrahedra: implementation of generalized vertical coordinate is straightforward inclined vertical stencil

4. Main features: Pressure(elevation) correction method (no barotropic velocity) Explicit (where possible) TG, FCT (TG-based) and GLS-stabilized advection schemes. Nonlinear free surface and nonhydrostatic options MPI parallelized, coupled to sea ice, realistic forcing (CORE) Examples – D. Sidorenko talk Main difficulties: Implicit vertical viscosity/diffusion is slow because of horizontal connections in CG Slope noise in GM in tetrahedral discretization Difficult to keep pressure and vertical velocity fully consistent In hydrostatic finite-element codes: elevation, w, tracers, and pressure should have the same horizontal discretization. w, pressure and density - same vertical discretization => Consistent w/p solution is difficult to obtain with CG tracers! CPU time: prismatic and tetrahedral codes behave very similar: Despite 3 times larger amount of elements each of them is less CPU-expensive.

5. Q. Wang

6. Ross Sea overflow, simulations by Q. Wang. Resolution 0.5 km – 30 km

7. Beobachtung Assimilation Modell Assimilation of ice drift velocities in a regional finite element sea ice-ocean model (SEIK filter) Drift fields derived from passive microwave satellite data m/s concentration R.Timmermann,K. Rollenhagen

8. P1nc-P1 branch Pressure correction method (without stabilization). Tracer part inherited from FEOM Advantages: no stabilization Diagonal mass matrix for velocity (on z-meshes). Difficulty: momentum advection (P1 re-projection is much more robust than true P1nc) Intercomparison between P1-P1 P1nc-P1 versions of FEOM and MITgcm suggests that NC version is marginally faster (about 10%); FEOM is about 10 times slower than MITgcm Temperature distribution in 1/6 degree, 16 layers channel after 3 years of evolution (100 m) Kinetic energy evolution in a baroclinically unstable channel flow

9. P1nc-P1: further developments Lon-lat free version (following ideas of Louvain group) Velocity: P0 in vertical (more consistent boundary conditions on z-coordinate meshes) CPU speed remains our major difficulty as our typical applications require weeks to be completed on available resources (64-256 cores). FV technologies have to be explored.

10. FV type of discretization (an analog of P0-P1; velocity vector at centroids, scalar fields – at nodes) Although scalar control volumes (dual median) look ugly, assembling RHSs is fast using the edge-based structure. Pressure (elevation) correction (semi-implicit), AB2 Coriolis, Implicit vertical diffusion, z-levels. Momentum and tracer advection is second-order (linear reconstruction) upwind. Main result: Compared to P1-P1 approach the code is a factor from 5 to 8 faster!

11. NA setup with focus on the Gulf Stream (1/5 degree), coarse otherwise.Total 0.7M 3D nodes. Time step 20 min, 1 year takes about 3 h on a single node (8 Power5 1.9 GHz processors) of IBM p575 Seemingly too diffusive tracer advection (upwind with linear reconstruction). Snapshot of pot. temperature at 175 m Sea surface elevation

12. Momentum advection is prone to noise on grid scales and demands additional dissipation (noisy vertical advection!). Leith-modified viscosity suggested by MITgcm is a very effective cure. It is relatively difficult to find a balance between damping and reproducing eddy dynamics Mesh-size Reynolds number is >100

13. Vertical velocity patterns: A very consistent global structure on coarse mesh and noisy pattern in well-resolved strong jets and eddies. Suppressing it is easy but affects dynamics. Resolving eddies brings about much stronger and noisy local vertical velocities than on coarse grids. Summary of FV approach: (i) Much faster than P1-P1. (ii) Requires better advection schemes for both momentum and tracer advection in eddy-resolving applications. There is no easy way of reaching this.

14. 1/6 degree mesh; Left: quadrilaterals are split in a regular way, Right: random splitting P1-P1 simulations

15. w representation and hence long-term tracer dynamics are sensitive to mesh quality. This becames a serious issue in eddy-resolving simulations. Summary of FV approach: (i) Much faster than P1-P1. (ii) Requires better advection schemes for both momentum and tracer advection in eddy-resolving applications. There is no obvious way of reaching this.

16. Spurious diapycnal mixing: Baroclinic instability in a channel Anomaly of sorted density Diagnosed mixing 1. Spurious mixing in FEOM advection scheme is not larger than in finite difference models as compared to Griffies et al., 2000. 2. FV 2nd order upwind advection scheme shows large dissipation and must be improved before it can be used in large scale ocean modelling. 3. Unstructured character of meshes does not necessarily imply increased mixing.

17. Some conclusions: P1-P1 setup is most robust (of tested by us) and allows us to run various applications at the current stage. FV approach suggests much higher CPU efficiency, however applying it to eddying regimes requires care with respect to advection schemes and dissipation operators. There are many promising element pairs, and there are many indications in favor of DG. How can we ensure a practical (sufficiently fast) approach?