1. SELFE: Semi-implicit Eularian- Lagrangian finite element model for cross scale ocean circulation Paper by Yinglong Zhang and Antonio Baptista Presentation by Charles Seaton All figures from paper unless otherwise labeled

2. Comparison of model types Structured grids, FD: ROMS, POM, NCOM: Good for ocean modeling, require small timesteps, not capable of representing coastline details Unstructured grids, FE (previous): ADCIRC, QUODDY: Archaic, don’t solve primitive equations Unstructured grids, FV: UNTRIM-like models: require orthogonality, low order SELFE: Unstructured grids, FE: higher order, does solve primitive equations, can follow coastlines

3. SELFE: equations coriolis Tidal force Atmospheric Horizontal viscosity Baroclinic barotropic Vertical viscosity Vertical and horizontal diffusion continuity

4. Turbulence Closure vertical diffusion, vertical and horizontal viscosity dissipation Length scale, 0.3, TKE, mixing length Stability functions Boundary conditions Model parameters

5. Vertical Boundary Condition for Momentum Surface Bed Bottom boundary layer velocity Stress in boundary layer Continued next slide

6. Vertical Boundary Condition for Momentum (continued) Constant stress = 0

7. Numerical methods Horizontal grid: unstructured Vertical grid: hybrid s-z Time stepping: semi-implicit Momentum equation and continuity equation solved simultaneously (but decoupled) Finite Element, advection uses ELM Transport equation: FE, advection uses ELM or FVUM

8. s-z vertical grid Can be pure s, can’t be pure z Allows terrain following at shallow depths, avoids baroclinic instability at deeper depths

9. Grid Prisms u,v elevation w S,T FVUM S,T ELM

10. Continuity

11. Depth averaged momentum Explicit terms Implicit terms Need to eliminate = 0

12. Momentum Viscosity Viscocity – implicit Pressure gradient – implicit Velocity at nodes = weighted average of velocity at side centers Or use discontinuous velocities Vertical velocity solved by FV

13. Baroclinic module Transport: ELM or FVUM (element splitting or quadratic interpolation reduces diffusion in ELM) FVUM for Temperature Stability constraint (may force subdivision of timesteps)

14. Stability From explicit baroclinic terms From explicit horizontal viscosity

15. Benchmarks 1D convergence 3D analytical test Volume conservation test Simple plume generation test

16. 1D Convergence With fixed grid, larger timesteps produce lower errors Convergence happens only with dx and dt both decreasing Changing gridsize produces 2 nd order convergence in SELFE, but produces divergence in ELCIRC (non-orthogonal grid)

17. 3D quarter annulus M2 imposed as a function of the angle SELFE ELCIRC velocity

18. Volume conservation River discharge through a section of the Columbia

19. Plume Demonstrates need for hybrid s-z grid

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