1. Non-hydrostatic algorithm and dynamics in ROMS Yuliya Kanarska, Alexander Shchepetkin, Alexander Shchepetkin, James C. McWilliams, IGPP, UCLA

2. UCLA ROMS UCLA ROMS  A parallel three-dimensional numerical oceanic model in vertical hybrid z-sigma and horizontal curvilinear coordinates with innovative algorithms for advection, mixing, pressure gradient, vertical-mode coupling, time stepping (Shchepetkin and McWilliams, 1998, 2003, 2005)  Non-hydrostatic capabilities (2005)

3. Where are non-hydrostatic effects important? steep waves on uneven bottom in coastal areas unbalanced flows, baroclinic barotropic instability steepening, breaking of internal waves of large amplitude generated by the tidally driven flows over steep topography steepening, breaking of internal waves of large amplitude generated by the tidally driven flows over steep topography gravity currents gravity currents deep convection in the open ocean deep convection in the open ocean….

4. Hydrostatic approximation: Governing Equations H/L<<1

5. Pressure decomposition Mahadevan et al. (1996), Marshall et al. (1997), Casulli and Stelling (1998) p=p h +q “Surface” “Hydrostatic”“Non-hydrostatic”

6. Core algorithm of the most non-hydrostatic models Core algorithm of the most non-hydrostatic models Mahadevan et al. (1996), Marshall et al. (1997), Casulli and Stelling (1998) Basic algorithm: Projection method (Chorin, 1968) Pressure decomposition + Projection method+ Implicit free surface

7. Non-hydrostatic effects are included in barotropic equations only as integrated 3D velocity from previous time step in 2D equations; Non-hydrostatic effects are included in barotropic equations only as integrated 3D velocity from previous time step in 2D equations; 2D depth integrated velocities and 3D baroclinic velocities are not agreed at each discrete time step; 2D depth integrated velocities and 3D baroclinic velocities are not agreed at each discrete time step; How we can improve and adopt the pressure decomposition technique in the case of explicit free surface calculations and mode splitting? How we can improve and adopt the pressure decomposition technique in the case of explicit free surface calculations and mode splitting?

8. Pressure-correction method Projection method Armfield, Street 2002

9. adopted from Shchepetkin, Mcwilliams 2005

10. Barotropic mode Pressure –correction step Tracers Provisional velocity field tracers (n-2,n-1,n,n+1) AM4 interpolation => momentum (n-2,n-1,n) AB3 extrapolation (n-2,n-1,n,n+1) AM4 interpolation

11. Barotropic mode Pressure –correction step Tracers Provisional velocity field tracers (n-2,n-1,n,n+1) AM4 interpolation => momentum (n-2,n-1,n) AB3 extrapolation (n-2,n-1,n,n+1) AM4 interpolation

12. Non-hydrostatic algorithm for ROMS model Non-hydrostatic algorithm for ROMS model Components Components  pressure decomposition on hydrostatic, non- hydrostatic (nh) terms  pressure correction method for nh pressure  mode splitting on barotropic and baroclinic components with explicit free surface treatment Algorithm Algorithm  includes non-hydrostatic terms in both barotropic and baroclinic modes  guarantees mass conservation properties and agreement between modes at each discrete time step

13. What new regarding boundary conditions in nh setup? Momentum equation and time splitting for w Kinematical boundary conditions for vertical velocity: Boundary conditions for velocity field are satisfied before correction step => Boundary conditions for velocity field are satisfied before correction step => Neumann conditions for q at rigid boundaries; Neumann conditions for q at rigid boundaries; q=0 at free surface. q=0 at free surface..

14. Poisson equation in curvilinear  - coordinate system L: L: 15 diagonal; non-symmetric; inseparable in horizontal and vertical directions inseparable in horizontal and vertical directions

15. MPI Massively parallel Elliptic solvers of large sparse matrix PETSC (Argonne National Laboratory) PETSC (Argonne National Laboratory) HYPRE (Lawrence Livermore National Laboratory) HYPRE (Lawrence Livermore National Laboratory) …? …?

16. HYPRE (Solvers and Preconditioners) HYPRE (Solvers and Preconditioners)  MPI domain portioning approach in the same way as in ROMS (in xy-plane) no decomposition in z-direction;  Using Structured grid interface of HYPRE

17. Preliminary results of the HYPRE implementation in ROMS CG GMRES SMG PFMG CG+SMG CG+ PFMG CG SMG PFMG CG+SMG CG+ PFMG 200x50x50 test case 100x100x100 test case 200x50x50 test case 100x100x100 test case (internal seiche waves in rectangular basin) (standing barotropic waves in deep basin) Testing of different solvers and preconditioners for 1 (red) and 4 (blue) processors  Multigrid converges quickly (1-4) iterations but requires significant execution time per iteration;  Krylov methods (CG, GMRES) converges for ~ 20 iterations but even for that number iterations generally it works faster;  Krylov methods with multigrid as preconditioner converges very quickly (1-5 iteration) and it is quite efficient

18. Internal seiche gravity waves simulations Horn et al. 2000 experiment

19. Hydrostatic vs. non-hydrostatic simulations with ROMS Hydrostatic Non-hydrostatic pressure distribution

20. Interface displacement in the center of tank

21. Standing surface waves in deep basin U-non-hydrostatic U-hydrostatic W-non-hydrostatic Non-hydrostatic pressure correction Dispersion relation Free surface oscillations

22. Density distribution in hydrostatic simulations KH baroclinic instability Density distribution in non-hydrostatic simulations Nh pressure correction Hydrostatic stable time step two times smaller then non-hydrostatic!

23. NLIW generation by interaction of barotropic tide with sill Dimensionless parameters Luzon strait sill: supercritical finite depth topography

24. Strong barotropic tide Temperature (C) U-velocity (cm/s) Non-hydrostatic pressure L=600 km H=2.5 km L SILL =80 km H SILL =1.8 km Resolution 2D 800x5x80

25. HydrostaticNon-Hydrostatic Temperature (C) U-velocity (cm/s)

26. Bernsten J., Furnes G. (2005) “Internal pressure errors in sigma coordinate ocean models- sensitivity of the growth of the flow to the time stepping method and possible non-hydrostatic effects” Non-hydrostatic  -error ROMS simulations with pressure-gradient Scheme (Shchepetkin, McWilliams 2003) Kinetic energy for seamount test

27. Future ROMS NH Algorithm Directions Further optimization of elliptic solver; Further optimization of elliptic solver; Optimization of the calculations of cross-derivatives terms in pressure equation; Optimization of the calculations of cross-derivatives terms in pressure equation; Simulations in complex domains and convergence testing; Simulations in complex domains and convergence testing; Studies and testing for larger number of processors and highly resolution problems. Studies and testing for larger number of processors and highly resolution problems.