Non-hydrostatic algorithm and dynamics in ROMS

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

2. 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 • gravity currents • deep convection in the open ocean • ….

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

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

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

7. 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; • How we can improve and adopt the pressure decomposition technique in the case of explicit free surface calculations and mode splitting?

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

9. adopted from Shchepetkin, Mcwilliams 2005

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

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

12. Non-hydrostatic algorithm for ROMS model 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 • 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 => Neumann conditions for q at rigid boundaries; q=0 at free surface. .

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

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

16. 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 200x50x50 test case 100x100x100 test case (internal seiche waves in rectangular basin) (standing barotropic waves in deep basin) CG+SMG CG+SMG PFMG CG+ PFMG GMRES SMG CG+ PFMG SMG CG PFMG CG 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 Non-hydrostatic pressure distribution

20. Interface displacement in the center of tank

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

22. KH baroclinic instability Density distribution in hydrostatic simulations 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 L=600 km H=2.5 km LSILL=80 km HSILL=1.8 km Resolution 2D 800x5x80 Temperature (C) U-velocity (cm/s) Non-hydrostatic pressure

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

26. Non-hydrostatic s-error 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” Kinetic energy for seamount test ROMS simulations with pressure-gradient Scheme (Shchepetkin, McWilliams 2003)

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