Comparisons of Numerical Aspects in POM and ROMS Tal Ezer Princeton University (in collaboration with H. Arango (Rutgers) & A. Shchepetkin (UCLA); Supported by ONR)

Part of an initiative to develop, evaluate and test an expert Terrain-following Ocean Modeling System (TOMS) Time stepping algorithms Time stepping algorithms Advection schemes Advection schemes Pressure gradient schemes Pressure gradient schemes Other numerical and configuration aspects Other numerical and configuration aspects Compare numerical elements and parameterizations in POM and ROMS/TOMS in terms of their numerical errors, numerical stability, computational cost etc.

AttributePOMROMS/TOMS Horizontal grid C-grid, curvilinear Vertical grid Sigma/generalS-coordinates Model type Free surface, primitive equations Vertical mixing Mellor-Yamada 2.5 M-Y2.5/KPP Model attributes- similarities and differences Code structure Stand-aloneModular Horizontal mixing Along-sigma,Smagorinsky Geopotential/isopycnal, Gent-McWil./others Advection schemes 2 nd order cent. (MPDATA alternative) 2 nd /3 rd /4 th Cent./upstream Pressure Gradient schemes Density Jacobian (high ord. alternatives) Density/Pressure Jac., Weighted/Polynomial Time stepping Standard Leap-Frog Predictor-corrector Code size ~3000 lines ~40,000 lines

Model configuration POM ROMS/TOMS

Seamount configuration test Very steep case h=4050m, w=50km s=0.36, r=14.2 moderately steep h=2700m, w=100km s=0.07, r=2.7 S=max(  H/2H) r=max(  grid=(64x64x20),  x=8km

Zonal flow Topography Sea surface height Topography Sea surface height -50cm 0 +50cm

Effect of advection scheme on model: Surface elevation anomaly ADV4- 4 th ord. cnt. ADV2- 2 nd ord. cnt. ADV3- 3rd ord. upst. POMROMS

Effect of advection scheme on model: Stream function anomaly ADV4- 4 th ord. cnt. ADV2- 2 nd ord. cnt. ADV3- 3rd ord. upst.

Advection Schemes in ROMS Second Order Centered Third Order Upstream Bias Fourth Order Centered V

Time-stepping schemes (split mode: baroclinic/internal and barotropic/external) POMROMS schemeLeap-Frog Predictor (LF) – Corrector (A-M)* Time-splitting filter Weights: (n-1, n, n+1) Asselin (  Adams-Moulton (-1/12, 2/3, 5/12) Internal-externalCoupling Once every internal time step Weighted, every external time step * Different terms in 3D ROMS may be treated differently

Coupling of barotropic (external) and baroclinic (internal) modes in ROMS DTI  U n+1 =   a m  U m  U n+½ =  b m’  U m’ DTE weights 1<m<N, N=DTI/DTE

Sensitivity to internal (DTI) & external (DTE) time steps DTI DTIDTE180s360s540s720s900s1080s 8s s s s s s s DTE180s360s540s720s900s1080s8s s s s ROMS POM UNSTABLE STABLE TDI/DTE CFL=13s

Computational cost for different models & parameterizations model modelfeatures CPU (ms)/ (Im*Jm*Km*n) CPU (s)/ 1 day 2 nd order cent. advection nd order upstream adv. Lin et al. (1994) 13.2 POM 3 rd order upstream adv. Smolarkiewicz (1984) th order PG (CPP) Chu & Fan (1998) Z-lev. Interp. PG scheme Kliem & Pietrzak (1999) nd order cent. Adv ROMS 4 th order cent. Adv rd order upstream adv. 20.0

The adjustment process in POM and ROMS: forced case (zonal flow)

Roms- sensitivity of adjustment proces to time step choices DTE=12s DTE=24s DTI=360s DTI=720s

Sensitivity to bottom topography Sensitivity to bottom topography T=85min T=111min T=85s T=111s

Pressure Gradient Schemes SchemeTypeReference POM-DJ Standard Density Jacobian scheme Mellor et al. (1998) POM-CCD Combined Compact Difference scheme (6 th ) Chu & Fan (1997) ROMS-FPJ Finite-Volume Pressure Jacobian scheme Lin (1997) ROMS-DJ Weighted Density Jacobian scheme (  0) Song (1998) ROMS-WDJ Weighted Density Jacobian scheme (  0.125) Song (1998) ROMS-PJQ Pressure Jacobian scheme with Quadratic Polynomial fit Shchepetkin & McWilliams (2001) ROMS-DJC Density Jacobian scheme with Cubic Polynomial fit Shchepetkin & McWilliams (2001)

Structure of  V (cm/s) in ROMS for different PG schemes (medium seamount case) R-DJ (Vmax=3.7) R-WDJ (Vmax=0.3) R-FPJ (Vmax=30) R-PJQ (Vmax=0.03) R-DJC (Vmax=0.06)

PG errors- moderately steep seamount

PG errors- very steep seamount

(preliminary) conclusions New numerical schemes show promising results in reducing numerical errors while saving computational costs. New numerical schemes show promising results in reducing numerical errors while saving computational costs. However, the behavior of these schemes may be more complicated than standard schemes, and require users for more careful choices of model parameterizations. However, the behavior of these schemes may be more complicated than standard schemes, and require users for more careful choices of model parameterizations. Therefore, communication between developers and users is important. Therefore, communication between developers and users is important. Further developments and testing of more elements for TOMS will continue. Further developments and testing of more elements for TOMS will continue.

And finally, no matter what car you drive (POM, ROMS, etc.) … … enjoy the ride as much as we enjoy building the car… THANK YOU