Non-hydrostatic algorithm and dynamics in ROMSYuliya Kanarska, Alexander Shchepetkin,James C. McWilliams,
IGPP, UCLA
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)
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
.
Pressure decompositionMahadevan et al. (1996), Marshall et al. (1997), Casulli and Stelling (1998)p=ph+qSurfaceHydrostaticNon-hydrostatic
Core algorithm of the most non-hydrostatic modelsMahadevan et al. (1996), Marshall et al. (1997), Casulli and Stelling (1998)
Basic algorithm: Projection method (Chorin, 1968)Pressure decomposition +Projection method+Implicit free surface
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?
Pressure-correction method Projection method Armfield, Street 2002
adopted from Shchepetkin, Mcwilliams 2005
Barotropic modePressure correction stepTracersProvisional velocity fieldtracers(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
Barotropic modePressure correction stepTracersProvisional velocity fieldtracers(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
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
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.
.
Poisson equation in curvilinear s-coordinate system
L: 15 diagonal;non-symmetric; inseparable in horizontal and vertical directions
MPI Massively parallel Elliptic solvers of large sparse matrixPETSC (Argonne National Laboratory)HYPRE (Lawrence Livermore National Laboratory)?
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
Preliminary results of the HYPRE implementation in ROMSCGGMRESSMGPFMGCG+SMGCG+PFMGCGSMGPFMGCG+SMGCG+PFMG 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) processorsMultigrid 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
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Internal seiche gravity waves simulationsHorn et al. 2000 experiment
Hydrostatic vs. non-hydrostatic simulations with ROMSHydrostaticNon-hydrostatic Non-hydrostaticpressuredistribution
Interface displacement in the center of tank
Standing surface waves in deep basinU-non-hydrostaticU-hydrostaticW-non-hydrostaticNon-hydrostatic pressure correctionDispersion relationFree surface oscillations
Density distributionin hydrostaticsimulationsKH baroclinic instabilityDensity distributionin non-hydrostaticsimulationsNh pressurecorrectionHydrostatic stable time step two times smaller then non-hydrostatic!
NLIW generation by interaction of barotropic tide with sillDimensionless parametersLuzon strait sill: supercritical finite depth topography
(advection speed)/(internal wave speed)=
, =
for tide flow
(maximum bottom slope)/(slope of characteristics)=
, where
(maximum topographic height)/(ocean depth)=
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,
,
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Strong barotropic tide
Temperature (C)U-velocity (cm/s)Non-hydrostatic pressure L=600 kmH=2.5 kmLSILL=80 kmHSILL=1.8 kmResolution 2D800x5x80
HydrostaticNon-HydrostaticTemperature (C)Temperature (C)U-velocity (cm/s)U-velocity (cm/s)
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 s-errorROMS simulations with pressure-gradient Scheme (Shchepetkin, McWilliams 2003)Kinetic energy for seamount test
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.
