Comparisons of Numerical Comparisons of Numerical Aspects in POM and ROMSAspects in POM and ROMS
Tal EzerTal EzerPrinceton UniversityPrinceton University
(in collaboration with H. Arango (Rutgers) & (in collaboration with H. Arango (Rutgers) & A. Shchepetkin (UCLA); Supported by ONR)A. Shchepetkin (UCLA); Supported by ONR)
Part of an initiative to develop, Part of an initiative to develop, evaluate and test an expert evaluate and test an expert Terrain-Terrain-
following Ocean Modeling System following Ocean Modeling System (TOMS)(TOMS)
• Time stepping algorithmsTime stepping algorithms• Advection schemesAdvection schemes• Pressure gradient schemesPressure gradient schemes• Other numerical and configuration aspectsOther numerical and configuration aspects
Compare numerical elements and Compare numerical elements and parameterizations in POM and ROMS/TOMS in parameterizations in POM and ROMS/TOMS in
terms of their numerical errors, numerical terms of their numerical errors, numerical stability, stability,
computational cost etc.computational cost etc.
AttributeAttribute POMPOM ROMS/TOMSROMS/TOMS
Horizontal gridHorizontal grid C-grid, curvilinearC-grid, curvilinear C-grid, curvilinearC-grid, curvilinear
Vertical gridVertical grid Sigma/generalSigma/general S-coordinatesS-coordinates
Model typeModel type Free surface, Free surface, primitive equationsprimitive equations
Free surface, Free surface, primitive equationsprimitive equations
Vertical mixingVertical mixing Mellor-Yamada 2.5Mellor-Yamada 2.5 M-Y2.5/KPPM-Y2.5/KPP
Model attributes- Model attributes- similaritiessimilarities and and differencesdifferences
Code structureCode structure Stand-aloneStand-alone ModularModular
Horizontal mixingHorizontal mixing Along-sigma, Along-sigma,
SmagorinskySmagorinskyGeopotential/Geopotential/
isopycnal, Gent-isopycnal, Gent-McWil./othersMcWil./others
Advection Advection schemesschemes
22ndnd order cent. order cent.
(MPDATA (MPDATA alternative)alternative)
22ndnd/3/3rdrd/4/4thth
Cent./upstreamCent./upstream
Pressure Gradient Pressure Gradient schemesschemes
Density JacobianDensity Jacobian
(high ord. (high ord. alternatives)alternatives)
Density/Pressure Density/Pressure Jac.,Jac.,
Weighted/Weighted/PolynomialPolynomial
Time steppingTime stepping Standard Leap-FrogStandard Leap-Frog Predictor-correctorPredictor-corrector
Code sizeCode size ~3000 lines~3000 lines ~40,000 lines~40,000 lines
Model configurationModel configuration
POMPOMROMS/TOMSROMS/TOMS
Seamount configuration testSeamount configuration testVery 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 flowZonal flow
Topography Sea surface heightTopography Sea surface height
-50cm
0
+50cm
Effect of advection scheme on model: Surface elevation anomaly
AD
V4-
4th o
rd.
cn
t.A
DV
2-
2n
d o
rd.
cn
t.
AD
V3-
3rd
ord
. u
pst.
POM ROMS
ROMS ROMS
Effect of advection scheme on model: Stream function anomaly
AD
V4-
4th o
rd.
cn
t.A
DV
2-
2n
d o
rd.
cn
t.
AD
V3-
3rd
ord
. u
pst.
Advection Schemes in ROMSAdvection Schemes in ROMS
Second OrderCentered
Third Order Upstream Bias
Fourth Order Centered
V
Time-stepping schemesTime-stepping schemes(split mode: baroclinic/internal and (split mode: baroclinic/internal and
barotropic/external)barotropic/external)
POMPOM ROMSROMS
schemescheme Leap-FrogLeap-Frog Predictor (LF) –Predictor (LF) –Corrector (A-M)*Corrector (A-M)*
Time-splitting Time-splitting filterfilter
Weights:Weights:
(n-1, n, n+1)(n-1, n, n+1)
AsselinAsselin
((
Adams-MoultonAdams-Moulton
(-1/12, 2/3, (-1/12, 2/3, 5/12)5/12)
Internal-externalInternal-external
CouplingCouplingOnce every Once every internalinternal time time stepstep
Weighted, every Weighted, every externalexternal time time stepstep
* Different terms in 3D ROMS may be treated differently
Coupling of barotropic (external) and Coupling of barotropic (external) and baroclinic (internal) modes in ROMSbaroclinic (internal) modes in ROMS
DTI
Un+1 =
amUm
Un+½ = bm’Um’
DTE
weights
1<m<N, N=DTI/DTE
Column 20
25
50
75
100
125
150
175
Relative Computation Time
POM 2nd order cent adv.
ROMS 2nd order cent adv.
ROMS 4th order cent adv.
ROMS 3rd order up-stream
%
CPU per Time Step
Sensitivity to internal (DTI) & external (DTE) time Sensitivity to internal (DTI) & external (DTE) time stepssteps
DTIDTI
DTEDTE180s180s 360s360s 540s540s 720s720s 900s900s 10801080
ss
8s8s 2222 4545 6767 9090 112112
12s12s 1515 3030 4545 6060 7575
16s16s 2222 3434 4545 5656
20s20s 1818 2727 3636 4545
24s24s 1515 2222 3030 3737
26s26s 1919 2525 3232
32s32s 1717 2222 2828
DTIDTI
DTEDTE180s180s 360s360s 540s540s 720s720s 900s900s 10801080
ss
8s8s 2222 4545 6767 9090
12s12s 1515 3030 4545 6060
16s16s 1111 2222 3434 4545
20s20s 99 1818 2727 3636
ROMS
POM
UNSTABLE
STABLE
TDI/DTE
CFL=13s
Computational cost for different Computational cost for different models & parameterizationsmodels & parameterizations
modemode
ll
featuresfeatures CPU (ms)/CPU (ms)/
(Im*Jm*Km*n)(Im*Jm*Km*n)CPU (s)/CPU (s)/
1 day1 day
22ndnd order cent. advection order cent. advection 12.512.5 21.321.3
22ndnd order upstream adv. order upstream adv.
Lin et al. (1994)Lin et al. (1994)13.213.2
POMPOM 33rdrd order upstream adv. order upstream adv.
Smolarkiewicz (1984)Smolarkiewicz (1984)17.117.1
66thth order PG (CPP) order PG (CPP)
Chu & Fan (1998)Chu & Fan (1998)207.4207.4
Z-lev. Interp. PG schemeZ-lev. Interp. PG scheme
Kliem & Pietrzak (1999)Kliem & Pietrzak (1999)40.040.0
22ndnd order cent. Adv. order cent. Adv. 17.317.3 21.621.6
ROMROMSS
44thth order cent. Adv. order cent. Adv. 19.219.2
33rdrd order upstream adv. order upstream adv. 20.020.0
The adjustment process in POM and The adjustment process in POM and ROMS: forced case (zonal flow)ROMS: forced case (zonal flow)
Roms- sensitivity of adjustment procesRoms- sensitivity of adjustment procesto time step choices to time step choices
DTE=12s
DTE=24s
DTI=360s DTI=720s
Sensitivity to bottom Sensitivity to bottom topographytopography
T=85min
T=111min
T=85s
T=111s
Pressure Gradient SchemesPressure Gradient SchemesSchemeScheme TypeType ReferenceReference
POM-DJPOM-DJ Standard Density Jacobian Standard Density Jacobian schemescheme
Mellor et al. (1998)Mellor et al. (1998)
POM-CCDPOM-CCD Combined Compact Combined Compact Difference scheme (6Difference scheme (6thth))
Chu & Fan (1997)Chu & Fan (1997)
ROMS-FPJROMS-FPJ Finite-Volume Pressure Finite-Volume Pressure Jacobian schemeJacobian scheme
Lin (1997)Lin (1997)
ROMS-DJROMS-DJ Weighted Density Jacobian Weighted Density Jacobian scheme (scheme (0)0)
Song (1998)Song (1998)
ROMS-WDJROMS-WDJ Weighted Density Jacobian Weighted Density Jacobian scheme (scheme (0.125)0.125)
Song (1998)Song (1998)
ROMS-PJQROMS-PJQ Pressure Jacobian scheme Pressure Jacobian scheme with Quadratic Polynomial with Quadratic Polynomial fitfit
Shchepetkin & Shchepetkin & McWilliams (2001)McWilliams (2001)
ROMS-DJCROMS-DJC Density Jacobian scheme Density Jacobian scheme with Cubic Polynomial fitwith Cubic Polynomial fit
Shchepetkin & Shchepetkin & McWilliams (2001)McWilliams (2001)
Structure ofStructure ofV (cm/s) in ROMS for different V (cm/s) in ROMS for different PG schemes (medium seamount case)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 seamountPG errors- moderately steep seamount
PG errors- very steep PG errors- very steep seamountseamount
(preliminary) conclusions(preliminary) conclusions• New numerical schemes show promising results New numerical schemes show promising results
in reducing numerical errors while saving in reducing numerical errors while saving computational costs.computational costs.
• However, the behavior of these schemes may be However, the behavior of these schemes may be more complicated than standard schemes, and more complicated than standard schemes, and require users for more careful choices of model require users for more careful choices of model parameterizations.parameterizations.
• Therefore, communication between developers Therefore, communication between developers and users is important. and users is important.
• Further developments and testing of more Further developments and testing of more elements for TOMS will continue. elements for TOMS will continue.
And finally, no matter what car And finally, no matter what car you drive (POM, ROMS, etc.) …you drive (POM, ROMS, etc.) …
… enjoy the ride
as much as we enjoy building the car…
THANK YOU