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