Sponsor Acknowledgment The development and application of the program has had many sponsors since They include the Geophysical Fluid Dynamics La

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1 USERS GUIDE for A THREE-DIMENSIONAL, PRIMITIVE EQUATION, NUMERICAL OCEAN MODEL George L. Mellor Program in Atmospheric and Oceanic Sciences Princeton University, Princeton, NJ ver ver ver ver subroutine ver Notes on the 1998 Revision 1991 Fortran tmean smean tclim sclim trnu trnv drx2d dry2d advuu advvv adx2d ady2d pom97.f bcond Notes on the 2002 revision pom2k.f Fortran John Hunte POM 1

2 Sponsor Acknowledgment The development and application of the program has had many sponsors since They include the Geophysical Fluid Dynamics Laboratory/NOAA, Princeton University, Sea Grant/NOAA through the New Jersey Marine Sciences Consortium, the Department of Energy, Minerals Management Services/DOI, the National Ocean Services/NOAA, the Institute of Naval Oceanography and the Office of Naval Research/DOD. 1 INTRODUCTION Alan Blumberg 1977 Leo Oey, Jim Herring, Lakshmi Kantha and Boris Galperin. Tal Ezer. POM WEB NOAA and Dynalysis of Princeton Blumberg and Mellor, 1987) POM HP the Princeton Ocean Model POM) Arakawa C 2 CFL 3 CFL Mellor, 1973) Tetsuji Yamada Mellor and Yamada,1974; Mellor and Yamada,1982) Mellor- Yamada turbulence closure model Rotta and Kolmogorov Level2.5 PROFQ ADVQ POM WEB Martin, 1985) Mellor and Blumberg 2004) Klein, 1980) 2

3 Mellor, 1985) Oey et al., 1985a, b) Zavatarelli and Mellor, 1995; Jungclaus and Mellor, 1996; Baringer and Price, 1996; Ezer and Mellor, 2004) Mellor and Wang, 1996) Arakawa C advt, advq, advct, advu, advv advave proft, profq, profu profv as of June 1996) bcond dxi, j) i dyi, j) j dx dy i j ) pom2k bcond FORTRAN Appendix B z pom2k ftp ftp.aos.princeton.edu anonymous guest login OK cd pub/pom ls get quit POM WEB pom2k.f pom2k.f pom2k.c netcfd pom2k.n netcdf 3

4 2 THE BASIC EQUATIONS 1 Pillips1957) Blumberg and Meller 1980, 1987) D x = x, y = y, σ = z η H + η, t = t 1) x y z D = H + y Hx, y) ηx, y, t) σ z = h σ = 0 z = H σ = 1 DU + DV + ω σ + η t = 0 2) UD + U 2 D t + UV D + Uω fv D 3) σ +gd η + gd2 0 [ ρ ρ o σ σ D ρ ] D σ dσ = [ ] KM U + F x σ D σ V D + V 2 D + UV D + V ω + fud 4) t σ +gd η + gd2 0 [ ρ ρ o σ σ D ρ ] D σ dσ = [ ] KM V + F y σ D σ T D T UD + + T V D + T ω t σ = [ ] KH T + F T R 5) σ D σ z SD + SUD + SV D + Sω t σ = [ ] KH S + F S 6) σ D σ q 2 D + Uq2 D + V q2 D + ωq2 t σ = [ Kq q 2 ] 7) σ D σ + 2K [ U ) 2 ) ] M V g ρ K H D σ σ ρ o σ 2Dq3 + F q B 1 l = [ Kq q 2 ] l 8) σ σ q 2 ld + Uq2 ld + V q2 ld + ωq2 l t σ [ U ) 2 ) ] KM V 2 +E 1 l + D σ σ + E 3 g ρ o K H ρ σ D ) Dq3 B 1 W + Fl 3 ω s W = ω + U σ D + η ) + V σ D + η ) + σ D t + η t ω = 1 + E 2 l/kl) L 1 = η z) 1 + H z) 1 ρ/ σ = ρ/ σ Cs 2 p/ σ A cs T A 3) ) BAROPG ρ ρ MEAN ρ ρ MEAN z POM Mellor 1994,

5 F x = Hτ xx) + Hτ xy) 9) F y = Hτ xy) + Hτ yy) U U τ xx = 2A M, τ xy = τ yx = A M + V ) V, τ yy = 2A M 10) F φ = Hq x) + Hq y) 11) φ q x = A H, q φ y = A H 12) f T S q 2 q 2 l σ Mellor and Blumberg1985) σ 12a.b) Smagorinsky q x q y 12a,b) T S T CLIM S CLIM Levitus s ρ MEAN Levitus T P RNI A H /A M 0.2 9a,b) 11) D H 2.1 The Smagorinsly Diffusivity A M = c x y 1 2 V + V )T V + V ) T = [ u/) 2 + v/ + u/) 2 /2 + v/) 2 ] 1/2 C HORCON Oey 1985a,b) C C A M A M 5

6 2.2 Vertical Boundary Conditions 2) ω0) = ω 1) = 0 13) ω0) 0 3) 4) K M D U σ, V ) = < wu0) >, < wv0) >), σ 0 14) σ 14a,b) K M D U σ, V ) = C z [U 2 + V 2 ] 1/2 U, V ), σ 1 σ [ κ 2 ] C z = MAX [ln1 + σ kb 1 )H/z)] 2, κ = 0.4 z0 14c,d,e) 1 + σ kb 1 )H/z 0 14e) ) 6) K H D T σ, S σ K H D ) = < wθ0) >), σ 0 15) T σ, S ) = 0, σ 1 σ 7) 8) ) q 2 0), q 2 l0) = ) = q 2 1), q 2 l 1) B 2/3 ) 1 u 2 τ 0), 0 B 2/3 ) 1 u 2 τ 1), 0 16) B 1 u τ 16a) 16c) pom97.f 16a) q 2 lσ 1 ) = q 2 σ 1 )κdσ 1 σ 1 k = 1 σ 2.3 The Vertically Integrated Equations Simons1974) Madala and Piacsek1977) 6

7 2) σ = 1 σ = 0 13a,b) 17) η t + ŪD + V D 3) 4) = 0 17) ŪD + Ū 2 D t + Ū V D F x f V D + gd η = < wu0) > + < wu 1) > +G x gd 1 V D t + V 2 D + Ū V D ρ o F η y + fūd + gd = < wv0) > + < wv 1) > +G y gd ρ o 0 0 σ σ [D ρ D ] ρ σ dσ dσ σ [D ρ D ] ρ σ dσ dσ σ 18) 19) 20) Ū = 0 1 Udσ 20) < wu0) > < wv0) > < wu 1) > < wv 1) > F x F y 21a) 21b) F x = [ H2ĀM [ F y = H2ĀM ] Ū + [ Ū HĀM ] V 22a) 22b) G x = Ū2D G y = V 2D + Ū V D + Ū V D + V ] + [ Ū HĀM + V ] F x Ū 2 D F y V 2 D UV D UV D + F y 21) + F x 22) A M 22a) 22b) F 18) 19) cum sole G x = G y = 0 3 THE NUMERICAL SCHEME Figure2 pom2k 3.1 External-Internal Mode Interaction MAIN el ua va u v t s 7

8 Figure3 t n 1 t n leap-flog integral involving Figure3 Feedback t n < t < t n+1 t n+1 dte leap-flog utf vtf utb vtb ua va ua va u utb utf v vtb vtf Care etb etf elf etf etb t s 0.5 egf + egbt n 1 t n+1 leap-flog egb t n 1 t n egf t n t n+1 el CFL 3.2 Structure of the Internal Mode Calculation 3 DT T + AdvT ) DifT ) = 1 D ) T K H R σ σ σ AdvT ) DifT ) D T D n 1 T n 1 = AdvT n ) + DifT n 1 ) 2 T advt D n+1 T n+1 2 T D T = 1 T n+1 ) D n+1 K H R σ σ σ 9 proft D T 3 T n 1 tb T n t T n+1 uf leap-flog weak filter Asselin 1992 T s = T n + 1 α T n+1 2T n + T n 1) T s α = 0.05 T s T n 1 T n+1 T n 3.3 Grid Arrangement Figure4 Figure5 3 pom2k prof adv Fortran 8

9 dx dy 2) 8) 17) 19) 5) 24) Adv AdvT )h x h y = δ x Dh y UT ) + δ y Dh x V T ) + h x h y δ σ ωt ) δσ δσ Dh y UT T δ x advt 3 U AdvU)h x h y = δ x Dh y UU) + δ y Dh x UV ) + h x h y δσωu) δσ f = V δ xh y ) h x h y Uδ yh x ) h x h y fv Dh x h y advct advu advv advu advv advct 3.4 Time Step Constraints CFL t E 1 1 C t δx /2 δy 2 C t = 2gh) 1/2 + U max U max CFL 90% 26) 29 t I 1 C T 1 δx /2 δy 2 C T = 2C + U max C T C gravest 2m/s U max T I / T E = dti/dte = isplit POM Ezer et al.2002) A = A M A = A H t I 1 4A 1 δx δy 2 t I < 1 f = 1 2Ω sin φ A H Ω φ 31 32)

10 6 program main and external mode pom2k iint advq profq advv profv advt proft dens iint 1 iend iint 9000 Imbedded isplit iext dti/dte = isplit advct advx advy advave advave ispadv ispadv 5 7 subroutine advave 18) 19) mode = 2 profu profv 8 subroutine advt 24 f advf) 26 d u f dy d v f dx dt d tclim 12ab Mellor and Blumberg1986) dx dy ff 9 subroutine proft 25) Richtmeyer Morton1967) ) null u v q2 q2l advt 25 σ D = D n+1 = dh D Fig.5 elevation f k f k = dti2 [ khk dh 2 f k 1 f k ) kh ] k+1 f k f k+1 ) dt2 [rad k rad k+1 ] dz k dzz k 1 dzz k dhdz k dz k = z k z k+1, dzz k = zz k zz k+1 f k k i, j 10

11 Solution Technique f k+1 a k + f k a k ) 10 subroutine baropg 3) 4) MEAN rmean ρ MEAN σ 3) 4) σ/ D 1 D/)σ ρ/ σ) 11 subroutine advct,advu and advv advct 27) advx advy 3) 4) adx2d ady2d advct advu advv 12 subroutine profu and profv proft 14c,d,e) 13 subroutine advq 14 subroutine profq 15 subroutine vertvl 2) wi, j, kb) ) ) wi, j, 1) vfluxi, j) 0 16 subroutine bcond dum dvm fsm 0 0 σ 10m d A 11

12 A-1) BC 0 A-4) 0 A-3) B B-2) B-1) B-2) B-2) U n+1 im = γu n im γ)u n im; γ = c i t i / x ci a)γ 1 ti b) c i H 0 < γ 1 17 subroutine dens Meller1991) UNESCO tbias sbias 32 tbias=10.0 sbias= dens 1025kg/m3 barog profq APPENDIX A 18 subroutine slpmin 19 Utility subroutine 20 program curvigrid POM grid. 4 X Y ORTHOG j=1 i=1 j=jm NB,NR NL 2 1 < j < jm) i i=1 y x ORTHOG x i,j y i,j 12

13 ) ) =, s j s i ) = s j ) s i 23) δ j x δ j s = δ iy δ i s, Fig.6 *) δ j y δ j s = δ jx δ j s 24) 46a) 46b) 47a),48b) x i,j x i,j 1 = δ js δ i s [y i+1,j y i 1,j + y i+1,j 1 y 1,j 1 ] 25) y i,j y i,j 1 = δ js δ i s [x i+1,j x i 1,j + x i+1,j 1 x 1,j 1 ] 26) CS 46a,b) 47a,b) ORTHOG j js,xi,j i is xi,j yi,j CS CS ORTHOG CS) 2 i= j POISSON yi,j CFL : FTP contrib code grid.f sepelli.f grid.f 13

mains.dvi

mains.dvi 8 Λ MRI.COM 8.1 Mellor and Yamada (198) level.5 8. Noh and Kim (1999) 8.3 Large et al. (1994) K-profile parameterization 8.1 8.1: (MRI.COM ) Mellor and Yamada Noh and Kim KPP (avdsl) K H K B K x (avm)