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1 0 MRI.COM Mellor and Kantha (989) - frazil ice MRI.COM Hunke and Dukowicz (997, 00) EVP (Elastic-Viscous-Plastic) MRI.COM hs hi snow ice ice QAI Qs QI QIO FTI T3 T T T0 T, S QAO T0L FTL WAI SI=4.0 S0 FSI WFR WIO WRO T, S S0L WAO FSL 0.: ( h I S I = 4.0 [psu]) ( h s ) 3 T 0 ;T ;T ;T 3 Q IO ;Q I ;Q S Q AI F TI (open leads) L 0.
2 0 0.: ( 0. ) MRI.COM MRI.COM h I hiceo h s hsnwo A (compactness) a0iceo T 3 tsfci T 0 ;T 0L - t0iceo,t0icel (, ) S 0 ;S 0L - s0,s0l (, ) Q IO ;Q AO fheati,fheat (, ) F TI ;F TL ftio,ftao (, ) F SI ;F SL sfluxi = F SI + F SL W wfluxi = W IO +W AO +W RO +W FR W AI ( ) snowfall W IO wio W AO wao W RO wrss,wrsi (, ) W FR frazil ice u I uice v I vice 0.. ( ρ I Ah I ) (u I ;v I ) ρ I t (Ah Iu I ) ρ I Ah I fv I = ρ I Ah I g h µ h µ + F µ(σ )+A(τ AIx + τ IOx ) (0.) ρ I t (Ah Iv I )+ρ I Ah I fu I = ρ I Ah I g h ψ h ψ + F ψ (σ )+A(τ AIy + τ IOy ) (0.) (0.3) h ( η h ) (F µ ;F ψ ) (σ ) τ AI ;τ IO t (Ah I)+A (Ah I )=D(Ah I )+ ρ o ρ I h A(W IO W AI )+( A)W AO +W FR i (0.4) A ; D
3 h I h A t + A (A) i = h I D(A)+ ρ o ρ I h Φ( A)W AO + ΨAW IO +( A)W FR i (0.5) open leads W AO Φ Φ > MRI.COM 4.0 Ψ ( ) MRI.COM Ψ = 0:05 Ψ = 0:0 Ψ = 0: Sh I, S S z, z (T freeze ) h I =(T freeze T ) z ρ o C po =ρ I L F (0.6) C po ρ I L F, open leads E(T;r) =r(l F +C po T )+( r)c pi T (0.7) r brine fractione(t;0) =C pi T E(T;) =L F +C po T C pi 3
4 0 MRI.COM C pi = 0 ( h s ) E(T;r) =r(l F +C po T ) (0.8) brine fraction (r = ) (r = 0) ( h I T ) -Q AI Q AI = Q SI + Q LI ( α I )SW LW + ε I σ (T :6) 4 (0.9) SW α I LW ε I σ Stefan-Boltzmann Q SI Q SI = ρ a C pa C HAI U 0 (T 3 T A ) (0.0) ρ a C pa C HAI -U 0 0m T A input T 3 Q LI Q LI = ρ a L s C HAI U 0 (q i q a ) (0.) L s q i (T 3 ) q a - Q AO 9 Q AO = Q SO + Q LO ( α o )SW LW + ε o σ (T :6) 4 (0.) LW O = LW + ε o σ (T :6) 4 (0.3) LW LW I = LW O ε o σ (T :6) 4 + ε I σ (T :6) 4 (0.4) 4
5 0.. LW I = LW + ε I σ (T :6) 4 (0.5) 0.5 Q S h s k s Q S = k s h s (T T 3 ) (0.6) Q I ;Q IO Q I = k I h I = (T T ) (0.7) k I Q S = Q I Q k I IO = h I = (T 0 T ) (0.8) W RO T 3 Q AI = Q S T 3 T 3 Q AI Q S h s = 0 W RO =(Q AI Q S )=(ρ o L 3 ) (0.9) L 3 [E(T 3 ;) E(T ;r )] (0.0) C po = C pi = 0r = 0 L 3 = L F -0. ffi C 0 ffi C h ρ I h I t E(T i ;r )+u I E(T ;r ) = Q IO Q I (0.) x i 5
6 0 (C pi = 0) (r = 0) 0 Q IO = Q k I I = (T 0 T ) (0.) h I ffl ffl W RO ffl Q AI Q IO ffl MRI.COM Q IO = Q I = Q S (h s = 0;T 3 = T ) (T 3 ) T 3 (0.9) (0.8) Q AI = Q IO T 3! T 3 +δt 3 Q SI + Q LI ( α I )SW LW + ε I σ ((T 3 + δt 3 )+73:6) 4 = k I h I (T 0 (T 3 + δt 3 )) (0.3) Q SI + Q LI ( α I )SW LW I + 4ε I σ (T :6) 3 δt 3 = k I h I (T 0 (T 3 + δt 3 )) (0.4) δt 3 T 3 δt 3 = Q SI Q LI +( α I )SW + LW I + k I h I (T 0 T 3 ) 4ε I σ (T :6) 3 + k I h I (0.5) T 3new = T 3old + δt 3 (0.6) T 3new T 3new (W RO )
7 0.. F T ;F S ;W IO :W AO (open leads) Mellor and Kantha (989) F TI = Q IO W IO ρ o L o (0.7) F TL = Q AO W AO ρ o L o (0.8) L o [E(T 0 ;) E(T ;r )](= L F ) (0.9) F T =(AQ IO +( A)Q AO ) W O ρ o L o (0.30) W O AW IO +( A)W AO (0.3) F SI = W IO (S I S) (0.3) F SL = W AO (S I S) (0.33) Mellor and Kantha (989) S 0 S ( ) S 0 F S =(AW IO +( A)W AO )(S I S) (0.34) 0 F S = AfWROice (S I S) W ROsnow Sg (0.35) z! 0 F TI =(ρ o C po ) = C Tz (T 0I T ) (0.36) F TL =(ρ o C po ) = C Tz (T 0L T ) (0.37) F SI = C Sz (S 0I S) (0.38) F SL = C Sz (S 0L S) (0.39) C Tz = u τ (P rt k ln( z=z 0 )+B T ) 7 (0.40)
8 0 u τ friction velocity u τ (τio x +τio ) =4 y ρ o = k von Karman s constant 0:4 z 0 roughness parameter(τ IOx ;τ IOy ) - z0 u τ = B T = b Pr =3 (0.4) ν Pr ν=α t = :9 Appendix B. u τ C Sz = (P rt k (0.4) ln( z=z 0 )+B S ) z0 u τ = B S = b Sc =3 ν Sc ν=α s = 43 roughness parameter z 0 (0.43) lnz 0 = Alnz 0I +( A)lnz 0L (0.44) z 0I = 0:05 h I h Ilim h Ilim = 3:0 [m] (0.45) z 0L = 0:06 ρ o u τ ρ a g MRI.COM (0.45) W O = 0 A = 0 T 0 = ms 0 A > 0 (0.46) (0.47) m (0.7) (0.3) (0.36) (0.38) (0.8) (0.33) (0.37) (0.39) S 0I S 0L S 0I = C S z S +(ρ o C po C Tz T Q IO )(S I S)=ρ o L o C Sz + ρ 0 C po C Tz m(s I S)=ρ o L o (0.48) S 0L = C S z S +(ρ o C po C Tz T Q AO )(S I S)=ρ o L o C Sz + ρ o C po C Tz m(s I S)=ρ o L o (0.49) (0.47) T 0I T 0L (0.36) (0.37) F TI F TL (0.38) (0.39) F SI F SL (0.7) (0.8) W IO W AO W AO W AO < 0 W AO = 0 (0.33) (0.39) S 0L = S (0.8) F TL = Q AO open leads T 0L = T (0.30) (0.34) 8
9 0.. z! 0: κ V T κ V z = F T (0.50) S κ V z = F S (0.5) W IO t > Ah I W IO h I = ρ o ρ I W IO t (0.5) F TI = W IO ρ o L o (0.53) 0..4 frazil ice frazil ice MRI.COM T S (freezing line) (supercooling) δt T S δt δs δt δs = F T F S (0.54) δt C δs = T z (T 0 T ) C Sz (S 0 S) (0.55) T 0 = ms 0 T = ms δt δs = mc T z C Sz (0.56) C Tz > C Sz m < 0 δt < mδs freezing line T f = ms + nz frazil ice W FR incremental mass of frazil ice γ = (0.57) total mass supercooled frazil ice fusion C po T + L o = ( γ)(c po T + L o )+γc pi T (0.58) S = ( γ)s + γs I (0.59) 9
10 0 γ T = T + γl=c po (0.60) S = S + γ(s S I ) (0.6) L = L F +(C po C pi )T T = ms + nz γ = ( Cpo (ms +nz T ) L C po m(s S I ) ms + nz T > 0 0 S + nz T < 0 (0.6) frazil ice W FR Z 0 W FR = δt γdz: (0.63) H ρ I t (Ah Iu I ) ρ I Ah I fv I = ρ I Ah I g h µ h µ + F µ(σ )+A(τ AIx + τ IOx ) (0.64) ρ I t (Ah Iv I )+ρ I Ah I fu I = ρ I Ah I g h ψ h ψ + F ψ (σ )+A(τ AIy + τ IOy ) (0.65) : - : τ AI = c a ρ a ju a j(u a cosθ a + k U a sinθ a ) (0.66) τ IO = c w ρ o ju w u I j[(u w u I )cosθ o + k (U w u I )sinθ o ] (0.67) U a U w c a c w --ρ a ρ o θ a θ o explicit implicit 0
11 0.3. Hunke and Ducowicz (997) viscous-plastic EVP elastic-viscous-plastic EVP constitutive low σ ij + E t η σ η ζ ij + 4ηζ σ kkδ P ij + 4ζ δ ij = ε ij (0.68) (i; j = ;) ζ η P E viscous-plastic constitutive low ε ij ε ij = uii u I j + x j x i divergence, tension, shear (0.69) D D = ε + ε ; D T = ε ε ; D S = ε (0.70) σ = σ + σ σ = σ σ σ E t E E σ P + ζ + ζ σ t σ σ + t η divergence, tension, shear = D D (0.7) + σ η = D T (0.7) = D S (0.73) D D = D T = h ψ h µ D S = h µ h ψ (h i µv) + ψ h µ v h ψ ψ h µ u h ψ v + ψ h µ h µ µ h ψ h (hψ u) h µ h ψ µ u µ h ψ (0.74) (0.75) (0.76) F σ (h ψσ ) µ = h µ µ + h µ h + h ψ µ h µh ψ ψ µ σ Λ F σ ψ = h ψ ψ (h µσ ) h µ h + h ψ ψ h µ h ψ µ ψ σ Λ (0.77) (0.78) (0.79)
12 0 ζ P = P η = e hd i = D + e (D T + D S ) = (0.80) (0.8) (0.8) P = P Λ Ah I eexp[ c Λ ( A)] (0.83) P Λ c Λ e e = E E = E oρ I Ah I t e min( x ; y ) (0.84) E o 0 < E o < t e x ; y U ν V V z ; = A ( A) (τ IOx ;τ IOy )+ (τ AOx ;τ AOy ) (0.85) z ρ o ρ o (τ IOx ;τ IOy ) ) ( σ E σ m+ σ m t σ m+ P + ζ m + ζ m = Dm D (0.86) σ m+ σ m+
13 0.4. u m+ I u m I ρ I Ah I t v m+ I v m I ρ I Ah I t = ρ I Ah I fv m+ I ρ I Ah I g h µ h µ + F µ(σ m+ )+Aτ AIx (0.87) +Ac w ρ o ju w u m I j[(u w u m+ I )cosθ o +(V w v m+ I )sinθ o ] (0.88) = ρ I Ah I fu m+ I ρ I Ah I g h ψ h ψ + F ψ (σ m+ )+Aτ AIy (0.89) +Ac w ρ o ju w u m I j[(v w v m+ I )cosθ o (U w u m+ I )sinθ o ] (0.90) τ AI τ IO n n+ m m+ m+ (u I ) n- (leap frog (U w ) τ IO = c w ρ o ju n w u m+ I j[(u n w u m+ I )cosθ o + k (U n w u m+ I )sinθ o ] (0.9) E (FA X ; 0. ) (FA Y ; 0. ) (h I h s ) (FA X ) i+ ; j = n U i+ +U i+ ; j o (A i; j + A i+; j) κ H (A i+; j A i; j)= x i+ y j (0.9) n (FA Y ) i = V i+ +V i o (A i; j + A i) κ H (A i A i; j)= y j+ x i (0.93) κ H [m s ] (A;Ah I ) (FA X ) i+ ; j A n+ i+; j = A n i+; j + t (FA X ) i+ ; j = S i+; j A n+ i; j = A n i; j t (FA X ) i+ ; j = S i; j (0.94) S i; j A i; j 0. 3
14 0 Ui-/, j+3/ Ui+/, j+3/ Ui+3/. j+3/ Ai, j+ Ai+, j+ Ui-/, j+/ Ui+/, j+/ Ui+3/. j+/ Ai, j Ai+, j Ui-/, j-/ Ui+/, j-/ Ui+3/. j-/ 0.: (A) (U) (h I h s ) A 0.4. ε σ T- 0.3 (divergence, tension, shear) T- (D D ) i; j = y i+ ; j x i; j y i; j + x i (v Ii+ (u Ii+ y i + u Ii+ ; j ) ; j (u Ii + u Ii ; j ) + v x i; j Ii ) (v Ii+ ; j + v Ii ; j ) (D h y i u Ii+ T ) i; j = u Ii y i; j u Ii+ + ; j u Ii ; j i x i y i+ y i x i; j y i+ ; j y i ; j h x i+ ; j v Ii+ v I i+ ; j x i; j v Ii v I i + ; j i y i+ ; j x i+ x i+ ; j y i; j x i x i ; j (D i S ) i; j = h y v Ii+ v Ii y i; j v Ii+ ; j v Ii + ; j i x i y i+ y i x i; j y i+ ; j y i ; j h x i+ + ; j u Ii+ u Ii+ ; j x i; j u Ii + u Ii ; j i y i+ ; j x i+ x i+ ; j y i; j x i x i ; j U- (F (σ ) i+ +(σ ) i+; j (σ ) i (σ ) i; j µ ) i+ = x i+ + y i+ [(σ ) i+ +(σ ) i+; j] y i [(σ ) i +(σ ) i; j] y i+ x i+ (0.95) + x i+ j+[(σ ) i+ +(σ ) i] x ; i+ j[(σ ) i+; j +(σ ) i; j] ; x i+ y i+ 4
15 0.5. (F ψ ) i+ (σ ) i+ +(σ ) i (σ ) i+; j (σ ) i; j = y i+ x i+ j+[(σ ) i+ +(σ ) i] x ; i+ j[(σ ) i+; j +(σ ) i; j] ; x i+ y i+ + y i+ [(σ ) i+ +(σ ) i+; j] y i [(σ ) i +(σ ) i; j] y i+ x i+ (0.96) Ui-/, j+3/ Ui+/, j+3/ Ui+3/. j+3/ σi, j+,εi, j+ σi+, j+,εi+, j+ Ui-/, j+/ Ui+/, j+/ Ui+3/. j+/ σi, j,εi, j σi+, j,εi+, j Ui-/, j-/ Ui+/, j-/ Ui+3/. j-/ 0.3: ffl Eular backward scheme ffl mkflux - ffl 0.5. siinit.f90: paramice.f90: 5
16 0 simain.f90: iaflux.f90: sidynevp.f90: mkstress.f90: stmrgni.f90: mkhisti.f90: writdt.f90: history, restart OGCM_ICE free drift /3 OGCM_SIDYN References Gill, A. E., 98: Atmosphere-Ocean Dynamics, Academic Press, 66pp. Hunke, E. C, and J. K. Dukowicz, 997: An Elastic-Viscous-Plastic Model for Sea Ice Dynamics, J. Phys. Oceanogr., 94, Hunke, E. C, and J. K. Dukowicz, 00: The Elastic-Viscous-Plastic Sea Ice Dynamics Model in General Orthogonal Curvilinear Coordinates on a Sphere Incorpolation of Metric Terms, Mon. Wea. Rev., 30, Mellor, G. L., and L. Kantha, 989: An Ice-Ocean Coupled Model, J. Geophys. Res., 94, 0,937 0,954. 6
17 0.5. Appendix A Gill(98) Appendix 4 e w ( [hpa]) log 0 e w (t) =(0: :03477t)=( + 0:004t) (0.97) e 0 w = f we w (0.98) f w = +0 6 p(4:5 +0:0006t ) (0.99) p [hpa] e i q log 0 e i (t) =log 0 e w (t)+0:004t (0.00) e 0 =p s = q=(ε +( ε)q) (0.0) ε m w =m a = 8:06=8:966 = 0:697 p s [hpa] q = εe 0 =(p s ( ε)e 0 ) (0.0) L s = : :6(T ) Jkg (0.03) 7
18 0 Appendix B B. SI SI MRI.COM Thermal ice conductivity k I = :04Jm s K CKI Thermal snow conductivity k s = 0:3Jm s K CKS Specific heat of sea water C po = 3990Jkg K CP Specific heat of air C pa = 004:67Jkg K CPAIR Specific heat of ice C pi = 0:0Jkg K Specific heat of snow C ps = 0:0Jkg K Stefan Boltzmann constant σ = 5: Wm K 4 STBL Albedo of open ocean surface α o = 0: ALBW Albedo of ice α I = 0:5 ALBI Albedo of snow α s = 0:75 ALBS Emissivity of ocean surface ε o = 0:97 EEW Emissivity of ice surface ε I = :0 EEI Emissivity of snow surface ε s = :0 EES bulk transfer coefficient C HAI = :5 0 3 CHAI Latent heat of fusion L F = 3: Jkg ALF Latent heat of sublimation equation (0.03) RLTH constants for fusion phase m = 0:0543 K=ppt XMXM equation: T f = ms + nz n = 0:000759Km XNXN? Ice roughness parameter z oi = 0:05h I =3 Z0 Empirical constant in eq. (0.5) Φ = 4:0 PHI Empirical constant in eq. (0.5) Ψ = 0:0 PSI Salinity of sea ice S I = 4:0psu SI von Karman s constant k = 0:4 XK Thickness/compactness diffusion of ice κ H = :0 0 3 m s AKH Scaling factor for κ H North : 3.0, South : 3.0 FKHDN, FKHDS Seawater kinematic viscosity ν = :8 0 6 m s ANU Seawater heat diffusivity α t = : m s AT Seawater salinity diffusivity α s = 6:8 0 0 m s AS Turbulent Prandtl number P rt = 0:85 PRT b in eqs (0.4),(0.43) b = 3:4 AB 8
19 0.5. B. MRI.COM Reference water density ρ o = 000kgm 3 RO Reference air density ρ a = :05kgm 3 ROAIR Reference ice density ρ I = 900kgm 3 RICE Reference snow density ρ s = 330kgm 3 RDSW e-folding constant for ice pressure c Λ = 0:0 CSTAR pressure scaling factor P Λ = : Nm PRSREF drag coefficient (air-ice) c a = :5 0 3 CDRGAI drag coefficient (ice-ocean) c w = 5:5 0 3 CDRGIW yield curve axis ratio e = :0 ELIPS scaling factor for Young s modulus E o = 0:5 EYOUNG water turning angle θ o = ±5 ffi (, ) WIANGL air turning angle θ a 9
20 30
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)
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