Presentation Slides for

Chapter 7of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

Stanford UniversityStanford, CA 94305-4020jacobson@stanford.edu

March 10, 2005

Vertical Model Grid___Model top boundary__ ˙

σ 1 2

= 0 , σ1 2

= 0 , pa , top

_ _ _ _ _ _ _ _ _ _ _ _ _ q1

, qv , 1

, u1

, v1

, pa , 1

_____________________ ˙ σ

1 + 1 2, σ

1 + 1 2, p

a , 1 + 1 2

_ _ _ _ _ _ _ _ _ _ _ _ _ q2

, qv , 2

, u2

, v2

, pa , 2

_____________________ ˙ σ

k − 1 2, σ

k − 1 2, p

a , k − 1 2

_ _ _ _ _ _ _ _ _ _ _ _ _ qk

, qv , k

, uk

, vk

, pa , k

_____________________ ˙ σ

k + 1 2, σ

k + 1 2, p

a , k + 1 2

_ _ _ _ _ _ _ _ _ _ _ _ _ qk + 1

, qv , k + 1

, uk + 1

, vk + 1

, pa , k + 1

_____________________ ˙ σ

NL

− 1 2, σ

NL

− 1 2, p

a , NL

− 1 2

_ _ _ _ _ _ _ _ _ _ _ _ _ qN

L

, qv , N

L

, uN

L

, vN

L

, pa , N

L

_Model bottom boundary_ ˙ σ

NL

+ 1 2= 0 , σ

NL

+ 1 2= 1 , p

a , σurf

Fig. 7.1

Estimate top altitude in test column (7.2)zbelow is altitude from App. Table B.1 just below pa,top

Estimating Sigma Levels

Estimate altitude at bottom of each layer in test column (7.3)

Find pressure from (2.41) --> sigma values (7.4)

ztop,test=zbelow+pa,below−pa,topρa,belowgbelow

zk+12,test=zsurf,test+ ztop,test−zsurf,test( ) 1− kNL

⎛ ⎝ ⎜

⎞ ⎠ ⎟

σk+12 =pa,k+12,test−pa,toppa,NL +12,test−pa,top

Estimate pressure at each layer edge (2.41)

pa,k+12,test≈pa,k−12,test+ρa,k−12gk−12 zk−12,test−zk+12,test( )

Estimating Sigma LevelsSigma values (7.4)

Model pressure at bottom boundary of layer (7.6)

Model column pressure (7.5)

σk+12 =pa,k+12,test−pa,toppa,NL +12,test−pa,top

pa,k+12 =pa,top+σk+12πa

πa =pa,surf−pa,topSigma thickness of layer (7.1)

Δσk =σk+12 −σk−12

Layer Midpoint Pressure

Example layers____________________________ p

a , k − 1 2 = 700 hPa

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ qv , k

= 308 K

____________________________ pa , k + 1 2

= 750 hPa

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ qv , k + 1

= 303 K____________________________ p

a , k + 3 2 = 800 hPa

Pressure at the mass-center of a layer (7.7)

pa,k =pa,k−12 +0.5 pa,k+12 −pa,k−12( )

Pressure where mass-weighted mean of P is located (7.10)When qv increases monotonically with height

Layer Midpoint Pressure

Mass-weighed mean of P (7.8)

Value of P at boundaries (7.9)

Consistent formula for qv at boundaries (7.11)

pa,k = 1000 hPa( )Pk1κ

Pk = 1pa,k+12 −pa,k−12

Pdpa,k−12

pa,k+12∫ pa = 11+κ

Pk+12pa,k+12 −Pk−12pa,k−12pa,k+12 −pa,k−12

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

Pk+12 =pa,k+121000 hPa

⎛ ⎝ ⎜

⎞ ⎠ ⎟

κ

θv,k+12 =Pk+12−Pk( )θv,k+ Pk+1−Pk+12( )θv,k+1

Pk+1−Pk

Arakawa C Grid

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2 Fig. 7.2

Prognostic equation for column pressure (7.12)

Continuity Equation For Air

First-order in time, second-order in space approx. (7.13)

Re2cosϕ ∂πa∂t

⎛ ⎝ ⎜ ⎞

⎠ ⎟ σ

=− ∂∂λe

uπaRe( )+ ∂∂ϕ vπaRecosϕ( )⎡

⎣ ⎢ ⎤ ⎦ ⎥ σdσ

01∫

Re2cosϕΔλeΔϕ( )i, jπa,t −πa,t−h

h⎛ ⎝ ⎜ ⎞

⎠ ⎟ i, j

=−uπaReΔϕΔλeΔσ( )i+12, j −uπaReΔϕΔλeΔσ( )i−12, j

Δλe

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

k=1

NL∑k,t−h

−vπaRecosϕΔϕΔλeΔσ( )i,j+12− vπaRecosϕΔϕΔλeΔσ( )i, j−12

Δϕ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

k=1

NL∑k,t−h

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

Horizontal fluxes in domain interior (7.15)

Prognostic Column Pressure

Fi+12, j,k,t−h =πa,i, j +πa,i+1, j

2 uReΔϕ( )i+12, j,k⎡ ⎣ ⎢

⎤ ⎦ ⎥ t−h

Gi,j+12,k,t−h =πa,i, j +πa,i, j+1

2 vRecosϕΔλe( )i, j+12,k⎡ ⎣ ⎢

⎤ ⎦ ⎥ t−h

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papapa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

Prognostic Column PressureHorizontal fluxes at eastern and northern boundaries (7.17)

FI+12, j,k,t−h = πa,I, j uReΔϕ( )I+12, j,k⎡ ⎣

⎤ ⎦ t−h

Gi,J +12,k,t−h = πa,i,J vRecosϕΔλe( )i,J +12,k⎡ ⎣

⎤ ⎦ t−h

πa,i, j,t =πa,i, j,t−h− hRe2cosϕΔλeΔϕ( )i, j

× Fi+12, j −Fi−12, j +Gi,j+12−Gi, j−12( )k,t−hΔσk⎡ ⎣ ⎢

⎤ ⎦ ⎥

k=1

NL∑Equation for column pressure (7.14)

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

Diagnostic equation for vertical velocity (7.19)

Diagnostic Vertical Velocity

Finite difference equation (7.20)

˙ σ πaRe2cosϕ =− ∂∂λe

uπaRe( )+ ∂∂ϕ vπaRecosϕ( )⎡

⎣ ⎢ ⎤ ⎦ ⎥ σ

dσ0σ∫ −σRe2cosϕ ∂πa

∂t⎛ ⎝ ⎜ ⎞

⎠ ⎟ σ

˙ σ πaRe2cosϕΔλeΔϕ( )i,j,k+12,t

=−uπaReΔλeΔϕΔσ( )i−12, j −uπaReΔλeΔϕΔσ( )i+12, j

Δλe

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

l=1

k

∑l,t−h

−vπaRecosϕΔλeΔϕΔσ( )i,j−12 − vπaRecosϕΔλeΔϕΔσ( )i, j+12

Δϕ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ l,t−hl=1

k

∑−σk+12 Re2cosϕΔλeΔϕ( )i, j

πa,t −πa,t−hh

⎛ ⎝ ⎜ ⎞

⎠ ⎟ i, j

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

Diagnostic Vertical VelocitySubstitute fluxes and rearrange --> vertical velocity (7.21)

˙ σ i, j,k+12,t =− 1πaRe2cosϕΔλeΔϕ( )i, j,t

× Fi+12, j −Fi−12, j +Gi,j+12−Gi, j−12( )l,t−hΔσl⎡ ⎣ ⎢

⎤ ⎦ ⎥

l=1

k

∑−σk+12

πa,t −πa,t−hhπa,t

⎛ ⎝ ⎜

⎞ ⎠ ⎟ i, j j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

Species continuity equation (7.22)Species Continuity Equation

Finite-difference form (7.23)

Re2cosϕ ∂∂t πaq( )⎡

⎣ ⎢ ⎤ ⎦ ⎥ σ

+ ∂∂λe

uπaqRe( )+ ∂∂ϕ vπaqRecosϕ( )⎡

⎣ ⎢ ⎤ ⎦ ⎥ σ

+πaRe2cosϕ ∂∂σ

˙ σ q( ) =πaRe2cosϕ ∇ •ρaKh∇( )qρa

+ Rnn=1

Ne,t

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Re2cosϕΔλeΔϕ( )i, jπa,tqt −πa,t−hqt−h

h⎛ ⎝ ⎜ ⎞

⎠ ⎟ i, j,k

+uπaqReΔλeΔϕ( )i+12, j,k,t−h− uπaqReΔλeΔϕ( )i−12, j,k,t−h

Δλe

+vπaqRecosϕΔλeΔϕ( )i, j+12,k,t−h− vπaqRecosϕΔλeΔϕ( )i,j−12,k,t−h

Δϕ

+ πa,tRe2cosϕΔλeΔϕ˙ σ tqt−h( )k+12 − ˙ σ tqt−h( )k−12

Δσk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ i, j

= πaRe2cosϕΔλeΔϕ ∇z•ρaKh∇z( )qρa

+ Rnn=1

Ne,t

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ i,j,k,t−h

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papapa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

Substitute fluxes --> final continuity equation (7.24)

Species Continuity Equation

Mixing ratios at vertical top and bottom of layer (7.25)

qi, j,k,t =πaq( )i, j,k,t−h

πa,i, j,t+ h

πa,tRe2cosϕΔλeΔϕ( )i, j×

Fi−12, jqi−1, j +qi,j

2 −Fi+12, jqi,j +qi+1,j

2+Gi, j−12

qi, j−1+qi, j2 −Gi, j+12

qi, j +qi, j+12

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ k,t−h

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

+ πa,tRe2cosϕΔλeΔϕ˙ σ tqt−h( )k−12 − ˙ σ tqt−h( )k+12

Δσk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ i, j

+ πaRe2cosϕΔλeΔϕ ∇z •ρaKh∇z( )qρa

+ Rnn=1

Ne,t

∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥ i, j,k,t−h

⎫ ⎬ ⎪

⎭ ⎪

qi, j,k−12 =lnqi, j,k−1−lnqi,j,k1 qi, j,k( )− 1 qi, j,k−1( )

qi, j,k+12 =lnqi, j,k−lnqi, j,k+11 qi, j,k+1( )− 1 qi, j,k( )

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papapa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

Thermodynamic Energy EquationContinuous form (7.26)

Final finite difference form (7.27)

Re2cosϕ ∂∂t πaθv( )⎡

⎣ ⎢ ⎤ ⎦ ⎥ σ

+ ∂∂λe

uπaθvRe( )+ ∂∂ϕ vπaθvRecosϕ( )⎡

⎣ ⎢ ⎤ ⎦ ⎥

+πaRe2cosϕ ∂∂σ

˙ σ θv( )=πaRe2cosϕ ∇ •ρaKh∇( )θvρa

+ θvcp,dTv

dQndt

n=1

Ne,h

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

θv,i, j,k,t =πaθv( )i, j,k,t−h

πa,i, j,t+ h

πa,tRe2cosϕΔλeΔϕ( )i, j

×Fi−12, j

θv,i−1, j +θv,i, j2 −Fi+12, j

θv,i,j +θv,i+1, j2

+Gi, j−12θv,i, j−1+θv,i, j

2 −Gi,j+12θv,i, j +θv,i, j+1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ k,t−h

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

+ πa,tRe2cosϕΔλeΔϕ˙ σ tθv,t−h( )k−12− ˙ σ tθv,t−h( )k+12

Δσk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ i, j

+ πaRe2cosϕΔλeΔϕ ∇z •ρaKh∇z( )θvρa

+ θvcp,dTv

dQndt

n=1

Ne,h

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎛

⎝ ⎜ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ⎟ i, j,k,t−h

⎫ ⎬ ⎪

⎭ ⎪

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

v v

vv

v

v

v

v

G

G

F F

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

ConservationKinetic energy (7.28)

Absolute vorticity (7.29)

Enstrophy (7.30)

KE=12 ρaV u2+v2( )

ζa,z =ζr + f =∂v∂x−∂u

∂y+f

ENST=12 ζa,z2

Fig. 7.3-2 10

-9

-1 10

-9

0 10

0

1 10

-9

2 10

-9

0 1200 2400 3600

Δζ /initial ζ

Δ ENST/initial ENST

Δ KE/initial KE

Relative error

Time from σtart (σ)

Rel

ativ

e er

ror

West-East Momentum Equation

Fig. 7.4j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum EquationContinuous form (7.31)

Re2cosϕ ∂∂t πau( )⎡

⎣ ⎢ ⎤ ⎦ ⎥ σ

+ ∂∂λe

πau2Re( ) + ∂∂ϕ πauvRecosϕ( )⎡

⎣ ⎢ ⎤ ⎦ ⎥ σ

+πaRe2cosϕ ∂∂σ

˙ σ u( )

=πauvResinϕ+πa fvRe2cosϕ−Re πa∂Φ∂λe

+σcp,dθv∂P∂σ

∂πa∂λe

⎛ ⎝ ⎜

⎞ ⎠ ⎟ σ

+Re2cosϕ πaρa

∇ •ρaKm∇( )u

West-East Momentum EquationTime-difference term (7.32)

ui+12,j,k,t =πa,t−hΔA( )i+12, jπa,tΔA( )i+12, j

ui+12, j,k,t−h + hπa,tΔA( )i+12,j

×⎧ ⎨ ⎪

⎩ ⎪

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papapa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum EquationColumn pressure multiplied by grid-cell area at u-point (7.38)

Grid-cell area (7.39)

πaΔA( )i+12, j =18

πaΔA( )i, j+1+ πaΔA( )i+1, j+1

+2 πaΔA( )i,j + πaΔA( )i+1,j⎡ ⎣

⎤ ⎦

+ πaΔA( )i,j−1+ πaΔA( )i+1, j−1

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

ΔA =Re2cosϕΔλeΔϕj+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papapa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum EquationHorizontal advection terms (7.33)

Bi, jui−12,j +ui+12, j

2 −Bi+1, jui+12,j +ui+3 2, j

2+Ci+12,j−12

ui+12, j−1+ui+12, j2 −Ci+12, j+12

ui+12, j +ui+12, j+12

+Di,j−12ui−12, j−1+ui+12, j

2 −Di+1, j+12ui+12, j +ui+32, j+1

2

+Ei+1, j−12ui+32, j−1+ui+12, j

2 −Ei,j+12ui+12, j +ui−12,j+1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ k,t−h

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum EquationInterpolations for fluxes (7.41-7.44)

Bi, j = 112 Fi−12, j−1+Fi+12, j−1+2 Fi−12, j +Fi+12, j( )+Fi−12, j+1+Fi+12, j+1[ ]

Di, j+12 = 124Gi,j−12+2Gi, j+12 +Gi, j+32 +Fi−12, j +Fi−12, j+1+Fi+12, j +Fi+12, j+1( )

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum EquationVertical transport of horizontal momentum (7.34)

+ 1Δσk

πa,tΔA˙ σ k−12,tuk−12,t−h−πa,tΔA˙ σ k+12,tuk+12,t−h( )i+12,j

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

ui+12,j,k+12,t−h =Δσk+1ui+12,j,k,t−h+Δσkui+12, j,k+1,t−h

Δσk +Δσk+1

U-values at bottom of layer (7.45)

West-East Momentum EquationInterpolation for vertical velocity term (7.40)

πa,tΔA ˙ σ k−12,t( )i+12,j =18

πa,tΔA ˙ σ k−12,t( )i, j+1+ πa,tΔA ˙ σ k−12,t( )i+1, j+1

+2 πa,tΔA ˙ σ k−12,t( )i, j + πa,tΔA˙ σ k−12,t( )i+1, j⎧ ⎨ ⎩

⎫ ⎬ ⎭ +πa,tΔA ˙ σ k−12,t( )i, j−1+ πa,tΔA ˙ σ k−12,t( )i+1, j−1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papapa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum EquationCoriolis and spherical grid conversion terms (7.35)

πa,i, jvi, j−12 +vi, j+12

2 fjRecosϕ j +ui−12, j +ui+12,j2 sinϕj

⎛ ⎝ ⎜

⎞ ⎠ ⎟

+πa,i+1,jvi+1, j−12 +vi+1, j+12

2 fjRecosϕ j +ui+12, j +ui+32,j2 sinϕ j

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ k,t−h

+Re ΔλeΔϕ( )i+12,j

2 ×

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papapa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum EquationPressure gradient terms (7.36)

−ReΔϕi+12, j

Φi+1,j,k−Φi,j,k( )πa,i,j +πa,i+1, j

2 + πa,i+1, j −πa,i, j( )

×cp,d2

θv,kσk+12 Pk+12−Pk( ) +σk−12 Pk−Pk−12( )

Δσk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ i, j

+ θv,kσk+12 Pk+12−Pk( )+σk−12 Pk−Pk−12( )

Δσk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ i+1, j

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ t−h

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum Equation

−ReΔϕI+12, j

Φ I, j,k,t−2h −Φ I, j,k,t−h( )πa,I , j,t−h+ πa,I , j,t−2h−πa,I, j,t−h( )

×cp,d θv,kσk+12 Pk+12 −Pk( )+σk−12 Pk −Pk−12( )

Δσk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ I, j,t−h

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

Boundary conditions for pressure-gradient term (7.46)

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

West-East Momentum EquationEddy diffusion terms (7.37)

+ πa,t−hΔA( )i+12, j∇z •ρaKm∇z( )u

ρa⎡ ⎣ ⎢

⎤ ⎦ ⎥ i+12, j,k,t−h

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎫ ⎬ ⎪

⎭ ⎪

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papapa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

vv

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

C

B

C

B

E

ED

D

South-North Momentum Equation

j+1

j+1/2

j

j-1/2

j-1

j-3/2i-3/2

i-1 i-1/2 i i+1/2 i+1

papa

pa

papa

pa

pa papa

v

u

uu

u

uu u

u

u

v

v

vv

v

v

v

v

j-1

j-1/2

j

j+1/2

j+1

u

u

u

vvv

i-3/2j+3/2 j+3/2i+1

i+3/2

i+3/2ii-1/2i-1 i+1/2

Q

R

Q

T

T

S

SR

Fig. 7.5

Vertical Momentum EquationHydrostatic equation

Geopotential at vertical center of bottom layer (7.61)

Geopotential at bottom of subsequent layers (7.62)

dΦ =−cp,dθvdP

Φi, j,NL ,t−h =Φi,j,NL +12 −cp,d θv,NL PNL −PNL +12( )[ ]i, j,t−h

Φi, j,k+12,t−h =Φi, j,k+1,t−h −cp,d θv,k+1 Pk+12 −Pk+1( )[ ]i, j,t−h

Φi, j,k,t−h =Φi, j,k+12,t−h −cp,d θv,k Pk−Pk+12( )[ ]i, j,t−h

Geopotential at vertical midpoint of subsequent layers (7.63)

Time-Stepping Schemes

Matsuno scheme (7.65-6)Explicit forward difference to estimate final value followed by second forward difference to obtain final value

Time derivative of an advected species (7.64)

Leapfrog scheme (7.67)

∂q∂t = f q( )

qest=qt−h+hf qt−h( ) qt =qt−h +hf qest( )

qt+h =qt−h +2hf qt( )L2 L4 L6

L3 L5M1

L2 L4 L6

L3 L5M1

Fig. 7.6