Presentation Slides for Chapter 7 of Fundamentals of Atmospheric Modeling 2 nd Edition
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Transcript of Presentation Slides for Chapter 7 of Fundamentals of Atmospheric Modeling 2 nd Edition
Presentation Slides for
Chapter 7of
Fundamentals of Atmospheric Modeling 2nd Edition
Mark Z. JacobsonDepartment of Civil & Environmental Engineering
Stanford UniversityStanford, CA [email protected]
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