presentation slides for chapter 7 of fundamentals of atmospheric modeling 2 nd edition mark z....
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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
, θv , 1
, u1
, v1
, pa , 1
_____________________ ˙ σ
1 + 1 2, σ
1 + 1 2, p
a , 1 + 1 2
_ _ _ _ _ _ _ _ _ _ _ _ _ q2
, θv , 2
, u2
, v2
, pa , 2
_____________________ ˙ σ
k − 1 2, σ
k − 1 2, p
a , k − 1 2
_ _ _ _ _ _ _ _ _ _ _ _ _ qk
, θv , k
, uk
, vk
, pa , k
_____________________ ˙ σ
k + 1 2, σ
k + 1 2, p
a , k + 1 2
_ _ _ _ _ _ _ _ _ _ _ _ _ qk + 1
, θv , k + 1
, uk + 1
, vk + 1
, pa , k + 1
_____________________ ˙ σ
NL
− 1 2, σ
NL
− 1 2, p
a , NL
− 1 2
_ _ _ _ _ _ _ _ _ _ _ _ _ qN
L
, θv , N
L
, uN
L
, vN
L
, pa , N
L
_Model botto m boundary_ ˙ σ
NL
+ 1 2= 0 , σ
NL
+ 1 2= 1 , p
a , surf
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 Levels
Sigma 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,top
Sigma thickness of layer (7.1)
Δσk =σk+12 −σk−12
Layer Midpoint Pressure
Example layers
____________________________ pa , k − 1 2
= 700 hPa
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ θv , k
= 308 K
____________________________ pa , k + 1 2
= 750 hPa
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ θv , 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 θv increases monotonically with height
Layer Midpoint Pressure
Mass-weighed mean of P (7.8)
Value of P at boundaries (7.9)
Consistent formula for θv at boundaries (7.11)
pa,k = 1000 hPa( )Pk1κ
Pk =1
pa,k+12 −pa,k−12Pd
pa,k−12
pa,k+12
∫ pa =1
1+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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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
=−∂
∂λeuπaRe( )+
∂∂ϕ
vπaRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥ σdσ
0
1
∫
Re2cosϕΔλeΔϕ( )i, j
πa,t −πa,t−hh
⎛
⎝ ⎜
⎞
⎠ ⎟ 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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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
2uReΔϕ( )i+12, j,k
⎡
⎣ ⎢ ⎤
⎦ ⎥ t−h
Gi,j+12,k,t−h =πa,i, j +πa,i, j+1
2vRecosϕΔλe( )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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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−h
Re2cosϕΔλeΔϕ( )
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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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
dσ0
σ
∫ −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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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 Velocity
Substitute fluxes and rearrange --> vertical velocity (7.21)
˙ σ i, j,k+12,t =−1
πaRe2cosϕΔλeΔϕ( )
i, j,t
× Fi+12, j −Fi−12, j +Gi,j+12−Gi, j−12( )l,t−hΔσl
⎡ ⎣ ⎢
⎤ ⎦ ⎥
l=1
k
∑
−σk+12πa,t −πa,t−h
hπa,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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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
+∂
∂λeuπaqRe( )+
∂∂ϕ
vπaqRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥ σ
+πaRe2cosϕ
∂∂σ
˙ σ q( ) =πaRe2cosϕ
∇ •ρaKh∇( )q
ρa+ Rnn=1
Ne,t
∑⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
Re2cosϕΔλeΔϕ( )i, j
πa,tqt −πa,t−hqt−hh
⎛
⎝ ⎜
⎞
⎠ ⎟ 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−12Δσk
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ i, j
= πaRe2cosϕΔλeΔϕ
∇z•ρaKh∇z( )q
ρa+ 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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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, j
qi−1, j +qi,j2
−Fi+12, jqi,j +qi+1,j
2
+Gi, j−12qi, j−1+qi, j
2−Gi, j+12
qi, j +qi, j+1
2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ k,t−h
⎧
⎨ ⎪ ⎪
⎩
⎪ ⎪
+ πa,tRe2cosϕΔλeΔϕ
˙ σ tqt−h( )k−12 − ˙ σ tqt−h( )k+12Δσk
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥ i, j
+ πaRe2cosϕΔλeΔϕ
∇z •ρaKh∇z( )q
ρa+ Rnn=1
Ne,t
∑⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ i, j,k,t−h
⎫
⎬ ⎪
⎭ ⎪
qi, j,k−12 =lnqi, j,k−1−lnqi,j,k
1 qi, j,k( )− 1 qi, j,k−1( )qi, j,k+12 =
lnqi, j,k−lnqi, j,k+1
1 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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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
+∂
∂λeuπaθvRe( )+
∂∂ϕ
vπaθvRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥
+πaRe2cosϕ
∂∂σ
˙ σ θv( )=πaRe2cosϕ
∇ •ρaKh∇( )θvρa
+θv
cp,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, jθv,i,j +θv,i+1, j
2
+Gi, j−12θv,i, j−1+θv,i, j
2−Gi,j+12
θv,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
+θv
cp,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
πaπa
πa
πaπa
πa
πa πaπa
v
u
uu
u
uu u
u
u
vv
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 start s
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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
D
West-East Momentum Equation
Continuous form (7.31)
Re2cosϕ
∂∂t
πau( )⎡ ⎣ ⎢
⎤ ⎦ ⎥ σ
+∂
∂λeπau
2Re( ) +∂∂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, j
πa,tΔA( )i+12, j
ui+12, j,k,t−h +h
πa,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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
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 =Re2cosϕΔλeΔϕ
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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
D
West-East Momentum EquationHorizontal advection terms (7.33)
Bi, jui−12,j +ui+12, j
2−Bi+1, j
ui+12,j +ui+3 2, j
2
+Ci+12,j−12ui+12, j−1+ui+12, j
2−Ci+12, j+12
ui+12, j +ui+12, j+12
+Di,j−12ui−12, j−1+ui+12, j
2−Di+1, j+12
ui+12, j +ui+32, j+12
+Ei+1, j−12ui+32, j−1+ui+12, j
2−Ei,j+12
ui+12, j +ui−12,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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
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 =124
Gi,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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
D
West-East Momentum EquationVertical transport of horizontal momentum (7.34)
+1
Δσkπa,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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
D
West-East Momentum EquationCoriolis and spherical grid conversion terms (7.35)
πa,i, jvi, j−12 +vi, j+12
2fjRecosϕ j +
ui−12, j +ui+12,j
2sinϕj
⎛
⎝ ⎜
⎞
⎠ ⎟
+πa,i+1,jvi+1, j−12 +vi+1, j+12
2fjRecosϕ j +
ui+12, j +ui+32,j
2sinϕ 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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
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,d
2
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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
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
πaπa
πa
πaπa
πa
πa πaπa
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
E
D
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
πaπa
πa
πaπa
πa
πa πaπa
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