Presentation Slides
for
Chapter 5
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
jacobson@stanford.edu
March 10, 2005
Altitude Coordinate Surfaces
Fig. 5.1
Equation for Nonhydrostatic Pressure
Decompose pressure into large-scale and perturbation term (5.1)
pa pa p
ˆ a
Large-scale atmosphere in hydrostatic balance (5.2)
1 pa
ˆ pˆ
ˆ a a g
ˆ a z
z
Decompose gravitational and pressure gradient term (5.3)
1 pa
ˆ a pa pa a pa a g
g g a ˆ ˆ
a z z z a
ˆ
Substitute (5.3) into vertical momentum equation (5.4)
w w w
w ˆ p
a ag
u v w a
t x y z z a
ˆ
Equation for Nonhydrostatic Pressure
Take grad dot the sum of (5.4), (4.73), and (4.74) (5.5)
ˆ a v v ˆ a fk v
vˆ a
t
2ˆ 2
a
z pa p g 2
a
z a
ˆ
Note that (5.6)
a
c v,d pa
v
a
ˆ ˆ
v c p,d paˆ
Remove local derivative from continuity equation (5.7)
--> Anelastic continuity equation
v 0
ˆa
Equation for Nonhydrostatic Pressure
Substitute (5.6) and (5.7) into (5.5) (5.8)
--> Diagnostic equation for nonhydrostatic pressure
cv,d p
2
p g
a ˆ a
c p,d z ˆ
a v v fk v
pa
ˆ a ˆ a
ˆ p
2ˆ
z pa g a
K m
ˆa v
z ˆ
p
Pressure Coordinate Surfaces
Fig. 5.2
Intersections of z and p Surfaces
Fig. 5.3
Change in mass mixing ratio over distance (5.9)
q 2 q 3 q1 q 3 p2 p1 q1 q 2
x2 x1
x2 x1 x2 x1 p1 p2
z to p Coord. Gradient Conversion
Change in mass mixing ratio over distance (5.9)
q 2 q 3 q1 q 3 p2 p1 q1 q 2
x2 x1
x2 x1 x2 x1 p1 p2
Approximate differences as x2-x1-->0, p1-p2-->0 (5.10)
q q2 q3 q q1 q 3
x z x2 x1 x p x 2 x1
pa p2 p1 q q1 q 2
x z x 2 x1 pa x p1 p2
Gradient conversion from the z to p coordinate (5.11)
q q pa q
x z x p x z pa x
z to p Coord. Gradient Conversion
General equations (5.12)
pa
x z x p x z pa x
Substitute time for distance (5.15)
pa
t z t p t z pa t
Gradient conversion altitude to pressure coordinate (5.13)
z p z pa
pa
z to p Coord. Gradient Conversion
Gradient conversion altitude to pressure coordinate (5.13)
z p z pa
pa
Horizontal gradient operator in the altitude coordinate (4.81)
z i j
x z y
z
Horizontal gradient operator in the pressure coordinate (5.14)
p i j
x p y
p
Geopotential Gradient
Take gradient conversion of geopotential
z p z pa and note that z 0
pa
Rearrange gradient conversion (5.16)
pa pa
z pa p p a p
gz
--> pressure gradient proportional to altitude gradient (5.17)
pa pa
a a
x z x p y z y p
p Coordinate Continuity Eq. for Air
Continuity equation for air in the altitude coordinate (3.20)
a
a v v a
t
Expand with horizontal operators (5.18)
a w a
a z vh v h z a w
t z z z
Gradient conversion of velocity (5.19)
v h
z vh p vh z pa
pa
Substitute gradient conversion and hydrostatic equation (5.20)
a
a p vh z pa vh v g w a
h z a a
t z pa pa
p Coordinate Continuity Eq. for Air
Vertical scalar velocity in the pressure coordinate (5.21)
dpa pa pa pa
wp v pa v h z pa w
dt t z t z z
Substitute ∂pa/∂z=-ag (+wp downward, +w upward) (5.22)
z
w p ag v h z pa wag
t z
Differentiate vertical velocity with respect to altitude (5.23)
w p a v h pa w a
g z pa v h z g
z t z z z z
Substitute dz=-dpa/ag (5.24)
w p a vh wa
a a z pa vh z a a g
pa t z pa pa
p Coordinate Continuity Eq. for Air
From two pages back (5.20)
a
a p vh z pa vh v g w a
h z a a
t z pa pa
From previous page (5.24)
w p a vh wa
a a z pa vh z a a g
pa t z pa pa
Add (5.20) and (5.24) (5.25)
w p
p vh 0
pa
p Coordinate Continuity Eq. for Air
Expanded continuity equation (5.26)
u v w p
0
x y p pa
Example 5.1
x = 5 km y = 5 km pa = -10 hPa
u1 = -3 (west) u2 = -1 m s-1 (east)
v3 = +2 (south) v4 = -2 m s-1 (north)
wp,5 = +0.02 hPa s-1 (lower boundary)
-->
1 3 m s1 2 2 m s1
w p,6 0.02 hPa s1
0
5000 m 5000 m 10 hPa
--> wp,6 = +0.016 hPa s-1 (downward)
Total Derivative in p Coordinate
Total derivative in Cartesian-altitude coordinate (5.27)
d
v h z w
dt t z z
Time derivative and gradient operator conversions (5.15, 13)
pa
t z t p t z pa t
z p z pa
pa
Substitute conversions into total derivative (5.28)
d pa
dt t p t z pa
vh p vh z pa
pa
w
z
Total Derivative in p Coordinate
Total time derivative (5.28)
d pa
dt t p t z pa
vh p vh z pa
pa
w
z
Vertical velocity in altitude coordinate from (5.21) (5.29)
pa
v h z pa w p
t z
w
ag
Substitute (5.29) and hydrostatic equation into (5.28) (5.30)
--> total derivative in Cartesian-pressure coordinates
d
dt t p
vh p w p
pa
p Coordinate Species Cont. Equation
Species continuity equation in the altitude coordinate
N e, t
dq a Kh q
dt
a
Rn
n1
Apply Cartesian-pressure coordinate total derivative (5.31)
Ne, t
q a Kh q
dq q
dt t p
vh p q w p
pa
a
Rn
n1
Convert mass mixing ratio to number concentration (5.32)
Nm
q
aA
p Coordinate Thermo. Energy Eq.
Thermodynamic energy equation in the altitude coordinate
Ne,h
d v aK h v v dQ n
c p,d Tv dt
dt a
n1
Apply Cartesian-pressure coordinate total derivative (5.34)
N
v aK h v
e,h
v
t p
v h p v w p
pa
a
v
c p,d Tv dt
dQ n
n1
p Coordinate Horiz. Momentum Eq.
Horizontal momentum equation in the altitude coordinate
dv h
fk vh
1
z pa
aK mv h
dt a a
Substitute from (5.16)
z pa a p
Apply Cartesian-pressure coordinate total derivative (5.35)
aKm vh
v h
vh p v h w p
t p
v h
pa
fk v h p
a
p Coordinate Vert. Momentum Eq.
Assume hydrostatic equilibrium
pa
a g
z
Substitute
g z pa a R v
T Tv v P
--> hydrostatic equation in the pressure coordinate (5.37)
R v
T Rv P R v p k
a
pa pa pa pa 1000 hP a
Substitute k=R’/cp,d for final hydrostatic equation (5.38)
p k
d c p,d vd
a c p,d v dP
1000 hP a
Geostrophic Wind in p Coordinate
Substitute (5.17)
pa pa
a a
x z x p y z y p
into (4.79)
1 pa 1 pa
vg ug
fa x f a y
--> Geostrophic wind in the pressure coordinate (5.39)
1 1
v g ug
f x p f y
p
Vector form (5.40)
1
v g iug jv g i j
f y
1 1
f x p f
k p
p
Geostrophic Wind on a Surface of
Constant Pressure
Fig. 5.4
Sigma-Pressure Coordinate Surfaces
Fig. 5.5
The Sigma-Pressure Coordinate
Definition of a sigma level (5.41)
pa pa,top pa pa,top
pa,surf pa,top a
Pressure difference between column surface and top
a pa,surf pa,top
Pressure at a given sigma level (5.42)
pa pa,top a
Intersections of p, z and -p Surfaces
Fig. 5.7
Change in mixing ratio per unit distance (5.51)
q1 q3 q 2 q 3 1 2 q1 q 2
x 2 x1 x 2 x1 x2 x1 1 2
Gradient Conversion p to -p Coord.
Change in mixing ratio per unit distance (5.51)
q1 q3 q 2 q 3 1 2 q1 q 2
x 2 x1 x 2 x1 x2 x1 1 2
Gradient conversion from p to -p coordinate (5.52)
q q q
x p x x p x
Generalize (5.53)
p p
Gradient Conversion p to -p Coord.
Gradient conversion (5.53)
p p
Gradient of sigma along surface of constant pressure (5.54)
1 p pa ptop
p pa ptop p
a a
a
p a
Where
p ptop 0 p pa 0
pa a z a
Substitute (5.54) into (5.53) (5.55)
p a
a
-p Coord. Continuity Eq. for Air
Continuity equation for air in the pressure coordinate
w p
p vh 0
pa
Gradient conversion from p to -p coordinate (5.55)
p a
a
Substitute gradient conversion and ∂pa/∂=a (5.56)
v h 1 w p
v h a 0
a a
p coordinate vertical velocity, where pa = pa,top+ a (5.58)
dpa d a d d a
wp a Ý
a
dt dt dt dt
-p Coord. Vertical Velocity
p coordinate vertical velocity (5.58)
dpa d a d d a
wp a Ý
a
dt dt dt dt
Sigma-pressure coordinate vertical velocity (+ is down)(5.57)
d
Ý
dt
-p Coord. Continuity Eq. for Air
Material time derivative in the -p coordinate (5.59)
d
vh Ý
dt t
p coordinate vertical velocity (5.58)
d a
wp Ý
a
dt
Total derivative of a (note that ∂a/∂=0)
da a
v h a Ý a
dt t
Substitute total derivative of a into (5.58) (5.60)
a
w p v h a a
Ý
t
Take partial derivative (5.61)
w p a v h
Ý
v h a a a
t
-p Coord. Continuity Eq. for Air
Partial derivative of vertical scalar velocity (5.61)
w p a v h
Ý
v h a a a
t
Gradient conversion previously derived (5.56)
v h 1 w p
v h a 0
a a
Substitute (5.61) into (5.56) (5.62)
--> continuity equation for air in -p coordinate
a
Ý
vh a a 0
t
Convert to spherical-sigma-pressure coordinates (5.63)
a
Ý
2
Re cos u a Re v aRe cos a Re cos 0
2
t e
Column Pressure
Continuity equation for air (5.62)
a
Ý
vh a a 0
t
Rearrange and integrate (5.64)
1 1 0
a d vh a d a d
0 t 0 0
Ý
Prognostic equation for column pressure (5.65)
a 1
v h a d
t 0
Analogous equation in spherical--p coordinates (5.66)
a 1
2
Re cos
t 0 e
u a Re v a Re cos d
Vertical Scalar Velocity
Continuity equation for air (5.62)
a
Ý
vh a a 0
t
Rearrange and integrate (5.67)
Ý
a d v h a d
0
Ý
0 0 t
a d
Diagnostic equation for vertical velocity (5.68)
a
Ý
a 0
vh ad t
Analogous equation in spherical--p coordinates (5.69)
a
Ý 2
a Re cos
e
0
u aRe va Re cos d Re cos
2
t
-p Coord. Species Continuity Eq.
Species continuity equation in Cartesian-z coordinates (3.54)
Ne, t
Rn
q 1
v q a K h q
t a
n1
Material time derivative is sigma-pressure coordinate
d
Ý
vh
dt t
--> Continuity equation in Cartesian--p coordinates (5.70)
N e,t
q a Kh q
Rn
dq q
v h q
Ý
dt t a
n1
-p Coord. Species Continuity Eq.
Combine species and air continuity equations (5.72)
a q Ne,t
Ý
q aK hq R
t
v h a q a
a
a n
n1
Apply spherical-coordinate transformations (5.73)
Re cos a q
2
u aqRe v a qRe cos
t e
N e,t
a Kh q
2
a Re cos
Ý
q a Re cos
2
a
Rn
n1
-p Coord. Thermodynamic En. Eq.
In Cartesian-altitude coordinates (3.76)
N e,h
v 1 dQ n
v v a K h v v
t a c p,d T dt
n1
Apply the -p coordinate material time derivative (5.74)
Ne, h
Ý v
a K h v
v
v dQ n
vh v
t a c p,d Tv dt
n1
-p Coord. Thermodynamic En. Eq.
Combine with continuity equation for air (5.75)
a v
v
Ý
v h a v a
t
Ne ,h
aKh v v
a
a c p,d Tv dQ n
dt
n1
Apply spherical-coordinate transformations (5.76)
Re cos a v ua v Re vav Re cos
2
t
e
Ne ,h
aK h v v
2
a Re cos
Ý
v aRe cos
2
a c p,d Tv
dQ n
dt
n1
-p Coord. Momentum Equation
In Cartesian-altitude coordinates (4.70)
dv 1 1
dt
fk v
a
2
pa a v
a a
a Km v
Material time derivative of velocity
dv h vh v
vh vh
Ý h
dt t
Apply to horizontal momentum equation (5.77)
v h v 1 aK mv h
v h vh
Ý h fk vh z pa
t a a
-p Coord. Momentum Equation
Pressure gradient term (5.78)
1
z pa p a
a a
Substitute into momentum equation (5.79)
v h v
v h vh
Ý h fk vh
t
a Kmvh
a
a a
Coupling Hor./Vert. Momentum Eqs.
Hydrostatic equation in the pressure coordinate (5.80)
a RTv a
a a
pa a
Re-derive specific density (5.82)
RTv kc p,d v P P c p,d v P
a c p,d v
pa pa pa a
Coupling Hor./Vert. Momentum Eqs.
Combine terms above with momentum/continuity eqs. (5.83)
v h a
Ý
vh vh a a vh v h a v h
t
P aK mv h
a fk vh a c p,d v a a a
Now horizontal and vertical equations consistent (5.38)
k
p
d c p,d vd
a c p,d v dP
1000 hP a
-p Coord. Momentum Equation
U-direction momentum equation (5.86)
Ý
Re cos au
2 2
a u Re auvRe cos aRe cos u
2
t e
2 P a
a uvRe sin a fvRe cos Re a c p,d v
e e
Re co s a a Km u
2
a
V-direction momentum equation (5.87)
Re cos a v
2 2
auvRe v a Re cos a Re cos v
2 Ý
t e
2 2 P a
a u Re sin a fuRe cos Re cos a c p,d v
Re co s a a Km v
2
a
Sigma-Altitude Coordinate
Sigma-altitude value (5.89)
ztop z ztop z
s
ztop zsurf Zt
Altitude difference between column top and surface
Zt ztop z surf
Altitude of a sigma surface (5.90)
z ztop Zts
Gradient Conversion
Gradient conversion between z and s-z coordinate (5.91)
z s z s
s
Horiz. gradient of sigma along const. altitude surface (5.92)
ztop z s
z s 2 z Zt Z z Zt
Zt t
Substitute into gradient conversion (5.93)
s
z s z Zt
Zt s
Conversions in -z Coordinate
Time-derivative conversion between z and s-z coordinate (5.94)
t z t s
Scalar velocity in the sigma-altitude coordinate (5.95)
ds s w
s
Ý v h z s w v h z s
dt z Zt
where
s 1
z Zt
Material time derivative in the sigma-altitude coordinate (5.96)
d
v h s sÝ
dt t s s
-z Coord. Continuity Eq. For Air
Continuity equation for air in the z coordinate
a w a
a z vh v h z a w
t z z z
Apply gradient conversion to horizontal velocity
v h
z vh s vh z s
s
Apply gradient conversion to air density
a
z a s a z s
s
Substitute these terms into continuity equation above (5.97)
a v h w a a
a s vh z s
t s
s
v h s a z s s w z
z
-z Coord. Continuity Eq. For Air
Rewrite vertical velocity w Zt v h z s s
Ý
Differentiate with respect to altitude (5.98)
w vh s s
Ý
Z t z s
vh z
z z z z
Substitute ∂s/∂z=-1/Zt (5.99)
w Ý
s vh 1
z s
z s
s
Zt
v h z Z t
Sub. w Zt v h z s s, (5.99), ∂s/∂z=-1/Zt into (5.97) (5.100)
Ý
a s 1
Ý
a s vh
t s
s Zt
vh z Zt v h s a Ý sa
s
Non/Hydrostatic Continuity Eq.
Substitute z Zt s Zt and compress --> (5.101)
Nonhydrostatic continuity equation for air in s-z coordinate
a 1
s v h a Zt Ý a s
t s Zt s
1 ua Zt v a Z t s a
Ý
Z t x y s
s
Hydrostatic equation in the s-z coordinate (5.102)
a 1 p
1 p a
a
g z Z t g s
Substitute into (5.101) --> Hydrostatic continuity eq. (5.103)
pa
p pa
a
s v h s
Ý
t s s s s
-z Coord. Species Continuity Eq.
Apply material derivative in the s-z coordinate to the continuity
equation for a trace species in the z coordinate (5.104)
N e,t
q a Kh q
dq q
vh s q s
dt s t s
Ý
s a
Rn
n1
-z Coord. Thermodynamic En. Eq.
Apply material derivative in the s-z coordinate to the thermodynamic
energy equation in the z coordinate (5.106)
Ne ,h
v a K h v
v
v dQ n
vh s v s
Ý
t s s a c p,d Tv dt
n1
-z Coord. Horiz. Momentum Eqs.
Horizontal equation in the z coordinate
dv h
fk vh
1
z pa
aK mv h
dt a a
Apply material time derivative of velocity
v h v 1 (5.107
z aKm z v h
) v h s v h s h fk vh
Ý z pa
t s s a a
Gradient conversion of pressure (5.108)
s pa
z pa s pa Z
Z t z t s
Substitute gradient conversion (5.109)
v h v
v h s v h s h
Ý
t s s
1 s pa
fk v h
a s pa Z z Zt s a Km vh
t
-z Coord. Vertical Momentum Eq.
Sub. ∂s/∂z=-1/Zt into z-coord. vertical momentum eq. (5.113)
w w w w 1 pa a K m w
u v Ý g
s
t x y s s Z t a s a
Substitute
w Zt v h z s s
Ý
Another form of vertical momentum equation (5.114)
s s 1 pa
u v s t u Z t v Z t s g
Ý Z Ý
t s
x s y s s x z
y z
Z t a s
1 s s
a
a Km Z t ux Z t v y Z t s
Ý
z
z