presentation slides for chapter 5 of fundamentals of atmospheric modeling 2 nd edition
DESCRIPTION
Presentation Slides for Chapter 5 of Fundamentals of Atmospheric Modeling 2 nd Edition. Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA 94305-4020 [email protected] March 10, 2005. Altitude Coordinate Surfaces. Fig. 5.1. - PowerPoint PPT PresentationTRANSCRIPT
Presentation Slides for
Chapter 5of
Fundamentals of Atmospheric Modeling 2nd Edition
Mark Z. JacobsonDepartment of Civil & Environmental Engineering
Stanford UniversityStanford, CA [email protected]
March 10, 2005
Altitude Coordinate Surfaces
Fig. 5.1
Δz5z5z6z4z3z2z1
Decompose pressure into large-scale and perturbation term (5.1)
Equation for Nonhydrostatic Pressure
Large-scale atmosphere in hydrostatic balance (5.2)
Decompose gravitational and pressure gradient term (5.3)
Substitute (5.3) into vertical momentum equation (5.4)
pa = ˆ p a + ′ ′ p a
1ˆ ρ a
∂ˆ p a∂z
=ˆ α a∂ˆ p a∂z
=−g
g +1ρa
∂pa∂z
=g+ ˆ α a + ′ ′ α a( )∂∂z
ˆ p a + ′ ′ p a( ) ≈ˆ α a∂ ′ ′ p a∂z
−′ ′ α a
ˆ α ag
∂w∂t
=−u∂w∂x
−v∂w∂y
−w∂w∂z
−ˆ α a∂ ′ ′ p a∂z
+′ ′ α a
ˆ α ag
Take grad dot the sum of (5.4), (4.73), and (4.74) (5.5)
Equation for Nonhydrostatic Pressure
Note that (5.6)
Remove local derivative from continuity equation (5.7)--> Anelastic continuity equation
∂∂t
∇ • vˆ ρ a( )=−∇• ˆ ρ a v•∇( )v[ ]−∇ • ˆ ρ a fk×v( )
−∇z2ˆ p a −∇2 ′ ′ p a +g
∂∂z
′ ′ α aˆ α a2
⎛
⎝ ⎜
⎞
⎠ ⎟
′ ′ α aˆ α a
≈′ ′ θ v
ˆ θ v−
cv,dcp,d
′ ′ p aˆ p a
∇ • vˆ ρ a( ) =0
Substitute (5.6) and (5.7) into (5.5) (5.8)
--> Diagnostic equation for nonhydrostatic pressure
Equation for Nonhydrostatic Pressure
∇2 ′ ′ p a−gcv,dcp,d
∂∂z
ˆ ρ a′ ′ p a
ˆ p a
⎛
⎝ ⎜
⎞
⎠ ⎟ =−∇• ˆ ρ a v•∇( )v[ ]−∇ • ˆ ρ a fk×v[ ]
−∇z2ˆ p a +g
∂∂z
ˆ ρ a′ ′ θ p
ˆ θ p
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ +∇ • ∇ •ˆ ρ aKm∇( )v
Pressure Coordinate Surfaces
Fig. 5.2
pa,2pa,1pa,3pa,4pa,5 pa,6
Δpa,1Δpa,1 pa,top
Intersections of z and p Surfaces Fig. 5.3
x
zp1p2z1
z2
x1 x2
q3 q2q1
Change in mass mixing ratio over distance (5.9) q2 −q3x2 −x1
=q1−q3x2 −x1
+p2 −p1x2 −x1
⎛
⎝ ⎜
⎞
⎠ ⎟
q1−q2p1−p2
⎛
⎝ ⎜
⎞
⎠ ⎟
Change in mass mixing ratio over distance (5.9)
z to p Coord. Gradient Conversion
Approximate differences as x2-x1-->0, p1-p2-->0 (5.10)
Gradient conversion from the z to p coordinate (5.11)
q2 −q3x2 −x1
=q1−q3x2 −x1
+p2 −p1x2 −x1
⎛
⎝ ⎜
⎞
⎠ ⎟
q1−q2p1−p2
⎛
⎝ ⎜
⎞
⎠ ⎟
∂q∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=
q2−q3x2 −x1
∂q∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p=
q1−q3x2 −x1
∂pa∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=
p2 −p1x2 −x1
∂q∂pa
⎛
⎝ ⎜
⎞
⎠ ⎟
x=
q1−q2p1−p2
∂q∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=
∂q∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+
∂pa∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z
∂q∂pa
⎛
⎝ ⎜
⎞
⎠ ⎟
x
General equations (5.12)
z to p Coord. Gradient Conversion
Substitute time for distance (5.15)
∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
=∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+
∂pa∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
∂∂pa
⎛
⎝ ⎜
⎞
⎠ ⎟
x
∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=
∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+
∂pa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z
∂∂pa
⎛
⎝ ⎜
⎞
⎠ ⎟ t
∇z =∇ p+∇z pa( )∂
∂pa
Gradient conversion altitude to pressure coordinate (5.13)
Horizontal gradient operator in the pressure coordinate (5.14)
z to p Coord. Gradient ConversionGradient conversion altitude to pressure coordinate (5.13)
∇ p =i∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+j
∂∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
p
∇z =∇ p+∇z pa( )∂
∂pa
∇z =i∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
+j∂∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ z
Horizontal gradient operator in the altitude coordinate (4.81)
Take gradient conversion of geopotential
Geopotential Gradient
and note that
Rearrange gradient conversion (5.16)
--> pressure gradient proportional to altitude gradient (5.17)
∇zΦ =∇ pΦ +∇z pa( )∂Φ∂pa
∇zΦ =0
∇z pa( ) =−∂pa∂Φ
∇pΦ =−∂pag∂z
∇ pΦ =ρa∇pΦ
∂pa∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=ρa
∂Φ∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p
∂pa∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
z=ρa
∂Φ∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
p
p Coordinate Continuity Eq. for Air
Expand with horizontal operators (5.18)
Gradient conversion of velocity (5.19)
Continuity equation for air in the altitude coordinate (3.20)
Substitute gradient conversion and hydrostatic equation (5.20)
∂ρa∂t
=−ρa ∇•v( )− v•∇( )ρa
∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=−ρa ∇z•vh +
∂w∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟ − vh•∇z( )ρa−w
∂ρa∂z
∇z•vh =∇p•vh +∇z pa( )•∂vh∂pa
∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=−ρa ∇p•vh+∇z pa( )•
∂vh∂pa
⎛
⎝ ⎜
⎞
⎠ ⎟ − vh•∇z( )ρa +ρag
∂ wρa( )∂pa
p Coordinate Continuity Eq. for Air
Substitute ∂pa/∂z=-ag (+wp downward, +w upward) (5.22)
Differentiate vertical velocity with respect to altitude (5.23)
Vertical scalar velocity in the pressure coordinate (5.21)
Substitute dz=-dpa/ag (5.24)
wp =dpadt
=∂pa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
+ v•∇( )pa =∂pa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
+ vh•∇z( )pa +w∂pa∂z
wp =− ρag∂z∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z+vh•∇z( )pa −wρag
∂wp∂z
=−g∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z+∇z pa( )•
∂vh∂z
+ vh•∇z( )∂pa∂z
−g∂ wρa( )
∂z
ρa∂wp∂pa
=∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
+ρa∇z pa( )•∂vh∂pa
+ vh •∇z( )ρa −ρag∂ wρa( )
∂pa
p Coordinate Continuity Eq. for Air
Add (5.20) and (5.24) (5.25)
From previous page (5.24)
ρa∂wp∂pa
=∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
+ρa∇z pa( )•∂vh∂pa
+ vh •∇z( )ρa −ρag∂ wρa( )
∂pa
∇ p•vh+∂wp∂pa
=0
∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=−ρa ∇p•vh+∇z pa( )•
∂vh∂pa
⎛
⎝ ⎜
⎞
⎠ ⎟ − vh•∇z( )ρa +ρag
∂ wρa( )∂pa
From two pages back (5.20)
p Coordinate Continuity Eq. for Air
Example 5.1
Expanded continuity equation (5.26)
Δx = 5 km Δy = 5 km Δpa = -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) -->
∂u∂x
+∂v∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
p+
∂wp∂pa
=0
−1+3( ) m s−1
5000 m+
−2−2( ) m s−1
5000 m+
wp,6 −0.02( ) hPa s−1
−10 hPa=0
--> wp,6 = +0.016 hPa s-1 (downward)
Total Derivative in p Coordinate
Substitute conversions into total derivative (5.28)
Time derivative and gradient operator conversions (5.15, 13)
Total derivative in Cartesian-altitude coordinate (5.27) ddt
=∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
+ vh•∇z( )+w∂∂z
ddt
=∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+
∂pa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z
∂∂pa
+ vh•∇p( )+ vh •∇z( )pa[ ]∂
∂pa+w
∂∂z
∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=
∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+
∂pa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z
∂∂pa
⎛
⎝ ⎜
⎞
⎠ ⎟ t
∇z =∇ p+∇z pa( )∂
∂pa
Total Derivative in p CoordinateTotal time derivative (5.28)
Vertical velocity in altitude coordinate from (5.21) (5.29)
Substitute (5.29) and hydrostatic equation into (5.28) (5.30) --> total derivative in Cartesian-pressure coordinates
ddt
=∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+
∂pa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z
∂∂pa
+ vh•∇p( )+ vh •∇z( )pa[ ]∂
∂pa+w
∂∂z
w =
∂pa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z+ vh•∇z( )pa −wp
ρag
ddt
=∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+ vh•∇p( )+wp
∂∂pa
p Coordinate Species Cont. Equation
Apply Cartesian-pressure coordinate total derivative (5.31)
Convert mass mixing ratio to number concentration (5.32)
Species continuity equation in the altitude coordinate
dqdt
=∇ •ρaKh∇( )q
ρa+ Rn
n=1
Ne,t
∑
dqdt
=∂q∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+ vh •∇p( )q +wp
∂q∂pa
=∇ •ρaKh∇( )q
ρa+ Rn
n=1
Ne,t
∑
q =NmρaA
p Coordinate Thermo. Energy Eq.
Apply Cartesian-pressure coordinate total derivative (5.34)
Thermodynamic energy equation in the altitude coordinate
dθvdt
=∇ •ρaKh∇( )θv
ρa+
θvcp,dTv
dQndt
n=1
Ne,h
∑
∂θv∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+ vh•∇ p( )θv +wp
∂θv∂pa
=∇•ρaKh∇( )θv
ρa+
θvcp,dTv
dQndt
n=1
Ne,h
∑
p Coordinate Horiz. Momentum Eq.
Apply Cartesian-pressure coordinate total derivative (5.35)
Horizontal momentum equation in the altitude coordinate
Substitute from (5.16)
dvhdt
=−fk×vh −1
ρa∇z pa( )+
∇ •ρaKm∇( )vhρa
∇z pa( ) =ρa∇pΦ
∂vh∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p+ vh •∇p( )vh +wp
∂vh∂pa
=−fk×vh −∇ pΦ+∇ •ρaKm∇( )vh
ρa
p Coordinate Vert. Momentum Eq.
--> hydrostatic equation in the pressure coordinate (5.37)
Assume hydrostatic equilibrium
Substitute
Substitute =R’/cp,d for final hydrostatic equation (5.38)
∂pa∂z
=−ρag
g =∂Φ ∂z pa =ρa ′ R Tv Tv =θvP
∂Φ∂pa
=−′ R Tvpa
=−′ R θvPpa
=−′ R θvpa
pa1000hPa
⎛
⎝ ⎜
⎞
⎠ ⎟
κ
dΦ =−cp,dθvdpa
1000hPa
⎛
⎝ ⎜
⎞
⎠ ⎟
κ⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
=−cp,dθvdP
Geostrophic Wind in p Coordinate
--> Geostrophic wind in the pressure coordinate (5.39)
Substitute (5.17)
into (4.79)
Vector form (5.40)
∂pa∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=ρa
∂Φ∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p
∂pa∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
z=ρa
∂Φ∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
p
vg =1fρa
∂pa∂x
ug =−1
fρa
∂pa∂y
vg =1f
∂Φ∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
pug =−
1f
∂Φ∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
p
vg =iug +jvg =−i1f
∂Φ∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
p+j
1f
∂Φ∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p=
1fk×∇ pΦ( )
Geostrophic Wind on a Surface of Constant Pressure
500 hPasurface Contour lineContour lineSouthNorth
West
East
510 hPa5.5 km500 hPa5.5 km
500 hPa5.6 km
Fig. 5.4
Sigma-Pressure Coordinate Surfaces
σ1=0
σ6=1σ5σ4σ3σ2
pa,topΔσ1, Δpa,1x Δσ1, Δpa,1y
Fig. 5.5
The Sigma-Pressure Coordinate
Pressure at a given sigma level (5.42)
Definition of a sigma level (5.41)
Pressure difference between column surface and top
σ =pa −pa,top
pa,surf −pa,top=
pa −pa,topπa
πa =pa,surf −pa,top
pa =pa,top+σπa
Intersections of p, z and σ-p Surfaces
Fig. 5.7x
z p1p2z1
z2
x1 x2q3
q2q1 σ1
σ2
Change in mixing ratio per unit distance (5.51) q1−q3x2−x1
=q2 −q3x2−x1
+σ1−σ2x2 −x1
⎛
⎝ ⎜
⎞
⎠ ⎟
q1−q2σ1−σ2
⎛
⎝ ⎜
⎞
⎠ ⎟
Gradient Conversion p to σ-p Coord.
Gradient conversion from p to σ-p coordinate (5.52)
Generalize (5.53)
Change in mixing ratio per unit distance (5.51) q1−q3x2−x1
=q2 −q3x2−x1
+σ1−σ2x2 −x1
⎛
⎝ ⎜
⎞
⎠ ⎟
q1−q2σ1−σ2
⎛
⎝ ⎜
⎞
⎠ ⎟
∂q∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p=
∂q∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
σ+
∂σ∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
p
∂q∂σ
⎛ ⎝ ⎜
⎞ ⎠ ⎟
x
∇ p =∇σ +∇ p σ( )∂∂σ
Gradient Conversion p to σ-p Coord.
Where
Substitute (5.54) into (5.53) (5.55)
Gradient of sigma along surface of constant pressure (5.54)
∇ p σ( ) = pa −ptop( )∇p1πa
⎛
⎝ ⎜
⎞
⎠ ⎟ +
∇p pa −ptop( )
πa=−
σπa
∇ p πa( )
∇ p pa( ) =0∇ p ptop( ) =0
∇ p πa( )=∇σ πa( ) =∇z πa( )
∇ p =∇σ −σπa
∇σ πa( )∂∂σ
Gradient conversion (5.53)
∇ p =∇σ +∇ p σ( )∂∂σ
σ-p Coord. Continuity Eq. for Air
p coordinate vertical velocity, where pa = pa,top+ aσ (5.58)
Continuity equation for air in the pressure coordinate
Substitute gradient conversion and ∂pa/∂σ=a (5.56)
∇ p•vh+∂wp∂pa
=0
∇σ •vh −σπa
∇σ πa( )•∂vh∂σ
+1πa
∂wp∂σ
=0
wp =dpadt
=σdπadt
+dσdt
πa =σdπadt
+˙ σ πa
∇ p =∇σ −σπa
∇σ πa( )∂∂σ
Gradient conversion from p to σ-p coordinate (5.55)
σ-p Coord. Vertical Velocity
p coordinate vertical velocity (5.58)
wp =dpadt
=σdπadt
+dσdt
πa =σdπadt
+˙ σ πa
Sigma-pressure coordinate vertical velocity (+ is down) (5.57)
˙ σ =dσdt
σ-p Coord. Continuity Eq. for Air
Take partial derivative (5.61)
Material time derivative in the σ-p coordinate (5.59)
Substitute total derivative of a into (5.58) (5.60)
ddt
=∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+ vh •∇σ( )+˙ σ ∂∂σ
wp =σ∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
σ+ vh•∇σ( )πa
⎡
⎣ ⎢
⎤
⎦ ⎥ +˙ σ πa
∂wp∂σ
=∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
σ+ vh•∇σ( )πa +σ∇σ πa( )•
∂vh∂σ
+πa∂ ˙ σ ∂σ
wp =σdπadt
+˙ σ πa
dπadt
=∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+ vh•∇σ( )πa +˙ σ ∂πa∂σ
p coordinate vertical velocity (5.58)
Total derivative of a (note that ∂a/∂σ=0)
σ-p Coord. Continuity Eq. for AirPartial derivative of vertical scalar velocity (5.61)
Substitute (5.61) into (5.56) (5.62) --> continuity equation for air in σ-p coordinate
Convert to spherical-sigma-pressure coordinates (5.63)
∂wp∂σ
=∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
σ+ vh•∇σ( )πa +σ∇σ πa( )•
∂vh∂σ
+πa∂ ˙ σ ∂σ
∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+∇σ • vhπa( )+πa∂ ˙ σ ∂σ
=0
Re2cosϕ
∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+∂
∂λeuπaRe( ) +
∂∂ϕ
vπaRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥ σ
+πaRe2cosϕ
∂ ˙ σ ∂σ
=0
Gradient conversion previously derived (5.56)
∇σ •vh −σπa
∇σ πa( )•∂vh∂σ
+1πa
∂wp∂σ
=0
Column Pressure
Prognostic equation for column pressure (5.65)
Continuity equation for air (5.62)
Analogous equation in spherical-σ-p coordinates (5.66)
Rearrange and integrate (5.64)
∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+∇σ • vhπa( )+πa∂ ˙ σ ∂σ
=0
∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
σ0
1
∫ dσ =−∇σ • vhπa( )dσ0
1
∫ −πa d˙ σ 0
0
∫∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
=−∇σ • vhπa( )dσ0
1
∫
Re2cosϕ
∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
=−∂
∂λeuπaRe( )+
∂∂ϕ
vπaRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥ σdσ
0
1
∫
Vertical Scalar Velocity
Diagnostic equation for vertical velocity (5.68)
Continuity equation for air (5.62)
Analogous equation in spherical-σ-p coordinates (5.69)
Rearrange and integrate (5.67)
∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+∇σ • vhπa( )+πa∂ ˙ σ ∂σ
=0
πa d˙ σ 0
˙ σ
∫ =−∇σ • vhπa( )dσ0
σ
∫ −∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
dσ0
σ
∫
˙ σ πa =−∇σ • vhπa( )dσ0
σ
∫ −σ∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
˙ σ πaRe2cosϕ =−
∂∂λe
uπaRe( )+∂∂ϕ
vπaRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥ σ
dσ0
σ
∫ −σRe2cosϕ
∂πa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
σ-p Coord. Species Continuity Eq.
--> Continuity equation in Cartesian-σ-p coordinates (5.70)
Species continuity equation in Cartesian-z coordinates (3.54)
Material time derivative is sigma-pressure coordinate
∂q∂t
+ v•∇( )q =1ρa
∇ •ρaKh∇( )q + Rnn=1
Ne,t
∑
ddt
=∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+vh•∇σ +˙ σ ∂∂σ
dqdt
=∂q∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+ vh•∇σ( )q+˙ σ ∂q∂σ
=∇ •ρaKh∇( )q
ρa+ Rn
n=1
Ne,t
∑
σ-p Coord. Species Continuity Eq.
Combine species and air continuity equations (5.72)
Apply spherical-coordinate transformations (5.73)
∂ πaq( )∂t
⎡
⎣ ⎢
⎤
⎦ ⎥ σ
+∇σ • vhπaq( ) +πa∂ ˙ σ q( )∂σ
=πa∇•ρaKh∇( )q
ρa+ Rn
n=1
Ne,t
∑⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
Re2cosϕ
∂∂t
πaq( )⎡ ⎣ ⎢
⎤ ⎦ ⎥ σ
+∂
∂λeuπaqRe( )+
∂∂ϕ
vπaqRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥ σ
+πaRe2cosϕ
∂∂σ
˙ σ q( ) =πaRe2cosϕ
∇ •ρaKh∇( )q
ρa+ Rn
n=1
Ne,t
∑⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
σ-p Coord. Thermodynamic En. Eq.
In Cartesian-altitude coordinates (3.76)
Apply the σ-p coordinate material time derivative (5.74)
∂θv∂t
+ v•∇( )θv =1ρa
∇•ρaKh∇( )θv +θv
cp,dTdQndt
n=1
Ne,h
∑
∂θv∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+ vh•∇σ( )θv +˙ σ ∂θv∂σ
=∇•ρaKh∇( )θv
ρa+
θvcp,dTv
dQndt
n=1
Ne,h
∑
σ-p Coord. Thermodynamic En. Eq.Combine with continuity equation for air (5.75)
Apply spherical-coordinate transformations (5.76)
∂ πaθv( )∂t
⎡
⎣ ⎢
⎤
⎦ ⎥ σ
+∇σ • vhπaθv( )+πa∂ ˙ σ θv( )
∂σ
=πa∇ •ρaKh∇( )θv
ρa+
θvcp,dTv
dQndt
n=1
Ne,h
∑⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
Re2cosϕ
∂∂t
πaθv( )⎡ ⎣ ⎢
⎤ ⎦ ⎥ σ
+∂
∂λeuπaθvRe( )+
∂∂ϕ
vπaθvRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥
+πaRe2cosϕ
∂∂σ
˙ σ θv( )=πaRe2cosϕ
∇ •ρaKh∇( )θvρa
+θv
cp,dTv
dQndt
n=1
Ne,h
∑⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
σ-p Coord. Momentum Equation
In Cartesian-altitude coordinates (4.70)
Apply to horizontal momentum equation (5.77)
Material time derivative of velocity
dvdt
=−fk×v−∇Φ−1ρa
∇pa +ηaρa
∇2v+1ρa
∇ •ρaKm∇( )v
dvhdt
=∂vh∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+ vh •∇σ( )vh +˙ σ ∂vh∂σ
∂vh∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+ vh•∇σ( )vh +˙ σ ∂vh∂σ
+ fk×vh =−1ρa
∇z pa( )+∇ •ρaKm∇( )vh
ρa
σ-p Coord. Momentum Equation
Substitute into momentum equation (5.79)
Pressure gradient term (5.78)
1ρa
∇z pa( ) =∇pΦ =∇σΦ −σπa
∇σ πa( )∂Φ∂σ
∂vh∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ σ
+ vh•∇σ( )vh +˙ σ ∂vh∂σ
+ fk×vh
=−∇σΦ +σπa
∇σ πa( )∂Φ∂σ
+∇ •ρaKm∇( )vh
ρa
Coupling Hor./Vert. Momentum Eqs.
Hydrostatic equation in the pressure coordinate (5.80)
Re-derive specific density (5.82)
∂Φ∂σ
=−πa ′ R Tv
pa=−
πaρa
=−αaπa
αa =′ R Tvpa
=κcp,dθvP
pa=cp,dθv
∂P∂pa
=cp,dθv
πa
∂P∂σ
Coupling Hor./Vert. Momentum Eqs.Combine terms above with momentum/continuity eqs. (5.83)
∂ vhπa( )∂t
⎡
⎣ ⎢
⎤
⎦ ⎥ σ
+vh∇σ • vhπa( )+πa vh•∇σ( )vh +πa∂
∂σ˙ σ vh( )
=−πa fk×vh −πa∇σΦ −σcp,dθv∂P∂σ
∇σ πa( ) +πa∇ •ρaKm∇( )vh
ρa
dΦ =−cp,dθvdpa
1000hPa
⎛
⎝ ⎜
⎞
⎠ ⎟
κ⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
=−cp,dθvdP
Now horizontal and vertical equations consistent (5.38)
σ-p Coord. Momentum Equation
V-direction momentum equation (5.87)
U-direction momentum equation (5.86)
Re2cosϕ
∂∂t
πau( )⎡ ⎣ ⎢
⎤ ⎦ ⎥ σ
+∂
∂λeπau2Re( ) +
∂∂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
Re2cosϕ
∂∂t
πav( )⎡ ⎣ ⎢
⎤ ⎦ ⎥ σ
+∂
∂λeπauvRe( )+
∂∂ϕ
v2πaRecosϕ( )⎡
⎣ ⎢
⎤
⎦ ⎥ σ
+πaRe2cosϕ
∂∂σ
˙ σ v( )
=−πau2Resinϕ−πafuRe2cosϕ−Recosϕ πa
∂Φ∂ϕ
+σcp,dθv∂P∂σ
∂πa∂ϕ
⎛
⎝ ⎜
⎞
⎠ ⎟ σ
+Re2cosϕ
πaρa
∇ •ρaKm∇( )v
Sigma-Altitude Coordinate
Sigma-altitude value (5.89)
Altitude of a sigma surface (5.90)
Altitude difference between column top and surface
s =ztop−z
ztop−zsurf=
ztop−z
Zt
Zt =ztop−zsurf
z=ztop−Zts
Gradient Conversion
Gradient conversion between z and s-z coordinate (5.91)
Substitute into gradient conversion (5.93)
Horiz. gradient of sigma along const. altitude surface (5.92)
∇z =∇s +∇z s( )∂∂s
∇z s( ) =−ztop−z
Zt2 ∇z Zt( )=−
sZt
∇z Zt( )
∇z =∇s −sZt
∇z Zt( )∂∂s
Conversions in -z CoordinateTime-derivative conversion between z and s-z coordinate (5.94)
Material time derivative in the sigma-altitude coordinate (5.96)
Scalar velocity in the sigma-altitude coordinate (5.95)
where
∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=
∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ s
˙ s =dsdt
= vh•∇z( )s+w∂s∂z
= vh•∇z( )s −wZt
∂s∂z
=−1Zt
ddt
=∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ s
+ vh•∇s( )+˙ s ∂∂s
-z Coord. Continuity Eq. For AirContinuity equation for air in the z coordinate
Substitute these terms into continuity equation above (5.97)
Apply gradient conversion to horizontal velocity
Apply gradient conversion to air density
∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
z=−ρa ∇z•vh +
∂w∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟ − vh•∇z( )ρa−w
∂ρa∂z
∇z•vh =∇s•vh +∇z s( )∂vh∂s
∇z ρa( ) =∇s ρa( )+∇z s( )∂ρa∂s
∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
s=−ρa ∇s•vh +∇z s( )
∂vh∂s
+∂w∂z
⎡ ⎣ ⎢
⎤ ⎦ ⎥ −vh• ∇s ρa( )+∇z s( )
∂ρa∂s
⎡ ⎣ ⎢
⎤ ⎦ ⎥ −w
∂ρa∂z
-z Coord. Continuity Eq. For AirRewrite vertical velocity
Sub. , (5.99), ∂s/∂z=-1/Zt into (5.97) (5.100)
Differentiate with respect to altitude (5.98)
Substitute ∂s/∂z=-1/Zt (5.99)
w =Zt vh•∇z s( )−˙ s [ ]
∂w∂z
=Zt ∇z s( )∂vh∂z
+ vh •∇z( )∂s∂z
−∂˙ s ∂z
⎡ ⎣ ⎢
⎤ ⎦ ⎥
∂w∂z
=∂˙ s ∂s
−∇z s( )∂vh∂s
+1Zt
vh•∇z( )Zt
w =Zt vh•∇z s( )−˙ s [ ]
∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
s=−ρa ∇s•vh +
∂˙ s ∂s
+1Zt
vh •∇z( )Zt⎡
⎣ ⎢
⎤
⎦ ⎥ − vh•∇s( )ρa −˙ s
∂ρa∂s
Non/Hydrostatic Continuity Eq.Substitute and compress --> (5.101)
Nonhydrostatic continuity equation for air in s-z coordinate
Hydrostatic equation in the s-z coordinate (5.102)
Substitute into (5.101) --> Hydrostatic continuity eq. (5.103)
∇z Zt( ) =∇s Zt( )
∂ρa∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
s=−
1Zt
∇s• vhρaZt( )−∂∂s
˙ s ρa( )
=−1Zt
∂ uρaZt( )∂x
+∂ vρaZt( )
∂y
⎡
⎣ ⎢
⎤
⎦ ⎥ s−
∂ ˙ s ρa( )∂s
′ ρ a =−1g
∂ ′ p a∂z
=1
Ztg∂ ′ p a∂s
∂∂t
∂ ′ p a∂s
⎛ ⎝ ⎜
⎞ ⎠ ⎟ =−∇s• vh
∂ ′ p a∂s
⎛ ⎝ ⎜
⎞ ⎠ ⎟ −
∂∂s
˙ s ∂ ′ p a∂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)
dqdt
⎛ ⎝ ⎜ ⎞
⎠ ⎟
s=
∂q∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ s
+ vh •∇s( )q+˙ s ∂q∂s
=∇ •ρaKh∇( )q
ρa+ Rn
n=1
Ne,t
∑
-z Coord. Thermodynamic En. Eq.Apply material derivative in the s-z coordinate to the thermodynamic energy equation in the z coordinate (5.106)
∂θv∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ s
+ vh •∇s( )θv +˙ s ∂θv∂s
=∇ •ρaKh∇( )θv
ρa+
θvcp,dTv
dQndt
n=1
Ne,h
∑
-z Coord. Horiz. Momentum Eqs.Horizontal equation in the z coordinate
Apply material time derivative of velocity (5.107)
Gradient conversion of pressure (5.108)
Substitute gradient conversion (5.109)
dvhdt
=−fk×vh −1
ρa∇z pa( )+
∇ •ρaKm∇( )vhρa
∂vh∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ s
+ vh•∇s( )vh+˙ s ∂vh∂s
+fk×vh =−1ρa
∇z pa( )+∇z•ρaKm∇z( )vh
ρa
∇z pa( ) =∇s pa( )−sZt
∇z Zt( )∂pa∂s
∂vh∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟ s
+ vh•∇s( )vh+˙ s ∂vh∂s
=−fk×vh−1ρa
∇s pa( )−sZt
∇z Zt( )∂pa∂s
− ∇ •ρaKm∇( )vh⎡
⎣ ⎢
⎤
⎦ ⎥
-z Coord. Vertical Momentum Eq.Sub. ∂s/∂z=-1/Zt into z-coord. vertical momentum eq. (5.113)
Substitute
Another form of vertical momentum equation (5.114)
∂w∂t
+u∂w∂x
+v∂w∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ s
+˙ s ∂w∂s
=−g+1
Ztρa
∂pa∂s
+∇•ρaKm∇( )w
ρa
w =Zt vh•∇z s( )−˙ s [ ]
∂∂t
⎛ ⎝ ⎜
⎞ ⎠ ⎟
s+u
∂∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟
s+v
∂∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ s
+˙ s ∂∂s
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Ztu∂s∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
+Ztv∂s∂y
⎛
⎝ ⎜
⎞
⎠ ⎟ z
−Zt˙ s ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
=−g+1
Ztρa
∂pa∂s
+1
ρa∇ •ρaKm∇( ) Ztu
∂s∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ z
+Ztv∂s∂y
⎛
⎝ ⎜
⎞
⎠ ⎟
z−Zt˙ s
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥