presentation slides for chapter 5 of fundamentals of atmospheric modeling 2 nd edition

Presentation Slides for

Chapter 5of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

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

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

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥