balances in the atmosphere - atmdyn.orgatmdyn.org/teaching/intro_dyn1/14_atmosbalance.pdf · ρ ~ 1...

Background: Equation of motion, Scale analysis

1. Geostrophic balance

2. Hydrostatic balance

3. Thermal wind balance

Pressure gradient Coriolis

Balances in the atmosphere

0. Equation of motion

∂w∂t+!u ⋅∇w+ g = − 1

ρ∂p∂z

+Fz

∂u∂t+!u ⋅∇u − fv = − 1

ρ∂p∂x

+Fx

∂v∂t+!u ⋅∇v+ fu = − 1

ρ∂p∂y

+Fy

horizontal momentum equation

vertical momentum equation

• Note: f (≡2Ωsinφ with angular velocity of the earth Ω ) f* (≡2Ωcosφ) terms are neglected (traditional approximation)

0. Equation of motion (scale analysis)

∂w∂t+!u ⋅∇w+ g = − 1

ρ∂p∂z

+Fz

∂u∂t+!u ⋅∇u − fv = − 1

ρ∂p∂x

+Fx

∂v∂t+!u ⋅∇v+ fu = − 1

ρ∂p∂y

+Fy

10,000 km

1,000 km

100 km

10 km

1 km

Planetary (AO, ENSO)

Synoptic (Cyclone, MJO)

Meso (Fronts, MCS, tropical cyclone)

Micro (Single thunderstorm, turbulence)


∂w∂t+!u ⋅∇w+ g = − 1

ρ∂p∂z

+Fz

∂u∂t+!u ⋅∇u − fv = − 1

ρ∂p∂x

+Fx

∂v∂t+!u ⋅∇v+ fu = − 1

ρ∂p∂y

+Fy

U/T U2/L fU δP/Lρ

Typical scale (in MKS unit) of synoptic phenomena in mid-latitude

U ~ 10 m/s W ~ 10-2 m/s T ~ 105 s (~1 day) L ~ 106 m f ~ 10-4 /s δP ~ 103 Pa (N/m2; kg/ms2) ρ ~ 1 kg/m3


∂w∂t+!u ⋅∇w+ g = − 1

ρ∂p∂z

+Fz

∂u∂t+!u ⋅∇u − fv = − 1

ρ∂p∂x

+Fx

∂v∂t+!u ⋅∇v+ fu = − 1

ρ∂p∂y

+Fy

U/T U2/L fU δP/Lρ~10-4 ~10-4 ~10-3 ~10-3 (m/s2)




∂w∂t+!u ⋅∇w+ g = − 1

ρ∂p∂z

+Fz

∂u∂t+!u ⋅∇u − fv = − 1

ρ∂p∂x

+Fx

∂v∂t+!u ⋅∇v+ fu = − 1

ρ∂p∂y

+Fy

U/T U2/L fU δP/Lρ~10-4 ~10-4 ~10-3 ~10-3 (m/s2)



• Coriolis force and Pressure gradient force (PGF) makethe first order balance


• Coriolis force acts to the right of the wind vector in the NH • It balances pressure gradient force (-∇hp)

http://www.ux1.eiu.edu/~cfjps/ 1400/pressure_wind.html

− fvg = −1ρ∂p∂x

; fug = −1ρ∂p∂y

!ug =1ρ fk×∇h p

%

&'

(

)*

L

L

H

1. Geostrophic balance (p-coords)

• Coriolis force acts to the right of the wind vector in the NH • It balances pressure gradient force (-∇hp)

http://www.ux1.eiu.edu/~cfjps/ 1400/pressure_wind.html

L

L

H

, , −fvg = − ∂Φ∂x

fug = − ∂Φ∂y ( ⃗u g = 1

fk × ∇h Φ)

2. Hydrostatic balance (scale analysis)

∂w∂t+!u ⋅∇w+ g = − 1

ρ∂p∂z

+Fz

∂u∂t+!u ⋅∇u − fv = − 1

ρ∂p∂x

+Fx

∂v∂t+!u ⋅∇v+ fu = − 1

ρ∂p∂y

+Fy

W/T UW/L g P/Hρ

U/T U2/L fU δP/Lρ~10-4 ~10-4 ~10-3 ~10-3 (m/s2)


U ~ 10 m/s W ~ 10-2 m/sT ~ 105 s (~1 day)L ~ 106 mf ~ 10-4 /sδP ~ 103 Pa (N/m2; kg/ms2)ρ ~ 1 kg/m3

P ~ 105 PaH ~ 104 m

2. Hydrostatic balance (scale analysis)

∂w∂t+!u ⋅∇w+ g = − 1

ρ∂p∂z

+Fz


U ~ 10 m/s W ~ 10-2 m/sT ~ 105 s (~1 day)L ~ 106 mf ~ 10-4 /sδP ~ 103 Pa (N/m2; kg/ms2)ρ ~ 1 kg/m3

P ~ 105 PaH ~ 104 m

W/T UW/L g P/Hρ~10-7 ~10-7 ~10 ~10 (m/s2)

∂u∂t+!u ⋅∇u − fv = − 1

ρ∂p∂x

+Fx

∂v∂t+!u ⋅∇v+ fu = − 1

ρ∂p∂y

+Fy

U/T U2/L fU δP/Lρ~10-4 ~10-4 ~10-3 ~10-3 (m/s2)

2. Hydrostatic balance (concept)∂p∂z

= −ρg

p+δp

p

gδm/S =gρ(δz)

s

δm

p+δp+ρgδz = p

δp/δz = -ρg(Straightforward!)

δz warm cold

Another application

1000 hPa

850 hPaδz

Physical concept

• Height of an air mass between two pressure levels is proportional to its temperature

δz/δp = -RT/pg (p=ρRT)

δz = -RTδp/pg

2. Hydrostatic balance (concept)

p+δp

p

gδm/S =gρ(δz)

s

δmδz

(P-coords)∂Φ∂p

= − RTp

δz/δp = -RT/pg (p=ρRT)

δz = -RTδp/pg• Height of an air mass

between two pressure levels is proportional to its temperature

p+δp+ρgδz = p

δp/δz = -ρg(Straightforward!)

warm cold 1000 hPa

850 hPaδz

Another applicationPhysical concept

3. Thermal wind balance

• Combination of hydrostatic and geostrophic balances

Relationship between vertical wind shear (du/dz) and horizontal temperature gradient (∇hT)

http://www.ux1.eiu.edu/~cfjps/ 1400/pressure_wind.htmlEQ Pole

J

J

PGF - Pressure Gradient Force Co - Coriolis Force

3. Thermal wind balance (derivation)• Combination of hydrostatic and geostrophic balances

ug = −1f ρ

∂p∂y

∂p∂z

= −ρg


ug = −1f ρ

∂p∂y

∂p∂z

= −ρg

• We use pressure coordinate for derivation (simple)

vg =1f ρ

∂p∂x

"

#$

%

&'

ug = − 1f∂Φ∂y

, using coordinate transform ∂p / ∂y( )z = ρg ∂z / ∂y( )p⎡⎣

⎤⎦

∂Φ∂p

= − RTp

, (where δΦ ≡ gδ z)



∂∂p

ug = − 1f∂Φ∂y

⎛⎝⎜

⎞⎠⎟

⇒ ∂ug∂p

= 1f

∂∂y

RTp

⎛⎝⎜

⎞⎠⎟

②①

ug = −1f ρ

∂p∂y

∂p∂z

= −ρg

vg =1f ρ

∂p∂x

"

#$

%

&'

ug = − 1f∂Φ∂y


⎤⎦

∂Φ∂p

= − RTp




∂∂p

ug = − 1f∂Φ∂y

⎛⎝⎜

⎞⎠⎟

⇒ ∂ug∂p

= 1f

∂∂y

RTp

⎛⎝⎜

⎞⎠⎟

②①

ug = −1f ρ

∂p∂y

∂p∂z

= −ρg

vg =1f ρ

∂p∂x

"

#$

%

&'

ug = − 1f∂Φ∂y


⎤⎦

∂Φ∂p

= − RTp


∂ug∂ln p

=Rf∂T∂y

∂ug∂z* = −

RHf

∂T∂y

, where δz* = −Hδ ln p#

$%

&

'(

Thermal wind balance log-pressure form (I love!)

3. Thermal wind balance (physical meaning)

http://www.ux1.eiu.edu/~cfjps/ 1400/pressure_wind.htmlEQ Pole

J

J

- ∂T/∂y > 0 thermal

∂u/∂z* > 0 wind

∂ug∂ln p

=Rf∂T∂y

∂ug∂z* = −

RHf

∂T∂y


$%

&

'(


PGF - Pressure Gradient Force Co - Coriolis Force

3. Thermal wind balance (real world)

W C

C

C

m/s

Subtropical jet

Polar night jet

∂ug∂ln p

=Rf∂T∂y

∂ug∂z* = −

RHf

∂T∂y


$%

&

'(


3. Thermal wind balance (real world)

Ug from T Not bad!

Subtropical jet

Polar night jet

m/s

∂ug∂ln p

=Rf∂T∂y

∂ug∂z* = −

RHf

∂T∂y


$%

&

'(


Recap

• Geostrophic balance Coriolis ≃ Horizontal pressure gradient

• Hydrostatic balance Gravity ≃ Vertical pressure gradient

• Thermal wind balance Geostrophic + Hydrostatic

Note: 1. Geostrophic and thermal wind balances work well

at mid-to-high latitudes

2. Hydrostatic balance works at all latitudes

∂ ⃗ug

∂ ln p= − R

fk × ∇h T

∂ ⃗ug

∂z* = RfH

k × ∇h T (wh ere, δz* = − Hδ ln p)

balances in the atmosphere - atmdyn.orgatmdyn.org/teaching/intro_dyn1/14_atmosbalance.pdf · ρ ~ 1...

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