Review 2: Balances in the atmosphere Equation of motion

Background: Equation of motion, Scale analysis 1. Geostrophic balance

2. Hydrostatic balance 3. Thermal wind balance

Pressure

gradient Coriolis

Review 2: Balances in the atmosphere

Equation of motion

horizontal

momentum equation

vertical

momentum equation

• Note: f (≡2Ωsinφ with angular velocity of the earth Ω ) 


f* (≡2Ωcosφ) terms are neglected (traditional approximation)

Du

Dt −fv = − ∂Φ

∂x +F_x Dv

Dt + fu = − ∂Φ

∂y + F_y

Dw

Dt = − 1 ρ

∂p

∂z −g +F_z

Equation of motion (scale analysis)

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)

In z-coords

Du

Dt −fv = − ∂Φ

∂x +F_x Dv

Dt + fu = − ∂Φ

∂y + F_y

Dw

Dt = − 1 ρ

∂p

∂z −g +F_z

Equation of motion (scale analysis)

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

U/T fU δP/Lρ U²/L

UW/H

In z-coords

Du

Dt −fv = − ∂Φ

∂x +F_x Dv

Dt + fu = − ∂Φ

∂y + F_y

Dw

Dt = − 1 ρ

∂p

∂z −g +F_z

Equation of motion (scale analysis)

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

U/T fU δP/Lρ

~10^-4 ~10^-3 ~10^-3 (m/s²)

In z-coords

Du

Dt −fv = − ∂Φ∂x +F_x Dv

Dt + fu = − ∂Φ∂y + F_y

Dw

Dt = − 1 ρ

∂p

∂z −g +F_z

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

, ,

−fv_g = − ∂Φ

∂x fu_g = − ∂Φ

∂y (u _g⃗ = 1f k× ∇hΦ)

2. Hydrostatic balance (scale analysis)

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

P ~ 10⁵ Pa H ~ 10⁴ m

U/T fU δP/Lρ

~10^-4 ~10^-3 ~10^-3 (m/s²)

In z-coords

W/T P/Hρ g UW/L

Du

Dt −fv = − ∂Φ∂x +F_x Dv

Dt + fu = − ∂Φ∂y + F_y

Dw

Dt = − 1 ρ

∂p

∂z −g +F_z

2. Hydrostatic balance (scale analysis)

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

P ~ 10⁵ Pa H ~ 10⁴ m

U/T fU δP/Lρ

~10^-4 ~10^-3 ~10^-3 (m/s²)

In z-coords

W/T P/Hρ g

~10^-7 ~10 ~10 (m/s²) Du

Dt −fv = − ∂Φ

∂x +F_x Dv

Dt + fu = − ∂Φ

∂y + F_y

Dw

Dt = − 1 ρ

∂p

∂z −g +F_z

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.html

EQ Pole

J

J

PGF - Pressure Gradient Force Co - Coriolis Force

3. Thermal wind balance (derivation)

• Combination of hydrostatic and geostrophic balances u_g = − 1

f ∂Φ

∂y , (v⃗_g = 1f k × ∇Φ)

∂Φ

∂p = − RT p

∂

∂p u_g = − 1 f

∂Φ

∂y

⎛

⎝⎜

⎞

⎠⎟ ⇒ ∂u_g

∂p = 1 f

∂

∂y RT

p

⎛

⎝⎜

⎞

⎠⎟

① ②

∂u_g

∂lnp = R f

∂T

∂y ∂u_g

∂z^* =− R Hf

∂T

∂y, where δz^*=−Hδlnp

#

$% &

'(

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

3. Thermal wind balance (physical meaning)

http://www.ux1.eiu.edu/~cfjps/

1400/pressure_wind.html

EQ Pole

J

J

- ∂T/∂y > 0 thermal ∂u/∂z* > 0

wind

∂u_g

∂lnp = R f

∂T

∂y ∂u_g

∂z^* =− R Hf

∂T

∂y, where δ^z*=−Hδ^lnp

#

$% &

'(

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

PGF - Pressure Gradient Force Co - Coriolis Force

3. Thermal wind balance (real world)

W C

C

C

m/s

Subtropical jet

Polar night jet

∂u_g

∂lnp = R f

∂T

∂y ∂u_g

∂z^* =− R Hf

∂T

∂y, where δ^z*=−Hδ^lnp

#

$% &

'(

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

3. Thermal wind balance (real world)

Ug from T Not bad!

Subtropical jet

Polar night jet

m/s

∂u_g

∂lnp = R f

∂T

∂y ∂u_g

∂z^* =− R Hf

∂T

∂y, where δ^z*=−Hδ^lnp

#

$% &

'(

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

Recap

• Geostrophic balance


Coriolis

≃

• Hydrostatic balance


Gravity

≃

• 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

∂u_g⃗

∂lnp = − R

f k× ∇hT ∂u_g⃗

∂z* = R

fHk× ∇hT (wh ere, δz* = −Hδlnp)