Effect of Stratification

Atmospheric dynamics 1

Effect of Stratification

Stratified fluid (atmosphere, ocean)

The Atmosphere and Ocean are good examples of 
 well stratified fluid

Image: wikiwand.com 
 ozonedepletiontheory.info

Stratified fluid (atmosphere, ocean)

The Atmosphere and Ocean are good examples of 
 well stratified fluid

Image: open.edu

Stratified fluid (atmosphere, ocean)

(Vallis, 2006; AOFD)

Fig. �.�

Boussinesq Approximation

• Notation (basic concept)

• Can consider two types

1) (Boussinesq Approximation)

for a stratified ocean

2) (Anelestic Approximation)

for a stratified atmos

˜

ρ ( ¯ ρ)

ρ = ρ

+ ρ(z) + ̂ ρ′ (x, y, z, t) δρ

ρ = ρ

+ δρ(x, y, z, t)

ρ = ¯ ρ(z) + ρ′ (x, y, z, t)

Boussinesq Approximation

• Basic Idea Assuming that

“Consider density perturbation only for the gravity term (ρg)”

and Replace with for the other terms

ρ

≫ δρ ρ ρ

ρ Dw

Dt = − ∂ p

∂z − ρg + ρν ∇

w

Boussinesq Approximation

• Momentum equations

Dw

Dt = − 1

ρ

∂δp

∂z − δρ

ρ

g + ν ∇

w

Du

Dt − fv = − ϕ

Dv

Dt + fu = − ϕ

Dw

Dt = − ϕ

+ b

Boussinesq Approximation

• Momentum equations

Dw

Dt = − 1

ρ

∂δp

∂z − δρ

ρ

g + ν ∇

w

Du

Dt − fv = − ϕ

Dv

Dt + fu = − ϕ

Dw

Dt = − ϕ

+ b

• In fact, mostly

(1) (2) (3)

Du

Dt − fv = − ϕ′

Dv

Dt + fu = − ϕ′

Dw

Dt = − ϕ′

+ b′

ϕ ≡ δp

ρ

b ≡ − δρ ρ

g

ϕ = ϕ(z) + ̂ ϕ′

b = b(z) + ̂ b′

Boussinesq Approximation

• Need two more equations

(4)

(5) or

u

+ v

+ w

= 0 Db

Dt = . . . Db′

Dt = . . .

Boussinesq Approximation

• Starting from the definition of b

where

N (Buoyancy frequency, 1/s)

Db

Dt = . . . b ≡ − δρ ρ

g

ρ = ρ(θ, q, p)

Dρ

Dt = − ρ

g

c

w ⇒ Db

Dt = g

c

w Db′

Dt = − (

d b ̂

dz − g

c

) w

≡ N

Adiabatic, no diffusion and displacement from 
 a well-balanced basic state (p is not disturbed)
 p = p₀(z)

Boussinesq Approximation

• Summary

Du

Dt − fv = − ϕ

Dv

Dt + fu = − ϕ

Dw

Dt = − ϕ

+ b u

+ v

+ w

= 0

Db

Dt = · b

• Adiabatic, no mixing case

Du

Dt − fv = − ϕ′

Dv

Dt + fu = − ϕ′

Dw

Dt = − ϕ′

+ b′

u

+ v

+ w

= 0 Db′

Dt = − N

w

Buoyancy frequency (ocean)

The Atmosphere and Ocean are good examples of 
 well stratified fluid

(Vallis, 2006; AOFD)

Depth(km)

2 1 0

3

mixed layer pycnocline

abyss

~10 mins.

~20 mins.

~2–3 hours

Density Buoyancy frequency N(z)

Fig. �.��