Large-scale motionsin the ocean and the atmosphere

case of stable stratification(ocean, stratosphere)

Characteristics of large-and meso-scale

ocean flows( 10 km < L < 6000 km ) :

Thin layer of fluid: H << L

Stable stratification

Importance of Earth’s rotationTrot < Tdyn Ro == Trot / Tdyn = U/2L<1

Apparent forcesin a rotating frame with =z

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Fapp = −2ρr Ω ×

r u tot − ρ

r Ω ×

r Ω ×

r r

Navier-Stokes equations in a rotating frame with =z

€

ru tot ≡ (u,v,w) ≡ (

r u ,w) , ρ, p

ρD

r u tot

Dt≡ ρ

∂

∂t+

r u ⋅∇ + w

∂

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟r u tot = −∇p − ρ g ˆ z − 2ρ

r Ω ×

r u tot − ρ

r Ω ×

r Ω ×

r r + ρ Du

Dρ

Dt+ ρ∇ ⋅

r u tot = 0

Dρ

Dt=

1

c 2

Dp

Dt

Navier-Stokes equations in a rotating frame with =z

€

ru tot ≡ (u,v,w) ≡ (

r u ,w) , ρ, p

ρD

r u tot

Dt≡ ρ

∂

∂t+

r u ⋅∇ + w

∂

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟r u tot = −∇p − ρ g ˆ z − 2ρ

r Ω ×

r u tot + ρ Du

Dρ

Dt+ ρ∇ ⋅

r u tot = 0

Dρ

Dt=

1

c 2

Dp

Dt

€

ru tot ≡ (u,v,w) ≡ (

r u ,w) , ρ, p

r u tot ≡ 0 ,

D

Dt= 0

∂p

∂z= −gρ

ρ = ρ 0 + ρ (z) + ρ '(x, y,z, t) , p = p0 + p (z) + p'(x, y,z, t)

ρD

r u tot

Dt≡ ρ 0 + ρ + ρ '( )

∂

∂t+

r u ⋅∇ + w

∂

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟r u tot = −∇p'−ρ 'g ˆ z − 2 ρ 0 + ρ + ρ '( )

r Ω ×

r u tot + ρ 0 + ρ + ρ '( ) Du

Dρ '

Dt+ w

dρ

dz+ ρ 0 + ρ + ρ '( )∇ ⋅

r u tot = 0

Dρ '

Dt+ w

dρ

dz=

1

c 2

Dp'

Dt+ w

dp

dz

⎛

⎝ ⎜

⎞

⎠ ⎟

Navier-Stokes in a rotating frame with =z

Boussinesq approximation

€

ρ(z) , ρ '(x, y,z, t) << ρ 0

ρ 0 ≈1.04 g/cm3 ρ ≈ 0.03 g/cm3 ρ ' ≈ 0.003 g/cm3

ρ ≈p

c 2≈

ρ 0gH

c 2⇒ ρ << ρ 0 if H <<

c 2

g≈ 200 km

p' ≈ ρ 'gH

1

c 2

Dp'

Dt≈

gH

c 2

Dρ '

Dt<<

Dρ '

Dtif H <<

c 2

g

Boussinesq approximation

€

ρ(z) , ρ '(x, y,z, t) << ρ 0 if H <c 2

g

ρ 0

Dr u tot

Dt= −∇p'−ρ 'g ˆ z − 2ρ 0

r Ω ×

r u tot + ρ 0 Du,0

∇ ⋅r u tot =∇ ⋅

r u +

∂w

∂z= 0

Dρ '

Dt+ w

dρ

dz=

w

c 2

d p

dz→

Dρ '

Dt− w

ρ 0N2(z)

g= 0

N 2(z) = −g

ρ 0

dρ

dz

⎛

⎝ ⎜

⎞

⎠ ⎟

Boussinesq approximation

€

H <c 2

g

ρ 0

Dr u tot

Dt= −∇p'−ρ 'g ˆ z − 2ρ 0

r Ω ×

r u tot + ρ 0 Du,0

∇ ⋅r u tot =∇ ⋅

r u +

∂w

∂z= 0

Dρ '

Dt− w

ρ 0N2(z)

g= 0

N 2(z) = −g

ρ 0

dρ

dz

⎛

⎝ ⎜

⎞

⎠ ⎟

Thin layer, stable stratification:hydrostatic approximation (L > 10 km)

€

Dw

Dt≈ 0

Dw ≈ 0

∂p

∂z= −gρ ⇔

∂p'

∂z= −gρ '

∂w

∂z= −∇ ⋅

r u ⇒ w ≈

H

Lu

Incompressible, hydrostatic fluid:primitive equations

€

ru tot ≡ (

r u ,w) , ρ, p

Dr u

Dt≡

∂r u

∂t+

r u ⋅∇

r u + w

∂r u

∂z= −

1

ρ 0

∇p'−2r Ω ×

r u tot( )

h+ Dh

∂p'

∂z= −ρ 'g

∇ ⋅r u +

∂w

∂z= 0

DB

Dt+ wN 2(z) = 0 , B = −

ρ 'g

ρ 0

Problem: A spherical Earth.Motion on a sphere: local tangent plane

€

r

€

z

€

y

€

φ0,φ

€

x = Rcosφ0 λ − λ 0( )

y = R φ − φ0( )

z = r − R

€

R

€

y =r Ω cosφ

Ωz =r Ω sinφ

€

z

€

x

€

y

Incompressible, inviscid fluidon a local Cartesian plane at midlatitudes

€

ru tot ≡ (

r u ,w) , ρ, p

Du

Dt−

uv tanφ

R+

uw

R= −

1

ρ 0

∂ p'

∂ x+ 2Ωv sinφ − 2Ωw cosφ

Dv

Dt−

u2 tanφ

R+

vw

R= −

1

ρ 0

∂ p'

∂ y− 2Ω usinφ

Dw

Dt−

u2 + v 2

R+

1

ρ 0

∂p'

∂z= −

ρ '

ρ 0

g + 2Ω ucosφ

∇ ⋅r u +

∂w

∂z= 0

DB

Dt+ wN 2(z) = 0 , B = −

ρ 'g

ρ 0

Problema:

l’approssimazione geostrofica

è diagnostica

abbiamo bisogno di una approssimazione prognostica

Primitive equationson a local Cartesian plane at midlatitudes

€

f = 2Ωsinφ

Du

Dt= −

1

ρ 0

∂p'

∂x+ f v

Dv

Dt= −

1

ρ 0

∂p'

∂y− f u

∂p'

∂z= −ρ 'g

∇ ⋅r u +

∂w

∂z= 0

DB

Dt+ wN 2(z) = 0 , B = −

ρ 'g

ρ 0

Primitive equationson the beta plane

€

2Ωsinφ = 2Ωsin φ0 + φ − φ0( )[ ] = 2Ωsinφ0 +2Ωcosφ0

RR φ − φ0( ) = f0 + β y

βL

f0

≈ 0.1

Du

Dt= −

1

ρ 0

∂ p'

∂ x+ f0v + β y v

Dv

Dt= −

1

ρ 0

∂ p'

∂ y− f0u − β y u

Primitive equations on the beta plane

€

f = 2Ωsinφ = f0 + β y

D

Dt=

∂

∂ t+ u

∂

∂ x+ v

∂

∂ y+ w

∂

∂ z

Du

Dt= −

1

ρ 0

∂p

∂x+ f0 + β y( )v

Dv

Dt= −

1

ρ 0

∂p

∂y− f0 + β y( ) u

∂p

∂z= −ρ g

∇ ⋅r u +

∂w

∂z= 0

DB

Dt+ wN 2(z) = 0 , B = −

ρ 'g

ρ 0