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Large-scale motionsin the ocean and the atmosphere
case of stable stratification(ocean, stratosphere)
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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
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Apparent forcesin a rotating frame with =z
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Fapp = −2ρr Ω ×
r u tot − ρ
r Ω ×
r Ω ×
r r
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Navier-Stokes equations in a rotating frame with =z
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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
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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
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€
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
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Boussinesq approximation
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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
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Boussinesq approximation
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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
⎛
⎝ ⎜
⎞
⎠ ⎟
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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
⎛
⎝ ⎜
⎞
⎠ ⎟
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Thin layer, stable stratification:hydrostatic approximation (L > 10 km)
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Dw
Dt≈ 0
Dw ≈ 0
∂p
∂z= −gρ ⇔
∂p'
∂z= −gρ '
∂w
∂z= −∇ ⋅
r u ⇒ w ≈
H
Lu
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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
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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
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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
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Problema:
l’approssimazione geostrofica
è diagnostica
abbiamo bisogno di una approssimazione prognostica
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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
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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
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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