12.003 Atmosphere, Ocean and Climate Dynamics Lecture VI ...

12.003Atmosphere, Ocean and Climate Dynamics

Lecture VIVertical Structure of the Atmosphere

Saturday, September 17, 2011

Lecture VI Outline

1. Physical properties of dry air2. Vertical distribution of temperature3. Hydrostatic balance4. Vertical distribution of pressure and density


Physical properties of dry air (p,ρ,T)• Ideal gas law

•

ρ(z) = ρ0 e−z/H (1)ρ0 = 1.35 kg/m3 (2)H = 6.8 km (3)

Ω =vr

=2πR/day

R=

2πday

= 7.27×10−5s−1 (4)

Ve =

2GM/R (5)

Vm =

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ρRg

mT = ρ RT (8)

Rg = 8.314 JK−1mol−1 (9)

R =Rg

ma= 287 Jkg−1K−1 (10)

p = ∑i

ρiRg

miT = 78%ρ

Rg

mN2

T +21%ρRg

mO2

T + · · · = ρRg

maT (11)

e = ρv Rv T (12)pd = ρd Rd T (13)p = pd + e≈ pd (14)

es = A eβT (15)

1

ρ(z) = ρ0 e−z/H (1)ρ0 = 1.35 kg/m3 (2)H = 6.8 km (3)

Ω =vr

=2πR/day

R=

2πday

= 7.27×10−5s−1 (4)

Ve =

2GM/R (5)

Vm =

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ρRg

mT = ρ RT (8)

Rg = 8.314 JK−1mol−1 (9)

R =Rg

ma= 287 Jkg−1K−1 (10)

p = ∑i

ρiRg

miT = 78%ρ

Rg

mN2

T +21%ρRg

mO2

T + · · · = ρRg

maT (11)


es = A eβT (15)

1

ρ(z) = ρ0 e−z/H (1)ρ0 = 1.35 kg/m3 (2)H = 6.8 km (3)

Ω =vr

=2πR/day

R=

2πday

= 7.27×10−5s−1 (4)

Ve =

2GM/R (5)

Vm =

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ρRg

mT = ρ RT (8)

Rg = 8.314 JK−1mol−1 (9)

R =Rg

ma= 287 Jkg−1K−1 (10)

p = ∑i

ρiRg

miT = 78%ρ

Rg

mN2

T +21%ρRg

mO2

T + · · · = ρRg

maT (11)


es = A eβT (15)

1

ρ(z) = ρ0 e−z/H (1)ρ0 = 1.35 kg/m3 (2)H = 6.8 km (3)

Ω =vr

=2πR/day

R=

2πday

= 7.27×10−5s−1 (4)

Ve =

2GM/R (5)

Vm =

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ρRg

mT = ρ RT (8)

Rg = 8.314 JK−1mol−1 (9)

R =Rg

ma= 287 Jkg−1K−1 (10)

p = Rg78%ρmN2

T +Rg21%ρmO2

T + · · · = ρRg

maT (11)

p V = ∑i

ni Rg T = ∑i

Mi

miRg T (12)


es = A eβT (16)

1

ρ(z) = ρ0 e−z/H (1)ρ0 = 1.35 kg/m3 (2)H = 6.8 km (3)

Ω =vr

=2πR/day

R=

2πday

= 7.27×10−5s−1 (4)

Ve =

2GM/R (5)

Vm =

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ρRg

mT = ρ RT (8)

Rg = 8.314 JK−1mol−1 (9)

R =Rg

ma= 287 Jkg−1K−1 (10)

p = Rg78%ρmN2

T +Rg21%ρmO2

T + · · · = ρRg

maT (11)

p V = ∑i

ni Rg T = ∑i

Mi

miRg T (12)


es = A eβT (16)

1

ρ Animation

Universal gas constant

Gas constant of a specific gas


http://legacyweb.chemistry.ohio-state.edu/betha/nealGasLaw/fr2.2.html

http://legacyweb.chemistry.ohio-state.edu/betha/nealGasLaw/fr2.2.html

Physical properties of dry air• Dalton’s law of partial pressures for dry air

ρ(z) = ρ0 e−z/H (1)ρ0 = 1.35 kg/m3 (2)H = 6.8 km (3)

Ω =vr

=2πR/day

R=

2πday

= 7.27×10−5s−1 (4)

Ve =

2GM/R (5)

Vm =

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ρRg

mT = ρ RT (8)

Rg = 8.314 JK−1mol−1 (9)

R =Rg

ma= 287 Jkg−1K−1 (10)

p = Rg78%ρmN2

T +Rg21%ρmO2

T + · · · = ρRg

maT (11)

p V = ∑i

ni Rg T = ∑i

Mi

miRg T (12)


es = A eβT (16)

1

ρ(z) = ρ0 e−z/H (1)ρ0 = 1.35 kg/m3 (2)H = 6.8 km (3)

Ω =vr

=2πR/day

R=

2πday

= 7.27×10−5s−1 (4)

Ve =

2GM/R (5)

Vm =

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ρRg

mT = ρ RT (8)

Rg = 8.314 JK−1mol−1 (9)

R =Rg

ma= 287 Jkg−1K−1 (10)

p = Rg78%ρmN2

T +Rg21%ρmO2

T + · · · = ρRg

maT = ρ R T (11)

p V = ∑i

ni Rg T = ∑i

Mi

miRg T (12)


es = A eβT (16)

1

ρ(z) = ρ0 e−z/H (1)ρ0 = 1.35 kg/m3 (2)H = 6.8 km (3)

Ω =vr

=2πR/day

R=

2πday

= 7.27×10−5s−1 (4)

Ve =

2GM/R (5)

Vm =

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ρRg

mT = ρ RT (8)

Rg = 8.314 JK−1mol−1 (9)

R =Rg

ma= 287 Jkg−1K−1 (10)

p = Rg78%ρmN2

T +Rg21%ρmO2

T + · · · = ρRg

maT = ρ R T (11)

p V = ∑i

ni Rg T = ∑i

Mi

miRg T (12)


es = A eβT (16)

1

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Dry air gas constant


Atmospheric temperature variations • Atmospheric temperature varies in horizontal and vertical• Vertical variations are similar everywhere

pattern set by radiation budget (thermodynamics)• Horizontal variations depend on height

pattern set atmospheric transport (dynamics)


Vertical variations of temperature Three hot spots:

1) thermopause =>

2) stratopause =>

3) surface =>Saturday, September 17, 2011

Thermosphere• Named from the Greek θερμός (heat) σφαιρα (sphere), begins about 90 km above Earth’s surface• Molecules are dissociated and electrons are ionized (no triatomic molecules)• Absorbs high-energy UV (λ < 0.1 μm) - heated from above• Temperature increases with height


Mesosphere• Named from the Greek μεσό (middle) σφαιρα (sphere), located from about 50 km to 80-90 km above Earth's surface• Composition is similar to troposphere and stratosphere (N2, O2, Ar, CO2) • Absorbs decreasing amounts of low-energy UV through ozone• Temperature decreases with height from 0oC to -100oC


Stratosphere• Named from the Greek στράτος (layered) σφαιρα (sphere), located from about 10 km to 50 km above Earth's surface• Contains most of the ozone in the atmosphere• Ozone absorbs medium wavelength UV (0.1 < λ < 0.35 μm) - heated from above• Temperature increases with height from -40oC to 0oC


Ozone layer• Ozone is byproduct of photodissociation of molecular oxygen• Ozone absorbs medium wavelength UV (important for life on Earth)


Troposhere• Named from the Greek τρόπος (turn) σφαιρα (sphere), located within 10 km of the Earth's surface• Contains 85% of the atmospheric mass• Absorbs infrared radiation through H2O and CO2 - heated from below• Temperature decreases with height from to 15oC -40oC


Troposphere• Atmospheric layer most dynamically active

weather we experience is confined to troposphere• Transport modifies temperature profiles


Hydrostatic balanced pdz

=−gρ (1)

FgGravitational force

+ FTTop pressure force

+ FBBottom pressure force

= 0 (2)

κ|∇T |2 (3)

uuueTe · ∇T (4)

(∂t +uuug · ∇) T +ΓT v+N2T w = DT (5)

(∂t +uuug · ∇) S +ΓS v+N2S w = DS (6)

(∂t +uuug · ∇) ρ +Γρ v+N2ρ w = Dρ (7)

(∂t +uuug · ∇) C +ΓC v = DC (8)

ρ = ρ0 [1−α (T −T0)+β (S−S0)] (9)

C = ρ0

1−

N2S

N2ρ

α (T −T0)+N2

TN2

ρβ (S−S0)

(10)

(11)

1

d pdz

=−gρ (1)




= 0 (2)

κ|∇T |2 (3)

uuueTe · ∇T (4)

(∂t +uuug · ∇) T +ΓT v+N2T w = DT (5)

(∂t +uuug · ∇) S +ΓS v+N2S w = DS (6)

(∂t +uuug · ∇) ρ +Γρ v+N2ρ w = Dρ (7)

(∂t +uuug · ∇) C +ΓC v = DC (8)

ρ = ρ0 [1−α (T −T0)+β (S−S0)] (9)

C = ρ0

1−

N2S

N2ρ

α (T −T0)+N2

TN2

ρβ (S−S0)

(10)

(11)

1


Vertical distribution of pressure• Isothermal atmosphere

• Non-isothermal atmosphere

d p

dz= −gρ (1)




= 0 (2)

p(z) = ps exp− z

H

H =

RT0

g(3)

p(z) = ps exp−

z

0

dz

H(z)

H(z) =

RT (z)g

(4)

uuueTe · ∇T (5)

(∂t +uuug · ∇) T +ΓT v+N2T

w = DT (6)(∂t +uuug · ∇) S +ΓS v+N

2S

w = DS (7)

(∂t +uuug · ∇) ρ +Γρ v+N2ρ w = Dρ (8)

(∂t +uuug · ∇) C +ΓC v = DC (9)

ρ = ρ0 [1−α (T −T0)+β (S−S0)] (10)

C = ρ0

1−

N2S

N2ρ

α (T −T0)+N

2T

N2ρ

β (S−S0)

(11)

(12)

1

d p

dz= −gρ (1)




= 0 (2)

p(z) = ps exp− z

H

H =

RT0

g(3)

p(z) = ps exp−

z

0

dz

H(z)

H(z) =

RT (z)g

(4)

uuueTe · ∇T (5)


w = DT (6)(∂t +uuug · ∇) S +ΓS v+N

2S

w = DS (7)

(∂t +uuug · ∇) ρ +Γρ v+N2ρ w = Dρ (8)

(∂t +uuug · ∇) C +ΓC v = DC (9)

ρ = ρ0 [1−α (T −T0)+β (S−S0)] (10)

C = ρ0

1−

N2S

N2ρ

α (T −T0)+N

2T

N2ρ

β (S−S0)

(11)

(12)

1


Vertical distribution of density• Isothermal atmosphere

• Non-isothermal atmosphere

d p

dz= −gρ (1)




= 0 (2)

p(z) = ps exp− z

H

H =

RT0

g(3)

p(z) = ps exp−

z

0

dz

H(z)

H(z) =

RT (z)g

(4)

ρ(z) =ps

RT0exp

− z

H

H =

RT0

g(5)

ρ(z) =ps

RT (z)exp

−

z

0

dz

H(z)

H(z) =

RT (z)g

(6)

uuueTe · ∇T (7)


w = DT (8)(∂t +uuug · ∇) S +ΓS v+N

2S

w = DS (9)

(∂t +uuug · ∇) ρ +Γρ v+N2ρ w = Dρ (10)

(∂t +uuug · ∇) C +ΓC v = DC (11)

ρ = ρ0 [1−α (T −T0)+β (S−S0)] (12)

C = ρ0

1−

N2S

N2ρ

α (T −T0)+N

2T

N2ρ

β (S−S0)

(13)

(14)

1

d p

dz= −gρ (1)




= 0 (2)

p(z) = ps exp− z

H

H =

RT0

g(3)

p(z) = ps exp−

z

0

dz

H(z)

H(z) =

RT (z)g

(4)

ρ(z) =ps

RT0exp

− z

H

H =

RT0

g(5)

ρ(z) =ps

RT (z)exp

−

z

0

dz

H(z)

H(z) =

RT (z)g

(6)

uuueTe · ∇T (7)


w = DT (8)(∂t +uuug · ∇) S +ΓS v+N

2S

w = DS (9)

(∂t +uuug · ∇) ρ +Γρ v+N2ρ w = Dρ (10)

(∂t +uuug · ∇) C +ΓC v = DC (11)

ρ = ρ0 [1−α (T −T0)+β (S−S0)] (12)

C = ρ0

1−

N2S

N2ρ

α (T −T0)+N

2T

N2ρ

β (S−S0)

(13)

(14)

1


12.003 Atmosphere, Ocean and Climate Dynamics Lecture VI ...

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