12.003 Introduction to Atmosphere, Ocean, and

Climate Dynamics Topic 6

Vertical Structure of the Atmosphere

Topic 6 Outline

1. Vertical distribution of temperature2. Hydrostatic balance3. Vertical distribution of pressure and density

As discussed in earlier lecture... • Ideal gas law

⇤(z) = ⇤0 e�z/H (1)⇤0 = 1.35 kg/m3 (2)H = 6.8 km (3)

� =vr

=2⇥R/day

R=

2⇥day

= 7.27⇤10�5s�1 (4)

Ve =�

2GM/R (5)

Vm =�

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ⇤Rg

mT = ⇤ RT (8)

Rg = 8.314 JK�1mol�1 (9)

R =Rg

ma= 287 Jkg�1K�1 (10)

p = ⇥i

⇤iRg

miT = 78%⇤

Rg

mN2

T +21%⇤Rg

mO2

T + · · · = ⇤Rg

maT (11)

e = ⇤v Rv T (12)pd = ⇤d Rd T (13)p = pd + e⌅ pd (14)

es = A e�T (15)

1

⇤(z) = ⇤0 e�z/H (1)⇤0 = 1.35 kg/m3 (2)H = 6.8 km (3)

� =vr

=2⇥R/day

R=

2⇥day

= 7.27⇤10�5s�1 (4)

Ve =�

2GM/R (5)

Vm =�

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ⇤Rg

mT = ⇤ RT (8)

Rg = 8.314 JK�1mol�1 (9)

R =Rg

ma= 287 Jkg�1K�1 (10)

p = ⇥i

⇤iRg

miT = 78%⇤

Rg

mN2

T +21%⇤Rg

mO2

T + · · · = ⇤Rg

maT (11)

e = ⇤v Rv T (12)pd = ⇤d Rd T (13)p = pd + e⌅ pd (14)

es = A e�T (15)

1

⇤(z) = ⇤0 e�z/H (1)⇤0 = 1.35 kg/m3 (2)H = 6.8 km (3)

� =vr

=2⇥R/day

R=

2⇥day

= 7.27⇤10�5s�1 (4)

Ve =�

2GM/R (5)

Vm =�

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ⇤Rg

mT = ⇤ RT (8)

Rg = 8.314 JK�1mol�1 (9)

R =Rg

ma= 287 Jkg�1K�1 (10)

p = ⇥i

⇤iRg

miT = 78%⇤

Rg

mN2

T +21%⇤Rg

mO2

T + · · · = ⇤Rg

maT (11)

e = ⇤v Rv T (12)pd = ⇤d Rd T (13)p = pd + e⌅ pd (14)

es = A e�T (15)

1

⇤(z) = ⇤0 e�z/H (1)⇤0 = 1.35 kg/m3 (2)H = 6.8 km (3)

� =vr

=2⇥R/day

R=

2⇥day

= 7.27⇤10�5s�1 (4)

Ve =�

2GM/R (5)

Vm =�

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ⇤Rg

mT = ⇤ RT (8)

Rg = 8.314 JK�1mol�1 (9)

R =Rg

ma= 287 Jkg�1K�1 (10)

p = Rg78%⇤mN2

T +Rg21%⇤mO2

T + · · · = ⇤Rg

maT (11)

p V = ⇥i

ni Rg T = ⇥i

Mi

miRg T (12)

e = ⇤v Rv T (13)pd = ⇤d Rd T (14)p = pd + e⌅ pd (15)

es = A e�T (16)

1

⇤(z) = ⇤0 e�z/H (1)⇤0 = 1.35 kg/m3 (2)H = 6.8 km (3)

� =vr

=2⇥R/day

R=

2⇥day

= 7.27⇤10�5s�1 (4)

Ve =�

2GM/R (5)

Vm =�

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ⇤Rg

mT = ⇤ RT (8)

Rg = 8.314 JK�1mol�1 (9)

R =Rg

ma= 287 Jkg�1K�1 (10)

p = Rg78%⇤mN2

T +Rg21%⇤mO2

T + · · · = ⇤Rg

maT (11)

p V = ⇥i

ni Rg T = ⇥i

Mi

miRg T (12)

e = ⇤v Rv T (13)pd = ⇤d Rd T (14)p = pd + e⌅ pd (15)

es = A e�T (16)

1

ρ

Universal gas constant Gas constant of a specific gas

⇤(z) = ⇤0 e�z/H (1)⇤0 = 1.35 kg/m3 (2)H = 6.8 km (3)

� =vr

=2⇥R/day

R=

2⇥day

= 7.27⇤10�5s�1 (4)

Ve =�

2GM/R (5)

Vm =�

2kT/m (6)

p V = n Rg T =Mm

Rg T (7)

p = ⇤Rg

mT = ⇤ RT (8)

Rg = 8.314 JK�1mol�1 (9)

R =Rg

ma= 287 Jkg�1K�1 (10)

p = Rg78%⇤mN2

T +Rg21%⇤mO2

T + · · · = ⇤Rg

maT = ⇤ R T (11)

p V = ⇥i

ni Rg T = ⇥i

Mi

miRg T (12)

e = ⇤v Rv T (13)pd = ⇤d Rd T (14)p = pd + e⌅ pd (15)

es = A e�T (16)

1

Dry air gas constant

Lines of constant temperature

Atmospheric temperature variations • Atmospheric temperature varies in horizontal and vertical• Vertical variations are similar everywhere (pattern mostly set by radiation budget)• Horizontal variations depend on height (radiation and atmospheric circulation)

Vertical variations of temperature Three hot spots:

1) thermosphere =>

2) stratopause =>

3) surface =>

Thermosphere • Named from the Greek (heat sphere), begins about 90 km above Earth’s surface• Molecules are dissociated, ionized • Absorbs high-energy UV (λ < 0.1 μm) • Temperature increases with height

Mesosphere • Named from the Greek (middle 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 (with height) amounts of UV through ozone• Temperature decreases with height from 0oC to -100oC

Stratosphere • Named from the Greek (layered 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)

Ozone hole over Antarctica in Spring (different issue from global warming!)

Troposhere • Named from the Greek (turn sphere), located within 12 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 15oC to -40oC

• Atmospheric layer most dynamically active- weather we experience is confined to troposphere

• Transport modifies temperature profiles

Troposhere

What determines pressure distribution with height?: Hydrostatic balance d p

dz=�g⌅ (1)

Fg⇧⌅⇤⌃Gravitational force

+ FT⇧⌅⇤⌃Top pressure force

+ FB⇧⌅⇤⌃Bottom pressure force

= 0 (2)

⇤|⇥T |2 (3)

uuueTe · ⇥T̄ (4)

(⇧t +uuug · ⇥) T +�T v+N2T w = DT (5)

(⇧t +uuug · ⇥) S +�S v+N2S w = DS (6)

(⇧t +uuug · ⇥) ⌅ +�⌅ v+N2⌅ w = D⌅ (7)

(⇧t +uuug · ⇥) C +�C v = DC (8)

⌅ = ⌅0 [1�� (T �T0)+⇥ (S�S0)] (9)

C = ⌅0

�1�

N2S

N2⌅

� (T �T0)+N2

TN2

⌅⇥ (S�S0)

⇥(10)

(11)

1

d pdz

=�g⌅ (1)

Fg⇧⌅⇤⌃Gravitational force

+ FT⇧⌅⇤⌃Top pressure force

+ FB⇧⌅⇤⌃Bottom pressure force

= 0 (2)

⇤|⇥T |2 (3)

uuueTe · ⇥T̄ (4)

(⇧t +uuug · ⇥) T +�T v+N2T w = DT (5)

(⇧t +uuug · ⇥) S +�S v+N2S w = DS (6)

(⇧t +uuug · ⇥) ⌅ +�⌅ v+N2⌅ w = D⌅ (7)

(⇧t +uuug · ⇥) C +�C v = DC (8)

⌅ = ⌅0 [1�� (T �T0)+⇥ (S�S0)] (9)

C = ⌅0

�1�

N2S

N2⌅

� (T �T0)+N2

TN2

⌅⇥ (S�S0)

⇥(10)

(11)

1

How close to isothermal?

Vertical distribution of pressure

• Isothermal atmosphere

• Non-isothermal atmosphere

d pdz

= �g⇤ (1)

Fg⌦ �↵Gravitational force

+ FT⌦ �↵Top pressure force

+ FB⌦ �↵Bottom pressure force

= 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 · ⇥T̄ (5)

(⌅t +uuug · ⇥) T +�T v+N2T w = DT (6)

(⌅t +uuug · ⇥) S +�S v+N2S w = DS (7)

(⌅t +uuug · ⇥) ⇤ +�⇤ v+N2⇤ w = D⇤ (8)

(⌅t +uuug · ⇥) C +�C v = DC (9)

⇤ = ⇤0 [1�� (T �T0)+⇥ (S�S0)] (10)

C = ⇤0

⇧1�

N2S

N2⇤

� (T �T0)+N2

TN2

⇤⇥ (S�S0)

⌃(11)

(12)

1

d pdz

= �g⇤ (1)

Fg⌦ �↵Gravitational force

+ FT⌦ �↵Top pressure force

+ FB⌦ �↵Bottom pressure force

= 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 · ⇥T̄ (5)

(⌅t +uuug · ⇥) T +�T v+N2T w = DT (6)

(⌅t +uuug · ⇥) S +�S v+N2S w = DS (7)

(⌅t +uuug · ⇥) ⇤ +�⇤ v+N2⇤ w = D⇤ (8)

(⌅t +uuug · ⇥) C +�C v = DC (9)

⇤ = ⇤0 [1�� (T �T0)+⇥ (S�S0)] (10)

C = ⇤0

⇧1�

N2S

N2⇤

� (T �T0)+N2

TN2

⇤⇥ (S�S0)

⌃(11)

(12)

1

Vertical distribution of density

• Isothermal atmosphere

• Non-isothermal atmosphere

d pdz

= �g⇤ (1)

Fg⌦ �↵Gravitational force

+ FT⌦ �↵Top pressure force

+ FB⌦ �↵Bottom pressure force

= 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)

⇤(z) =ps

RT0exp

�� z

H

⇥H =

RT0

g(5)

⇤(z) =ps

RT (z)exp

⇤�

⌥ z

0

dz⇤

H(z⇤)

⌅H(z) =

RT (z)g

(6)

uuueTe · ⇥T̄ (7)

(⌅t +uuug · ⇥) T +�T v+N2T w = DT (8)

(⌅t +uuug · ⇥) S +�S v+N2S w = DS (9)

(⌅t +uuug · ⇥) ⇤ +�⇤ v+N2⇤ w = D⇤ (10)

(⌅t +uuug · ⇥) C +�C v = DC (11)

⇤ = ⇤0 [1�� (T �T0)+⇥ (S�S0)] (12)

C = ⇤0

⇧1�

N2S

N2⇤

� (T �T0)+N2

TN2

⇤⇥ (S�S0)

⌃(13)

(14)

1

d pdz

= �g⇤ (1)

Fg⌦ �↵Gravitational force

+ FT⌦ �↵Top pressure force

+ FB⌦ �↵Bottom pressure force

= 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)

⇤(z) =ps

RT0exp

�� z

H

⇥H =

RT0

g(5)

⇤(z) =ps

RT (z)exp

⇤�

⌥ z

0

dz⇤

H(z⇤)

⌅H(z) =

RT (z)g

(6)

uuueTe · ⇥T̄ (7)

(⌅t +uuug · ⇥) T +�T v+N2T w = DT (8)

(⌅t +uuug · ⇥) S +�S v+N2S w = DS (9)

(⌅t +uuug · ⇥) ⇤ +�⇤ v+N2⇤ w = D⇤ (10)

(⌅t +uuug · ⇥) C +�C v = DC (11)

⇤ = ⇤0 [1�� (T �T0)+⇥ (S�S0)] (12)

C = ⇤0

⇧1�

N2S

N2⇤

� (T �T0)+N2

TN2

⇤⇥ (S�S0)

⌃(13)

(14)

1