Atmospheric Convection

A Primer

Buoyancy and EntropySpecific Volume: α = 1ρSpecific Entropy: s

α =α ( ,p s)

( ) ∂α ∂Tδα δ= = s δ sp s p ∂ ∂p

( )s

δαB p g T ∂ ∂ T= =g s δ = − δ δα α p z ≡

∂ ∂s s

Homogeneous Compressible Gases

∂α TMaxwell : = ∂s p s

s sΓ

p∂∂

The adiabatic lapse rate:

( )

( )

( )

First Law of Thermodynamics :dsrev dT dαQ T= = cv + pdt dt dt

dT d pα dp= +cv −α

dt dt dtdT dp

= +c Rv −αdt dt

dT dp= −cp α

dt dtAdiabatic : 0cpdT − =αdp

Hydrostatic : 0cpdT + =gdz

dT g→ = − ≡ −Γdz c ds p

gΓ = cp

Earth’s atmosphere: Γ =1K100m

Model Aircraft Measurements(Renno and Williams, 1995)

Image courtesy of American Meteorological Society.

Prandtl’s (1925) problemQ

∫∞0 Qdz = F

F L~ 2 3t−

( )1

→w F~ z 3

1

B ~ F 2 3 z

Above a thin boundary layer, most atmospheric convection involves phase change of water:

Moist Convection

Moist Convection

• Significant heating owing to phase changes of water

• Redistribution of water vapor – most important greenhouse gas

• Significant contributor to stratiform cloudiness – albedo and longwave trapping

Water VariablesMass concentration of water vapor (specific humidity):

Mq ≡ H O2

MairVapor pressure (partial pressure of water vapor): e

Saturation vapor pressure: e*

Relative Humidity:

( )

H ≡ ee*

17.67 T−273

C-C: e h* 6= .112 Pa e T+30

The Saturation Specific Humidity

Ideal Gas Law:

e R*T=ρv mv

p R*T= ρ m

q ρv m e= = vρ m pm ev *q*= m p

Phase Equilibria

Bringing Air to Saturation

e q= pm m v

e e* *= (T )

1. Increase q (or p)

2. Decrease e* (T)

When Saturation Occurs…

• Heterogeneous Nucleation• Supersaturations very small in atmosphere• Drop size distribution sensitive to size

distribution of cloud condensation nuclei

00

Perc

enta

ge o

f clo

uds

with

ice

parti

cle

conc

entra

tions

abo

ve d

etec

tabl

e le

vel

-5 -10 -15 -20 -25

20

4010

15

1233

25 25

23

2014

16 168

315

4

7 1

1 1

2 1 1 1

60

80

100

Cloud top temperature (oC)

ICE NUCLEATION PROBLEMATIC

Figure by MIT OCW.

Precipitation Formation:

• Stochastic coalescence (sensitive to drop size distributions)

• Bergeron-Findeisen Process• Strongly nonlinear function of cloud water

concentration• Time scale of precipitation formation ~10-

30 minutes

StabilityNo simple criterion based on entropy:

T p s cd p= −ln Rd ln

T p0 0

α =α (sd , p) T p qs c= −ln R ln + L − qR ln ( )Hp d T p v T v 0 0

α =α (s, ,p qt )

Virtual Temperature and Density Temperature

Assume all condensed water falls at terminal velocity

V Vα a c+=Md v+ +M Mc

pV n= R*T

R T* M MV d va = + ,p m md v

1md ≡ 1 Mi

M ∑d ii m

R Td M→ =V M va d + ,

p ε

wheremε ≡ ≅v 0.622md

R R*d ≡ md

V V+= =a c RdT q qα 1 1− + c ρq a

M Md v+ +Mc p t ε 1− qc ρcR Td q≅ −1 qp ( )t + ε

Density temperature:

T T ( qρ ≡ −1 qt + ε )

RdTα = ρ

p

Mv c+M Mq q, vt ≡ ≡

M M

( ) +

Trick:

Define a saturation entropy, s* :

s* ,≡ s T( p,q*)α =α (s*, p q, t )

We can add an arbitrary function of qt to s* such that

α ≅α (s*', p)

Stability Assessment using Tephigrams:

-40 -30 -20 -10 0 10 20 30 40 50

100

200

300

400

500

600700800900

1000

Temperature (C)

Pre

ssur

e (m

b)

Stability Assessment using Tephigrams:

Convective Available Potential Energy(CAPE):

CAPE pii p≡ −∫pn ( )α αe dp

= −∫pip R Td ( )ρ ρp e

T d ln ( p)

Other Stability Diagrams:

“Air-Mass” Showers:

This image is copyrighted by Oxford, NY: Oxford University Press, 2005. Book title is Divine Wind: The History and Science of Hurricanes. ISBN: 9780195149418. Used with permission.

Image courtesy of American Meteorological Society.

Precipitation Effects:

Buoyancy Reversal:

Summary of Differences Between Dry andMoist Convection:

• Possibility of metastable states• Strong asymmetry between cloudy and

clear regions• Typically, only thin layers near surface are

unstable to upward displacements• Mixing can cause buoyancy reversal• Large potential for evaporatively cooled

downdrafts• Separation of buoyancy from displacement

can lead to propagating convection

• Buoyancy of unsaturated downdrafts depends on supply of precipitation

• Entropy produced mostly through mixing, not dissipation

• Internal waves can co-exist with unstable convection

Tropical Soundings

Annual Mean Kapingamoronga:

Radiative-Moist Convective Equilibrium

Precipitating Convection favors Widely Spaced Clouds (Bjerknes, 1938)

This image is copyrighted by Oxford, NY: Oxford University Press, 2005.Book title is Divine Wind: The History and Science of Hurricanes. ISBN: 9780195149418. Used with permission.

Properties:

• Convective updrafts widely spaced• Surface enthalpy flux equal to vertically

integrated radiative cooling•• Precipitation = Evaporation = Radiative

Cooling • Radiation and convection highly interactive

c TM p ∂θ

= −Qθ ∂z

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

)ya (m

m/d

niotait

ipceP

r

0 1 2 3 4 5 6 7 8 9 10Time (days)

This image is copyrighted by Academic Press, NY, 1999. It is in a chapter entitled "Quasi-equilibrium Thinking" (author is Kerry Emanuel) that appears in the book entitled General Circulation Model Development: Past, Present and Future. Edited by D. Randall. ISBN: 9780125780100. Used with permission.

This image is copyrighted by Academic Press, NY, 1999. It is in a chapter entitled "Quasi-equilibrium Thinking" (author is Kerry Emanuel) that appears in the book entitled General Circulation Model Development: Past, Present and Future. Edited by D. Randall. ISBN: 9780125780100. Used with permission.

This image is copyrighted by Academic Press, NY, 1999. It is in a chapter entitled "Quasi-equilibrium Thinking" (author is Kerry Emanuel) that appears in the book entitled General Circulation Model Development: Past, Present and Future. Edited by D. Randall. ISBN: 9780125780100. Used with permission.

This image is copyrighted by Academic Press, NY, 1999. It is in a chapter entitled "Quasi-equilibrium Thinking" (author is Kerry Emanuel) that appears in the book entitled General Circulation Model Development: Past, Present and Future. Edited by D. Randall. ISBN: 9780125780100. Used with permission.

Recovery from mid-level specific humidity perturbation

1 2 3 4 5 6 7 8 9 10-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

Specific humidity perturbation (g/Kg), from -5.479 to 2.645

Time (days)

P (m

b)

-100

-200

-300

-400

-500

-600

-700

-800

-900

-1000

)bm

P (

1 2 3 4 5 6 7 8 9 10

T Perturbation, from -6.87 to 0.848v

Time (days)

Robe and Emanuel, J. Atmos. Sci., 1996

Image courtesy of American Meteorological Society.

Islam et al. Predictability Experiments

Robe and Emanuel, J. Atmos. Sci., 2001

Image courtesy of American Meteorological Society.

Fovell and Ogura, J. Atmos. Sci., 1988, 1989

Image courtesy of American Meteorological Society.

Rotunno, Klemp and Weisman, J. Atmos. Sci., 1988

Image courtesy of American Meteorological Society.

Speculative Regime Diagram

2-D Shear-ParallelConvection

3-D Convection

2-D Shear-PerpendicularConvectionE, zb

/∆U/

Figure by MIT OCW.

Non-equilibrium Convection

Gulf of Mexico

Desert

Low θeHigh θe

Figure by MIT OCW.

Klemp and Wilhelmson, J. Atmos. Sci., 1978

Image courtesy of American Meteorological Society.