Cloud Physics
• What is a cloud?
Cloud Physics
• What is a cloud?
– Water droplets or ice crystals in the air.
Cloud Physics
• What is a cloud?
– Water droplets or ice crystals in the air.
– Why important?
Cloud Physics
• What is a cloud?
– Water droplets or ice crystals in the air.
– Why important?
• Precipitation
• Solar radiation
Cloud Physics
• What is a cloud?
– Water droplets or ice crystals in the air.
– Why important?
• Precipitation
• Solar radiation
• What do we want to learn?
– Formation of clouds
– Development of precipitation
Cloud Physics
• What is a cloud?
– Water droplets or ice crystals in the air.
– Why important?
• Precipitation
• Solar radiation
• What do we want to learn?
– Formation of clouds
– Development of precipitation
• Methods?
Cloud Physics
• What is a cloud?
– Water droplets or ice crystals in the air.
– Why important?
• Precipitation
• Solar radiation
• What do we want to learn?
– Formation of clouds
– Development of precipitation
• Methods?
– Cloud microphysics
– Cloud dynamics
Cloud Physics
• Understanding the properties of clouds
– What clouds are (why are they different)
– How they develop in time
– How they interact and affect the energy balance of the
planet
– Development of precipitation, rain, hail, and snow
– Role in general circulation of the atmosphere
Cloud Physics
• Understanding the properties of clouds
– What clouds are (why are they different)
– How they develop in time
– How they interact and affect the energy balance of the
planet
– Development of precipitation, rain, hail, and snow
– Role in general circulation of the atmosphere
• These subjects are important to
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Radar meteorology
Weather modification
Severe storms research
Global energy balance (greenhouse effect)
Overview
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Thermodynamics of dry air
Water vapor and its thermodynamic effects
Parcel buoyancy and atmospheric stability
Mixing and convection
Observed properties of clouds
Formation of cloud droplets
Droplet growth by condensation
Initiation of rain
Formation and growth of ice crystals
Severe weather
Atmospheric composition
Atmospheric composition
• Permanent gases
• Variable gases
• Aerosols
Atmospheric composition
• Permanent gases
– Nitrogen, oxygen, argon, neon, helium, etc.
• Variable gases
– Water vapor, carbon dioxide, and ozone.
• Aerosols
– Smoke, dust, pollen, and condensed forms of
water (hydrometeors).
Review
• Zeroth law of thermodynamics
Review
• Zeroth law of thermodynamics
• Concept of thermometer
• Charles’ Law
Review
• Zeroth law of thermodynamics
• Concept of thermometer
• Charles’ Law
•  /T = R/p = f(p)
• Define temperature, K = ?
Review
• Zeroth law of thermodynamics
• Concept of thermometer
• Charles’ Law
•  /T = R/p = f(p)
• Define temperature, K = ?
• Boyle’s Law
Review
• Zeroth law of thermodynamics
• Concept of thermometer
• Charles’ Law
•  /T = R/p = f(p)
• Define temperature, K = ?
• Boyle’s Law
• p  = RT = g(T)
• Avogadro’s law (ideal gas)
Review
• Zeroth law of thermodynamics
• Concept of thermometer
• Charles’ Law
•  /T = R/p = f(p)
• Define temperature, K = ?
• Boyle’s Law
• p  = RT = g(T)
• Avogadro’s law (ideal gas)
• p  /T = R* / m =R (for individual gas or R’ for dry air)
• m: molecular weight = ?
Review
• Zeroth law of thermodynamics
• Concept of thermometer
• Charles’ Law
•  /T = R/p = f(p)
• Define temperature, K = ?
• Boyle’s Law
• p  = RT = g(T)
• Avogadro’s law (ideal gas)
• p  /T = R* / m
• m: molecular weight = ?
• 1st law of thermodynamics
Review
• Zeroth law of thermodynamics
• Concept of thermometer
• Charles’ Law
•  /T = R/p = f(p)
• Define temperature, K = ?
• Boyle’s Law
• p  = RT = g(T)
• Avogadro’s law (ideal gas)
• p  /T = R* / m =R (for individual gas or R’ for dry air)
• m: molecular weight = ?
• 1st law of thermodynamics
• dq = du + dw = du + p d = dh - dp
• Work-heat relation (1 cal = ? J)
Review
• Zeroth law of thermodynamics
• Concept of thermometer
• Charles’ Law
•  /T = R/p = f(p)
• Define temperature, K = ?
• Boyle’s Law
• p  = RT = g(T)
• Avogadro’s law (ideal gas)
• p  /T = R* / m =R (gas constant for individual gas or R’ for dry
air )
• m: molecular weight = ?
• 1st law of thermodynamics
• dq = du + dw = du + p d = dh - dp
• Work-heat relation (1 cal = ? J)
• Dalton’s law
Review, cont.
• Specific heats:
Review, cont.
• Specific heats: c = dq/dT
– cv =(q/T)
– cp= ( q/T)p
– cp = c v + R
Review, cont.
• Specific heats: c = dq/dT
–
–
–
–
cv =(q/T)
cp= ( q/T)p
cp = c v + R
Equipartition of energy
Review, cont.
• Specific heats: c = dq/dT
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cv =(q/T)
cp= ( q/T)p
cp = c v + R
Equipartition of energy
degree of freedom: f
u = fRT/2
Review, cont.
• Specific heats: c = dq/dT
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cv =(q/T)
cp= ( q/T)p
cp = c v + R
Equipartition of energy
degree of freedom: f
u = fRT/2
• Entropy (3 meanings)
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Review, cont.
• Specific heats: c = dq/dT
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cv =(q/T)
cp= ( q/T)p
cp = c v + R
Equipartition of energy
degree of freedom: f
u = fRT/2
• Entropy
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d = dq/T
Irreversible processes: entropy change is defined by
that in reversible processes.
Review, cont.
• Specific heats: c = dq/dT
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cv =(q/T)
cp= ( q/T)p
cp = c v + R
Equipartition of energy
degree of freedom: f
u = fRT/2
• Entropy
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d = dq/T
Irreversible processes: entropy change is defined by
that in reversible processes.
• 2nd law of thermodynamics
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Review, cont.
• Specific heats: c = dq/dT
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cv =(q/T)
cp= ( q/T)p
cp = c v + R
Equipartition of energy
degree of freedom: f
u = fRT/2
• Entropy
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d = dq/T
Irreversible processes: entropy change is defined by
that in reversible processes.
• 2nd law of thermodynamics
–
d  system + d  environment  0.
Review: Processes
Isochoric:
Review: Processes
Isochoric: dq = du, dq = cv dT
Isobaric:
Review: Processes
Isochoric: dq = du, dq = cv dT
Isobaric: pV0 = const, dq = cp dT
Isothermal:
Review: Processes
Isochoric: dq = du, dq = cv dT
Isobaric: pV0 = const, dq = cp dT
Isothermal: pV1 = const, du = 0,
dq = -  dp = pd = dw
Adiabatic:
Review: Processes
Isochoric: dq = du, dq = cv dT
Isobaric: pV0 = const, dq = cp dT
Isothermal: pV1 = const, du = 0,
dq = -  dp = pd = dw
Adiabatic: pV = const, dq = 0
cp dT =  dp, cv dT =- pd
where  = cp / cv = 1+2/f
Polytropic:
Review: Processes
Isochoric: dq = du, dq = cv dT
Isobaric: pV0 = const, dq = cp dT
Isothermal: pV1 = const, du = 0,
dq = -  dp = pd = dw
Adiabatic: pV = const, dq = 0
cp dT =  dp, cv dT =- pd
where  = cp / cv = 1+2/f
Polytropic: pVn = const.
(adiabatic) Free expansion:
Review: Processes
Isochoric: dq = du, dq = cv dT
Isobaric: pV0 = const, dq = cp dT
Isothermal: pV1 = const, du = 0,
dq = -  dp = pd = dw
Adiabatic: pV = const, dq = 0
cp dT =  dp, cv dT =- pd
where  = cp / cv = 1+2/f
Polytropic: pVn = const.
(adiabatic) Free expansion: q= u= T = 0, 0
Review: Processes
Isochoric: dq = du, dq = cv dT
Isobaric: pV0 = const, dq = cp dT
Isothermal: pV1 = const, du = 0,
dq = -  dp = pd = dw
Adiabatic: pV = const, dq = 0
cp dT =  dp, cv dT =- pd
where  = cp / cv = 1+2/f
Polytropic: pVn = const.
(adiabatic) Free expansion: q= u= T = 0, 0
Homework: 1.1, 1.2, and 1.3, 1.5* due on ?
Diagrams
• P-V diagram:
Diagrams
• P-V diagram: work pd,
Diagrams
• P-V diagram: work pd,
– u: state function, remains same in a cycle.
– ∮dw=∮𝑑𝑞.
Diagrams
• P-V diagram: work pd,
– u: state function, remains same in a cycle.
– ∮dw=∮𝑑𝑞.
• P-T diagram:
Diagrams
• P-V diagram: work pd,
– u: state function, remains same in a cycle.
– ∮dw=∮𝑑𝑞.
• P-T diagram:
– Where is each state, triple point
– phase transitions.
Diagrams
• P-V diagram: work pd,
– u: state function, remains same in a cycle.
– ∮dw=∮𝑑𝑞.
• P-T diagram:
– Where is each state, triple point
– phase transitions.
• e- diagram:
Diagrams
• P-V diagram: work pd,
– u: state function, remains same in a cycle.
– ∮dw=∮𝑑𝑞.
• P-T diagram:
– Where is each state, triple point
– phase transitions.
• e- diagram:
– e: vapor pressure
– phase transitions, isotherm.
Diagrams
• P-V diagram: work pd,
– u: state function, remains same in a cycle.
– ∮dw=∮𝑑𝑞.
• P-T diagram:
– Where is each state, triple point
– phase transitions.
• e- diagram:
– e: vapor pressure
– phase transitions, isotherm.
• Stüve (p –T) diagram: adiabatic
• T/ = (p/1000mb) , potential temp, =R/cp
Diagrams
• P-V diagram: work pd,
– u: state function, remains same in a cycle.
– ∮dw=∮𝑑𝑞.
• P-T diagram:
– Where is each state, triple point
– phase transitions.
• e- diagram:
– e: vapor pressure
– phase transitions, isotherm.
• Stüve (p –T) diagram: adiabatic
• T/ = (p/1000mb) , potential temp, =R/cp
• Diagrams: area of a closed path
Diagrams, cont.
• Emagram:
• Work: V is difficult to measure for a p-V diagram.
Diagrams, cont.
• Emagram:
•
•
•
•
Work: V is difficult to measure for a p-V diagram.
dw = pd = R’dT-  dp = R’dT – R’T dp/p
∮ dw = -R’ ∮ T d(lnp)
energy-per-unit-mass diagram (R’=R*/m)
Diagrams, cont.
• Emagram:
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•
•
•
Work: V is difficult to measure for a p-V diagram.
dw = pd = R’dT-  dp = R’dT – R’T dp/p
∮ dw = -R’ ∮ T d(lnp)
energy-per-unit-mass diagram (R’=R*/m)
• Tephigram:
Diagrams, cont.
• Emagram:
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•
•
•
Work: V is difficult to measure for a p-V diagram.
dw = pd = R’dT-  dp = R’dT – R’T dp/p
∮ dw = -R’ ∮ T d(lnp)
energy-per-unit-mass diagram (R’=R*/m)
• Tephigram:
• T  d = dq: heat
• dq = T d  = cp T d(ln) (note do not need closed path int.)
•  is measured by T and p.
Diagrams, cont.
• Emagram:
•
•
•
•
Work: V is difficult to measure for a p-V diagram.
dw = pd = R’dT-  dp = R’dT – R’T dp/p
∮ dw = -R’ ∮ T d(lnp)
energy-per-unit-mass diagram (R’=R*/m)
• Tephigram:
• T  d = dq: heat
• dq = T d  = cp T d(ln) (note do not need closed path int.)
•  is measured by T and p.
Homework: 1.6, due on ?
Water Vapor and Its
Thermodynamic Effects
• Equation of state for water vapor: (not for water
or ice)
• e = vRvTv=vR’Tv/ R’=R*/m for whole (dry)
air
Water Vapor and Its
Thermodynamic Effects
• Equation of state for water vapor: (not for
water or ice)
• e = vRvT=vR’T/ R’=R*/m for whole (dry)
air
• Ratio of molecular weights
•  = R’/Rv=mv/md= 0.622 ~ 18/29
Water Vapor and Its
Thermodynamic Effects
• Equation of state for water vapor: (not for
water or ice)
• e = vRvT=vR’T/ R’=R*/m for whole (dry)
air
• Ratio of molecular weights
•  = R’/Rv=mv/md= 0.622 ~ 18/29
• Thermal equilibrium
Water Vapor and Its
Thermodynamic Effects
• Equation of state for water vapor: (not for
water or ice)
• e = vRvT=vR’T/ R’=R*/md for whole (dry)
air
• Ratio of molecular weights
•  = R’/Rv=mv/md= 0.622 ~ 18/29
• Thermal equilibrium (Tvapor=Tdry)
• Saturated water vapor
Water Vapor and Its
Thermodynamic Effects
• Equation of state for water vapor: (not for
water or ice)
• e = vRvT=vR’T/ R’=R*/m for whole (dry)
air
• Ratio of molecular weights
•  = R’/Rv=mv/md= 0.622 ~ 18/29
• Thermal equilibrium
• Saturated water vapor
• Saturation pressure, es
Phase Change and Latent Heats
•When heating a piece of ice at a constant
rate, temperature increases in steps.
Phase Change and Latent Heats
•When heating a piece of ice at a constant
rate, temperature increases in steps.
•Molecules absorb energy and increase their
internal energy.
Phase Change and Latent Heats
•When heating a piece of ice at a constant
rate, temperature increases in steps.
•Molecules absorb energy and increase their
internal energy.
•Latent heat: heat supplied or taken away
from a substance during a phase change,
while the temp. remains constant.
L12=q=u2-u1+es(2-1)
Phase Change and Latent Heats
•When heating a piece of ice at a constant rate,
temperature increases in steps.
•Molecules absorb energy and increase their internal
energy.
•Latent heat: heat supplied or taken away from a
substance during a phase change, while the temp.
remains constant.
L12=q=u2-u1+es(2-1)
Heat and entropy (at constant temp)
L12=q= T  = T (2- 1)
Or
u1+es1- T1 = u2+es2- T2
Phase Change and Latent Heats
•When heating a piece of ice at a constant rate, temperature
increases in steps.
•Molecules absorb energy and increase their internal energy.
•Latent heat: heat supplied or taken away from a substance
during a phase change, while the temp. remains constant.
L12=q=u2-u1+es(2-1)
Heat and entropy (at constant temp)
L12=q= T  = T (2- 1)
Or
u1+es1- T1 = u2+es2- T2
•Gibbs function
G = u+es - T
State Functions
• Internal energy: u = fRT/2 (isothermal)
State Functions
• Internal energy: u = fRT/2 (isothermal)
• Enthalpy: h = u + p (isobaric)
State Functions
• Internal energy: u = fRT/2 (isothermal)
• Enthalpy: h = u + p (isobaric)
• Entropy: d = dq/T = (du + pd)/T
(adiabatic:   0)
State Functions
• Internal energy: u = fRT/2 (isothermal)
• Enthalpy: h = u + p (isobaric)
• Entropy: d = dq/T = (du + pd)/T
(adiabatic:   0)
• Free energy function: F= u - T
(isothermal: F  0)
State Functions
• Internal energy: u = fRT/2 (isothermal)
• Enthalpy: h = u + p (isobaric)
• Entropy: d = dq/T = (du + pd)/T
(adiabatic:   0)
• Free energy function: F= u - T
(isothermal: F  0)
• Gibbs function: G = U - T + pV
dG = du + des  + es d - dT  - T d
=  des -  dT
State Functions
• Internal energy: u = fRT/2 (isothermal)
• Enthalpy: h = u + p (isobaric)
• Entropy: d = dq/T = (du + pd)/T
(adiabatic:   0)
• Free energy function: F= u - T
(isothermal: F  0)
• Gibbs function: G = U - T + pV
dG = du + des  + es d - dT  - T d
=  des -  dT
Isobaric, isothermal: G = 0
Isobaric: G  0,
free enthalpy, thermopotential, chemical potential.
The Clausius-Clapeyron Equation
The Clausius-Clapeyron Equation
C-C equation: phase transition.
The Clausius-Clapeyron Equation
C-C equation: phase transition.
latent heat – pressure – temperature
The Clausius-Clapeyron Equation
C-C equation: phase transition.
latent heat – pressure – temperature
Boiling point:
The Clausius-Clapeyron Equation
C-C equation: phase transition.
latent heat – pressure – temperature
Boiling point: ambient pressure = es.
The Clausius-Clapeyron Equation
C-C equation: phase transition.
latent heat – pressure – temperature
Boiling point: ambient pressure = es.
Boiling temp. as function of pressure.
The Clausius-Clapeyron Equation
C-C equation: phase transition.
latent heat – pressure – temperature
Boiling point: ambient pressure = es.
Boiling temp. as function of pressure.
Clausius-Clapeyron equation:
des/dT = L12/[T(2-1)]
des/es = (mvL12/R*)(dT/T2)
ln es = - (mvL12/R*T) + const.
C-C Equation, cont.
• C-C equation: es (T) = A exp(-B/T)
A and B are different for water and ice.
C-C Equation, cont.
• C-C equation: es (T) = A exp(-B/T)
A and B are different for water and ice.
• Ratio of saturation pressures for water and ice:
es/ei = exp[(Lf/RvT0)(T0/T-1)]
C-C Equation, cont.
• C-C equation: es (T) = A exp(-B/T)
A and B are different for water and ice.
• Ratio of saturation pressures for water and ice:
es/ei = exp((Lf/RvT0)(T0/T-1))
• Near 0°C
es/ei  (273/T)2.66
(T < 273, es > ei )
C-C Equation, cont.
• C-C equation: es (T) = A exp(-B/T)
A and B are different for water and ice.
• Ratio of saturation pressures for water and ice:
es/ei = exp((Lf/RvT0)(T0/T-1))
• Near 0°C
es/ei  (273/T)2.66
(T < 273, es > ei )
• Partial pressure: e
– Saturated: e = es
– Unsaturated: e < es
– Supersaturated: e > es
C-C Equation, cont.
• C-C equation: es (T) = A exp(-B/T)
A and B are different for water and ice.
• Ratio of saturation pressures for water and ice:
es/ei = exp((Lf/RvT0)(T0/T-1))
• Near 0°C
es/ei  (273/T)2.66
(T < 273, es > ei )
• Partial pressure: e
– Saturated: e = es
– Unsaturated: e < es
– Supersaturated: e > es
• When T<273, e = es (saturated to water)
e > ei (supersaturated to ice)
Moist air: its vapor content
Vapor pressure, e: partial pressure associated with H2O.
Saturation vapor pressure, es: maximum vapor pressure
before condensation occurs, specified by C-C equation.
Absolute humidity, v: density of water vapor.
Mixing ratio: w = Mv/Md = e /(p- e) = e /p
Saturation mixing ratio: ws = es /(p- es) = es /p
Specific humidity: q = Mv/(Mv + Md ) = e /p
Relative humidity: f = w/ws= e/es
Virtual temp.: Tv = T(1+ w/)/(1+w)
=(1+0.609w)T
Thermodynamics of UNsaturated moist air
• Keep quantities in equations for dry air but
include moisture (w) – change definitions of
parameters.
• Gas constant:
Rm = R’ (1+0.6w)
• Specific heat:
cvm = (dq/dT)v = cv (1+wr)/(1+w)  cv (1+w)
cpm = (dq/dT)p  cp (1+ 0.9 w)
r = cvv /cv = 1410/718 = 1.96
• Adiabatic constant:
Rm/cpm  k (1- 0.26w)
HW: 2.2 and 2.5, *2.5 for grads
• Ways of Reaching Saturation: Want e => es
• Ways of Reaching Saturation: Want e => es
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–
–
–
three measured quantities, T, p, w
Cooling
Decreasing pressure
Adding moisture
• Ways of Reaching Saturation: Want e => es
–
–
–
–
three measured quantities, T, p, w
Cooling
Decreasing pressure
Adding moisture
• Dew-point temp,
• Ways of Reaching Saturation: Want e => es
–
–
–
–
three measured quantities, T, p, w
Cooling
Decreasing pressure
Adding moisture
• Dew-point temp, Td: when cooling to Td holding p and w
constant so that w = ws .
Td = B/ln(A/wp)
• Ways of Reaching Saturation: Want e => es
–
–
–
–
three measured quantities, T, p, w
Cooling
Decreasing pressure
Adding moisture
• Dew-point temp, Td: when cooling to Td holding p and w
constant so that w = ws .
Td = B/ln(A/wp)
• Frost-point temp: Tf < 273K, when w=wi
• Ways of Reaching Saturation: Want e => es
–
–
–
–
three measured quantities, T, p, w
Cooling
Decreasing pressure
Adding moisture
• Dew-point temp, Td: when cooling to Td holding p and w
constant so that w = ws .
Td = B/ln(A/wp)
• Frost-point temp: Tf < 273K, when w=wi
• Wet-bulb temp.: Tw,
• Ways of Reaching Saturation: Want e => es
–
–
–
–
three measured quantities, T, p, w
Cooling
Decreasing pressure
Adding moisture
• Dew-point temp, Td: when cooling to Td holding p and w
constant so that w = ws .
Td = B/ln(A/wp)
• Frost-point temp: Tf < 273K, when w=wi
• Wet-bulb temp.: Tw, temp to which air may be cooled by
evaporating water into it holding p const.
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–
–
–
–
Evaporation: adding w while taking away heat.
T decreases and w increases.
Heat loss equals latent heat consumed: cpdT = -Ldw.
However: decrease in T reduces es.
Adding w increases e =>es; w = e /p
• Equivalent temp.: Te, temp a sample of moist
air would attain if all the moisture were
condensed out at constant pressure.
Te = T + Lw/cp
Te > T.
• Equivalent temp.: Te, temp a sample of moist
air would attain if all the moisture were
condensed out at constant pressure.
Te = T + Lw/cp
Te > T.
• Isentropic condensation temp.: Tc, temp.
cooling adiabatically to saturation holding w
const while T and p change.
Tc = B/ln [A/wp0(T0/Tc)1/k]
Tc/To = (pc/p0)k
Pseudoadiabatic Process
Adiabatic: dQ = 0
Dry/ unsaturated/ saturated air.
Pseudoadiabatic Process
Adiabatic: dQ = 0
Dry/ unsaturated/ saturated air.
Adiabatic cooling of a moist air:
condensation =>
Latent heat release =>
less cooling than in dry air or unsaturated air.
Pseudoadiabatic Process
Adiabatic: dQ = 0
Dry/ unsaturated/ saturated air.
Adiabatic cooling of a moist air:
condensation =>
Latent heat release =>
less cooling than in dry air or unsaturated air.
Adiabatic process:
water droplets/ice crystals remain suspended in the air.
dQair=dMair=0, reversible
Pseudoadiabatic Process
Adiabatic: dQ = 0
Dry/ unsaturated/ saturated air.
Adiabatic cooling of a moist air:
condensation =>
Latent heat release =>
less cooling than in dry air or unsaturated air.
Adiabatic process:
water droplets/ice crystals remain suspended in the air.
dQair=dMair=0, reversible
Pseudoadiabatic process:
condensation to water droplets/ice crystals and precipitation occurs.
dQair = Ls > 0, dMair < 0
irreversible.
Pseudoadiabatic process, cont.
• Energy conservation.
Pseudoadiabatic process, cont.
• Energy conservation.
dQ = Cp dT – Vdp
-L dws = cp dT -  dp
Pseudoadiabatic process, cont.
• Energy conservation.
dQ = Cp dT – Vdp
-L dws = cp dT -  dp
• Pseudoadiabatic equation.
dT/T = k dp/p – (L/Tcp) dws how to obtain dws ?
Pseudoadiabatic process, cont.
• Energy conservation.
dQ = Cp dT – Vdp
-L dws = cp dT -  dp
• Pseudoadiabatic equation.
dT/T = k dp/p – (L/Tcp) dws
dws = (BdT/T2 – dp/p) (A/p)e-B/T
Pseudoadiabatic process, cont.
• Energy conservation.
dQ = Cp dT – Vdp
-L dws = cp dT -  dp
• Pseudoadiabatic equation.
dT/T = k dp/p – (L/Tcp) dws
dws = (BdT/T2 – dp/p) (A/p)e-B/T
• Tephigram.
Pseudoadiabatic process, cont.
• Energy conservation.
dQ = Cp dT – Vdp
-L dws = cp dT -  dp
• Pseudoadiabatic equation.
dT/T = k dp/p – (L/Tcp) dws
dws = (BdT/T2 – dp/p) (A/p)e-B/T
• Tephigram.
• Water mixing ratio:  (= 0 before saturation)
d = -dws
: weight percentage of liquid form of H2O
Pseudoadiabatic process, cont.
• Energy conservation.
dQ = Cp dT – Vdp
-L dws = cp dT -  dp
• Pseudoadiabatic equation.
dT/T =  dp/p – (L/Tcp) dws
dws = (BdT/T2 – dp/p) (A/p)e-B/T
• Tephigram.
• Water mixing ratio:  (= 0 before saturation)
d = -dws
: weight percentage of liquid form of H2O
• Total water density .
• HW 2.1, 2.2, 2.3, 2.5, *2.4
Reversible saturated adiabatic
process
• Total H2O mixing ratio: Q = ws +
Reversible saturated adiabatic
process
• Total H2O mixing ratio: Q = ws +
• Entropy of cloudy air
 = d + wsv + w
Reversible saturated adiabatic
process
• Total H2O mixing ratio: Q = ws +
• Entropy of cloudy air
 = d + wsv + w
But v = w + L/T
 = d + w Q + (L/T) ws
Reversible saturated adiabatic
process
• Total H2O mixing ratio: Q = ws +
• Entropy of cloudy air
 = d + wsv + w
But v = w + L/T
 = d + w Q + (L/T) ws
• Isentropic
dd = 0 = dd + d(w Q) + d(L ws /T)
Reversible saturated adiabatic
process
• Total H2O mixing ratio: Q = ws +
• Entropy of cloudy air
 = d + wsv + w
But v = w + L/T
 = d + w Q + (L/T) ws
• Isentropic
dd = 0 = dd + d(w Q) + d(L ws /T)
(cp + Q cw) d(ln T) – R’d(ln pd) + d(Lws/T) = 0
Problem 2.3
Household humidifiers work by evaporating water
into the air of a confined space and raising its
relative humidity. A large room with a volume of
100 m3 contains air at 23°C with a relative
humidity of 15%. Compute the amount of water
that must be evaporated to raise the relative
humidity to 65%. Assume an isobaric process at
100 kPa in which the heat required to evaporate
the water is supplied by the air.
instability analysis
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
• Looking for the solution for d2x/dt2 = -kx
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
• Looking for the solution for d2x/dt2 = -kx
• Assuming x = A sin t + B cos t, or Csin(t + )
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
• Looking for the solution for d2x/dt2 = -kx
• Assuming x = A sin t + B cos t, or Csin(t + )
Obtain: 2 = k, where  is the frequency.
(t + ) is called the phase.
This is NOT a general solution, i.e. Ct+D is also a solution
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
• Looking for the solution for d2x/dt2 = -kx
• Assuming x = A sin t + B cos t, or Csin(t + )
Obtain: 2 = k, where  is the frequency.
(t + ) is called the phase.
• In exponential notation
C (cos (t + ) + i sin (t + ) ) = C ei (t + )
x = Cei (t + )
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
• Looking for the solution for d2x/dt2 = -kx
• Assuming x = A sin t + B cos t, or Csin(t + )
Obtain: 2 = k, where  is the frequency.
(t + ) is called the phase.
• In exponential notation
C (cos (t + ) + i sin (t + ) ) = C ei (t + )
x = Cei (t + )
The real part is cosine and imaginary part sine.
If  = -i , x is not oscillatory. (k <0)
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
• Looking for the solution for d2x/dt2 = -kx
• Assuming x = A sin t + B cos t, or Csin(t + )
Obtain: 2 = k, where  is the frequency.
(t + ) is called the phase.
• In exponential notation
C (cos(t + ) + i sin (t + ) ) = C ei (t + )
x = Cei (t + )
The real part is cosine and imaginary part sine.
If  = -i , x is not oscillatory. (k <0)
If  > 0, x decreases => damping.
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
• Looking for the solution for d2x/dt2 = -kx
• Assuming x = A sin t + B cos t, or Csin(t + )
Obtain: 2 = k, where  is the frequency.
(t + ) is called the phase.
• In exponential notation
C (cos (t + ) + i sin (t + ) ) = C ei (t + )
x = Cei (t + )
The real part is cosine and imaginary part sine.
If  = -i , x is not oscillatory. (k <0)
If  < 0, x decreases => damping.
If  > 0, x increases => unstable or growth.
Mathematical background for
instability analysis
• A small perturbation near equilibrium.
• Looking for the solution for d2x/dt2 = -kx
• Assuming x = A sin t + B cos t, or Csin(t + )
Obtain: 2 = k, where  is the frequency.
(t + ) is called the phase.
• In exponential notation
C (cos (t + ) + i sin (t + ) ) = C ei (t + )
x = Ce  i (t + )
The real part is cosine and imaginary part sine.
If  = i , x is not oscillatory. (k <0)
If  < 0, x decreases => damping.
If  > 0, x increases => unstable or growth.
Oscillatory solution condition: k > 0.
(stable solution)
Unstable condition
k < 0.
Vector Analysis
• Gradient: pointing to the direction with
maximum increase in value of a scalar field
• p = p/x i + p/y j + p/z k
• Pressure force per volume is -p
• In general, pressure is a tensor!
• Navier-Stokes equation.
Recap: Instability Analysis
• Deriving the equilibrium condition (/t = 0)
0 = ma = S F(x = 0)
• Taking a small perturbation x near the equilibrium.
ma = S F(x) = D1x + D2 x2 + …
• Deriving the linear momentum equation d2x/dt2 = -kx
• Its general solution is
x = C ei(t + )
• Oscillatory solution condition: k > 0.
2 = k, where  is the frequency.
(t + ) is the phase.
• When k < 0,
 = i , x is not oscillatory.
For e-it
If  < 0, x decreases => damping.
If  > 0, x increases => unstable or growth.
• Instability is a mechanism that converts other types of energy into
kinetic energy
Pressure Force
• Force:
• Thermal pressure force? pd
– A simplification (no time dependence
– A force is associated with pressure difference
• Pressure gradient force
• Gradient: pointing to the direction with maximum increase
in value of a scalar field
• p = p/x i + p/y j + p/z k
• Pressure force per volume is -p
• In 1-D, -p = -dp/dz k
• pd is time integrated total energy change of the volume
Hydrostatic Equilibrium
• Atmospheric stratification (1-D approximation).
Lx, Ly >> H
Hydrostatic Equilibrium
• Atmospheric stratification (1-D approximation).
Lx, Ly >> H
0 = ma = S F = F1 + F2 + …
Hydrostatic Equilibrium
• Atmospheric stratification (1-D approximation).
Lx, Ly >> H
0 = ma = S F = F1 + F2 + …
• 1-D steady state equation (force balance).
dp/dz = - g
dp/p = - (g/R’Tv)dz
Hydrostatic Equilibrium
• Atmospheric stratification (1-D approximation).
Lx, Ly >> H
0 = ma = S F = F1 + F2 + …
• 1-D steady state equation (force balance).
dp/dz = - g
dp/p = - (g/R’Tv)dz
• Dry adiabatic equilibrium (dq = 0)
0 = cpdT – (R’T/p) dp
dT/dz = - g/cp  -
Hydrostatic Equilibrium
• Atmospheric stratification (1-D approximation).
Lx, Ly >> H
0 = ma = S F = F1 + F2 + …
• 1-D steady state equation (force balance).
dp/dz = - g
dp/p = - (g/R’Tv)dz
• Dry adiabatic equilibrium (dq = 0)
0 = cpdT – (R’T/p) dp
dT/dz = - g/cp  -
• Dry adiabatic lapse rate
  1K/100m
Hydrostatic Equilibrium
• Atmospheric stratification (1-D approximation).
Lx, Ly >> H
0 = ma = S F = F1 + F2 + …
• 1-D steady state equation (force balance).
dp/dz = - g
dp/p = - (g/R’Tv)dz
• Dry adiabatic equilibrium (dq = 0)
0 = cpdT – (R’T/p) dp
dT/dz = - g/cp  -
• Dry adiabatic lapse rate
  1K/100m
• The adiabatic lapse rate describes:
• Steady state temp. adiabatic height profile
Hydrostatic Equilibrium
• Atmospheric stratification (1-D approximation).
Lx, Ly >> H
0 = ma = S F = F1 + F2 + …
• 1-D steady state equation (force balance).
dp/dz = - g
dp/p = - (g/R’Tv)dz
• Dry adiabatic equilibrium (dq = 0)
0 = cpdT – (R’T/p) dp
dT/dz = - g/cp  -
• Dry adiabatic lapse rate
  1K/100m
• The adiabatic lapse rate describes:
• Steady state temp. adiabatic height profile
• Temp of an air parcel adiabatically moving in height (although no
flow is allowed)
Parcel Buoyancy and Atmospheric Stability
• Buoyancy force: the net force that a parcel of air with a small density
difference  from ambient air feels is
FB = -g /0 = g T/T0 = d2z/dt2
- =  - 0, T = T- T0
Parcel Buoyancy and Atmospheric Stability
• Buoyancy force: the net force that a parcel of air with a small density
difference  from ambient air feels is
FB = -g /0 = g T/T0 = d2z/dt2
- =  - 0, T = T- T0
• In air of lapse rate 
At Z0 + z, Tair = T0 -  z
Parcel Buoyancy and Atmospheric Stability
• Buoyancy force: the net force that a parcel of air with a small density
difference  from ambient air feels is
FB = -g /0 = g T/T0 = d2z/dt2
- =  - 0, T = T- T0
• In air of lapse rate 
At Z0 + z, Tair = T0 -  z
• An air parcel adiabatically moves from Z0 to Z0 + z
Tparcel = T0 -  z
Parcel Buoyancy and Atmospheric Stability
• Buoyancy force: the net force that a parcel of air with a small density
difference  from ambient air feels is
FB = -g /0 = g T/T0 = d2z/dt2
- =  - 0, T = T- T0
• In air of lapse rate 
At Z0 + z, Tair = T0 -  z
• An air parcel adiabatically moves from Z0 to Z0 + z
Tparcel = T0 -  z
• Buoyancy force exerted on the parcel
T = Tparc- Tair = - (-  ) z
d2z/dt2 = FB = g T/T = - (g/T)(-  )z
Parcel Buoyancy and Atmospheric Stability
• Buoyancy force: the net force that a parcel of air with a small density
difference  from ambient air feels is
FB = -g /0 = g T/T0 = d2z/dt2
- =  - 0, T = T- T0
• In air of lapse rate 
At Z0 + z, Tair = T0 -  z
• An air parcel adiabatically moves from Z0 to Z0 + z
Tparcel = T0 -  z
• Buoyancy force exerted on the parcel
T = Tparc- Tair = - (-  ) z
d2z/dt2 = FB = g T/T = - (g/T)(-  )z
When  < : stable
When  = : neutral
When  > : unstable
Parcel Buoyancy and Atmospheric Stability
• Buoyancy force: the net force that a parcel of air with a small density
difference  from ambient air feels is
FB = -g /0 = g T/T0 = d2z/dt2
- =  - 0, T = T- T0
• In air of lapse rate 
At Z0 + z, Tair = T0 -  z
• An air parcel adiabatically moves from Z0 to Z0 + z
Tparcel = T0 -  z
• Buoyancy force exerted on the parcel
T = Tparc- Tair = - (-  ) z
d2z/dt2 = FB = g T/T = - (g/T)(-  )z
When  < : stable
When  = : neutral
When  > : unstable
• Air of smaller lapse rate is stable: cloudy day.
• Air of larger lapse rate is unstable: clear day.
HW #1, #5
Schedule
•
•
•
•
•
•
•
HW 3: Oct 30
HW 4: Nov 8
Chapters 5 and 13 presentations: Nov13, 15 (10%)
Exam II: Dec 4 (30%).
Project presentations: Dec 6 (two hours) (10%).
Project reports due Dec 21 (10%).
Report format: title, author, affiliation, abstract,
introduction, body of study, discussion,
conclusions/summary, acknowledgments,
references.
Recap: Atmospheric Stability Analysis
• Deriving the equilibrium condition (/t = 0)
– Assume a given lapse rate of ambient air 
(Tair = T0 -  z)
• Taking a small perturbation x near the equilibrium.
– Assume a parcel moving adiabatically in height, lapse rate 
– Buoyancy force: FB = -g /0 = g T/T0 .
• Deriving the linear momentum equation d2x/dt2 = -kx
– d2z/dt2 = - (g/T)(-  )z
– k = (g/T)(-  )
• Oscillatory (stable) solution condition: k > 0, 2 = k,  is
the frequency.
–  < , frequency  = [(g/T)(-  )]1/2 , 8 min (Brunt-Vaisala)
• Non-oscillatory (unstable) solution condition: k < 0,
–  > . (colder air on top: has to be colder than adiabatic cooling)
Stability criteria for dry air
• d/ = dT/T – dp/p
Stability criteria for dry air
• d/ = dT/T – dp/p
• (1/)/z = (1/T)T/z – (/p)p/z
= (1/T)(-)
Stability criteria for dry air
• d/ = dT/T – dp/p
• (1/)/z = (1/T)T/z – (/p)p/z
= (1/T)(-)
• Stable:  <  => /z > 0
• Unstable:  >  => /z < 0
• Neutral:  =  => /z = 0
Stability criteria for moist air
• Pseudoadiabatic lapse rate
dT/dz = (T/p) dp/dz – (L/cp)dws/dz
Stability criteria for moist air
• Pseudoadiabatic lapse rate
dT/dz = (T/p) dp/dz – (L/cp)dws/dz
s  - dT/dz
= [1+Lws/R’T]/[1 + L2ws/R’cpT2]
Always: L/cpT > 1, => s < 
Stability criteria for moist air
• Pseudoadiabatic lapse rate
dT/dz = (T/p) dp/dz – (L/cp)dws/dz
s  - dT/dz
= [1+Lws/R’T]/[1 + L2ws/R’cpT2]
Always: L/cpT > 1, => s < 
• Conditions
–
–
–
–
–
Absolutely stable:  < s
Saturated neutral:  = s
Conditionally unstable: s <  < 
Dry neutral:  = 
Absolutely unstable:  > 
Nonlinear Effects
• Terms of 2nd order or higher in the perturbation
equation:
Nonlinear Effects
• Terms of 2nd order or higher in the perturbation
equation:
– Acting along with the instability: more unstable
– Acting against the instability: saturation.
Nonlinear Effects
• Terms of 2nd order or higher in the perturbation
equation:
– Acting along with the instability: more unstable
– Acting against the instability: saturation.
• Convective Instability: finite size of parcel.
Nonlinear Effects
• Terms of 2nd order or higher in the perturbation
equation:
– Acting along with the instability: more unstable
– Acting against the instability: saturation.
• Convective Instability: finite size of parcel.
– Mass conservation: stretching in height, z when lifted.
Nonlinear Effects
• Terms of 2nd order or higher in the perturbation
equation:
– Acting along with the instability: more unstable
– Acting against the instability: saturation.
• Convective Instability: finite size of parcel.
– Mass conservation: stretching in height, z when lifted.
– Into region of lower ambient temp/pressure
Nonlinear Effects
• Terms of 2nd order or higher in the perturbation
equation:
– Acting along with the instability: more unstable
– Acting against the instability: saturation.
• Convective Instability: finite size of parcel.
–
–
–
–
Mass conservation: stretching in height, z when lifted.
Into region of lower ambient temp/pressure
Unstable (requires lower amb. temp) less easy
Stable (requires higher amb. temp) less easy
Nonlinear Effects
• Terms of 2nd order or higher in the perturbation
equation:
– Acting along with the instability: more unstable
– Acting against the instability: saturation.
• Convective Instability: finite size of parcel.
–
–
–
–
–
Mass conservation: stretching in height, z when lifted.
Into region of lower ambient temp/pressure
Unstable (requires lower amb. temp) less easy
Stable (requires higher amb. temp) less easy
Acting against linear effect
Nonlinear Effects
• Terms of 2nd order or higher in the perturbation
equation:
– Acting along with the instability: more unstable
– Acting against the instability: saturation.
• Convective Instability: finite size of parcel.
– Mass conservation: stretching in height, z when lifted.
– Into region of lower ambient temp/pressure
– Unstable (requires lower amb. temp) less easy
– Stable (requires higher amb. temp) less easy
– Acting against linear effect
– Moist air: condensation occurs at the low temp end
– Temp of the parcel increases => more unstable
– Stable => abs. or cond. unstable.
HW 3.3
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
• Acceleration in inertial frame of reference
a = a’ + /t  r - 2r + 2v’
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
• Acceleration in inertial frame of reference
a = a’ + /t  r - 2r + 2v’
• Acceleration in rotating frame of reference
a’ = a - /t  r + 2r - 2v’
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
• Acceleration in inertial frame of reference
a = a’ + /t  r - 2r + 2v’
• Acceleration in rotating frame of reference
a’ = a - /t  r + 2r - 2v’
• Forces in rotating frame of reference
ma’ = F - m/t  r + m2r – 2mv’
Inertial force, centrifugal force, Coriolis force.
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
• Acceleration in inertial frame of reference
a = a’ + /t  r - 2r + 2v’
• Acceleration in rotating frame of reference
a’ = a - /t  r + 2r - 2v’
• Forces in rotating frame of reference
ma’ = F - m/t  r + m2r – 2mv’
Inertial force, centrifugal force, Coriolis force.
• Coriolis parameter: (projection on horizontal plane)
f = 2sin
: lat.
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
• Acceleration in inertial frame of reference
a = a’ + /t  r - 2r + 2v’
• Acceleration in rotating frame of reference
a’ = a - /t  r + 2r - 2v’
• Forces in rotating frame of reference
ma’ = F - m/t  r + m2r – 2mv’
Inertial force, centrifugal force, Coriolis force.
• Coriolis parameter: (projection on horizontal plane)
f = 2sin
: lat.
• Horizontal force balance (ug: in x, east; and vg in y, north) (2-D)
p/x = fvg
p/y = -fug
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
• Acceleration in inertial frame of reference
a = a’ + /t  r - 2r + 2v’
• Acceleration in rotating frame of reference
a’ = a - /t  r + 2r - 2v’
• Forces in rotating frame of reference
ma’ = F - m/t  r + m2r – 2mv’
Inertial force, centrifugal force, Coriolis force.
• Coriolis parameter: (projection on horizontal plane)
f = 2sin
: lat.
• Horizontal force balance (ug: in x, east; and vg in y, north) (2-D)
p/x = fvg
p/y = -fug
• Geostrophic wind (steady state): along isobars.
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
• Acceleration in inertial frame of reference
a = a’ + /t  r - 2r + 2v’
• Acceleration in rotating frame of reference
a’ = a - /t  r + 2r - 2v’
• Forces in rotating frame of reference
ma’ = F - m/t  r + m2r – 2mv’
Inertial force, centrifugal force, Coriolis force.
• Coriolis parameter: (projection on horizontal plane)
f = 2sin
: lat.
• Horizontal force balance (ug: in x, east; and vg in y, north) (2-D)
p/x = fvg
p/y = -fug
• Geostrophic wind (steady state): along isobars.
• Geostrophic wind shear: variation of geo. wind with height. (3-D)
Effects of Horizontal Motion (3-D Equilibrium)
• Coriolis force
• Acceleration in inertial frame of reference
a = a’ + /t  r - 2r + 2v’
• Acceleration in rotating frame of reference
a’ = a - /t  r + 2r - 2v’
• Forces in rotating frame of reference
ma’ = F - m/t  r + m2r – 2mv’
Inertial force, centrifugal force, Coriolis force.
• Coriolis parameter: (projection on horizontal plane)
f = 2sin
: lat.
• Horizontal force balance (ug: in x, east; and vg in y, north) (2-D)
p/x = fvg
p/y = -fug
• Geostrophic wind (steady state): along isobars.
• Geostrophic wind shear: variation of geo. wind with height. (3-D)
• Thermal wind: difference between the geo wind at two levels
HW 3.3
2-D Instabilities
• Slantwise displacement (perturbation equation)
– Parcel Temp:
– Ambient Temp:
T par  T   T  p / p
T am b  T  T  /    T  p / p
2-D Instabilities
• Slantwise displacement (perturbation equation)
– Parcel Temp:
– Ambient Temp:
– Temp difference:
T par  T   T  p / p
T am b  T  T  /    T  p / p
T par  T am b   T  /   

  
T   

z


y





  z 
 y 

2-D Instabilities
• Slantwise displacement (perturbation equation)
– Parcel Temp:
– Ambient Temp:
– Temp difference:
– Vertical force:
T par  T   T  p / p
T am b  T  T  /    T  p / p
T par  T am b   T  /   
FB  

  
T   

z


y





  z 
 y 


  
g   

z


y





  z 

y



2-D Instabilities
• Slantwise displacement (perturbation equation)
– Parcel Temp:
– Ambient Temp:
– Temp difference:
– Vertical force:
– Horizontal force:
T par  T   T  p / p
T am b  T  T  /    T  p / p
T par  T am b   T  /   
FB  

  
T   

z


y





  z 
 y 


  
g   

z


y





  z 

y



 u g
u g 


FH  f 
z f 
 y
y 

 z

2-D Instabilities
• Slantwise displacement (perturbation equation)
– Parcel Temp:
– Ambient Temp:
– Temp difference:
T par  T   T  p / p
T am b  T  T  /    T  p / p
T par  T am b   T  /   
– Vertical force:
FB  
– Horizontal force:
– Perturbation equation:
d 

  
T   

z


y





  z 
 y 


  
g   

z


y





  z 

y



 u g
u g 


FH  f 
z f 
 y
y 

 z

2
dt
2
 F B sin   F H cos 
 
 u g
u g 



  
g   

z


y
sin


f

z

f


y

 cos 







  z 

y

z

y







2-D Instabilities
• Slantwise displacement (perturbation equation)
– Parcel Temp:
– Ambient Temp:
– Temp difference:
T par  T   T  p / p
T am b  T  T  /    T  p / p
T par  T am b   T  /   
– Vertical force:
FB  
– Horizontal force:
– Perturbation equation:
d 

  
T   

z


y





  z 
 y 


  
g   

z


y





  z 

y



 u g
u g 


FH  f 
z f 
 y
y 

 z

2
dt
2
 F B sin   F H cos 
 
 u g
u g 



  
g   

z


y
sin


f

z

f


y

 cos 







  z 

y

z

y







• 1-D instability: y = 0,  = 90°
• Baroclinic instability: FH = 0
• Symmetric instability: FB = 0
2-D Instabilities, cont.
• Baroclinic inst.
z
 / y 
 g  
 

y
sin







z

y


/

z




d 
2
dt
2
 slope < parcel slope : stable
=
neutral
>
• Symmetric inst.
unstable
d 
2
dt
2
u g   z
f  u g / y
 f  y cos 



z   y
u g / z
 surface slope < abs. vort. slope : stable
=
neutral
>
unstable
• Geopotential
gpm ( z ) 
 (z)
g0

1
g0

z
0
gdz ' 
g
g0
z




Geopotential
gpm ( z ) 
 (z)
g0

1
g0

z
gdz ' 
0
HW 3.1, 3.2, 3.3 and 3.5, *3.6, *3.8
g
g0
z
Mixing and Convection
• Isobaric mixing: same pressure
– simple mass-weighted mean of humidity, q, mixing
ratio, w, vapor pressure, e, temp., T, poten. temp, total
mixing ratio
Mixing and Convection
• Isobaric mixing: same pressure
– simple mass-weighted mean of humidity, q, mixing
ratio, w, vapor pressure, e, temp., T, poten. temp, total
mixing ratio
– C-C equation
Mixing and Convection
• Isobaric mixing: same pressure
– simple mass-weighted mean of humidity, q, mixing
ratio, w, vapor pressure, e, temp., T, poten. temp, total
mixing ratio
– C-C equation
– Possibility for condensation: breath in cold weather
Mixing and Convection
• Isobaric mixing: same pressure
– simple mass-weighted mean of humidity, q, mixing
ratio, w, vapor pressure, e, temp., T, poten. temp, total
mixing ratio
– C-C equation
– Possibility for condensation: breath in cold weather
– Latent heat release dq = -Ldws
– Temp and saturation vapor pressure increase
– Isobaric: de/dT  -pcp/L
Mixing and Convection
• Isobaric mixing: same pressure
– simple mass-weighted mean of humidity, q, mixing
ratio, w, vapor pressure, e, temp., T, poten. temp, total
mixing ratio
– C-C equation
– Possibility for condensation: breath in cold weather
– Latent heat release dq = -Ldws
– Temp and saturation vapor pressure increase
– Isobaric: de/dT  -pcp/L
• Adiabatic mixing: different pressures
– Adiabatic: Potential temp: mass-weighted mean
Mixing and Convection
• Isobaric mixing: same pressure
– simple mass-weighted mean of humidity, q, mixing
ratio, w, vapor pressure, e, temp., T, poten. temp, total
mixing ratio
– C-C equation
– Possibility for condensation: breath in cold weather
– Latent heat release dq = -Ldws
– Temp and saturation vapor pressure increase
– Isobaric: de/dT  -pcp/L
• Adiabatic mixing: different pressures
– Adiabatic: Potential temp: mass-weighted mean
HW 4.1
Presentations of Observations
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
• Sunlight heats the ground. Temp gradient reverses.
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
• Sunlight heats the ground. Temp gradient reverses.
• When the temp gradient is greater than the adiabatic lapse
rate, the instability occurs.
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
• Sunlight heats the ground. Temp gradient reverses.
• When the temp gradient is greater than the adiabatic lapse
rate, the instability occurs.
• Cold air convects down, until the adiabatic lapse rate is
reached in a “mixing layer”.
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
• Sunlight heats the ground. Temp gradient reverses.
• When the temp gradient is greater than the adiabatic lapse
rate, the instability occurs.
• Cold air convects down, until the adiabatic lapse rate is
reached in a “mixing layer”.
• As the surface temp further increases, the mixing layer
thickens.
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
• Sunlight heats the ground. Temp gradient reverses.
• When the temp gradient is greater than the adiabatic lapse
rate, the instability occurs.
• Cold air convects down, until the adiabatic lapse rate is
reached in a “mixing layer”.
• As the surface temp further increases, the mixing layer
thickens.
• This process is equivalent to an upward heat propagation
(from the surface).
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
• Sunlight heats the ground. Temp gradient reverses.
• When the temp gradient is greater than the adiabatic lapse
rate, the instability occurs.
• Cold air convects down, until the adiabatic lapse rate is
reached in a “mixing layer”.
• As the surface temp further increases, the mixing layer
thickens.
• This process is equivalent to an upward heat propagation
(from the surface).
• The total heat added equals the area between the original
and final temp profiles.
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
• Sunlight heats the ground. Temp gradient reverses.
• When the temp gradient is greater than the adiabatic lapse
rate, the instability occurs.
• Cold air convects down, until the adiabatic lapse rate is
reached in a “mixing layer”.
• As the surface temp further increases, the mixing layer
thickens.
• This process is equivalent to an upward heat propagation
(from the surface).
• The total heat added equals the area between the original
and final temp profiles.
• Condensation occurs at intersection of constant ws line and
temp profile.
Convective Condensation Level (CCL)
• Before sunrise: the surface temp is low (stable)
• Sunlight heats the ground. Temp gradient reverses.
• When the temp gradient is greater than the adiabatic lapse
rate, the instability occurs.
• Cold air convects down, until the adiabatic lapse rate is
reached in a “mixing layer”.
• As the surface temp further increases, the mixing layer
thickens.
• This process is equivalent to an upward heat propagation
(from the surface).
• The total heat added equals the area between the original
and final temp profiles.
• Condensation occurs at intersection of constant ws line and
temp profile.
• CCL: condensation height: bases of cumulus clouds.
Elementary Parcel Theory
• What happens when heated from Earth’s surface?
• Instability and vertical convection motion
– Instability: converts potential E to kinetic E.
– Convection vel: kinetic E.
Elementary Parcel Theory
• What happens when heated from Earth’s surface?
• Instability and vertical convection motion
– Instability: converts potential E to kinetic E.
– Convection vel: kinetic E.
• Elementary parcel theory
– Force equation: d2z/dt2 = gB where B = -/0 = T/T0
– Vertical velocity U = dz/dt
U dU  gB dz
U
U
2
2
p
 U  2 g  B ( z ) dz
2
0
p0
p
 U  2 R '  (T  T ') d (ln p )
2
0
p0
Elementary Parcel Theory
• What happens when heated from Earth’s surface?
• Instability and vertical convection motion
– Instability: converts potential E to kinetic E.
– Convection vel: kinetic E.
• Elementary parcel theory
– Force equation: d2z/dt2 = gB where B = -/0 = T/T0
– Vertical velocity U = dz/dt
U dU  gB dz
U
U
2
2
p
 U  2 g  B ( z ) dz
2
0
p0
p
 U  2 R '  (T  T ') d (ln p )
2
0
p0
HW 4.1, 4.2, 4.5
Correction to Elementary Parcel Theory
• Acceleration by buoyancy force is too fast.
Correction to Elementary Parcel Theory
• Acceleration by buoyancy force is too fast.
• Burden of condensed water:
– Condensation: volume decreases/density
increases.
– Buoyancy force: decrease (less buoyant)
Correction to Elementary Parcel Theory
• Acceleration by buoyancy force is too fast.
• Burden of condensed water:
– Condensation: volume decreases/density
increases.
– Buoyancy force: decrease (less buoyant)
– However, condensation=>latent heat release=>
temp increases =>more buoyant.
Correction to Elementary Parcel Theory
• Acceleration by buoyancy force is too fast.
• Burden of condensed water:
– Condensation: volume decreases/density
increases.
– Buoyancy force: decrease (less buoyant)
– However, condensation=>latent heat release=>
temp increases =>more buoyant.
– Should the condensation reduce or increase the
upward speed in the EPT?
Correction to Elementary Parcel Theory,
cont.
• Compensating downward motions
– Mass conservation: when warm air goes up,
cooler air has to go down to fill the void.
Correction to Elementary Parcel Theory,
cont.
• Compensating downward motions
– Mass conservation: when warm air goes up,
cooler air has to go down to fill the void.
– Downward air is heated
– This will cause differences only when the
upward cooling and downward heating occur at
different rates.
Correction to Elementary Parcel Theory,
cont.
• Compensating downward motions
– Mass conservation: when warm air goes up,
cooler air has to go down to fill the void.
– Downward air is heated
– This will cause differences only when the
upward cooling and downward heating occur at
different rates.
– Slice method:
• Ascending => Pseudo-adiabatic rate s
• Descending => dry adiabatic rate d
• Since d > s Ts > T
Correction to Elementary Parcel Theory,
cont.
• Dilution by mixing
– Ambient air: cooler and drier
– Entrainment: mixing/transfer through boundaries
Correction to Elementary Parcel Theory,
cont.
• Dilution by mixing
– Ambient air: cooler and drier
– Entrainment: mixing/transfer through boundaries
• Aerodynamic resistance (when a volume of high
speed hotter gas move)
– Entrainment: cooler near the boundaries
– Cool air descends around it
– Aerodynamic resistance: when downward cooler air
moves against the upward warm air
Correction to Elementary Parcel Theory,
cont.
• Dilution by mixing
– Ambient air: cooler and drier
– Entrainment: mixing/transfer through boundaries
• Aerodynamic resistance
– Entrainment: cooler near the boundaries
– Cool air descends around it
– Aerodynamic resistance: when downward cooler air
moves against the upward warm air
– Atmospheric thermals
– Development of cumulus in the presence of a wind
shear
Formation of Cloud Droplets
•
When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
Formation of Cloud Droplets
•
•
When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
Pure water vapor condenses when f ~ nx100%.
Formation of Cloud Droplets
•
•
•
When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
Pure water vapor condenses when f ~ nx100%.
Phase transition in free space in equilibrium: latent heat only.
Formation of Cloud Droplets
•
•
•
•
When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
Pure water vapor condenses when f ~ nx100%.
Phase transition in free space in equilibrium: latent heat only.
Formation of small droplets: surface tension force, free energy barrier (diving).
Formation of Cloud Droplets
•
•
•
•
•
When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
Pure water vapor condenses when f ~ nx100%.
Phase transition in free space in equilibrium: latent heat only.
Formation of small droplets: surface tension force, free energy barrier (diving).
pdroplet = pamb + 2/r, where : tension force and r: curvature
(in the case of balloon, pin > pamb).
Converting to vapor pressure, we have es = es + 2/r
Formation of Cloud Droplets
•
•
•
•
•
•
•
When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
Pure water vapor condenses when f ~ nx100%.
Phase transition in free space in equilibrium: latent heat only.
Formation of small droplets: surface tension force, free energy barrier (diving).
pdroplet = pamb + 2/r, where : tension force and r: curvature
(in the case of balloon, pin > pamb).
Converting to vapor pressure, we have es = es + 2/r
Given that es is independent of r, es is larger with smaller r.
When r is 10-6 – 10-7 cm, the required es is significantly larger than es for a flat
surface given by the C-C equation: supersaturation.
Formation of Cloud Droplets
• When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
• Pure water vapor condenses when f ~ nx100%.
• Phase transition in free space in equilibrium: latent heat only.
• Formation of small droplets: surface tension force, free energy barrier
(diving).
• pdroplet = pamb + 2/r, where : tension force and r: curvature
(in the case of balloon, pin > pamb).
Converting to vapor pressure, we have es = es + 2/r
• Given that es is independent of r, es is larger with smaller r.
• When r is 10-6 – 10-7 cm, the required es is significantly larger than es for a
flat surface given by the C-C equation: supersaturation.
• For small r, es (r) is a function of r, des  - es (2/r2)dr,
e s ( r )  e s  exp(2 / rR  L T )
Formation of Cloud Droplets
•
•
•
•
•
•
•
•
When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
Pure water vapor condenses when f ~ nx100%.
Phase transition in free space in equilibrium: latent heat only.
Formation of small droplets: surface tension force, free energy barrier (diving).
pdroplet = pamb + 2/r, where : tension force and r: curvature
(in the case of balloon, pin > pamb).
Converting to vapor pressure, we have es = es + 2/r
Given that es is independent of r, es is larger with smaller r.
When r is 10-6 – 10-7 cm, the required es is significantly larger than es for a flat
surface given by the C-C equation: supersaturation.
For small r, es (r) is a function of r, des  - es (2/r2)dr,
e s ( r )  e s  exp(2 / rR  L T )
•
•
“Seeds”, condensation nuclei r~10-5 cm, help to increase r.
Small droplets are seeds and grow bigger (coalescence).
Formation of Cloud Droplets
• When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
• Pure water vapor condenses when f ~ nx100%.
• Phase transition in free space in equilibrium: latent heat only.
• Formation of small droplets: surface tension force, free energy barrier
(diving).
• pdroplet = pamb + 2/r, where : tension force and r: curvature
(in the case of balloon, pin > pamb).
Converting to vapor pressure, we have es = es + 2/r
• Given that es is independent of r, es is larger with smaller r.
• When r is 10-6 – 10-7 cm, the required es is significantly larger than es for a
flat surface given by the C-C equation: supersaturation.
• For small r, es (r) is a function of r, des  - es (2/r2)dr,
•
•
•
•
e s ( r )  e s  exp(2 / rR  L T )
“Seeds”, condensation nuclei r~10-5 cm, help to increase r.
Small droplets are seeds and grow bigger (coalescence).
Coalescence: forming bigger ones through collisions.
Cascading: tendency for bigger droplets to break into smaller pieces.
(breakup oil drops more easily than make bigger ones)
Formation of Cloud Droplets
• When air ascends, es decreases. Droplets should form when
e = es, or f = 100%.
• Pure water vapor condenses when f ~ nx100%.
• Phase transition in free space in equilibrium: latent heat only.
• Formation of small droplets: surface tension force, free energy barrier
(diving).
• pdroplet = pamb + 2/r, where : tension force and r: curvature
(in the case of balloon, pin > pamb).
Converting to vapor pressure, we have es = es + 2/r
• Given that es is independent of r, es is larger with smaller r.
• When r is 10-6 – 10-7 cm, the required es is significantly larger than es for a
flat surface given by the C-C equation: supersaturation.
• For small r, es (r) is a function of r, des  - es (2/r2)dr,
•
•
•
•
e s ( r )  e s  exp(2 / rR  L T )
“Seeds”, condensation nuclei r~10-5 cm, help to increase r.
Small droplets are seeds and grow bigger (coalescence).
Coalescence: forming bigger ones through collisions.
Cascading: tendency for bigger droplets to break into smaller pieces.
(breakup oil drops more easily than make bigger ones)
• Droplets in clouds r~1.8 x 10-3 cm. (stable to cascading)
Thick cloud
Haze: r1
visible
Droplet growth by condensation
• Size of droplets: larger ones tend to grow
Smaller ones tend to evaporate.
Droplet growth by condensation
• Size of droplets: larger ones tend to grow
Smaller ones tend to evaporate.
• Critical size: (~ 1 m) separate the two situations
(survive in severe weather, first settlement).
Droplet growth by condensation
• Size of droplets: larger ones tend to grow
Smaller ones tend to evaporate.
• Critical size: (~ 1 m) separate the two situations
(survive in severe weather, first settlement).
• Diffusion: controlling process before reaching critical size.
n/t = D2n
n: number density of interested molecules,
D: diffusion coefficient [L2/t]  VL
Smell, Brownian movement
Droplet growth by condensation
• Size of droplets: larger ones tend to grow
Smaller ones tend to evaporate.
• Critical size: (~ 1 m) separate the two situations
(survive in severe weather, first settlement).
• Diffusion: controlling process before reaching critical size.
n/t = D2n
n: number density of interested molecules,
D: diffusion coefficient [L2/t]  VL
Smell, Brownian movement
• Mass change:
dM/dt = 4rD (namb – nr)m0 = 4rD (vamb - vr)
Sub “r”: at the boundary
vamb > vr: droplet grows
vamb < vr: droplet evaporates
vamb : determined from air conditions
vr: depend on size, chemical composition and temp.
Droplet growth by condensation, cont.
• Condensation can occur when Tvamb > Tr : cooler seeds since esr < esamb
(cool drink containers)
Droplet growth by condensation, cont.
• Condensation can occur when Tvamb > Tr : cooler seeds since esr < esamb
(cool drink containers)
• Condensation: latent heat release => Tr > Tamb
Can condensation still occur?
Droplet growth by condensation, cont.
• Condensation can occur when Tvamb > Tr : cooler seeds since esr < esamb
(cool drink containers)
• Condensation: latent heat release => Tr > Tamb
Can condensation still occur?
• Heat transfer: dQ/dt = 4rK (Tr – Tamb)
K: thermal conductivity
Droplet growth by condensation, cont.
• Condensation can occur when Tvamb > Tr : cooler seeds since esr < esamb
(cool drink containers)
• Condensation: latent heat release => Tr > Tamb
Can condensation still occur?
• Heat transfer: dQ/dt = 4rK (Tr – Tamb)
K: thermal conductivity
• Heat budget:
Gain: latent heat LdM
Loss: conduction dQ
Change in state function: enthalpy mcpdT
M = (4/3)r3L
(4/3)r3L cp dT = L dM – dQ
Droplet growth by condensation, cont.
• Condensation can occur when Tvamb > Tr : cooler seeds since esr < esamb
(cool drink containers)
• Condensation: latent heat release => Tr > Tamb
Can condensation still occur?
• Heat transfer: dQ/dt = 4rK (Tr – Tamb)
K: thermal conductivity
• Heat budget:
Gain: latent heat LdM
Loss: conduction dQ
Change in state function: enthalpy mcpdT
M = (4/3)r3L
(4/3)r3L cp dT = L dM – dQ
• Steady state: no change in temp, heat gain = heat loss
(vamb - vr) / (Tr – Tamb) = K/LD
Droplet growth by condensation, cont.
• Condensation can occur when Tvamb > Tr : cooler seeds since esr < esamb
(cool drink containers)
• Condensation: latent heat release => Tr > Tamb
Can condensation still occur?
• Heat transfer: dQ/dt = 4rK (Tr – Tamb)
K: thermal conductivity
• Heat budget:
Gain: latent heat LdM
Loss: conduction dQ
Change in state function: enthalpy mcpdT
M = (4/3)r3L
(4/3)r3L cp dT = L dM – dQ
• Steady state: no change in temp, heat gain = heat loss
(vamb - vr) / (Tr – Tamb) = K/LD
• To have a growth in the droplet:
vamb > vr, Tvamb < Tr (counter-intuitive? For steady state, not dynamic stage)
• Condensation can occur when Tr > Tamb only under supersaturation.
Growth of Droplet Populations
• Droplets grow through diffusion when small (limited seeds)
Growth of Droplet Populations
• Droplets grow through diffusion when small (limited seeds)
• Collisions become important when droplets are big enough,
producing more seeds.
Growth of Droplet Populations
• Droplets grow through diffusion when small (limited seeds)
• Collisions become important when droplets are big enough,
producing more seeds.
• Maximum of condensation:
Supersaturation increases as droplets ascend from cloud base.
Little moisture left in the air at even higher altitudes
Growth of Droplet Populations
• Droplets grow through diffusion when small (limited seeds)
• Collisions become important when droplets are big enough,
producing more seeds.
• Maximum of condensation:
Supersaturation increases as droplets ascend from cloud base.
Little moisture left in the air at even higher altitudes
• Haze droplets (< 1 m) : maximum condensation is less
than the critical supersaturation to form cloud droplets
– Condensation  evaporation
– No net visible clouds form
– Can condensation nuclei help?
Initiation of Rain (one type: cumulus)
• Unstable air (for air not for vapor) forms warm, moist air updraft
Initiation of Rain (one type: cumulus)
• Unstable air (for air not for vapor) forms warm, moist air updraft
• Condensation occurs at convective condensation level (CCL)
Initiation of Rain (one type: cumulus)
• Unstable air (for air not for vapor) forms warm, moist air updraft
• Condensation occurs at convective condensation level (CCL)
• Cooler air comes down lowering es, mixing, lowering CCL
Initiation of Rain (one type: cumulus)
•
•
•
•
Unstable air (for air not for vapor) forms warm, moist air updraft
Condensation occurs at convective condensation level (CCL)
Cooler air comes down lowering es, mixing, lowering CCL
Supersaturation occurs above CCL
Initiation of Rain (one type: cumulus)
•
•
•
•
•
Unstable air (for air not for vapor) forms warm, moist air updraft
Condensation occurs at convective condensation level (CCL)
Cooler air comes down lowering es, mixing, lowering CCL
Supersaturation occurs above CCL
Moisture (droplets, not vapor) stays in the clouds while drier air
continues to go up
Initiation of Rain (one type: cumulus)
•
•
•
•
•
Unstable air (for air not for vapor) forms warm, moist air updraft
Condensation occurs at convective condensation level (CCL)
Cooler air comes down lowering es, mixing, lowering CCL
Supersaturation occurs above CCL
Moisture (droplets, not vapor) stays in the clouds while drier air
continues to go up
• New moist air continues to dump the moisture
Initiation of Rain (one type: cumulus)
•
•
•
•
•
Unstable air (for air not for vapor) forms warm, moist air updraft
Condensation occurs at convective condensation level (CCL)
Cooler air comes down lowering es, mixing, lowering CCL
Supersaturation occurs above CCL
Moisture (droplets, not vapor) stays in the clouds while drier air
continues to go up
• New moist air continues to dump the moisture
• Cloud droplets (heavier) grow and are suspended by the updraft
Initiation of Rain (one type: cumulus)
•
•
•
•
•
•
•
•
•
Unstable air (for air not for vapor) forms warm, moist air updraft
Condensation occurs at convective condensation level (CCL)
Cooler air comes down lowering es, mixing, lowering CCL
Supersaturation occurs above CCL
Moisture (droplets, not vapor) stays in the clouds while drier air
continues to go up
New moist air continues to dump the moisture
Cloud droplets (heavier) grow and are suspended by the updraft
Coalescence becomes more important when droplets get bigger
Large droplets become heavier than that the updraft can support and
start falling
Initiation of Rain (one type: cumulus)
•
•
•
•
•
•
•
•
•
•
Unstable air (for air not for vapor) forms warm, moist air updraft
Condensation occurs at convective condensation level (CCL)
Cooler air comes down lowering es, mixing, lowering CCL
Supersaturation occurs above CCL
Moisture (droplets, not vapor) stays in the clouds while drier air
continues to go up
New moist air continues to dump the moisture
Cloud droplets (heavier) grow and are suspended by the updraft
Coalescence becomes more important when droplets get bigger
Large droplets become heavier than that the updraft can support and
start falling
On the way falling down, collisions form bigger droplets (rain drops)
Initiation of Rain (one type: cumulus)
•
•
•
•
•
•
•
•
•
•
•
•
Unstable air (for air not for vapor) forms warm, moist air updraft
Condensation occurs at convective condensation level (CCL)
Cooler air comes down lowering es, mixing, lowering CCL
Supersaturation occurs above CCL
Moisture (droplets, not vapor) stays in the clouds while drier air
continues to go up
New moist air continues to dump the moisture
Cloud droplets (heavier) grow and are suspended by the updraft
Coalescence becomes more important when droplets get bigger
Large droplets become heavier than that the updraft can support and
start falling
On the way falling down, collisions form bigger droplets (rain drops)
20 min
PM storms
Droplet Terminal Fall Speed
• Do large drops fall faster, or smaller ones, or same?
Droplet Terminal Fall Speed
• Do large drops fall faster, or smaller ones, or same?
• Free-fall speed:
v = gt, H =  vdt = gt2/2, v =(2gH)1/2
H = 10 km, g = 9.8 m/s2, v = 400 m/s,
Cs =343 m/s
Droplet Terminal Fall Speed
• Do large drops fall faster, or smaller ones, or same?
• Free-fall speed:
v = gt, H =  vdt = gt2/2, v =(2gH)1/2
H = 10 km, g = 9.8 m/s2, v = 400 m/s,
• Friction force FR = (/2)r2u2CD
Cs =343 m/s
CD: drag coefficient, : density of ambient air
(skydive)
Droplet Terminal Fall Speed
• Do large drops fall faster, or smaller ones, or same?
• Free-fall speed:
v = gt, H =  vdt = gt2/2, v =(2gH)1/2
H = 10 km, g = 9.8 m/s2, v = 400 m/s,
Cs =343 m/s
• Friction force FR = (/2)r2u2CD
CD: drag coefficient, : density of ambient air
(skydive)
• Gravity: FG = (4/3)r3g(L-) = (4/3)r3gL
Droplet Terminal Fall Speed
• Do large drops fall faster, or smaller ones, or same?
• Free-fall speed:
v = gt, H =  vdt = gt2/2, v =(2gH)1/2
H = 10 km, g = 9.8 m/s2, v = 400 m/s,
Cs =343 m/s
• Friction force FR = (/2)r2u2CD
CD: drag coefficient, : density of ambient air
(skydive)
• Gravity: FG = (4/3)r3g(L-) = (4/3)r3gL
• Steady state: FR = FG
u2 = (8/3)rg (L/)/ CD
Low speed CD = 1/ur: u = k1 r2
High speed CD = const: u = k2 r1/2
Droplet Terminal Fall Speed
• Do large drops fall faster, or smaller ones, or same?
• Free-fall speed:
v = gt, H =  vdt = gt2/2, v =(2gH)1/2
H = 10 km, g = 9.8 m/s2, v = 400 m/s,
Cs =343 m/s
• Friction force FR = (/2)r2u2CD
CD: drag coefficient, : density of ambient air
(skydive)
• Gravity: FG = (4/3)r3g(L-) = (4/3)r3gL
• Steady state: FR = FG
u2 = (8/3)rg (L/)/ CD
Low speed CD = 1/ur: u = k1 r2
High speed CD = const: u = k2 r1/2
• Larger drops: faster, Smaller drops: slower
• Larger drops overtake and collide with smaller one
Droplet Terminal Fall Speed
• Do large drops fall faster, or smaller ones, or same?
• Free-fall speed:
v = gt, H =  vdt = gt2/2, v =(2gH)1/2
H = 10 km, g = 9.8 m/s2, v = 400 m/s,
Cs =343 m/s
• Friction force FR = (/2)r2u2CD
CD: drag coefficient, : density of ambient air
(skydive)
• Gravity: FG = (4/3)r3g(L-) = (4/3)r3gL
• Steady state: FR = FG
u2 = (8/3)rg (L/)/ CD
Low speed CD = 1/ur: u = k1 r2
High speed CD = const: u = k2 r1/2
• Larger drops: faster, Smaller drops: slower
• Larger drops overtake and collide with smaller one
• Terminal speed: < 10 m/s ,
freefall from H=5 m
Collision Efficiency
•
•
•
•
E(R,r) = x02/(R+r)2
Table 8.2: E when r peak at R0.5mm
x0: effective radius
Falling raindrops: collect all droplets within radius
x0.
– Increase the size of the drop
– Slow down by the collisions
– Prolong the interaction time: small horizontal motion of
droplets.
Formation and Growth of Ice Crystals
Formation and Growth of Ice Crystals
• Ice formation
– Freezing from water
– Sublimating directly from vapor
Formation and Growth of Ice Crystals
• Ice formation
– Freezing from water
– Sublimating directly from vapor
• If a first ice piece exists: easy to grow
– ei < es (near 0°C: es/ei  (273/T)2.66 ) equ. (2.16)
– Conditions saturated to water are supersaturated to ice
Formation and Growth of Ice Crystals
• Ice formation
– Freezing from water
– Sublimating directly from vapor
• If a first ice piece exists: easy to grow
– ei < es (near 0°C: es/ei  (273/T)2.66 ) equ. (2.16)
– Conditions saturated to water are supersaturated to ice
– In equilibrium, when droplets and crystals co-exist: condensation continue
to occur on crystals, droplets continue to evaporate =>
– Ice crystals grow and droplets shrink and disappear, Bergeron process.
Formation and Growth of Ice Crystals
• Ice formation
– Freezing from water
– Sublimating directly from vapor
• If a first ice piece exists: easy to grow
– ei < es (near 0°C: es/ei  (273/T)2.66 ) equ. (2.16)
– Conditions saturated to water are supersaturated to ice
– In equilibrium, when droplets and crystals co-exist: condensation continue
to occur on crystals, droplets continue to evaporate =>
– Ice crystals grow and droplets shrink and disappear, Bergeron process.
• Overcome surface tension: very difficult
– Require f ~ 20 for sublimation, water droplets form before it
Formation and Growth of Ice Crystals
• Ice formation
– Freezing from water
– Sublimating directly from vapor
• If a first ice piece exists: easy to grow
– ei < es (near 0°C: es/ei  (273/T)2.66 ) equ. (2.16)
– Conditions saturated to water are supersaturated to ice
– In equilibrium, when droplets and crystals co-exist: condensation continue
to occur on crystals, droplets continue to evaporate =>
– Ice crystals grow and droplets shrink and disappear, Bergeron process.
• Overcome surface tension: very difficult
– Require f ~ 20 for sublimation, water droplets form before it
– Freezing of water droplets occur at –5 > T > -40° C (droplet-ice
combination in this range)
Formation and Growth of Ice Crystals
• Ice formation
– Freezing from water
– Sublimating directly from vapor
• If a first ice piece exists: easy to grow
– ei < es (near 0°C: es/ei  (273/T)2.66 ) equ. (2.16)
– Conditions saturated to water are supersaturated to ice
– In equilibrium, when droplets and crystals co-exist: condensation continue
to occur on crystals, droplets continue to evaporate =>
– Ice crystals grow and droplets shrink and disappear, Bergeron process.
• Overcome surface tension: very difficult
– Require f ~ 20 for sublimation, water droplets form before it
– Freezing of water droplets occur at –5 > T > -40° C (droplet-ice
combination in this range)
– Vertical temp profile: colder on the top (but may have no moisture)
Formation and Growth of Ice Crystals
• Ice formation
– Freezing from water
– Sublimating directly from vapor
• If a first ice piece exists: easy to grow
– ei < es (near 0°C: es/ei  (273/T)2.66 ) equ. (2.16)
– Conditions saturated to water are supersaturated to ice
– In equilibrium, when droplets and crystals co-exist: condensation continue
to occur on crystals, droplets continue to evaporate =>
– Ice crystals grow and droplets shrink and disappear, Bergeron process.
• Overcome surface tension: very difficult
– Require f ~ 20 for sublimation, water droplets form before it
– Freezing of water droplets occur at –5 > T > -40° C (droplet-ice
combination in this range)
– Vertical temp profile: colder on the top (but may have no moisture)
• Nucleation
–
–
–
–
Water condensation on cold surface then frozen
Ice crystal surface: easy to form more lattice
Nuclei: aerosol particles and icy crystals (formed at higher altitudes)
Different seeds nucleate at different temp (table 9.1), few ~ teens negative
degrees.
Formation and Growth of Ice Crystals, cont.
• Diffusional growth of ice crystals
– Diffusion equation
– Solutions depend on shape
dm
dt
 4  C D (  v   vr )
C : capacitance, a function of shape
Formation and Growth of Ice Crystals, cont.
• Diffusional growth of ice crystals
–
–
–
–
–
dm
Diffusion equation
 4 C D (    )
dt
Solutions depend on shape
C : cap acitan ce, a fu n ctio n o f sh ap e
Latent heat to warm up the crystal
 
K

Growth depends then on temp/pressure T  T L D
Ambient conditions determine also the shape (all hexagonal)
v
v
vr
r
s
vr
Formation and Growth of Ice Crystals, cont.
• Diffusional growth of ice crystals
–
–
–
–
–
dm
Diffusion equation
 4 C D (    )
dt
Solutions depend on shape
C : cap acitan ce, a fu n ctio n o f sh ap e
Latent heat to warm up the crystal
 
K

Growth depends then on temp/pressure T  T L D
Ambient conditions determine also the shape (all hexagonal)
v
v
vr
vr
r
s
• Further growth by accretion
– General: Accretion: larger one captures smaller ones
– Special: Accretion: ice crystal captures supercooled droplets
Formation and Growth of Ice Crystals, cont.
• Diffusional growth of ice crystals
–
–
–
–
–
dm
Diffusion equation
 4 C D (    )
dt
Solutions depend on shape
C : cap acitan ce, a fu n ctio n o f sh ap e
Latent heat to warm up the crystal
 
K

Growth depends then on temp/pressure T  T L D
Ambient conditions determine also the shape (all hexagonal)
v
v
vr
vr
r
s
• Further growth by accretion
–
–
–
–
–
–
–
General: Accretion: larger one captures smaller ones
Special: Accretion: ice crystal captures supercooled droplets
Liquid-to-liquid: coalescence
Aggregation: ice crystals form snowflakes
Fast freezing: coating of rime => rimed crystals, graupel
Slow freezing: denser, hail
Free-fall speed is slower for less dense structures: forming even
bigger structures
Global Convection
• Coriolis force:
– Mathematical form, physical meanings, examples
• Forces in rotating frame of reference
ma’ = F - m/t  r + m2r – 2mv’
Initial force, centrifugal force, Coriolis force.
• Geostrophic wind:
– cyclones, anticyclones
• Thermal wind: (geostrophic wind as function of height)
– westerlies
• Global atmospheric convection patterns
• Global oceanic surface convection patterns
Mixing and Convection
• Mixing:
– mass-weighted mean, T-p diagram
• Condensation due to mixing:
– C-C equation, breath in cold weather
• Convective Condensation Level:
– processes, schematic, physical meanings
– bases of cumulus.
• Elementary parcel theory:
– potential energy=> kinetic energy
• Burden of condensed water:
– should the condensation reduce or increase the upward speed in the
EPT?
• Development of cumulus: with wind shear
Formation of Clouds and Rain
• Formation of cloud droplets:
– tension force, supersaturation, seeds,
coalescence/cascading
• Growth of droplets:
– critical size, diffusion, heat conduction
• Growth of droplet populations:
– Collisions, maximum of supercondensation, haze
• Initiation of rain:
– Convection, condensation, collisions, collection
– Terminal fall speed:
– Collision efficiency