Cloud Physics What is a cloud?.

Cloud Physics What is a cloud?

Cloud Physics What is a cloud?
Water droplets or ice crystals in the air.

Water droplets or ice crystals in the air. Why important?

Water droplets or ice crystals in the air. Why important? Precipitation Solar radiation

Cloud Physics What is a cloud? What do we want to learn?
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? What do we want to learn? Methods?
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? What do we want to learn? Methods?
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 Radar meteorology Weather modification Severe storms research Global energy balance (greenhouse effect)

Overview 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

Permanent gases Variable gases Aerosols

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 Charles’ Law
Concept of thermometer Charles’ Law

Review Zeroth law of thermodynamics Charles’ Law
Concept of thermometer Charles’ Law  /T = R/p = f(p) Define temperature, K = ?

Review Zeroth law of thermodynamics Charles’ Law Boyle’s Law
Concept of thermometer Charles’ Law  /T = R/p = f(p) Define temperature, K = ? Boyle’s Law

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)

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 = ?

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

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)

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 = cv + R

cp = cv + R Equipartition of energy

cp = cv + R Equipartition of energy degree of freedom: f u = fRT/2

Review, cont. Specific heats: c = dq/dT Entropy (3 meanings)
cv =(q/T) cp= ( q/T)p cp = cv + R Equipartition of energy degree of freedom: f u = fRT/2 Entropy (3 meanings)

Review, cont. Specific heats: c = dq/dT Entropy cv =(q/T)
cp= ( q/T)p cp = cv + R Equipartition of energy degree of freedom: f u = fRT/2 Entropy d = dq/T Irreversible processes: entropy change is defined by that in reversible processes.

Review, cont. Specific heats: c = dq/dT Entropy
cv =(q/T) cp= ( q/T)p cp = cv + R Equipartition of energy degree of freedom: f u = fRT/2 Entropy d = dq/T Irreversible processes: entropy change is defined by that in reversible processes. 2nd law of thermodynamics

Review, cont. Specific heats: c = dq/dT Entropy
cv =(q/T) cp= ( q/T)p cp = cv + R Equipartition of energy degree of freedom: f u = fRT/2 Entropy 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:

Isochoric: dq = du, dq = cv dT Isobaric:
Review: Processes Isochoric: dq = du, dq = cv dT Isobaric:

Review: Processes Isochoric: dq = du, dq = cv dT
Isobaric: pV0 = const, dq = cp dT Isothermal:

Isobaric: pV0 = const, dq = cp dT Isothermal: pV1 = const, du = 0, dq = -  dp = pd = dw Adiabatic:

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:

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:

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

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, P-T diagram:
u: state function, remains same in a cycle. ∮dw=∮𝑑𝑞. P-T diagram:

Diagrams P-V diagram: work pd, P-T diagram:
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, P-T diagram: e- diagram:
u: state function, remains same in a cycle. ∮dw=∮𝑑𝑞. P-T diagram: Where is each state, triple point phase transitions. e- diagram:

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.

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

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: Tephigram:
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:

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.

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

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= ~ 18/29

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= ~ 18/29 Thermal equilibrium

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= ~ 18/29 Thermal equilibrium (Tvapor=Tdry) Saturated water vapor

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= ~ 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.

When heating a piece of ice at a constant rate, temperature increases in steps. Molecules absorb energy and increase their internal energy.

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)

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

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)

Enthalpy: h = u + p (isobaric) Entropy: d = dq/T = (du + pd)/T (adiabatic:   0)

Enthalpy: h = u + p (isobaric) Entropy: d = dq/T = (du + pd)/T (adiabatic:   0) Free energy function: F= u - T (isothermal: F  0)

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

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

C-C equation: phase transition.

C-C equation: phase transition. latent heat – pressure – temperature

C-C equation: phase transition. latent heat – pressure – temperature Boiling point:

C-C equation: phase transition. latent heat – pressure – temperature Boiling point: ambient pressure = es.

C-C equation: phase transition. latent heat – pressure – temperature Boiling point: ambient pressure = es. Boiling temp. as function of pressure.

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.

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)]

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) (T < 273, es > ei )

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) (T < 273, es > ei ) Partial pressure: e Saturated: e = es Unsaturated: e < es Supersaturated: e > es

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) (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)  ( w)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 ( w) r = cvv /cv = 1410/718 = 1.96 Adiabatic constant: Rm/cpm  k ( w) HW: 2.2 and 2.5, *2.4 for grads

Ways of Reaching Saturation: Want e => es

three measured quantities, T, p, w Cooling Decreasing pressure Adding moisture

three measured quantities, T, p, w Cooling Decreasing pressure Adding moisture Dew-point temp,

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)

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

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,

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. Evaporation: adding w: w = e /p so that e =>es Heat loss equals latent heat consumed: cpdT = -Ldw. T decreases and w increases. Drier air: more water vapor is needed to saturate=>larger T However: decrease in T reduces es. T=Tw -T= -L/cp [(A/p)e-B/Tw-w]

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.

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: 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

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.

Energy conservation. dQ = Cp dT – Vdp -L dws = cp dT -  dp

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 ?

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

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.

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

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 +

Total H2O mixing ratio: Q = ws + Entropy of cloudy air = d + wsv + w

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

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)

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.

A small perturbation near equilibrium. Looking for the solution for d2x/dt2 = -kx

A small perturbation near equilibrium. Looking for the solution for d2x/dt2 = -kx Assuming x = A sin t + B cos t, or Csin(t + )

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

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 + )

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)

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.

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.

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 > (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-igt 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

Atmospheric stratification (1-D approximation). Lx, Ly >> H 0 = ma = S F = F1 + F2 + …

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

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  -

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

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

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

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

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: 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

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

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: Nov10 HW 4: Nov 17
Chapters 5 and 13 presentations: Dec 6 (10%) Exam II: Dec 8 (30%). Project presentations Dec 15 (1.5 hours) (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

d/ = dT/T – dp/p (1/)/z = (1/T)T/z – (/p)p/z = (1/T)(-)

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

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 < 

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

Coriolis force Acceleration in inertial frame of reference a = a’ + /t  r - 2r + 2v’

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’

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 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.

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

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.

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)

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:

Parcel Temp: Ambient Temp: Temp difference:

Parcel Temp: Ambient Temp: Temp difference: Vertical force:

Parcel Temp: Ambient Temp: Temp difference: Vertical force: Horizontal force:

Parcel Temp: Ambient Temp: Temp difference: Vertical force: Horizontal force: Perturbation equation:

Parcel Temp: Ambient Temp: Temp difference: Vertical force: Horizontal force: Perturbation equation: 1-D instability: y = 0,  = 90° Baroclinic instability: FH = 0 Symmetric instability: FB = 0

2-D Instabilities, cont. Baroclinic inst. > unstable
 slope < parcel slope : stable = neutral > unstable Symmetric inst.  surface slope < abs. vort. slope : stable = neutral > unstable Geopotential

Geopotential HW 3.1, 3.2, 3.3 and 3.5, *3.6, *3.8

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

simple mass-weighted mean of humidity, q, mixing ratio, w, vapor pressure, e, temp., T, poten. temp, total mixing ratio C-C equation

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

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

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

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)

Before sunrise: the surface temp is low (stable) Sunlight heats the ground. Temp gradient reverses.

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.

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”.

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.

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).

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.

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.

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.

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

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 HW 4.1, 4.2, 4.5

Correction to Elementary Parcel Theory
Acceleration by buoyancy force is too fast.

Acceleration by buoyancy force is too fast. Burden of condensed water: Condensation: volume decreases/density increases. Buoyancy force: decrease (less buoyant)

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.

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.

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.

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

Dilution by mixing Ambient air: cooler and drier Entrainment: mixing/transfer through boundaries

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

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%.

When air ascends, es decreases. Droplets should form when e = es, or f = 100%. Pure water vapor condenses when f ~ nx100%.

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.

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).

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

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.

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,

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, “Seeds”, condensation nuclei r~10-5 cm, help to increase r. Small droplets are seeds and grow bigger (coalescence).

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, “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)

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, “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.

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).

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

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)

Condensation can occur when Tvamb > Tr : cooler seeds since esr < esamb (cool drink containers) Condensation: latent heat release => Tr > Tamb Can condensation still occur?

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

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

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

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)

Droplets grow through diffusion when small (limited seeds) Collisions become important when droplets are big enough, producing more seeds.

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

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

Unstable air (for air not for vapor) forms warm, moist air updraft Condensation occurs at convective condensation level (CCL)

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

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

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

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

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

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

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)

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?

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

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)

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

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

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

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

Ice formation Freezing from water Sublimating directly from vapor

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

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.

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

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)

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)

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

Diffusional growth of ice crystals Diffusion equation Solutions depend on shape Latent heat to warm up the crystal Growth depends then on temp/pressure Ambient conditions determine also the shape (all hexagonal)

Diffusional growth of ice crystals Diffusion equation Solutions depend on shape Latent heat to warm up the crystal Growth depends then on temp/pressure Ambient conditions determine also the shape (all hexagonal) Further growth by accretion General: Accretion: larger one captures smaller ones Special: Accretion: ice crystal captures supercooled droplets

Diffusional growth of ice crystals Diffusion equation Solutions depend on shape Latent heat to warm up the crystal Growth depends then on temp/pressure Ambient conditions determine also the shape (all hexagonal) 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: Condensation due to 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

Cloud Physics What is a cloud?.

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