cloud physics what is a cloud?. cloud physics what is a cloud? –water droplets or ice crystals in...

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


• Precipitation• Solar radiation

Cloud Physics



• What do we want to learn?– Formation of clouds– Development of precipitation

Cloud Physics




• Methods?

Cloud Physics




• 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


• Concept of thermometer

• Charles’ Law



• Charles’ Law /T = R/p = f(p) • Define temperature, K = ?




• Boyle’s Law




• Boyle’s Law • p = RT = g(T)

• Avogadro’s law (ideal gas)





• Avogadro’s law (ideal gas)• p /T = R* / m =R (for individual gas or R’ for dry air)• m: molecular weight = ?





• Avogadro’s law (ideal gas)• p /T = R* / m • m: molecular weight = ?

• 1st law of thermodynamics





• 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 = ?


• 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


– cv =(q/T) – cp= ( q/T)p – cp = cv + R– Equipartition of energy


– cv =(q/T) – cp= ( q/T)p – cp = cv + R– Equipartition of energy – degree of freedom: f– u = fRT/2



• Entropy (3 meanings)–



• Entropy– d = dq/T– Irreversible processes: entropy change is defined by

that in reversible processes.





• 2nd law of thermodynamics–





• 2nd law of thermodynamics– d system + d environment 0.

Review: ProcessesIsochoric:

Review: ProcessesIsochoric: dq = du, dq = cv dT

Isobaric:


Isobaric: pV0 = const, dq = cp dT

Isothermal:



Isothermal: pV1 = const, du = 0,

dq = - dp = pd = dw

Adiabatic:




dq = - dp = pd = dw

Adiabatic: pV = const, dq = 0

cp dT = dp, cv dT =- pd

where = cp / cv = 1+2/f

Polytropic:




dq = - dp = pd = dw




Polytropic: pVn = const.

(adiabatic) Free expansion:




dq = - dp = pd = dw





(adiabatic) Free expansion: q= u= T = 0, 0




dq = - dp = pd = dw





(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,


– u: state function, remains same in a cycle.– ∮dw= .∮𝑑𝑞



• P-T diagram:



• P-T diagram: – Where is each state, triple point– phase transitions.




• e- diagram:




• e- diagram: – e: vapor pressure– phase transitions, isotherm.





• Stüve (p –T) diagram: adiabatic• T/ = (p/1000mb) , potential temp, =R/cp





• 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.


• 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:



• 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.



• 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= 0.622 ~ 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


• Thermal equilibrium (Tvapor=Tdry)

• Saturated water vapor




• 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.



•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+es1- T1 = u2+es2- T2

Phase Change and Latent Heats•When heating a piece of ice at a constant rate, temperature increases in steps.


•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+es1- T1 = u2+es2- T2

•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)


(adiabatic: 0)

• Free energy function: F= u - T(isothermal: F 0)


(adiabatic: 0)


• Gibbs function: G = U - T + pVdG = du + des + es d - dT - T d

= des - dT


(adiabatic: 0)


• Gibbs function: G = U - T + pVdG = du + des + es d - dT - T d

= des - dTIsobaric, isothermal: G = 0Isobaric: G 0, free enthalpy, thermopotential, chemical potential.

The Clausius-Clapeyron Equation

The Clausius-Clapeyron EquationC-C equation: phase transition.


latent heat – pressure – temperature



Boiling point:



Boiling point: ambient pressure = es.




Boiling temp. as function of pressure.




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.



• Ratio of saturation pressures for water and ice:es/ei = exp[(Lf/RvT0)(T0/T-1)]



• 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





• 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


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



• Dew-point temp,



• 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,





• Wet-bulb temp.: Tw, temp to which air may be cooled by evaporating water into it holding p const. – 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 ProcessAdiabatic: dQ = 0Dry/ 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 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 < 0irreversible.

Pseudoadiabatic process, cont.• Energy conservation.


dQ = Cp dT – Vdp

-L dws = cp dT - dp


dQ = Cp dT – Vdp

-L dws = cp dT - dp

• Pseudoadiabatic equation.dT/T = k dp/p – (L/Tcp) dws how to obtain dws ?


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


dQ = Cp dT – Vdp

-L dws = cp dT - dp



• Tephigram.


dQ = Cp dT – Vdp

-L dws = cp dT - dp



• Tephigram.• Water mixing ratio: (= 0 before saturation)

d = -dws : weight percentage of liquid form of H2O


dQ = Cp dT – Vdp

-L dws = cp dT - dp

• Pseudoadiabatic equation.dT/T = dp/p – (L/Tcp) dws


• 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 + wsv + w



= d + wsv + w

But v = w + L/T = d + w Q + (L/T) ws



= d + wsv + w

But v = w + L/T = d + w Q + (L/T) ws

• Isentropicdd = 0 = dd + d(w Q) + d(L ws /T)



= d + wsv + w

But v = w + L/T = d + w Q + (L/T) ws

• Isentropicdd = 0 = dd + 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 + )



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



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)




• 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.





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.




• In exponential notation C (cos (t + ) + i sin (t + ) ) = C ei (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 = Fx = 0)

• Taking a small perturbation x near the equilibrium.ma = Fx) = D1x + D2 x2 + …

• Deriving the linear momentum equation d2x/dt2 = -kx • Its general solution is

x = C ei(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-it

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


Lx, Ly >> H

0 = ma = F = F1 + F2 + …


Lx, Ly >> H

0 = ma = F = F1 + F2 + …

• 1-D steady state equation (force balance).dp/dz = - g

dp/p = - (g/R’Tv)dz


Lx, Ly >> H

0 = ma = F = F1 + F2 + …



• Dry adiabatic equilibrium (dq = 0)0 = cpdT – (R’T/p) dp

dT/dz = - g/cp -


Lx, Ly >> H

0 = ma = F = F1 + F2 + …




dT/dz = - g/cp -• Dry adiabatic lapse rate

1K/100m


Lx, Ly >> H

0 = ma = F = F1 + F2 + …





1K/100m

• The adiabatic lapse rate describes:• Steady state temp. adiabatic height profile


Lx, Ly >> H

0 = ma = F = F1 + F2 + …





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 isFB = -g /0 = g T/T0 = d2z/dt2

- = - 0, T = T- T0



- = - 0, T = T- T0

• In air of lapse rate At Z0 + z, Tair = T0 - z



- = - 0, T = T- T0


• An air parcel adiabatically moves from Z0 to Z0 + zTparcel = T0 - z



- = - 0, T = T- T0



• Buoyancy force exerted on the parcel T = Tparc- Tair = - (- ) z

d2z/dt2 = FB = g T/T = - (g/T)(- )z



- = - 0, T = T- T0




d2z/dt2 = FB = g T/T = - (g/T)(- )zWhen < : stableWhen = : neutralWhen > : unstable



- = - 0, T = T- T0




d2z/dt2 = FB = g T/T = - (g/T)(- )zWhen < : stableWhen = : neutralWhen > : 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


• 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 ratedT/dz = (T/p) dp/dz – (L/cp)dws/dz



s - dT/dz

= [1+Lws/R’T]/[1 + L2ws/R’cpT2]

Always: L/cpT > 1, => s <



s - dT/dz

= [1+Lws/R’T]/[1 + L2ws/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:


equation:– Acting along with the instability: more unstable– Acting against the instability: saturation.



• Convective Instability: finite size of parcel.



• Convective Instability: finite size of parcel.– Mass conservation: stretching in height, z when lifted.



• Convective Instability: finite size of parcel.– Mass conservation: stretching in height, z when lifted.– Into region of lower ambient temp/pressure



• 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



• 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 + 2v’



a = a’ + /t r - 2r + 2v’

• Acceleration in rotating frame of referencea’ = a - /t r + 2r - 2v’



a = a’ + /t r - 2r + 2v’


• Forces in rotating frame of referencema’ = F - m/t r + m2r – 2mv’Inertial force, centrifugal force, Coriolis force.



a = a’ + /t r - 2r + 2v’



• Coriolis parameter: (projection on horizontal plane)f = 2sin : lat.



a = a’ + /t r - 2r + 2v’




• Horizontal force balance (ug: in x, east; and vg in y, north) (2-D)p/x = fvgp/y = -fug



a = a’ + /t r - 2r + 2v’





• Geostrophic wind (steady state): along isobars.



a = a’ + /t r - 2r + 2v’





• 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 + 2v’





• 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 levelsHW 3.3

2-D Instabilities• Slantwise displacement (perturbation equation)

– Parcel Temp:– Ambient Temp:

/

/ /

par

amb

T T T p p

T T T T p p


– Parcel Temp:– Ambient Temp:– Temp difference:

/

/ /

/

par

amb

par amb

T T T p p

T T T T p p

TT T T z y

z y



– Vertical force:

/

/ /

/

par

amb

par amb

B

T T T p p

T T T T p p

TT T T z y

z y

gF z y

z y



– Vertical force: – Horizontal force:

/

/ /

/

par

amb

par amb

B

g gH

T T T p p

T T T T p p

TT T T z y

z y

gF z y

z y

u uF f z f y

z y



– Vertical force: – Horizontal force: – Perturbation equation:

2

2sin cos

sin cos

B H

g g

dF F

dt

u ugz y f z f y

z y z y

/

/ /

/

par

amb

par amb

B

g gH

T T T p p

T T T T p p

TT T T z y

z y

gF z y

z y

u uF f z f y

z y



– Vertical force: – Horizontal force: – Perturbation equation:

• 1-D instability: y = 0, = 90°• Baroclinic instability: FH = 0• Symmetric instability: FB = 0

2

2sin cos

sin cos

B H

g g

dF F

dt

u ugz y f z f y

z y z y

/

/ /

/

par

amb

par amb

B

g gH

T T T p p

T T T T p p

TT T T z y

z y

gF z y

z y

u uF f z f y

z y

2-D Instabilities, cont.• Baroclinic inst.

slope < parcel slope : stable = neutral

> unstable

• Symmetric inst.

surface slope < abs. vort. slope : stable = neutral > unstable

• Geopotential

2

2

/sin

/

yd g zy

z y zdt

2

2

/cos

/g g

g

u f u yd zf y

z y u zdt

00 0 0

( ) 1( ) '

zz ggpm z gdz z

g g g

Geopotential

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

00 0 0

( ) 1( ) '

zz ggpm z gdz z

g g g

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



– C-C equation– Possibility for condensation: breath in cold weather



– 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





• Adiabatic mixing: different pressures– Adiabatic: Potential temp: mass-weighted meanHW 4.1Presentations 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.


rate, the instability occurs.• Cold air convects down, until the adiabatic lapse rate is

reached in a “mixing layer”.



reached in a “mixing layer”.• As the surface temp further increases, the mixing layer

thickens.




thickens.• This process is equivalent to an upward heat propagation

(from the surface).





(from the surface).• The total heat added equals the area between the original

and final temp profiles.





(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– Force equation: d2z/dt2 = gB where B = -/0 = T/T0

– Vertical velocity U = dz/dt

0

0

2 20

2 20

2 ( )

2 ' ( ') (ln )

p

p

p

p

UdU gBdz

U U g B z dz

U U R T T d p




• 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

0

0

2 20

2 20

2 ( )

2 ' ( ') (ln )

p

p

p

p

UdU gBdz

U U g B z dz

U U R T T d p

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)




increases.– Buoyancy force: decrease (less buoyant)– However, condensation=>latent heat release=>

temp increases =>more buoyant.




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.



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.



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




• 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




• 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%.


e = es, or f = 100%.

• Pure water vapor condenses when f ~ nx100%.


e = es, or f = 100%.


• Phase transition in free space in equilibrium: latent heat only.


e = es, or f = 100%.



• Formation of small droplets: surface tension force, free energy barrier (diving).


e = es, or f = 100%.




• 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


e = es, or f = 100%.






• 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


• 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,( ) exp(2 / )s s Le r e rR T


e = es, or f = 100%.






• 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).

( ) exp(2 / )s s Le r e rR T





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)






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

visible

Haze: r1

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 = D2n

n: number density of interested molecules,

D: diffusion coefficient [L2/t] VL

Smell, Brownian movement




• Diffusion: controlling process before reaching critical size.n/t = D2n

n: number density of interested molecules,

D: diffusion coefficient [L2/t] VL

Smell, Brownian movement

• Mass change: dM/dt = 4rD (namb – nr)m0 = 4rD (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: latent heat release => Tr > Tamb

Can condensation still occur?






• Heat transfer: dQ/dt = 4rK (Tr – Tamb)K: thermal conductivity







• Heat budget: Gain: latent heat LdMLoss: conduction dQ

Change in state function: enthalpy mcpdT

M = (4/3)r3L

(4/3)r3L cp dT = L dM – dQ









M = (4/3)r3L


• Steady state: no change in temp, heat gain = heat loss(vamb - vr) / (Tr – Tamb) = K/LD









M = (4/3)r3L


• 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)



• Collisions become important when droplets are big enough, producing more 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


• 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








• 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)


• 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


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)r2u2CD

CD: drag coefficient, : density of ambient air (skydive)


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



• Gravity: FG = (4/3)r3g(L-) = (4/3)r3gL


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




• 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


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





u2 = (8/3)rg (L/)/ CD



• Larger drops: faster, Smaller drops: slower• Larger drops overtake and collide with smaller one


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





u2 = (8/3)rg (L/)/ CD



• 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 R0.5mm• 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



• 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



• 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





• 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)






combination in this range)– Vertical temp profile: colder on the top (but may have no moisture)






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 4 ( )

: capacitance, a function of shape

v vr

dmCD

dtC



– 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)

4 ( )


v vr

v vr

r s

dmCD

dtC

K

T T L D







• Further growth by accretion– General: Accretion: larger one captures smaller ones

– Special: Accretion: ice crystal captures supercooled droplets

4 ( )


v vr

v vr

r s

dmCD

dtC

K

T T L D







• 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

4 ( )


v vr

v vr

r s

dmCD

dtC

K

T T L D

Global Convection

• Coriolis force:– Mathematical form, physical meanings, examples

• Forces in rotating frame of referencema’ = F - m/t r + m2r – 2mv’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

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