what did we learn in ch. 1? final exam 2013: 6...
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Final Exam 2013: 6 Questions
1. First/Second Laws Question 2. Definitions Question 3. Clausius-Clapeyron (with q/H/e) Question 4. Köhler, Clouds, and Stability Question 5. Forcing/Feedback Question 6. Term Project (Answer 2 of 4 Choices)
Same as midterm
What did we learn in Ch. 1? • What P, T, U are for a fluid • What an ideal gas is • How P, T, v relate for an ideal gas (and we call
this relationship an equation of state) • What chemical components constitute the
atmosphere (for homosphere <110 km) • What the hydrostatic balance is • How p, T vary with z for observed, “standard,”
isopycnic, isothermal, constant lapse-rate atmospheres
Key Combined 1st+2nd Law Results • 1st Law: du=dq+dw; u is exact Eq. 2.8 • du=dqrev-pdv (expansion only) p. 56
• Define Enthalpy: H=U+PV Eq. 2.12 • dh=du+pdv+vdp (dh=vdp-Tdη)
• 2nd Law: [dqrev/T]int.cycle=0 Eq. 2.27 • Define Entropy: dη=dqrev/T Eq. 2.25a • Tdη=dqrev • du=Tdη-pdv
• Define Gibbs: G=H-Tη Eq. 2.33 • dg=dh-Tdη-ηdT=(du+pdv+vdp)-Tdη-ηdT • dg=du-(Tdη-pdv)+vdp-ηdT=vdp-ηdT p. 58
• (δp/δt)g=η/v Eq. 2.40
What did we learn in Ch. 3? • Radiative transfer definitions
– Diffuse vs. direct – From all directions (irradiance F) or one (radiance I) – Absorption coefficient and optical thickness – Blackbody radiation
• Radiative transfer equations – Kirchoff’s law (gaseous molecules): Eλ = Aλ – Planck’s radiation law: F = fcn(λ, T) – Wien’s displacement law: λ ~ 3000/T – Stefan-Boltzmann law (black body: Fbb = σT4)
What did we learn in Ch. 4? • Phase equilibrium definitions
– Criteria of phase equilibria (thermal, mechanical, chemical)
– Degrees of freedom reduced by phases – Phase diagram of (pure) water
• Clausius-Clapeyron equation (dp/dT=Llvp/RvT2) – Strong dependence of esat on temperature (and Llv)
• Doubles every 10C – There are two ways to saturate, i.e. H=e/esat=1
• Increase water vapor in parcel (e) • Decrease temperature (and hence esat)
Adiabatic!
First Law!
Reversible!
Internal Energy!
Ideal Gas!p1v1T1
= R = p2v2T2
Δu = cvdT
€
dw = −pdv
Δu =Q +W
Q = 0
€
T2T1
=P2P1
⎛
⎝ ⎜
⎞
⎠ ⎟
Rcp
thick walls!
Low P, High T
Frictionless
Reversible, Adiabatic!
(mass is conserved)!
€
−pdv = cvdT
−RTv
⎛
⎝ ⎜
⎞
⎠ ⎟ dv = cvdT
− R1
2
∫ dvv
= cv1
2
∫ dTT
2
Reversible, Adiabatic
Expansion of an Ideal Gas
€
−pdv = cvdT
−RTv
⎛
⎝ ⎜
⎞
⎠ ⎟ dv = cvdT
− R1
2
∫ dvv
= cv1
2
∫ dTT
−R lnv2 − lnv1( ) = cv lnT2 − lnT1( )
ln v2v1
⎛
⎝ ⎜
⎞
⎠ ⎟
−R
= ln T2T1
⎛
⎝ ⎜
⎞
⎠ ⎟
cv
T2T1
⎛
⎝ ⎜
⎞
⎠ ⎟ =
v2v1
⎛
⎝ ⎜
⎞
⎠ ⎟
−R cv
T2T1
⎛
⎝ ⎜
⎞
⎠ ⎟ =
RT2p2
⎛
⎝ ⎜
⎞
⎠ ⎟
RT1p1
⎛
⎝ ⎜
⎞
⎠ ⎟
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
−R cv
=T2T1
⎛
⎝ ⎜
⎞
⎠ ⎟
−R cv p2p1
⎛
⎝ ⎜
⎞
⎠ ⎟
Rcv
T2T1
⎛
⎝ ⎜
⎞
⎠ ⎟
1+R cv=
p2p1
⎛
⎝ ⎜
⎞
⎠ ⎟
Rcv
T2T1
⎛
⎝ ⎜
⎞
⎠ ⎟ =
p2p1
⎛
⎝ ⎜
⎞
⎠ ⎟
RR +cv
=p2p1
⎛
⎝ ⎜
⎞
⎠ ⎟
Rcp
Potential Temperature
Hydrostatic Balance
• Applicable to most atmospheric situations (except fast accelerations in thunderstorms)
€
g = −1ρ∂p∂z
∂p = −pgRdT
∂z
Curry and Webster, Ch. 1
Special Cases of Hydrostatic Equilibrium
• 1. rho=constant (homogeneous) – H=8 km =RT/g=scale height eq. 1.39
• 2. constant lapse rate (e.g. if hydrostatic, homogeneous, and ideal gas) – -dT/dz=constant=-g/R=-34/deg/km
• 3. isothermal T=constant (and ideal gas) – p=p_0*exp(-z/H)
Homogeneous Atmosphere
• Density is constant • Surface pressure is finite • Scale height H gives where pressure=0
€
p0 = ρgH
H =pρg
=RdT0g
€
g = −1ρ∂p∂z
dp = −ρgdz
dpp0
0∫ = − ρgdz
0
H∫
0 − p0 = − ρgH − 0( )
Curry and Webster, Ch. 1
Hydrostatic + Ideal Gas + Homogeneous
• Evaluate lapse rate by differentiating ideal gas law
€
p = ρRdT∂p∂z
= ρRd∂T∂z
−1ρ∂p∂z
⎛
⎝ ⎜
⎞
⎠ ⎟ = Rd −
∂T∂z
⎛
⎝ ⎜
⎞
⎠ ⎟
g = −1ρ∂p∂z
€
Γ = −∂T∂z
=gRd
= 34.1oC/km
Density constant
Ideal gas
Hydrostatic
Curry and Webster, Ch. 1
3
Hydrostatic Equilibrium Example���(Constant Lapse Rate) Water Saturation Pressures
es doubles with every 10C!
T(C) eS (hPa)
10 12.3
20 23.4
30 42.4
40 73.8
(this is one consequence of Clausius-Clapeyron’s equation)
Water Vapor Metrics
Mixing ratio Specific humidity Relative humidity
Water vapor by mass
Water vapor by partial pressure
Water saturation
Virtual temperature
Virtual potential temperature
€
qv =mv
md +mv=
wv1+ wv
€
wv =mv
md
=ρvρd
€
qv = 0.622 ep − 1− 0.622( )e
⎛
⎝ ⎜
⎞
⎠ ⎟
€
ws = 0.622 esp − es
⎛
⎝ ⎜
⎞
⎠ ⎟
€
H ≈wv
ws
€
H =ees
€
θv = T 1+ 0.608qv( ) p0p
⎛
⎝ ⎜
⎞
⎠ ⎟
Rdc pd
€
Tv = T 1+ 0.608qv( )€
wv = 0.622 ep − e
⎛
⎝ ⎜
⎞
⎠ ⎟
€
H = 1
€
qv = 0.622 esp − 1 − 0.622( )es
⎛
⎝ ⎜
⎞
⎠ ⎟
€
MvMd
⎛ ⎝ ⎜ ⎞
⎠ ⎟ = Rd
Rv⎛ ⎝ ⎜ ⎞
⎠ ⎟ = 0.622
€
RvRd
⎛ ⎝ ⎜ ⎞
⎠ ⎟ − 1 = 0.608
Virtual Temperature
Terminology Review • Isotropic
– Same in all directions, such that F=πI • Reflection
– Change in direction but not energy or wavelength • Isentropic
– Adiabatic+reversible • For adiabatic, ideal:
– p determines T and vice versa • Potential temperature
– temperature that air would have if raised/lowered to a reference pressure.
What you need to know in Ch. 12 • 0 and1-layer Earth radiation balance model
– Earth’s actual energy imbalance (ocean sink) • Detailed energy streams (Kiehl and Trenberth)
– Atmospheric window – Latent heat
• Major heat-driven features of Earth’s circulation – Hydrological cycle (evaporation-precipitation) – Latitudinal differences in heating – Meridional heat transfer (equator to pole) – Zonal heat transfer (Walker, monsoonal)
4
€
T
€
z
On Temperature Axis: Simplified
€
dTdz
= 0
€
dθvdz
= 0
€
dθedz
= 0 stable
unstable
€
T
€
z
€
dTdz
= 0
On Skew-T Axis: Simplified
€
dθvdz
= 0
€
dθedz
= 0
stable
unstable
Ch. 8: Main Cloud Types 1. Cirrus (Ci) 2. Cirrocumulus (Cc) 3. Cirrostratus (Cs)
4. Altocumulus (Ac) 5. Altostratus (As) 6. Nimbostratus (Ns) 7. Stratocumulus (Sc) 8. Stratus (St)
9. Cumulus (Cu) 10. Cumulonimbus (Cb)
All high clouds
Middle clouds
Grayish, block the sun, sometimes patchy
Sharp outlines, rising, bright white
Low clouds
GFDL AM2p5 vs NCAR CAM2
B. Soden
Ch. 8: Cloud Types and Drop Sizes
• Frequency distributions of the mean cloud droplet size for various cloud types
Ch. 12 Simplified Climate Model • Atmosphere described as one layer
– Albedo αp~0.31: reflectance by surface, clouds, aerosols, gases
– Shortwave flux absorbed at surface
• Earth behaves as a black body – Temperature Te: equivalent black-body temperature of
earth – Longwave flux emitted from surface
Curry and Webster, Ch. 12 pp. 331-337; also Liou, 1992
FS=0.25*S0(1- αp)
FL=σTe4
5
FL Emitted from sphere
surface 4πr2 FS
Incident on projected disc πr2
FS = FL
Simplified Climate Model
• Incoming shortwave = Outgoing longwave • Energy absorbed = Energy emitted
FS = 0.25*S0(1- αp) FL = σTe4
Simplified Climate Model • At thermal equilibrium (why?)
• Observed surface temperature T = 288K • What’s missing?
FS = FL 0.25*S0(1- αp) = σTe
4
Te = [0.25*S0(1- αp)/σ]0.25
Te ~ 255K
Add an Atmosphere! • Atmosphere is transparent to non-reflected portion
of the solar beam • Atmosphere in radiative equilibrium with surface • Atmosphere absorbs all the IR emission
Fsurf
FS
TOA: FS = Fatm
0.25*S0(1- αp) = σTatm4
Tatm = 255K
Fatm
Fatm Atmos: Fsurf = 2Fatm
σTsurf4
= 2σTatm4
Tsurf = 303K
Section 13.3 Water Vapor Feedback
• Key points:
• This is a strongly positive feedback, nearly doubling the effect of carbon dioxide alone.
• Relative humidity seems to be approximately constant under climate change (p. 359).
• Relatively dry regions, such as the upper troposphere and polar regions, are especially sensitive (p. 362).
. Section 13.4 Cloud-radiation
Feedback Key points:
• Clouds affect both shortwave (low clouds) and longwave (high clouds). Present climate has cloud cooling dominating cloud warming (pp. 368-369).
• Many different mechanisms, including those involving aerosol-cloud interactions, may be important, but the sign and magnitude of cloud feedbacks is still largely unknown (p. 374). Clouds have big effects in models
Section 13.5 Snow/Ice-albedo Feedback
Key points:
• This feedback is large and positive in high northern latitudes.
• Observations show that this effect is occurring now.
• Melting ice on land has another large effect, unrelated to albedo: it causes sea level to rise.