METR 24133 March 2004
ThermodynamicsThermodynamics
IV IV
Review
First law of thermodynamics: conservation of energydu = dq – dw
dq = cv ΔT + p Δα = cp ΔT - α Δp = cp ΔT – Δp/ρ
Adiabatic process, dq = 0, no external energy input to parcelDiabatic process, radiation or latent heating
Adiabatic lapse rate
Entropy dS = dQ/T remains constant or increases
pcdz
dT g
Adiabatic temperature variations
Consider an adiabatic process againdq = 0 = cp dT – dp / ρ
Then cp dT = R T dp / p using ideal gas law
Divide by cp T gives
So
Integrating from initial level pi to final level pf gives
p
dp
cT
dT
p
R
)(lnR
)(ln pdc
Tdp
pc
R
i
f
i
f
pi
f
p
p
p
p
cT
T
lnln
Rln
Adiabatic temperature variations
So with κ = R/cp = 0.286
Given an initial pressure and temperature, we can calculate the final temperature Tf at pressure pf for adiabatic motion.
Since p decreases with height, T also decreases with height for dry adiabatic temperature variations (as we have shown before).
i
fc
R
i
f
i
f
p
p
p
p
T
T p
i
fif p
pTT
Potential temperature
We define the potential temperature θ to be the temperature an air parcel would have if was raised or lowered under dry adiabatic motion to pressure level of 1000 hPa.
Setting pf = p0 = 1000 hPa, we obtain an equation for the potential temperature of an air parcel with temperature T at pressure p; for adiabatic motion.
The potential temperature of an air parcel is constant for adiabatic motion.This is one of the most important concepts in meteorology!
pT
p
pT
10000
Atmospheric stability
Potential temperature θ constant corresponds to a dry adiabatic lapse rate and a neutrally stable layer.
A stable layer has the temperature decrease with height smaller than Γd and θ increasing with height.
An unstable layer has the temperature decrease with height greater than Γd and θ decreasing with height.
10
100
-60 -40 -20 0 20 40 60
T or T_d in C
P i
n k
Pa
Thermo Diagram –Dry Adiabats
-40 -20 0 8020 60 θ in °C
40
Dry adiabats are lines of constant potential temperature
Atmospheric stability
Potential temperature θ constant corresponds to a dry adiabatic lapse rate and a neutrally stable layer.
A stable layer has the temperature decrease with height smaller than Γd and θ increasing with height.
An unstable layer has the temperature decrease with height greater than Γd and θ decreasing with height.
10
100
-60 -40 -20 0 20 40 60
T or T_d in C
P i
n k
Pa
Thermo Diagram –Moist Adiabats
-40 -20 0
80
20 60 θL in °C
40
Moist adiabats show the temperature variations of a saturated air parcel that is rising through the atmosphere. The temperature decreases less quickly with height than a dry adiabat due to latent heat relase from condensation
Atmospheric stability
Potential temperature θ constant corresponds to a dry adiabatic lapse rate and a neutrally stable layer.
A stable layer has the temperature decrease with height smaller than Γd and θ increasing with height.
An unstable layer has the temperature decrease with height greater than Γd and θ decreasing with height.
0, dz
d
dz
dTd
dpcdz
dT
g
0, dz
d
dz
dTd
Boundary layer
Atmospheric boundary layer is region of turbulent motion due to heating from by the ground or strong winds.
Heating of the ground by solar radiation causes heating of the air close to the ground. This air will warm until the temperature gradient is unstable, causing dry convection to occur (if there is not too much moisture around).
Well-mixed boundary layer has adiabatic lapse rate and constant potential temperature with height.
Usually topped by a strong temperature inversion ( temperature increase with height) and a very stable layer.
Maximum temperature forecast
MAXT = estimated maximum afternoon temperature
Most relevant when using morning sounding
Most accurate on days with clear skies and moderate winds
Assumes mixing depth of planetary boundary layer is ~150 mb
To determine MAXT:Note surface pressureFind sounding temperature 150 mb above the surfaceFrom the temperature 150 mb above surface, follow the dry adiabat down to the surface
Maximum temperature forecast
Once the planetary boundary layer mixes to dry adiabatic lapse rate, further warming is slow
- this is one of the reasons why temperatures tend to increase most rapidly in the first half of the day and more slowly in the second half of the day
Temperatures may be higher if wind is lightTemperatures may be lower if wind is strong (wind strength affects depth of atmospheric mixing)
Number of daylight hours affects accuracy (more accurate in warm season
Technique does not work well near fronts or in cases of strong advection
Technique does not work well in regions with complex topography, or in coastal areas
CAPE
CAPE = Convective Available Potential Energy
On the skew-T, CAPE is indicated by the area where a rising air parcel would be warmer than the environment
CAPE gives an indication on the stability of the atmosphere. In general, the higher the CAPE value, the more unstable the atmosphere is.
To find the CAPE from a skew-T thermodynamic diagram, simply locate the area on the diagram where the parcel sounding is warmer than the atmosphere sounding.
CAPE
The white region is called the "positive energy" region. The size of the positive energy region gives an indication on how buoyant, and hence unstable, a parcel is.
CAPE
CAPE values can be used to objectively determine how convective the atmosphere is. CAPE has unites of Joules per kilogram. Use the following scale to determine convective potential (from Sturtevant, 1994):
CAPE valueConvective potential
< 300 Little or none
300-1000 Weak
1000-2500 Moderate
2500-3000 Strong
A CAPE value above 3000 would indicate a potentially highly unstable atmospheric condition, and storms will build vertically very quickly.
CAPE
CAPE > 2500 J/kg hail potential increases (large hail requires large CAPE)
CAPE > 2000 J/kg expect isolated regions of very heavy rain, perhaps accompanied by strong downdrafts
CAPE > 2000 J/kg will typically produce storms with intense lightning
Caveats:Storms will only form if low level capping inversion is broken
CAPE magnitude can rise or fall very rapidly
CINH
Convective Inhibition (CINH) – basically anti-CAPE
CINH is defined as the amount of energy beyond the normal work of expansion need to lift a parcel from the surface to the Level of Free Convection (LFC).
Increasing amounts of CINH indicate more energy is needed to lift the parcel
CINH
On a skew-T diagram, the CINH is the area bounded by the temperature sounding on the right and the Dry/Saturated adiabats on the left (dry if below the LCL, wet if above the LCL).
CINH
CINH area is generally called the "negative energy region“ the more CINH in the sounding, the greater the atmospheric stability and the less chance of vigorous convectionThe top of the CINH area is the Level of Free Convection (LFC), which is the first level in the atmosphere where the parcel can continue to rise on it's own, without any outside energy contribution.
CINH may also be referred to as a “capping layer” – must be broken before a parcel can move into a region of CAPE and develop into deep convection
CINH
Like CAPE, units of CINH are Joules/kilogram
CINH will be reduced by:1) Daytime heating2) Synoptic upward forcing3) Low level convergence4) Low level warm air advection
CINH index is only relevant to the lower planetary boundary layer convectionIf there is no CAPE, CINH index is meaningless
CINH Value Cap Strength0 – 50 weak51 – 199 moderate200 + strong
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