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METR 2413
3 March 2004
Thermodynamics
IV
Review
First law of thermodynamics: conservation of energy
du = dq – dw
dq = cv ΔT + p Δα = cp ΔT - α Δp = cp ΔT – Δp/ρ
Adiabatic process, dq = 0, no external energy input to parcel
Diabatic process, radiation or latent heating
dT
g

dz
cp
Entropy dS = dQ/T remains constant or increases
Adiabatic lapse rate
Adiabatic temperature
variations
Consider an adiabatic process again
dq = 0 = cp dT – dp / ρ
Then cp dT = R T dp / p
Divide by cp T gives
So
using ideal gas law
dT R dp

T cp p
R
d (ln T )  d (ln p)
cp
Integrating from initial level pi to final level pf gives
 Tf
ln 
 Ti
 R  pf 
 pf 
  ln 
  ln 

 c p  pi 
 pi 
R
cp
Adiabatic temperature
variations
So
 pf 

 
Ti  pi 
Tf
R
cp
 pf 

 
 pi 

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.
 pf 

T f  Ti 
 pi 

Since p decreases with height, T also decreases with height for
dry adiabatic temperature variations (as we have shown before).
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.

 p0 
 1000 

  T    T 
 p
 p 

The potential temperature of an air parcel is constant for
adiabatic motion.
This is one of the most important concepts in meteorology!
Atmospheric
stability
Thermo
Diagram –Dry Adiabats
10
Potential temperature θ constant corresponds to a dry adiabatic
lapse rate
and a neutrally stable layer.
P in kPa
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
0
20 height.
40
60
80
greater than-40Γd and-20θ decreasing
with
100
-60
-40
-20
0
20
T or T_d in C
Dry adiabats are lines of constant
potential temperature
40
60
θ in
°C
Atmospheric
Thermostability
Diagram –Moist Adiabats
Potential temperature θ constant corresponds to a dry adiabatic
lapse rate
and a neutrally stable layer.
10
P in kPa
A stable layer has the temperature decrease with height smaller
than Γd and θ increasing with height.
80
An unstable layer has the temperature decrease with height
0
20
40
60
greater than-40Γd and θ-20decreasing
with height.
100
-60
-40
-20
0
20
40
θL in
°C
60
T or T_d in C
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 dT   g   and a neutrally stable layer.
d
dz
cp
A stable layer has the temperature decrease with height smaller
than Γd and θ increasing with height.
dT
d
 d ,
0
dz
dz
An unstable layer has the temperature decrease with height
greater than Γd and θ decreasing with height.
dT
d
 d ,
0
dz
dz
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 pressure
Find sounding temperature 150 mb above the surface
From 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 light
Temperatures 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 value
Convective 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
convection
The 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 Value
0 – 50
51 – 199
200 +
Cap Strength
weak
moderate
strong
CINH will be reduced by:
1) Daytime heating
2) Synoptic upward forcing
3) Low level convergence
4) Low level warm air advection
CINH index is only relevant to the lower planetary boundary layer convection
If there is no CAPE, CINH index is meaningless