1. Stability and Introduction to the Thermodynamic Diagram
SO 254 – Spring 2017
LCDR Matt Burich

2. What happens to its temperature?
Before beginning to look at the thermodynamic diagram, there are two remaining terms we need to define
For the first, imagine grabbing an unsaturated air parcel from an altitude of about 2 km (roughly 800 mb) where the environmental temperature is 8℃ and moving it adiabatically toward the ground
What does this mean?
It exchanges no heat with the environment around it
What happens to its temperature? 8℃
increases
800 mb
850 mb
925 mb
1000 mb
At what rate?
Γ 𝑑
9.8℃ / km
The temperature it acquires upon reaching the level of 1000 mb is referred to as its potential temperature which we assign the Greek lower-case letter theta 𝜃
27.6℃

3. Potential Temperature
Mathematically, the potential temperature of an air parcel is given by:
𝜃=𝑇 𝑝 0 𝑝 𝑅 𝑐 𝑝
𝑇= parcel temp (K)
where
𝑝 0 = 1000 mb
𝑝= parcel pressure
𝑅=287 J K −1 kg −1
𝑐 𝑝 =1004 J K −1 kg −1
Significantly, potential temperature is a conserved quantity (remains constant) for an air parcel that moves around dry adiabatically
Since most atmospheric motion is close to dry adiabatic (if phase changes of water are not occurring), 𝜃 becomes a great “tracer” of atmospheric motion
From the formula, it is evident that the farther a parcel is initially from 1000 mb, the bigger the difference between 𝑇 and 𝜃 8℃
−26℃
𝜃=27.6℃
1000 mb
925 mb
850 mb
800 mb
700 mb
500 mb 8℃
−26℃
These two parcels have the same potential temperature and will lie along the same dry adiabat (curve of constant 𝜃) on our thermodynamic diagram
27.6℃

4. Potential Temperature
So what if phase changes of water are occurring? (clearly we’re interested in what goes on in clouds!)
In this case we have to take into account that heat is being added (condensation) or removed (evaporation) from the parcel
With the aid of the 1st Law of Thermodynamics, we can mathematically quantify this effect by applying a “correction factor” to 𝜃 to account for water vapor:
𝜃 𝑒 ≅𝜃 exp 𝐿 𝑣 𝑤 𝑠 𝑐 𝑝 𝑇
𝜃 𝑒 ≅𝜃× 𝑒 𝐿 𝑣 𝑤 𝑠 𝑐 𝑝 𝑇 or
𝑇= parcel temp (K)
where
𝑤 𝑠 = saturation mixing ratio
𝑐 𝑝 = 1004 J K −1 kg −1
𝐿 𝑣 = latent heat of vaporization (2.25× J kg −1 )

5. Potential Temperature
What is happening to 𝑤 𝑠 as condensation is occurring?
increasing decreasing remaining constant
What happens when 𝑤 𝑠 goes to zero?
𝜃 𝑒 =𝜃
𝜃 𝑒 ≅𝜃 exp 𝐿 𝑣 𝑤 𝑠 𝑐 𝑝 𝑇
𝜃 𝑒 ≅𝜃× 𝑒 𝐿 𝑣 𝑤 𝑠 𝑐 𝑝 𝑇 or
𝑇= parcel temp (K)
where
𝐿 𝑣 = latent heat of vaporization (2.25× J kg −1 )
𝑤 𝑠 = saturation mixing ratio
𝑐 𝑝 = 1004 J K −1 kg −1
𝜃 𝑒 (“theta-e”) is thus the temperature a parcel would acquire if we lift it to a point where it has condensed (and removed) all of its water vapor and then lower it from there to 1000 mb. We refer to this as equivalent potential temperature
As dry adiabats will be curves of constant 𝜃 on our thermodynamic diagram, moist adiabats or pseudoadiabats will be curves of constant theta-e
Significantly, theta-e is conserved for both dry adiabatic and moist adiabatic (or “pseudoadiabatic”) processes (where phase changes of water are occurring) making it a particularly good tracer for air motion

6. Thermodynamic Diagram
We will now put together the measures of 𝑝, 𝑇, 𝑤, 𝜃, and 𝜃 𝑒 to construct our thermodynamic diagram. Starting with 𝑝…
Lines of constant pressure (isobars) are plotted horizontally every 50 mb across the diagram from high ~1050 mb at the bottom (near surface level) to low 100 mb at the top
lower stratosphere
The isobars are spaced logarithmically to account for the non-linear decrease of pressure with height
near surface

7. Thermodynamic Diagram
We next add temperature to the diagram
Lines of constant temperature are skewed to slope upward and to the right at a precise angle to produce the proper lapse rate for a parcel of air being “lifted” (drawn) upward from the bottom of the diagram
-40
-30
-20
-10
The skewed rendering of the isotherms together with the logarithmically decreasing pressure from bottom to top produce the name skew-𝑻 log-𝒑 for the diagram 10 20 30 40 °C
(skew-𝑇 for short)

8. 𝜃=0℃ Thermodynamic Diagram
We next look at the curves of constant theta (dry adiabats) on the diagram
These curves slope up and to the left from the bottom of the diagram
𝜃=0℃
-40
-30
Their value is read by observing where they intersect the 1000 mb isobar and reading down the isotherm to the temperature scale at the bottom of the diagram
By definition of potential temperature
Their value is read by observing where they intersect the 1000 mb isobar and reading down the isotherm to the temperature scale at the bottom of the diagram
-20
-10 10 20 30 40 °C

9. Thermodynamic Diagram
Now we look at the curves of constant theta-e (pseudo-adiabats) on the diagram
These dashed curves slope upward and eventually leftward with increasing height
-40
-30
-20
As with theta, their value is read by observing where they intersect the 1000 mb isobar and reading down the isotherm from that point to the temperature scale at the bottom of the diagram
-10 10 20 30 40 °C

10. Thermodynamic Diagram
Lastly, we look at the lines of constant mixing ratio 𝑤 (or saturation mixing ratio 𝑤 𝑠 …if a parcel is saturated) on the diagram
These dashed lines slope upward and rightward and are labeled by a separate scale at the bottom of the diagram in units of g/kg
-40
-30
-20
-10 10 20 30 40 °C

11. Thermodynamic Diagram
Combining the curves for 𝑝, 𝑇, 𝑤, 𝜃, and 𝜃 𝑒 together creates the full skew-𝑇
The dry adiabats (which represent the change in temperature of a dry air parcel being lifted) are increasingly curved upwards with height to ensure that the dry adiabatic lapse rate Γ 𝑑 remains a constant −9.8℃/km (since 𝑝 varies logarithmically)
-40
-30
-20
-10 10 20 30 40 °C

12. Thermodynamic Diagram
Combining the curves for 𝑝, 𝑇, 𝑤, 𝜃, and 𝜃 𝑒 together creates the full skew-𝑇
The pseudoadiabats (which represent the change in temperature of a saturated air parcel being lifted) are quite different from the dry adiabats at low-levels but become parallel to them at upper-levels
-40
-30
-20
-10 10 20 30
Why is this? 40 °C

13. Thermodynamic Diagram
Combining the curves for 𝑝, 𝑇, 𝑤, 𝜃, and 𝜃 𝑒 together creates the full skew-𝑇
𝜃 𝑒 ≅𝜃 exp 𝐿 𝑣 𝑤 𝑠 𝑐 𝑝 𝑇
Latent heat release slows the rate of cooling for the saturated parcel (which is condensing water vapor)
as condensation occurs  𝑤 𝑠 decreases  condensation rate slows  magnitude of latent heat release diminishes
-40
Note that because the parcel is saturated (and following a pseudoadiabat as it continues to rise), the diagonal dashed lines are now interpreted as constant values of 𝑤 𝑠 vice 𝑤
-30
-20
-10 10 20
…once 𝑤 𝑠 goes to zero, parcel cools at the dry adiabatic lapse rate as it continues to rise 30 40 °C

14. Thermodynamic Diagram
Combining the curves for 𝑝, 𝑇, 𝑤, 𝜃, and 𝜃 𝑒 together creates the full skew-𝑇
Warm air being less dense supports higher values of 𝑤 which supports greater condensation for saturated parcels and greater latent heat release
Notice at low temperatures that pseudoadiabats and dry adiabats are nearly parallel through the depth of the troposphere as cold air (being very dense) can hold less water vapor and is associated with correspondingly low values of 𝑤
-40
-30
-20
-10 10 20
Of course the complete opposite is true at warm temperatures 30 40 °C

15. Thermodynamic Diagram
Combining the curves for 𝑝, 𝑇, 𝑤, 𝜃, and 𝜃 𝑒 together creates the full skew-𝑇
Lastly, notice the relationship between the isotherms and lines of constant 𝑤
-40
-30
-20
𝑤 𝑠 = 𝑒 𝑠 𝑝
Recall that for a saturated parcel at constant temperature:
-10 10 20
𝑤 𝑠 is inversely proportional to pressure 30 40 °C

16. Tracing a Parcel with the Skew-𝑇
Begin with a near-surface 𝑇 and 𝑇 𝑑
Parcel initially unsaturated  draw a curve going up parallel to a dry adiabat (parcel’s theta)
This point is the lifting condensation level (here ~870 mb) where the parcel becomes saturated and condensation begins
LCL
From 𝑇 𝑑 , parallel a line of constant 𝑤 until intersecting the parcel’s theta curve
From the LCL, draw a curve going up parallel to a pseudo- adiabat
How would you determine the parcel’s theta-e?
Recall…
𝜃 𝑒 (“theta-e”) is the temperature a parcel would acquire if we lift it to a point where it has condensed (and removed) all of its water vapor and then lower it from there to 1000 mb.
𝑇=20℃
𝑇 𝑑 =10℃

17. Finding a parcel’s theta-e
𝜃 𝑒 (“theta-e”) is the temperature a parcel would acquire if we lift it to a point where it has condensed (and removed) all of its water vapor and then lower it from there to 1000 mb.
Where pseudoadiabat becomes parallel to dry adiabat
𝜃 𝑒 ≅𝜃 exp 𝐿 𝑣 𝑤 𝑠 𝑐 𝑝 𝑇
Here, parcel’s 𝑤 𝑠 ≅7.5 g/kg at the LCL
𝜃 𝑒 =40℃
Makes sense that 𝜃 𝑒 >𝜃 here since 𝑤 𝑠 >0
LCL
𝑇 𝑑 =10℃
𝑇=20℃

18. Assessing Stability with the Skew-𝑇
Here, parcel 𝑇 is everywhere to the left of (colder than) the environmental 𝑇
The mb layer is absolutely stable
If we stop lifting the parcel at 500 mb and release it, where will it come to rest?
~600 mb ~850 mb ~1000 mb
environmental 𝑇 (as measured by rawinsonde)
think,
pair,
share
Parcel immediately becomes unsaturated as it begins to descend and warm (dry adiabatically) until its 𝑇 becomes equal to the environmental 𝑇
𝑇 𝑑 =10℃
𝑇=20℃

19. Assessing Stability with the Skew-𝑇
Here, parcel 𝑇 is everywhere to the left of (colder than) the environmental 𝑇 for dry adiabatic motion, but once condensation begins occurring parcel 𝑇 eventually becomes warmer than environmental 𝑇 above ~750 mb
The mb layer is conditionally unstable
The “condition” here needed for instability is for a parcel to be lifted above 750 mb
The point at which the parcel first becomes warmer than the environment is referred to as the level of free convection (LFC)
LFC
LCL
𝑇 𝑑 =10℃
𝑇=20℃

20. Assessing Stability with the Skew-𝑇
the word “convection” refers to overturning motion (mixing) in the atmosphere…caused above the LFC by the fact that the parcel is warmer than its environment and thus positively buoyant
The point at which the parcel first becomes warmer than the environment is referred to as the level of free convection (LFC)
LFC
LCL
𝑇 𝑑 =10℃
𝑇=20℃

21. Assessing Stability with the Skew-𝑇
Here, parcel 𝑇 is everywhere to the right of (warmer than) the environmental 𝑇
The mb layer is absolutely unstable or superadiabatic
In nature, absolutely unstable layers are usually fleeting and shallow since any slight vertical displacement to air parcels in such a layer would immediately release the instability and adjust the profile to adiabatic or pseudoadiabatic
LCL
𝑇 𝑑 =10℃
𝑇=20℃