1. ISENTROPIC ANALYSIS using Theta and Theta-E (Equivalent) surface
Special Thanks to Prof James Moore, Saint Louis Univ COMET
Introduction of isentropic analyses
Montgomery streamfunction
Geostrophic balance
Vertical Motion in isentropic analysis
Advantages and Disadvantages of isentropic vertical coordinates
The temperature,pressure, and moisture characteristics of the atmosphere arise in large part from the continuous exchange of energy and water vapor near the surface. When energy inputs exceed energy losses, the temperature of the air increases. In the same way, when there is more evaporation than precipitation, the moisture content of the atmosphere increases. But because the heat and water are not uniformly distributed across the globe, the cooling and warming of the atmosphere vary from place to place, as does the net input of water vapor. Thus, air over the tropical Pacific, for example, takes on different characteristics from air over northern Canada. [Understanding Weather and Climate, 2001]. 1

2. Parcel does not exchange heat with its surroundings
Conduction
Convection
Temperature Advection
Latent Heating
Adiabatic (heating-cooling)
Radiative Heat Transfer
Latent Heat release
Atm Avg Lapse rate ~6.5 °C/km
Expansion cooling  Compression warming
Parcel does not exchange heat with its surroundings 2

3. Visualizing Static Stability – Vertical Gradients of 
Vertical changes of potential temperature related to lapse rates:
U = unstable
N = neutral
S = stable
VS = very stable

4. Neutral-Superadiabatic Lapse Rates SA=superadiabatic and N=neutral

5. Vertical Gradient is a Function of Static Stability
LS = less stable (weak static stability) and VS = very stable (strong static stability)

6. vertical motion & diabatic processes
layer
perspective
parcel
perspective

7. Slope of Isentropes WARM COLD COLD WARM
Isentropes slope toward cold air
CONTRARY to isobaric surface, isentropic surfaces are further from the earth’s surface
when cold (dense) air is located between the isentrope and earth’s surface.
COLD
WARM 7

8. Three-Dimensional Isentropic Topography
cold
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warm 8

9. Isentropes near Frontal Zones

10. Thermal Wind Relationship in Isentropic Coordinates
Isentropic surfaces have a steep slope in regions of strong
baroclinicity. Flat isentropes indicate barotropic conditions
and little/no change of the wind with height.
Frontal zones are characterized by sloping isentropic surfaces which are vertically compacted (indicating strong static stability).
In the stratosphere the static stability increases by about one order of magnitude.

11. Isentropic Mean Meridional Cross Section

12. Advantages of Theta Analysis
Under adiabatic conditions, air parcels remain on isentrophic surfaces. The conserve their potential temperature
Air parcel trajectories on theta surfaces are more accurate because they capture the adiabatic component of the vertical motion.

13. Elements of Isentropic Chart
Winds on theta surface are 3D in Pres Height
Pressure contours on theta surface are Isotherms
Montgomery stream function – used to compute geostrophic wind (similar to Heights on P surface)
Mixing ratio (g/kg) moves on Theta/ThetaE surfaces
Pressure difference between theta surface (measure of static stability)

14. Montgomery Streamfunction Similar to Height Field in pressure coordinates
“M” is the Montgomery streamfunction is equal to the dry static energy of a parcel
“CpT” is the enthalpy and “g0 Z” is the potential energy of a parcel
Dry static energy is conserved following a parcel in adiabatic and frictionless motion
In absence of horizontal motion, vertical velocity would be negative if a cold dome of
air arrived at a given location (Term A negative) 14

15. Geostrophic Balance “M” is the Montgomery streamfunction
In absence of horizontal motion, vertical velocity would be negative if a cold dome of
air arrived at a given location (Term A negative)
“M” is the Montgomery streamfunction
“f0” is the Coriolis parameter at a single latitude (constant) 15

16. Advection of Moisture on an Isentropic Surface
Moist air from low levels on the left (south) is transported upward and to
the right (north) along the isentropic surface. However, in pressure coordinates water vapor appears on the constant pressure surface labeled p in the absence of advection along the pressure surface --it appears to come from nowhere as it emerges from another pressure surface. (adapted from
Bluestein, vol. I, 1992, p. 23)

17. Moisture transport and lift along an isentropic surface6 8
Isentropic mountain
Warm moist air

18. Cross section of isentropes and mixing ratios: 12 UTC 12-24-02

19. X . A B

20. Subjective Analysis for 12 UTC 24 December 2002

21. 294 K G-R streamlines/isobars: 12 UTC 24 Dec 2002

22. 294 K G-R omega: 12 UTC 24 Dec 2002
Advective component of Vertical Motion

23. System-Relative Flow on Isentropic Surfaces
A key assumption is that the weather system translates horizontally without change in shape or intensity (anticyclone or cyclone changes only slowly over synoptic time scales).
We subtract the speed of the system, C, from the ground-relative wind, V. V – C then becomes the system-relative flow on the isentropic surface, i.e., it is the flow relative to the moving system and takes into account the change in the isentropic topography.
C is computed by tracking a feature on the isentropic surface, e.g., a vorticity maximum, a short-wave trough axis, or some identifiable feature that can be reliably tracked.
Streamlines of system-relative wind are identical to trajectories relative to the moving system!

24. Computing Isentropic Omegas
Essentially there are three approaches to computing isentropic omegas:
• Ground-Relative Method (V • ∇P) :
• Okay for slow-moving systems (∂P/∂t term is small)
• Assumes that the advection term dominates (not always a good
assumption)
• System-Relative Method ( (V-C) • ∇P ) :
• Good for situations in which the system is not deepening or filling rapidly
• Also useful when the time step between map times is large (e.g.,
greater than 3 hours)
• Brute-Force Computational Method (∂ P/ ∂ t + V • ∇P ):

25. System-Relative Flow: How can it be used?
Carlson (1991, Mid-Latitude Weather Systems) notes:
Relative-wind analysis reveals the existence of sharply-defined boundaries, which differentiate air streams of vastly differing moisture contents. Air streams tend to contain relatively narrow ranges of Theta and Thetaw peculiar to the air stream’s origins.
Relative-wind isentropic analyses for synoptic-scale weather systems tend to show well-defined air streams, which have been identified by the names – warm conveyor belt, cold conveyor belt and dry conveyor belt.
Thus, a conveyor belt consists of an ensemble of air parcels having nearly the same Thetaw (or Thetae) value, starting from a common initial location, which travel over synoptic-scale time periods (> 18 h).

26. 294 K G-R streamlines/isobars: 20 UTC 24 Dec 2002

27. 294 K S-R streamlines/isobars: 20 UTC 24 Dec 2002
Subtract C = 243.1deg at m s-1
Tracking the circulation center
Streamlines in System Rel Coord are like Trajectories

28. 294 K G-R omega: 20 UTC 24 Dec 2002
Advective component of Vertical Motion

29. 294 K S-R omega: 20 UTC 24 Dec 2002

30. Choosing the “Right” Isentropic Surface(s)
The “best” isentropic surface to diagnose low-level moisture and vertical motion varies with latitude, season, and the synoptic situation. There are various approaches to choosing the “best” surface(s):
Use the ranges suggested by Namias (1940) :
Season Low-Level Isentropic Surface
Winter K
Spring K
Summer K
Fall K

31. Isentropic Analysis: Advantages
For synoptic scale motions, in the absence of diabatic processes, isentropic surfaces are material surfaces, i.e., parcels are thermodynamical bound to the surface
Horizontal flow along an isentropic surface contains the adiabatic component of vertical motion often neglected in a Z or P reference system
Moisture transport on an isentropic surface is three- dimensional - patterns are more spatially and temporally coherent than on pressure surfaces
Isentropic surfaces tend to run parallel to frontal zones making the variation of basic quantities (u,v, T, q) more gradual along them.

32. Disadvantage of Isentropic Vertical Coordinates*
Atmosphere is not completely adiabatic
Regions of strong vertical mixing or convection
Isentropic surfaces may intersect the ground
Isentropic surfaces are not quasi-horizontal
Meteorologists are unaccustomed to interpreting isentropic weather maps
*Carlson (1998) “Mid-Latitude Weather Systems” 32