1. Isentropic Analysis James T. Moore
Cooperative Institute for Precipitation Systems
Saint Louis University
COMET COMAP Course May-June 2002

2. Now entering
a no pressure
zone!

3. Thetaburgers
Served hot and juicy at the Isentropic Café!
Boomerang Grille
Norman, OK

4. Utility of Isentropic Analysis
Diagnose and visualize vertical motion - through advection of pressure and system relative flow
Depict 3-Dimensional advection of moisture
Compute moisture stability flux - dynamic destabilization and moistening of environment
Diagnose isentropic potential vorticity
Diagnose dry static stability (plan or cross-section view) and upper-level frontal zones
Diagnose conditional symmetric instability
Help depict 2-D frontogenetical and transverse jet streak circulations on cross sections

5. Theta as a Vertical Coordinate
q = T (1000/P)k , where k= Rd / Cp
Entropy =  = Cp lnq + const
If  = const then q = const, so constant entropy sfc = isentropic sfc
Three types of stability, since dq/ dz = (q /T) [d - ]
stable:  < d, q increases with height
neutral:  =  d, q is constant with height
unstable:  >  d, q decreases with height
So, isentropic surfaces are closer together in the vertical in stable air and further apart in less stable air.

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

7. Theta as a Vertical Coordinate
Isentropes slope DOWN toward warm air, UP toward cold air – this is opposite to the slope of pressure surfaces: since q = T (1000/P)k, as P increases (decreases), T increases (decreases) to keep  constant (as on a skew-T diagram).
Isentropes slope much greater than pressure surfaces given the same thermal gradient; as much as one order of magnitude more!
On an isentropic surface an isotherm = an isobar = an isopycnic (const density); (remember: P = RdT)
On an isentropic surface we analyze the Montgomery streamfunction to depict geostrophic flow, where:
M = y = Cp T + gZ ;

8. 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.

10. Advection of Moisture on an Isentropic Surface

11. 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)

12. Relative
Humidity
305K surface
12 UTC
RH>80% = green
Pressure analysis
305K surface
12 UTC
Benjamin et al.

13. Relative
Humidity
at 500 hPa
RH > 70%
=green

14. Isentropes near Frontal Zones

15. Surface Map for 00 UTC 30 December 1990

16. Surface Map for 12 UTC 30 December 1990

17. Sounding for Paducah, KY 30 December 1990 12 UTC
898 hPa
+14.0 C
962 hPa
-1.3C

18. Cross Section Taken Normal to Arctic Frontal Zone: 12 UTC 30 December 1990

19. Three-Dimensional Isentropic Topography
cold
warm

20. Isentropic Analysis: Advantages
Atmospheric variables tend to be better correlated along an isentropic surface upstream/downstream, than on a constant pressure surface, especially in advective flow
The vertical spacing between isentropic surfaces is a measure of the dry static stability. Convergence (divergence) between two isentropic surfaces decreases (increases) the static stability in the layer.
The slope of an isentropic surface (or pressure gradient along it) is directly related to the thermal wind.
Parcel trajectories can easily be computed on an isentropic surface. Lagrangian (parcel) vertical motion fields are better correlated to satellite imagery than Eulerian (instantaneous) vertical motion fields.

21. Thermal Wind Relationship in Isentropic Coordinates
Usually only the wind component normal to the plane of the cross section is plotted; positive (negative) values indicate wind components into (out of) the plane of the cross section.
With north to the left and south to the right:
when isentropes slope down, the thermal wind is into the paper, i.e, the wind component into the cross-sectional plane increases with height
when isentropes slope up, the thermal wind is negative, i.e., the wind component out of the cross-sectional plane increases with height.

22. 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.

23. Cross section of  and normal wind components; dashed (solid) yellow = out of (into) the cross-sectional plane.
24 h Eta forecast valid 00 UTC 29 November 2001

24. Isentropic Mean Meridional Cross Section

25. Isentropic Analysis: Disadvantages
In areas of neutral or superadiabatic lapse rates isentropic surfaces are ill-define, i.e., they are multi-valued with respect to pressure;
In areas of near-neutral lapse rates there is poor vertical resolution of atmospheric features. In stable frontal zones, however there is excellent vertical resolution.
Diabatic processes significantly disrupt the continuity of isentropic surfaces. Major diabatic processes include: latent heating/evaporative cooling, solar heating, and infrared cooling.
Isentropic surfaces tend to intersect the ground at steep angles (e.g., SW U.S.) require careful analysis there.

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

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

28. Surface Map for 00 UTC 27 November 2001
DTX
DDC

29. Sample Cross section for 00 UTC 27 November 2001

30. Radiational Heating/Cooling Disrupts the Continuity of Isentropic Surfaces
As time increases solar heating causes the 300 K isentropic surface to become “redefined” at higher pressures
Namias, 1940: An Introduction to the Study of Air Mass and Isentropic Analysis, AMS, Boston, MA.

31. 304 K Isentropic Surface for 12 UTC 2 May 2002

32. 304 K Isentropic Surface for 00 UTC 3 May 2002
Note loss of data in SE U.S. and in Texas…304 K surface went underground

33. Isentropic Analysis: Disadvantages
The “proper” isentropic surface to analyze on a given day varies with season, latitude, and time of day. There are no fixed level to analyze (e.g., 500 hPa) as with constant pressure analysis.
If we practice “meteorological analysis” the above disadvantage turns into an advantage since we must think through what we are looking for and why!

34. 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

35. Choosing the “Right” Isentropic Surface(s)
BEST METHOD:
Compute a cross section of isentropes and isohumes ( mixing ratios) normal to a jet streak or baroclinic zone in the area of interest.
Choose the low-level isentropic surface that is in the moist layer, displays the greatest slope, and stays hPa above the surface.
A rule of thumb is to choose an isentropic surface that is located at ~ hPa above your area.

36. Using an Isentropic Cross Section to Choose a  Surface: Isentropic Cross Section for 00 UTC 05 Dec 1999

37. Isentropic Moisture Parameters
Lifting Condensation Pressure (LCP): The pressure to which a parcel of air must be raised dry-adiabatically in order to reach condensation. Represents moisture differences better than mixing ratio at low values of mixing ratio. Condensation pressure on an isentropic surface is equivalent to dew point on a constant pressure surface.
Condensation Difference (CD): The difference between the actual pressure and the condensation pressure for a point on a isentropic surface. The smaller the condensation difference, the closer the point is to saturation. Due to smoothing and round off errors, a difference < 20 hPa represents saturation. Values < 100 hPa indicated near saturation. Condensation difference on an isentropic surface is equivalent to dew point depression on a constant pressure surface.

38. Isentropic Moisture Parameters
Moisture Transport Vectors (MTV):
Defined as the product of the horizontal velocity vector, V, and the mixing ratio, q. Units are gm-m/kg-s ; values typically range from , depending upon the level and the season.
Typically, stable precipitation due to isentropic upglide falls downstream from the maximum of the moisture transport vector magnitude in the northern gradient region. The moisture transport vectors and isopleths of the magnitude of the moisture transport vectors are usually displayed.
Note that the negative divergence of the MTVs is equal to the horizontal moisture convergence, since

39. Mass Continuity Equation in Isentropic CoordinatesB C D
Term A: Horizontal advection of static stability
Term B: Divergence/convergence changes the static stabil-
ity; divergence (convergence) increases (decreases) the static
stability
Term C: Vertical advection of static stability (via diabatic
heating/cooling)
Term D: Vertical variation in the diabatic heating/cooling
changes the static stability (e.g., decreasing (increasing)
diabatic heating with height decreases (increases) the static

40. Term A: Horizontal Advection of Static Stability
Very stable (50 hPa/4K)
Decreased static stability
Less stable (100 hPa/4K)
Term B: Divergence/Convergence Effects
Increased static stability
Divergence
Term C: Vertical Advection of Static Stability
Increased static stability
Latent Heating
Term D: Vertical Variation of Diabatic Heating/Cooling
Decreased static stability
Evaporative Cooling
Latent Heating

41. Horizontal Mass Flux
Vertical Mass Flux

42. Moisture Stability Flux
Where q is the average mixing ratio in the layer from to q + Dq, DP is the distance in hPa between two isentropic surfaces (a measure of the static stability), and V is the wind.
The first term on the RHS is the advection of the product of moisture and static stability; the second term on the RHS is the convergence acting upon the moisture/static stability.
MSF > 0 indicates regions where deep moisture is advecting into a region and/or the static stability is decreasing.

43. Computing Vertical Motion
A B C
Term A: local pressure change on the isentropic surface
Term B: advection of pressure on the isentropic surface
Term C: diabatic heating/cooling term (modulated by the dry static stability.
Typically, at the synoptic scale it is assumed that terms A and C are nearly equal in magnitude and opposite in sign.

44. Local pressure
tendency term
computed over 12, 6 and 3 hours by Homan and
Uccellini, 1987
(WAF, vol. 2, )

45. Example of Computing Vertical Motion
1. Assume isentropic surface descends as it is warmed by latent heating (local pressure tendency term):
P/ t = 650 – 550 hPa / 12 h = +2.3 bars s-1 (descent)
2. Assume 50 knot wind is blowing normal to the isobars from high to low pressure (advection term):
V   P = (25 m s-1) x (50 hPa/300 km) x cos 180
V   P = -4.2 bars s-1 (ascent)
3. Assume 7 K diabatic heating in 12 h in a layer where  increases 4 K over 50 hPa (diabatic heating/cooling term):
(d/dt)(P/ ) = (7 K/12 h)(-50 hPa/4K) = -2 bars s (ascent)

46. Understanding System-Relative Motion

47. Isentropic System-Relative Vertical Motion
Define Lagrangian; no -
diabatic heating/cooling
System tendency
Assume tendency following system is = 0; e.g., no deepening or filling of system with time.
Insert pressure, P, as the variable in the ( )

48. System-Relative Isentropic Vertical Motion
Defined as:
~ (V – C)   P
Where  = system-relative vertical motion in bars sec-1
V= wind velocity on the isentropic surface
C = system velocity, and
P = pressure gradient on the isentropic surface
C is computed by tracking the associated vorticity maximum on the isentropic surface over the last 6 or 12 hours (one possible method); another method would be to track the motion of a short-wave trough on the isentropic surface

49. System-Relative Isentropic Vertical Motion
Including C, the speed of the system, is important when:
* the system is moving quickly and/or
* a significant component of the system motion is across the isobars on an isentropic surface, e.g.,
if the system motion is from SW-NE and the isobars are oriented N-S with lower pressure to the west, subtracting C from V is equivalent to “adding” a NE wind, thereby increasing the isentropic upslope.

50. When is C important to use when compute isentropic omegas?
Vort Max at t1
In regions of isentropic
upglide, this system-rela-
tive motion vector, C,
will enhance the uplift (since C is subtracted from the Velocity vector),
Vort Max at to

51. Computing Isentropic Omegas
Essentially there are three approaches to computing isentropic omegas:
Ground-Relative Method:
Okay for slow-moving systems (P/ t term is small)
Assumes that the advection term dominates (not always a good assumption)
System-Relative Method:
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)
S-R velocity vectors are useful in computing S-R MTVs
Brute-Force Computational Method ( P/ t + V  P ):
Best for situations in which the system is rapidly deepening or filling
Good approximation when data are available at 3 h or less interval, allowing for good estimation of local time tendency of pressure

52. Which Term is Important?
We chose four cases: two cases were non-developing systems with weak or little cyclogenesis, and two were developing, dynamic systems with moderate cyclogenesis.
We ran simulations of these cases using the MASS model (version ); model results were viewed using GEMPAK.
Our focus was on those regions on lower to mid-tropospheric  levels where the relative humidity was >99% and for which the model had generated precipitation > 0.5 mm during the preceding hour.

53. Which Term is Important? (cont.)
We computed isentropic omegas using all three terms noted earlier:
for the local pressure tendency term a simple 2 h time centered difference was used
the pressure advection was computed using a centered finite difference
the diabatic term was computed by first computing the diabatic heating/cooling on the isentropic surface and then multiplying by the static stability centered on the isentropic surface in question.
System-relative vertical motion was also computed, using a system speed, C, estimated subjectively using the trough motion on the isentropic surface previous to the map time.

54. Computing Diabatic Heating/Cooling in Isentropic Coordinates
Approach developed by Keyser and Johnson (1982, MWR)
Derived from the continuity equation in isentropic coordinates
Vertically integrate the stability flux from the level of interest ()
to an isentropic surface near the tropopause (t) + the difference
between the pressure tendencies at the same two levels
Note: Diabatic heating is modulated by the static stability

55. Four Precipitation Cases used for Study
21 UTC 16 January 1994
non-developing system associated with long-wave trough well to west
weakly-defined surface system associated with a weak-moderate cold front with inverted trough, produced banded heavy snow over Kentucky with amounts exceeding 60 cm
15 UTC 10 April 1997
non-developing system associated with a weak ridge over MO
light snow (~10 cm) fell in a band across central MO ahead of a weak west-east oriented stationary front in southern MO
00 UTC 6 April 1999
strong S/W trough associated with moderate cyclogenesis in central Plains
strong baroclinic system with 996 hPa low in KS; movement to NE; strong mid-level jet streak
21 UTC 15 April 1999
strong S/W trough and moderate cyclogenesis in Ohio Valley
extensive precipitation shield, 994 hPa low with occlusion; movement to NE, strong mid-level jet streak

56. Table 1: Statistical Analysis for 21 UTC 16 January 1994

57. Table 2: Statistical Analysis for 15 UTC 10 April 1997

58. Table 3: Statistical Analysis for 00 UTC 6 April 1999

59. Table 4: Statistical Analysis for 21 UTC 15 April 1999

60. Conclusions
Local pressure tendency and diabatic term do NOT generally offset one another
The advection term alone accounts from 30-60% of the total omega and agrees in sign
The sum of the local pressure tendency + advection term account from 50-90% of the total omega (i.e., this product is a better approximation to omega than just using the advection term alone
System-relative omega approximation can exceed the sum of the local pressure tendency + advection term, other times it was about > 80% of their sum. It is also from 50-70% of the total omega.
It you have the data it is worthwhile computing the local pressure tendency term using a small time difference, otherwise it is best to use the system-relative omega method.

61. Early Season Snowstorm in the Upper Midwest
Case Study: November 2001
Early Season Snowstorm in the Upper Midwest

62. Courtesy of MKE, WI

64. Lifting condensation pressure and condensation difference on the 296 K surface
Moist

65. 700 hPa e for 12 UTC 26 November 2001

66. Ground-relative streamlines and isobars on the 296 K surface

67. Ground-relative omegas on the 296 K surface

68. System-relative streamlines and isobars on the 296 K surface
C = 5.5 m s-1 at 314.5°

69. System-relative omegas on the 296 K surface

70. Ground-relative moisture transport vectors on the 296 K surface

71. System-relative moisture transport vectors on the 296 K surface