Isentropic Analysis

James T. Moore

Cooperative Institute for Precipitation SystemsSaint Louis University

COMET COMAP Course May-June 2002

Now entering a no pressurezone!

Thetaburgers

Served hot and juicy at the Isentropic Café!

Boomerang Grille

Norman, OK

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

Theta as a Vertical Coordinate

• = T (1000/P) , where = Rd / Cp

• Entropy = = Cp ln + const

• If = const then = const, so constant entropy sfc = isentropic sfc

• Three types of stability, since / z = ( /T) [d - ]

– stable: < d, increases with height

– neutral: = d, is constant with height

– unstable: > d, decreases with height

• So, isentropic surfaces are closer together in the vertical in stable air and further apart in less stable air.

Vertical changes of potential temperature related to lapse rates:

U = unstable

N = neutral

S = stable

VS = very stable

Visualizing Static Stability – Vertical Gradients of

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 = T (1000/P)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 = = Cp T + gZ ;

V

k

fgeo

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.

Advection of Moisture on an Isentropic Surface

Advection of Moisture on an Isentropic Surface

Moist air from low levels on the left (south) is transported upward and tothe 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 fromBluestein, vol. I, 1992, p. 23)

RelativeHumidity305K surface12 UTC 3-17-87RH>80% = green

Pressure analysis305K surface12 UTC 3-17-87

Benjamin et al.

RelativeHumidityat 500 hPaRH > 70%=green

Isentropes near Frontal Zones

Surface Map for 00 UTC 30 December 1990

Surface Map for 12 UTC 30 December 1990

Sounding for Paducah, KY 30 December 1990 12 UTC

898 hPa+14.0 C

962 hPa-1.3C

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

Three-Dimensional Isentropic Topography

warm

cold

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.

Thermal Wind Relationship in Isentropic Coordinates

V C

f

Pk P

p

[ ]( )

1

1 0 0 0

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.

Thermal Wind Relationship in Isentropic Coordinates

• Isentropic surfaces have a steep slope in regions of strongbaroclinicity. Flat isentropes indicate barotropic conditionsand 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.

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

Isentropic Mean Meridional Cross Section

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.

Neutral-Superadiabatic Lapse Rates

SA=superadiabatic and N=neutral

Vertical Resolution is a Function of Static Stability

LS = less stable (weak static stability) and VS = very stable (strong static stability)

Surface Map for 00 UTC 27 November 2001

DDC

DTX

Sample Cross section for 00 UTC 27 November 2001

Radiational Heating/Cooling Disrupts the Continuity of Isentropic Surfaces

Namias, 1940: An Introduction to the Study of Air Mass and Isentropic Analysis, AMS, Boston, MA.

As time increases solar heating causes the 300 K isentropic surface to become “redefined” at higher pressures

304 K Isentropic Surface for 12 UTC 2 May 2002

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

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!

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 290-295 K

– Spring 295-300 K

– Summer 310-315 K

– Fall 300-305 K

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 50-100 hPa above the surface.

• A rule of thumb is to choose an isentropic surface that is located at ~700-750 hPa above your area.

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

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.

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 50-250, 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

M C O N V q

w here V q M T V

:

t

PV

P PV

dd t

P P dd t

( ) ( ) ( ) ( ) ( )

A B C D

Mass Continuity Equation in Isentropic Coordinates

Term A: Horizontal advection of static stabilityTerm B: Divergence/convergence changes the static stabil-ity; divergence (convergence) increases (decreases) the staticstabilityTerm C: Vertical advection of static stability (via diabaticheating/cooling)Term D: Vertical variation in the diabatic heating/coolingchanges the static stability (e.g., decreasing (increasing) diabatic heating with height decreases (increases) the staticstability

Term A: Horizontal Advection of Static Stability

Increased static stability

Increased static stability

Decreased static stability

Decreased static stability

Term B: Divergence/Convergence Effects

Term C: Vertical Advection of Static Stability

Term D: Vertical Variation of Diabatic Heating/Cooling

Very stable (50 hPa/4K)

Less stable (100 hPa/4K)

Divergence

Latent Heating

Latent Heating

Evaporative Cooling

V

P PV

PV( ) ( ) ( )

Horizontal Mass Flux

dd t

P P dd t

dd t

P

( ) ( )

Vertical Mass Flux

Moisture Stability Flux

M S F q P V

M S F V q P q P V

[( ) ]

[ ] [ ]

Where q is the average mixing ratio in the layer from to + , P 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.

d pd t

Pt

V Pdd t

P( )

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.

Computing Vertical Motion

Local pressuretendency termcomputed over 12, 6 and 3 hours by Homan andUccellini, 1987(WAF, vol. 2, 206-228)

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-1

(ascent)

Understanding System-Relative Motion

Isentropic System-Relative Vertical Motion

d

d t tV

d

d t tC V C

t tC

() ()()

() ()() ( ) ()

() ()()

0

d

d tV C

dp

d tV C P

()( ) ()

( )

Define Lagrangian; no -diabatic heating/cooling

Assume tendency following system is = 0; e.g., no deepening or filling of system with time.

System tendency

Insert pressure, P, as the variable in the ( )

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

( ) V C P

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.

In regions of isentropicupglide, this system-rela-tive motion vector, C, will enhance the uplift (since C is subtracted from the Velocity vector),

Vort Max at to

Vort Max at t1

When is C important to use when compute isentropic omegas?

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

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 5.10.1); 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.

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.

P d

d tPV d

Pt

Pt

t

t( ) ( ) [( ) ( ) ]

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

• 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

Four Precipitation Cases used for Study

PrecipiationArea

PrecipitationArea

PrecipitationArea

N Areal Mean StandardDeviation

P/t 80 -1.95 1.87 V P 80 -2.45 2.50(P/)d /dt 80 -2.76 2.91 (V-C) P 80 -3.70 2.10

Table 1: Statistical Analysis for 21 UTC 16 January 1994

PrecipiationArea

PrecipitationArea

PrecipitationArea

N Areal Mean StandardDeviation

P/t 51 -0.53 1.19 V P 51 -1.59 1.55(P/)d /dt 51 -2.24 2.49 (V-C) P 51 -3.10 1.30

Table 2: Statistical Analysis for 15 UTC 10 April 1997

PrecipiationArea

PrecipitationArea

PrecipitationArea

N Areal Mean StandardDeviation

P/t 87 -0.40 3.50 V P 87 -3.30 3.90(P/)d /dt 87 -2.90 4.00 (V-C) P 87 -4.70 5.10

Table 3: Statistical Analysis for 00 UTC 6 April 1999

PrecipiationArea

PrecipitationArea

PrecipitationArea

N Areal Mean StandardDeviation

P/t 116 -0.90 2.40 V P 116 -2.00 3.20(P/)d /dt 116 -0.40 2.30 (V-C) P 116 -2.90 4.50

Table 4: Statistical Analysis for 21 UTC 15 April 1999

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.

Case Study: 26-27 November 2001

Early Season Snowstorm in the Upper Midwest

Courtesy of MKE, WI

Lifting condensation pressure and condensation difference on the 296 K surface

Moist

700 hPa e for 12 UTC 26 November 2001

Ground-relative streamlines and isobars on the 296 K surface

Ground-relative omegas on the 296 K surface

C = 5.5 m s-1 at 314.5°

System-relative streamlines and isobars on the 296 K surface

System-relative omegas on the 296 K surface

Ground-relative moisture transport vectors on the 296 K surface

System-relative moisture transport vectors on the 296 K surface