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Research ArticleThe Application of Diabatic Heating in π-Vectors
forthe Study of a North American Cyclone Event
Katie L. Crandall,1,2 Patrick S. Market,1 Anthony R. Lupo,1
Laurel P. McCoy,1,3
Rachel J. Tillott,1 and Justin J. Abraham1,4
1Department of Soil, Environmental, and Atmospheric Science,
University of Missouri, 302 E ABNR Building,Columbia, MO 65211,
USA2National Oceanic and Atmospheric Administration, National
Weather Service, Operations Proving Ground, 7220 NW 101st
Terrace,Kansas City, MO 64153, USA3National Weather Service
Portland Weather Forecast Office, 5241 NE 122nd Avenue, Portland,
OR 97230-1089, USA4The Weather Channel, 300 Interstate North
Parkway, Atlanta, GA 30339, USA
Correspondence should be addressed to Patrick S. Market;
[email protected]
Received 14 November 2014; Revised 7 May 2015; Accepted 25 May
2015
Academic Editor: Klaus Dethloff
Copyright Β© 2016 Katie L. Crandall et al. This is an open access
article distributed under the Creative Commons AttributionLicense,
which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properlycited.
An extended version of the π-vector form for the π-equation that
includes diabatic (in particular latent) heating in the
π-vectoritself is derived and tested for use in analyzing the
life-cycle of a midlatitude cyclone that developed over the central
United Statesduring 24β26 December 2009. While the inclusion of
diabatic heating in the π-vector π-equation is not unique to this
work, theinclusion of diabatic heating in theπ-vector itself is a
unique formulation. Here it is shown that the diabaticπ-vector
gives a betterrepresentation of the forcing contributing to the
life-cycle of the Christmas Storm of 2009 using analyses derived
from the 80-kmNAM.
1. Introduction
Formore than 60 years, quasigeostrophic theory (QG theory)has
provided the underlying basis for techniques that explainthe
existence or evolution of a wide range of atmosphericphenomena
including cyclones (e.g., [1β3]) and blockinganticyclones (e.g.,
[4β8]). QG theory has also been usedand continues to be used, as a
guiding principle in weatheranalysis and forecasting (e.g., [1,
9β11]). QG theory representsa scaling of the primitive equations
using the concept ofapproximate equality between the horizontal
pressure gradi-ent force and Coriolis (inertial) force in a
two-dimensionalatmosphere. QG theory also represents an
acknowledgementthat atmospheric circulations are three-dimensional
but thatthe vertical components of these circulations are
muchweaker than their horizontal counterparts, and these serveto
restore geostrophic and hydrostatic balance. In the
earlierapplications of QG theory (e.g., [1]), the atmosphere
isassumed to be adiabatic, which neglects the role of
diabaticprocesses (such as latent heat) and surface friction. Also,
in
model simulations or in diagnostic studies, large-scale
verti-cal motions are typically calculated subject to an initially
QGbalanced environment (e.g., [12]).
Given advances in the understanding of the lifecycles
ofatmospheric phenomena as well as advances in computingpower,
there are many studies which have examined the roleof forcing
neglected in earlier studies using QG equationsin midlatitude
cyclones (e.g., [13β18]). These have includedprocesses such as
boundary layer friction, boundary layersensible heating, or latent
heat release. These have even beenincluded in the study of
large-scale phenomena such as block-ing anticyclones (e.g.,
[19β21]), more specifically midlatituderidging due to lower
tropospheric diabatic heating. By includ-ing processes or forcing
such as diabatic heating or frictionin the study of these
phenomena, a better understanding oftheir life-cycle evolution has
been gained.This has resulted inbetter model formulations of, for
example, convection (e.g.,[22]), as well as providing better
guidance to operationalforecasters.
Hindawi Publishing CorporationAdvances in MeteorologyVolume
2016, Article ID 2908423, 11
pageshttp://dx.doi.org/10.1155/2016/2908423
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2 Advances in Meteorology
While many of the diagnostic tools used in meteorolog-ical
analysis were developed in full form, such as the omegaequation
(e.g., [23]) or the Zwack-Okossi vorticity tendencyequation (e.g.,
[5, 16]), they have primarily been used inQG form in many studies
(e.g., [24β27]). An increase incomputing power has been one factor
in inspiring the useof more complete forms of these diagnostic
equations (e.g.,[4, 13, 16, 20, 28]), and these publications, as
well as others,initially refer to the QG-equations that include
latent heatand/or friction as βextendedβQG forms. In recent years,
theseβextendedβ equations are referred to as QG forms, droppingthe
βextendedβ notation. The increase in computing powerhas also led to
the development of new expressions, suchas the ageostrophic
vorticity tendency equation (e.g., [18]).These new diagnostic
quantities have allowed the user toexamine the role of traditional
atmospheric forcing processesusing a more complete framework.
The role of diabatic heating in forcing ascent (e.g.,
latentheating) has long been included by researchers for use in
thediagnostic equations cited in examples above. Other exam-ples
include the studies of [29], who developed and [30] laterused a
π-vector form of the π-equation. Then [30] furtherincluded latent
heating on the right hand side of the equationas an additional
forcing process. In [30], a cyclone caseoccurring over the Iberian
Peninsula was studied, and thelatent heating was included in their
π-vector equation inorder to examine the role of this process in
the cyclone.Both studies demonstrated that the inclusion of latent
heatingprovided a better estimate for the divergence field that
isassociated with vertical motion. Thus, in both of their work,the
divergence ofQwas calculated separately from the Lapla-cian of
latent heating in order to determine the role of thisprocess in
cyclone development.
The standard π-vector equation [25] is used heavily
inoperational analysis and forecasting as it amalgamates impor-tant
QG forcing processes into one variable (e.g., [17, 25, 26]).In the
standard formulation of the π-vector, the differentialvorticity
advection and the Laplacian of the temperatureadvection are
combined to form a term that contains theadvection of the
temperature by the gradient of thewind field.This has some
important advantages, such as eliminatingβoverlapβ between the
differential vorticity advection and theLaplacian of the
temperature advection and the fact that theforcing is Galilean
invariant (e.g., [31]).
Diabatic heating of any kind is not included regularly inany
formulation of Q known to the authors. Since diabaticheating (in
particular latent heating) has been shown to bea contributor to the
development of phenomena such asmidlatitude cyclones, it is often
included in diagnostic studiesusing, for example, the omega
equation. Since there is a π-vector form of the omega equation, it
would be useful todevelop a π-vector expression that includes the
contributionof diabatic heating inside the one variableQ. Thus,
this is thegoal of the work here, and the details will be presented
inSection 2. In Sections 3 and 4, we perform a case study of
asynoptic-scale cyclone where diabatic heating (in
particulartropospheric latent heating) was an important process
and
demonstrate the effectiveness and utility of the diabatic
π-vector. Section 5 will summarize the work done here andpresent
our conclusions.
2. Data and Methods
2.1. Derivation of an Extended π-Vector Form of the π-Equa-tion
and a Diabatic π-Vector. The derivation of an extendedπ-vector is
similar to the QG version found in [32]. Thisderivation begins with
a quasigeostrophic form of theNavier-Stokes equations ((1a), (1b),
and (1c)) here:
ππ’π
ππ‘π= ππVππ +π½π¦Vπ +
βπΉπ₯ (1a)
πVπππ‘π
= βπππ’ππ βπ½π¦π’π +βπΉπ¦ (1b)
π
ππ‘π=
π
ππ‘+ π’π
π
ππ₯+ Vπ
π
ππ¦, (1c)
where π’ and V are the zonal and meridional wind compo-nents,
respectively. The subscripts βπβ and βππβ represent ageostrophic or
ageostrophic wind, respectively. The Coriolisparameter is ππ and
the change in the Coriolis parameter inthe meridional direction is
represented by π½. Friction is rep-resented by βπΉ.β The complete
derivation of the π-vector canbe found in [10].
Then, following, for example, [10, 26, 30] and others, wearrive
at (2a) and (2b) the final extended π-vector form:
π1π =ππ
ππ₯βπ2
π
π
ππ’ππ
ππ
= β 2π
ππ(πβππ
ππ₯β
βπ)+
π
πππ½π¦
ππ
ππ¦
βπ
ππ
π
ππ₯(οΏ½ΜοΏ½
ππ)β
ππ
π
π
ππ
πβπΉπ¦
ππ₯
(2a)
π2π =ππ
ππ¦βπ2
π
π
πVππππ
= β 2π
ππ(πβππ
ππ¦β
βπ)β
π
πππ½π¦
ππ
ππ₯
βπ
ππ
π
ππ¦(οΏ½ΜοΏ½
ππ)+
ππ
π
π
ππ
πβπΉπ₯
ππ¦,
(2b)
where
βπ = π1ππ +π2ποΏ½ΜοΏ½. (2c)
In (2a), (2b), and (2c), π» is used for the diabatic
heatinginstead of the traditional notation of π in order to
avoidconfusion with the π-vector. Taking the partial derivative
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of (2a) and (2b) with respect to π₯ (π¦), adding the two
partstogether, and using continuity in the form
β β
βπ = (
ππ’ππ
ππ₯+πVππππ¦
) = βππ
ππ, (3)
result in (4a), (4b), and (4c). Consider the following:
(β2+π2π
π
π2
ππ2)π = β 2β β
βπβββββββββββ
Aβ
π
πππ½ππ
ππ₯βββββββββββββββB
βπ
ππβ2 οΏ½ΜοΏ½
ππβββββββββββββββC
+ππ
π
π
πποΏ½ΜοΏ½ β
(β Γ
βπΉ)
βββββββββββββββββββββββββββββββββD
,
(4a)
whereQ is now comprised of the traditional π-vector:
π1 = βπ
ππ(πβππ
ππ₯β
βπ) (4b)
π2 = βπ
ππ(πβππ
ππ¦β
βπ) . (4c)
The result is a traditional form of the π-vector
relationship((4b) and (4c)) derived by [25] originally and found in
[32]in QG form, respectively. In (4a), Term A is the divergenceofQ,
Term B is the βBeta Termβ (meridional difference in theCoriolis
parameter), Term C is the diabatic heating term, andTermD is the
friction term. Further, for the βextendedβ formsderived by those
cited in section one, they also state that theobserved winds can be
used in the calculation instead of theirgeostrophic values.
However, one of the results of the [25]π-vector derivationwas
that the differential vorticity and the Laplacian of thetemperature
advection terms in the π-equation were com-bined. Since the
Laplacian operator is the divergence of thegradient operator, or (β
β
β), (4a), (4b), and (4c) can berewritten as;
(β2+π2π
π
π2
ππ2)π = β 2β β
βοΏ½ΜοΏ½ β
π
πππ½ππ
ππ₯+ππ
π
π
πποΏ½ΜοΏ½
β
(β ΓβπΉ) ,
(5a)
where οΏ½ΜοΏ½ now has the components
οΏ½ΜοΏ½1 = βπ
ππ(πβππ
ππ₯β
βπβ
12
π
ππ₯
οΏ½ΜοΏ½
ππ) (5b)
οΏ½ΜοΏ½2 = βπ
ππ(πβππ
ππ¦β
βπβ
12
π
ππ¦
οΏ½ΜοΏ½
ππ) , (5c)
and the diabatic heating term is now clearly part of the
π-vector formulation ((5b) and (5c)) and the divergence of Qand not
a separate term as in [29, 30]. Thus, the goal of thisstudy will
not be to demonstrate primarily the importance
of diabatic heating in contributing to cyclone development,since
that has been established by many researchers, butthe utility of a
diabatic π-vector in an operational context.Nonetheless, it will be
compared to the traditional π-vectoras a point of reference. It is
also noted here that this study onlyexamines the latent heating in
the diabaticπ-vector and sen-sible and radiational heating/cooling
process are neglected.
2.2. Data and Analysis. Numerical output from the
NorthAmericanMesoscale (NAM) Etamodel was used tomake
thecalculations of the diabatic π-vector and its components.
Inparticular, a thinned, 80 km grid was employed for the
QGdiagnostics as suggested by [33].While this is relatively
coarseresolution compared to what is used operationally today,these
data were readily available in-house. The operationalresolution at
the time of this stormwas 12 kmout to 84 h.Out-putwas taken from
the run initialized at 1200UTC24Decem-ber 2009. At this time, a
well-developed cyclone was alreadypresent over the southern United
States, as suggested byboth the surface analysis (showing a mature
cyclone) and thesatellite signature of a comma cloud in the
southern plains(see Figure 1).
Given its initial position, the cyclone was well sampledby the
observing network for the center of the circulationespecially over
the land areas of the United States. Grids fromthe NAM solutions
were thus selected at 12 hours (0000 UTC25 December 2009) and 24
hours (1200 UTC 25 December2009) into the run. This approach allows
time for the NAMEta model run to become dynamically balanced and
still beearly enough to prevent serious departures from reality. As
aresult, the vertical motions that are produced by the modeldo not
include subgrid-scale process. Thus the verticalmotions would be
result of larger-scale QG processes andparameterizations of latent
heating and friction (documen-tation for the NAM can be found on
the website suchas
http://www.srh.noaa.gov/ssd/nwpmodel/html/nam.html).Nonhydrostatic
versions of the model with finer resolutionare able to produce
verticalmotions with latent heat included.
2.3. Latent Heating Calculation. GEMPAK [34] was used toprocess
the NAM model grids and to format and produceestimates of οΏ½ΜοΏ½1 (5b)
and οΏ½ΜοΏ½2 (5c). Consistent with theformulations of these
expressions, it is important to note thatthese π-vectors were
constructed using a single level only(500 hPa), requiring fewer
calculations. This is an advantageover techniques that would
require multiple levels for acalculation (e.g., [16]). Moreover,
the static stability is heldconstant so that the entire leading
term, asβπ
/ππ, is constant;this also promotes amore direct
comparison to the traditionalπ-vector. Additionally, the actual
temperature, π, in οΏ½ΜοΏ½1 andοΏ½ΜοΏ½2 is replaced with the potential
temperature, π, which alsois more in keeping with traditional
formulations of the π-vector and facilitates use of the method
employed effectivelyby [35] and derived by [36] and for estimating
the diabatic(specifically latent) heating:
οΏ½ΜοΏ½ =ππ
ππ‘= π(
ππ
ππβπΎπ
πΎπ
π
ππ
πππ
ππ) . (6)
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Figure 1: Maps and analyses for 1200 UTC 24 December 2009, where
(a) is the mean sea level pressure (solid, every 4 hPa) and
1000β500-hPageopotential thickness (dashed, every 60 gpm) from NAM
Eta initialization, (b) the 500 hPa geopotential heights (solid,
every 60 gpm), and(c) the GOES-12 water vapor image at 1215
UTC.
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is the equivalent potential temperature, and the rest of
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found in this work. This formulation is also similarto that used in
[30].This computation uses the model verticalmotion to calculate
latent heating.Thenew calculated verticalmotion can be then used to
recalculate latent heating itera-tively (e.g., use the new vertical
motion in the latent heatingand recalculate vertical motion);
however, subsequent calcu-lations do not improve the result
significantly.This strategy isused often in latent heat
calculations (e.g., [16, 18, 20]).
3. Synoptic Analysis
The cyclone to be examined here occurred during Christmas2009
and brought a variety of weather to the mid-MississippiValley
region aswell as the Southern Plains ofNorthAmerica.This included
41 tornadoes, 24 high wind reports, and 10 hailreports in the
southern Mississippi Valley region, especiallyLouisiana, as well as
heavy rain and snow to points further
north (http://www.spc.noaa.gov/).As of 1200UTC23Decem-ber 2009
(not shown), the surface cyclone was a weak andunorganized area of
low pressure located over the SouthernPlains near Texas and
Oklahoma with the minimum pres-sures around 1001 hPa. At 500 hPa,
there was a strong troughand strong meridional flow over the
western portion of theUnited States, and the strongest winds at 300
hPawere locatedon the upstream side of the trough. However, the
coldest airwas still located over western Canada at this time and
wasbeginning to move southward into the Plains region.
By 1200 UTC 24 December 2009 (the time of NAMmodel
initialization used here), the surface low was locatedover eastern
Texas and the surface pressure was 998 hPa(Figure 1(a)). Blizzard
conditions existed over Oklahomanear this time and even included
reports of thundersnow,though thundersnow events are not
geographically favoredin this region [37].There was
strongmidlevelmeridional flowover thewesternUnited States extending
deep into Texas, andthe trough itself (with a closed center over
Texas) had notmoved appreciably eastward; instead, the trough and
ridgeover the United States amplified substantially and the
trough
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(a) is themean sea level pressure (solid, every 4 hPa) and
1000β500-hPageopotential thickness (dashed, every 60 gpm) from NAM
Eta initialization, (b) the 500 hPa geopotential heights (solid,
every 60 gpm), (c)the 300-hPa geopotential heights (solid, every
120 gpm) and the wind speeds (dashed, every 10msβ1, at 35msβ1,
45msβ1, and 55msβ1), and(d) the Doppler radar base reflectivity
mosaic.
narrowed (Figure 1(b)). Satellite imagery suggested a
well-developed,mature cyclone, with a comma signature featuringtwo
southern cloud bands suggesting complex flows withinthe warm
conveyor belt (Figure 1(c)).
Over the next 24 h, the surface cyclone deepened(Figure 2(a))
concurrent with the intensifying 500 hPalow (Figure 2(b)) and
attendant upper-level jet structure(Figure 2(c)), more specifically
the exit region of the cycloni-cally curved jet (e.g., [38]). The
precipitation shield at 0000UTC 25December 2009 (Figure 2(d))
emphasizes thematur-ity of the system. By 1200UTC 25December 2009,
the surfacecyclone had reached βΌ990 hPa (Figure 3(a)). The cyclone
hasalso occluded, based upon Bergeronβs rule of analysis, that
afrontal wave is occluded if the pressure difference betweenthe
cyclone center and the last closed isobar is 15 hPa orgreater [39].
At 500 hPa (Figure 3(b)), the upper level lowdeepened significantly
(more than 120m in 24 hr) over thecentral United States and had
become a closed system withfour closed contours. At 300 hPa (not
shown, deepened about200m), the trough was located over the same
region andmirrored the evolution of the 500 hPa trough over the
same24 hr period but has only one closed contour by this time.
Up to this time, the trough is tilted westward with height inthe
lower troposphere indicating baroclinic development.Generally,
cyclones that were studied in the referencesabove show strong
surface development in terms of rapidcentral pressure falls. This
storm showed midtroposphericdevelopment, without strong pressure
falls in the surfacecyclone (only 8 hPa in 24 hr). It is suspected
here that themidlevel deepening was due in part to latent heating
abovethis level based on previous studies (e.g., [16, 40, 41]),
thougha study of the entire heating profile would be needed
toverify this scenario. Nonetheless, [40] show that mid andupper
level latent heating would have little impact on
surfacedevelopment. Also, the impact of diabatic (latent)
heatingdiminishes with distance from the maximum (e.g., [41]).
At 0000 UTC 26 December 2009, the satellite perspectiveshows
that this stormwas of large enough scale to have drawnin moisture
from the southeast Pacific, the Gulf of Mexico,and the Atlantic
(Figures 1(c) and 4). In fact, an examinationof the 950β500 hPa
relative humidity (not shown) showed abroad area of saturation
(values greater than 80%) across theupper midwest and plains states
within the cyclone region(Figure 4) in the low- to midtroposphere.
While high relative
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2009.
humidity values are often used to show where clouds andlatent
heating reside, it should be cautioned that these valuesdo not
reflect actual moisture values.
By 1200 UTC 26 December 2009, the surface cyclonebegan to fill;
there would have been more low level conver-gence present than
upper level divergence at the time, forexample, [16, 40], and the
pressures rose. The upper-leveltrough had become a closed low
located overWisconsin.Theheaviest precipitation was located to the
north and east of thesurface cyclone, and a significant dry air
stream had wrappedup into the Great Lakes region near the cyclone
center. Afterthis time, over the next 48 h, the surface andmidlevel
cyclonewould decay and fill before being swept out of the region
bythe next transient system (not shown).
4. Dynamic Analysis and Discussion
The traditional π-vector (Figure 5(a)) and diabatic
π-vector(Figures 5(a) and 5(b)) were both used in order to
examinethe forcing for vertical motion associated with this
cyclone
at 500 hPa. The shaded regions here are the 6-h
precipitationgreater than 4mm. Precipitation is shown here since it
tendsto correlate with latent heat release; however, there may
besome differences as not all latent heat release results in
precip-itation. Latent heat release can also be associated with
cloudformation, but significant latent heat release is
associatedwithclouds that produce precipitation. Recall that
precipitationis the result of moisture, lift, and instability, and
these arerepresented (6) shown in section two. In this section,
theverticalmotion is examined using the diabaticπ-vector equa-tion,
which is a diagnostic equation. Thus, this section willillustrate
the relative contribution of latent heating only in thediabatic
π-vector.
At 0000 UTC 25 December 2009 (Figure 5(a)), the 12 hNAM model
solutions show that the traditional π-vectordivergence included
only relatively strong forcing due to thevorticity and temperature
advections over the region fromArkansas and surrounding areas and
weak forcing elsewherein the figure. This would only partially
explain the devel-opment of the midtropospheric low over extreme
southwestArkansas (Figure 2(b)) by this time. Inclusion of latent
heat-ing (presumablymaximizing above the 500 hPa level, e.g.,
[16,20]) in the π-vector produced only slightly weaker forcingin
the π-vector during this time (Figures 5(b) and 5(e)).The region of
forcing is a single, globular region even in thediabaticπ-vector
field, broadly mimicking the comma shapeseen in the satellite
imagery. Presumably, by this time theforcing due to latent heat
release has yet to fully materialize.
The divergence of π (Figure 5(c)) showed a patternconsistent
with that of theπ-vectormagnitudes (Figure 5(a));as the magnitudes
of the π-vectors were small, the π-vectorconvergence was weaker.
However, the divergence of the dia-baticπ-vector field (Figure
5(d)), while similar to the bandedpattern in the field of diabatic
π-vector magnitude, revealsnicely the expected displacements of the
divergence maximainto the gradients of the raw vector magnitudes.
Moreover,the divergence of the standard π-vector field revealed a
cir-cular area of divergence (and forcing for descent) acrossmuchof
Missouri, where precipitation was actively falling bothin the model
and in reality (Figure 2(d)). Yet, the adiabatic
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Figure 5:π-vector diagnostics from theNAMmodel run initialized
at 1200UTC24December 2009 and valid at 0000UTC25December
2009(12-hour solutions), including (a) standard 500-hPaπ (hPamβ1
sβ1), (b) the diabatic form of the 500-hPaπ (hPamβ1 sβ1), (c) the
divergenceof the standard 500-hPa, β β
οΏ½βοΏ½ (every 20 Γ 10β15 hPamβ2
sβ1; values
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8 Advances in Meteorology
30
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β100 β90 β80(β10ββ10)091225/1200V024 500MB
(a)
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(f)
Figure 6: As in Figure 5, but from the NAMmodel run initialized
at 1200 UTC 24 December 2009 and valid at 1200 UTC 25 December
2009(24-hour solutions).
diagnosis misses the stronger precipitation areas
centeredoverwestern Illinois and centralAlabama.Thedifference
fieldof the divergence of standard Q is less the divergence of
thediabatic Q (Figure 5(e)) and reveals a band of relatively
largepositive values across southeastMissouri, western
Tennessee,and into northern Mississippi. Such values result from
near-zero (positive or negative) values of standard Q
divergence
having large values of diabatic Q convergence subtractedfrom
them. The regions of greatest positive differences runalong and
immediately adjacent to the most intense 6-hourmodel precipitation
accumulations in the ensuing hoursbetween 1200 UTC and 1800 UTC on
25 December 2009.The Laplacian of the π field (Figure 5(f)) also
matches nicely(pattern and magnitude) with the divergence pattern
in both
-
Advances in Meteorology 9
the traditional and diabatic Q field (Figure 5(d)) as well asthe
precipitation region, confirming the importance latentheating may
have later and the calculation of the diabatic π-vector.
By 24 hours into the NAM simulation (1200 UTC 25December 2009)
the presence of a deep cyclone is reflectedin both the standard
π-vector field (Figure 6(a)) and thediabatic π-vectors (Figure
6(b)). Indeed the vector fields aremuch more similar over Missouri
and Illinois, nearer thecenter of the circulation. To the east over
Ohio, Kentucky,andWest Virginia is where the larger differences
occur at thistime, associated with the warm sector and warm
conveyorbelt of the cyclone. The divergence of the standard
π-vectorfield is unrevealing where the model precipitation is
accu-mulating (Figure 6(c)), with divergence values close to
zeroacrossmuch of Indiana andOhio.The addition of the
diabaticheating term to Q provides a divergence field (Figure
6(d))with a clearly better correlation to the actual
precipitationfield that is accumulating in the model. The
differencefield between the divergences in the standard π-vectorand
the diabatic π-vector (Figure 6(e)) reveals maximumdifferences
well-correlated with areas of stronger modeledprecipitation over
Ohio and southeastward into the Caroli-nas, further suggesting the
limited utility of the standardπ-vector in diagnosing upward
motions at this state in thecycloneβs lifecycle. However, even in
the Illinois region wherepresumably dynamic forcing is dominant,
the diabatic π-vector is stronger (Figure 6(e)). Moreover, the
Laplacian ofπ (Figure 6(f)) suggests ascent over locations such as
Ohioand South Carolina, where precipitation was occurring inthe
model. Finally, the differences between the traditionaldivergence
ofQ calculations and the diabatic π-vectors werelarger in Figure 6
which is later in the cyclone life-cycle thanin Figure 5. This is
because the gradients in the diabaticheatingwere larger in Figure 6
as implied by the divergence ofQ calculations which include the
gradient of diabatic heating.
5. Summary and Conclusions
An extended version of the π-vector is developed here
thatincludes diabatic heating as a forcing term in the diver-gence
of Q itself, amalgamated with the differential vorticityadvection
and the Laplacian of the temperature advectionterms.While this
version of theπ-vector could include latentheat release, sensible,
or radiational heating/cooling, onlylatent heat release is included
in this study for the purposeof demonstration. A strong cyclone,
which developed overthe central United States during 24β26 December
2009,was studied using output from the NAM Eta model witha
resolution of 12 km, thinned to 80 km, in order to showthe utility
of this technique. This cyclone was different frommany of the
cyclones studied by those cited in section one inthat the
development in the middle troposphere leading tothe formation of a
closed cyclone was studied. Also, in thereferenced studies, a
strong surface development occurred,which did not happen in this
case. Additionally, this eventwas of such scale that it drew
inmoisture from the threemajormoisture sources bordering the United
States.
The dynamic analysis showed that the diabatic π-vector(in this
case latent heating only) revealed stronger midtropo-spheric
forcing associatedwith this storm than the traditionalπ-vector.
Thus, this form of the π-vector would have beenmore useful in this
case from an operational perspective asit would have indicated the
stronger forcing aloft, especiallylater in the lifecycle of the
storm following the strongerprecipitation associated with the
event. Further study testingthe utility of this technique will be
performed with morecyclone events.
The traditional π-vector identified the general regionwhere
there was forcing that was favorable to cyclone devel-opment acting
to generate upward motions and height fallsbut was significantly
weaker overall. The diabatic π-vectorwas stronger, especially later
in the lifecycle and identifiedπ-vector convergence in the warm
moist air ahead of thecyclone that was not identified using the
traditional formu-lation. While the importance of diabatic
processes has beenpreviously established, this comparison was made
in orderto demonstrate the utility and ease of calculating a
diabaticπ-vector. Presumably, the diabatic π-vector would
possessall of the advantages of the traditional model including
thenonlinear interaction between the three forcing terms
andGalilean Invariance. This technique also allowed the use
ofanalyses produced from coarser resolutionmodel output andusing
only one pressure level for the computations.
Conflict of Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
Acknowledgments
The authors are indebted to Dr. Scott Rochette for hisinsightful
review of this paper. Dr. LouisUccellini also offereduseful
comments during early phases of this work. Theauthors also wish to
thank the anonymous reviewers formaking this paper a stronger
contribution.
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