an
**A p~ds
AxVzq
4 *ao4Op*
US tAPMfk"At~gevejo4"*A0~
0T
2HZ 1 'ITIE~dso
(run)
otnwn
Mopw
p
.weAz g
'.8.
u'dm
A IRLuopurII
Aft
IIt
TMc
w"=uoo
wunr
V 01111
vZAof Xv Q'lWIM M'~
4*
TRE WIND
WIL AT
LEELS DU
wm11
iG A
COLD OUTSU"#
cw
by
Attn7r A. NpUPa
uhistted to the Deart
of Noteorology on Janay 16, 1961
is Nrtatal Nifflttasat of tbe Zeqlflaate
for the Degree of Master of Scios
lhe ability of
nrent dynsical modela to prediot
In the vinnity of a colt front is asrkedly weak,
%y
te
treat
'atel models
lfor lO(DWb. *ndiate
eight changes
The forseasts yefld4d
that
oest of cold
Is actually Sigpitisntly geatr than predoted by those model.
the ara of greatest 1000-*.b* forecast rrPrS by the dynioaIa
odls is in the Plaise state.
frequnt cold outbeke he
Te PLains States is also an area of
Cnaa
The rVIdae sems to
aestia by the lrrtatiozal
gat tbat the proesas of lawee ve4
(dinergat)
wind field plays a large
2negh role, Uduing a cOld utbrak, to produce a signficaut
soureo of
erWr L the l000-ab. dycasloal foreaste.
In tis study te
adveetions of temperature, vrtictty, and lapse
rat by tme zaocDdritrgent and irrotatie4
during a sold abarak,
ith teir
se-dinergt
The irwetatloa
wind field# aft evaluated
adrections ae ten
counterparts, to determine wtehbr
ompared
eomission
ot
the Tortatonal adreotima
by the dPend'tmOdels is a signat
saturo af error for dyMMiatl predictioans, in the vicinitir t a o.
outbreak 1
ma ee
in
a
is that the role of th, irrotatiemal vid 'field is
large enuAgh, in a eold outreak, to
3.OOO-eb. predictioat erdnaal
by the inotational wad
Ibeats gaperviar
ttt*
produ
iflat
ewers In
inodels whiob aegleot adrectionw
ieL.
Frederick Szdern
Assento Professor of St.rology
*
on aoaad
*Asses
" owp nuttsx) asg mu * unswwrra so tomwaodao o*Ar
n 01 V" 'stna nim aq eawq tn eow 3aysmesd uy nouvsew
aq sqg
.2"Iag
asTrnmo
To TreoqT e4 n'to en Iseow ftam'sqo t
#1*g1
eg%
"r
*a4erat--s
-dead sq, 3wap oenefm seq e
P4fl474V
*Apf#
Syn 4fl
q*uaafl
U$
uap"tm
oyfAj zo*;4 01
easengasyssag ensgan n qun
V)
uT
-goi "Lq
pawTO
0".
";
n"eft
is
""o Me~wxa
01 pnsennd eto t
e P
too-f
nDYee
S.Ue
flS en "ae
flqSa .tqtiw OVA
nia on
IrsrV
44w4.
TAMAE Of 000irO
Pp
TAME OF 00W210 S
1.
4
Im2
U.
P8OCEU
V* D
V.
O00UM0N AND MSOW-8IQN
0
so
25
usTor fOIMS
Fig*
ag
Surface
ap, 00001 Nowaber 17, 1199
2
Chart illustratng grid ad frontl position's
5
Ohart Illustrating streaainae and velocity
potential fields at 20004t. on 0000Z
NOT'
14
17. 199
4a
F7e4d of vorticity adveeted by non-divergent
wind at 2000aft. at 00002, Not, 17, 1969
40
Field of vntioity adoted by irrotatUal
wind at 20004%t, at 0000,
Ga &
6b
2
ow 17,1960
Graphs illustrating variation of ratios of
noe-divrget advetions to irretational
adveotiona, ith alUtude,
1
.
16
18
I
studies of te
IfKWDUOTION
results of dpamioal preditieon edela shoy that
"old frontal situations are poorly handled by these adela.
"M4
In the
air insadiately behind the front, the actual heiht rises ane
aigificantly larger tan those give by the 1000..b, dryeatoul pre.
4*fta
eodei.(l)
U the United 8atest,
tbn area of largest 1000.abo errors by te
dynauleal sedael is in the Plins States, ()
tih
ay be responsible for this reult,
do not
Adeguately
Ummb
Sere are two tastOrs
pirst, the ynUa models
take into ascowit the terrain effects of the Rocky
p and seond,
e Plaine States is an area of frequent strong
*old outreaks, ad the poor bandlng of the cold frontal situaten
Is nesponsiblb*
Ue second of these two consideration
is
mind in this study.
se poer handling of the cold outreak is believed to be caused by the
msien of the advetins of signifinot elements by te
tand field
ot 000,
the strong
0ld
NOv, 17, 1959 (se.
irrotaional
athrnak, in the Plains States area,
tgol.) was selected for study because
the wid fiel appeared to be fairly irrotational in the old air,
eth the non-diergst and irrotatonal wind fields were determined
fret the observed wind field.
mso
teir
The adeotiems of te
tmen eapted, and the irrotaional advetonUs
signifiant
wer eeapared
a
ith
non-divergent counterparts,
Altough the exact effect Inclusion of the omitted ir'tatioal
adrectien
would hare bad on the 1000-..b. dyamical prediction was ret
-2-
A.-
(N
N
\
-
-I
\.A-
-7
V-p
Fri
C2
1.
to*
o did *how tMt to* irava
ere o*f
ftttet
pndttiaa It
al advo"toes
npitade to produ4. a sigpit"aatly better 1000o ..
hey ew In&Lwle in
tme eld tretal uitntiea.
XU, 200U
t a pre
Tee gderiatom
ati equatin for a dyaau
uwdel paraflele ver 4losly the notes for a course an numerical
watdbar prediotte by Prateseor 5, Piilipe of 5.1,?, (2),
at te derivatios U
formas,
La
siiaase of the adveotena at
to sbe the
ttetty, tanpnature,
and lapse rate La obtaining 4dnaaiol
. obtain a
the purpwae
tedel
pdequation, censider te besic equation,
prWSUnSe eW41dtatfl, below,
is e simplified vrflity oquate sad .quatia(t.)
(14)
sen
equaio
e = r(g)",
MeCo
whre
/(a
C
.e my be replaced by -7 in te
had st
4. +V l- 7,47'
sosdpsi
,asr
ftirt two torms on te riglh
ot egeaa (1.), and (4.)
7-
"
an tman be writtea
=o(a
+0
whe diabeticbeting Is omitted,
fl te .qu.atiom
af stae and the
=-=
7
on aW be mitte;
-(-and
,
droottio oqua-ang
thre #0
y
,eo that
W a bstution egtztion
(Si,)
440.
the tean 4
* can be ocasidered to be a fansoUs of presm e
asWg since its tariatmes (4m a coastant pressure surface) fros a oliaaslegiesl serage are
=mal,
Since each derivative in the produet above
to alUot *lwa* neAtve, it oan be written,
S -
5
-As
(r)
-
As an fiustratan of the gsmaral method of developing a prediction
wind is asasad to be geestropbie, in a barootnic atmoe.
te
and neglectag abges in
,
,
it as be sea that,
so that (1$) can be writtsa;
-T95
+
tmitaattoA
of
e
F
S
( 5.)
-
between (4.) and (4$) givegj
i
Since
,.
) -A>
Or
is a slowLy varyng function of pressure, the left band
side of (8.) am be written to resemble a three disesional laplacian,
d with the subsituon
" =- 8A,
equation (6.) bo
pnsion of the last term on the right band side of (7.) gives;
u4a
then ay a~r tw be
two n U6 the1"
kan~T,
fltt* jtstiflntian
for
U.lst
)mW$
a t=I
6Pp
FIs(80W)
apw
ob
sa
1f
* so a qeffIcoat, The treatmet bore Is
in Supp
J10
gialar t Um tesawmt
at f is tme a4uti4 vnttity .'aaqata,
a A
4. d of (I),
Za the first tea a tbe rigbt
(ntuf-())
ast Ue ditterenuabad
and *van tahou
tAe variaios
t
my be sea-Us
en Signfianty
M
rater
apar#
retritetd to rutivly .masl valnn.
eansd is aaWa
the Ir
any ta.
in f
1+
and tars m the right had aids of (1) f
auituda. nu the
s7 sea
fnea et
F
-r, T
=o
by the toraXl
wiad elatiao4up,
=
W-f 7,(:yj+f>)-,
W
so
(74) b.nPseg
.fatia
(r.
£Q
26 ald b stresed- that equatia
sm% the most
aes A d
Mgt h
tr, that
as4 oer dsamiea
0e4rd
ntios
an
(06) does show, hno"
edeotia of vrti44y,
Ia equation (s.) de
my and lapse ate
Iva uen
(#,) is "t izteaded to repro-
tis equatin.
nerM
t9 +RI$9)
have In sensn with
pretion models La that trza abor mentioned adveat
o4
Iy awtAd Celd ;oan ivergenes L .i4e-r zero at
sot reprewataUvre et the wAu1 dIWa*rg(*found in the. lowar levla,
Ee 4uatter *bob
Woe4inot Avaqti
a reade, "
Aang
h d produwe te sae trdiniae in Lho graditfl
aquation (S.)
.s a rodl,
it
n
saOe p4aial
9
.a b een that ponitive advteta
Using
of
teperature and vrtiitys and the adveticn of more stable lapse rates
weulA all tend to make the rigt hand side more positive* Sine in a
region here the Lapla of a quanmtity is positive, the quantity itself
is
egative, the above mationed advections would tend to produce height
fals. Coversely, when the adveetiona of teperature snd vorticity
are negative, ad when less stable lapse rates are advected, the right
hand aide becsee negative, and
beaomes positive so that height rises
are predicted,
£nediately hda
s adveeted are negative so
ad temperaft
and -IV .7r *Mke
t
gWaty
*
a cold fret in the Plains States, the vorticities
*
-
;- (x +1)
e right bad side of (9.) sore negative,
is
iFegatve
ehi
Also, the
old treat, so all three
a md
terms act in the same directimn in the cOld air* Mis direction is to
produce height risesHowever, as was stated above, dynamical models earry out these
adteetions with a wAnd fi2ld that is essentially ncsadivergant,
In this
study the non-divrgent and irretational advetimos are coaputed at four
levels, sear the ground and in the vicinity of a cold front* The irrotatiaul advectins are then cowpared with their uao-divergent counterparts,
e sigfcare of the nmisiou
dynamical
of the irrotatinal advections by the
del is then evalated*
"a".
111, PhQODURZ
A group af aarfae aps was inspected and the map of 0000 , Nov. 17,
1960 was selected baonse the flow is the cold air appeaed to have a
The sext task was to determine the
,
fairly strong irrotatiaal
ampeste of tia wind field in order to
aona-dvergat ad irrotatnal
evaldate the advetmis by each ompoent.
of vector analysis (5), any vector feld with, zero
Sy a thoas
as be separated into twe
atinuity (pieceuk.e dif i reatiable)
arder
vector fields, one field stristly nn.-divergeat, and te
other purely
Aeang the wind fields at 2OO, WO00 4000, and
irtatiole
OO.fnt
into this category, this meae that the velocity at say point
to fll
o each level ca be represented by a veterj
IV
where
and
\%1 j
+
-VWmor
lMJ z Vj0, V-.
IVM4Or =
rx
(where hJ means
a,o# and
r.iv,1 itx..,. = '
I
=
7
r
ncfludivergeat)
147
, and 7k Wnyer = *
This *eas that at each of the four levels chosen for investigation,
there mut exist fields of stream funsti
free
(y), and veloity potential (x),
hih the na-divergaat and irrotational wind fields can be determined,
A rectangular grid with unfters distance of one halt inch a a map
Of scale 1:10,000,000 (o8*
nautial miles) was employed over an area
eovering most of the U.8. (see
ig.24)
Strealine and isotach analyses
were made at each of the four levels of interest# At points where a level
of interest fl
below the terrain, the wind at the first level wtia
-1-
FOO
A.
al0.
was above the terrain was ued to represent the wind at all levels below,
the atealine and isotach fielde rere interpolated to give wind directions
and speeds at each grid point of each Level.
Te wind directions were then adjusted to that they would be
interpreted with repect to colun orientatie ramter than true north,
(This was convenient for computatin of divergence and vorticity,) The
correated vales of wind direction and the values of wind speed were then
punched a cards and served as input to a computer routine developed
by the Dpanioal Weather Prediction Project at M.I.T.
The program first determined fields of divergence () and vortieity
(Y)from the observed wind data, by the relationabips
+
,
Next the program solved te two Poison equationa
T= TV
and
CT
'-or
fields of Y ad j
thecputed fieldAs of S and J as known quantities.
using
Boaver1 to
determine unique solutions to these Poison equatioas, boundary conditions
bad to be established, The boundary conditions used were,
T
V and X
=4 o
S
wher
repreeto the diretio along the boundary and
to the boundary,
the direction normal
Tese conditions serve to maimize the kinetic energy
of the rotational wind field, (4)
Output of the program was fields of S,
F,
J
and 7X
Pields of temperature (7), at all four levels of interest, and
;
at
50004ft. were determined by analysing values obtained directly from robs,
Band oaloulatons ad a an aL sk calculator were used to deteraoa
frtlf of
of
a
ext
AT at all four levels, sad also fielde
-
r and a
at the SOOe*tt, lavol,
Rand "alcutio
were again used to detersine the, aon-divergent
and trrotatiual adectims of temperature, vortioity, and lapse rate
the expressions used were;
flib.dlflrgat to 1 iity
edvoc~aa
irretattesnvritg adves
=
-
Is-
-
7S
rr
7
-E
T
-
= -7
a.a
eat
ge~i temperature adnetes =-ff4s
Mie
-
vrr
expreesionse were used to determtne the lapse mte advetion
at 000ftt
It should be noted that no adveotioas were computed in areas where
to 1evel of intereat fll
below the terrain,
fieds were computed and apared in te
Advetions by both wind
entire area of te
the leAel of Interest was above the groud.
grid where
To obtain the sawecoom
partsoa of advootoae in an ares more nearly allied with the cold air,
ratio of mnedtvergnt to irretatiemal adveotions were determined In
te
inwe restritted ara of res
isantiatwo,
through 12, and colums 9 through 14
(tee 7igg)
Once te fields of the adveoted quantities were dtermied, a
eoparisen of the nondivergnt and irretational advectios was mado
SlmA
the min area of lterest, and tin atrnget winds, was in the
area behind the *old front, ad the adeatioas of temperature and
v*rtdIty in this area an nagtiwve, only average negative advectien
were compared,
At attempt was ad., by Sa
i
M0
is
aps of the advowivo
fields, to ne
we" eeters of negatin temperature or Vorbioity advortia
WhIk e4ul4 be ideettd from ace level to the adjacant levels, but the
ptterns Were not consistent enough in detail to giv a somaistent pattvn,
Tiw ratios of average negative
trrtatianw
on..odivergent to arage negative
advctions were made for both temperature and vwrtity.
(On* tor each area at all levels*)
The ouly area where average non.
divergat lapse rate advettoa was compared to avrage irrotatienal laps,
rate advootten was in the restrioted area, behind the cold freut, and ten
this ratie was dtermined only at 30004ft.
We seaga
the l-hour frontal displaement yielded by same dynamical
mod4U wit te
ta ges
actual 12-heur dispacemeant, the front was moved with bait
ph i%4hd .t 500-u.k.
A the de
breoliia
n
(This method
f displacemnt is
Sonsistent
predietia givrn by the barotropic ad two level
wodel.(f))
Position Do in Fig.2 is the displaaemsn* an..
ticipated by the dynaaio models,
Position C is the actual 1-bour
It earms that the large unpredieted movement of the front into the
Gulf of Meico should be due at least in part, to advections by a
northerly irrotational wind field within the oold air mass (see Fig. 1).
IV
The fields of strea
Rt8ULTS
fmtimon and velocity potential at the 2OOmufi
level are illustrated 1% ig44
The magnitude of thetrrotatianl wind
is proprtional to the grediet of the velocity potential lnAe (whieb
are dashed) snd the directimn of the irrtational wind is norml to
these linen
Th. solid lines in figs5 are lime. of constant stream
fUctimn vales.
The megnitud. of the nsmodivergent wind is proportional
to the gradient of tee
lines and the direction of the
one-divergent
wind is parallel to these lines (with lower stream fumntion values to
th lett),
The adveotion of verticity by Us
Q000-.ft, mn-divergent
aind field
is shos In Fig4 while the irrotational vortieity advection is
ItlMstrated by Fig,4.,
The differene betwen successive contour lines
n mape 4a ad 0 is the same.
The ratios of the average negative vorticity and negative teprature
adveeted by tbe na-4ivergent wind to those advoeted by the irretational
wind are tabulated.
-/4'-
SOLID
LINES -
2000'
STREAM
FUNCTION
FIELD.
DOTTED
LINES- 2000' VELOCITY POTENTIAL FIELD.
THE DIFFERENCE
BETWEEN ISOLINES IS THE\
SAME
.bor7'Zb
ro.i,
ARi,'ow
a4rrowUs
/--Jr'!rT
/LLUSr4Arg
.br/qcr/o,
Dacrio'r
FOR BOTH FUNCTIONS
og >oAT/oArnt.
OP
A*'A/'"~A
W(tr.D
-/i--
.4VV--"4dr&*J
ON
J'.f0t..IAol4F-r
J'OV'OM4'b
as
&Y
-10- "'
XCc-a
-/6-
2000'
0
aa
0
-3
-70
0
FIG.
48
2000'
IRROTATIONAL
VORTICITY ADVECTION
FIELD
og
/ SO'/!ocS
.foyd.D
G5
s
*y
g
-- '
SE
.
-17For the entire area (where above ground)
-,ATIO
/AVG.
MEG. M.D.
EL Av
Jaco-Pr
ZRo7VoR
9VG.
IVZG.
TAeMp
AV D.
9& ,VG. AEG. xRRo:
/.-4(=
/5
1.79
./4
/=
Sooo-Pr
/
AD
wEG&.
000-p7
4looe-
VORT.
. 3.
.. 413
3.5#
4. 70
AVG.
ADV
ADV.
VoRr?
.
For the smaller area (rows 6-9, cols. 9-14, Fig.2.)
(/
s
LEVEL
AVG.
ME.
AVG.
'VEG.
ZMEoG
26oo-pr
_____
.
V ooo-r
19 VG,
ADv
VORr R.
(AVG.
/VGo.
AVG.
WEG.
M.D.
rTER
R
_
_
93
__.__
2,0
AD)
/.75
2.
3.20
2.28
_.__
ADv.
TsMR
413
-__
Scoo-r
voRw
m..
_
_
S
-/I-LEVEL
5000 FT-
4000 FT -
3000 FT
I-
2000 FT I-
(AVG.
___a____I____IRATIO
1
2
FIG. 5 A.
3
(LARGER
4
5
NEG. N.D. ADV.
AVG. NEG. IRROT. ADV
AREA)
LEVEL
5000FT r
4000 FT
I-
3000 FT I-
2000FT
Sm
1RATIO
I
I
2
FIG. 5B.
3
(SMALLER
4
5
AREA)
IN BOTH FIGS. VORT ADV. RATIO - SOLID
TEMP. ADV. RATIO - DASHED
AVG. NEG. N.D. ADV )
(AVG.
NEG. IRROT. ADV./
The result. tabulated above are shows graphically in Figs, 5% and
Sb.
The ratio of average son.divergent positive lapse rate adnctioa
to average Srrtatenal positive lapse rate adventou na 1.47
Finally, as may be seen
Fig,2, the actual U hour froutal
displacement (position C) is swb greater than the displaemet given
W half the ormal oo
(position C)*
e4 of the geostrophic wind at 500-m4b,
eg0-
7. DISCUSSION AND CONCLUSIOi
As would be sormally expected, the ratios of the aOn.divergent
to irrotational adveotie
of both temperature ard vortlaity inorease
with height (with the eaptIon of the teperature advnotion ratio
betweep 5000Ctf. and 4000ftt.),
are all rather sml
The isportat fact is that the ratios
The arerag ratie for all levels indicates that
*ore than twenty fttn per-omt of the height riss in the cold air at
low levels behad a aold front are attributable to advections by the
irrotationel wind fielA
An toission of this sise by dy"Amical models
is certainly sigificeat,
Thle
g
rati obtained for the adretion of lapse rate elso
tee that the omisiown of ta irrotatoial adrectia should also
be as (proportioatedy) aplfiacnt as are the irrotatiema
taaparatre
and vortiity adveetiens,
The surprisingly large displaceent of the fromt in it's
southernmwt positimns an be attributed to the relatively strong
iroatattna4wicd field is this area.
(mw FIgt5)
The purpoe of obtaining raMos in two areas was to compare the
results of adveetisa over a less restricted area to those in an area
more clos&y allied with the cold air,
The di! erene in ratios ho1
twean the entire area (where te level was above ground) and the area
restricted to the *old air was not significant.
onever, the greatest
of all negative tewperature and negative vrticity advetims lie in
cad air along with the greater number of points, se the ratios are
largely determined by the behavior of the advectitas in the o*d air*
t wuid probabig bate ben suspeated that the ratios yielded
should bhe. bee greater tn the larger area Anse the wiad should
have bee sore irnotatiasai tere,
uind
Hwever, the analysis of e
fields was subjeot to greater error tn tho Gulf oC Vezico beAus of
the Sparsity of data i
thi area,
enalysis
the greamtr error tn 4sd
was s
prebay reultt I a wind field hoe irrtationalep
(Pecause of the inureed
great as its no-dijvergent **%powt,
n eewould tend to wweal-
yandoe" et the sld field,) This
tetOal4y dereese the raties Is the larger area,
I^ both areas,
greater thn bOeSe for vrtiity
resuit is
VId fields
advetions,
It is belired that this
brOgt about by the different e-lignante of te
sad vortdity field
are
he ratis for the temture adreotins
osa.divnrget and irrotatiocal
with ospest to the
Rowever, no atem
temperature
to inveStgate thi
was made
feature,
It should be stated that the ratios would have bees smaller if it
was not
to latrodne. assmptions of he nature of
an the boundary of the grid,
The boaeary coaditiss oboe
f
s/ d7
served to
minimise the kinetic aergy of the irrotaional wind field,
An iaterestiag featur, is illustrated by Fig4
5, 4a aAd 4b
A large senter of Fsitive vrOUeity was located near the freptal
sirfan at
ea.dtvergst
wind field,
Th presenc
either ide of the trot a Fig 4b, i
thi
ceater as advnsted br te
f000l "-go 4.shows how tis
of tie egative centers cn
the reslt of a 'pinching'
eater by te irre tatmlwiud field*
of
Vi
SOURES OF ERROR
Some error M have entered in the original streamline and ieotah
aalysis, sad in the interpolation of wind speed and direction values
to the grid poiat,
A small error probably resulted in the determination of the temp.
erature field at each of the four levels of interest,
Bere the temp-
natures were taken directly frwo the raobs in the area of interest.
the temperatures at each level were determined from the standard
atamphere values of height versus pressure, and no attempt was mad*
to get a more accurate temperature at each level.
Aother source of error probably resulted from the fact that there
was only one digit available, in the output of the electronic computer,
to nxpress a value for the velocity potetial, and only two digits to
expess values for stream fuaction.
It was feared that such poor
resolution of these fields may have introduced some error.
42&.
(1) katen,
4,
P
, L
00WAhrSlea Reewh aer
, and G#7, Jenkins, 1966;
No-* 46V AFORG4TR.6*200,
GOWPhrIos0 Researdk Direetorste, Bedford,
(a) smnups 1*1950 M*1
es., P.49, P.80
Gourse Notes, Lecture Notes an
Mumertcal Weather Prediattoa, Uapubflshed, PP. S&4,
(5) NA5 Phillips, 195 ;Law As&"- Joh Wiley and Sons
Se Zork, Po4M
(4) A *Iasa,
196; U,04A.4 Course Notes, Lectures an Phsioal
Weather Preditton, bpablishsd, P,48
(8) RSBee4, 1980
Vol# 8, Number 4,
uthly Weather bev.,
P#MM
Jun. 1980,