STUDY OF AGliSTROPHIC FLOW IN A POLAR OUTfMEAI
by Charles W. C. Rogers
B.S., Massachusetts Institute of Technology
(19T8)
WT
SUMID IN PARTIAL FULYILLUWNT OF TaE REQUIRDEnNTS
FM TO agE OF MASTER OF SCIENCE
at the
MASSACHIETTS IISTITWI OF TZCMLOGY
January 1961
Signature of Author
Departmert of ieteoroloi,
is January 1061
Certified by
IThesis Super isor
Accepted by
Chairman, Departmental Coaittee on
Graduate Students
STUDY OF AGROBTROPHIC FLOW IN A POLAR OUTBREAK
by
ChARLEg W. C. Room
Submitted in Partial Fulfillment
of the Requirements for the
Degree of Master of Science
ABSTRACT
Strong low-level northerly flow occurs behind the cold front
associated with polar outbreaks in the central plains of the United
States. This northerly flow has large cross-Isobar components on
the sea-level chart and large cross-contour components at levels a
few thousand feet above the ground. The cross-isobar flow decreases
northward from its maximum value located immediately behind the cold
front. Examination of the procedure used for reduction of station
pressure to sea level shows that use of a twelve-hour mean surface
temperature in this procedure produces a fictitious element in the
sea-level pressure pattern. This fictitious element makes part of
the surface flow appear more highly ageostrophic than it actually is.
It is found that use of this twelve-hour mean temperature in the reduction computations accounts for some of the north-south variation
of the cross-isobar flow.
The dynamics of the cross-contour flow are studied by compartoon of observed geostrophic departures and geostrophic departures
computed from a ten-level model. The vector difference between these
two geostrophic departures is found to be large. Examination of the
various approximations made in obtaining the ten-level model indicates
that the observed difference results primarily from the approximation
made concerning the eddy stress: Estimates of the actual values of
the various components of the geostrophic departure are obtained from
observed data. Comparison of these values indicates that the eddystress component predominates in producing the observed geostrophic
departures.
Thesis Advisor: Frederick Sanders
Title:
Associate Professor
of Meteorology
--
F-
TAL
OF 001I'1tT
Page
I
Abstract
Table
f nasteUnts
1
iii
List of Figures
1.
Ii.
111.
aeeral Introduction
1
Agostrophic Flow on the Sea-Level Chart
2
A.
Introduction
B.
Theory of Reduction of Station Pressure to
Sa Level.
3
C.
Prooedure
4
0.
Discussion of Results
5
Agostrophi
14
A.
Introduction
14
5.
Theory for Computed Goostrophie Departures
14
C. Procedure for Computation of T* at 910 ab
21
V.
IV.
Fleow Above the Ground Surtae
Discussion of Results
Summary and Conclusions
24
3$
Acknoeodgements
40
Referenees
41
I
LIST OF f1==h8
Page
ad Burts
ea,-3evel Pressures
tor 1200 00T
18,
Pobrary
Temperatures
195?
910-ab Caat for 1300 OT 18 Fbrury 1959
*2
Obervod ad computed Geostrophlc Depwtures
910 ub 1300 OT 18 february 199
23
Vip0 (V H 0)
alb
,
p
(
F =o
) atOC, AW!,
and AC0
28
v'- Compo
-
t DI
am tor OKC
couipemt Diagam
#- Compm
32
for ABI
33
Dlagwra tor A
Stress omponts for V' tor OEC, AS,
34
ad ACW
36
1.
GHN6RAL INTRODUCTION
Polar outbreaks occurring in the central plains of the Uhited
States are characterized by a cold front moving rapidly southward
behind which there is a strong low-level northerly flow.
Observation
shows that this low-level flow is highly agmostrophic on the sea-level
chart and is centered behind the fastest-moving portion of the cold
front.
On the sea-level chart this flow extends some distance back
into the cold air, but it tends to be most pronounced itasdUtely
behind the cold front.
The height to which this flow extends varies
from one case to another, but, generally the flow is absent at 83O mb.
At the higher levels this flow occurs over a small area located
primarily just to the rear of the cold front.
The present investigation is a case study of the ageostrophic
flow in the polar outbreak of 1200 OCT 18 February 1959 and is concerned with gaining an insight into the dynamics and spatial variation
of this flow.
II.
A.
AGEOSTROPHIC FLOW ON THE SEA-LIVEL C(ART
Introduction
The flow on the sea-level chart appears to be highly ageostro-
phic because the strong northerly surface winds blow at large angles
across the sea-level isobars.
In the region where this ageostrophic
flow is found, the land elevation is appreciable and increases toward
the west.
Therefore it is possible that the pressure field associated
with the surface flow is not well represented by the fictitious seslevel pressure field and that this flow is more geostrophic than it
appears to be.
Methods have been presented for obtaining a pressure field
which better describes the surface flow.
These methods enable one
to represent the field of the horizontal pressure-gradient force in
non-isobaric surfaces.
Phillips (1957) introduced a coordinate sys-
tem which uses the ratio of pressure to ground pressure as the independent variable.
Sangster (1960) uses a coordinate system which
is similar to Phillips' between the earth's surface and 500 ub, but
which has coordinate surfaces equal to isobaric surfaces at and
above 500 ab.
These methods eliminate the fictitiousness of the pressure
field used to describe the surface flow by completely doing away
with the reduction of station pressure to sea level.
-2-
This procedure
uses a twelve-hour mean of the surface air temperature to determine
the vertical mean temperature, T, in the fictitious air column extending to sea level.
Because of the large twelve-hour temperature
changes occurring in the cold outbreak region, it is possible that
this procedure would introduce a large fictitious element into the
sea-level pressure field.
The following computations were carried
out to investigate this effect.
a. Theory of Reduction of Station Pressure to Sea lavel
The sea-level pressure reported by United States stations
located above sea level is obtained by reducing the station pressure
to sea level by means of the hypsometric formula.
This formula may
be written in the form
sea-level pressure
where
station pressure
r
station elevation above sea level in feet
2
being the assumed lapse rate in
the fictitious air column extending to sea level
T'
the assumed surface temperature
In the tited States Weather Bureau computations, T is taken equal
-3-
mr
to Ts, the average of the present surface temperature, T
one twelve hours earlier, T&_
,
,
and the
in degrees Klvin1 .
We wish to compute new values of see-level pressure while holdsIng rj- constant but changing To
to To*
In the actual reduction procedure, tables (differing from one
station to another) are used which contain a semi-empirical relationship between ( and To .
assume that
(for
T'
Since we do not have these tables we will
Is the same as e for
T
.
Uder these assump-
tiass we obtain the formula
4
where
new sea-level pressure.
p_i
For stations where
that
C.
pO/
(2)
-
-
4 1000 feet
e
is assumed to be equal to sero so
is computed from (1) with T. replacing T*
Procedure
Values of
were computed for a number of stations located
in the region of strong aseostrophic flow on the sea-level chart for
1200 GCT 18 ftbruary 199.
Values of
surface wind were plotted for each station.
I
and
Sea-level pressure and
1 anual of Surface Observations (WIAN) CIRCAR W, Seventh dition,
October 1987. Uhited States Government Printing Office, Washington,
D. C.,
pg. 72C.
-4-
surface wind were plotted for stations for which no new sea-level
pressure was computed.
Isobars were then drawn forAand
The
analysis and plotted values are shown in Figure 1.
D.
Discussion of Results
The contention here is that the use of T, in determining T
introduces a fictitious element into the sea-level pressure pattern
which makes the surface flow look highly ageostrophic.
T
produces another set of sea-level isobars.
The use of
This set is
itself
fictitious, as are all sets of sea-level isobars obtained by reduction
of station pressure to sea level; however, it does not contain any fictitious element introduced by the use of T
.
Therefore this latter
set may be used as a standard for determining what fictitious element
is introduced into the sea-level isobars when T. is used.
Fig. 1 shows that between BB' and AA',
region I,
the surface
flow appears highly ageostrophic with respect to the T.-isobare because
of the large angle between the winds and the isobars (this angle will
be called the crossing angle).
These large angles are in turn a re-
sult of the pronounced southward bulge in the T.-isobars in this region.
The surface flow appears less highly ageostrophic with respect to the
T;-isobare because of the smaller crossing angles which result from
the absence of a southward bulge in the T'-isobars.
Therefore some of
the highly ageostrophic character of the flow in region I may be attributed to the fictitious bulge in the TO-isobars.
4
4~w
413*
*aI
r - sah .s
D*
*4'.,
wMW
To the west of BB', region II, the surface flow appears highly
ageostrophic with respect to the T5-isobars because of the large
crossing angles in this region.
These large angles are caused by the
northwest to southeast orientation of the T.-isobars.
The flow appears
more highly ageostrophic with respect to the T-isobars because of the
large crossing angles caused by the north-northwest to south-southeast
orientation of the T -isobars.
Therefore the orientation of the To-
isobars gives a less highly ageostrophic character to the flow than
the T;-isobars do.
The use of Ta in determining T produces a fictitious element in
the sea-level isobars which increases the highly ageostrophic character
of the flow in region I but decreases it in region 11.
These different
changes are a result of the different effect which the use of T. produces on the sea-level pressure pattern.
IMing the T'-isobars as a
5
standard we can investigate how this fictitious element is produced
in going from the T- to the T -isobars.
Let us consider region I
first.
gince there is only one station in the interior of region I,
the pressure field there is determined by the pressure values at the
boundary stations.
It is seen from Fig. I that the change in the
pressure field in going from the T'- to T -isobars is caused by the
sea-level pressure decreasing more at the western boundary stations
that at the eastern ones.
In order to examine the factors governing
-.
7-
this change we take the partial derivative of (1) with respect to To
and introduce finite differences. This procedure leads to the equation
(3)
0J.,
0@@
where
AT
-
TO-T
and T is considered asa mean between T. and T.
This equation can be simplified by considering the magnitude
of the various terms in the denominator.
Y a 1/330 0 C/feet we find that
of T = 280K and very large value of
(j)
Using a very small value
will be the same order of magnitude as ()
75,000 feet.
when Z is approximately
Similarly when Z is approximately 150,000 feet (3)will
be the same order of magnitude as (D.
Therefore, for our purposes (3)
may be written as
/d
L
=
(4)
7-
From (4) we see that the change in the seam-level pressure
pattern in region I described above occurs when the western stations
have a value of Z&T larger than the eastern ones.
graphy of region I
% is larger than Zg (where the subscripts refer to
western and eastern stations).
(WT)
Because of the topo-
Thereforej (&\T)w will be larger than
for all cases in which (AT)
is larger than (0T) and for some
cases where (WT) is smaller than (6T)g . An example of the former
cse Is found in Fig. 1 by comparison of Amarillo, Texas (ANA) and
Comparison of Midland, Texas (MY)
Oklahoma City, Oklahoma (Og1).
and Abilene, Texas (AB)
shows an example of the latter case.
latter comparison demonstrate
stations to values of AT .
This
the greater sensitivity of the western
It can be seen from (4) that this greater
sensitivity is caused by the higher elevation of the western stations.
Therefore we conclude that the fictiousness of the sea-level pressure
pattern in the southern part of region I is caused by the greater sensitivity of the western boundary stations to the use of a
determined
from T
9
The oemparison of AMA and OKC show that the fictitiousness in
the sea-level pressure pattern is caused by the larger positive value
of 6T at the western station.
A large positive value AT occurs when
the twelve-hour temperature change is large; this change is large for
a station which is in the warm air immediately adjacent to the front
twelve hours prior to observation time.
Such a situation gives a very
warm value of Tt- 1 2 and then allows the station to be far back in the
cold air for its value of T. .
An example of this situation in Fig. I
is AMA with Ttg 12 W690? and T 0
2 ?.
In these cold outbreaks) the cold front in the western part of
region I has a west-northwest to east-southeast orientation.
This
orientation results from slower movement of the front along the east
-9-
slopes of the Rocky Mountains than in the cntral plains.
Such an
orientation leads to a larger value of CT at the western station.
Therefore the To fictitious element of the sea-level pressure pattern
in the northern part of region 1 is ultimately caused by the slow
movement of the cold front along the eastern slopes of the Rocky
Mountains.
Synoptic experience shows that the southard bulge in the T"
Therefore it
isobars is characteristic of these cold outbreaks.
seems reasonable to assume that the ?'-isobars for each cold outbreak
will be similar to those shown in Fig. 1, and that variations in the
Te-isobarn will be caused by variations in the field of AT .
Since
northerly surface flw possessing a large latitudinal extent is also
characteristie of cold outbreaks, variations in the crossing angle
qre associated with variations in the 7,-isobars.
Uing the T'-isobsri
as a standard we can then discuss the spatial and temporal variations
of this characteristic bulge and the large crossing angles associated
with it.
From the previous discussion e find that a necessary condition
for the formation of this bulge is the existence of a large positive
value of AT at the western boundary station, As pointed out, this change
is largest for a station which is in the warm air immediately adjacent
to the front twelve hours prior to observation time.
Thus one espects
the bulge and the acoampanying large crossing angles to be most pronoun-
-10-
ced imediately behind the cold front sad to decrease northward from
there.
This is true for the case shown here but many other oases show
the bulge and crossing angles Just as pronowed well back of the cold
front as they are immediately back of it. Synoptic experience shows
that this situation usually occurs when there is strong cold advoection
well back in the cold air such that AT is appreciable there.
Synoptic
experience also indicates that strong cold adveetion lamediately behind
the cold front produces the most pronounced bulges and large crossing
angles.
One would expect the frequency of these large crossing-angle
siftations to be higher in winter than in summer because of their dopendense on the magnitude of AT .
An examination of eight years of
sea-level charts (October 1931-December 1939) shows that this phenomenon occurs almost exclusively in the months October through April.
As in region 1, the pressure pattern in region It is determined
by the pressures at the boundary stations.
The change in the orients'
tion of the isobars is caused by a larger dorease in pressure at the
eastern stations compared to the western ones.
In region 11
Z
is
larger than ZZ , while the larger values of AT occur at the eastern
stations.
The magnitude of AT at the eastern stations is so much
larger than the one at the western stations that it offsets
of g%> Sg in the product (MAT).
the effect
Therefore there is a larger doerease
in pressure at the eastern stations than at the western ones.
Compari--
son of Roswell, Now Mexico (ROW) and Lubbock, Texas (W)
gives an example of this situation.
in Fig. I
The larger value of AT at the
western station is caused by the orientation of the cold front disccussed in connection with region 11.
It has been found that the f ictitious element introduced into
the sea-level pressure field by the use of T, in determining
V
makes
the surface flow appear highly ageostrophic in both regions I and It.
Removal of this fictitious element increases the highly ageostrophie
flow in region IU, but decreases it in region 1.
hven after the remo-
val of this fictitious element in region Ij considerable ageostrphic
flow remains.
This residual ageostrophic flow in the two regions could
also be fictitious because of the fictitiousness of the sea-level
pressure field obtained by the use of V5 .
In order to examine this
possibility one could apply Sangster's method to this cold outbreak.
This is not necessary since gangster (1960) applied his method to the
cold outbreak of Ig30 OCT 9 January 1957.
The results show that the
surface flow is highly ageostrophic Imediately behind the cold front.
We conclude then that the surface flow does have a highly ageostrophic
character which is real.
We wish to investigate the dynamics of this highly ageostrophic
flow.
Investigation of the dynamics of this flow at the ground would
entail obtaining the pressure field associated with this flow by gangster's method.
This would be a very time consuming job.
-12-
Observation
shos
that this highly agsostrophic flow algo exists in isobwaric
*eaoe
up to 3000 feet above the grond.
to obtain the presur
were 0t4
Sin8e It is
kaeasier
field assooiated with this flow, the dynamics
at these levels.
.Z.
A,
AGEOSTROPRIC FUW ADMVg THE
01WD SIMWAC
Introduction
The ageostrophic flow above the ground surface was studied by
comparing computed and observed geostrophic departures.
The computed
departures depend in part on the results of high-speed electronie
machine computations carried out for Phillips's ten-level model (hereafter referred to a P model).
In the following diseussions those
features of this model which are pertinent to computation of the geoThe model is fully described by
strophic departures are described.
Sanders, Wagner, and Carlson (1960).
B.
Theory for Cosputed Gsostrophic Departures
The equation of motion an be written in the usual atmospheric
coordinate system of x, y, p, and t so
cZ-
where:
V -
-Y(
-7
horisontal velocity
Coriolis parameter
-2-
? -
-
2/L sin (latitude)
geopotential
horisontal stress
=V
is taken along a surface of constant pressure
-14-
(5)
nit vector in (-.) direotlon
k =
and g
a aeeleration due to pavlty.
upon defining
', the g
Y , the geostrophie velocity, and
trophie
departure, as
-~
(7
(*)
F
(6)
- (b)
(5)
we obtala fie
a In the epamssi
Isgieettag
Y)
Ia
-
owa
~ 7~
=7V
Order to
for
Wpute T' frm
(8) VW intidiMe
the following gostrphlo
appuoaUatloss
(a)
(9)
-15-
Putting (9) into (8) gives
V Ar~t'~-.1z
1 y~,ru(10)
rp
which is used to compute V'.
--
Veais of
o'
Vw
--
=
a be obtained from the observed contour field
for the appropriate pressure level.
the P mdel.
are computed in
Vaes of
it we use then computed tendeacies then the tora of? '
in (10) should be consistent with the equations used in the P model.
Tahking the divergence of V* we have from (10)
where
-
tant, to0
geostrophic vorticity and I he
, in the divergence of ?'
.
been onsidered a cons-
From (6) we have
-v=
(12)
where I has been allowed to vary in the divergence of V.
Putting (12) into (11) we have finally
ytn
-- >
-
-t
-16-
2
Z(13)
-I
which is the quasi-geostrophic vorticity equation used in the P model.
Thus we can use the tendencies obtained from the P model to compute V'
from (10).
The final term in (10) is the frictional term
.
We ob-
tan values for this term by considering how friction is taken into
account in the P model.
The effect of friction is incorporated into the P model in the
boundary condition for 4(
*) at 1000 mb (taken as the base of the
This boundary condition is written as
atmosphere).
(h
c"4A- =
-?
'i)
(14)
where the subscripts refer to pressure levels in decibars.
cal velocity,
W1o,
is given by
Zo
where (L
,
The verti-
oc
~
'(15)
) is the flow through the top of the friction layer due
to frictional divergence within the layer, and (
Voe'/
) is the
advoction of the land elevation by the 1000-mb wind. The introduction
of friction in this manner follows Qiarney and Eliassen (1949).
Because of the finite differencing scheme used in the vertical
in the P model,
W100
appears only in the quasi-geostrophic vorticity
-17-
equation for 950 ab.
where Ap
-
This vorticity equation takes the form
100 ab.
Aquation (16) doe not give any explicit inter-
nation about the torm or vertical distributioa of the frictional stress
in P model.
The qui-esgetrophic vortioity equation for 950 ub obtained
from detting the unit vector k into the ourl of (5) gives
where finite diteemoes have been introdueed in the divergence and
frict ion toneo,
ad (4
has been neglected in the expansion of
It we sae the following assmptions about the frictional
stres.
V
(a)
(18)
and nae the boundary oondition
(19)
-18-
we find that (17) becomes mathematically identical to (16).
Thus the
physical model set up by (18) and (19) is mathematically equivalent
to the one set up by (14) and (15).
We therefore use (18) to evaluate
the frictional term in (10).
In our case study we are interested in obtaining computed values
of V' at 910 mb.
In the P model we have oomputed values for (
and hence for V*,
at 850 mb and 950 mb.
-),
Assuaing a linear variation of T' between 950 ub and 850 ab we have
(20)
In actual computation of
values
f
were used instead
of the linear combination indicated in (20) since the 910 mb contours
were available.
Also computation of the former was less time comau-
ming than that of the latter.
appromimated by V
.
In the machine computations V -O
was
Putting these approximations in (20) and
simplifying we have
4(21)
-19-
000ff
C.
Procedure for Cmutations of V
at 910 ab
The basic work of obtaining input data ter the machine *omp&tatioms consisted of making a careful and detailed analysis of the
contour field at 1000 ab, 850 ab, y00 xb, 600 ab, 400 ub, 300 sb,
280 mb, 150 ub, sad 50 ab.
A more detailed description of how the
input data is prepared has been given by Sanders, Wagner, and Carlson
(1960).
The horisontal derivatives in (21) ware approximated by centered
finite differenoes taken over the interval of 89 n.
This interval
corresponds to the one used in the machine computations.
Following (21) the field of the 910m'-b tendency, was formed from
the machine-ceputed fields of the 960-mb tendency and 850-mb tendency.
Values of
'ivf
were computed from the 910-mb contour field.
Jbights of the 910-mb surface were obtained for the radiosonde stations
by subtracting the thickness of the layer from 910 sb to 850 sb from
the reported 850-mb height.
The above thickness was calculated from
the hypeomtric formula where T was determined from the radiosonde
sounding.
The 910-mb centours are shown in Fig. 2.
The analysis of
these contours was performed before the winds ware plotted in order to
minimise the bias introduced by the normal procedure of trying to make
the hsight field correspond to the wind field.
Values of
were obtained from the 950-mb contour field used in
-20-
the machine computations.
This field was computed by interpolation
between the 1000-mb and 850-mb input heights.
Fields of u'- and v *- components were computed separately.
Values of these quantities were rounded to the nearest tenth of a knot
and were plotted at grid points.
These grid point values were then
analysed to obtain values for stations for *hich observed values of V
were available.
The components were then combined by graphical means
to give the computed value of V' at a station to the nearest degree and
nearest knot.
Observed winds were recorded from the synoptic data.
Since in
the lower levels winds are reported only at the 1000-foot levels above
sec level,
the reported wind nearest the analyzed 910-mb height was
used as the 910-mb wind.
These reported winds are given to the near-
est ten degrees and the nearest knot.
GOeotrophic winds were computed for each station from the analysed
contours and recorded to the nearest degree and nearest knot.
The
value of 1' observed was obtained graphically and recorded to the
nearest degree and nearest knot.
-21-
....
......
*
+
.
$k*
300T
31*6
+
.40
)
47
.eOle
.00
jK
jrq
4V4
.WO
7
41
*o
ft
Aw
#'sist
s'VAM:
Fi ?0-obC*pt 4 ptfFekdi
fr/Z]cT
C
....
.......
4
Z*
n
4o
++
to4
*
*
*
*
y
4$
*
"it
+...
+
... +.
-PN
*l+t
40%
ro
p.VA
+F
*RR111
D.
Discussion of Results
In Fig. 2 there is marked cross-contour flow at OKC, AB,
ACV?
and
In Fig. 3 the speeds of the observed gostrophic departure,
,
at these stations are 20, 16, and 23 knots respectively, the
The magnitudes of the vector error
largest values observed at 910 ab.
between the observed geostrophic departure ad the computed geostrophic
departure, V'c
for these stations are 12, 10, and 18 knots respec-
These discrepancies are caused either by errors in obtaining
tively.
values of
V'ep.
,
'
or failure of the P model to compute correct values of
Lot us examine the effect of errors in V'ob
The values of V'o6,
geostrophic wind.
and speed of V
first.
depend on the observed wind, V, and the
It can be seen from Fig. 2 that both the direction
at OC are fairly accurate.
However, the accuracy of
the direction of V;- at ABI and ACV is more questionable.
These direc-
tionsare detrathod by the shape and location of the trough in the
910-sb contours in the region of AI and ACF.
gince the characteristic
dimension of this trough is smaller than the distance between radiosonde stations, it is difficult to determine the shape and location
of the trough from the reported 910-mb heights.
There does not appear
to be any way of determining what the actual location and orientation
of this trough are.
the error in V
Therefore no accurate quantitative estimate of
can be obtained.
However, the 910-mb contours in
Fig. 2 could be drawn so that the direction of Vy?. at ABM and ACV would
2 ACF is Fort Worth, Texas.
-24-
be closer to the direction of V. This different direction of V
would
reduce the magnitude of V.
We can examine quantitatively the error in VUb
caused by an
error in V. The direction of V is coded to the nearest ten degrees.
For any coded wind we know the ten-degree interval within which the
direction of the uncoded wind must lie.
The maximum possible error
in V./, occurs when the direction of the uncoded equals that of the
wind at the appropriate end point of the ten-degree interval.
The
appropriate end point Is the one whose wind direction makes a smaller
angle with the direction of V than the coded wind's direction does.
The appropriate uncoded winds for OKC, ABI, and ACF were used to compute new values of VU
vector error V4/,
OC, ABI, and ACF.
-V
.
The new values of the magnitude of the
were 10, 9, and 17 knots respectively for
Comparison of these values with the original ones
showe that use of the coded wind direction of the observed wind does
not contribute much to the discrepancy between Y,
and Vito
Therefore.)it seems that the major part of this discrepancy is caused
by failure of the P model to compute correct values of V'.
This
failure is caused by the P model equations not being a good approximation to the equations governing the cross-contour flow.
We proceed
now to an examination of the approximations made in obtaining the
P-model equations.
The approximations made in obtaining (21) are
(a)
(b)
(28)
(a)
where 2 means
is approximately equal to."
The approximation (22b)
depends on the computed values of the tendenaoes at 950 ab and 850 ab.
The differential equations which are sowved for these tendencies are
-
7
~
eaV '-
t E
t/V
L f //
~~
(a
. 7(23)
(b)
where A through P ae positive Constan.
(22) which are used in obtaining (23) are
~26-w
Approximations other than
I
(ad
(24)
9,o,,(b)
The values of Y' computed from (21) depend on horisontal finite
differences of the 910-mb contours, the 980-ab contours, and the 910-mb
tendencies.
The area, Region III, over which these finite differences
The
were taken for the stations OKC, ABI, and ACW is shown in Fig. 3.
discussion of the approximations will be concerned only with Region III.
We are not interested in how good the approximations themselves
are, but rather how the approximations affect the values of V
investigate the effect of a given approximation on Vbf
,
.
To
the values
used as the approximations of a quantity are replaced by values based
on observed data.
New values of Va
of the new and old values of Ve
on Ve
.
are then computed.
Comparison
shows the effect of the approximation
Let us look at approximation (24b) first.
Comparison of the 950-mb and 1000-nb charts shows that (V )has the same direction as (V
),,
, but the speeds are less.
Thus
the right-hand approximation in (24b) has little effect on the values
of V'
To examine the left-hand approximation in (24b) we need values
-27-.
....... ....
...!9
+.
;
;
*
;
A6;
#q;
tix
koll~;
I~
C
jo
..
..
. x.
;
a
r
;T
of
V-oo
.
Since in Region III the 1000-mb surface is below the ground,
we use the surface wind for (V),o .
The surface wind is predominantly
from the north while the 950-mb geostrophic wind veers from northeast
to southeast as one goes from east to west across the region.
In the
western half of the region, the gradient of N is fairly large and
toward the west.
Thus in this region the values of ()
Y.
? N are
fairly large and positive while the values of (V)
,
mately aero.
for (V1q. ), - in (24b)
The effect of substitution of (V),,
would be to make the term (V9
- V H equal to zero in (23a).
result can be obtained by setting
the winds.
H are approxi-
The same
H equal to zero but not changing
H
Therefore tendencies were computed from the P model for
the cose 9H equal to zero. The geostrophic departures computed from
these tendencies are shown in Fig. 4.
nitude of the vector error, Y,,
duced in the ease V?
The resulting values of the mag-
-
not equal to zero.
,
are larger than those pro-
Consequentlyelimination of
approximation (24b) increases rather than decreases the magnitude of
the vector error V'00
- V'ce
The approximation (24a) estimates the actual thickness advection
by the geostrophic thickness adveotion.
From the hydrostatic relation
and the equation of state we can write (24a) as
-
/r-
7 ,K7
-
OW29-
(28)
where 1 is the gas constant for dry air.
We are interested in approwi-
mation (25) at 900 ab and 800 mb. Since 900 ab is close to 910 ub we
can use the isotherms at 910 ab for examining (25) at 900 ab; the isotherms for 910 ub are shown in Fig. 2.
that
xamination of Fig. 2 show
' T is less positive than V 7 T in the semicircle north of the
three stations.
From (23) we see that this discrepancy would produce
more positive values of the 950-tendency and more negative values of
the 850-mb tendency in this northern semicircle.
Ubing the 850-mb
chart to approximate the 800-ab one we find that (25) is a good approximation at 800-mb.
new values of V
Extensive computations are necessary for obtaining
in the case where VV T is used in (23).
These com-
putations were not carried out in the present study.
The remaining approximations caninot be studied by comparison of
new and old values of Vb , because the new values cannot be computed.
However, these remaining approximations can be studied by considering
what happens to the vector error V.,
are eliminated individually.
-
Y'c
when the approxmations
The total effect of elimination of all
these remaining approximations must be the reduction to zero of the
vector error
-
cf
(V
(Y'oh
for the case
7g equal to mero.)
The reduction of the magnitude of this error is a necessary but not sufficlent requirement for correspondence of the computed and observed
geostrophic departures.
To investigate approximation (22b) observed 910-ab tendencies
~1
Tendencies
were computed from non-centered finite differences in time.
centered in time at 0600 GCT 18 February 1959 were used to approximate
the ones at 1200 OCT 18 February 1959.
The components of V'ce produced
by the computed and observed tendencies are shown in Fig. 5.
From Fig. 5 we see that the two components along Y
-Y'
are essentially the same for OKC and ACT while at ABI there is an increase
of 2.5 knots.
16,
1
- ve,/
The values of /V,
obtained from Fig. 4 are
and 21 knots for OKC, ABI, ACF respectively.
,
Thus changing
approximation (22b) would make a very small contribution toward reduction of the original value of the error.
Values of
were needed for investigation of (22a).
These
values were obtained by the same procedure as that used to obtain obser-
ved values of the tendency.
The component of V'
produced by
shown in Fig. 5. Comparison of the components along (Vht, - Yt
x
that for ABI the component of
the component of
X(
.
is
) shows
is 4.5 knots greater than
The total contribution by approxima-
tions (22a) and (22b) toward reduction of the original error would be
7.0 knots.
This would be a large reduction compared to the original
value of 13.5 knots.
This change may be somewhat large because of the
way in which the value ( $-)
At ACF the (
component.
-
was obtained.
)-component is 2 knots larger than the ( p-e )01
This 2 knot increase is the total contribution by approxima-
tions (22a) and (22b) to zeduction of the original error.
-31-
This value is
m-l6m4.
,
-
+i
*
4
-4
*
XL%**
*a
I
R
+ *ON
b
44
$
'Y'
I
;i
e
&w
. ........ .*
*
;r
ya
*o
MIT
-
Am
*
>
t-
a
y
-
*
1+
. -
.-
}Al+.*l
t-t
-
Comparison of the
small compared to the original value of 21 knots.
same two components at OKCJas at ACFI shows that the former contributes
an increase of 1.5 knots to the original error.
Thus changing both
of the approximations (22a) and (22b) produces no systematic change in
the original error of the three stations.
To study approximation (22c) values of V
V V were computed from
the observed wind field at 910 ub. The components of V' produced by
this term and its corresponding approximation
-
V Y- are shown in
Fig. 8. We see that changing approximation (22c) would contribute to
an increase in the magnitude of VAf,
-
V'c{'
is needed to examine approximation
The actual value of
(22d). This actual value was estimated by the vector difference between
V(j,
and the sum of the vectors4
(
_
and4
y({7
The latter two vectors were obtained from observed data.
X
of Fig. 6.
VA&,
(see (8)).
The estimate
and its approximation obtained from (22d) are plotted in
Comparison of the components of these vectors along the error
- V
shows that use of this estimate of the stress term would
contribute reductions of the error of 22, 8, and 18 knots reopectively
for OKC, ABI,
and ACT.
These reductions are a large percentage of the
original error; in fact the reduction at OgC is larger than the original
error.
Thus it appears that a major portion of the error in the machi-
ne-computed geostrophic departures comes from the approximation made
concerning the eddy stress.
--35-
ft
*
VON"*ol+w
Sl
Ao*$46+w
+Soo"4
7.7+-
Ai
4
4+
+
+
r
S4a
tit+
7+
Jim
M
*-
Wn-
4e
Comparison of the magnitude of the above reductions and the
magnitude of the observed geostrophic departures suggests that there
is a large contribution to the geostrophic departure from the stress
effect.
The estimated values of the various contributions to V' in (8)
are shown in Table 1.
The values of V' are also presented in Table 1.
The first three digits of an entry in Table 1 are the directions in
degrees, the last two the speed in knots.
eta.
09C
10801
15505
31425
311*0
ABI
34008
16403
29610
30916
ACT
30704
14904
28922
28523
Comparison of the above values indicates that the stress contribution
predominates in producing V'.
-37-*
NI
IV.
SUIIIARY AND CONCLINIONS
It was found that in the region over which the highly ageostrophic flow occurs on the sea-level chart, the use of a twelve-hour mean
surface temperature in the reduction of station pressure to sea level
produces a fictitious element in the sea-level pressure field.
In the
eastern part of the above region this fictitious element makes the surface flow appear more highly ageostrophic than it actually is. This
result indicates that the highly ageostrophic character of the flow could
be caused by the sea-level pressure field not being a good representation of the pressure field associated with the surface flow.
It was
found from Sangster's work that the surface flow immediately behind
the cold front was still highly ageostrophic even when compared with
the appropriate pressure-gradient force.
Thus it is fairly certain
that the northerly surface flow immediately behind the cold front is
highly ageostrophic.
The dynamics of this flow could be studied by measuring the different terms in (8) from observed data.
Evaluation of one of these
terms would require approximating an instantaneous time derivative by
a finite difference in time.
Since meteorologists have had to resort
to these finite differences in time for so long, they are always eager
to use methods which utilize instantaneous time derivatives.
Phillips's
ten-level model produces instantaneous time derivatives of the geopotential.
Since (8) could be put into a form, (10) which could use these
-38-
instantaneous tendenoies, the dynamics of the flow were studies trom
(10) rather than (8).
The dynamics of the surface flow could not be
studied from (10) because (10) applies only to isobaric surfaces.
Therefore the dynamics of this highly ageostrophic flow were studied
at 910 ub.
A large error was found between the geostrophic departures conputed from the P model and the observed geostrophic departures.
It was
found that this large error is caused by the approximation made for
the eddy stress in the P model.
Calculations from (10) indicate that
the eddy stress term predominates in producing the geostrophic departure.
Therefore any model which does not properly take account of the
eddy stress cannot produce computed gsostrophio departures which compare
well with observed ones.
The instantaneous local rate of change of the wind seems to have
a small effect on the geostrophic departure of this highly ageostrophic
flow.
Therefore, this flow could, and should, be investigated using
finite differences in time to evaluate the instantaneous local rate
of change of the wind.
-.39-
ACK
I PI
InII
On I
The machine computations for Phillips's ten-level model
were performed an the IBM 7090 computer at the MIT Computation
Center.
he author is indebted to Prof . Norman Phillips for
the helpful discussions held with him during the course of
this work.
The author wishes to express his heartfelt thanks
to Prof . Frederick Sanders for the encouragement sad many suggestions given to the author during the course of this study.
Finally, I wish to express my gratitude to my wife,
arlene,
for her patience and understanding during these past few months.
-40-
Charney, J. C., ad A. Eliassen, 1949: A atrical method tr
predietiag the perturbatlens of the middle latitude westerlies.
Ls. 1, 38-84.
Phillips, 3. A., 1967: A coordinate system hwing see special advantages for nimerical toreasting. J. l og., 14, 184-168.
sanders, F., A. James Wagner, and Toby N. Carlsn, 1900: Spetiation
of ocudiness sad precipitation by multi-level dynamical models.
Icietifit Report No. 1 A? 19(604)-349, Pept. of Mteer.,
.I.T.,
111 pp.
angster, W. ., 190: A method of representing the horisental pressure
to"e without rOstion of station pressure to sea level, J.
noter., 17, 166-176.
-41-