M a y 1971 393
ANALYTIC SOLUTIONS OF THE NONLINEAR OMEGA AND VORTICITY EQUATIONS
FOR A STRUCTURALLY SIMPLE MODEL OF DISTURBANCES
IN THE BAROCLlNlC WESTERLIES
UDC 661.611.32:661.613.1:~1.616.11
FREDERICK SANDERS
Department of Meteorology, Massachusetts Institute of Technology, Cambridge, Mass.
ABSTRACT
1.
2.
3.
4.
5.
6.
7.
8.
Exact solutions of the nonlinear quasi-geostrophic omega and vorticity equations yield three-dimensional
fields of vertical motion and geopotential tendency for a structurally simple model of cyclonic and anticyclonic dis-
turbances in the baroclinic westerlies. Information derived from these fields is consistent in many respects with the
results of classical analyses of baroclinic instability, but is more detailed. The present results seem entirely reasonable
synoptically. New findings are (1) that in most instances a surface cyclone should move to the right (and an anti-
cyclone to the left) of the upper flow over the center and (2) that the rate of deepening provided by the present
physical theory probably does not account for observed instances of explosive cyclogenesis.
CONTENTS We assume that the flow is governed by the quasi-
Introduction. ...................................... 393 geostrophic vorticity equation
a@
Field of geopotential tendency at- aP
Structural model.. .................................. 394
.... (1) Field of vertical motion.. --- ai- V*Vq+qo-
Motion of geopotential features.. .....................
Intensification of geopotential features. ................
Field of temperature tendency. .......................
Concluding comments. .....
399
402
404
Acknowledgments 407
References ....................................... 407
and by the thermodynamic equation
................................. (2)
I. INTRODUCTION
In textbook treatments of the structure and dynamics
of extratropical cyclones and anticyclones, assumptions
are often numerous, arguments often qualitative, and
explanations often partial (e.g., Willett and Sanders 1959,
pp. 256-259; Petterssen 1956, pp. 320-329). These con-
siderations can be supported by numerical calculations
for particular synoptic cases, but these are tedious to
prepare and lack generality.
Here, we wish to follow a middle path. That is, we will
show that a wealth of realistic diagnostic information can
be obtained from a structurally simple model of the
temperature and geopotential fields with rather few
assumptions. Specifically, we will obtain analytic solutions
for the three-dimensional fields of vertical motion and
geopotential tendency, which are interesting in themselves
and permit quantitative determination of the rates of
displacement and of intensification of significant features
of the geopotential and temperature fields. The relation-
ship of these fields and rates to the basic structural
parameters of the model is often quite explicit. As an
illustrative case, we shall obtain the model parameters
from the situation of Mar. 26, 1970, shown in figure 1
and denoted as case C.
where qo is a constant value of i- + f, the absolute vor-
ticity; I$ is the geopotential; and the stability parameter
~(p ) 3 (d+/dp) (d lne/dp) is a function only of pressure.
When the, geostrophic relationship is used to express the
vorticity,
i- = v2+l.fo ;
and eq (1) becomes
Here, fo is a constant value of the Coriolis parameter.
When eq (3) and (2) are combined to eliminate at#+%, we
obtain the omega equation
which has been widely discussed in this form or in related,
simplified versions. The solution of this equation yields
values of aw/ap, which enable a solution of eq (3) for the
field of geopotential tendency. We shall obtain such
solutions for the structurally simple model.
394 MONTHLY WEATHER REVIEW Vol. 99, No. 5
FIGURE 1.-Surface maps for (A) 0000 GMT and (B) 1200 GMT on
Mar. 26, 1970. The dashed lines are isopleths of thickness of the
layer from 1000 to 500 mb and are labeled in dekameters. Sea-
level isobars are labeled in millibars. The encircled crosses denote
the positions of perturbation centers of temperature and 1000-mb
height, used as a basis for estimating the model parameters.
2. STRUCTURAL MODEL
Aiming for a maximum of flexibility and realism without
paying an excessive price in the form of mathematical
complexity, we assume the following stmcture of the
temperature field :
A 2n X(ay+T cos - L 2 cos ?E L y) (5)
where the x and y axes are directed eastward and north-
ward, respectively; pressure is expressed in millibars; and
the base of the model atmosphere is the 1000-mb level.
We use the 8-plane approximation. This model tempera-
ture field at 1000 mb for case C is shown in figure 2. The
vertical structure of the temperature perturbation
-- it2 0 L I Z
FIUURE %--Case C 1000-mb isotherms a t intervals of 2OC,
labeled as departure from map average. The values of parameters
are a= 1.12X 10-5 OC?.m-l, T=7.25 O C , and L=2900 km.
is governed by the factor, 1-a In(lOOQ/p), which allows
for damping with elevation and reversal of sign at a
pressure level determined by the value of a, simulating
the effect of the tropopause. The reversed perturbation
becomes unrealistically large near the top of the atmos-
phere, but the effects of this artificiality are probably
small in the lower troposphere. When a=Q.722, the
tropopause reversal occurs at the 250-mb level, as in
middle latitudes in the real atmosphere, while the strength
of the temperature perturbation at 500 mb drops to one-
half its 1000-mb value. This decrease through the tropo-
sphere seems fairly representative of continental condi-
tions, but is excessive for maritime regions, in which there
appears to be little change in the intensity of horizontal
temperature contrasts through much of the troposphere.
However, the benefit of a more realistic vertical variation
than that used in eq (5) is probably smaller than the cost
of the additional mathematical complexity.
Since the stabilty factor u is independent of z and y, it
will be associated with T,, the average over a wavelength
in 2 and y on a constant-pressure surface. It follows from
the definitions of u and of potential temperature 0 and
from the hydrostatic equation that
and
where K is the ratio of the gas constant for dry air to the
specific heat at constant pressure. Let us define
It is evident that Y=K corresponds to an isothermal lapse
rate, while r=O is easily shown to correspond to a dry-
adiabatic lapse rate. We will regard y as independent of
pressure, thus underestimating the stability in the
stratosphere, where the model becomes unrealisic anyway.
We will also adopt a constant value of temperature, To,
where it appears as a coefficient in the expression for u,
May 1971 Frederick Sanders 395
L 14 I
_---_ - --- c I
.
I
- L /2 0
FIGURE 3.-Case C isopleths of 1000-mb geopotential at intervals
of 600 m2-s-z. The values of parameters are $,,,=1020 m2+-2
and A= 0.25L.
A
so that
At 1000 mb, we assume the simple distribution of
geopo tential given by
(7)
A 2n 27F
L L +(x, y, 1000)=+10 cos - (x+X) cos - y
where X is the phase lag of the 1000-mb geopotential field
relative to the temperature field. It is measured westward
either from the high temperature perturbation center to
the 1000-mb Low center or from the cold center to the
1000-mb High center. Thus, X=O corresponds to a system
of warm Lows and cold Highs while X=L/2 yields cold
Lows and warm Highs. A sample pattern for our selected
case with X= L/4, representing the typical intensifying
situation in the real atmosphere, is shown in figure 3.
Note that the ((map average" value of GI,, is arbitrarily set
equal to zero.
In this model, the 1000-mb geopotential centers and
the centers of the temperature perturbations are at a
common latitude. In the real atmosphere, we often
observe approximately this alinement, particularly while
the disturbance is intensifying; but in the later stages of
development, the surface cyclone tends to move to the
north of the temperature perturbation centers, while the
anticyclone drifts to the south.
At pressure levels above the 1000-mb level, the geo-
potential field is found by hydrostatic integration. Thus,
where
-[. ln -- 'Oo0 (Ra/2) (In ?>'I P
X(ay+T A cos - 2s z cos 2y) (8)
L L
I --- --
I --
Lf4 I
I
0-
120 0
-L14
-L /2 0 L 1 2
FIGURE 4.-Case C isopleths of 300-mb geopotential at intervals of
600 r n 2 c 2 , labeled as departures froni map-average value. The
value of QL is 0.722.
is the map-average value of C#J(P>. The field at 300 mb for
case C is shown in figure 4. The transition from the lower
to the upper level is quite realistic.
3. FIELD OF VERTICAL MOTION
We must first evaluate the forcing function of eq (4) for
the structural model described above. Geostrophically,
and
From these relationships and from eq (6) and (8), the
omega eq (4) becomes
. ~(1 -a In Eo) sin zy. 4n '
P (9)
It is interesting t o trace the identity of the various
parts of the right side of eq (9). The lengthy first term
represents the effect of vorticity advection in that part of
the flow attributable hydrostatically to the temperature
field, the first part referring t o advection of relative vor-
ticity and the second to advection of earth vorticity. The
second term is composed of two identical contributions,
one from advection of the part of the relative vorticity
due to the 1000-mb geopotential field by the part of the
flow due to the mean meridional temperature gradient and
the other from advection of this temperature gradient by
3% MONTHLY WEATHER REVIEW V O l . 99, No. 5
the part of the flow due t o the 1000-mb geopotential field.
The third term represents the effect of advection of the
perturbed part of the temperature field by the part of the
flow attributable to the 1000-mb geopotential field. The
first term vanishes at the "longitude" of the temperature
perturbation centers, while the second is similarly phased,
but with respect to the 1000-mb geopotential centers.
The third term is independent of 5 and vanishes at the
"latitude" of the 1000-mb centers.
The solution of this equation is less unwieldy and
perhaps more meaningful if it is undertaken piecemeal.
That is, let
2* x sin r; (s+X) cos L y, (llb)
and
x p (I-a In Eo) sin 4r y. (11c)
P
Taking the first of these equations, we require w1 to vanish
in the horizontal where the forcing function vanishes.
Thus, we are led to
A 27r 2u
L L wl=wl(p) sin - z cos - y
from which
2.A 27r
L L ~~w ~=-2 (2 7 r /~)~w ~= -2(27r/~)2& sin - x cos - y,
so that eq (lla) becomes
We require, as further boundary conditions, w to vanish
a t the base and at the top of the atmosphere. The same
conditions apply to w1 and thus to because there will be
values of z and y at which w1 is the only nonzero component
of W. The solution of eq (12), subject to these boundary
conditions, can be written as
-[2+sak+Wk(k+l)]p In - 1000
P
P +(a k +l )p [l-(&Y-']-aP In -' 1000
P
folio , 2(2~/L)~RToy k=
and
p - 1 =) (1 + 4/k)'" - ).
Proceeding similarly with eq (llb) and (llc), we have
w2=w2(p) A sin 27r - (x+X) cos 217 y L and
The solutions for c2 and :a, subject to the boundary condi-
tions that each should vanish at the bottom and at the top
of the atmosphere, are
A
02=KgPg where
and
~ ' S
where
Pg1=ap In -- ' P
and
r-l=+ (l+8/k)1/2-$-
May 1971 Frederick Sanders 397
P (r n b ) I I I I I I I I I I
FIGURE 5.-Case C profiles of various contributions to vertical
motion. The values of the parameters are r=0.114, fo=$o=
0.92X 10-4 s-1, and To=250K.
Each component of the vertical motion, as measured by
the product of a KuPu and the appropriate function of x
and y, can be physically identified with the same process
as was the corresponding part of the forcing function.
Thus, for example, GIF'il sin(4a/L)y is the vertical
motion induced by the advection of the perturbation part
of the temperature field.
Profiles of the (KuPw)'s for our sample case appear in
figure 5. Note that K;,PY2 opposes K;,P;, and is much
smaller. I n the troposphere, w1 contributes ascent from the
cold trough to the downstream warm ridge. The largest
component in this case is w2, as shown by the profile of
IC&,. It contributes ascent from the 1000-mb Low
eastward to the 1000-mb High. The profile of K;;Pil
shows that the remaining component, w3, which consti-
tutes a meridional circulation, gives ascent to the north of
the latitude of the perturbation centers and descent to the
south under normal circumstances when the 1000-mb Low
lies between the cold trough and the downstream warm
ridge. Note that this component vanishes when X=O or
L/2 (ie., when the 1000-mb geopotential centers coincide
with the temperature perturbation centers). The profiles
themselves show maxima in the vicinity of 600 or 700
mb, with a reversal near the 250-mb tropopause level. The
largest divergence in the low troposphere is associated
with w2.
Patterns for case C at 800 and at 400 mb are shown in
figure 6 , while a vertical cross section along y=O appears
in figure 7. The patterns of vertical motion slope westward
with elevation in the troposphere, but not as strongly as
the geopotential troughs and ridges. Note that the trough
line below about 750 mb lies in a region of ascent; above
this level, descent occurs in the trough up to the tropo-
pause. At all levels, however, a w J a p is positive a t the
FIQURE 6.-Vertical motion for case C a t (A) 800 mb and (B) 400
mb. The isopleths are labeled in units of 10-4 mb.s-1. The posi-
tions of the 1000-mb Lows and Highs are indicated by letters.
200
looL-L 0
0
FIGURE 7.-Case C vertical cross section of vertical motion along
y=O. The isopleths are labeled in units of 10-4 mb&. The
position of the geopotential trough is shown as a dashed line.
trough, implying that horizontal convergence is leading to
intensification.
The patterns of vertical motion, though of course not
amenable to verification by direct observation, seem to
be consistent with our generally accepted notions with
respect to both magnitude and distribution. Above the
tropopause, our results are not to be trusted since the
forcing functions become unrealistically intense and are
too highly correlated with tropospheric patterns. The
tropospheric maxima QCCW at a somewhat lower elevation
than one might expect. This elevation in the model, though
subject to some variation with wavelength and static
stability, is determined principally by the choice of tropo-
pause level (which tends to act as an upper lid) and by the
variation of the temperature field with pressure. The low
maximum is associated with the rapid upward decrease of
magnitude of the temperature gradient, which as dis-
cussed above may be unrealistic in some instances. The
magnitude of the vertical motion is doubtless insufficient
to account for heavy rainstorms of either large or small
scale; but latent heat release, which plays a most impor-
tant sole in these situations, is absent in the model.
Once the field of vertical motion has been obtained, the
most straightforward way of obtaining the geopotential
tendencies is to solve the vorticity eq (3). The vorticity
advection is evaluated from eq (8) and awlPp by differ-
entiating the solution (10). Then eq (3) becomes,
schematically,
where x = &p/&.
meal solution. That is, let
As with the omega equation, we will undertake a piece-
x= XI + xa+ x3
where 2.R 27r
E vzxl=Fl(p) sin x cos - y*
2s 21r v%a=Fdp) sin 1 (%+A) cos y,
and 4s v?t3=Fa(p) sin y.
Assuming that each component of x vanishes where the
corresponding forcing function vanishes in the horizontal,
we have
A . 27r 27r
L H, xl=Xl sin - x cos - y
where
2 1 = - Fl (p>/2( 2lr/L)2.
Similarly,
and
A 2n 27r x,=x, sin - (%+A) cos L y,
A 47r
L x,=x, sin - y.
When the F(p)'s are evaluated, we find
where
,=Cn P (1-4'.
Vo!. 99, No. 5
where
May 1971 Frederick Sanders 399
0
100
200
300
400
500
600
700
800
900
Oo0 -60 -40 -20 0 20 40 60
FIGURF~ &-Case C profiles of various contributions to geopotential
tendency.
where
Again, we can trace out the physical identity of the
various parts of the geopotential tendency. Thus, for
example, K:lPt,, K:2P:2, K& P;l, and K& represent
L14
I \
-L I Z 0 L l e
L14,
11 B -L14
-L I Z
I
0
FIGURE S.-Case C geopotential tendency at (A) 1000 mb and (B)
300 mb. The isopleths are labeled in units of 10-3 mh-8. The
positions of the trough and ridge a t 300 mb are indicated by
dashed lines. The positions of the 1000-mb Low and High are
shown by letters.
for the moderate success of mid-tropospheric barotropic
prediction even when applied to highly baroclinic situa-
tions. In the upper troposphere, vorticity advection
dominates, but divergence exerts a strong modifying
influence.
Patterns of geopotential tendency a t 1000 and at 300
mb are illustrated in figure 9. In synoptically more famil-
iar terms, the maximum at the former level corresponds
to a change in sea-level pressure of about 5.2 mb/3 hr and
at the latter level to a 12-hr change of height of 280 m,
both quite reasonable. Qualitatively, it appears from a
comparison of figure 9 with figures 3 and 4 that the
1OOO-mb Low is deepening and moving east-northeast-
ward and that the High is building and moving east-
southeastward, while the trough and ridge at 300 mb are
moving eastward and amplifying.
5. MOTION OF GEOPOTENTIAL FEATURES the effects of vorticity advection, while the remaining
terms represent effects of horizontal divergence. The reader
can work out the identity of each term further by examin-
ing the structure of the (Kx)s.
Profiles of the (KxPx)s and of the ;s appear in figure 8.
Near 1000 mb, we see that divergence effects dominate;
and in this case, the one attributable to the interaction of
the 1000-mb flow and the meridional temperature gradient
is by far the most important. Between 700 and 600 mb,
divergence effects are at a minimum (though there is no
single level of nondivergence in the model), accounting
Petterssens (1956, pp. 48-49) formulas can be used to
evaluate exactly the motion and intensification of features
of the geopotential field. Consider, for example, the trough
at y=O. For the eastward speed of this feature, we may
write
(21)
where x=xT is the position of the trough axis. For finding
xT, we differentiate eq (8) with respect to 5, set the result
4P)= -&&? axaz (XT, 0, PI
4 W -3 &1 ,0 -7 L -5
400 MONTHLY WEATHER REVIEW
equal to zero, and look for a solution between x =O and x=L/2--X. Thus, we find
VOl. 99, No. 5
The appropriate expression for cz(p) is found by differentiating eq (17) once and eq (8) twice with respect to x, leading to
Results for our sample case are given in table 1. For
comparison, the observed 12-hr displacements at 1000 and
at 500 mb were at rates of 11 m/s and 12 m/s, respectively,
in reasonably good agreement with the theoretical values.
The first prominently unrealistic result of our calculations
is the relatively slow trough speed in the lower troposphere
above 1000 mb. The real atmosphere avoids the develop-
ment of inteimediate-level trough lag by means which are
evidently not contained in our theory.
The northward speed of the 1000-mbLow is given by
~,(iooo)=-- ax/* (--A, L 0, 1000). (24)
a%/ay= 2
Proceeding as with eq (21), we find that
For case C, the computed value of c, (1000) is 9.1 m/s,
while the observed 12-hr displacement rate of the surface
Low is 7.5 mls.
At 1000 mb, the eastward speed of the Low as given
by eq (23) can be expressed in terms of the basic param-
TABLE 1.-Case C eastward speed of trough
Pressure (mb) ZT (% of A) c. (m/s)
14
10
10
11
12
14
16
19
22
25
13.6
18.1
17. 1
14.9
12. I
10.9
9.9
11.4
13.7
ia o
eters of the model as
cz(looo)=c~l cos (27rX/L)+czz (26)
where
and
represents the effects of x1 and u, respectively. Ordinarily,
cZ1 is distinctly smaller than cZ2. For example, in the
sample case where czz is 13.7 m/s, czl is 4.7 m/s. Of course,
in this case (since X=L/4) it has no effect. With X=Q)
(warm Lows, cold Highs), cZ1 would retard eastward speed
of the geopotential centers, while with X=L/2 (cold
Lows, warm Highs) it would have an enhancing effect.
This result seems opposed to observational experience
that warm Lows tend to move more rapidly than cold
ones; but no extensive test of eq (26) has been performed,
and the difference in observed behavior of warm and
cold systems may be due to other factors, principally
differences in the meridional temperature gradient.
The response of cZz to variation of parameters of the
model is illustrated in figure 10. Isopleths are shown for
three values of the vorticity stability factor, qo/Toy.
The intermediate value corresponds to intermediate
atmospheric values, say,
qo= 10-4~-1, To=25QoK, r=O.P44,
May 1971 Frederick Sanders 401
I I I I I I I I I 1 I
I 2 3 4 5 6 7 8 9 IO I1
L ( lo3 km )
FIQURE lO.-Eastward speed cZa of the 1000-mb center as a function
of wavelength and meridional temperature gradient for selected
values of the vorticity-stability parameter. The dashed, solid,
and dotted lines are for values of 1.4, 2.8, and 5.6X10-8 s-1,
respectively. The intermediate value most nearly represents
average atmospheric conditions. The isotachs are labeled in
meters per second.
a vertical temperature gradient halfway between iso-
thermal conditions and the dry-adiabatic lapse rate. The
smaller Salue might be regarded as appropriate for anti-
cyclonic conditions (small vorticity and large stability) ,
while the larger might be thought of as representing
cyclonic conditions. Only quaIitative inferences at best
could be drawn from these distinctions because our theory
is based on horizontal uniformity of this parameter over
both the cyclonic and anticyclonic portions of the pattern.
The range of wavelengths presented in figure 10 is extrava-
gant because our p-plane geometry is inadequate for the
largest, while the smallest is below the limits of size
regarded as appropriate for geostrophic analysis. With
these disclaimers in mind, we note in figure 10 that the
zonal motion of centers is eastward unless the wavelength
is too large or the meridional temperature gradient is too
small. This motion has a maximum in the wavelength
range from 2000 t o 6000 km where our theory is most
appropriate and where the values look synoptically
reasonable. This maximum speed increases somewhat
more than linearly with the meridional. temperature
gradient. As the vorticity-stability factor increases,
eastward speed increases and the wavelength of maximum
speed decreases through most of the region of interest.
I I I
I 2 3 4 5 6 7 8 9 IO I1
L (IO3 km I
FIQURE 11.-Northward speed cu3 of the 1000-mb center a8 a
function of wavelength and vorticity-stability parameter for
$= 1. The isotachs are labeled in meters per second.
We can examine the meridional motion of centers by
writing
cu(looo)=cu~ sin (27rA/L) (27)
where
Note that this motion is always northward for centers
(usually cyclonic) between the cold trough and the
downstream warm ridge, southward for centers (usually
anticyclonic) between this trough and the upstream
ridge. This finding is consistent with Petterssens (1956)
evidence for the systematic meridional motion of surface
cyclones and anticyclones, but suggests that our model,
with centers at a common latitude, can have only a
transitory close resemblance to real life.
The behavior of cu3 in our model is illustrated in figure
11 for a nominal value of T=l. Note that the value of
cy3 is directly proportional to the magnitude of the
temperature perturbation. For a given value of T, we
see from figure 11 that cy3 increases with decreasing
wavelength and with increasing value of the vorticity-
stability parameter and that the values are synoptically
realis tic.
I t is interesting to examine the steering concept in the
context of our theory. By (steering we mean the time-
honored belief that the motion of a surface cyclone or
anticyclone is controlled by the flow aloft over the surface
center. We have shown here that the motion of 1000-mb
centers is due to divergence at this level rather than t o
transport of some quantity by the flow at upper levels.
The divergence producing zonal motion is associated
primarily with that part of w due t o the meridional tem-
perature gradient, while the meridional motion is attrib-
utable to the temperature perturbation; but the structure
of the relationships is different. Thus, the relationship
between the displacement velocity of the center and the
overlying wind velocity is far from obvious.
A
A
402 MONTHLY WEATHER REVIEW Vol. 99, No. 5
For orientation, consider the flow at 500 mb for the
case of h = L/4. Over the 1000-mb center, its components
are
From eq (8), we find for this case that
and
For case C we have US = 18.1 m/s and US = 25.4 m/s,
while C~ and cy3 are 13.7 and 9.1 m/s, respectively, so
that the 1000-mb Low is moving to the right of the 500-mb
flow. Specifically, the former is toward 056' at 16.0 m/s
while the latter is toward 035' at 31.2 m/s.
The properties of the model are examined with respect
to steering in figure 12. Note that the ratio, C ~/U ~, dis-
plays a maximum at a wavelength between 2000 and
6000 km, depending on the vorticity-stability parameter
and the meridional temperature gradient. The ratio for
a given wavelength varies directly with these two param-
eters and can achieve values greater than unity. The ratio
cV3/vs (which is independent of T ) is more modest, in-
creasing with wavelength and with the vorticity-stability
parameter. For the case of X=L/4, the Low moves to the
sight of the 500-mb flow when c z Z /~~5 >c ~3 /~5 . Observe in
figure 12 that this condition prevails for conditions under
which most cyclones are obsyved. (The amount of devia-
tion depends additionally on T and is not easily portrayed.)
As h approaches zero or L/2, both the motion and the
500-mb wind approach a west-to-east direction.
These results provide an explanation for Austin's
(1947) empirical findings that surface cyclones move to
the right of the upper flow and that there is little correla-
tion between cyclone speed and speed of the flow aloft.
Of course, our theory is symmetric: we find also that
1000-mb anticyclones move to the left of the upper flow,
but we are not aware of any empirical studies that might
be used as a test of this result.
A
6. INTENSIFICATION OF GEOPOTENTIAL FEATURES
The deepening of Lows and the building of Highs is
a s important as their motion and can be readily examined
with our model. Consider the 1000-mb Low. The deep-
ening rate is given by the geopotential tendency at the
center, which from eq (17) can be expressed as
sin (2rXIL). (28)
We see immediately thah deepening, if occurring at all,
will be a maximum when h=L/4 (i.e., when the cold
trough lies to the west and the warm ridge to the east of
the surface center, a synoptically familiar necessity for
(A I
I I I I I I
I 2 3 4 5 6 7 8 9 IO I I
L (1o3krn)
01 I
5.6
2.8
I .4
B
I I I 1 I
I 2 3 4 5 6 7 8 9 ,1 0 1 1
L I IO' km)
FIGURE l2.-Ratio of (A) cSa to the zonal component of a 500-mb
wind &s a function of wavelength and meridional temperature
gradient and (B) cv3 to the meridional component of a 500-mb
wind as a function of wavelength and vorticity-stability param-
eter. No isopleths are shown where the ratios are negative. In
(A), the dashed, solid, and dotted lines are as explained in figure
10. The heavy dashed lines are loci of maximum wavelength for
which the 1000-mb Low moves to the right of the upper level
flow.
Low coincides with either the warm ridge or the cold
trough; the latter presumably representing the c1assical
occluded cyclone stage.
From eq (18) we find, since P:, = P:, = 0, that
{ [2k+6&k+l)+6012k2(k+2)](p-l) 2(WQ Tor 2, (1000) =
-- .. cyclogenesis). No -deepening occurs when the 1000-mb - -~ - - L--- - > ~ For niir samole case. in which h=L/4. the vdue of
May 1971 Frederick Sanders 403
A
~~(1 0 0 0 ) is -109XlO-* m2[sa, which corresponds to a
deepening rate for sea-level pressure of about -6.4
mb(12 hr)-l, while the observed 12-hr deepening was -8
mb. The agreement is not as good as it appears, however,
for we have taken no theoretical account of the filling
effect of surface friction, which we shall see is far from
negligible.
Despite the complex structure of eq (29) we cas draw
some inferences from it. Both terms on the right side
represent effects of divergence associated with wlJ the
component of vertical motion arising from the tempera-
ture field alone. This vertical motion, as discussed above,
is due to vorticity advection. The first term on the right
side of eq (29) is associated with advection of relative vor-
ticity, while the second is associated with advection of
earth vorticity, as indicated by the presence of (aj/*), in
its coefficient. Qualitatively, the first produces upward
motion, and 1000-mb convergence, east of the cold per-
turbation, while the second produces a counteracting
descent with divergence at 1000 mb. Since the 1000-mb
Low ordinarily lies between the cold trough and the
downstream warm ridge, the first term represents the
active deepening mechanism, while the second acts as a
brake. This finding is entirely consistent with Petters-
sen's (1956, p. 337) view of the conditions for cyclone
development.
Further examination of eq (29) suggests because of the
exponent of I; that the braking action of the second term
will increase with increasing wavelength more rapidly
than the deepening action of the first term, so that there
will be a limiting wavelength above which no net intensi-
fication will occur. This finding is tentative for the moment
because k and p are also functions of wavelength. The
occurrence of intensification is also controlled by the
meridional temperature gradient, since for any wavelength
there is a minimum required value of a for net deepening.
The temperature perturbation T and the vorticity-sta-
bility parameter vo/Tor have no effect on the occurrence
or nonoccurrence of deepening but are directly propor-
tional to its magnitude.
These findings are confirmed and made quantitative in
figure 13. If we regard as a measure of the instability of
the baroclinic system, we see that there is a wavelength of
maximum instability in the region from 1500 to 3500 km,
which increases with the meridional temperature gradient
and with the value of the vorticity-stability parameter.
The anticipated longwave cutoff is clearly indicated, but
there appears to be no limiting short wavelength for
intensification.
The preferred sizes of observed cyclones and anti-
A
[deg. ClIOOkm-'~]
1 I I I I
I 2 3 4 5 6 7 8 9 IO II 0 '
L (IO' km)
A FIQURE 13.-Deepening rate x of the 1000-mb cyclone as a function
of wavelength and meridional temperature gradient for selected
values of the vorticity-stability parameter and for $'= 1C.The
isopleths are labeled in units of 10-3 m2.s-3. The dashed, solid,
and dotted lines are as explained in figure 10. The dashed lines
are loci, of the wavelength of maximum deepening rate.
As mentioned above, we have overestimated the
theoretical deepening rates because of neglect of surface
friction. Let us now suppose, somewhat artificially, that
the 1000-mb surface lies at the top of the surface friction
layer. Frictional divergence or convergence within the
layer produces a vertical motion a t the top which we will
regard as a lower boundary value ala.
For evaluating this effect, we start with Petterssen's
(1956, p. 83) calculation that, with an angle of 25Obetween
the surface isobars and the wind at anemometer level,
the mass transport across the isobars within the friction
layer is 14 percent of the geostrophic transport along the
isobars. Neglecting vertical variations of density and of
geostrophic wind in a friction layer of depth h, we can
express this result as
V,dz=0.14 hkxV,
cyc1ones and do to the Of where V, is the wind component normal to the geostrophic maximum instability. We are unaware of observational
evidence to confirm the computed relationship between
preferred wavelength and meridional temperature gradient.
We can argue, however, that the relationship between the
wind, V,. If variations off are neglected (so that V Vg=O)
as well as horizontal variations of density and h, then
the equation of continuity yields
preferred wavelength and the vorticity-stability param-
eter is supported by observational experience that Highs
are larger than Lows and that small hydrostatic stability
tends to favor small synoptic scale disturbances.
wh=-v Vndz=-0.14hV kxV,=0.14 htlo.
Since wl0 = -qploWh, and with an assumed friction layer
404 MONTHLY WEATHER REVIEW VO?. 99, No. 5
depth of 1 km, we find approximately that
wl0= - 14 Ilo mb/s. (30)
In the light of eq (7) and since r10=V2t$1~/fo, we have
010=w10 cos - (s+A) cos L (31)
A 27r 2u y
where
A 28(2~/L)'&.
w10=
f o
This result yields a frictional updraft over the surface
cyclone and a downdraft over the anticyclone.
For obtaining the distribution of this vertical motion
above the 1000-mb level, we may simply define an addi-
tional component of vertical motion, w4, which satisfies
(v 2 +f z 2 gj-) w4=0
subject to boundary conditions that w@) =O and
w4(1000)=o10. We shall assume that UP is distributed
horizontally as wl0, so that
V2w4= -2(27r/L)'w4.
The solution of eq (32) is then
Since the exponent in eq (33) is positive, this frictional
vertical motion decreases with elevation, so that we find
divergence over the surf ace cyclone and convergence
over the surface anticyclone, tending in both instances
to weaken the circulation features.
For obtaining the quantitative effect, we add x4, a
frictional contribution, to the geopotential tendency eq
(17). This contribution satisfies
a@,. v2x4=foso - aP
If we assume that the horizontal distribution of x4
that of o4 and use eq (33) and (31), we arrive at
A 2u 27r
L L
x4=x4 cos - (23-A) cos - y
matches
(34)
where
8 (27r/L) 'RT0 7
lo00
A 7f$io (I+ f o l l o
1000
X4=
A
At the 1000-mb level in our sample case, x4=4OX10-'
IIP-S-~, corresponding to filling of the central sea-level pres-
sure at the rate of about 2.3 mb (12 hr)-I. From our previ-
QUS calculation of the frictionless deepening rate, we see
that the net rate with the frictional effect included is -4.1
mb (12 hr)-', only about half of the observed rate.
A
It is clear from the definition of x4 that, for given values
of the other parameters, the frictional filling rate increases
as the wavelength decreases. Since we have seen that the
frictionless deepening rate decreases with decreasing wave-
length when the length is sufficiently small, the frictional
mechanism provides a shortwave cutoff for intensification
provided the disturbance is of finite amplitude (ie., zl0 #O). It also shifts the wavelength of maximum intensifi-
cation to somewhat larger values. The amount of this
shift and the exact position of the shortwave cutoff depend
on virtually all the parameters of the model and cannot
be easily summarized.
that friction establishes a l y t i n g intensity of t? circu-
lation in any event because x1 is independent of t$lo while
the case of A=L/4 is
Moreover, it is evident from a comparison of x1 A and x4 A
x4 A is directly proportional to it. This limiting intensity for
which for our sample case is 2.78X lo3 rn2.s2. This value
corresponds to a range of sea-level pressure between low
and high of about 74 mb, somewhat high but not hope-
lessly unrealistic. The discrepancy may be due in part t o
the linear variation of wl0 with cl0 shown in eq (30). Per-
haps, the frictional vertical motion should vary more
strongly with relative vorticity. Moreover, we traditionally
regard intensifkation as terminating with occlusion (i.e.,
X+L/2) rather than with a frictional balance, but we are
not aware of any observational studies bearing on this
point.
Finally, we have the impression that this model, espe-
cially with the inclusion of frictional effects; is incapable
of accounting for the really explosive cyclogenetic in-
stances that, for example, characterize the synoptic me-
teorology of the winter oceans. These intense storms must
depend crucially on some form of diabatic heating, either
sensible heat transfer from the sea or release of latent
heat of condensation. The failure of current numerical
prediction techniques to predict this phenomenon ade-
quately, though due perhaps t o large grid distances em-
ployed operationally, confirms our suspicions.
7. FIELD OF TEMPERATURE TENDENCY
The pattern of local rate of change of temperature will
complete our picture and can be obtained in either of two
ways: directly from the thermodynamic eq (2) or by differ-
entiating the geopotential tendency eq (17). We chose the
latter approach, actually to check the correctness of the
work on which eq (17) depends.
From the equation of state and the hydrostatic equation,
In obtaining tixldp from eq (17), we differentiate the
(Px)'s and rearrange the coefficients and the resulting
May 1971 Frederick Sanders 405
pressure functions, arriving at Ptmb)
, , I I
I
aT @ 2a 2a
-&J sin x cos - L y
49
2= (9, L
2u A1
aT
+(%>a+ sin (x+~) cos y+ - sin - y
where
P?,=[k+3&(k+l) +3(w2k2(k+2)] [q (&Oy-l-l]
(35)
(36)
KT PT (10-4deg. G . 5 -' 1
FIQURE 14.-Case C profiles of various contributions to temperature
tendency.
- [k+3ak2+3a2k2(k+ l)] - (3& +k2k2) (1 - In P
- (3/2)a2k In =-(h P ?>21;
P$=l-a ln-- P
(37)
1-1 &
P&=(a %+1) (&o) -F.
At the 1000-mb level (since wlO=O), this expression should
and does reduce to the expression for the temperature
advection
dT,o=-V,o vTl0 at
n Profiles of the (K*PT)'s and of the (aT/at)'s for case C
appear in figure 14. In (aT/at)l, produced entirely by n
the adiabatic heating or cooling associated with wl, the
contribution arising from (aflay), is an order of magnitude
smaller than that arising from advection of relative
vorticity. This component indicates eastward progression
of the cold troughs and warm ridges except at 1000 mb
where the features are stationary and above the 75-mb
level where the results are not to be trusted. The con-
tribution (a$$$)2, due to advection of the zonally averaged
temperature and the adiabatic effects due t o w2, is
/\
relatively complex. Since h=L/4 in this case, (aT/at),
indicates intensification of the temperature perturbations
below the 700-mb level and weakening above. The final
term (aT/at),, arising from advection of the perturbed
part of the temperature field and from wa, yields warming
north of the latitude of the perturbation centers and
cooling to the south up to the 450-mb level, above which
the sense reverses. At the lowest levels, (6$& and
/\
(aT/at), tend to dominate because of the contributions
due to advection; but above 800 mb, the adiabatic
heating and cooling produced by w1 make (aT/&), the
largest individual term.
Patterns of temperature tendency at the 900- and
500-mb levels for our sample case are shown in figure 15.
The advection-dominated pattern at the lower level is
quite different from the vertical motion-dominated
pattern at the higher level, but both seem synoptically
reasonable, with respect to both general pattern and
magnitude.
If we take the difference between the tendencies at the two
levels, moreover, we find maximum destabilization oc-
curring approximately in the path of the cyclone and
stabilization ahead of the anticyclone, consistent with the
characteristic observed differences in tropospheric lapse
n
/\
406 MONTHLY WEATHER REVIEW V O l . 99, No. 5
I
0
- L/2 0 L f 2
I
I I 0 L 12
-L/4
-LIZ
FIGURE 15.-Temperature tendency for case C at (A) 900 mb and
(B) 500 mb. The isallotherms are labeled in units of lO-'"C.s-*.
The positions of the 1000-mb Low and High are indicated by
letters. The boxed values are extrema in the difference X!'/at at
500 mb-al'/at at 900 mb.
rate associated with these two circulation systems. The
rate, however, at the points of peak stability change is
about 3.5OC(3 hr)-l over the layer of 400-mb depth. This
value seems excessive for systems of synoptic scale and
would tend after operating for a short time to cast doubt
on the modeling assumption that the hydrostatic stability
parameter u is uniform on a constant-pressure surface.
Thus, we find another inconsistency in the model. Evi-
dently, the physical mechanism by which the atmosphere
controls its hydrostatic stability is not en tirely satisfactorily
represented here.
Whether or not occlusion is occurring, or more specifi-
cally whether or not the cold trough is overtaking the
1000-mb cyclone, is a rather ambiguous question. Since
the cold trough is stationary at 1000 mb in our sample
case, the geopotential Low is obviously escaping from it.
At higher levels, however, kinematic calculations show an
appreciable rate of overtaking. Perhaps a more meaningful
(but not conclusive) measurement is the eastward speed
of the 250-mb trough; if it is overtaking the 1000-mb
trough, then we would expect that the tropospheric mean
temperature trough is doing likewise. For our case, the
data in table 1 indicate that the 250-mb trough is over-
taking the 1000-mb counterpart at the rate of about 5 m/s.
To explore the occlusion properties of the model, even
through this approach, is not simple because from eq
(22) and (23) it is evident that the 250-mb trough speed
depends on every parameter of the model. Without per-
forming an exhaustive study, we have made some sample
[deg. CUOO km-I)]
A
calculations for the case of T=5"C, $10=103 m2.s-2, and
r10/T0~=2.8 X s-l. The results are shown in figure 16
where we see that, for wavelengths shorter than about
3800 km, the upper troposphere trough is overtaking the
1000-mb system. For wavelengths approximating the
value for maximum deepening (fig. 13), the overtaking
rate is close to 5 m/s for a wide range of values of the
meridional temperature gradient. From a comparison of
figures 13 and 16, however, occlusion does not seem to
be a prerequisite for deepening.
A final consideration is the intensification rate of the
extremely important temperature perturbation. For ex-
amining the effect in terms of the tropospheric mean
temperature, we need merely to obtain at the origin the
tendency.of the thickness of the layer from 1000 to 260
mb :
x(0, 0, 250) -x(O, 0, 1000) =&(250) -%(lOOO)] sin
From the definition of &, we find that
2* h.
(39)
From eq (39), we see that intensification (or weakening)
of the temperature perturbation cannot occur if the 1000-
Frederick Sanders 407
[deq. C(IOOkml-]
A A
FIGURE 17.-Difference x , a t 250 mb minus X , a t 1000 mb for
A= LJ4 as a function of wavelength and meridional temperature
gradient for selected values of the vorticity-stability parameter.
The dashed, solid, and dotted lines are as explained in figure 10.
The isopleths are labeled in units of 10-2 m2.s-3. This difference
represents the tropospheric mean temperature tendency a t the
cold perturbation center.
mb geopotential centers coincide with the centers of the
temperature perturbation. Further, we see from eq (40)
that there will probably be a critical wavelength below
which intensification will not occur, because of the different
dependence upon L shown by the two terms on the right
side. It also is evident from eq (40) that, whether inten-
sification or weakening is occurring, its intensity will be
linearly proportional to a and to C$lo. These observations
are confirmed in figure 17^that refers to the calculations
with a= C .m-, lo3 m2.s- , and dT,r=
2.8X s-l. We see that, for wavelengths smaller than
about 3500 km, the temperature perturbations are weaken-
ing, provided that O<X<L/4 (the usual state of affairs).
In particular for the preferred wavelength shown in figure
13, the temperature perturbations are weakening at a rate
somewhat in excess of lOC(12 hr)-l. These results are not
strongly sensitive to variations in the vorticity-stability
parameter. We are therefore left with something of a
dilemma: the temperature perturbation is necessary for
the deepening of the geopotential center, but this deepen-
ing simultaneously annihilates the perturbation, the origin
of which is not clear. The theory is not much help, and we
are not aware of any relevant observational studies.
A
8. CONCLUDING COMMENTS
In appraising our work, we have the impression that
we have rediscovered baroclinic instability because our
results are remarkably similar in many respects to those
of Fjortoft (1950), Charney (1947), and Eady (1949).
In these classical studies, time-dependent solutions of
linearized equations are obtained, while here we have
only instantaneous solutions, but ones that pertain to
the nonlinear situation. We are at a disadvantage because
we must postulate the structure of the disturbance and
cannot explain convincingly how it came to be. But we
have the advantage of being able to compute detailed
properties of the model, which appear realistic in most
respects. We find, for example, an explanation for the
tendency of surface cyclones to move t o the right of the
upper flow and the suggestion of an explanation for the
failure of numerical prediction methods t o forecast
explosive cyclogenesis. It is true that we ha,ve found also
a number of internal inconsistencies in the model, but
these probably arise from nonlinear effects and would
thus not be revealed by the linear analyses.
For some years now, the specific study of cyclones and
anticyclones has been almost abandoned with the feeling
that all questions would be answered in due course by
generalized numerical prediction. As we find points of
puzzlement in our theory, which underlies numerical
prediction either explicity or implicitly, we wonder
whether the time is not ripe for re-examination of
disturbances of synoptic scale from other points of view.
We feel, however, that the main significance of this
work lies in the bridge it provides between the concepts
that the synoptic meteorologist has traditionally employed
and what appears to him t o be the more esoteric notions
of the theoretical meteorologist.
ACKNOWLEDGMENTS
The author wishes to express his gratefulness to Miss Colleen
Leary of MIT (Massachusetts Institute of Technology) who
programmed the numerous calculations and to Maj. H. J. E.
Fischer, USAF (U.S. Air Force) who performed many of the
computations. The work mas supported in part by the Atmospheric
Sciences Section, National Science Foundation, NSF Grant GA-
10426.
REFERENCES
Austin, J. M., An Empirical Study of Certain Rules for Forecasting
the Movement and Intensity of Cyclones, Journal of Meteorology,
Vol. 4, No. 1 Feb. 1947, pp. 16-20.
Charney, Jule G., The Dynamics of Long Waves in a Baroclinic
Westerly Current, Journal of Meteorology, Vol. 4, NO. 5, Oct.
Eady, E. T., Long Waves and Cyclone Waves, Tellus, Vol. 1,
No. 3, Stockholm, Sweden, Aug. 1949, pp. 33-52.
Fjortoft, Ragnar, Application of Integral Theorems in Deriving
Criteria of Stability for Laminar Flows and for the Baroclinic
Circular Vortex, Geofysiske Publikasjoner, Vol. 17, NO. 6, Oslo,
Norway, Mar. 1950,52 pp.
Petterssen, Sverre, Weather Analysis and Forecasting, 2d Edition,
Vol. 1, McGraw-Hill Book Co., Inc., New York, N.Y., 1956,
428 pp.
Willett, Hurd Curtis, and Sanders, Frederick, Descriptive Mete-
orology, 2d Edition, Academic Press, New York, N.Y., 1959,
1947, pp. 135-162.
355 pp.
[Received June 6,1970; rewised August SI, 19701