)DYNAMIC AND THERMODYNAMIC
PHERIC MODELS
i Lorenz
,h College
niversity
.tute of Technology
'ULFILIMENT OF THE
'HE DEGREE OF
CIENCE
|
i
TE OF TECHNOLOGY
Signature of Author... -----------------------...........
Departmenk of Me.eoroiogy, Can. 9, 1948
Certified by.
a0a.
... aee
.... ..
Chairman, Department(yO~m
v
.0.,
tee
. .**....**
*S
, Thesis Supervisor
on Graduate Students
r
TABLE OF COTINTS
page
List of Symbols
1
I.
4
Introduction
II.
A Tensor Notation for 2,eteorological Problems
III.
Power Series Solution of the Hydrodynamic
and Thermodynamic Equations
IV.
V.
"lotion of a Point
Vertically Isothermal iviodels and Polynomial
VI.
Cyclones with Gradient Wind Fields
VIII.
XI.
30
Simple Cyclone M•odels
VII.
X.
Deviation from the Gradient Wind Field
Linear Cyclone Coefficients
Quasi-gradient Wind Fields
Conclusion
37
51
62
70
92
107
Appendix
114
Abstract
125
References
129
Acknowledgment
131
Biographical Note
132
1~9
_
15
25
Models
IX.
9
I~_~____
_
r-ctziý
_·~__
- 1-
LIST OF SYMBOLS
Many symbols appearing in this list are in standard use
in meteorology.
The meanings of symbols in this list not in
standard use, and the meanings of auxiliary symbols not in
this list, are explained in detail in the following chapters.
The significance of the subscripts in the symbols for vectors
is explained in Chapter II.
A
o•.,
C), k
constants describing the pressure field
of a linear cyclone
velocity, acceleration, --- of a pressure
center
C
the binomial coefficient
base of natural logarithms
acceleration of gravity
Coriolis parameter
natural logarithm
pressure
b
a standard pressure
n!
We"Ifti
__g_
-
_- -~-
I
I
I
I
I
- 2 radius of curvature of an isobar
gas constant for dry air
absolute temperature
time
AA
A,
WAr
components of wind velocity parallel to
x, y, z axes, respectively
geostrophic wind velocity
AA .
magnitude of geostrophic wind
gradient wind velocity
magnitude of gradient wind
wind velocity
Wit.1o),.-.
velocity, acceleration, --- of a wind
center
1 (I 1
+ Zt
2z)
rectangular space coordinates directed,
eastward, northward, upward respectively
ratio of gas constant for dry air to specific heat of dry air at constant pressure
"a-mmmii
-3ratio of specific heat of dry air at constant pressure to specific heat of dry air
at constant volume
JA
/0
ratio of gradient wind to geostropnic wind
density
latitude
rr~LiEi·_
MiMOMOMMMM
-4I.
INTRODUCTION
If the distribution of all the meteorological elements
throughout space at a given time is known, and if the effect
of influences outside the atmosphere is known, it is theoretically possible to determine the distribution of the meteorological elements at any future time by means of a set of
differential equations, which express the laws of hydrodynamics and thermodynamics.
This observation has led to a
number of attempts to forecast the weather by the direct
application of these equations.
These attempts have encountered two major difficulties.
First, the differential equations are very complicated.
Second, it is not possible at present to determine the distribution of the meteorological elements at a given time
with sufficient accuracy.
Perhaps the most detailed of these attempts was made
by Richardson (1922).
By replacing the partial derivatives
appearing in the differential equations by finite arithmetical
differences, Richardson obtained a set of algebraic equations.
On the basis of these equations he constructed a set of forms
for computing forecasts.
He computed a forecast for one
particular situation.
The first difficulty was only partially overcome by
Richardson's method, for although filling in the forms was a
matter of simple arithmetic, the time consumed was excessive.
-L -
1_1__1__
"AN ~
- 5Richardson estimated that it would require 64,000 persons
to compute the weather over the whole globe as fast as the
weather occurred.
With the introduction of electronic
computing devices into meteorology, this difficulty may be
overcome in the near future.
Richardson found that his forecast was unsatisfactory,
and attributed its failure to errors in the data for the
initial wind distribution.
There is no indication that
sufficiently accurate meteorological observations will be
made at a sufficiently dense network of stations in the
near future.
This study presents a method of applying the hydrodynamic and thermodynamic equations not to actual meteorological situations but to idealized models of meteorological situations.
In such models the distributions of the
meteorological elements can be expressed by equations.
The
difficulty caused by an insufficient knowledge of the initial conditions is thus automatically eliminated.
Instead
there is the problem of constructing models which are so
simple that tangible results may be obtained by applying the
equations to them, and which resemble actual situations so
closely that the results obtained are valid in the situations which the models resemble.
If in a model the atmosphere is piezotropia,, the set
of equations to be applied to the model consists of three
equations of motion, the equation of continuity, and the
i
•T
-6physical equation.
These five equations contain four inde-
pendent variables and five dependent variables.
If the
curvature of the earth is neglected, the independent variables nmay be chosen as t,
x, y, and z,
and the dependent
variables may be chosen as u, v, w,p , and p.
The three
equations of motion may be chosen as those expressing the
accelerations parallel to the x, y, and z axes.
The third equation of motion states that the hydrostatic equation is approximately true.
It is shown in
Chapter III that if the third equation of motion is replaced by the hydrostatic equation, the new system of five
equations may be reduced to a system of three partial differential equations with the dependent variables p, u, and
v.
Mioreover, if the values of p, u, and v are known through-
out space at the initial time t = 0, the equations may under
certain conditions be solved for the initial values of
0/0
and ý
, and from these quantities, expressions
for the initial values of
, and
.
be obtained for all positive integral values of n.
may
Under
suitable conditions, the pressure at any later time t is
given by the power series
ft) -t
A*- =
-
_
)(t
o)
ii
--
-7Similar series give the components of the wind velocity
at time t.
It is therefore possible to construct an atmospheric
model by choosing functions of x, y, and z for the initial
values of p, u, and v.
The remainder of the model is com-
pletely determined by these functions.
Since more is known about typical pressure fields than
about typical wind fields, it is convenient to choose the
initial pressure field first and tne accompanying horizontal wind field afterward.
Since there is probably no unique
wind field corresponding to a given actual pressure field,
and since some features of the wind fields which can accompany a given pressure field are not understood, it is often
desirable to construct several models with identical pressure
fields but with different wind fields.
The process of choos-
ing the pressure field before the wind field does not imply
that the wind field is a result of the pressure field.
In the following chapters, a routine procedure is
established for computing the power series expressing the
pressure and wind fields in models defined by initial pressure and wind fields.
models.
The procedure is applied to simple
Physical conclusions are drawn from a study of
the models.
Comparison of the changes of the pressure and
wind fields to changes of actual pressure and wind fields
constitutes a test of the resemblance of the models to
actual situations.
This resemblance in turn constitutes
__
-8a measure of the validity of the physical conclusions.
An
estimate of the feasibility of the method presented in this
study is based upon the success obtained in applying the
method to simple models, and upon the apparent applicability
of the method to more complicated models.
-9II.
A TENSOR NOTATION FOR METEOROLOGICAL PROBLEMS
Whenever the hydrodynamic and thermodynamic e.quations
are applied to atmospheric models, much repetitious writing
and computation may be avoided by the use of a notation
designed especially for problems of dynamic meteorology.
In this chapter two simplifications are introduced.
First, many partial derivatives appear in the hydrodynamic and thermodynamic equations and in expressions
derived from them.
It is therefore convenient to use a
more compact notation for partial differentiation.
The
notation presented here appears in some textbooks on the
calculus (cf. Berry et al., 1945).
Second, both three dimensional and two dimensional
vectors, which will be called three-vectors and twovectors respectively, occur in meteorology.
For example,
the wind velocity is a three-vector, but the velocity of
a pressure center is a two-vector, for a pressure center
has no meaning in three dimensions.
It is therefore
convenient to use symbols both for three-vectors and for
two-vectors.
The vector notation presented here is a
modification of a notation frequently used in tensor
analysis (cf. Eisenhart, 1940).
In this study, partial differentiation with respect
to x, y, or z is indicated by a subscript x, y, or z respectively.
Partial derivatives of higher order contain
- 10 -
several subscripts.
A subscript with an exponent n may
be used instead of n identical subscripts.
Partial dif-
ferentiation with respect to t is indicated by a prime.
A superscript n in parentheses may be used instead of n
primes.
For example,
A
At
means
---
A'
means
AA
or
A
A
.
means
2
a'a2
Dt'
Any term containing no Roman capital letter subscript
and no Greek letter subscript denotes a scalar.
A term with a single Roman capital subscript denotes
a three-vector, whose components are obtained by replacing
the subscript in turn by x, y, and z. A letter with a
single Greek subscript denotes a two-vector, whose components are obtained by replacing the subscript in turn by
x and y.
Since subscripts x, y, and z also denote partial
differentiation, any subscript representing a space variable and not denoting partial differentiation is placed in
parentheses.
A
For example,
is a scalar
(.z)
is a three-vector with components
(3) is a two-vector with components
13 t ) , 3~, ,
13),
A . is a three-vector with components a A
)
j3
A
C) I>3
--
AA
9
_i/_____ ~·
- 11 -
A term containing n different Roman capital subscripts,
each exactly once, and m different Greek subscripts, each
exactly once, has
of order m + n.
3nx 2m
components, and is called a tensor
A scalar is a tensor of order zero,
a vector is a tensor of order one.
and
Each component is ob-
tained by replacing each Roman capital subscript by x, y,
or z, and each Greek subscript by x or y. For example,
13
-)
has the six components
a 13tv
9 ('2o
The summation convention, frequently used in tensor
analysis, states that if the same subscript representing
a space variable appears twice in one term, the term denotes
the sum of the terms obtained by replacing the subscript
in turn by x, y, and z if it is a Roman capital subscript,
or by x and y if it is a Greek subscript.
The same Roman
capital or Greek subscript never appears more than twice
in one term.
For example,
J3(zC
C)
to)
B.*) C
B
+8
C
R/ -_(e)'If
B~P&
+ 8ita C (a
8(4
0 -P
-
-
12 -
A2AZ)a'A
-A
II
AA
A,
A
The above quantities are scalars.
_DA
a'A
a-L
They are the scalar
product of two vectors, the (horizontal) divergence of a
vector, and the Laplacian of a scalar, respectively.
The order of any tensor is therefore 3nx 2 m , where n
is the number of Roman capital subscripts which appear exactly once, and m is the number of Greek subscripts which
appear exactly once.
B
For example,
13 1)C(
is the two-vector with components
Dt
c,c.* +
C (1,.)
CC))
a
1
40
ca)
;.)
Bc)
C
---
-)>
A circumflex (^') placed over a Greek subscript
in-
dicates that the order of the two quantities obtained by
replacing the subscript by x and by y is to be reversed,
and that the sign of the quantity in which x replaces the
subscript is to be changed.
In particular, a circumflex
placed over the Greek subscript which appears exactly once
in a two-vector has the effect of rotating the vector clockwise through 900.
A circumflex is never placed over a
Roman capital subscript.
3•
(
h,
For example,
is the two-vector with components
4
Bu(),
-
,
-
(C()
13 -
has the four components
3
B
BC
B
The last two quantities, which are scalar
standard vector notation as the z-components o.
product of two vectors and of the curl of a ve
Some important identities follow:
1 €3
13
-
3
C (),
=0
and, because the order of partial differentiat:
versible,
A
0
The space variables x, y, and z are themsi
components of a three-vector, denoted by X I.
pressions Xx, Xy, and Xz are identical with x,
respectively.
Parentheses about the subscript!
sary, since the space variables may be treated
derivatives of the scalar
X
01 ( *%
+
A
÷
- 14 It
follows that the tensors
X,J
and ).
are equiva-
lent to the Kronecker deltas frequently used in tensor
analysis.
X
X
X
That is,
1
xx =
X
= 0
X
yx
zx
xy
0
=0
=0
'
1 ,
Xzy = 0
zy
'
X
xz
=0 0
Xyz =0
Xzz = 1 .
zz
Some important identities follow:
XrII
X '*Y- xjý
X
-- X-
X-
X
XI
X,, -
3
=
b
XXi,
The use of a notation resembling the one presented in
this chapter is recommended to whoever wishes to study
problems of dynamic meteorology with a minimum of repetition.
_____ ··
.__
_·L
·
- 15 III.
POWER SERIES SOLUTION OF THE
HYDRODYNAMIC AND THERMIODYNAMIC EQUATIONS
If the curvature of the earth is neglected, the three
equations of motion may be written
assumIed
atmosphere are
of the
changes
of state
If the* A&+MC+A
+ A
to
of
if the
In particular,
1are is
a constant.
where
of changes
the specific
, the ratio
=
dry-adiabatic,
state of
1941).
Haurwitz,
dry
air (cf. P+r
heats
+
T 10AA F
4 A PA+
In
the
changes
of
of
studyate the
are
assumed to
simplificationshere
atfollowing
written
equation
thermodynamic may
phhydrodynamicInand
uponpic,
bepolytrsed
changes of
ifbe the
particular,
athe
constant.
where
)is
the ratio of the specific
,
state are dry-adiabatic'
heats of dry air (cf.
iaurwitz, 1941).
and upon the models to which they are applied:
(1) The curvature of the earth is neglected.
(2) The earth's surface is assumed to be horizontal.
j
L
- 16 (3) The latitudinal variation of f
(4) All variations of g
is neglected.
are neglected.
(5) The atmosphere is assumed to contain no water
vapor in any phase.
(6)
The changes of state in the atrmosphere are as-
sumed to be dry-adiabatic.
(7) The external forces F(x), F(y), and F(z) are
assumed to be absent.
(8)
The third equation of motion is replaced by the
hydrostatic equation
(9)
The term
yml
Ap-
in the first equation of mo-
tion, which is usually small compared to the terms mr '
and
-XA
, is omitted.
With these simplifications, the hydrodynamic and thermodynamic equations may, in the notation of Chapter II, be
written
V, V)
/•V•),
p'-V(2
/&I + V(.11\p.,
-(3-1)
(3-3)
z0
• (
;s -ý-
V-1)1,
o
(3-4)
- 17 Equation (3-4) is obtained by (total) differentiation of
the physical equation with respect to t.
These equations will be solved for p' and V()
terms of p and V(,) and their space derivatives.
in
Equation
(3-4) is first replaced by the simple equation
p'
-' Vq- f
-ŽP Vtz
(3-5)
which is a combination of (3-3) and (3-4).
When P
is
eliminated from (3-1) and (3-3) by means of (3-2), these
equations become
V-) V(")I
(.)
-
t
eV-
(3-6)
- P V( I
V ( Pz
(3-7)
A suitable expression for V(z) must be substituted into
(3-5) and (3-6) to make these expressions the desired solution for p 'and V(k)
A second exoression
?I r-
v\
-V
VI
pCZ
p
P
-
(
(3-8)
\13
for pz' is obtained by differentiating (3-5).
Elimination
of pz' from (3-7) and (3-8) yields the expression
( ; A V ,,
- /'O:Vvx
-
X
( 0r)t
"
P'-VX•
/Oz Vt. .0)
(3-9)
- 18 and integration yields the expression
0O
V ,:(t'V))
-'6 VEIA)K
Pa
C1
(3-10)
in which each quantity to the right of the integral sign is
evaluated at z =
.
The limits of integration follow if
it is assumed that
Since
V(,A.
V(2)
÷ VL')-L
= V'(X]
S( W
(3-11)
The limits of integration follow because the wind velocity
is horizontal at the earth's surface.
When (3-10) and
(3-11) are substituted into (3-5) and (3-6), the desired
expressions for p 'and
Equation (3-10)
V(')
is
are obtained.
equivalent to an equation obtained
by Richardson (1922), who also assumed the existence of
hydrostatic equilibrium.
Equations (3-5) and (3-6) are
equivalent to simplifications of Richardson's expressions
for pressure and wind changes.
It is convenient to substitute for p the variable
where p
is a constant with the dimensions of pressure.
Then (3-5) and (3-6) are replaced by the expressions
-
V(T A
A' - - )ýV\(r
V,.'
-
19 -
A V, -)
- Vc
(3-12)
V>V<
(3-13)
,A( AV7t) z- Az-V~sov) J'3
(3-14)
-A
S
s
where
•.)•
V(Z)
I ,-A
-
i"
i:
2
( V(T)
V(W) I
-
b)
are obtained by
Higher time derivatives of A and V ()
Thus
differentiating the expressions for A' and V()
-iKC.,
V/(A&t
'1)
t IL)
Here
C.
Vt,:)) A1(i
sk -A)
-T)
Ch
V
VtC)COVA 2A
(0)
1)
(3-15)
(A.)
-
7
C
V,)
and
I)
Vt.),
(3-16)
n'
i:(n-i) '.
is the binomial coefficient
Suitable expressions for
CA)
C4 VC,
(A;' A.)
must be
substituted into (3-15) and (3-16).
Expression (3-9) may be written
(V(V,,7r2 + AA/Vwi2)
= - A Vt.~)
+
A
2
V00
Differentiation, with some rearrangement of terms, shows
that
- 20 -
AA)e
I ,aI
,AAV
(3-17)
z
Integration of (3-17),-yielding an expression for
, is similar to integration of (3-9)
and hence for V\%
if it is assumed that
tv~v,
A(V")
C
The expression
)
A2 (A - C
(A2 A4
(3-18)
is obtained by differentiating the identity
Az(Az'
Ax) = A
, and rearranging the terms.
With the aid of equations (3-15)
to (3-18)
it
is pos-
sible to present a routine procedure for computing any
desired number of coefficients of the power series for A
and V x
when the initial values of A and V(,) are known.
To obtain
A•A)
and
V,,
by this procedure it is neces-
sary to perform n sets of twelve steps each.
The jth step
in the nth set of steps will be indicated by the pair of
numbers
[n,j]
in square brackets.
Each step involves
quantities determined in previous steps, so that with
slight variations the steps must be performed in the fixed
order
`~"`~
""~`'
I
S:.
- 21 -
placed before any expression
In,j]
The pair of numbers
indicates that the expression is obtained by applying step
n,j], to expressions obtained previously.
The steps follow.
[n+l,l)
Compute
A•
In+1,21
Compute
AA
tn+1,31
Compute
9
)
A
by differentiating
A
(Au. A.)
by differentiating
A.)
In+l, 5
Compute
[n+l,61
Compute
(A.A.)
by differentiating
Compute
.
from the expression
A. -A C A
(A;' A.)"- A
In+1, 4
by differentiating
V•/e)
(.
C'.'
by setting
o1
\t
-- e
in V(0\),
and summing.
[n+l, 7
A) ,Ve
Compute -A(V()I
from the expression
(A()
(A
AVtz;)l
*X
(A)
m Vj.Io-w-'Oi
-A
CA A( V
A:6-A~~qte
In+l,8] Compute
~w
by integration:
AV(:,
I
VS/
AI(Al
LL
LA.-A)'J
- 22 -
V("-'
In+1,9] Compute
[n+l,lO]
by subtracting V, *)b
Compute V( 1 )(
from V():
by integration:
(..)
2
CtC1)Sb
n+1, ill
VC(l
A
Compute
from the expression
A +')
.()
fro
= - / V(,)
(+
V
In+1,121
Compute
V(A+V)
teVr
A,
I)
from the expression
(.1)
A
(A
(i-)
V,.,,
VC.-V
The above steps make it possible to compute any desired
number of coefficients in the power series
oA
(3-19)
,4-O
I1
VIA•)
:: Y
^1
A-V CO"
V
4)
0%
4)u
(týo)
(3-20)
Power series for imoortant quantities other than A and
V (, are incidentally determined by steps
[n,j] .
these quantities are the horizontal divergence V
Among
and the
vertical speed V(Z), whose coefficients appear in step
and
[n,lO1
.
[n,6]
Coefficients for the geostrophic wind U(C)
are determined by steps
[n,3] , for the wind is
geostrophic
when the pressure gradient force and the Coriolis force are
.._.._..__ .
I1·_
23 -
in equilibrium, and so, according to (3-13),
S'A "7A
(3-21)
If the power series (3-19) and (3-20) are to represent
the desired solutions for A and V(.), they must converge.
Power series in a given model may be examined for convergence after the coefficients have been computed.
If a
series fails to converge, no solution is obtained, but a
divergent series should not lead to any false conclusions.
On the other hand, it is possible for a series to converge and yet fail to represent the desired solution for A
or V
For example, suppose a model contains a cold fron-
tal surface treated as a mathematical discontinuity in temperature and wind velocity.
If the vertical column L
through a point P in the warm air mass does not intersect
the frontal surface, the initial values on L of A and V()
and their space derivatives are not affected by the presence
of the cold front at a distance.
Since the procedure pre-
sented above, applied at P, makes no use of the initial
values of A and V)
at points not on L, the presence of the
front cannot affect the values of the coefficients A
VV
(
at P.
and
In other words, according to the power series
solution the front can never pass P.
The failure of the power series to represent the desired solutions results from the fact that the initial
values of A and V
are not continuous.
If a discontinuity
values of A and V(•)
are not continuous.
If
a discontinuity
- 24 -
in some atmospheric quantity is moving through space, the
function expressing the quantity must be discontinuous in
t at all points which the discontinuity crosses.
It is conceivable that in certain models the presence
It
of a discontinuity may lead to no false conclusions.
seems desirable however to require that in all models the
initial values of A and V ()
be analytic functions of
(x,y,z).
In
such models fronts appear as narrow zones of large
temperature gradient and wind shear rather than as mathematical discontinuities.
Since an analytic function is
completely determined by the coefficients in its power
series expansion, some of the space derivatives of A and
V(.) on L are affected by the presence of the cold front
at a distance, no matter how far away the front may be.
Therefore some of the coefficients A
and V
are
affected, and the model may describe the passage of a front.
I
r
-r--
- 25 IV.
iMOTION OF A POINT
In certain meteorological problems it
is
desirable to
know the path of a particular point, such as a surface low
It
pressure center.
is
sometimes possible to determine
this path without computing the complete pressure and wind
fields at any future time.
Let P be a point moving in a horizontal plane and let
the coordinates of P satisfy the equations
x = fx)(t)
y = f(y)(t)
where f(x) and f(y) are analytic functions.
ponents of the velocity of P be C,c,
Let the corn-
and Cc)
.
Then
in vector notation,
XOL ,,(t
(4-1)
(4-2)
d-t
Let
1
(x, y, t) be any analytic function of (x, y, t)
and let
at the moving point P.
c4.
CX .-
Then
at 3v.
J.
-L
4CIL
(4-3)
Let P be defined explicitly by the expression
(4-4)
E~r:s
g
26 -
-
which represents a pair of equations.
Expressions (4-1)
and (4-4) are equivalent, though they may differ explicitly.
at the moving point P does not
The value of F ()(x,y,t)
vary with time.
So, according to (4-3),
F (f)
i(o)
F(tv
=o
.
(4-5)
Expression (4-5) represents a pair of linear equations
F(x)
F(x)y
F
F
F(y)
C(v)
and
C-,c*)
in the two unknowns
. Its solution is
)
()
F
F(y)y
x F(y)
(y)x
(y)
F(y)x
FWy)
C(.V) C
C IC,")
F(x)x F(x)y
F(y)x
or,
F(y)y
in texsor notation,
F()
C
(4-6)
-
FC)y F(,)•
provided that
F(Py F(~,y%
the solution of (4-5)
is
O
.
If
FP
F) )=
indeterminate.
A power series expression for the path of P involves
not only the velocity
C~,,
but also the acceleration and
higher time derivatives of the position of P.
n 1 O, let
For all
- 27 Cs(4-7)
If the power series (3-19) and (3-20) represent the desired
values of A and V(~),
the coordinates of P satisfy the
equation
o0
t=O
-,
A.
provided that each coefficient
The quantity CI,)
C.(,,
is evaluated at t = 0.
has been defined only at P.
means of equation (4-6), the definition of
CI.),
By
may be
extended over the entire plane, except for the points where
F( y F(A )
C-,.c)
=
.
When this extended definition of
is used,
C•c.,. = C.•-,
", C
determine in turn CL)
,
the definition of each C-.Y
Cl,)
(4-9)
Expression (4-9) makes it possible to
according to (4-3).
Although
I (at)
3Ca
)
, ---
, and also extends
over the entire plane.
may be defined over the whole plane,
it is sometimes inconvenient to determine its value explicitly at points other than P, and to obtain its space and
time derivatives.
It is therefore desirable to obtain al-
ternative expressions for the acceleration Cx.,)
for higher time derivatives C.,)
of P and
of the position of P.
p
I··
- 28 -
F:
Differentiation of (4-5) with respect to t and X
yields the
expressions
I f
I
FtC
*F(Oc)
l
I
C,)
F(.d,,I
Cr(,.
c,,,
's
+ F.voy C, ) + F(at
Expression (4-9),
C;(O)
,
with n =
=~C (e
+ C_
(4-10)
(4-11)
C, t3)y
, may be written
(e)-y
(4-12)
Cry)
and C.,Py
Elimination of C-,c,
= 0
from (4-10),
(4-11),
and
(4-12) gives the equation.
II
,
F
st ýF(t)t
C..~ t- F
CIoC
y3,,I+-(',
C)(0 )
0.
(4-13)
Expression (4-13) represents a pair of linear equations
C.(_,)
in the two unknowns
sembles that of (4-5).
and C(4 )
.
An expression of the same form as
(4-5) and (4-13) may be obtained for each
general expression is
Its solution re-
rather complicated,
C, ()
a
The
and is not pre-
sented here.
Suppose that the point P under consideration is a pressure center.
At such a point
A,
=
.
Therefore the velocity and the acceleration of a pressure
center are given by the following special cases of (4-5) and
(4-13)
- 29 -
A
i-
A.
A,
,)
o(4-14)
A
+1AD' *Aeo
A"
ctoe)
,.
(:+-l*
t Am.p C-> 0
C-nts) ,fcm
In the following chapters, the symbols
Cos(-
(4-15)
) refer only to
the motion of a oressure center.
The symbols
are used instead of
W•(.)
)
to refer
At such a point
to the motion of a wind center.
V
c-
0
The velocity and the acceleration of a wind center are
therefore given by the equations
NA)
),-
,
tAI
t IV(PC
*~:;
W
0+\A
U)
*VsyW,~,)~ y+*VjW)B~(p)-
(4-17)
(4-17)
Equations (4-6) and (4-12) are generalizations of formulae obtained by Petterssen (1940) for the velocity and the
acceleration of pressure centers and other points.
The
method of derivin- the equations in this chapter is similar
to the one used by Petterssen.
[i
- 30 V.
VEliTICAl.LY ISOTHERiM"AL 1MODELS AND POLYNTOIAL MODELS
In the procedure for computing the power series for A
and V(,,),
steps
In,8
and
[n,10]
tion along every vertical column.
each require integraIn an arbitrary vertical
column, the initial values of A and V(,)
derivatives are functions of z.
In,81
The ease with which steps
may be performed depends upon the nature
[n,10]
and
and their space
of these functions of z
The space derivative A z is considered first.
Since
A = In p/po
0
AA
L
-
-
-
~
-.
LA
So A z is a negative quantity whose value depends only upon
the temperature.
quantity, -Az'
The negative of the reciprocal of this
, has the dimensions of distance, and is
equal to the height of the homogeneous atmosphere if the
temperature is 0O
C. (cf. Willett, 1944).
In an arbitrary vertical column the simplest possible
function which can be chosen for the initial value of A z is
a negative constant,
A
Z
= -B
O
The constant B10 need not be the same in different columns,
so that initially
A
(x,y)
zo = -B o
(4
(5-1)
- 31 A model in which (5-1)
is satisfies everywhere initially will
be called a vertically isothermal model.
Integration of
shows that initially
(5-1)
A = A0
(x,y)
-
B
(xv)
'--,s
(5-2)
in a vertically isothermal model.
Step
In,8]
requires the evaluation of an integral
of
SA&%~,
the form
ao
-A (*),'L)
&-)
A-
p, 9) F ,
)5 5I
'7
In a vertically isothermal model,
o0
0SO
F
If expression (5-3)
(5-3)
a/
is integrated by parts n+l times, the
expression
S= B, F
+ (3
- (,,t,+
+ 3;FF2 +
B.- 3 Fx
- (,, ) 84
Bb
FIZ&V
7
0o
-
SOS
F
zL+
(5-4)
is obtained, provided that
,u (- -B6
4('(';)f
O
Suppose that F (x,y,z) is
z, say
.,,', )!
for i
=
0,---,
0
(5-5)
a polynomial of degrss n in
- 32 Then
h.r
(5-6)
( .C'4(q3s)2~
In particular, f #+I
= 0 .
Moreover,
nomial in z, expression (5-5)
holds.
is a poly-
since
Hence,
according to
(5-4),
S-,
(5-7)
Therefore G (x,y,z) is also a polynomial of degree n
say
(',t ) 2
(5-8)
Expression (5-6) may be substituted into (5-7).
Cormparison
of the resulting expression with (5-8) shows that
OZ0
"-•
-,~
t1 (
(5-9)
It should be noted that an integral of the form (5-3)
also appears in
the expression for the isobaric mean value
of a quantity.
The isobaric mean wind has been discussed by
Petterssen (1945) and by James (1945).
If H is any quantity,
the isobaric mean value H of H in a given vertical column is
defined by the expression
I--
r
I
H/ P
H - -where p, is the pressure at the base of the column.
If
the
r
1
- 33variable of integration is changed from pressure to height,
AAAH
H
d•
(5-10)
If the column extends upward from the surface, the lower
limit of integration is zero.
A vertically isothermal model in which the initial value
of V(0)
is
a polynomial in z, with functions of x and y for
coefficients, will be called a polynomial model.
With the
aid of (5-7) it is easily verified that in a polynomial
model the quantity computed in eacn step
nomial in z.
In,j]
is a poly-
In such a model no difficulties arise from the
necessity to interrate in
steps
[n,8J
and
[n,10
.
The above conclusions depend upon the assumption, made
in Chapter III, that
00
V)
O
for all n=0 .
(5-11)
It is possible to prove expression (5-11) in the case of a
polynomial model.
The proof uses mathematical induction.
Assume that for some particular value of n,
Then all the steps
[n+l,l1
= o
V(t)
Az•
to
li,JJ
[n+1,71
for i = 0, --- , n-l. (5-12)
for i = O,
---
, n
, are valid, and the quantity computed
in each of these steps is a polynomial in z.
[n+1,91
to
[n+1,12]
, and steps
Also steps
are valid, though the quantities com-
puted in these steps are not necessarily polynomials unless
i2211MG99-
- 34
step
In+ll,8
is valid.
-
Equation (3-17) is equivalent to
the expression
X(A
)I-
F,
SA
V(
where Fn is a polynomial in z.
Integration yields the ex-
pression
S-
,J
)
(A#-
--
(5-13)
V(-x:r
where Gn is a polynomial in z defined by an expression similar to (5-7), and hn deuends upon the boundary conditions.
Steps
[n+1,91
[n+l,11
to
CA.)
In+1,91
= P 1,
In+l, 101
-.
+ k, .
0 )
.., ,*lt
=
V2
are now applied.
)
,
.
I\
P&%3
ýn+l,1]
Here P ni'
Pn~
and P
Z) +
2f,,
A,-
are polynomials in z.
(
.-
(A.-13 *2)
Since
tpo e,
P,=
~1'00
Aý'
and, in general,
s, -•
A (•
.q,),
where Qi is a polynomial in A', A" ,
(A)
A
'
So
A,)
)
- 35 -
P A.(A02)(q
"
PA 3)
+ (-ý~)
*.
k
is a polynomial in z for i = 1, --- , n , Qn is
Since A
a polynomial in z.
Therefore
Z -too
The physical nature of atmospheric pressure requires that
not only p but also every time derivative of p must approach
zero as
Z-i•o,
Therefore
hn (x,y)
-
0,
and according to (5-13),
At^%
(A AV
V
) )
0
(5-14)
If n = 0 , no assumption is made in (5-12).
must hold when n = 0 , i.e.,
So (5-14)
(5-12) holds when n = 1.
There-
fore, by induction, (5-12) and (5-14) hold for all values of
n.
It follows that all the steps
[n,j]
are valid for
polynomial models.
The resemblance between vertically isothermal models
and actual situations must be considered.
Ordinarily the
temperature varies considerably with elevation.
But in a
given vertical column, at least in the lowest thirty kilometers, the temperature appears never to differ from the
surface temperature To by more than 1/2 To .
A vertically
isothermal model is probably preferable to one in which the
temperature varies by a factor of 5 in the lower atmosphere.
-36Vertically isothermal models should prove useful in the
study of phenomena where the important feature of the temperature field is a pronounced horizontal temperature gradient.
The motion of a cyclone in temperate latitudes may be a phenomenon of this sort.
Vertically isothermal models cannot be used for studying phenomena which depend upon the presence of steep lapse
rates, such as convective phenomena, nor phenomena which
depend upon the presence of warm air above cold air, such as
frontal phenomena.
Neither can they be used to study phe-
nomena depending upon horizontal variations of lapse rate,
or,
equivalently upon vertical variations of horizontal
temperature gradient.
The choice of a polynomial in z for the wind field
places essentially no restriction upon the model, since
throughout low levels any analytic function may be closely
approximated by a polynomial.
At very high levels, say 150
kilometers, virtually nothing is known about the wind, and
there can be no basis for saying that the wind field in any
particular model resembles an actual wind field at such levels.
-
VI.
37 -
SIMPLE CYCLONE MODELS
A problem which can perhaps be studied with vertically
isothermal models is that of the motion of a cyclone in a
baroclinic atmosphere.
It seems likely that some features
of cyclone motion, such as the phenomenon of steering
(cf. Austin, 1947), depend more upon the large scale horizontal temperature field than upon the lapse rate.
Although any simple analytic function doubtless differs
greatly from the function representing the instantaneous
pressure field of any actual cyclone, it is not difficult
to describe many of the important features of the initial
pressure field by a simple function A.
to find analytic functions V(x
)
It may also be easy
and V(y) describing the im-
portant features of the wind field of a cyclone.
But it is
not easy to determine a wind field which can reasonably
accompany a given pressure field.
If
the pressure and wind
fields in a cyclone model are incompatible, results which do
not agree with anything ever observed on synoptic weather
maps may be obtained.
For example, pressure tendencies of
1000 millibars per three hours may occur.
In the present chapter, simple cyclonic pressure fields
are chosen, and simple wind fields are chosen to accompany
them.
Velocities and accelerations of the pressure centers
are obtained.
The resemblance of these velocities and ac-
celerations to those of actual cyclones constitutes one test
- 38 of the compatibility of the pressure and wind fields.. In
Chapters VII and VIII, the same pressure fields are used.
The wind fields appear to be more compatible with the pressure fields, but they are more complicated.
Special attention is given to the problem of steering.
For the purposes of this study, a cyclone is said to be
steered if the velocity vector
C•a•)
of the surface pres-
sure center is parallel to the isobaric mean geostrophic
wind vector
above the center.
jlIu)
In a vertically isothermal model,
A
=
A
- B
z
according to (5-2), so
(6-1)
(-A0~i I~z
A"f' R
according to (3-21), and at the surface pressure center,
where Ao
r O ,
U
a-
according to (5-11).
IDA3~
9. ~(6-2)
At a given latitude,
pletely determined by the temperature field.
L
is comIt should be
noted that in a vertically isothermal model, a cyclone which
is steered moves parallel to every isobar and every isotherm
above the surface pressure center, since all of these lines
are parallel to
,)
*
In the models used in this study, Cl~ot
direction and magnitude to
wind
VI
.
U,)
is compared in
, and to the isobaric mean
Austin's empirical study (1947) indicates
- 39 that the majority of cyclones, although by no means all of
them, move in approximately the direction of the isobars
and isotherms at various upper levels.
No connection is
observed, however, between the speed of the cyclones and the
speed of the upper level winds, or of the upper level geostrophic winds.
The complete physical explanation of these
empirical results cannot be obtained from the study of special models only.
But this study suggests a possible physi-
cal explanation.
Three simplifications are imposed upon the cyclonic
pressure fields considered in this chapter.
(1)
The models are vertically isothermal.
(2)
The surface isobars are concentric circles.
(3)
The surface isotherms are straight lines parallel
to the x-axis.
In the most general model satisfying these three conditions
A (x,y,z) = A o
(x,y) -
A o (x,y) = F(R')
B (x,y) = G(y)
Here R =
x
+ ya
Bo
(x,y) z ,
,
(6-3)
.
is the distance from the z-axis.
Let
- 40 Then
An l, =nAn+ X
B
Steps
[1,11
to
1,1
Az = -B o
1,2'
AK = AX
1,3 1
AAAA
-1
z
= B n +X
[1,31
may now be applied
-B1 X
-1
=B 0
z
(-A
B XyV(-Az)
The simplest possible initial wind field is a null wind
field i. e., a field where V(,)
= 0 everywhere.
Experience
with synoptic weather maps suggests that a null wind field
never accompanies a well developed cyclonic pressure field.
But it is interesting to consider the null wind field, in
order to observe the results which are possible when the
wind and pressure fields are known to be incompatible.
Another simple wind field, which may appear to be more
compatible than the null wind field, is the geostrophic wind
field U(), defined by (6-1).
If the wind field is either
[n,j]
null or geostrophic, steps
are easy to apply when n
is small, because many terms vanish.
The null and the geostrophic wind fields may be treated
as special cases of the wind field
C4
•_
U (4)
where c is a constant.
8
-A,'(AX- ii-x,XS. )
(6-4)
The present chapter deals with cy-
clone models in which the wind field is given by (6-4).
- 41 Values of A
V(,.
[1,4]
I
I
are now determined.
and V(,)
Sc. f' 3
C'A~c.
[1,5]
[1,7
V
C1,lo]
L
V,
To
t3,A ,, -B.AA X,X••
,A,A 4
-•\/(:o, +A,
2V
11,8]
oA,X,p
x,
+ (8.Ba - ,)X,,, X,,,
V\/,=-•(c
[1,6]
2
X)
.
P " g;B'
-
'
=o
+ ,A,
+C1t 9 -BA, v.,
+ 3,,A
A
The expression for A' is simple because the wind field
has been so chosen that many terms vanish.
surface, A
At the earth's
= 0 , so
CI t)
= 0.
On the other hand,
According to convention the null vector
-o,.)
is parallel
-
42
-
to any given vector, and in particular to 5U,
, but for
practical purposes the cyclone is not steered.
However,
since not all actual cyclones are steered, and since quasistationary cyclones are observed, it cannot be concluded
immediately that the wind field is incompatible with the
pressure field.
The expression for V)
is rather complicated.
Very
can be expected,
complicated expressions for A' and V
since it is necessary to differentiate V
to obtain them.
It therefore seems impractical to compute many terms of the
power series for A and V(.) when the pressure field is de-
fined by (6-3).
The simplest special case of (6-3) occurs when F and G
are linear functions of g RI
and y respectively.
In this
case the model is called a linear cyclone model.
If Po , which is arbitrary, is chosen equal to the
pressure at (0,0,0), the pressure field of the most general
linear cyclone model may be conveniently expressed by the
function
SA=
In (6-8),
O-•
-
\
ItcI
)
(6-6)
c9 may be treated as the reciprocal of a horizon-
tal distance which depends upon the curvature of the surface
pressure profile along a line through the center, while b is
the reciprocal of a vertical distance, comparable to the
height of the homogeneous atmosphere.
The dimensionless
F%
'9
43-
k-
constant k is included to make the model general.
(6-6),
by (6-6),
d3efined by
When AA is defined
When
a R
A
A
An
B, =kab
a
=
= (l+kay)b
B
,
0
for
n.
2
,
B
rl
= 0
for
n
=
2
II
I/
becomes
defined by
by (~-4)
(6-4) therefore
The wind
wind field
therefore
becomes
defined
field
The
8
- 9
Vt•)
VI,,
Sl`lcB~IO·(·X~-k~LX
-(c
X f"--k
.
X1 c·')*z)
(6-7)
(5-7)
simplest
the sir~plest
(6-6) appears
appears to
to define
define tl~ie
Expression (~-~)
~xpression
possible
possible
pressure field
field of
of aa cyclone
cyclone in
in aa baroclinic
baroclinic zt;nosphere,
atmosphere, and
and
pressure
The
remainder of
of
The remainder
simplest
includes the
(6-7) includes
wind fields.
fields,
t~e
simPlest wind
(t-7)
/
the present
present
the
by (~-6)
(6-6)
defined by
with rrodels
models defined
chapter deals
deals with
cha.t~ter
and
and
(6-7).
(6-7).
In performing
the steps
steps
In
performing the
[n,jl
Cnjl
,, it
it
is
is
convenient to
to
convenient
combination
quantity aa combination
each quantity
of each
factor
the first
first factor
as the
include as
of
factor
include
as
of
include
A
and R
b, and
of a,a, b,
of
b, and
R
of a,
quantity.
that quantity.
as tiiat
di~iensions as
same diiuensions
having ·the
l·ii;ving
as that
qu~ntity.
t;he same
same dii:;lensions
h~vin~-~ tile
by
avoided by
may be
be avoided
writing ~~y
of writing
deal of
great deal
purpose aa great;
this purpose
For this
~reat
For
~urpose
m~y
the dimensionless constant
the introduction
introduction of the
h
i?
=
g
gRc~a'a
gR"a'
(6-8)
(t;-s)
(6-~)
b
b-)
b-l
dimensionless variable
variable
and the dimensionless
Y =kay
=i;ay
Y
kay
quantities
these quantities
of these
terms of
In terms
In
i
F
A
A·
A =
R'
R' --
V
R
R Cr·
Cr· C-
t4
(l+
~> ·G z
(R+Y)
((+
~)
klX·r.r)
~1
1
9
(6-9)
(G-9)
-
-
are now repeated for the linear case.
l,j]
Steps
44
AZ = -b(l+Y)
[1,21
A, = a(aX,-
[1,
A'
3
-b a(l+Y)
A,-
[l,3]
kbXyg z)
- kbXy CK
(aX
=
'bchk(l+Y)' Xy,
V(,)z = •
[1,41
[1,51
=
1,6]
1[1,7
I
chk(1+Y)
= -
+ kaX.
-(1+Y)X -t it
ch(l+Y)
"'
' XYee
- k bX
X
x'. ye'
ax
= 0
- \ (V( )z + AzV( )I )
= 0
[1,9]
= t chk(l+Y) 2 ax
V(z)z
_%I
%
, =t 6chk(l+Y)
11.101
[1,11]
A' =
[1,12]
V'()
abxz
kchk(l+Y)' abxz
= 1 a (l+Y)-
-
ch +k
(c-)h(X
- c'ka
'
(1+Y)(aX
It is convenient
Frequent use is
=- -
-kbXy ,)
X -x
is now determined.
The value of A
to use the constant
a
+Y))
made of
the identity
(6-10)
= X v Ra
[2,11
[2,21
[2,41
A
'
= t bchk(l+Y)-' ax
=
V(')z =
achk(l+Y)-
bz((l1+Y)Xx
aabk(l+Y)
c2
- kaXy x)
(c-l)h(l+Y)
Xy
- 45 -
E--ý,Q~~Vv)u=
clhl 3 tY-Y a)-3)
tA2V,)
I
-
k(
v<=
wa
A R'
Y)-3
(I
(C -")k
(I+Y)k'
42)
(I+
? T - >1
(
Vý (\t I)
-t
*·(-t)(
It1LO
Y)k2A2
t(I
-t
-o
-O)k(it y)(1+k t
C
;,
~1V•2=,A
+ k:-L 2)
(1
y
2 R- Y
CI, "1
* - ((I+Y)*
h 2)
-+
(C-1) k(I+
Y)' (-.I+Y4+" 2)
·.ei~·I·
Ev]1
a
y
t-(s*Y)kA2)
lot I
Y)JýJ
,(tt)*
, (- -,~k) ' -Y
/I
F
- 46 -
K
Sl
(1 +
-~LRK
(-
k~ o-'*$ 2
\41'
[*I
lo VN--1"k-, (It
?
VP)-.'
I
V
(·-I,
C
r:
t (-1-2xW) V-
h(~+Y)
. 4-
y
(t4)
-q )
/(-:+
+
Yl) CL
y~3Y
y- Y
Y)
It-
+
y 'lJý'JrA2
I+k
+YY
(6-11)
f
I1
S•1('+Y)
jk-
+(<-) ( . 'i)
.
Y +y),C
Y-.+Y
' L)u
(,4zy
-"H
~-
- 47 'I
Although the expression for A
is very complicated,
the expression for the surface value of C3(c.)
simple.
At the surface pressure center (0,0,0) ,
[3,2§
Ac
Also
A•
Since
C(
C,(3L)
is relatively
=
akXy
ch
(8+16 k0) + (c-l)h (4+6kai)
= 0 , it follows from (4-15) that
=
0
=
-Q
(6-12)
C; ()
k(cah(8+16
C,(.)
The acceleration
k'a )
+ (c-l)h (4+6k2
)
is therefore parallel to the y-axis.
It is difficult to appreciate the meaning of (6-11) and
(6-12) without considering the numerical values of the quantities involved.
The values of the physical constants may
be taken as
0.98 x
0=
10
km sec
= 2.87
10 "1 km2 sec
=
=
l0fo " sec
0.99
=- 1.405
,
= 0.288
.
The value of A
degree
,
,
(6-13)
corresponds to latitude 42.80
N.
Representative values of the constants in the expression
for A are now introduced.
p = 1000 mb at (0,0,0)
Let
,
,
p = 1020.2 mb at (0,800,0)
T = 273.20 abs at (0,0,0)
,
~-C^-- ----
I
- 48
-
T = 270.50 abs at (0,200,0)
The coordinates of the above points are expressed in kilometers.
a =
b =
Then
4000
1
km
km
-|
,
(6-14)
h=
1
k =-
5
,
The pressure field is thus defined by
A =
R
00
-
i +
0
y
z
It
provided that R, y, and z are expressed in kilometers.
is evident that the surface pressure increases without
limit with increasing distance from the center, and that
there is no upper or lower limit to the temperature.
There
is also no upper limit to the wind speed if the wind field
is given by V()
=
cU()
, and if c *
O.
It follows that
a linear cyclone model can resemble an actual cyclone only
in a restricted region.
In the above example, when R = 2000
kilometers, p = 1133.1 millibars.
The pressure and temperature remain within limits actually observed if only the portion of the model within
1000 kilometers of the z-axis is considered.
region,
Within this
- 49 1000 mb
-
p •
1031.7 mb
at the surface, and
287.60
abs I
T 1
260.20 abs
With somewhat different values of po,
a,
b, and k, rea-
sonable values of pressure and temperature still occur
within a restricted region.
According to the discussion at the end of Chapter III,
the exclusion of the portion of the model where R is large
in no way decreases the effect of this portion of the model
upon the portion near the z-axis.
It
is
therefore impor-
tant to determine that any conclusions based upon linear
cyclone models do not depend upon the extreme values of
pressure, temperature, and wind at great distances from the
z-axis.
Values of
Cact'
c are listed below.
corresponding to various values of
They are obtained from (6-12)
with the
numerical values (6-13) and (6-14), and are expressed in
kilometers per hour per hour.
C.
C.)
0.0
215
0.2
163
0.4
94
0.6
7
0.8
-97
1.0
-219
- 50 For values of c near 0.6,
the acceleration is small.
The compatibility of the wind and pressure fields in such
cases is considered in Chapter IX.
For other values of c,
the acceleration is much larger than accelerations actually observed.
In the geostrophic case, where c=l, the
cyclone, starting from rest, is moving rapidly southward
after one hour.
With a null wind field initially, the cy-
clone is soon moving rapidly northward.
Thus, as might have
been anticipated, the null and geostroohic wind fields are
both incompatible with the pressure field defined by (6-6).
Moreover, the geostrophic wind field appears to be no more
satisfactory than the null wind field.
- 51 VII.
CYCLONES WITH GRADIENT WIND FIELDS
It is often claimed that in the free atmosphere the
gradient wind is a good approximation to the actual wind.
In the present chapter the pressure field defined by (6-3)
is considered again, and the gradient wind field U)
is
chosen to accompany it.
According to the usual definition, the wind is gradient if the pressure gradient force, the Coriolis force,
and the centrifugal force are in equilibrium.
tude u
The magni-
of the gradient wind is therefore the positive
root of the equation
u
+
2
ru
-(ruo = 0
,
(7-1)
where r is the radius of curvature of the trajectory, assumed to be positive if the curvature is cyclonic, and u
is the magnitude of the geostrophic wind (cf. Petterssen,
1940).
Petterssen (1944) has shown that in the free atmosphere any wind moving parallel to the isobars is equal to
the gradient wind.
There is therefore no unique gradient
wind field corresponding to a given pressure field.
For
example, the wind field cU(W) of the previous chapter is a
gradient wind field.
A unique gradient wind field can be defined, however,
if r is required to be the radius of curvature of the isobar.
In practice, such a definition is often used when the
16"
__
trajectory is not known.
In this study, r will always mean
the radius of curvature of an isobar.
of r, the magnitude u
With this definition
of the gradient wind will be defined
by (7-1).
/
Let
4
If the wind field is gradient,
1
The gradient wind field V(,) =,MU()
field V(,)
c U(,)
l
III
i :~
n 1L)
,r~
(7-2)
differs from the wind
of the previous chapter because
varies with x, y, and z.
poUYI1JIlyno
+ B,Xo
B*.' (-A,%XY
8'
/4
th
Since /4
,,,
iielutl,
is not necessarily a
L
al)hIougi
is not necessarily polynomial.
,m
,,~,7,
vlerically
~
,
isu
1,,,
ithermal,
Expressions containing inte-
gral signs may be obtained for A' and V()
, but if
,A
varies
with z in a complicated manner it may be impossible to perform the integrations.
An expression containing integral signs is now obtained
for the velocity
[1l61 Vi.,
[1)7]
C~K
of the surface pressure center.
A t t ,/ 1..1-46-1ko
-m ;.X" Be-'
,,,
N(V,.,,,=-,
--AlV. ) =.
,,-;· "
°'
a
- 53 At the surface,
A'
-o
si;ccMA
· ~i'·* I0
4
1
(A výA,A
•"
4:- A
[2,2A
A
AOwt~~ij
At the surface pressure center (0,O,0),
A
=0 j
A,,c - A, X(,
Hence according to
(4-14)
AAto
po. A"tmA )ýjG(7-3)
-o
Further simplification of (7-3) depends upon the beand its derivatives on the
havior of the functionM
axis.
According to
'A r
I
AA*.
(7-1),
V"oA
IMLet
(7-4)
Then
(7-5)
The positive root of (7-5)
is
/AA = - V 4- VIL + 2,V
-54
iLet
1
In terms of /
,Ap
Then
9
,
/ A,
"
)
Next,
expressions for
V
,
y*
V~'~)L~`
+
/411 0 %%/A
V Y,
, and
terms of the pressure field are obtained.
(
))
in
The expressions
are valid in all vertically isothermal models.
If a curve
is defined by the equation
F (x,y) = constant ,
the radius of curvature r, which is the reciprocal of the
curvature, is given (cf. Berry et al., 1945) by
(F
r =
F
xx F y
+ F
y
x
- 2F F F
xy x y
+ F
F
yy x
If the curve is an isobar on a horizontal surface, A
stant along the curve.
= con-
The radius of' curvature of an isobar
is therefore given, in tensor notation, by
a:·\
- 55 -
(As. Ao,) -'*
Also,
according
Ao
Therefore
to (3-21),
8:
g
+y)
( A., A.,)
/
8o
(7-6)
A.AA, A-s
The sign is to be chosen so that V > o
is
cyclonic.
Differentiation shows that
(v"v,) =
B, B,X,
(Ay Ay) a
A+y
Ar Av
- B23
,
if the curvature
( A,AsA
Ai)
A- As A--
,It
Y
(Ay AY)p (A3 S~v
A , A,
(A Ay)
A•A
(AyA, As ),
AAcA5-
(AvAAc•),(AyAsAA•;)
(AI A,A.)
56-
-
When A I'slivev by (6-3)
Aa A)(. - , X,
A.,
2
A, X, * AXx,.X -8ON..
AAo = A
)(,YX-A,18, (X,
AA,. A2 R' A., A, A-,P
while (
X
= As (AVAy )p
+
)(,KX.)2
+1,
2"
2'
(Av Ay)
+ A ,Y
*
B; ו, X,,
2a
- A B•a,
+ (-A3%,+ A, 13';%z
Thus V
X,,
4
8,
3A,*2
SA(A.A
(A, AsA),
that .
,
+- (-A"
8a,2 -A,1312
2) Xit
is the quotient of two quadratic expressions in z,
,)
is
, m
, and
somewhat tmore complicated.
."
The model is therefore
and the expression for A
integrable.
Along the z-axis,
however,
2
AeA.A = 83
z
A A AA A = A(A
(A, A, AYr
follows
are in general complicated irra-
tional algebraic functions of
not polynomial,
It
A •)
=,A, (AyAY),
j
is not readily
- 57 -
(A
ýAs A-ys),O
= A, (AyAy),
t AR Xea.
8,-z+ A38,
+ A(-tA
C AV)
2)
Therefore, along the z-axis,
S= 11 .r8
8. Al
"v, '
8, XI
(V" 9)wk
. (8' 8,
- 8." ) XX'
, a,- A As
BRass'" -A,"'A,) (,y X.O
-+ a.(A,
So, along the z-axis,
,(
v' ) Aa = o
e
A
or-
AAo
s
PI(- At8,.81' +3a1, Ar.Aa'
P/ A' ,
a)
(v`'y, Asa
/A,
A,
-,18,
,13, 618
A
X
•.
The above expressions make if possible to evaluate the
integral
-(qILI)
in (7-3).
A1
Thus
so
-A
S
I
A
l-
r3B, AAJX,7
__
r"
j~M
QI
" ::.
- 58 -
C
- ,"A,
) ý'c'
14AlLBB;B
Bb, x.,,
Comparison with (6-5) shows that
(+
-3AlxAI) U<.o
(7-7)
Therefore the cyclone is steered.
The speed of the cyclone is now considered.
The linear
cyclone model, defined by the pressure field (6-6), is considered first.
/4
In this case (7-7) becomes
Lot
and (6-5) becomes
Equation (7-6) reduces to
At (0,0,0), with the numerical values (6-13) and (6-14),
ý
= -
h-l
h
= 0.732
/4%= -1+2/1
Therefore
U(x)
V(x)
= 0.155
143 km per hour
=
-
U(x) =
104 km per hour
22 km per hour
59 -
-
The cyclone therefore moves eastward with a speed comparable to the speeds of actual cyclones.
models,
then,
In linear cyclone
a gradient wind field seems to be more comcJ(,) .
patible with the pressure field than a wind field
and /l
It should be noted that V , and hence also /
, are
Therefore a linear cyclone model with a
independent of z.
gradient wind field is a polynomial model.
The above results show that it is unnecessary to assume
any deviation from the gradient wind to account for the fact
that cyclones move.
In actual cyclones, however, signifi-
cant deviations from the gradient wind are undoubtedly
present.
Expression (7-7) is
models.
not limited to linear cyclone
A pressure field which closely resembles the linear
field (6-6) for small values of R, but which is not linear,
is given by
(l
)
-
(-I
7
A-
)
(7-8)
The pressure field (7-8) appears to be more realistic than
the field (6-6), because both the surface pressure and the
temperature remain within reasonable limits as
power series expansion of (7-8) is
4A R
(~+ko
3t
'0"".
4;0
+1
-4 co
.
The
----- -ii-
7
r
- 60 -
So at (0,0,0),
0
,
AA
= a
Bo = b
,
B
= kab
A
As R-, o
, p-lO000
,
A
= -a & m
B,
= 0
at the surface, so a reasonable
value for m is 0.04.
Thus
and
A,
Cae_)
A,
= -m
= -25
,
= 76/jU(x) = 1680 km per hour
In contrast to the linear cyclone, the cyclone defined
by (7-8) moves with a speed much greater than the speed of
actual cyclones.
It must be concluded that except in spe-
cial models, such as linear cyclone models, a gradient wind
field is incompatible with the pressure field.
For the pur-
pose of computing pressure changes, the gradient wind is
usually not a sufficiently close approximation to the actual
wind.
In this connection it should be noted that in a linear
cyclone model the isobars in any horizontal plane are concentric circles.
In the more general model defined by (6-3)
the isobars are usually circular only at the surface.
In
the pressure field (7-8), for example, at sufficiently high
levels some of the isobars are curved anticyclonically a
short distance from the z-axis, since for x 4
sign with elevation.
0, r changes
-
61
-
It is beyond the scope of this study to find suitable
val-ues of the wind when the isobars are neither straight nor
circular.
jected.
The geostrophic
and gradient
winds have been re-
In the remaining chapters only 1inear- cyclone models
are studied.
- 62 VIII. DEVIATION FROM THE GRADIENT WIND FIELD
The results of Chapter VII show that in a linear cyclone model with a gradient wind field, the surface pressure
center moves with a velocity resembling the velocities of
actual cyclones.
But nothing is said in Chapter VII about
the wind center.
Experience with synoptic weather maps
indicates that in actual cyclones the pressure and wind
centers move with approximately equal velocities.
It is evident that a surface wind center must be stationary at any instant when it is also a pressure center.
At such a point
A o=
Step [1,12]
V
,
V
) = 0
V(Z) = 0
shows that
'
=0
whence, according to (4-16)
,
Therefore, if the pressure and wind fields are compatible,
the surface pressure and wind centers cannot coincide.
The
distance between the centers is probable small, since frequently two distinct centers are not observable on synoptic
weather maps.
In a theoretical study, Hesselberg (1915) found that
the angle, measured counterclockwise from the direction of
motion of the pressure center to the direction of the wind
I:ii
- 63 -
center fron
from the
the ~ressure
pressure center,
center, should
should lie
lie between
between 900
900
center
and 1800
180O
and
in
the Northern
Northern Hemisphere.
Hemisphere.
in the
In ti.ie
the absence
absence of
of
In
friction, the
the ani7le
anle should
should equal
equal 900
900.
friction,
Hesselberg
Hesselberg (1915)
(1915)
then studied
studied 55
56 cyclones
cyclones appearing
appearing in
in the
the United
United States
then
States
Weather Bureau
Bureau maps
maps for
for 1906.
Weat~er
lg0~.
His
findings appeared
appeared to
to
nis findin~s
be verified,
verified.
.be
If aa cyclone
cyclone is
is moving
moving eastward,
eastward, the
the surface
If
surface wind
wind
center may
may be
at some
some point
center
be expected
expected to
to lie
lie
at
point
of the
the surface
pressure center
center (0,0,0).
(0,0,0).
of
surface pressure
A = ~ a'R"
a R
AE
--
(0,a
,O) north
(O, crO)
north
If
If
(l+kay), Rz z
(l+kzy)
the gradient
gradient wind
wind field
field is
given by
by
the
is given
dL
~,,,
o~
1rb(~+r)~'(-~X
tkbX**~)
wind field
field which
which resembles
resembles the
the gradient
gradient wind
wind field
field rather
rather
kA wind
closely
which has
has aa wind
(0,6
closely but
but whic~i
wind center
center at
at
(0,6,0j ,0)
~(~
Y -
o(
is given
given by
is
by
X -+
X, +
kf, X,1
+~-aX,.
+kLX,,L)
(8-1)
(s-i)
The present
present chapter
chapter deals
cyclone
defined
The
deals with
with linear
linear
cyclone models
Inodels defined
by (6-~)
(6-6) and
(8-1).
by
and (8-1).
computed.
are computed,
Surface values
values of
of
Surface
It is
is then possible
possible to
to
It
exists aa value
of
exists
val~e
of
rQ` for
for which
which
Since
Since
Since IA
is
Since
r
is
y
h~l(
C,(.)
Ccr,
and
and WaI(a
\FJI(,,
determine whether
deterrrine
whether there
there
W•
I(a)
~J I(a)
( i+Y)
+)
'+Y)
independent of
of z,
independent
z,
the model
model isis
the
polynomial.
polynomial.
- 64
Computations follow.
IA,ý7
= - t- ((+
[1,2]
Aa
[1,3]
A.' A.
[1,43
V
(CX- - kIM
CL
[1,810
"c'a (l+
Y
- -
11,6
[1,7]
Y)
4 t(-/ * A)
I./A, I
-, ( V,,),, +A,V,,,
Vt
--= A , ,
[1,91
J (=xUV9 = ),
[1),.o]
V,,, = I
)
Y)'
II+
(+ Y) Ik -
1o'-2'
~~~'
kC7L
xI (ctt'(rkaG ~
(/-'X-
1"/4-RMKAk
-1
[1,'11
A' = -
(+ Y)
x = 0 .
At the wind center at any le-
At such a point,
1,12
V
Lo')dh
IL
At the pressure center at an'
At such a point,
+.
X <...
2)
- 65 -
[2,)1
A%
I
I Y1
/a,I+'
At the surface pressure center,
o.? X ,
Av,,
z -sI
AI
so
Cut,=
OL
('M,k +
~1 +j
kI a a-)
(()
(8-2)
ci,
since
At the surface wind center
:- - I"
Vista
V(
k
I+Y-I
I
,
~"k(·t
(8-3)
In (8-2.),/
and 1,
are evaluat
A is evaluated at (0, G-,0)
.
small, ,M(0,-,0) may be replac
ing the value of
Wi t .%
apprec
are approximately equal when at
The solution for
- , in
terms o
while in (8-3),
- 66 -
S%
Thus
/4(
6
C I(VL./)~
(8-4)
With the numerical values of the previous chapter,
UtJc = 143
=
•CIY)
km per hour
30.12 km per hour
The distance between the centers is
P- = 61,9
km
So at (0,~6,0)
r = 1.0031 ,
r
J,(,)
= 0.7325 ,
=
30.10 km per hour .
For practical purposes,
- C, .
.
AJ,, the pressure
At
center,
/
In
(o,o,o)
this model,
= 8.1 km per hour
therefore, the cyclone is steered.
The
pressure and wind centers move together at a speed consistent
with speeds of actual cyclones.
centers is less thi
an
The distance between the
the usual distance betwveen adjacent
stations on a synoptic weather map,
and the wind speed at
- 67 the pressure center is small enough to be obscured by local
fluctuations.
The model seems reasonably satisfactory.
The isobaric mean wind V()
now considered.
over the pressure center is
Since
V
'ý4 +
a-
.j
It follows that
U
I(
Expression (8-5)
.
)
(8-5)
states that the speed of the cyclone
equals the speed of the isobaric mean geostrophic wind minus
the speed of the mean wind.
This result contradicts state-
ments sometimes heard to the effect that a cyclone moves
with the speed of the upper level winds, or of the upper
level geostrophic winds.
Expression (8-4) states that the speed of the cyclone
varies linearly with the isobaric mean geostrophic wind,
and also in a more complicated manner with the curvature of
the surface pressure profile.
Values of significant quan-
tities, corresponding to various curvatures of the surface
pressure profile, appear in Table I.
is identical to that of (6-6).
The temperature field
F·
68 -
-
TABLE I
Significant quantities in linear cyclone models where
k x "0)
and where
2
waJ I
=0.99x10 sec ,
•)
1ka
1
=, ka-20000
k-ni
R (km)
400
600
800
1000
a (kmn')
1/2000
1/3000
1/4000
1/5000
0.10
0.15
0.20
0.25
2.00
0.89
0.50
0.32
0.25
0.56
1.00
1.56
0.50
0.64
0.73
0.80
6-
U(x)
143
(km/hr)
95
Vx)
(x) (km/hr)
CII+)
62
67
(km)
14it3
143
143
105
113
119
(km/hr)
V(x)(0,0,0)
(km/hr)
R = distance from surface pressure center (1000 millibars)
to 1020.2 millibar isobar.
- 69 The above figures do not apply to the general cyclone,
and they apply to linear cyclones only when the wind field
is defined by (8-1).
But they at least suggest a physical
explanation of the empirical results obtained by Austin
(1947).
The steering of cyclones may be a result of a tend-
ency of the pressure and wind fields to be "balanced",
in
the sense that the pressure field and the gradient wind
field are balanced in a linear cyclone model.
This study
does not attempt to explain vihy the pressure and wind fields
are balanced.
The lack of a relation between the speed of
a cyclone center and the speed of the upper winds is a result of the dependence of cyclone speeds upon additional
factors.
It is conceivable that a study based on synoptic
maps might yield an empirical relation between the speed of
a cyclone,
the upper level geostrophic wind speed,
and some
third quantity, such as the curvature of the surface pressure profile.
The model defined by (6-6) and (8-1) is admittedly not
entirely satisfactory.
Some of its faults have been brought
out in the discussion of linear models in Chapter VI.
An-
other fault is that the pressure and wind centers at upper
levels move with decidedly different velocities, even though
the surface centers move together.
Nevertheless, it is be-
lieved that many features of cyclone motion are correctly
described by this model.
- 70 IX.
LINEAR CYCLONE COEFFICIENTS
The choice of the wind field (8-1) in Chapter VIII,
and the conclusions drawn from the study of the model, are
based upon a consideration of the first time derivatives of
A and V( ).
A complete study of the model requires consid-
eration of higher time derivatives, since the power series
for A and V,)
are not well represented by linear functions
The complexity of the expression (6-11) for A'
of t.
sug-
gests that it may be difficult to obtain these derivatives.
It is possible to simplify expression (6-11) by an
approximation.
tion that k is
The approximation is based upon the assumpa small quantity.
Although k is a constant in any particular model, it
may assume different values in different models.
(6-11)
is treated as a general expression for A
When
in a number
of different linear cyclone models, it becomes a function
not only of x, y, and z, but also of
k, a, b, h, k, and c.
It is therefore possible to expand (6-11) in a power
series in k, with coefficients containing the remaining
quantities used to define the model.
An approximate expres-
sion for A" consists of the constant and the linear terms
of the expansion of A"' in a power series in k. Thus,
approximately,
- 71 -
(-2+8kay)
cl h
+(-l+2kay)a R
+(-2k+(2+6K)kay)
bz
A =•
(9-1)
(-2+4kay)
+ (c-1)h
+(-l+kay)a% R'
+\( -2+(1+2K)kay) bz/
From (9-1), the approximate expression
C()= - Iak(8lc•h+4(c-l)h)
(9-2)
for the acceleration of the surface pressure center is obtained.
Expression (9-2) may be treated as an approximation
for (6-12).
An estimate of the accuracy of the approximations may
be obtained from a study of Table II.
c
=
For the extreme values
0 and c = 1 corresponding to the null and geostrophic
wind fields, the error introduced by the approximation is
small compared to the quantity under consideration.
For
such wind fields, the approximation appears to be justified.
For intermediate values of c,
when the quantity being com-
puted is small, the relative error involved is of little
significance, and the absolute value of the error must be
considered.
- 72 TABLE II
Exact and approximate values of
and
at the origin, for linear cyclones with wind fields cU(4)
(h= I
, k=
)
0.000
0.500
0.18
0.732
1.000
A/i ~(exact)
1.040
0.385
0.191
-0.011
-0.540
• (approx.)
AI/x
1.000
0.375
0.191
-0.040
-0.010
(exact)
2.120
0.520
(approx,)
2.000
0.500
-0.120
-0.020
error
C_1 ('eV)/
.
h
/1**- k
C
error
0
-0.015
0
'0.015
0
-0.500
+0.011
+0.040
-0.590
-2.160
-0.536
-2.000
+0.054
+0.160
When the pressure is approximately 1000 millibars, a
change of 0.001 in A means a pressure change of about one
millibar.
The error +0.011 in Al"/A
corresponding to
c = 0.732 therefore means that the approximation causes an
error of about 5.5 millibars in the power series expansion
for A in t, whien t t = 1, that is, when t is about three
hours.
Such an error is so large that the use of the ap-
proximation for computing pressure changes is not justified.
The numerical value of a:` K in Chapter VI is 800 kilo-
meters.
The error +0.015 in C•/)/ I 0 "'k
corresponding
- 73 to c = 0.618 therefore means that the approximation causes
an error of about 6 kilometers in the position of the cyclone center when t is
tA
~r
thcl
l\f~
unjustified.
It
~h
about three hours.
~
f~~
Such an error is
~
hC+1~~~~-l\~:*,.C:
rr
~f
u-
should be observed that the principal error
in the computation of A l is common to every point, so that
only minor errors affect the pressure pattern relative to a
given point.
Since the approximation is made by omitting terms of
second and higher degrees in k, the errors are much smaller
when k is small.
The above errors of 5.5 millibars and 6
kilometers become errors of 1.4 millibars and 0.8 kilometers
if the value of k is changed to
.
It may be concluded
that the approximation is always justified if k is sufficiently small.
In addition to simplifying the computation, the approximation yields an expression (9-1) for A" which has the
same form as the expression (6-6) for A.
Both expressions
are of the form
K
+ KaR a'
where K,,
+ K& bz + kay(K. + K a Ra + Ksbz)
Ka,
---
are dimensionless constants.
,
(9-3)
On the other
hand, the exact expression (o-11) for A" involves terms in
a2R 2 bz and b1 z.
This observation suggests the possibility
of obtaining a general expression resembling (9-3) for A
by means of a similar approximation.
- 74 When the model is defined by (6-6) and (6-7), approxiand V (
mate general expressions for A
if the quantity computed in each step
may be obtained
[n,j]
is replaced
by the constant and linear termis of the expansion of that
quantity in a power series in k.
The expressions are estab-
lished by mathematical induction.
Assume that for i = 0, ---
, n-l,
there exists a set of
dimensionless constants
E
V
Q
s
H
'''3
¾
each defined for a finite number of values of j, such that
the following eight expressions, nu-mbered (9-4) to (9-11),
are true:
(Expressions obtained by steps
with the assumptions (9-4) to (9-11).
In,jl
are listed
Each expression
follows directly from the one or two assumptions just stated.)
A
~i(E)26
E
+A! &(E
( •)=
a
+
E )A
RE
+ KX (E
R1E-1 +A-A
E;S
- 75 -
(Al A .);
+,
X, (
F)
+
X,(F +
+dR'7R'FA+~czF~)
( - s)
~sXS(F;
(
V(al
+
- (5.))l
[2i+1, 4
[2i+1,5]
)
Q:6
X'( QL
.-X
.,c)
(4-6)
(Q6
k
"'l'B
- a o~ R'q, i Az
k
444;)
V(,,•, -
X-+
,(Qdh
+ko~q:
XKO
(G.
[2i+1,61
...V~sI
V(,,
(.) I
u (c
(VI.
+
+
A
A . Vt'•,,IV(),
I 1
[2i+1,10]
(;,dh)
V(I)
=(,;,,
I
-~>
(~)I
a
2
4
K.-,';
I
kht
jA*
[2i+1,91
t'
X; X,, Q,+
5
~'
)
iS)
O&a.
PR
XaAL4~
R6
LX(P'qs
dL
+ 1;
I
LxtgxQ~
I
-
q(2
-qp;
RZ G
+ ~z
+D-r)R1)
G;\
81
- 76 -
A%.
i) *.
L2i+2, 1"
i+
A (.b.,L
k+)
12i+2,2]
·-" X,~ G-~
X,,(c; +.
AE +
-(3·"tl)l
R' •0'e
, - Ca;))
(t
-X;
G
-1 ++ c R + +, 2
(A; A.)
(>+r'
A
h
ZX~
k
~iI
't
X,< (Hi + ' R'H
V
114t+%
-t
+,
*)
*
4+
1A
a.
Jt,:)
f
~~25*(c(-1
2 i+2, 41
ULLl)
Ai
[2i+2, 51
L
t.(k) S.
&;
B
Vot) a
+C,
( +41)
i
V(%ca
12i+2,6]
(V
(a441)
()la;
= (o.-,+))1
(36iL
*)
+A2VtI )l
tI
~o
=(zit·I)!1Z"+i
X. )Xs
,- xlx
1
i&
Si
SA (3 Sss + s
:+kq ( , .) (,-,,)
(-r ITr
Ia (T'ý -tT-4)
- 77 -
[2i+2,81
-- (1·t\,
(
%'AT T
!a
* k
TG6
V(26;
[2i+2,91
(a;tl)!
Vt) IL
B
T))
2i+2,
101
,
V
(
/h~S
6+I)
v(4)
+
(~I;+I)!R
=
+(-s-S)TT)
42r
In the following equations, much repetitious writing
may be avoided if each expression of the form
z EjFr
is replaced by the abbreviated expression
The number in square brackets is the sum of the superscripts of the two factors in each product, and the limits
of summation are the limits of the superscript of the first
factor.
The equation in
xA V,-
step
3,6--
L2n,ll
may be written
(VW)
(,.ý.-I-.)
C, Vc
) A
(.&4)
+ A- Vc:
(~c~l~~i)
- 78 Substitution of expressions appearing in the inductive
assumptions shows that assumption (9-4) is
verified for
i = n, and that
El
_•
r
O
'
T LC~
rrc~)
,-.-,] (-• E, S.o
E-,-, E3(4s
0-
b
(iK)T,)
-T^(at
+ 4--,
f~1
I
So -E SI -CQ4dG
)
I--- (-- E,S.- E,S- ?Es,S,) -a6
0I~
+E(E
(s ,6- (I- X)T,) - East
c.
3 SS + S,- (I-T-,)
x)
The equation in step
(A-A.
Az A> -
't~1AI
Assumption (9-5)
[2n+1,3]
- Q. G-
may be written
.C, Az(A;A ,)
C A)
A, (A A)
is verified for i
=
n, and
-
- 79 -
+ Xj E3 F.
F,= E7 -EC*
-+E vi-E,
~c(EFs,+ E,
-- E. F,
+EE-"
l
E,F,
1
-E; -ji
t~
E3 F, .
The equation in step
[2n,12]
F,
-I
-
F.- E, F,
Vc<o
-+
z.C6
V(T
1
may be written
(A;' A
V,
MI%)
SV() V.i-)
"
Assumption (9-6) is verified for i = n, and
q7
(S,
o
H, - --. ] (S
o
s-
3h 5,4
s + q. 5,
+
I~5
ZiE--'] ( q6s
*+QS.
0
q> '
(s -'
Ir3
IQS +q.7, --ýQ,ý
ce--t (3
si)
(-QsS
+Q.S + 0(-Id)Q OT01)
- 80 The equation in
step
[2n+1,7
may be written
"(I-)2'~
%)-1
AILV(ol6
IILA7V
: A l ;;
S,-
Ot V
1
A- Vn) 1
A3l
-~'i-))
-
V
-
-I
> AI VWr1
*.
4
is verified for i = n,
Assumption (9-7)
+
~
-I
(-S
The equation in step
-
V1m$^
-
and
TC-i(Epqt±
,
[^-I E
I
A'
A2
c.,
P.S.
+
- E,,.,)
T
[2n+1,11
may be written
•I -
(,V,
,%.-LA)
Z2a
-c , A
)
Assumption (9-8) is verified for i = n, and
ar'.
+~~
' (-G S.)
+
i
(- Eq
E Q.)
,
~
rC~ ·
G7
~h+~
cA-.
Gt
- t -
(C~
r-- (-CSO)t
-Eaql
't
E. oQ))
[.,--, I , ( ,,- ,I-k7 T,)
E;V-(-ExQS + E1
('' ")E3
Rc+
Es9,-E3
d
r
C
rt·
Ir
- 81 The equation in step
[2n+2,31
may be written
A. - C, A" (A'A,)
(A;*"'A`V(
Sa A c) A
ri
o
Assumption (9-9) is verified for i = n, and
Afto
+
4I
H~~
?" =
-&,
*
,I:']E•H
+
S
Th,
- Gequation in ste
The equation in
V(Q)
step
'~
).A
L1w+1,121
.•.
Vlot
may be written
(A;' Aa.)
*
= ) V()
A,;0
may be written3H
)c
Assumption (9-10) is verified for i
--
I• C*
=
n, and
VI
V(K
VZX)
)
ar
- 82 -
,I
c-bl
POQ
1-
*k Fs
-q +
S
- q+
s'> )+~(
-130
O
S
(-
S.)
4
c.
(395Q5))
0
E-[ Q6, Q
o
I-S., S)4
0
&7ý ( Q
:---] (SaS.-(t-Y)SJT.)-e
ST3
Q^+WFv
-
--
k F" *"
0
The equation in step
+
L2n+2,7]
A, V-()
A
1 ry
rV
+r
S )_s+1
(St
Z
(2) Ax
4%
0
I
is verified for i = n, and
0
|
-
V .)
(Sh+**A)
(gIVE
Assumption (9-11)
may be written
)
:4(
T,~
-q (-3ss) t
4
-q ^+ F
1
TI-
+)
--,3[-5.s(O
.~F±
q[+ h F," +:
~
E S.)
~L-~J'V.
S. - E,TtsT s
.- E,(3s, +si) t Ext)
9,Q?.5 C.)
qQb)
- 83 Assumptions (9-4) to (9-11) are therefore established
for i = n.
Assumption (9-4) is true when i = 0, and com-
parison of (9-4) with (6-6) shows that
(9-12)
Assumption (9-6) is true when i = 0, provided that the
constant and linear terms of the power series expansion of
V ()
in k are of the form
(9-13)
= cU( ) , and
Expression (9-13) holds when V(,)
.-- r,0
C,
0
5
0
6
Q '.?
0
q 0
The remaining assumrtions, when i = 0, follow directly
from assumptions (9-4) and (9-6).
The set of assumptions
is therefore established by mathematical induction for all
values of n.
It is thus no longer necessary to treat ex-
pressions (9-4) to (9-11) as assumptions.
The dimensionless constants
called linear cyclone coefficients.
,
, --- will be
The symbol
to refer to any linear cyclone coefficient.
is used
3f
The subscripts
j have been chosen so that only those coefficients Y,
with
F
- 84t 3 appear in the constant terms of the power series in k.
These coefficients will be called linear cyclone coefficients
The remaining coefficients will be called
of order zero.
coefficients of order one.
The expressions for some of the linear cyclone coefficients may be simplified with the aid of (9-12).
sions for
T
,
, and T&
The expres-
may be written
LC-l E,(", - , s o
E-1((
E,(Rn-* ;,)/+ER.r
+.E--,G-,(T
Tro
--s.)- o
TheE3
thes eqain so
+firstht E'S
The first of these equations shows thiat
M
T, = aSo
for all n.
The terms containing the factor T,in the second and third equations.
S.,
The three equations may
be used to simplify the expressions for
Mp
, G^
therefore vanish
E
pt It8
and S
The simplified set of equations follows.
~~
, ET
iui
- 85 Coefficients of order zero:
(9-14)
r
E.
-C'
EaSo
0
]L
E
' :-
F.
Q~- ,
(' "
[-,-I]
5o
=-"
(a•Q . S)
(-S.s)
(5-
F*
0o3"1(9+
A0
[A-]c- 9
S.
T, =
Coefficients of order one:
E
E
M.
(9-15)
L -- ,] ( E(S.* ES.
5;
(•
-A ('c-ZO
-E,
+Q
6
(s3E, S.4 Ea S,* E, S,+ 9.
r--1((I* ) E,56c3 Ex F, *
I
0
cc7
o
- 86 -
F; --E" _c-3 E,F,
EtZeK
Ez F,
FIA
t
-
*c-
E3 F,
F
9
',WH, - •--' (3qs.+ 3q.s,)
-
-'
(3q, S.
H.
- IE--
+,,
-I
-
16 = -
Q.1 -
0
Q
(
,S +
i
cb1
E3 R,
-M+
.@-
0
C--IJ
1---3 (3 G-6,
*k C-]
(E,Q,- Ec
r.s.
so
+
I
b
+
-GG
3. -t 9. S5S
E,(q,- q,) t E.•,
5L-r
s
OW
H
I C--I·
1-
0*~
i
-(-h
-
I~t
G
1r
1;-
A,.
Q.s,1)
(-xEsR 1,- E,Q.o)
-·1E, H, - Ec- (-F. G,C
Fi
____ 1
iimim
- 87
Hi
Zc-~
E -s Fi
o
E H
c tH
- -
-
CI+
u~l
S
I
Z
~r,
1.I
S
Y.
- Tc--IJ
o
t H****
to
±
-
t
r~ I
S
IS-
+ ' - -~( , (
-
T&1
(3SS.) I
1E, T,
S-
so)
Z
~-r~ (~-yK)
(E
(-3SS-
Expressions for A, V(&),
0
c-1 (3q,SQ
~
csc~
*tr~ss-spj
SS°
S,)
q, Q.)
tO0
Ea S
and other quantities as power
series in t involve the power series
Y.
, defined as follows:
00
Y#1
**O
Y^~ (.t)
Y, 00
Co
=
-t
z
According to (3-20),
(9-4),
field at time t is given by
if
Y= E,
if
Y=
and
(9-8),
F,
Q, R,
G, H, S3,
the pressure
I:_
M
- 88 -
C
+It
Cr+
AE
&s')
E 1+A
if the initial pressure and wind fields are given by (9-12)
At points near the z-axis, the terms contain-
and (9-13).
ing k a
y Ra
and k a3 x R a
remaining terms, unless G
are much smaller than the
excessively large.
or G., is
At (0,800,0), with the numerical values (6-14),
I a a Ra
ka
= 0.02
y = 0.04
RA k a y
a
=
If the terms containing G,
E,
+
'- 0
0.0008
and E
are neglected,
+,-+L
K
,
EL ((C6,
L
E
II +
(9-16)
Expression (9-16) represents the pressure field of a
linear cyclone,
provided that
E,>o
and E
3
<
.
So
within the limits of the approximation used to obtain (9-16),
a linear cyclone with a wind field (9-13) remains a linear
- 89 cyclone as t increases, until such time as
E 3 = 0, or one of the power series Gj
or E
= O,
0 or
fails to
converge.
(9-6), and (9-10), the wind field
According to (3-21),
at time t is given by
O+
X
+
xV - =
(9-17)
+ k
Except when S,
,+
R'Q,
= 0, the wind field (9-17) is more compli-
cated than the initial wind field (9-13), since its horicontains the term
zontal divergence V(W)W
dependent of k.
2S, which is in-
The horizontal divergence of the initial
wind contains k as a factor.
The surface pressure center in (9-16) is given by
(9-18)
C-I' E
E,
The pressure at the center is given by
- 90-
A = E,
(9-19)
In view of (9-19) it is unnecessary to obtain individual
values of C.(,)
to determine the paths of the cyclones
under consideration.
Figure 1 shows the paths of linear cyclones with wind
fields cU(W)
Table I.
corresponding to values of c appearing in
The computations, based on equations (9-14),
(9-15), and (9-18),
pendix.
appear in Tables III to IX in the ap-
The numerical values (6-13) and (6-14) are used.
Each curve in Figure 1 shows that the corresponding
cyclone, initially at rest, is soon moving much more rapidly than actual cyclones move.
The complete lack of re-
semblance between these curves and the paths of actual
cyclones may be accepted as final evidence that, except
for values of c near 0.6, wind fields cU(,)
are incompat-
ible with the pressure fields of linear cyclone models.
Values of c near 0.6 must still be considered.
case c = 0.618,
The
omitted from Figure 1, is included in
Tables V, VIII, and IX.
During the first three hours the
surface pressure center moves about 12 kilometers.
This
distance has the same order of magnitude as the errors introduced by the approximation.
It can be concluded that the
cyclone is quasi-stationary, but the details of the path are
not certain.
Since quasi-stationary cyclones exist, a fur-
ther test of the compatibility of the pressure and wind
--
*
-~~~~
-----------
P
-7-71
I
1 .1
-----I-
2K
24
~~~i----
`
--
V Vi
~
~
~
I__ii*~
__
--
----~tl-~
i
i---: '-::
I
f-I!--
-
J-i
I
L4.-
.---
4--;
i
i1 :
I
V4
1.
1
t
i I
5:
I
j
1
!-
i
i
I
7
'2
Il_,_
A
-
77:
4.·
III
i_
- i
I
1
1
t
i.
r21
22
i7
. T
* 12
-t
T-
--17T-
A--i2-
-;
-
.--.
I"
- -t.
I~il :
i
---
'
I-· ?
---·-
K--
4
IL:
41
t·
· -t
crtr
o
,;i i
IT·· -I
I
:Tjf .i
I!
T1-i
f
· ·1
777
-vi
,--------II,
---
·i i
-1
i_
,·
44~
-:7·
77
t
f
i `:
t i
lit
9
17
=77
.tT
N.I-ti
44
Il if
li Lt
ffii
idti
rjrl
· ' 'i-1
i I
F~Fi
,IA
2''
77i
2L
irt
r--i
ci
ulL
r
4i
iI
1 ii'
-·C-L
POh
-j ·t:~i
t,'
cA·T1YidF
it~
4,,i
Eli
Lt 2
iqL-t-i
TI? IE IN
IiI--'
TT--
i-
I--
V:BE
i_-_L11
7 ----
i;
1t;
*--
CL
L-I--f--.-
-I
----·-
RVEi
-- F-1.7 2
-- L
-i
-A-t
I-
·-.- C --
TT
*
a. 4
-
ItI
NALUtS_ -00 _CQC4T~
O
'-K
-1--t
i.
-t-
-
;7
i
V-
--
,....
I
L.·
77
-----
I
2
_______
-.I
r
,,
L
--- i-C---i-:
I1
V
tj*
__
{-
1~
-.
-
Li
WYIN
SI____
LL-
Fi1
iii
Siii
F1TT
owfl
I-
ii
'-I-'
LLL
Il
-
91 -
fields seems desirable.
The figures in Table V show that
t = 3 hours.
E,
= 0.096 when
According to (9-19),
P
=
A
= 1101 millibars
This excessively high pressure indicates that the pressure
and wind fields are incompatible.
iT W-IbI
FNIELDS
hold for all wind fields
-15)
in k have constant and linear
le gradient wind field (7-3)
is
ie power series expansion
+>OX
hOO2
jL
of
,
and /4,
(10-1)
at the origin.
.3) shows that
-hm
=
0.
hat every linear cyclone coef.ishes unless n = 0.
Thus
____._
Fmý
__
MMMMý
- 93 equations
(9-15)
for the coefficients Y
of
order one may
be simplified to the following set, referred to collectively
as (10-2).
;.II
**^
-
5'1
-"
S 4 d"- -I,e
*^ - ' t
T4
4
(KT ,
E.
41
F,-=
1")
Ss
~4~LCG;
-
-E?
4"
=
-
E1
; E
F,
%& ((1*
4,
P.(
.3 k U%)
-
(I+
S
I
Q3
'I
)"
krrc) S;
il···-)
S
+ •He
+
,- ,•"3
SAt
sot t
H;il)
'
HhI,
- 94 -
--c
*c(*~
'5
*
~(L.
H~
~r
4
I
-
c.
G-;+
8
?
H
hi
G
c~cS
S
m
S
L
--
*
S
(+
(6 -
-a-
c
-
.
S( -.
-
3s
Q+AF
+i~rF;)
e;
.t.-
(-~-)q
+LF;)
Qs
tS
*Sr
thF-j
5 -
-
Equations (10-2) hold not only for gradient wind fields
but for all other wind fields of the form (9-13) in which
Such wind fields will be called quasi-gradient
o = -hm.
wind fields.
; is a linear
In equations (10-2) each coefficient
combination of coefficients
Y7
.
or
Examination shows
that if
-IeKI
for Y = E, F, Q, R and for all j, then
Y
;I ~ ;;?;;
for Y = G, H,
for Y = E,
series
F,
K,
T and for all j, and
S,
Y.L i
(SclSL~
|
<
(s+lt
Q, R and for all j.
)(3te
)
Therefore each power
yi converges for all values of t.
The power series £y
containing cyclone coefficients
of order zero reduce to constants when the wind field is
quasi-gradient.
Thus
EF>
El: o
qoS
- ~~Ve ,
3'
Sco
d
-t
J
- 96 -
Equations (9-17) and (9-18) therefore become
-(
(c. -
Iz
±+
Ec,))
C4 (ca + k~
+
X,
(10-3)
LCZ
sC +
-- •R~'(5,+s,) +
,, 4(S
S~.)
Equation (10-3) represents the pressure field of a
linear cyclone which is exactly like the initial pressure
field except that the pressure center is at
(10-5)
and the isotherms are parallel to the line
Gs"
+
EK -
instead of to the x-axis.
= o
The pressure and temperature
changes at the center are of the order of magnitude of the
terms neglected in the approximation.
The wind field (10-4) resembles the initial
in
that its divergence V(,))
However,
the wind
wind field
contains k as a factor.
- 97 -
at the surface pressure center is no longer parallel to the
isotherms unless
G-, Q, + E-t S
=
o
and the vertical wind shear
V ,4- = AC"' Lik (Xa QS + X,, S,)
is
no longer parallel to the isotherms unless
Qi
(s
'+
EsS,
= o
The wind center is at
St2 --S
-------
Q&
)
(10-6)
Q(.+
42
OS
The wind field (8-1) of Chapter VIII
wind field in which
Q .5=
k ( 44%
Q
Sh nm kkl
0-
0
'I
9"8
-,
kV\Am
(kI- ,, -•r
)
a quasi-gradient
Y____
__
:___
1
- 98A consideration of first time derivatives alone indicates that the wind field (8-1) is fairly satisfactory.
Consideration of higher time derivatives reveals some of
its deficiencies.
Figures 2 and 3 show the paths of centers of linear
cyclones whose wind fields are given by (7-2) and (8-1),
respectively.
The computations, based on equations (10-2),
(10-5), and (10-6), appear in Tables X, XI, and XIII in
the appendix.
The numerical values (6-13) and (6-14) are
used.
The paths of the surface pressure centers in Figures
2 and 3 resemble each other closely.
In Figure 3, although
the cyclone is steered at tne initial time, it is not steered
at other times.
For example, when t = 2 hours, the surface
pressure center is moving approximately toward the S E.
The
axis of the cyclone'at t = 2 hours, which joins the pressure
centers at different levels at this time, leans approximately to the N i W.
The cyclone would have to move 900
right of this direction, or toward the E N E,
if
to the
it
were
,and
Q
being steered.
By altering the values of
slightly,
Q
,
Q4
,
it may be possible to find a model whose wind
field resembles the field (8-1) very closely, but which continues to be steered as t increases from zero.
Since the
steering direction is 900 to the right of the temperature
gradient, the angle e
from the positive x-axis to the
r
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FIE*
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i
- ·
I~ L~-ii
_·
iwm
- 99 steering direction is given by
-A .
SAz-
.
Thus
t~b3A 2
JA_2 ý 2.a +
39
A29
-I--
A2
A2
Initially,
Azx =0
A
= -kab
FA AZi
=
A Ixat
a
= kab
kab
Go
So
at
The angle 4
from the positive x-axis to the direc-
tion of motion of the surface pressure center is given by
JA;ý
y
TC- 'M1ý
ýCO
C (,)
At the moving center,
+ Cli
C1)
p)
dt C1(o)
).
U--'---~
iim
_
-· ~-·~" ~-·-
.·· ~ --
100 -
-
Initially,
:'i ost
Cyl,)
C
If the cyclone is steered
C,(~)
~
So
Thus
.F.
-/,E
d z
Cr
If the cyclone continues to be steered as t increases,
OtN
j
whence
2 E'
:GAD 6
(10-7)
According to (10-5) and (10-6), the surface pressure and
wind centers move parallel to the x-axis with the same
speed if
S--
r G-~
(6-
Condition (10-8)
(i0-8)
is of course satisfied when the wind field
is given by (8-1).
-
-101
(10-2) show thrat
Equations
when the wrind field is
quasi-gradient,
-4kC
o
tat wtne
surfae
tlempratur
zonaladetion.(07
(1-9
If
Unes
9
changesdu
t
aiabti
sstsidapoiaeyi
109
holds,
,
nls
the sucrface
.e.
initiallystatizontary
hat
pressraure
aibter
det cen~e
this vac luediffersfromte
vmalueom
isa
I__ -_
immmý_
Iw
_
- 102 -
which occurs in the wind fields (7-2) and (8-1).
A quasi-gradient wind field which satisfies (10-10) but
which otherwise resembles the field (8-1) very closely is
defined by the equations
(10-11)
eb =
In such a wind field the values of
(,
and
0
Q
,
,
,
G
,
& are identical with thle values of these quan-
tites in the field (8-1).
Figure 4 shows the paths of centers of the cyclone
whose wind field is defined by (10-11).
The conmputations
appear in Tables XII and XIII in tnie appendix.
Again the
numerical values (6-13) and (6-14) are used.
Figure 4 shows tlhat tihe surface pressure center follows
the steering direction very closely for about three hours,
at the end of which time it becomes almost stationary.
It
then regains speed and moves approximately parallel to the
isotherms, but in the direction opposite to the steering
I
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.
4-
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Ll r
-
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....
----7---
-i-
--
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4-- !
,........
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I
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__
- 103 direction.
The surface wind center moves in a somewhat
similar path, but the chane of direction is less abrupt.
The failure of this cyclone to move in thie manner of
actual cyclones except for small values of t is
very likely
a result of the failure of linear cyclones to resemble actual cyclones except for small values of x and y.
pointed out in
Chapter III that if the initial
It
is
pressure and
wind fields are analytic functions of x, y, and z, every
feature of these fields, no matter how far from the origin,
affects some of the coefficients A
and V
at the
origin, and hence affects the values of A and V(,)
origin for sufficiently large values of t.
It
is
at the
shown in
Chapter VI that linear cyclone models must be restricted in
horizontal extent.
It
now seems desirable to require that
linear cyclone models be restricted in time.
The wind field (10-11) has been chosen to satisfy the
two conditions (10-7) and (10-8).
bitrary constants
Q.
Since there are four arit
in a quasi-gradient wind field,
should be possible to make such a wind field satisfy four
physical conditions.
For example, the vertical wind shear
may be required to remain parallel to the isotherms,
or the
pressure and wind centers at some upper level may be required to have equal initial velocities.
lWind fields sat-
isfying such conditions are not considered in this study,
since the wind field (10-11) is fairly satisfactory for
small values of t, and since the use of linear cyclone
-
104 -
models with large values of t may lead to false conclusions.
Within the time interval
-2 hours <
t
<
+2 hours,
the model defined by (6-6) and (10-11)
appears to describe
correctly many features of cyclone motion.
The velocities
of the surface pressure and wind centers are consistent with
each other and with the velocities of actual cyclones.
The
pressure does not vary at the pressure center, and the shape
of the cyclone is unchanged.
The steering properties have
of course been "forced" into the model by the choice of the
wind field.
The initial vertical velocity
is positive to the east of the y-axis, and negative to the
west, at all levels above the surface.
Such a distribution
of vertical velocity is consistent with the presence of
stormy weather ahead of a cyclone and fair weather behind,
frequently observed in actual cyclones.
The initial horizontal divergence V(),
is
given by
Thus there is convergence to the east of the y-axis and divergence to the west, at all levels.
Horizontal convergence is usually observed at low levels
in advance of actual cyclones.
The extreme value of such
-
105 -
convergence indicates the presence of horizontal divergence
It is sometimes stated that the diver-
at higher levels.
gence aloft must be great enough to overcompensate for the
low level convergence, if the cyclone is to move as it does.
In the above model there is
divergence at all.
no compensating horizontal
This observation suggests that in actual
cyclones, although compensating horizontal divergence aloft
exists, there is no necessity for overcomoensation.
The pressure fall in the model is
vection of warm air.
at the surface.
explained by the ad-
Near the origin,
According to (10-2)
0
With the numerical values used in Figure 4, the contribution
is
of a horizontal convergence to
-
QQ
= +0.268
The contribution of advection to
- •-
&6
is
= -0.374
Thus the convergence tends to produce a surface pressure
rise, but the tendency of advection to produce a surface
pressure fall is greater.
The principal deficiencies of the model are the deficiencies common to all linear cyclone models.
The main
__
~
~
1
-
fault is
106 -
the limitation of the model in
space and time.
Other faults result from the lack of variation of temperature with height, and the accompanying lack of vertical
variation of such quantities as divergence.
The numerical values of the errors in Table II suggest
the possibility that the approximation has introduced errors
into Figures 2, 3, and 4 which are large enough to make some
of the details entirely incorrect.
It should therefore be
noted that if k is decreased while the other constants of
the model are not changed,
tains its shape,
each curve in these figures re-
but decreases in size in
proportion to k.
On the other hand, the errors introduced by the approximation decrease in proportion to k
.
So all the details of
the curves are accurate for sufficiently small values of k,
even though some of them may be incorrect when k =-.
- 107 -
XI.
CONCLUSION
The preceding chapters have developed and demonstrated
a method of using the hydrodynamic and thermodynamic equations to study meteorological phenomena.
In Chapter III
these equations are solved for A, thle logarithm of trie presthe horizontal wina velocity.
sure, and V(),
of the equations are power series in t.
The solutions
The coefficients in
these power series are functions of the initial values of A
and V(,,
and their space derivatives.
This solution may
theoretically be used to study a wide variety of meteorological phenomena.
In Chapters VI to X several models are used to study
the motion of cyclones in a baroclinic atmosphere.
sions for A and V(,)
Expres-
as functions of t are special cases of
the general solutions obtained in Chapter III.
The study of
these models brings out some of the difficulties involved in
the use of the method.
The method appears to be more readily applicable to
linear cyclone models,
cyclone models.
defined in
Chapter VI,
than to other
In general the pressure and wind variations
in linear cyclones can be expected to resemble actual pressure and wind variations only for small values of t.
If a
knowledge of the behavior of cyclones throughout a short
period only,
may be used.
say two hours,
is desired, linear cyclone models
A fairly satisfactory model is defined by (6-6)
- 108 and (10-11).
Since linear cyclone models are vertically isothermal,
such important phenomena as instability in cyclones cannot
be studied.
If the wind fields are simple enough to make
the algebraic equations (9-14) and (9-15) applicable, such
phenomena as the presence of divergence above convergence
cannot be studied.
Linear cyclone models therefore give at
best an incomplete picture of cyclone behavior.
If
it
is
a reasonable twenty-four hour "forecast" is
desired,
necessary to use a model in which the surface value of
A always lies between two fixed numbers, whose difference
can hardly exceed 0.1.
(7-8).
Such a pressure field is given by
The difficulty of finding a suitable wind field to
accompany (7-8) is discuissed in Chapter VII.
wind field is entirely unsatisfactory.
wind field can be found,
Even if a conmpatible
the work required to obtain more
than one or two' time derivatives of A and V(q)
large.
A gradient
is extremely
The best hope seems to be that of obtaining a set of
equations analagous to equations (9-4) to (9-11)
to accom-
pany a pressure field resembling the field (7-8) and an associated wind field.
The method may then yield the desired
results without an excessively large amount of labor.
It is
not at all certain that the desired set of equations can be
found.
It therefore appears possible, but hardly probable,
that reasonable twenty-four hour "forecasts" can be made
from non-trivial models by the method of Chapter III, with-
__
_~
out a prohibit
It
should
is given by (7
sults can be o
plete.
Complete
only with mode
in such models
occuring in st
Vertical disco:
not occur in a
tical temperat
Al
where a, b, an
case
T
where a,,
-
b,,
therefore decr,
in the stratos,
occurs
tne
at
occurs at the
m
-
110 -
In any column where (11-1) holds,
and
It may be difficult if not impossible to find a wind field
which will make the exoressions (11-2) easy to integrate
whenever they occur.
Other non-constant vertical tempera-
ture distributions lead to similar difficulties.
The possibility of making a complete study of a meteorological entity by the method of Chapter III therefore
seems rather remote.
The above discussion has mentioned
cyclone models specifically, but the conclusions apply
equally well to models of other meteorological entities,
such as open troughs.
Even if
results are obtained by the method, their va-
lidity must be questioned in view of the nine assumptions
appearing in Chapter III.
The assumption of hydrostatic
equilibrium is discussed by 'Richardson (1922).
assumption it
is
Without this
necessary to define a model by the initial
values of the five quantities p, F , u, v, w instead of
three quantities.
It is difficult to justify any model in
which these initial values do not approximately satisfy the
hydrostatic equation.
The assumption of hydrostatic equi-
librium does not appear to lead to false conclusions.
The
-
111 -
other alteration of the equations,
term
2
e.4
A4r
from the first
made by omitting the
equation of motion,
ap-
pears to have little effect upon the equations.
The greatest restrictions upon the models are the assumptions that the changes of state are dry-adiabatic and
that the external forces vanish.
These assumptions exclude
from the models two important controlling factors of atmospheric motion.
The forces which lead to almost all atmos-
pheric motion appear to result from solar heating, which is
non-adiabatic.
On the other hand, friction appears to be
the force which prevents the atmospheric motions from assuming a different order of magnitude.
Nevertheless, it is
possible that the pressure and wind changes during a restricted period of time may be only slightly affected by the solar
heating and the friction occuring during that period.
Another important case in which the changes of state
are not dry-adiabatic occurs when the air is saturated.
Such phenomena as cyclogenesis may be related to the occurrence of evaporation and condensation, in which case they
cannot be studied by equations (3-1) to (3-4).
Derivatives
of temperature with respect to time and elevation often
show abrupt changes at points of transition from unsaturated
to saturated air.
Since it is awkward, although not impos-
sible, to represent these abrupt changes by analytic functions, the method of Chapter III is not readily applicable
to models in which both dry-adiabatic and moist-adiabatic
- 112 changes occur.
The' remaining assumiptions are frequently made in meteorology.
Somewhat different results are obtained if the
f
latitudinal variation of
is
considered.
For example, if
the initial wind field is geostrophic, the surface pressure
systems are not stationary initially, but are moving westward (cf. Haurwitz, 1941).
However, it is reasonable to
expect that results reseimbling those of the previous chapters may be obtained if
It
the wind fields are altered slightly.
should be mentioned that the conclusions based on a con-
sideration of first
tinie derivatives of A and V(,) are still
valid in the vicinity of the north pole.
The assumptions that the curvature of the earth and the
variations of e
may be neglected appear to have little ef-
fect in a restricted region,
earth's surface is
while the assumption that the
horizontal obviously limits the study to
regions free of mountains.
If this study had been made earlier in the present century, the writer would have believed that it would be worth
a few months, or even years, of somebody's time to make a
complete study of integrals of the form (11-2),
or of some
analogous form, and then to use the method of Chapter III in
a systematic study of cyclones,
or some other meteorological
entities, wvith vertical temprerature distributions represented
by (11-1), or by som:e analogous equation.
It appears at pres-
ent, however, that a complete study of the behavior of a
-
113 -
meteorological entity can be much more satisfactorily made
by means of electronic computing devices.
of such devices to atmospheric models is
On the other hand,
is
The application
recoimamended.
the method presented in
Chapter III
reconmmended for the study of certain details of meteor-
ological phenomena.
APPENDIX
- 114 -
TABLE III
Linear cyclone coefficients,
(C = 0,
h = 0.5)
1
Ej
Ei
E3
F.
QO
so
T1
H
H6
H-
wind field cU
0.000
1.000
-1.000
-1.000
0.000
-0.500
-1.000
0.500
0.500
0.144
-0.644
0
0
-1.000
-1.000
1.000
0
0
1i000
0
0
0
0
2
0.054
0.179
0.005
0.006
0.054
-0.036
0.000
-0.097
-0.027
-0.001
-0.002
-0.656
-0.277
-0.557
-0.394
-0.245
-0.352
-0.074
1.682
1.000
0.500
1.579
0.800
0.629
0.538
0.328
1.026
0.397
0.445
0.124
0.250
0
0
0.250
0.310
0.042
-0.062
0.044
0.219
0.019
-0.057
0.010
0
0
-0.167
-0.168
0
0
0
0
0
0.075
0.037
0.062
0.044
0.026
0
0
00
0.083
HS
-0.250
-0.107
-0.214
-0.500
0.083
0
-0.167
-0.083
-0.276
-0.090
-0.018
0.051
0.150
-0.075
-0.262
-0.063
0.500
0
0
0.447
0.167
0.500
0.042
0.225
0.088
0.175
0.018
2.000
1.955
1.058
0.083
- 115 TABLE IV
Linear cyclone coefficients, wind field cU(4 )
(c
=
0.500, h = 0.5)
3
n
0
1
2
E,
0.000
0.188
-0.009
-0.002
E,
Eg
1.000
-1.000
0.188
0.054
0.008
-0.004
-0.002
0.000
-1.000
-0.242
-0.017
0.003
QO
S,
T,
-0.250
-0.188
-0.375
-0.141
0.018
0.036
-0.006
0.006
0.012
0.002
0.000
0.000
Fo
-0.250
-0.020
0.008
E7
Eg
0
-1.000
-0.125
-0.081
-0.033
0.015
0.001
0.005
Fs
F(
F,
F,
1.000
0
0
1.000
0.502
0.250
0.125
0.135
0.078
0.034
0.040
-0.011
Qs
Q(
Q7
Q3
0.250
0
0
0.250
0.188
0
-0.062
0.065
0.030
0.005
0.014
-0.007
R
0
-0.125
-0.004
0.062
0.010
E(
Gi
0
0
G
0.031
0.008
Gf
0.250
0.072
0.003
HS
He
H7
Hg
0.250
0
-0.500
-0.250
0.176
-0.062
-0.323
-0.085
0.040
-0.013
-0.072
-0.007
SS
0.062
-0.050
-0.024
S4
0
0.042
0.004
SI
ýF
0.125
0.188
0.125
-0.011
0.031
-0.002
0.500
0.009
-0.046
~1
- 116 -
TABLE V
Linear cyclone coefficients, wind field cU(Y)
(c = 0.618,
1
O0
-0.009
-0.004
-0.003
0.004
0.004
0.001
0.002
0.000
-0.001
0.000
0.002
0.001
0.000
0.000
0
0
0.028
0.011
0.000
0.001
0.000
1.000
0
0
1.000
0.123
0
0
0
-0.026
0.309
0
0
0.030
-0.013
0.o96
o0.0o96
0.000
1.000
-1.000
-1.000
-0.309
-0.096
-0.191
-0.123
-0.077
0.018
0.036
0
0
-1.000
FF1
F1
Qs
F
Q7
Qt
S3
S
S,
h = 0.5)
0.028
0
0.005
0.013
-0.011
-0.005
-0.016
0
0.007
-0.004
0.309
0
0
0.013
0
0
0.309
0
-0.003
-0.002
0.027
-0.003
0.309
0
-o.618
-0.309
0.074
0
-0.147
-0.036
-0.002
-0.096
0
0.191
0.096
-0.031
0
0.040
-0.003
0.001
-0.001
-0.004
0.000
-0.044
-0.001
0.003
0.002
0.003
__ ___~
___
~
- 117 TABLE VI
lear cyclone coefficients, wind field
(c = 0.732, h = 0.5)
0.000
1.000
-1.000
-1.000
-0.366
0.000
0.000
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.268
-0.039
0
-1.000
0.134
0.144
-0.020
-0.015
1.000
0
0
1.000
-0.278
-0.268
-0.134
-0.144
0.035
0.039
0.020
0.015
0.366
-0.190
0
0
0
0.035
0
0.098
-0.003
-0.040
-0.092
0.011
0.196
-0.064
0
0
0.366
-0.098
-0.049
-0.036
0.016
0.009
0.005
0.366
-0.085
0.098
-0.016
-0.732
-0.366
0.121
0.036
-0.019
-0.005
0.366
0
0
0.014
0.087
-0.011
0
-0.045
0.005
0.268
-0.113
0.018
0.018
-0.001
0.165
-0.016
-0.268
0
-0.536
B
I
s, wind field cU
0.5)
2
n
3
E,
Ez
E3
Fo
Q.
S.
T,
0.067
O.098
0.017
-0.138
-0.098
0.051
0.102
-0.017
-0.036
-0.004
0.055
0.036
-0.020
-0.040
E,
E
Ej
-0.564
-0.386
-0.221
0.341
1.010
0.636
0.422
0.277
-0.674
-0.404
-0.311
-0.144
-0.582
-0.339
0.862
0
-0.690
0.119
1.000
-1.255
0.929
-0.500
-0.250
-0.275
0.382
0.264
0.144
-0.213
-0.157
-0.722
0.500
1.194
0.311
0.625
-0.418
-0.970
-0.174
-0.708
-0.250
0.919
-0.167
-1.042
0.130
-2.000
1.925
-1.575
FrF,
F
Fy
1.000
0
O1.000
1.000
-1.360
-1.000
0.500
0
0
0.500
-1.000
Q1
Q
Qk
0
0
0.500
Hs
H7
H,
H,
Sg
S4
S7
Sk
0.500
0
-1.000
-0.500
-0.750
0
0.500
-0.500
-0.538
0
0.500
0.080
0.763
-0.041
0.273
0.111
-0.008
0.500
-0.041
-0.058
119 -
-
TABLE VIII
urly values of
E
, and E, for linear
, G
cyclones with wind fields cU
(h = 0.5)
.000
t
0.500
.016
1.0
.067
1.5
.159
2.0
.308
2.5
3.0
1.006
1.024
1.054
1.097
1.152
1.224
1.0
1.5
2.0
2.5
3.0
.000
.001
0.000
.004
0.009
.020
0.003
0.024
0.050
0.090
2.5
3.0
1.003
1.000
0.992
0.969
1.012
1.000
1.027
1.048
1.000
1.000
1.000
1.000
0.935
0.000
0.000
0.000
-0.001
-0.002
-0.001
-0.003
-0.014
-0.021
-0.062
-0.033
-0.132
-0.005
-0.098
1.074
1.102
0.894
-0.004
-0.060
values of
t (hours)
0.5
1.0
1.5
2.0
1.000
values of G"
t (hours)
0.5
0.732
values of E-
(hours)
0.5
0.618
.033
-0.008
.714
-0.074
-0.131
.139
.346
-0.032
-0.208
-0.300
0.000
0.000
0.001
0.003
0.007
0.015
0.009
0.033
0.074
0.126
0.191
0.260
0.031
0.119
0.248
0.408
-
120 -
TABLE IX
Half hourly positions of surface pressure centers
(b2
of linear cyclones with wind fields cUo(~)
4
1
-1
1
h1 ksec -/
km
,
b=~j
km~
2'
k5
=0.99K10
0.000
-1
-3
-12
1.0
2.0
2.5
3.0
1
2
3
17
11
26
53
11g
-35~
-59
y (kilometers)
t (hours)
0.5
1.000
x (kilomleters)
t (hnours)
0.5
1.0
1.5
2.0
2.5
3.0
0.732
0.500
26
104
239
4·37
6
25
56
144·
196
-7
-26
-1
-2
-5
-11
-59
-101
-153
-208
-25
-212
-
366
X
uasi-4radient wind fields
0.732)
2
IO x
-3
0
I0
x
13
-1
-18
10
18
-9
3
0
18
-13
1
-10
13
-9
-58
38
-30
25
95
-54
52
-20
1.4
6.0
-1.6
-3
-4
0
4.4
-1.4
-2.8
1.6
-4
3
4
0
-3.6
16.2
10.2
-24.2
-1057
3
-23.6
2
-8
-6
10
O10
122-
-
TABLE XI
Linear cyclone coefficients, quasi-gradient wind fields
(h
-4'
EJ
E
E•
=
0.5, in= 0.732)
/0
x
0
528
0
0
-10000
7r31
F-
10000
0
-52E8
F7
0
10000
-7$1
2886
283
0
3660
-250
0
112
-E~59
774
-497
Q7
H1
-1057
0
34-37
100
-37
1057
-100
133
-3437
SI.
87
-1057
-387
2114,
0
-1057
4.5
-112
261
-v
17
-15
15
0f
13
-42
-30
-1
i0
-9
-17
1
-10
-9
30
25~
-54
-20
-1.0
-2
6.0
-1.6
-4
4~.4
-4
0
1.0
2
1.6
4
-3.6
13.0
-24.2
2
-7
-6
10
-23.6
i0
10.2
- 123 TABLE XII
Linear cyclone coefficients, quasi-gradient wind fields
(h = 0.5, m = 0.732)
n
0
1
2
3
E6
E,
El
0
0
-10000
-195
-362
573
87
73
19
-83
-27
-15
21
10
6
Fs
F.
F,
F,
10000
0
0
10000
-211
195
362
-573
-92
-87
-73
-19
42
83
27
15
-16
-21
-10
-6
Q,
Qt
Q,
Q,
2678
283
0
3452
211
0
-154
-665
-80
-35
77
69
53
38
-80
-37
-21
-16
34
9
RL
774
-1030
226
-170
64
G4
G7
Gj
-1057
0
3437
367
95
29
-45
-21
-14
23
13
8
-6
-4
-2
HSHI
H,
3437
1057
-6874
124
-367
-153
-35
45
49
21
-23
-29
-6
6
8
Hf
-3437
-29
14
-8
2
-618
-387
1960
284
-183
33
182
208
24
1
-40
-21
-13
-1
23
8
4
2
-7
-2
390
-159
13
s
St
S3
S,
4
- 124 TABLE XIII
itions of centers of linear cyclones
th quasi-gradient wind fields
c , a= 4 0
0
0
km , b
2
1
h= , k= , m=0.732
km,
3
4
5
6
ace pressure center, cyclone I
0
0
22
-4
42
-18
61
-36
77
-63
90
-100
ace pressure center, cyclone II
0
30
57
80
100
0
-5
-22
-47
114
-128
-82
sure center at one kilometer, cyclone II
0
17
32
42
48
49
00
94
74
ace wind center
0
31
66
62
62
60
44
3
-51
cyclone II
110
161
214
51
32
-9
ace pressure center, cyclone III
0
29
50
59
55
0
2
6
10
9
41
-2
21
-21
sure center at one kilometer, cyclone III
0
16
DO
25
21
5
101
103
103
97
-22
78
ace wind center, cyclone III
0
30
57
81
99
52
62
60
53
=0.2886,
:=0
,
=0.286,
=0.26,
q=0.0283,
=0,
Q
=0,)
,=0.2678, C=0.0283,
50268
.
=0,,
Q7
52
105
36
C
-53
8
Q(
99
-34
=0.3660
=0.3660
q
=0.3452
m
- 125 AB STAC T
A number of attempts have been made to forecast the
weather by direct application of the equations of hydrodynamics and thermodynamics.
Perhaps the most detailed of
these attempts was made by Richardson (1922).
i'ichardson
obtained a solution of the equations by replacing the partial derivatives by finite arithmetical differences.
Richardson found that his method was unsatisfactory in
practice, because the initial conditions could not be observed with sufficient accuracy.
This study presents an alternative method of applying
the hydrodynamic and thermodynamic equations.
cable not to actual meteorological
It
is
appli-
situations but to ide-
alized models of meteorological situations.
If
the chanTes of state in the atmosphere are poly-
tropic, the variations of pressure, density, and wind velocity may be expressed by a system of five differential
equations, consisting of three equations of motion, the
equation of continuity, and the physical equation.
If a
condition of hydrostatic equilibrium is assumed, these
equations may be replaced by a set of three equations in
which the dependent variables are A, the logarithm of the
pressure, and V(0), the horizontal wind velocity.
A tensor notation is
introducea to facilitate the
application of these equations.
iiME
- 126 -
On the basis of the hydrodynamic and thermodynamic
equations, a routine procedure is established for computing
the initial values of the nt h partial derivatives A
and
w~) respect to time t. If the inwith
of A and V
V (a)
itial values of A and V(,) are analytic functions of the
space variables x, y, and z, the derivatives A
and V(a)
will under suitable conditions be contained in the power
series expansions of A and V(4)
in t.
It is assumed that
the changes of state in the atmosphere are dry-adiabatic,
that there are no external forces,
and that the latitudinal
variation of the Coriolis parameter may be neglected.
The procedure may be applied to models of atmospheric
situations, defined by the initial values of A and
V 4).
It may be most conveniently applied to "oolynomial" models,
in which the temperature does not vary with height initially,
and in which V(0)
is a polynomial in z initially.
The prodedure is applied to polynomial models of circular cyclones in a baroclinic atmosphere.
The application
is least complicated in the case of "linear cyclone models",
in which the initial surface value of A is a linear function
of
OP
-
, and in which the initial value of
linear function of y.
is a
In linear cyclone models the isobars
are circular at all levels.
A linear cyclone model can re-
semble an actual cyclone only in a restricted region about
the surface pressure center.
If the initial wind field is a constant times the
;;ý
I' c
-
127 -
geostrophic wind field, the initial velocity of the surface
pressure center is zero, and in general the initial acceleration is unreasonably large.
If the initial wind field is gradient, the cyclone is
steered, i. e., the surface pressure center moves parallel
to the upper level isobars and isotherms.
The speed of the
center is consistent with the speed of actual cyclones if the
cyclone model is
linear, but the speed is
unreasonably large
if the upper level isobars are not circular.
With a wind field which deviates slightly from the gradient w;ind field, the cyclone is
steered, and the surface
pressure and wind centers move with the same velocity.
speed varies linearly with the temperature gradient,
The
and
varies in a more complicated manner with the curvature of
the surface pressure profile through the center.
The speed
differs considerably from the isobaric mean wind speed above
the center.
A possible explanation of the phenomenon of
steering is suggested.
By means of an approximation,
a set of al 'ebraic equa-
tions is obtained for the initial values of A
in
linear cyrclore
&)odeis
.ith special wind fields,
ing the geostrophic and gradient wind fields.
mation is
justified if
and V
includ-
The approxi-
the temperature gradient is
not too
large.
The paths of centers of several linear cyclones are
computed from the algebraic equations.
W;hen the wind field
- 128 Istant times the geostrophic wind field, the paths
resemblance to the paths of actual cyclones.
:n the wind field is
approximately gradient,
a lin-
Lone remains a linear cyclone, and the pressure at
;er does not vary.
Ahead of the cyclone there is
lotion and horizontal convergence.
The tendency of
rergence to produce a surface pressure rise is overthe tendency of advection to produce a surface pres1.
In certain cases the cyclone is steered for about
urs.
For large values of t,
linear cyclones do not
ike actual cyclones.
is
concluded that the use of models of atmosoheric
ns, defined by initial values of A and V(,),
and
edure for obtaining expressions for A and V( ) as
ries in t,
constitute a workable method of studying
details of meteorological phenomena,
but that the
s too complicated for a comolete study of any phe-
i
-
129 -
REFERENICES
Austin, J. M., 1947:
An empirical study of certain rules
for forecasting the movement and intensity of cyclones.
Journ. iNleteor.,
Berry, F.
4, pp. 16-20.
A., Bollay, E.,
of meteorology.
and Beers, N. R.,
1945:
Handbook
New York, ',cGraw-Hill Book Co.,
1068 pp. (see pp . 199, 203).
Eisenhart, L. P., 194 0:
An introduction to differential
geometry, with u se of the tensor calculus.
Princeton,
Princeton Univ. Press, 304 pp.
Haurwitz, D.,
Dynamic meteorology.
1941:
Hill Book Co.,
New York, MKcGraw-
365 pp. (see pp. 129-131, 163).
Hesselberg, T., 1915:
Ober die Beziehung zwischen Luftdruck
und Wind im nichtstationaren Fall.
Ver5ff.
- e ophys.
Inst. Univ. Leipzig, 2 Serie, 1, IHeft 7, pp.175-205.
James,
R. W.,
1945:
Miovement in meteorology.
Quart. Journ.
Roy. meteor. Soc., 71, pp. 74-81.
Petterssen, S.,
1940:
bieather analysis and forecastin.
New York, M4cGraw-Hill Book Co.,
505 pp. (see pp.214,
378-387).
Petterssen, S.,
1944:
Computation of winds in the free
atmosphere (unpublished).
Petterssen, S.,
variations.
pp. 56-73.
1945:
Contribution to the theory of pressure
Quart. Journ. Roy. meteor. Soc., 71,
-
Richardson,
L.
process.
Willett, H. C.,
F.,
1922:
Cambridge,
1944:
130 -
Weather prediction by numerical
Cambridge Univ. Press,
Descriptive meteorology.
Academic Press, 310 pp. (see p. 12).
236 pp.
New York,
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·
li_
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ACK NikOWVLELDGMLENT
The author wishes to express his gratitude
to Professor J. 1Mi.Austin, his advisor, for his
counsel and criticisms during the preparation of
this thesis.
Thanks are also due to Professor
V. P. Starr and Dr. A. J. Abdullah for the helpful suggestions which they offered.
-
132 -
AOGRAP
hICL
Born in West Hartford,
Conn.,
Attended Dartrmouth College,
EOTE
i:iay 23,
1917.
1934-38.
Received A. B. degree in Ii'athemstics (magna cui
lkue)
1938.
Attended Graduate School of Arts and Sciences, Harvard
University, 1938-42.
Received A. 1M.degree in Mathematics, 1940.
Awarded John harvard Fellowship, 1940-41.
Teaching Fellow in mathematics,
1941-42.
Served in Army Air Corps, 1942-46.
Attended Mleteorology Course at Milassachusetts Institute of
Technology,
1942.
Stationed at !Miassachusetts Institute of Technology, 1942-44.
Research Assistant in Meteorology, 1943-44.
Received S. i'M.degree in M'eteorology, 1943.
Stationed in Pacific Theater as Forecaster,
1944-46.
Reappointed ihesearch Assistant in IMeteorology,
1946.
Attended Graduate School at Massachusetts Institute of
Technology,
1940-48.
Awarded Richard DuPont
eriemorial Fellowship, 1946-47.