NINE CHAPTERS (XI-XIX) FROM THE BOOK ENTITLED 'DYNAMIC METEOROLOGY'

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000100220017-9
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 360
• Document Creation Date: December 22, 2016
• Document Release Date: March 23, 2012
• Sequence Number: 17
• Publication Date: February 28, 1952
• Content Type: REPORT

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STAT
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 V. A. BELINSKIY
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'
na anva3aw,so,...,..
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CHAPTER XI
VARIATIONS IN GEOSTROPHIC WIND1ITH HEIGHT
The geostrophic wind equations show that, due to altitude
variations in atmospheric density and in the pressure gradient force,
the velocity and direction of the geostrophic wind are functions of
altitude. This conclusion remains true even when the horizontal
component o1' wind velocity is constant.
In the follow rig paragraphs the regularities, to whl.ch the
variations of ieostrophic wind w_i th height are subject will be
stu.da_ecl.
Assuming that 1 = 2 ~) sin p , we write the geostrophic wind
equation as fol?aows:
1t v1p.
. y ' I)
Eliminating the density with the aid of the equation of state
we obtain:
~ ate, V ~ a/np___ a/np
J y 7__J ax T - '~ 2z c2>
Differentiating the first two equations (2) by Z , and
the last one by J/ and , ., respectively, and eliminating the com-
pound derivatives, we find:
_/vI 1 ?7f),
z: ( ' r:J ? . I y I 7 1" ~z ( 7/ J C Ti'
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precise formulas, which determine the variations of the velocity
components of the geostrophic wind with changes in altitude.
iL~r 7_
'- ?zJry . .
r r
Completing thef1ifferenti_ation and reducing by T, we obtain the
aZ r aZ Tyr ~K
Let us ascertain the condition, under which the geostrophic
wind does not change with height, that is, under what condition
u_ =
v =0
0
z ~
_z
as per (1), we obtains
It follows from (~) that, in the absence of change in the
u=_ i2:7-_)PT:
~1 -tp (fir az az ay
)v 2r
geostrophic wind with height the following condition must be satis-
: ayr. ar
?-,?.P_ar a
~K y'aZ-~a'='
This means that the isobaric surfaces must, at the same time,
be ,isothermi.c surfaces, and therefore, also isosteric surfaces, i. Ce
the air must be barotropic If this condition does not exist, the
velocity of the geostrophic wind will vary with height. Thus, the
variation of the geostrophic wind with height is a result of the baro-
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Equations (9) show that the pressure gradient force at a given
upper level can also be considered as consisting of two components:
(a) a pressure gradient force conditioned upon the pressure distri-
bution at the lower level, and (b) a pressure gradient force conditioned
upon the temperature distribution in the given atmospheric layer.
By analogy with the "thermic" and "baric" winds, we will call
the last item the "thermic" component of the pressure gradient
force, and will designate it by ( I&TY)
the first item =-- the.'tbaric'l component, which we will designate
B
upper level.
. Obviously:
is the pressure gradient force at the
Supposing first, that in the atmospheric layer, within which
the variations of the geostrophtc wind with height are being con-
sidered, the horizontal temperature gradient is absent,
that as, we wall first contemplate the variation of the ttbarictt wind
with altitude,
termined by:
7) and (8), the velocity of the "baric" wind is de-
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from where:
that is:
This means, that the direction of the 11baric" wind (determined
by the angle , which is formed by the wind with the x-axis) does
~
not change with altitude, coinciding all the time with the direction
of the geostrophic wind at the lower level, but the only thing that
changes is the absolute value of velocity.
we obtain:
Thus with the reduction of temperature with altitude (O ),.
the o the "baric" wind is, diminished (Q c < d }
velocity -
zn the isothermic layer (=0)it remains unchanged (Q = 0 ) 9 and
~
in the inversion layer ( yO and r> Q , Q >O Thus, we arrive at the
hemisphere,
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VARIATIONS OF TP ERNIO PhD ;JITH ALTITUDE
egrees per 100 kilometers
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I degree per 100 kilometers
2 degrees per 100 kilometers
1L4.6 10.0 7.8
3 degrees per 100 kilometers L3.2 - 21.9 15.0 11.7 ??9 8.7 8.1 7.8
1~.Ls 7,3 5.0 3.9 3.3 2.9 2.7 2.6 2..
72.2 36.7 2;.0 19.5 16.3 11.5 13.3
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goo
latitudes only. ? 1n the low latitudes, and, in to presence of higher
temperature gradients, even in the high latitudes, the value of the
"thermic"wind is not to be overlooked, since in this case, its value
is o.f the same order of magnitude, as the velocity of the geostrophic
;+ kilometers, can the value of the
"thermic" wind be overlooked (in the formula), and this for high
As per egpression (9) in Section 1 and expression (2) in this
section, the thermal component of the pressure gradient force is
determined by equations
x
L
7"
Equations (6) show, that the thermal components of the pressure
gradient :rorce Q & is directed along the temperature gradient'
its absolute value is in direct ratio to the absolute value of the
temperature gradient
pheric layer, as depicted below in Table 11.2, and computed from the
we obtain the following values for the variation of the ttthermictt corn-
ponent of the pressure gradient force within the 1 - kilometer atmos-?
_i
de reCs
per second, T - 273 degrees Centigrade, B. = 287 M2 Sec~2
per 100 kilometers, Qz in kilometers, and assuming that g = 10 meters
r
Expressing Q&in millibars per 100 kilometers, in degrees
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wa~~..aw:m..u..~a.+e..w.t.+?F3.1+~Ya~9~-i~~.ie+i ~a.3'd.".+~W~Ya.e..1:._~,..uYG~:.k.,fl.,r..cY).'~3f`.C"a/-,~KY,YA~.._w..?'1,~w.tX* ~' _ i _. .-.
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millibars
`7 RIATIONO IN TH1L TFiE t L COMPONENT OF THE
1 degree per 100 kilometers
G.46
0.41
0.32
0.23
0.14
0.09-
0.05
2 degrees per 100 kilometers
0.9
0.8
0.6
0.5
0.3
0b2
0.1
3 degrees per 100 kilometers
1.4
1.2
0.9
0.7
C.h
0.3
0.1
5 degrees per 100 kilometers
2.3
2.0
1.6
1.1
0.7
0.5
0.2
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pressure gradient force is ever more increasing, in conformance
With increase in altitude, the thermal component of the
with which there is a steeper inclination of the isobaric surface,
as determined by equation
It follows from (8) that the isobaric surfaces, with reference
to the horizon, will always have a considerable smaller angle of
inclination, than the corresponding isothermic surfaces. These angles
~rrherePr is the angle of inclination of the isothermic surface, and
zT is the temperature differential at the boundaries of the contem-
plated atmospheric layer.
Section 3.
Variation in Geostrophic ~Ji nd with Height in. Relation to
the a~utual ~ispos~:tian of the ire; sure Gradient Force and the dermal
In the presence of the ressure gradient force and the thermal
gradient, the variations of the geostrophic wind with height may be
diverse, with relation to the mutual ciisposition of the above gradients/,
or in other ?ords, with relation to the mutual disposition of the iso-
bars and isotherms.
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v.euor t;C, i. e, the geostropha..c wind below an'
a
pointed ou, above, the velocity vector. y ctor of the geostrophic
wind at an altitude, can be considered as a
product of two vectors:
.
,
vec for J c , 1.
the nthermic'wind, wh_~eh is predicated on the mean -
therma]. gradient
*
'in
the conteznlaced atmos)he
~ rjc layer
vector C0 is determa.ned in magnitude and direction bY the
pressure gradient force vector (j
9
and vector c~ is ci.etermined in magnitude and direction by the the
y rural
gradient vector / , it being known that
vec for
Finally vector C is determined by the pressure gradient
force
Thus, the presure gradient force vector ( can be com:;
1 u ted
gradient forced accordance with (7) of Section 2 which s
~" ecla,als
which is
mul t~.pl i ed by --? times
, nd the therri~al component of the pressure
a
as the vector slum of the pressure gradient force
P
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The greater the thickness of the contemplated atmospheric
layer, the less the effect of the first items in equations (1) and
and the more the effect of the second items. In confornian
c e with
this, the cL.rect?on of vector C, with altitude, approximates more
and
more the direction C , and the di.rec Lion of vector apZ ,
roxzla to s
the direction of vector f . Thus, t~rith increLse in altitude, the
direction of the geostrophic wind apYroxirnates the. direction of the
isotherm, and the direction of the pressure gradient force approximates
the direction of the thermal gradient.
Figures 98 and 99 are a diagranurlatic presentation of the
variation, with altitude, of the velocity of the geostro hic
b p ~.nd and
the pressure gradient force. 'they show clearly that the altitud.inal
changes in the velocities of the wind tiLll vary, with relation to the
disposition of the pressure gradient force and the temperature gradient.
Figure 98. The geostrophic wind, with higher altitudes
more
and more approximates the isotherm
altitude.
Figure 99'. Variations in the pressure gradi..ent force,
with
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Above, we discussed the altitudinal variation of the ttthermict'
wind vector4c This variation determines the altitudinal change
r
in the geostrophic wind. For practical purposes, the wind is con-
with relation to the angle O between the pressure and the thermal
gradients. .
variations in the modulus of velocity and the direction of the wind,
the angle determining the direction of the air current, tii!e will
derive the corresponding formulas, which express the altitudl.nal
scalar magnitudes, the velocity raodulus of the motion of the air and
sidered in meteorology not as a vector, but as an aggregate of the two
vector sum ~a 1,LIC.1.. , Let us designate the angle formed by
velocity c with the x - axis (the isobar at the lower level) as
It is obvious that
the angle 4lwr r with the isobare The wind velocity c will be the
dire
Let us the x - axis along the isobar, and the y - axis
along the pressure gradient force at the lower level (Figure 100).
The ''thermic" wind L~C will be directed along the isotherm, forn~ng
Figure 100. Attitudinal var:i..ation in .the velocity and direction
of the geostrophic wind, with relation to the angle between the
pressure and temperature gradients
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Differentiating equation (6), we derive:.
Q u- = 4 c was B
effect of to pressure component of the gradient,
7L:: ZL
therefore,
whe,ice
,Q c szBtc cos ELI
B.
Since we are disregarding the change in the ti_nd under the
L Dr
C cos 6 - c2.c/rt L:1e = = 4 c caS O)
T (9)
, after solving (9) for and L 6
. C .: 0 c:os (G(
or, taking into consideration (3)
1_ .._,_ .._ tr cosi' -,
40
4 ,QT c
Equations (ii) show that the altitudinal change in the
geostrophic wand depends substantially upon the angle (O ),
formed between the i.sotherins and the isobars.
then (QC - )
is C. when the isobars coincide with tine isotherms, then
uz
there as no rotation of the wind with higher altitude. z 0
and
also in the case when (d) = 180 degrees. If (d- ~ ) = ? 90 degrees,.
L1 c w
- 0, i, e. the wand changes direction only, without a change in
Liz .
the magnitude of its velocity, it being the case, that when (OC`-~ )30,
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0,
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there is left-hand rotation of the wind,. and when (' 6) 0 there
is right-hand rotation. Generally, the smaller the angle (d? )
the more rapid the change in the magnitude of the wind velocity, and
the slower the change in its direction. h~hen the thickness of the
contemplated atmospheric layer is not too great, the angle (- ) can
be considered as the angle that j-s formed by the thermal gradient with
the p~ e;sure gradient force at the lower level. hen the thickness of
the contemplated atmospheric layer is considerable, the angle 0
)
is an angle that is formed by the pressure gradient force with the
thermal gradient at the center level of the atmospheric layer.
y y
the same token, the magnitude of C, in this case, is to be its
magnitude at the center level of the contemplated atmospheric layer.
Table L3 characterizes the altitudinal variations in the
velocity modulus of the geostrophic wind for various values of the
angle (-e )
~ee page 20 for Table ) 7
The variations in the direction of the wind within a l
kilometer atmospheric layer, for various values of (-e) and C
are given in Table L2.I.
ee page 21 for Table L7
The computations were made on the assumption that -
O - 60 degrees,
T = 273 degrees Centigrade, = degrees per 100
kilometers, by the
formula 4 __ (-6)
83 ? / ,
A
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and r . The Table is c om~.uted. for T = 273 degrees
Centigrade; Cf 60 degrees;
~. = 1 kilometer, by she forrriula --
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TABLE 143
G~+~ziA?7onV ac v~~ocirr 4opv~.u5 o~ GEcasrRMic W'rvD
10 20 34 L.0 50 60 70 80 90
1 degree per 100
kilometers 2.9 2.9 2.7 2.5 2.2 1.9 1.1~. 1.0 0.5 0.0
2 degrees per 100
kilometers 5.8 5.7 5.L. 5?Q 4.1~ 3.7 2.9 2.0 1.0 0.0
N
O
3 degrees per 100
kilometers 8.7 8.6 8.1 7.5 6.6 5.6 1t,3 3.0 1.5 0.0
degrees per 100
kilometers 1~.5 11.3 13.6 12.6 11.1 9.3 7.2 5.0 2.5 Az0
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.20 r sec
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.CHANGE IN D I PJI a'ION OF GEOSTROPHIC y iNTD
29 Jl.1 59 63 72 78 82 83
3 6 8 11 13 1!t 1Z 16 - 17
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The above Table shows that in the case of high wind velocities
(i., e. in the presence of great preEsure gradient forces), the change
of direction in the wind, with higher altitudes, is small. Inversely,
in the case of mild winds, the change in. direction, with higher
altitude, is sharp. or example, within the 1 - kilometer air layer,
a wind with a velocity of 20 meters per second, will tern only Li.
degrees, while a 3 meter per second wind, under the same conditions.,
will turn 28 degrees. Low-velocity Uainds, irrespective of their
direction at lower level, will rapidly approximate the isotherm at
higher altitude.. in the presence of great velocities, the change in
direction is slower. However,
at great altitudes, in a homogeneous
air mass, the air still has a tendency to move in the direction of
the isotherms, since accretion of velocity in this case is great, but
the direction coincides with the isotherm. ince the actual wind at
great atmos
:ieri.c altitudes frequently is not much distinct from the
geostrophic wind, the predominance of the general westerly current,
which is observed in the troposphere of the temperate latitudes,
may be explained by the diminution in tomperatiire from the equator to
the pole through out the troposphere. In the stratosphere, the western
component of the c..rind diminishes with altitude, in conformance with
the inversion of the thermal gradient.
The above considerations make it l:~os?ble to reverse the order
of comput..tion: by the altitudinal change in the velocity and direction
of the wind, to determine the horizontal temperature distribution.
For example .?- acord:ing to pilot balloon observations, bhe
-22
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velocity of the id_nd at the lower level equals C0 (Figure ioi), and
at the upper level, it equals C. By determining vectorially, ~C=e- Q ,
we derive all the required values for the determination of the thermal
gradient / in the contemplate: atmospheric layer. Indeed, the
thermal gradient f' is perpendicular to 4 and directed to the
left of it (in the northern hemisphere), and its magnitude is de-
termined by the equation:
fL1z
(12)
These considerations can be utilized in the extrapolation of
synoptic charts. Given a network of pilot ballon observation points,
it is possible to get :adequately corn; fete data on the pressure and
temperature distribution beyond the cut-off of the synoptic chart.
However, in the practical application of these deductions, it
must be remembered, that all the above said relates to the geostr.ophic
wind, and therefore, it holds true only for 'homogeneous atmospheric
layers, located outside the ground turbulent zone, i. e. for atmos-
pheric altitudes in excess of 500 - 1000 meters.
Figure 101. ~etermi.nation of the horizontal temperature gradient
by the attitudinal changes in the gelstrophic wind Lean be studied from
original texg.
23
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Section ).. Variations of Temperature and Pressure in Tine with
Relaton?to the Altitudinal Change of Direction of the Geostrophc
Wind.
The variation in the temperature of. the air at a given point
can be expressed by:
a T = dT_ ?T _ aT aT
at dtf u ~y- v a y - zu ~Z
(1)
Assuming that the air.masses are shifting without a change in
dT I
temper Lure, i. C. -= 0 . Then, due i to the geostrophic
wind being horizontal:
- ? ~
---- -- -- & --j ------ C ~Gos~ /)
Converting; to finite increments, we derive {
where is the temperature variation for a unit of time, and (9)
is the angle between the thermal and the pressure gradients (or be-
tween the isobar and the isotherm). The angle is counted off from
the pressure gradient to the thermal gradient.
From (2) it can be seen that the change in temperature with time,
which is taking place at a given point, on account of the inflow of
air with a different temperature, is in direct ratio to the thermal
gradient and wind velocity, and also to the sine of the angle formed
by the wind.direction and the isotherm. The direction of the wind is
. determined by the mutual disposition of the gradients: if the thermal
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Assuming at a certain height h, the pressure Ph being constant,
we will, from the barometric equation
which is true in the case of the geostrophic wind, derive by
differentiating by t:
Ar
or, by substituting for --, as per equation (3) :
Equation (6) shows that, in the case of a right rotation of the
d9
wind with height (- 0 - v < Q
2
Southern hemisphere o > 0 t - v~ > 0
' < o fa? oc
Figure 110 shoves a diagram of the values of the geostrophio wind
at a front. We see that the discontinuity of the wind has cyclonic
characteristics both for the warm and the cold fronts. We thus
obtain a result which is in conformity withthe behavior of the iso-
bars in the vicinity of a front. It must be kept in mind, however,
that the rule derived for the isobars was derived with complete
rigidity. It is, therefore, true for all fronts without exception
while the rule for the wind was derived only after a series of
simplifkying assumptions, and is, therefore, less rigid.
In individual cases it may occur that the horizontal accelera-
tions are so significant that they will change the cyclonic rotation
of the wind at the front into an anticyclonic rotation. However,
such cases are rare and of short duration.
. Thus, any front may be detected by the cyclonic rotation of
the wind, irrespective of the circumstance as to What type of aix
mass lies to, the right of the front, and what type of air mass lies
to the left of it. In determining the relative displacement of air
masses at a front the following rule may be followed
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'If the observer facing the front is being displaced together with
one air mass,the second air mass lying
late is true in the case of any air mass and any front,
displaced,,with relation to the observer, to the left
Let us finally note that during
direction of the wind at a given point changes
"the wind turns to the right1'. (Figure 114)
Figure 114. During the passing of any front the wind with tins
turns to the right.
However, even though a front is accompanied by a wind rota-
Lion, it does not follow that any rotation line of the wind is a
front. A line of rotation of the wind will be a front only in that
case when there is present a locality of discontinuity of density
what 1L4)79,
or, is the sameA a discontinuity of temperature.
Section 6._ Change in Gradientind with Hei ht, in the Area of a
celeration in passing from a cold mass into a warm mass is in
with a sudden shift, it being the case that the vector of wind ac-
wind distribution along a vertical line. This formula shows that
in its ascent through a frontal surface the gradient wind changes
Equation (1) in Section 5 may be applied to the analysis of
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direct ratio to the incline of the front fan d( and to the jump in,
temperature at the front and is directed parallel with the,line of
the front in such a manner that the wedge of cold air remains, to
the left and the warm air remains to the right. This rule follows
from the analysis of the problem on the rotation of the gradient
wind; with height in a homogeneous air mass.(Chapter XI).
In the case of a warm front tai 7? 4 (Figure 115, A),, and,.
consequently4V=U --'dos i.e?, before a warm front the gradient
wind sustains a discontinuity in such a way that the rotation of
the wind with height in the area of a front is to the right
(Figure 115, A). In the case of a cold front, &D1 Q O, and
LI
=
Therefore, behind a cold front the rotation of the wind with height
at the frontal surface is to the left (Figure 115, C).
Figure 115. Change in wind with height in the vicinity of a front
(a) warm front, (b) warm air mass; (c) cold front. .
, dp,ps '
Whirr -- -,_
Fl ure // # tr, utro~ o f wind ctf 'hc ra~va a vs~',
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If the .front is stationary, the gradient wind with height
becomes decelerated with a sudden shift, retaining the: direction
of the cold current,or becomes accelerated with a
sudden shift, or,
finallyreverses its direction with a sudden shift.
Equation (1) in Section 5.shows that sharply accentuated
fronts with a significant discontinuity of temperature and a steep
incline are accompanied by a considerable change in the gradient
wind with height. And, inversely, fronts mildly accentuated in `a
temperature field, and with small angles of incline are accompanied
by a mild shift in wind with height. .
Thus, radiosonde observations can be used for? the determinate
tion of the location of fronts in the free atmosphere. But,
together with that, as already mentioned above a sudden shift in wind
-will indicate the presence of a front only when a discontinuity
in density (or temperature) occurs. Therefore, radiosonde observa-
tions can only be used as a supplement to temperature sounding.
We shall also note that when the wind with height rotates to
the right the height of the frontal.. surface is increased in the
direction of the wind (Figure 115, A). When the wind with height
rotates to the left the height of the frontal surface decreases in
the direction of the wind (Figure 115, C).
face of the first order with relation to density and the pressure
gradient. Therefore, the tropopause is a discontinuity surface of
the first order, also with relation to wind velocity, i.e., the velo-
We have seen above that the tropopause is a discontinuity sur-
city of the wind at the tropopause will change continuously.
,~, .~~. a R
r.
t r., 14
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As is known, the tropopause under normal conditions is
inclined to the Pole. Figure 116 shows the possible distribu~
Lion of the vuind T~vith height ' in the vicinity of the tropopause.
Disposition of a Surface of 'Separation in a Non-Stationary Wind
Fielde
that at an initial moment a certain surface of
separation is inclined to the horizon more steeply than a stationary
surface of separation, the inclination of which is determined from.
the formula (8) in Section 3, i.e., in Figure 117.
Fure 11.7? Evolution of ;accelerations in the vicinity of a non
stationary surface of separation, with c > c(a ??
Let us assume in addition that no perturbations are acting
upon the currents. An elementary but cumbersome analysis shows
that in this case a discontinuity of the acceleration components,
normal to the front, occurs, it being the case that
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Ze i'.5 ,5S*d'1
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7;~r4?tiei- ?a); phis r~edr' #; b'1 ~~r QCC' 1'/,ere aJ~G vcc J' 5
"
Cara e ?'
'iscanr~inuI y oa
dis-
placed greater
7%2>>U
o>2.>j
of c.G~
d h
>a>z < .
z ~
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(-?IaPt
. >u>O; ?>>O;
for in . I
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ascending currents developing in both the cold and the warm air
(the ascending warrc. air overtakes the ascending cold air).
The emerged normal components of velocity induce tangen~
tial accelerations which are predicated on the rotation of the
earth. These supplerr~ntary accelerations in the first case have
the same direction with the absolute value of the warm mass ac-
celeration greater than that of the cold mass and with the warm air
remaining to the right with relation to the cold air.
In case No. 2, the air masses being considered are being dis-
hoed toward each other, i.e=, both masses are moving toward the
p
cold air .
front. It being understood that. the front proper will not undergo
displacement only in the case when the accelerations of the air
masses in their absolute values are equal to each other. If the
accelerations in their absolute values are not the same, the front
will underga~ displacement in the direction of the cold mass (when
a =
and in the direction of the warm mass (when ~~ ) ?
In conformity with this the front may be stationary (when ._- 4 )
warns ~wh ~a I1>1u21J or cc/d (h I uz j) _
In this case regardless of the nature of the front an ascend-
ing motion will develop in the warm air and a descending one in the
Accelerations which are tangent to the front are of opposite
directions and they contribute to the evolution of cyclonic motion
at the front
Finally.in case No 3, both air masses will be displaced in
the direction of the warm rriass with the cold-mass ' sustaining a
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II'
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a
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cold mass.
The accelerations analyzed above result in that (1) either
the warm air will overtake the receding cold air and rise at a
greater vertical velocity along the wedge of the simultaneously
rising cold air. Together with this the existing discontinuity in
velocity at the front, will increase which circumstance entails' a
greater angle of?inclination of the stationary surface of separa-
tion. (2) Or the warm air will flow into the wedge of the descend-
ing cold air encountered by it. In this case the present disoon-
tinui-by in velocity will increase which circumstance also entails a
greater value in the angle of inclination of the stationary surface
of separation. (3) Or the cold air vdll flow under the receding
warm air with the warm air in its receding . lagging in its vertical
motion behind the. more rapidly descending cold air. The discon~
tinuity in the tangential velocities will also become greater,
which circumstance will entail for purposes of equilibrium a steeper
angle of inclination of the surface of separation.
greater (in absolute value) acceleration than the vuarm mass.
This motion will be accompanied by descending currents developing
in both air masses, it being the. case that the cold mass will be
descending more rapidly than the warm mass.,
The accelerations which. are tangential to the front w111
again bu in the ~same direction with the acceleration of the warm.
mass in absolute value being 1o~uer than the acceleration of the
In all cases as a result of accelerations the angle of in-
cllnation of the surface of separation will be diminished while the
q! ..
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intensified discontinuity in the tangential oornponent of wind
t'~'GessiN/
velocity is increased so thatT' low-sloping surface of separation
(Figure' 118) is already not in conformity with conditions of
equilibrium. In such a case accelerations having an inverted sign
begin to develop at the surface of separation which results in that
either (i) both masses will be displaced in the direction of the
warm mass descending at the same time with the warm mass, moving
more rapidly and overtaking the descending cold mass, or (2) both
masses will flow in different direotionsywith the warm mass descend-.
ing along the wedge of the ascending cold mass, or (3) both the
warm and cold mass will move in the direction of the cold mass, as-
tending simul Eaneously)with the cold mass receding more rapidly,
overtaking the warm mass along a vertical line.
Together with this in all the above cases the tangential oom-
ponen-ts of the generated accelerations have anticyclonic character-
istics~ as a result of which, with time, the discontinuity in velo-
city at the front is gradually diminished, and may even change its
sign. The angle of inclination of the surface of separation will
increase while at the same time for a stationary surface the re-
quisite angle of inclination will already be smaller, Thus, we
returned to the initial postulate from which we began our analysis.
A more detailed investigation shows that a surface of separa-
tion oscillates somewhat about the position of equilibrium which
is not coincident with the position of a stationary surface but,
deflected from the latter in the same direction as is the initial
position of a surface of separation.
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Fiell84 Evolution of aocelerations in the vicinity of a non-
stationary surface of separations with c as
Section 8. Classification of Fronts.
The above described characteristics of fronts relating to
their position in a pressure field, wind field, and temperature
field, are applicable to both shifting fronts and stationary (mo-
tionless) fronts. By these charaoteristics the presence or absence
of fronts in a synoptic chart or in a vertical section may be judged.
However, with relation to the movement of fronts and to the
relative position of the warm and cold air masses and their stabi-
lity it is possible to establish a series of additional character-
istics for any given type of front.
Thus, it becomes possible to classify fronts by their
characteristics. Such a classification relegates a contemplated
front to its proper type.
Fronts may be classified by their geographic index. There
are four basic types of air masses: arctic, polar, tropical and
equatorial air,
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are usually characterized by great horizontal' homogeneity in tem'-
perature as a result of which no secondary fronts are evolved with'-
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Besides the general frontal characteristics, such as disconw
tinuity in temperature discontinuity an the tangential oomponent
of wind velocity and others which were discussed in the preceding
g
text and which are applicable to all fronts, there are no other
particular` characteristics which would determine the geographical
classification of the fronts. Nevertheless, the usefulness of such
a classification is obvjous. Since the main fronts predominate over,
the secondary ones, by the frequency of their emergence, by the
vastness 'of the area occupied and by the duration of their exist
ence,the clarification of weather data that is tied in with the ex-
istence of fronts for the purpose of a synoptic analysis is con-.
corned, first of all, with the primary fronts. Only in the case
when a particular weather feature under analysis cannot be explained
's
by the existence of the main front attention to be drawn to the
secondary fronts. In so doing it is recommended to pay particular
attention to the clarification of the processes promoting the for-
mation, as well as the disappearance, of these secondary fronts.
The most essentialclassification of fronts is their classificaw
tion with relation to the characteristic of the novement of the
front and the migration of the air masses. Within this classifi-
cation there are 4.types of fronts:
(1) A warm front, i.e. a front moving in such a way that
the warm air displaces the cold air;
(2) A cold front, i.e. a front moving in such a way that
the cold air displaces the warm air;
A stationary front, i.e. a front that is motionless;
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ULLG vY Lj. t~1L ueang roreea out after a more rapidly moving cold
front reaches a slowly moving warm front. .n occluded front is
generated vuhen between the cold air receding before a warm front
and the cold air advancing behind the cold front there exists a
discontinuity in temperature.
It can be shown that the movement of a front is determined
by the formula
When the vertical velocities w are very small it can be as-
sumed with approximation that the velocity of the movement of the
front is equal to the wind velocity component which is normal to
the front. It is, of course, understood that in this case we are
considering not the velocity of the surface wind but the velocity
of the geostrophic wind which is observed above the friction 1eve1.
Consequently a fast moving front must cross the isobars at an
almost right angleawith the isobars themselves running adequately
close to each others
Stationary fronts must be parallel to the isobars. Strictly
stationary 'fronts occur very rarely in the atmosphere. Very slowly
moving fronts are encountered more frequently. Such a slow moving
front called a quasi stationary front,greatly complicates the pro-
blem of weather forecasting, since its subsequent movement is
in this case there is no displacement of one air mass by anothex;
(4) n occluded fronts i.e. a front generated as a result
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Therefore, the classification of fronts or surfaces of
separation is done with relation to the behavior of the warm air
mass
There are surfaces of uppslope sliding or anafronts when
0 and surfaces of dovanslope sliding or katafronts, when
w;L < 0 . Differentiating also warm (~ ,> D) , cold (c,c d) , the isobars are d splaced along the pressure
worce (C. 0) , and in the area of negative barometric ten-
gradient .L
If
s (T dA'L , the Ri number asymptotically approximates the
Finally, Obukhov computes theoretically the change in the
coefficient of turbulence' K with heights
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But -- - -? , consequently ~- --
z
. -the behavior of function , basically speaking, is known;
therefore it is simple to investigate the variations of k(z)
with re1ativr~ly small (as cornpax'od v/.'1,h L. ),and relatively great
values of z.
therefore
when --> , then, with L ,
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Further: 1(e)- r
z~L
when c_>? f ?t ; consequently,
Thus, Obukhov arrived theoretically at the following result:
with love values of z, the coefficient of turbulence K increases
linearly with height, and with high values of z, the coefficient
of, turbulence asymptotically approximates its ultimate value
/ uL
If the scale L is eliminated from (30) with the aid of
(17), k00 will be expressed thus;
Obukhov computed the following Table of ultimate values
...,~ --1
/4 gram/centimeter second (Table 48).
as
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Calories
cm2 mmn
0.01 0.02
0.05
0.31 degrees centigrade/100 meters,
grace/ Oo meters.
0.1
0.15
0.2
0.3
-
As seen from the above 'fable, the values of the parameters
of turbulence theoretically derived have the same order of values
of these parameters, established from observations. .
Obukhov cites the following example to show hove ~ve11 the
theoretical values of the turbulence parameters coincide with their
observational values:
When uo~ 5 meters/second, u* 0,25 meter/second, and q
0.1 clorie/s uare centimeter, K~ ' 1.82 square meters/second,
a ~.
Air, a 22.4 grams/centimeter second L 18.2 rosters,
-0.69 degrees centi-
Table 48. Ultimate value of the coefficient of turbulent exohange
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gram centimeter~1 second ~~ for a stable atmosphere.
u
me
se
20 u*
ter S/
cond
u
sec
0.5
2.5,
0.0224
Oe0112
1
5
O?3567
0.1722
2
10
5.7195
2.8536
3
15
28.905
14.514
.
4
20
91.512
45756
5
25
.11.684
30
10
50
0.00443
0.00221 0.OO15
0.0010
0.000738
0.0738
0.0369 0.0246
0.0123
0.0123
1.1439
0.5658 0.3813
0.2829
0.1845
5.781
2.829 1.968
1.476
0.98
18.327
9.102 6.150
4.551
3.08
44.895
22386 14.883
11.95
7.38
93 .l 11
46.371 30.873
23.124
15.38
X238.743
178.719
119.19
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This example shows that the value of the temperature gra~
client computed by the theoretically derived value of Kam, also
conforms well to observational magnitudes. Let us take note of
the fact that in his computations Obukhov assumed a value for Ri
ultimate that was obtained by Sverdrup from observations, namely;
I
(Ri) ultimate
Section 3. Variation of Wind with Heig t and Turbulent Exchange
in .the Planetary Boundary _Layer.
One of the raost known manifestations of the effect of turbuw
lence in the atmosphere is undoubtedly the variation. of the velo-
city and direction of the wind with height in the lowest atmospheric
layer. In a homogeneous air mass the increase in the velocity of
the wind vrith height is manifested with perfect clarity, also its
rotation to the right in the northern hemisphere, and to the left
in the southern hemisphere.
Such a change in the velocity of the wind begins at a height
of 1020 meters, and extends in our latitudes to a height of about
1 kilometer. Below the level of 1020 meters the direction of the.
wind remains constant and the velocity magnitude varies logarithiM
mically or in accordance with the law of exponents.
The change in the direction of the wind with height in the
higher atmospheric layers is obviously stipulated by the combined
effect of the deflecting force of the earth's rotation and turbu-
fence .
The layer in which the wind rotates with height, the
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fW~
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rotation stipulated by the combined effect of turbulence and the
deflecting force of the earth's rotation, is called the planetary
boundary layer.
`The most substantial characteristic of this layer is the
angle , which is formed by the direction of the wind 'Z with the
direction of the pressure gradient force (Figure 130)0 This angle
"angle of deflection", . gl~
cc; rt/? 1 of is .k .own as the ~ s
d;rzhi of /e ''
fI)e u? /e ?jD d /e~h M ;s ~ail~d the :9
6f 1/ 71hc
Figure 130. Angle of deflection and angle of inclination 9
of the wind,
1~;fhin the limits of the planetary boundary layer the angle
of inclination of the wind is more or less substantially different
from zero and the angle of deflection is always an acute angle.
The magnitude of the angle of deflection is a function of the height
As shown by numerous observations the angle of inclination
of the wind depends not only on the height, but changes also with
time (annual and diurnal cycles), latitude and orography of a
locality. Thus, for example at the, earth's surface the angle of
inclination of the wind is usually. smaller in the summer than in
the winter, smaller in the higher latitudes than in the lower ones,
u~
ip,t
k;li
gar
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1 7-9
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smaller over the surface of the `sea' than it is over land.
ire will now analyze theoretically the . change in the velo-
city and direction of the wind with height under the effect of
turbulence and the deflecting force of the `eartht s rotation.
d~
Assuming the motion"horizontal and the field of velocities
u d
along a horizontal '"`homogeneous, we will proceed from the averaged
equations of motion of the atmosphere which can be written down as
follows
dz 1 4 au
d T : + O) Z) fit` z
P r /)
(1)
c -
dL %o ' r ~z: (- ?zJ
Let us also make the following simplifying assumptions
(1) The motion of the air is homogeneous and rectilinear,
therefore
d d~
d
(2) The pressure gradient does not change with height.
Therefore, directing the x-axis along the isobar, and the y-axis
along the gradient, we have:
Density does not change with height, i.e. /? M coast.
(4), The coefficient of turbulent exchange 'does not change
with height, i.e. A M const.
._.~- ~. Q
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Then, the equations of motion will assume` the following
form:
dz 2
,I
Now we f o rmul ate the boundary conditions:
(1) At the earths surface there takes place adhesion of
air so that
lzs/7e.1 Z =
(2) At great heights the wind becomes a geostrophic wind
(U , ), consequ.ently, with Z-- cx --~
d ~
z'- r 0
The problem of integrating the system of equations (2) with
two unknown functions Ua V , oan be reduced to the problem of
integrating one equation but with a complex unimown. function W
= zF:iv To accomplish this we multiply the second one of
equations (2) by , and add the result to the first equation.
~ ar2~v
dz
a
?G
(3)
Thus, we obtain an ordinary linear differential equation of
the second order with constant coefficients which can be re-written
as follows:
4.
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Consequently, the final solution of equation (4) is like this:
7.- {L,
We now separate the imaginary and the' essential parts of
solution (12).
since
consequently '
Applying the Euler formula
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r
we find
15)
and finally
e ('cos (fix c /i; aZ)
-a C45 G Z~
: a
c-'
1 -?--~ 2Le,e s?rr a
Thus, we derived the components of wind velocity as func-
tions of height z. We now find the absolute value of velocity
2 z and the angle of inclination
if
_ e - c( cos qZ
(17)
(18)
Formulas (17) and (18) show that with the increase in height
z, the velocity value increases and the angle of inclination diniin-
fishes so that the wind approximates the geostrophic wind in magni-
tude as well as direction.
greater the value of GC's
This 'process is the more rapid, the
71Li.e. the lower the turbulent
exchange and the higher the latitude of a locality.
From (18) it follows that at some heights the wind. assumes
the direction of the ~eo str ophi e wind so that 00 with
~,
The minimum he ight
51;9
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at which the `wind assumes the direction of the geostrophic wind,
is called the level of friction. Although with subsequent in-
crease in height the wind is somewhat deflected from the direction
of the geostrophie wind, these deflections are so small that practw
ically they cannot be detectedy as can be seen from Table 50.
Therefore, the level of. friction which is determined by equation
(19) can be. accepted as the upper limit of the planetary boundary
layer. Table 49 gives the thickriess of the planetary boundary layer
for various values of the coefficient of turbulent exchange and
-3 -3
latitude vuith 1.25 x l0 gram centimeter ,
Table 49. Level of friction with coefficient of turbulent exchange
constant.
100
300
50? 600
9Q0
1 -1.
-
sec
10 gram cm
790
470
380 350
330
50
1770
1040
840 790
740
100
2500
1470
1190 1120
1040
150
3060
1800
1460 1370
1280
As can be seen from formula (19) and Table 49, the level of
friction is the higher, the greater the coefficient.of turbulent
exchange and the lower the latitude of a location.
The velocity value at the level of friction Which is ex-
pressed by formula
e J>Z4;$) ???(20)
Irj''11h''
byp
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When Z >00
-- ' t
When z= O formula (18) results in an undeterminate form
of the type Evaluating it in accordance with the LtIEIospital's
y
rule we derive s
d -~z
.Y
d.#.~... (e sir- c(z)
~-ar G ~ aZ
z o d..... (J-eos!)
Thus, at the earth's surface the arngle of deflection under all
conditions equals 45 degrees
height in the planetary boundary layer.
4 ~a
-'J/
is greater than the velocity of the geostrophic wind since
is an essentially positive magnitude. Thus, the velocity of the
wind attains in absolute magnitude the velocit3r "value of the geo-
strophic wind somewhat below the level of friction.
Assigning definite values to OC. it becomes possible to eom-
4ji(17,hz 2a(/~) C
puce the corresponding values for and -- . Assuining that
9 : 450, :: 10- gram,/em3, A " 25.2 gram crn sec and, con-
sequently O( = 0.004775 m, we derive the following Table of
values for and' for various heights (Table .50)?
and the wind becomes geostrophica
10 20 40 100 200 400 . 800 1200 m
0.075 0.145 0.286 0.584 0.893 1.068 1.005 0.999
43 42 39 31 20 5 -1 0
Table 50s Variation in the velocity and direction of the wind with
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In constructing by table. 50 a diagram of the, wind velocities
as function of the height z, we derive a spiral which is known in
geophysics under the name of the Eckman spiral (in honor of Eck-
direction of ocean currents under the effect of turbulence and the
logarithmic spiral with an angle of 45 degrees.. It passes through
the origin of coordinates and approaches asymptotically the point
Figure 151. The Eckman spiral.
shown` that under the combined effect of the Coriolis force end turn
bulence, the velocity of the wind increases with height, simultane-
ously turning to the right (in the northern hemisphere). At some
height the wind becomes equal to the geostrophic wind, first in
magnitude and then continuing to increase at some greater height
becomes equal to the geostrophic wind in direction. At the earth's
surface the angle of deflection is always equal to 45 degrees.
Accumulated empirical data shows that the factual mean.dis-'
tribution of 'the wind by height at an adequate distance from the
earth's surface., concurs well with the theoretical distribution,
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but in the :lowest dekameters there is no such concurrence. Thus,
at the earth's surface the angle of inclination of the wind is
considerably less than 45 degrees and in addition:it does not rem
main oonstant, but varies within rather wide limits, decreasing
over the sea during the summer and during the day and increasing
over land during the winrber and during to night The factual
magnitude of the wind velocity at the anemoscope :i,evel is consi-
derably`above its theoretical value
The subsequent efforts of the soientists were directed toward
the further development of the theory relating to the change in the
wind with height, the beginning of which was laid by the work of
Eckrnan and Okkerblor. .
Gessel'berg and Sverdrup took into account the change with
height in the pressure gradient forces considering this change as
taking place in accordance with the linear rule, but assumed, as
before, than
P
and A are constants. However, they completely
excluded from consideration the layer below the anernoscope levely
assuming justly that conditions of turbulent exchange predonr.nat-
ing there are different from those in the layers above.
With the indicated assumptions, Gessel'berg and Sverdrup
laid out a model of Wind distribution with height, which concurred
well with results of observations. Figure 132 shows a rectified
(in accordance with Gessel'berg and Sverdrup) Eckman spiral for
the atmospheric layer above the level of the anemometer. For
comparison purposes the same Figure shows a spiral constructed as
a result of developing the data obtained by 99 observations
in
Lindenberg. Figure 133 shows the distribution by height of. the
l";f
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absolute magnitude of velocity, obtained both theoretically and
Figure 132. The Eckman spiral for the layer above the anemoreter.
The solid line is the mean of all observatjons9 the dotted line -
the mean of all computations4
Figure 133. The absolute magnitude of wind velocity as a funetio?n
of height above the anemometer. The solid line is the observation
our, the dotted line M the computation curve.
The research by Gessel 'berg and Sverdrup substantially im-
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proved the Eckman-Okkerblorn theory. However, the solution they
arrived at is by far not an exhaustive one. The change in the angle
of inclination with relation to the time of the year and orography
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A considerable contribution to the solution of the con-
,-em fated problem Was made by the labors of Soviet scientists,
which we are going to analyze in the Section irnnedia e1y follow-
ing.
Section 4. Variation of Wind with Height iii the hePesence of a
Variable Coefficient of Turbulent 'exchange.
The state of turbulencet any point is determined by one
number- the coefficient of exchange. Having assumed in the pre-
g .. paragraph that the coefficient of exchange was the same for
cedin
all heights, we have by the virtue of this accepted the state of
turbulence as the same for all heights.
Yet, the state of turbulence and, consequeltlY, the coeffi-'
dent of exchange in the atmosphere are subject to considerable
variations. The principal factors affecting the value of the co-
efficient of exchange are : the velocity of the ge o strophi c wind
( or the pressure gradient that determines it), the roughness of
the underlying surface, the stability of atmospheric stratifioar
Lion vertical temperature gradient), the height above the earth? $
surface, Since the coefficient of exchange is not measured three'
ly, but is computed on the basis of measurements of other moteoro-
loggical elements, it follows that a correctly assigned character
of variation of the coefficient of exchange is very 'important to
such computations. The assumption of constancy of the coefficient
of exchan e by height, in, some cases, does not allow the correct
g
evaluation from observations of even the order of rnagnx.tude of the
coefficient of exchange.
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On the other hand the assumpbiofl that the coefficient of
exchange in the entire planetary boundary layer increases accord-
ing to the linear l.aw leads to incorrect results, namely; to ex-
cessively large values of the coefficient of exchange.
?In their latest works, Soviet' scientists use one of these
two computation patterns, outlining the variation of the coeffi-
cient of exchange with height.
l~atbern A; the coefficient of turbulent exchange increases
with height asymptotically approaching the constant value.
. e_m) (1)
Y ~
where #)' 6 is molecular viscosity. This pattern was suggested
by B. I. Izvekov.
Pattern B: the coefficient of exchange first increases
with height according to the linear law up to a certain height h,
after which it remains constant, i.e.
: . ch, whz}h, ~
Pattern A is deficient in the sense that its use leads into
more complex computations. Pattern B is more convenient for corn-
utntions although it leads to the solution of a two-layer problem.
p
In this Section we will analyze the problem of the varia-
tion of wind with height, 'accepting, pattern B as the coefficient of
exchange. This pattern was first suggested by M. E. Shvets and
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M., I. Yudin who derived an adequately simple and precise solution
of the contemplated problem.
Yudin and Shvets, assuming the motion to be homogeneous,
proceed from the following equations
Multiplying the second equation by i, subtracting result
froxrL first, and interpolating the complex velocity
the two equations (3) are reduced to one
Equation (.5) is rewritten in the following form;
height.
I?t. is assumed that the pressure gradient does not change with
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This is one of the forms of the well known Bessel r s equa-
tion, which is integrated by the use of special sopcalled cylindri-
cal functions which have been extensively studied:
gration constants, which are determined from the boundary conditions;
Io is a Bessel function of the zero order; No is a Neumann function
of the zero order; ~,~~~^ the eostro-
?2c::)@Y
phic wind.
For the layer above level h we derive the Okkerblom solu-
tion already analyzed before:
where A. and B are arbitrary integration constants, which are deter-
2
om the bo
min
d f
d
d
t
e
r
un
ary con
i
ions and =
Yudin and.Shvets assume the following boundary conditions:.
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(3;
At infinity the force of friction is finite:
C'(a~
~z . )/
it being the case that the value of arbitrary constant is determined
from condition (27).
Shvets and Yudin computed several examples of the distribu..
Lion of wind with height, assuming the following values for the
parameters:
imaginary part we find:
ir
j z
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3.88
5.75
12.8
20
6.62
0.1
0.1
0.1
0.1
0.1
0.077
3.44
3.82
42.3
20.9
10.52 m/sec
190 lvi
1.77 m2/sec
The analysis of Tables compiled by the authors led them to
formulate the following deductions resulting from theoretical re-i
search and agreeing well with observations
(1) The angle of inclination of the wind is increased with the
increase in the roughness of the underlying surface.
(2) Turbulent viscosity is increased with an increase in the
pressure gradient.
(3 ) ,The velocity of the vlind at the height of the anemometer
is about one half of the geostrophic wind velocity.
(4) yith increased height the wind in magnitude and direction
approximates the geostrophic wind, it being the case that the velocity
magnitude attains the geostrophic value before the direction does.,
(5) lAlith greater turbulent viscosity the angles of inclina-
tion are smaller.
(6) In the case of winds of great force the height at which
the wind attains the geostrophic values is greater than in the case
of mild winds.
meter height and the velocity of the geostrophic wind, is tied in
(7) The ratio between the velocity of the wind at the anemo.
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vlith this velocity, roughness of the underlying surface and the
angle of inclination. With a decrease in any of these 3 values,
the above ratio is increased,
(s) At some heights the value of the velocity of the wind
exceeds the value of the velocity of the geostrophic wind at the
earths surface.
Section 5. Diurnal Variation in the Velocity of the Wind.
The diurnal variation in the velocity of the Wind is tied in with the
At the earthts surface under conditions prevailing over a
plain the velocity of the wind has a clearly pronounced diurnal
variation unless this variation is disrupted by perturbations,
which are stipulated by changes in v'eather. The wind is at its
lnaxirnurn velocity soon after middayd At night there is a wide
minimum. The angle of deflection of the wind from the isobar is
at its minimum during the day, and at its maximum during the night,
season of the year, the underlying surface, and the air mass,
the surface of the sea this diurnal variation is almost completely
absent0
summer when turbulence is greater , the height of rotation attains 300
on the season of the year and the force of the wind. During the
Above a certain level the velocity of the wind has an inver-b
ed diurnal variation: the maximum is observed at night and the
minimum during the day. The height of rotation of the diurnal vari.a-
tion of the velocity of the wind varies within wide limits, dependent
meters, while during the winter it is only 30-50 meters,
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A computation shows that the diurnal variation in the velo-city of the wind cannot be explained by the diurnal variation in
pressure since wind velocities induced by the diurnal variation in
pressure do not exceed 10 centimeters per second, while actually
observed diurnal wind velocity ranges are tens of titles greater
than the above indicated value 0
On the other hand, the enumerated pecul.iar?iti.es of the di-
urnal variation in the wind indicate clearly that the _di.urnal vari-
ation in the velocity of the wind is tied in yvith the diurnal van-' ?
~.
ation in turbuience a
The first attempt to explain the diurnal variation in the
velocity of the wind with the aid of the equations of hY? drozrnechan'
zcs
was made by B. I. Izvekov, He considered the kinematic coefficient
cf turbulent viscosity as non-varying with height, but as a periodi-
cal function of time.
where is the mean diurnal value of tlie kinematic coefficient
G
of turbulent viscosity, 6 is the diurnal variation amplitude of
this coefficient and W is the angular velocity of the earth.
However, having assumed the coefficient of turbulent viscosjM
ty as non-varying with height, Izvekov arrived at rather unsatis-
factory results. The velocity of the wind, as per Izvekov9 does not
approximate with height the velocity of the gradient wind. The
condi'taon cif the air adhering to the earth's. surface also remains
unsatisfied, Finally, the rotation of the diurnal variation of
wind velooity at the upper levels of the boundary layer is also
absent,
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 1
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Pzm=Ae
-2wxc
G~?
where A is the arbitrary integration ccnstant.
It was also shown there that after determining the arbitrary
constants of integration there remain two equations tying in three
magnitudes s'C(t) /L&), o (, where G(. is the angle between the
direction of the wind and the direction of the isobar at the earth's
surface .
These equations can be ~vritten like this:
1J2"n t
~L ~a z.
J J2
I~(
zo %oo
J
..-4 _. z I. - -....~.-. rn e
4jz
where
-J
~r i j
a
= (k.
Jz ~d Z a
nz t c
when Z k) (10)
wk ) LIN0'(yf;I)
(12)
Eliminating from equation (ii) angle 0Xwe derive one equa-'
tion tying in ) and k('
4 0
2 ,, ' z_(G ) Z (13)
(2L::L ,J.2 ,. ( *
Jt iR Accepting as the coefficient of turbulent viscosity ) at a
height above the level z ^ h,? the condition
= c(M(&,)= (ZtEs.1n )2),
Shvets determines c(t
and h(t) from equations (13) and (14),
14)
then by the Tables he constructed he derives the distribution of
the wind with height,
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matic characteristics of rule (it),
Irrespect%ve of the sche
. iciont of turbulent viscos~.ty with
the vara.ations a~' the coef f
b~ ~hvats, will agree wa:th obser-
time, as derived theoret~.cally :r
vatianal values.
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CHJ PTER XTII
THE ENERGY OF ATMOSPHERIC 1VIOTION3
Section le Transformation of .ergy in the 'Atmosphere?
For the atrnospheric processes under study in dynamic meteor-
ology, the following forms of enemy are of" essential significance:
(1) Radiation energy (R);
. (2) Kinetic energy of averaged large scale currents of the
order of general circulation or currents in cyclones and anticy-
clones (E);
(3) Kinetic energy of the turbulent motions which is. tied
in with the presence in the averaged currents of additional veto-
cities (Es). .
(4) Potential energy of th,e air masses (TI);
Internal energy of ai r (U).
We are not concerned here with. electric and magnetic energy,
which have considerable significance for some individual phenomena
in the atmosphere, but for the processes under study in dynamic
meteorology, are not essential.
The potential and internal energy of the air are in direct
ratio to temperature T; therefore, as will be shcwn below there is,.
a close and direct relationship betvueen there to to 'effect'that
the increase in the internal energy of the atmosphere is ?always
accompanied by an increase` in its potential energy and vice versa.
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A continuous transformation of energy from one form into
r one is continuously taking place in the atmosphere The
anothe
basic mechanisms for wuch trasfo rn~ata on are absorption and radix'
tin, mechanical work and the dissipation of mechanical energy into
heat .
The following transformations of energy are possible in the
atrosphere
9 - >(Ufr E >.(u E '- (U
(Ut ) ,
(u~ ) -> E_
(U*7T)a
Radian on energy cannot be converted into kinetic energy
directly. It has no significance for the atmosphere until such
time as the atmosphere will absorb it, i.e. until such time as it
Will be transformed into internal energy! It being the case that
in atmospheric conditians (in a gravitational field) an increase
in internal energy is always accompanied by an increase in poten-
tial energy, therefore, I'LT, I77)
Internal energy can
again be transformed into radiation energy by way of atmospheric
radiation, it being the case that a decrease in internal energy is
alway$ accompanied by a corresponding decrease in potential energy
The internal and potential energies, accumulated as a result
of absorption of solar radiation, may subsequently be transformed
into kinetic energy Which sometimes assumes the form of kinetic
scale motions (J#7T).-*E, or the form of kinetic
energy of large
..
w u4.~~~ r
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'
4
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energy of turbulent motions (U)E. The transformation
of potential and inbernal energies into kinetic energy is one 'of
the most important aspects of modern _dynamic meteorology, since
the general circulation of the atmosphere and the evolution of
cyclones and anticyclones are specifically' tied in with such trans'
formation. In its turn kinetic energy may be transformed into pa-
tential and internal energies either by vray of direct transition
of kinetic energy into potential
E y (C/t T1~.
by way of dis-
sipation of mechanioal energy into thermal energy. In the latter
case, the gradual process of degradation of kinetic energy by vray
of the breaking up of vortices takes place. The kinetic energy of
large currents is converted into kinetic energy of progressively
smaller vortices E'E " until such time as it is dissipated into
heat energy under the effect of viscosity ,f -f?
..
The process of transformation of kinetic energy of large
currents into turbulent kinetic energy is an irreversible processe
A process which would lead to the straightening of the lines
of flow and to the increase in total velocity of a current at the
expense of a diminution in the velocities of the additional motions
is not possible in nature.
.ection 2,. Equation of the Ener balance of an Individual Air
Particles
Let us develop the equation of the energy balance 'of an in-
dividual air particle having a' mass equal to unity. We.. will assume
that frict ofl is completely absent, and the air is in motion as the
ideal liquid. In this case the equations can be written 'down this
way:
;2I
"
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Multiplying equation (1) respectively by t Z/ W and adding
the results we derive the equa b.on of the mechanical energy balance
or the equation of live forces;
dt
(2) is simplified,
dt P a~
dv - s aP
2 w Stn
E'p
where c is the velocity of the wind, is the component of the
pressure ascendent in the direction of velocity.
. 'iVe note that the deflecting force of the earth' s rotation
did not orr er the equation of live forcoS,,vahioh was to be expected
since the amount of work performed by the force of inertia is iden-
tically equal to zero, due to the direction of velac?ty and the
force of inertia being perpendicular to each other,
Uthen the motion is horizontal (fz)C , an equation
d~ .
d
d(zJ ../o ~5
It follows from equation (3) that if pressure drops in the
a
r is increased
of t she motion >ororlni..ts m, Izvekov, and Shvets repeated the com-
putat,:i.ons for theconstruction of the Kochin model of zonal circulation,
proceeding from a more perfect sir tern of equations and basing themselves
on new data for temperature and pressure distribution. They succeeded
in proving that, in the case of zonal circulation, there are present
in the atmosphere 3 basic "rings" of a distinctly pronounced circulation
in a vertical plane. The tropical and the polar rings are of thermal
origin. Teynoperate" as simple thermodynamic machines. The heaters
are below at high pressure, the refrigerators are above at low pressure.
The middle ring of the temperate zones is of dynamic origin. At the belt
of high pressure (in he horse latitudes), the meridional currents di-
verge, and the loss of mass is comInsated by descending currents. In the
I
low prey ure belt (in the subarctic) the convergi A currents are compen-
sated by ascending currents.
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frad
t _7/ $r
d -
,
I
,O
F- 2/ vJ . ( i)
.Get us remove the vortex from both parts of this equation.
Section 9. T'he_Hydrorl., ,arni c ~l~eory of pressure eaves and A-~mos~here;
E. N. Bliriova analyzed the problem of long tend forecasting
of pressure, temperature, and. wind, proceeding from the equations of
hydromechani_c s.
The results/ which she obtained consti tu:t,e a valuable contra
bution in the co3tructio1 of mathernaticaT theoxr of the general
circulation of the atmosl:here.
The earth is consider :d as a globe having a radius c~ .
The gravitational force and the Coriolis force at upon the atmosi'here,
1
!hc force of friction is assufi~ed. not to exist. A5 the initial equation
the Fridman equation (16) from Section 1, Chapter IX is utilized. It
deternnnes the change in the vortex vector of a given particle of 1?-
quid. In order to derive this equation, we take Luler's equation in
its vectorial form
Assuming tai V , we derive the following equation, which
is the Fric1 an ecuati.on:
- rya' T2 Tr ()
7w
Converting to coord:i hate equations, Clinova interpolates the
fol1owi.ng spherical coorclinatesc r -. the dista
f
nce
rom the center
of the ear'ti1
- the co-latituc e ( at the no.: tli Polo, nn
at the south. pole ) , anti - the longitude of locat:i on .
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The problert~ considers the motion of the ,atmosphere as consisting
of Lhe basic motion, w}ich is the zonal circulation, and the
perturbations of this motion herefore, the component velocities along
and )- , and pressure can be taken as
P-p(B~tp(B
Let us note that here, as well as i n the :[urther procedure, all
axes 8
values are averaged with relation to height. All the elements of
are considered. small.
Assuming that
where is the angular velocity of the air in its motion relative to
the earth, but not depend on . The ratio. is called
the circulation index.
Considering ?r,h.e v1ind as a gradient wind, we obtain the follotiling
formula for pressure
Mlle temperature formula is taken as foli..ows
r= 1- r ltT?~a,~, t~
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where ?is the mean (monthly or seasonal) temperature distri-
bution, stip1ated by radiation, turbulence, and di_Sposi t.ion of conti-
nents and oceans. This distribution is considered. as fixed, with
the zonal temperature given by the formula which is analogous with
formula (5) for zonal pressure:
77)T-)-4f5;2O (7)
,
where and iW are constant magnitudes, 7it is the departure
from steady temperature di.stributione 1agnitudes rand r ich
express the perturbations of the zonal t.em .erature, are considered
small.
Assuming that there are no vertical velocities, and disregarding
the variatiGns in density, t~linova interpolates the stream function ) ,
proceeding from the equation of ontinuity
It is obvious that the stream function ' for a small perturbation
may be presented as the sum of 2 star'earn functions
O( A1 ) =- '( A) -1 92 " t,)
is a stream function, which is not dependent on time,
and it expresses the steady part of the perturbaT,ion0of~r zonal circulation.
'Ihi.s function is stipulated by the presence of the steady perturbation.
of the zonal temperature di stri1:uti.,on, expressed by term A in
1
equality (6), and
is the stream function expressing the non-
steady perturbation of zonal circulation.
ny~ .
,
1.._. , ?- t:
', ,r. ,.. 1. .., .., ,.f , . ,: W s Y Fv. :L'
,.~. .ti ., ~,.., ..- v. '... r. r,n , ~.. ,. ,. ,,:,.. ... ,. _, , ..,.;... ,, .. .,. s .r..,. ,. i. r ~. -t _..b.m
. ~i? t, -. .. ~,. ,.n: ,;XI,a v. k. .. ~ 1:'. .~a., ., .. ... .. ...... .,.., ,. .. .. h .4. .r ,. r:. 4C
re. .~ f....k. 4. ., _., F-e~
.... ,. ~,. ~,~. .. v. .... .' ... . , u .~. ~ ,.. .._ '
. r 'l. ' 1. . .. n,. t. - w,.: . f W. I. aY.C.
,Cs ,. r?. R. ~:., d,, .,w. ,, .c ,a -.,,i' w, ,,. Ar,. .. ~:~.... , .r ,w t. ,. 6...e r .,n.?qs ~ .:>...c.N, .rk.Mw.,,
..~,,ux.e hl~,a9sr:r s ,u*.. Pl+aSw.,, ..v ...a, roz rdu `P'.am.4nYt,u~v,ur P4:.J,,,,PA P9~~kaxrl#,d.~~M kaz. d.l~iW:~lFrt3 ..w?r...
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2 (
~) $a~ B ~ ~J t 2,~aC ~Gt~) e'en 8
The equation of motion, taken for axis
, has the form of
Finally, l linova takes the equation of heat inflow in the form of
Dy the same token, the pressure perturbation t is taken
f 1
as the sum of two components: which is the steady
perturbation of zonal pressure di. stributio~rt, and t~
t), ih ich is thy'
/ e
non-steady perturbation:
By projecting the Fridman equation (2) onto axis r, by substi-
tutang the velocity components with the respective derivatives of the
stream function, and by disregarding the small terms, J3linova deri..ves
0 ?he followJ..ng basic equation of the i. rcblen
She simplifies somewhat this equation by d.isreg