MONTHLY WEATHER REVIEW Editor, EDGAR W. WOOLARD
VOL. 67, No. 5
W. B. No. 1267 MAY 1939 CLOSED JULY 3, 1939 ISBTJEIJ AUGUST 1939
THE READJUSTMENT OF CERTAIN UNSTABLE ATMOSPHERIC SYSTEMS UNDER
CONSERVATION OF VORTICITY ’
By VICTOR P. STARR
[Weather Bureau, Boston, Mass., February 19391
In his memoir “On the Energy of Storms”, Margules
devised methods for the calculation of the maximum kinetic energy of atmospheric systems that can result
from the rearrangement of two air masses of differing
properties, from an initial unstable arrangement to one possessing stability. Although his results show that the
kinetic energy made available by these processes is suffi- cient to account for observed wind velocities, no attempt
was made by Margules to deduce the probable distribu-
tion of this kinetic energy within the air masses wdiich comprise the systems studied. In order to secure some idea of the character that this distribution might be ex-
pected to assume, a problem similar in certain respects to
M
earth’s rotatmion on all the motions involvecl in the readjust-
ment? Also, what will be the distribution of velocities
genera ted by the action of pressure gradients and of
Coriolis forces within each mass of fluid? In outline the changes that take place upon the transi-
tion to a state of equilibrium may be described as follows:
A portion of the denser fluid located in the region A in the initial state will tend to be displaced to the right. Sinco
this displacement is accompanied by a Coriolis force ac ting
a t right angles to the direction of motion, a transverse
horizontal velocity will be generated in this portion of fluid. In n steady state this velocity must be accompanied
by a gradient of pressure which in this case is supplied by a
-
A
P
B
P‘
FIQunE 1.-Two fluid maqses in the initial stnte.
some of the cases treated by Margules will be formulated
and solved making use of a principle which may be alluded to as that of conservation of absolute vorticity. Because
of the great complexities which arise in an attempt to deal
with a compressible atmosphere, attention will be COP-
h e d to a system composed of two homogeneous and in-
compressible fluids of slightly differing densities. It will
also be assumed in the treatment that all frictional effects
are absent. Let it be supposed that in the initial state two masses of
fluid of differing densities lie side by side, a t rest relative to the earth, separated by a vertical plane as in fi ure 1,
thus formed is considered as being of uniform thickness
and extencling an infinite distance to the right and to the left. The problem for which a solution is sought may be stated as follows: Assuming no mixing of the fluids, what
will be the shape of the free surface and of the internal
boundary, if the system is allowed to come infinitely slowly to equilibrium, taking into consideration the effect of the
with the lighter fluid to the right. The horizonta 7 layer
1 This paper is a report 011 an investigatlon which. has been conducted at the Massa- chusetts Institute cl,Technology in cooperation with the Weather Bureau under the BankheadJones Special Besearch Fund.
161835-39-1
mutunl adjustment of the slope of the free surface and of
the internal boundary separating the liquids. Likewise a portion of the lighter fluid originally situated at B will
tend to move to the left, bringing about a transverse
velocity in the opposite sense. The requisite pressure gradient in this case is produced by the deformation of the
free surface alone.
Due to the fact that continuity of depth must be
preserved throughout the layer, it must be assumed that
some lateral movement with resulting transverse velocities must occur in the regions farther to the left and to the
right, decreasing to zero a t great distances from the
original discontinuity. It is now possible to schematically represent the final state as in figure 2.
It thus appears that for the purpose of analysis the system may be divided into three regions, regions I and
111 in figure 2 consisting of a single la er, while region II consists of two superimposed layers. 4 he distances c and a
are to be obtained as a result of the investigation. In the above diagrnm the positive y direction is taken to the left
and the positive 2 direction as perpendicular to the plane of
the figure away from the reader, while the origin is located
125
126 MONTHLY WEATHER REVIEW MAY 1939
at, 0. For convenience, the diagram is taken 111 the plane
of a meridian with tlie positive y direction estending north-
ward. In the Northern Hemisphere this leads to easterly winds in the regions marked E and westerly winds in the
of t8he axes, this choice of coordinates will not produce any
lacli of generality in the results. Restricting our att<ention for the present to region I, let it be assumed that, a column of fluid which in the initial
state was located a t a point yo1 and had a width dyo' and
a heiglit Do is iiow locntetl a t a point y, has a width d y
nnd a height D'. The principle of continuity of mass gives
I n the above p,, may be taken as the pressure a t the sur-
face, for which the value gpD' niay be substituted. Sub-
stituting also for u' from (5),
(7) region marked TI'. Since the equations of motion used in meteorology are independent of the azimuthal orientation
0 = -f'(y - y"') - 9- dD' . tlY
dD1 Substituting for - froni (3), dY
Qo d"ol o= (y-yyo') +-- 7-
f' dy' (8)
Since tlie quantities g , Do niidf are considered constant in
m I I I
E
FIOIJBE 2 --Schematic diagram of the flnal state.
Y'O
(I) this whole discussion and since t,he factor 9 recurs
frequently. a quantity X having the dimensions of a length
will be defined as follows, and substituted in the equations:
(9)
pDodyo'=pD'dy
DI=D d&' (2)
O dY or
where y is the acceleration of gravity and p is the density.
By differentiation, A=--- JBno .. -7 (3) Equation (S) now becomes,
(10) Indicating velocities in the x direction by u' and velocities d'yo' I
dY2 7-+p(y-yyo')=o. in the y direction by v the equation of motion for the transverse velocity of the element, since no net esternal
forces nre present, is The solution of this equation is easily shown to be
where f is the Coriolis parameter 2 Q sin4.
between the limits yo' and y
Integrating
ur =f(y- YO'). (5 )
This implies that the velocities are constant with elevation
and therefore the pressure gradien ts must be independent of elevation--a result which agrees wit8h hydrostaticnl
considerations. Since a steady state is assumecl, in the final condition the gradient wind equation niust be fulfilled. For straight
line flow, this w-ill be
In this equation Qr and Q"' nre constants of integration
which must he fised hp boundary conditions while e is
the Napierian base. Obviously Q"' must be zero in region
I since the liquid is to be undisturbed a t great distances to the left. This is only possible if yI0=y for large values
of y. The value of Q' on the other hand cannot be determined a t this point, but niust be evaluated later, siniultaneously with the determination of other similar
integration consttLnts for the remaining two regions. The fundamental equation for region I, giving the lateral
change of position of a particle from the initial to the
final state niay be written,
MAY 1939 MONTHLY WEATHER REVIEW 127
From this equation the ve.locity distribution niay be
obtained by the aid of equation (5) and is found to be,
Y -- (13) ul= -fpe x
By differentiating (13) with respect to y and combining
the result with (2) an espression for the shape of the free
surface results,
(14)
Considering now the conditions in region 111, an ex-
actly analogous line of reasoning may be followed as wis
used above, and an equation comparable to (11) niay
be set down,
(1 5 )
It is clear, however, that now the constant Q' must be
zero since the fluid is not' disturbed a t great distances to
the right, and the fundamental equation for region 111
becomes
Y
y2II = y + &"'ex. (16)
The constant &I1' will be evaluated later. Espressions for the velocity distribution and for the
shape of the free surface may be written imrnediiLtely us
in the case of region I.
and
These are respectively,
Y
(17)
(1%
Turning our attention to region 11, the situation is more
complicated due to the presence of a double layer. Re- taining the convention of indicating quantit'ies referring
to the initid state by the subscript zero, and designating
quantities applying to the upper layer by prinies, an elemental vertical column extending from the surface to the top will now be considered. Corresponding to equa-
tion (l), which specified the requirement of continuity of mass, there are now two such equations:
p == - f QIIIe5;,
D1I1=D0 .["I 1 +,@"ex .
p Do'dy~I' = p'D"'dy, (19)
for the upper layer and
pDdt~O"= pDI'dy (20)
for the lower layer. I n these two equations a distinction
is made between the quantities Do and Dof. Although according to the statement of the problem made pre- viously these are equal, this simplification will not be
made a t this point, in view of the possible application of
the equations derived to more general problems than the
one dealt with a t present. Solving (19) and (30) for the depths in the final state,
(21)
(32)
and
(23)
Corresponding to equation (5) there are two espressions for velocity, namely
and
We also have two gradient wind equations:
for the upper layer and
(27)
(28)
for the lower. A question now arises as to the interpreta-
tion of pa' in equation (27). Since (27) involves only the gradient of pressure and not the absolute value of the
pressure, the derivative appearing may be evaluated by forming an expression for the pressure distribution along
any level surface in the upper layer, and differentiahg. This can be done conveniently by considering the pressure
at a level surface situated a t an arbitrary elevation 2.
Thus
Differentiating and changing to ordinary derivatives,
p = gp' (DIIf + D" - 2). (29)
For tlie lower layer,
and p a = gp'D" ' + gpD" (31)
Substitmuting in equations (27) and (38) from (25) and
(26) and also from (30) and (32)
Subst,ituting in the above two equations for the deriva- tives of the depths tlie values given by (23) and (24),
we get,:
and
We thus finally have a pair of second order simultaneous
differential equat'ions involving the single independent Differentiating the above with respect to y
128 MONTHLY WEATHER REVIEW MAY 1939
variable y and two dependent variables yo1*’ and yo”, which must be solved. Customary methods of separating
the dependent variables lead to two fourth order equations whose solutions contain eight constants of integration. Rather than to pursue this course the separation of variables will be efl’ected as follows: Let the first equa-
tion be multiplied by an umcletermined constant factor a,
so that the equations have the form,
d2yo11’ - g [( :+ a ) Do’ dy” + (1 + a) D $F ] - (39)
Rearranging,
0 = f*[yo’: + a y y - (1 + alYI
The solutions of (48) and (49) may be set down h m e -
diately in the form
and
in which K,, K,, Q,, and Qz are constants of integration
and K~ and K~ are given by
A1= (l +~~~)y +K ,e ~l ~+Q ~e -~~~ (50)
d2= (1 + a2)y+K2e‘au+ Q2e-N3v, (51)
1
K2= x J l f a a (53)
Reverting to the original dependent variables, yd” and
yo11, we have from (44) and (45)
aid
L41 -A2= a1y:”- azY:”=~~I’(a~-aZ), (54)
(55)
111 -A1-A2 Yo -=&-
and
Let the value of a be now defined SO as to make the coeffi-
cient of y:” in the first brackets equal to the coefficient Substitutblg ill (5 5 ) and (57) from (50) and (51),
of y,,II‘ 111 the second brackets. Ths gives the relation yoII’= y + - [ KleclY - I<2~Kau + Q,e-Klv- Qre-‘av]; (58) al - a?
(41) and a=- P ‘+P a n,l P 0 +4 Do, Yo”=Yf- [a2Kle~lu- alK2e‘tY+ a2Qle-x1u- a l ~2 e -x ~v ~. (59)
Let the simplification now be made that Po=D0.
(60)
(61)
which has two roots, az-ai
(yl= -~ Do-Do’+5d 1 (Do-Do’)’ D; += 4p‘Do‘ (42) From (42) and (43) it is seen that 2 0 ,
and
a2= - Do- 2Do Do’ - ’J 2 (Do io?’)’ +r 4 PI Do’ (43)
Let two dependent variables A1 and Az be defined by the
following:
AI = yoI’ + (YlYOII’, (44) In other words we may say that
A 2 = 3:’ + a2ygl”. (45) (@)
Introducing these into (40) two equations in which the variables are separated are obtained: The final form of the fundainental equations for regioii I1
thus becomes,
1
2a y:”=y+, [KleKlv-IC2e1~~+ Qle-K1~-Q2e-x~*], (63) (46) gD0 d2Al. 0 = Al - (1 + al)y -- (1 + a1)- 2
f’ dY2
(47)
Introducing and rearranging, By the aid of (25) tlnd (26) espressions for the velocities
in the two layers are obtained:
(48)
(491
d’d, 1
(PL4, 1
dy’ (1 -I -a 2 )X 2
-[A - (1 + a1) yl = 0 ;
[Az- (1+adyl=O.
-- dy2 (1fa1)XZ - - - - [K ,A ~I ~-K ~~*~~+&~B -~~~-&~~-~I (65)
9 u”= -~[K1e+‘+&eKaY-~ Qle-Klu+ Q2e-c9v]. (66)
2 1Y
__
I 1 This procedure was suggested to the writer by Prof. C. Q. Rossby.
MAT 1939 MONTHLY WEATHER REVIEW 129
By means of (22) and (64) the position of the internal
boundary may be specified:
The thickness of the upper layer can be obtained in a
similar manner from (21) and (63):
DII'=Do 1 +~(KIKlexlY-KZKZex~'- &IKle-'IY+ Q2~2e-rsu)]- (68)
c 1 The shape of the free surface is given by the sum of D"'
and Dxl for each value of y. Having derived the fundamental equations (12) and (16)
for regions I and 111, respectively, and (63) and (64)for region
111, it is next necessary to determine simultaneously the
six integration constants Q, PI1, Kl, K,, Q1, and Qa and also the constants c and a. In order to accomplish this it is necessary to have eight equations not involving the independent and dependent variables. By examining
the requirements a t the two vertical boundaries separating the three regions it will be seen that there exist exactly
eight independent boundary conditions which can be inter-
preted in the form of the eight required equations. Taking first the vertical a t y=O in figure 2, the fluid in
the upper layer of region I1 a t this point originally was located a t the discontinuity, hence the condition exists
that when
Continuity of displacements gives the relation that when
J
y=-c
yolII * (75)
Finally again c0ntinuit.y of depth gives the condition
y=-c 1
Imposing these eight boundary condit'ions in the order
given upon the fundamental equations (12), (16), (63), and (64), there restdt the eight equations written below:
- aa= ,[IC1 - Kz + &I - Q21 (77) 1
1
2 - 1 =-[Kl~le-x~C+K~2e-K~C- &l~leriC-Q&z~zeKaC] (82) (69)
upper layer of region I1 becomes zero, hence mcording to
dyt'' . (70)
Also it was assumed t h ~t a t this point the thickness of the
(21) when ,&IIIe =-[K,e-~~~-K~e-xs~+ 2 Qlexlc- Q,e~f]
c --
(83) A 1
1 -; 1 aQ"I_xe =jj[KlKle-'1'- K2K2e-'YC- &IKlf?'lC+ &2~ZexsC] (84) -- dY '/=OI -0
Since the fluid in region I is in contrlct, with the fluid in the lower layer of region I1 a t y=O, there must be continuity
in the disp1accnient.s a t this point so that
(71) Yo1 = Y,'I
Finally, since the depth of the fluid in the lower layer of region I1 must, be the same as that of region I a t this point,
we have from (2) and (22)
(72)
At the vert4icrll houiirla.r;v between regions 11 and 111
there exist, four additional conditions which arc coni- parable to the four already given.. Espressing the dis-
placement of the lower layer of region I1 a t t'his point, we
have the condition
I-
y= -c yo"=-a (73)
The Fanishing of the thickness of the lower layer gives the condition that
It will be noted that a11 the constants enter these equatinns linearly except c, which is contained both linearly and
exponentially. It is therefore possible to eliminate the
six constants of integration and a algebraically, the result being a single esponcntinl equation in c which can be solved by trial for nny particular numerical example.
When this has been accomplished. the reninining con- stants may be determined without difficulty. The actual process of carrying out the elimination beconies rather involved, however, and it would not be practical to
reproduce the necessary steps in this discussion. From the relation expressed by (62) i t is seen tlint the
solution of the entire problem depentls in n o way upon
the absolute values of the densities but only upon their
ratio d. Taking a figure of 0.960,788 for this ratio,
a, K ~, and K' have the following values: P
a=0.980,198
0.710,634
x
7.106,335
x
K1=
K2
P'
(74) P This particular choice of - makes K~ exactsly t,en times
larger than K~ which simplifies the calculat.ion slightly.
130 MONTHLY WEATHER REVIEW MAY 1939
Corresponding to these conditions the eight constants to be determined assume the numerical values given below in terms of A. It w-ill be found that these figures satisfy
the last, eight equations sufficiently closely for practical purposes.
C= $0.336,10X
a= +0.159,07X
Q'= +O.O09,08X
p= +O.O10,7SX
KI= -O.'i69,48X
K z = +0.150,86X
Q1= +0.622,61 X
Q 2 = +0.014,20X
Inserting these figures in the espression for t81ie depths
discussed previously, and assuniing a value of Do equal
right of the free surface and the generntion of mild easterly winds as shown in the velocity diagram. In region I11 there is also a shift of the fluid to the right amounting
to 0.00771X at, y=-0.3361X and decreasing to zero for large negative values of y. This causes a piling up of
the fluid in this region so that the free surface again
slopes downward t'o the right and provides the necessary
Coriolian pressure gradient for the mild easterly winds indicated on the velocity chart. In region I1 the original
discontinuity M is located a t y=-O.l5907X. In t'he
lower layer there exist easterly winds which a t the
boundary y=O are equal-in intensity to those in region
I at this point, since continuity of displacement exists at
the boundary. Farther to the right these increase rap- idly and reach a maximum at the point y=-0.3361X.
The Coriolian pressure gradient, as already mentioned,
is here provided by the inclination of the internal boundary and of the free surface. In the upper layer the lateral displacement of the verti-
cal columns may be thought of as the sum of two effects.
-.2692X I k?
&"
.20-
VELOCITIES UPPER LAYER
.05--
0
:05--
-.IO--
--.E-- VELOCITIES LOWER LAYER
720-
FIGURE 3.-Readjustmtnt of two air masses under consprration of vorticity.
h = J== 4 (2800 km)
to the height of the homogeneous atmosphere which is
taken to be eight kilometers, the shape of the free surface and of the internal boundary- may be plotted. In this way the diagram reproduced in figure 3 is obtained. In middle latitudes tlie vnlue of A is approsimately 2,500
km., which gives a horizontal scale for the diagram.
Likewise the velocities imy now be obtained. These are also plotted in figure 3, the unit of velocity beirig-fh which in niiddle latitudes hns the approsininte value of
280 meters per second. The directions are taken :is
corresponding to the Nortlieni Hemisphere.
The general features of the final state may now- be
described as follows: Beginning with region I, there is here a shift of the fluid to the right amounting to 0.009OSX
a t th\e point y=O, and decreasing with increasing posi-
tive values of y. This leads to a slope downward to the
In t,he first, place, since there is a shift of the fluid in re.gion I11 t.o t'lie right, and since t,here must' be continuity
of displacement a t the. vert'icd boundary, the liquid in t,he upper 1ape.r in tlie vicinity of the boundary must take part in this motion to t,he right. Secondly, the
vert,ical shrinking of the fluid columns from the original
height Do tmo that shown on figure 3 must lead to a dis- placement t,o the left in order t,o preserve continuity of volume. Because these t,wo eff e,cts oppose one another,
t,he,re must e,sist a vertical in the upper layer of region
I1 at which the displacement, the velocitry, pressure
gradient and slope of the free surface a.re all zero. By set,t,ing yO1I'=y in equation (63) this vertical is found to
be at y=-0.269%. It is also apparent that the eleva-
tion of the free surface must reach a masimum a t this point. This turns out to be 8.064 km., or 64 met'ers
MAY 1939 MONTHLY WEATHER REVIEW 131
above the initial depth of the fluid layer. The velocities
for the upper layer are as shown, being mild easterly to the right of this vertical and increasing westerly to
the left. The free surface has an increasing slope down- ward to the left,, the depth reaching a minimum a t y=O
where the value is 7.927 km. or 73 meters below the
initial depth. Having obtained a description of the equilibrium state,
it becomes a matter of interest to esamine the energy changes that must occur during the transition. Initially
the system possesses no kinetic energy, but has a certain amount of potential energy of mass distribution. In the
final state a certain kinetic energy of transverse circulation
exists, while the potential energy is less than originally.
It is now proposed to make a quantitative calculation of
these energy changes.
The change of potential energy between the two states in region I may be expressed in the form of the following
integral; energy rendered available being reckoned as positive :
(85) J U \- -/
Upon substitution for D' from (1 4 ) and integration be-
tween the limits indicated it is found that this quantity
assumes the value of 0.004,519 pgDo2h. Taking cognizance
of the fact that one dimension of length is suppressed, it will be seen that this quantity has the dimensions of
energy. The corresponding integral for region I11 gives the
result that the change in potential energy is -O.O03,S66
In region I1 it is convenient to calculate the change in
potential energy for each layer independently. In the lower layer an integral of the form of (S5) may be applied
directly, although the integra tion is more laborious than
in regions I and 111. The result is that the change in
potential energy is equal to 0.O3Ol442~gU~'X. For the
upper layer the integral has the form
p'gD,2X or -O.OO3,714pgD~X.
s" -0.3361X ~-(n "+~)]p ' g D r r r d y . (86)
The evaluation of this quantiby gives as a result, -0.031,-
174pfgDo2X or -0.030,953pyD,sX.
Summing for the three regions the net change in potential energy turns out to be +0.001,29.5~gDn2X,
The kinetic energy of the transverse circulat,ion for region I may be expressed in the form of the integral,
(Si)
which turns out to have R value of O.C)00,030 Qf2Doh'.
The unit of energy appearing here will be seen to be equivalent to that occurring ili the calculation of potential
energy if the value of he substituted for Xz. ('or-
responding integrals may be set up for the remaining regions. giving the result tliat, the kinetic energy is
0.000,150plf?DoX3 and 0.000,357pf'DoX3 for the upper and lower layers of region 11, respectively, and o.oon,o29
efZDOh3 for region 111. Summing for the three regions, the total kinetic energy of transverse circiilation for the system beconies O.OOC),~%Q~~D~X~.
Comparing this figure with that for the potential energy
liberated during the readjustment,. it is seen that the latter
17D"
.f
is about three times as large as tlie former, and the ques-
tion arises as to the physical meaning of the lnrge differ-
ence between these quantities. E t will be recalled that in the original statement of the problem the assumption was made that the transition from the initial to the final state proceeds infinitely slowly. Under these conditions it is
permissible to neglect all accelerations in the y direction
so that the system arrives a t the equilihrium state without any finite velocities in the y direction.
Had this restriction not been made, but instend the
readjustment allowed to take place a t its natural rate, the solution obtained above would no longer describe com- pletely tlie conditions when the system arrives a t the
equilibrium state. In addition to the transverse velocities
in the x direction there would also exist finite velocities in the y direction. By virtue of these velocities the read-
justment would proceed beyond the equilibriuin state, then reverse its direction and the system would continue
to oscillate about the equilibrium position. It appears that the fraction of the potential energy liberated which
is not used to establish a transverse circulation represents energy of inertia oscillations whose character, however,
cannot be in any way determined from this investigation. I t remains to be pointed out, however, that in general
differences in density in the atmosphere are set up largely by gradual radiative processes, nnd that rendjustment of actual atmospheric systems takes place simultnneously
with the slow establishment of density differences. For this reason readjustment processes must sutoninticnlly
take place on a nearly quasistatic basis, passing through an infinite number of equilibrium stages, and arriving a t
the final state witliout appreciable energy of oscillation. Under these conditions the amount of lieat energy re- quired to bring about the change in densit;v is less than
would he required if the henting and expansion took place prior to the readjustment of mass distribution.
As a variation of the above problem it would be of
interest to determine tbe changes that would be produced if a third inass of relatively lighter fluid overlies the two
overturning air masses. Strictly spenliing, since there
would be upon rearljustnient a deformatioii of the internal boundary between the two Inyers thus formed, Interwl convergence ancl divergence accompanier1 by the qe11e~q-
tion of weak transverse circulation und also a defonnntion
of the free surface should be expected in the upper layer.
If, however. the upper layer is relatively deep, these
effects would be sufficiently sninll to he neglectetl, tlie upper layer being considered at' rest with a level surface
in the final state as well as initially. An examination of the conditions imposed by tlie ncldi-
tion of such a layer shows that dmost the same fund:+
ment:il equations as were previously derived will serve to specify the solution of a problem of this type, provided
that for each set of densities chosen, the initid tlepths of
tlie two air masses comprising the lower layer he:lr a
given fixed ratio to each other. We may thus represent
the initial (clashed lines) and final stntes schematically
as in figure 4.
In region I the pressure a t tlie surface niay be expressed
as follo\\-s:
where p , is t,he pressure a t the level DO'. differentiated with respect' to y,
\\lien this is
- "
132 MONTHLY WEATHER REVIEW MAY 1939
P"
II m Yo I
FIGURE .I.-Srhernatic representation of fiwl h t e 01 system containing three alr m s w .
1ncorporat)ing this expression for the pressur.e grnclient in
cquation (6) and proceeding as in the prenous problem,
the fundamental equation for this region is fount1 to be,
yoI = y + Q'e -5 (90)
in which A' is now defined as
Esactly similar consiclera tions lend to the following
y0111=y+ &kIIe& (92)
fundamenttil equation for region 111,
where AI1' is given by
dgDof py
(93)
In region I1 the pressure gmdieiits may be oht,zir.ed
XI" =
.f
from the followng expressions for ps' nnd pu:
p,' =gp' (D" + D"' - 2) + gp" (Do' - I)" - D"') + p ,, (041
(95) p o =gpD" +gp'D"' + gp" (Do' - D'I - 0"') + pc.
Differentiating these
Again defining a by equating the coefficients of yor1' and
solving, the following equation results:
If now the fundamental equations for region I1 :ire to have the same form as (63) and (64), it is necessary that the roots of this equation differ only as to sign, as was the
case previously. This imposes the condition that
-- Do' P ' (P -P f f )=O .
Do P(P'-P")
Retaining the same rnhio of densities in the lower layer as was used in the first problem and assuming that p"=
Do
0 .8 0 ~~ the value of the ratio DO'
~
t.urns out to be 1.195,098.
The quantity CY now becomes equa.1 to
constants K ], K ~, and A are defined as follows:
1
I'=
P -P 'I P ,P (101)
A- J i a
Carrying out similar operations as previously, an equation f
comparable to (40) is obtained: The condition (100) also causes X I to be equal to VI1.
MAY 1939 hlONTHLY ITEATHER REVIEW 133
boundary conditions can be done in almost' esmtly the same manner as before, leadilig to the nunicricnl i-ti,lues
given below:
The determinat.ion of the constants to be fised by BUMMARY
1. If, on a rotating eftrth, a layer of incoIilpressible fluid
of uniform depth and infinite horizontal estent composed
of two motionless fluid masses of differing densitv a.nd sepa,rate,d initially by a plane vertical partition, is allowed
to readjust itself slowly upon removal of the pa.rtition, the system will come to a state of equilibrium in which the two
masses a,re se,para.ted by an inclined boundary sloping
downwa,rd toward the lighter fluid mass, and in which the free surface is deformed in such a nianne,r that the maxi-
mum depth occurs a t a point above the internal boundary and the minimum depth a t t,he point where the thicl- \ness
of the lighter fluid becomes zero. In obtaining this result
n.nd those that follow, all frictional effects as well as mixing
n t the mutual boundary of the fluids, are assumed to be
absent. The Coriolis parameter is not a.ssumed to vary
formed as previously, and if the results are plotted on t,he within the dimensions of the system. For an initial
C= +0.335,15X
a= +0.180,20X
Q"'= -O.O27,73X
K1= - 0.12 5,19 X
IC2= +0.171,17X
Q1= +0.324,97X
The expressions for the depths and for the yelocities are
&I= -0.00'7,13X
Q 2 = +O.O14,81X
N
.20-
VELOCITIES UPPER LAYER
.05--
-.05--
A=--= (2800 km.)
% --.15--
-.20-
FIQURE 5.-Readjustmcnt of three air masses under conservation of vorticity.
.20-
VELOCITIES UPPER LAYER
.05--
-.05--
A=--= (2800 km.)
% --.15--
-.20-
FIQURE 5.-Readjustmcnt of three air masses under conservation of vorticity.
same scale as before the diagram shown in. figure 5 is obtained. It is seen that now a weali west wuid prevails
in regions I and I11 and also in the left portion of the lower
layer of region 11. The upper layer nom has n continuons distribution of westerly winds, while the right hand por-
tion of the lower layer in region I1 has easterly winds with velocities as shown. The initial state is shown by the
dashed lines. In concluding, it might be remarked that the above analyses depend essentially upon the fact that in an in- compressible fluid of uniform density, that is, in a homo-
geneous and incompressible fluid, there can be no variation of gradient wind with elevation. This is also true for a
homogeneous and compressible fluid such as an adiabatic
atmosphere. It should therefore be ossible, n t least, in
composed of two air masses each hnving a constant poten-
tial temperature.
theory, to carry out a similar calcu .p ntion for a system
depth of 8 kni. and a ratio of densities equal to 0.96 the
shape of the free surface and internal boundary is given
by figure 3.
2. Due to the action of pressure gradients and Coriolis forces during the readjustnient of the above system, hori-
zontal velocities parallel to the original discontinuity will
be set up. The distribution of these velocities will be such that if in the Northern Hemisphere the discontinuity
is imagined, for the purpose of reference, as extending from east to west with the lighter fluid to the south,
easterly velocities will appear in the denser fluid, decreas-
ing to zero at great distances to the north and increasing southward to a niasimum a t the point where tlic thickness
of the heavier fluid becomes zero. In the lighter fluid the velocity will be zero a t great
distances to the south and also along the vertical a t which
the elevation of the free surface is a maximum. To the south of this vertical the velocities will be mild easterly,
134 MONTHLY WEATHER REVIEW MAY 1939
having a maximum above the point where the internal
boundary intersects the lower surface of the layer. To the north of this vertical, westerly velocities increasing northward will prevail, retiching a maximum a t the point where the thickness of the lighter fluid hecomes zero.
Within each n:ass of fluid there will be no variation of
velocity with elevation. A graph of the velocities for a
density ratio of 0.96 and an initial depth of 8 km. is given in figure 3. The unit of velocity used is a function of
geographical latitude and amounts to about, 2S0 meters
per second in niiclclle latitudes.
3. I n the above system there will be a decrease of
potential enercy of mass distribution during the read- justment. Also, there will exist in the fianl state a certain
amount of kinetic energy represented by the relocity dis-
tribution already described. Computation on the ex- ample already cited shows that the amount of potential
energy set free is nearly three times as great as the kinetic energy represented by the circiilaiton set up. The
difference between these two quantities is accounted for by the fact that the readjustment was assiinied to take place
slowly on a quasistatic basis, implying the existence of an
external retarding agency which nhsnrbs a portion of the potential energy rendered available.
If the process is considered as taking place a t its natural
rate, the system will then arrive a t the equilibrium point with a certain amount of kinetic energy of oscillation. If this additional kinetic energy is included, the suni of
potential and kinetic energy for the system remains con- stant a t all stages of the process. The character of tlie
oscillations perfornied by the system cannot he detcr- mined from this analysis. However, it is prohnhle that in
the actual atmosphere density differences are generally set
up gradually, so that redjust’ment proce,sses automatmi-
cally proceed in an almost qua.sistat,ic manner, aad energy
of oscillat,ion does not a.ppear t,o a significant ext,e,nt>.
4. A problem comparable to the above but involving
the presence of a third ma.ss of fluid overlying t,he system can he treat,ed by a.lmost exn.ct,ly t>he same niat,hematical
set-up sa,ve for numericd vdues as was used in the first andqsis, provided that t,he clepth of t.he third mass is
re.latively large in comparison with the ot’her two, and that
the initia.1 dept,hs of t,he first two masses bear the following rehtion to the densit,ies chosen:
D0’- P! (P- P”) ,
a- - P (P’ - P” 1
where Do and Do’ are the clepths of the havier a,nd lighter fluids respect’ively a.nd p and p’ their densit’ies, while p” is
the density of t’lie third mass. Taking the same value for the rat’io of densities in the
lower two masses and assuming a value of p” equa.1 to
0.80 p, and a va.lue of 8 km. for Do, the result shown in
figure 5 is obtaine,d. The dia,gra.m is to he interpreted in the same. manner a,s figure 3.
The writer wishes to acknowledge his indebtedness to
Prof. (2. G. Rossby of the Massachuset’ts Institute of Technology for suggesting t,liis investmigation, a.nd for the
continual assist.a,nce and encouragement, which he so
gladly gave. The underlying principles involved in this work form a. part of t8he niaterinl c.overetl by Professor Rossby in his lectures at8 the Inst,itute which the author
ha.s had the ple,asure t,o a.ttend. Also, the author desires
to express his apprec.iat,ion of the he,lp rendered by R. D. Fletcher in re,ading the ma,nuscript and checking the
numerical work.
BIBLIOGRAPHY
[RICHMOND T. ZIXH. in Charge of Library1
By AMY P. LESHER
RECENT ADDITIONS
The following have bee,n selecte,cl from among t,he tit,les of books rece,nt,ly received a.s represent>inp t81iose, most)
likely to be useful to Wea,t.her Bureau officials in tbrir meteorological work and st>udies:
Aliverti, G.
Sugli elementi elettrici del clima in rapport0 alla terapfa cli- matica. Pavia. 1936. 9 p. illus. 3034 cm. (Estratto dal
Bollettino del Comitato per la geodesla e la geofisica del Consiglio nazionale delle ricerche. Serie 11, anno VI, n. 1-2, gennnio-aprile 1936.)
Auer, Reinhold.
uher die allseitige Intensitfit. der Ultrastrahlung in der Atmos- phiire. Berlin. 1939. p. 559-587. tables, diagrs. 26f$ cm. (Reprint from Zeitschrift fur Physik. 111. Band. 9. und 10.
Heft.)
Bergeiro, Jose M.
Climatologfa agrfcola. Montevideo. 1937. 15 p. maps, tables, diagrs. 28 cm.
Bhar. J. N.
Stratification of the ionosphere and the origin of the EI layer.
Calcutta. [1938.] p. 363-386. tables, diagrs. 27 cm. (Reprinted from the Indian journal of physics, v. 12, pt. 5. November 1938.)
Brazil. Divislo de meteorologia.
Formas symbolicas e codigos internacionaes, para despachos
niet,eorologicos cifrados. Rio de Janeiro. 1937. 130 p.
tables. 24 cm.
Carrier corporation, Syracuse, New York.
Air conditioning. Refrigeration. Heating. Syracuse. 1938. 15p. illus. 305cm.
Carton, P.
Cartes plrivioniCtriques moyennes mensueIIes e t annuelles, annee nioyenne 1907-1924, du Tonkin et du Nord-Annam, de la C‘ochinchine e t dn Camhodge. Hanoi. 26 maps. 30 em. (Supplhment au Bulletin 6 c o ~~~!q u ~ {e l’Indochine, 1935.)
Castrill6n, Manuel Alvlrez.
Estudio estadfstico de las caracteristicas meteoroMgicas, seghn
la direcci6n del viento. Barcelona.,. 1936. 40 p. tables, plates. 30 cm. (Mem. del’acad. de cienc. i arts de Barcelona. Tercera Bpoca. v. 25, no. 11.)
Christensen, Lars.
Ma derniPre expedition aus regions antarctiques (1936-37).
Avec un apercu des recherches faites au cows des expeditions entreprises de 1927 A 1937. Oslo. 1938. 16 p. front. (port.), plates. 23% cm.
Dove, Leonard P.
The dust storm of January 1921. [n. p.] 1921. p. 24g252.
plate, table. (Reprint from the Quarterly Journal of the University of North Dakota. v. 11, No. 3. April 1921.)
Le osservazioni anemologiche eseguite nella Libia e nell’ Impero italiano in relazione alla circolazione atmosferica generale
del continente africano. Perugia. 1937. 14 p. maps, tables, diagrs. 31f.L cm. (Estratto dalla Rivista “La Mete- orologia Pratica,” Anno 18, num. 3.
_-- Le correnti aeree a1 suolo e a quote a Tripoli. Perugia. [1936.] 11 p. tableb, diagrs. 30% cm. (Estratto dalla Ri-
vista “La Meteorologia Pratica,” Anno 17, num. 6. 1936.)
Kew England hurricane, a factual, pictorial record, written and
compiled by members of the Federal writers’ project of the Works progress administration in the New England states.
Boston. [c 19381. 222 p. illus. 26 cm.
Eredia, Filippo.
1937.)
Federal writers’ project.