MONTHLY WEATHER REVIEW
Editor, W. J. HUMPHREYS
CLOSED JUNE 3, 1935
ISSUED AUQIIST 12, 1935
VOL. 63, No. 4 APRIL 1935 W.B. No. 1156
MATHEMATICAL THEORY OF THE GRAPHICAL EVALUATION OF METEOROGRAPH
SOUNDINGS BY MEANS OF THE STWE (LINDENBERG) ADIABATIC CHART
By LOUIS P. HARRISON
~-~ -
[Weather Bureau, Washington. May 19351
INTRODUCTION
The use of airplanes by iiieteorological servkes to obtain
pressure, te,mperature, and humidity values in the free air
necessitated the adoption of rapid methods for the evalua-
tion of the records, in order that the data.might be com- municated as promptly as possible to clesignated centers for use in air-mass analysis and daily forecasting. Pre-
viously, when kites, and captive and sounding balloons were the principal menns of iiinking such soundings, it was
customary in some services to use A more or less laborious arithmetical method, based on the. Laplace fiypsomet8ric
equation, for coniput,ing the heights of the successive levels attained by the, recording instrume,nt. This was
unsatisfactory because of the time consumed in t,he
computations; and several semigraphical or graphical
methods of obtaining essentially tflie same re,sults were devised. One suc,h method is that of V. Bjerknes (l), which necessitates the construction of a diagram with
virtual temperature plotted against the logarithm of
the barometric pressure, and the use of four auxiliary tables in the computation of heights. A more straight-
forward graphical method is that of Stiive (2), which is an adaptation of the older Hertz-Neuhoff (3, 4) adiabatic
diagram, with a t least one feature (graphical humidity
correc,tion) apparently adopted from the Bjerknes virtual- temperature diagram. The resulting diagram (fig. l),
commonly known as the adiabatic chart, also embodies
several new and valuable features, and facilitates greatly the evaluation of aerologicnl soundings with an accuracy
comparable with that of the observational data. It
therefore was adopted by the United States Weather Bureau and other services several years ago.
It is our purpose in this paper: (l), to describe the
groundwork coordinate system that forms the adiabatic
chart; (2), to show how an airplane meteorological sound-
ing, emepting the determination of altitude, is evaluated
through its use; (3), to develop the inathematical theory that underlies the graphical computation of altitude by
means of the chart; and (4), to indicate the various
graphical methods of different degrees of approximation which may be used on the &art for determining the
altitudes of the successive levels of a sounding.
I. GROUNDWORK OF THE ADIABATIC CHART
1. Temperature and pressure.-The network of vertical
and horizontal lines on the chart (fig. 1) represents a rectangular Cartesian coordinate system, with a uniform
scale of temperature for the abscissas, and a logarithnlic
1 For the debition of this term see the next section of this article (p. 125).
90-1
scale of barometric pressure for the orclinate,s. The verti- cal lines, separating unit distanc,e.s on the abscissa scale,
are drawn for each whole degree of temperature in degrees
centigrade, with values ordinarily ranging from +35O C. on the right to -45O C. on the left. The horizontal lines extending entirely across the chart and cutting t,he loga-
rithmic ordinate scale are drawn for every 10 millibars of pressure, with values ordinarily ranging from 1,050 mb
at t,he bottom to 400 mb at the top. The indicated ranges of pressure and temperature suffice for airplane ascenbs made to elevations of 17,000 feet in continent,al United
States. In the actual construction of the coordinate system of
the adiabatic chart, it is customaiy in the United States
to use a space of 4.5 mm to represent an interval of 1 C. It is also customary to c,onstruct t,he horizontal pressure lines by measuring from the line for 1,000 mb as a datum
and drawing the remainder of the lines at distances there-
from in accordance with the relationship 75 loglo -L-
cm, where P is pressure in mb. The number 75 is here merely an arbitrary modulus. A smaller scale which is convenient for the construcbion of larger c.harts used in
the evaluation of sounding balloon observations is two- thirds of that just stated, viz, 3.0 mrn per lo C., and 50
for the modulus of the pressure scale.
On this coordinate system, free-air temperatures ob- tained from an aerological sounding may be plotted
against the corresponding barometric pressures.
2. Potential temperature.-The family of curves that
run in the general direction from the upper left hand corner of the chart to the lower right hand corne,r are
curves of constant (or equal) potential temperature
called adiabats. The potential temperature, 0 in
O K., of air containing an average amount of water vapor (disregarding effects of possible condensation of water
(l Po)
vapor), for any given temperature and pressure is com- 1000 0.288
puted according to the equation e=T (F ) 7
where T is in degrees absolute (Kelvin) and P fs in mb ,n \
and the exponent is the value of (-5- in which J is
the mechanical equivalent of heat, and R and CP are respectively the gas constant and specific heat at constant
pressure for the humid air. The figures associated with
: 1 millibar (mb) represents a pressure of 1,000 dynes per square centimeter, or 1 mb
: The potential temperature of absolutely dry air may be de6ned as the temperature it would assume if brought adiabatically, i. e., without gain of heat from the environment or loss thereto, from its given initial conditions of temperature and pressure to a Bnal constant reference pressure, usually (and bere) chosen to be 1,000 mb.
of mercury under standard conditions of temperature and gavity.
123
124 MONTHLY WEATHER REVIEW APRIL 1935
the adiabats and running nearly diagonally in the direc-
tion from the lower le.ft t'o t,he upper right hand corne,rs of the chart represent the potential temperatures corre-
sponding to the given adiabats. It is obvious that an
adiabat may be construcked by selecting for e an arbi- trary value, and solving for the values of T corresponding
to a series of values assumed for P, or alternatively one
might solve for the values of P c.orresponding to a seiies
of values assunied for T. Since 0 is merely a function of T and P, the potential
temperature corresponding to any given temperature
humidity. Acc.ordingly, free-air relative humidities may
be plotted against barometric pressures (ordinates) for corresponding levels of a sounding.
4. Vapor pressure.-The partial pressure of aqueous
vapor in the atmosphere, or vapor pressure, is computed from the relative humidities and temperatures observed
in the sounding according to the following fundame,ntal relation:
e =T . e,@).
where e = vapor pressure; r =relabe hunlidity (expressed
as a decimal); e,@) =partial pressure of saturated aqueous
- 40 -30 - 20 - 10 0 10 30 (OC.)
PH-T-uw-h.lpln
m t, ............................ T-rernpd-
V..Lllrl by ._.................. ...... XK-R.luir. h d L V
l".pndbl ......................... vF-v.pm p-y.
FIGURE ].-Adiabatic chart showing representation of meteorograph sounding made at Omaha, Nebr., March 9, 1934.
and pressure may be immediately read from the family of curves under consideration.
3. Relative humidity.-There is no special scale printed
on the chart for plotting relative humidity against baro-
metric pressure; however, it is customary to select some space to the left of the chart, and regard a horizontal
interval corresponding to loo of temperature as repre-
senting an interval from 0 to 100 percent of relative
vapor (over water, where the hair hygrometer is used) a t t,he given temperature of the air, t . Values of e ,(t ) are
given in the Smit,hsonian Meteorological Tables and other
standard works. The vapor pressure, obtained in the
manner just shown, may be plotted (on a horizontal
scale) against barometric pressure (on the vertical scale)
in the allotted space on the farthest right-hand side of
the chart. The scale for this purpose, indicated over the
APRIL 1935 MONTHLY WEATHER REVIEW 125
upper right-hand corner of the chart, is such that a hori- zontal distance corresponding to 1' of temperature repre-
sents a vapor pressure of 1 mb. The zero of this scale coincides with the fart8hest right-hand margin of the chart,
and increasing values of vapor pressure are represented to the left of this.
5 . Spec-jic humidity.-The fanlily of curves printed in
the form of dashed lines on the right-hand side of tbe chart
represent curves of constant (or equal) specific humidity.' The specific humidity of a space that contains the normal
proportion of ordinary atmospheric constituents plus water vapor is computed from the equation
where s=specific humidity in grams of water vapor per kg of moist air; e=vapor pressure; p=barometric pressure. The two arguments necessary for this equation,
viz. vapor pressure and barometric pressure, are respec-
tively represented by the horizontal vapar pressure scale on
the extreme right side of the chart referred to in the last paragraph, and by the vertical logarithmic scale of baro-
metric pressure forming the ordinates of the chart. The curves corresponding to specific humidities of 5 , 10, 15,
and 20 g/kg are indicated by figures associated with the
appropriate curves. These figures are near the upper
portions of the dashed curves (see fig. 1 ). These curves enable one to ascertain the specific humidities that cor-
respond to given values of vapor pressure and barometric
pressure.
on the absolute scale, which is used as an auxiliary in com- puting altitudes by means of the chart, is given by the
equation
6 . Virtual temperature.-The virtual temperature
T
e 1-0.377- P
To=
where Tis the temperature of the air in the absolute scale
(degrees Kelvin), T= (273.2+t), t in O C. The difference between the virtual temperature and the actual tempera-
ture of the air is given by
1
If the space under consideration were saturated, the vapor pressure, e, in the above equations would take on a
limiting value e,(T>, that is, the saturation vapor pressure
which is a function of the air temperature alone. Hence the virtual temperature would take on a corresponding
limiting value which we will call the saturation virtual
temperature represented by the symbol T,v. Then
obviously
(Tau - T ) = T ( 1 e m -1 ) 1-0.377- P
Thus ( Tsu- T) is a function of T and P alone. The
magnitudes of the differences (Tsn-T) are indicated on
4 The speciflc humidity of a given s p m may be defined as the ratio of the ma%? of water vapor contained in the space to the total mas8 of moist air (including water vapor or other gases) in the same space. The unit customarily used to represent speciflc humidity is: Grams of water vapor per Bilogram of moist air, i. e. g k g , which is a unit 1/1000th as large BS that given by the definition.
8 The virtual temperature of (moist) air at a given barometric pressure is thetempera- ture at which dry air would possess the same density as the oven (moist) air when at the same barometric pressure.
the chart by the horizontal distances between the short vertical dashes shown on the horizontal lines correspond-
ing to the multiples of 100 inb. of pressure. The value of T upon which is based each of the differences so indicat,ed
is that corresponding to the center of the respective interval . The last two equations given above may be simplified
by performing the indicated divisions on the right-hand sides and neglecting terms containing second or higher
powers of 0.37'ie or 0.34 I -. This is permissible since
the latter rarely exceeds 0.02, and thus squares and higher
powers are negligible in comparison with the first power. Therefore
rrc
P P
(T,-T) , approximately;
and
approximately. (T,,-T)=T(0.377 P
Since e=r.es(T), (see section 4, where r=relativehumidity
expressed as a fraction), we have
(Tn- T)=r.(T,,- T), approximately.
T, = T+ ( T,- T ) = T+ r e ( T,, - T).
Hence if for a given pressure we wish to deternline the horizontal position which corresponds to the virtual tem-
perature under the given conditions of actual temperature
and relative humidity, we merely find the position that corresponds to the actual temperature and make a dis-
placement to the right equal to the fractional part T of the
value (Tau-T). The latter is indicated by the hori-
zontal distance between the nearest short vertical dashes
on the given (horizontal) pressure line. For example, if the relative humidity is 75 percent, the virtual tempera-
ture is to the right of the actual temperature by an amount equal to ji (Ts0-T); etc. For pressures intermediate between the multiples of 100
mb. upon the lines for which t,he short vertical dashes
representing values of (Tau- T ) are printed, it is obviously
necessary to interpolate values of (Tan-- T ) vertically from the printed dashes to obtain values appropriate to
the given pressures. 7. Height.-A special scale for indicating heights is not printed on hhe chart; however, as will be proved later,
the temperature scale may be made to serve this purpose.
A broken line, called the pressure-height curve (indicated
by PH in fig. 1)) is constructed according to a procedure to be described in a later section (IV), and by its aid is
obtained the height that corresponds to any pressure
observed in hhe sounding. This is accomplished by first going (vertically) along the pressure scale until one lo-
cates on the PH curve a point a t the pressure for which
one desires to find the corresponding height. Taen as the theory developed in section I11 shows, the height (in meters) a t which this pressure exists aboae Ihe station ele- vation is obtained when the constant' 102 is multiplied
by the horizon,tal projection (measwed in degrees centi- grade) of the PH curve in passing from its lower extremity
to the given point. To facilitate the computations of height, it is customary to enter near the foot of the chart a height scale, employing a temperature interval of 10' C. to represent a height interval of 1,000 m. However, as stated above, the projections of the PH curve must be multiplied by 102 to give heights in meters, so that a
Thus
126 MONTHLY EWEATHER REVIEW APRIL 1935
loo C. projection really represents 1,0201neters, or 1 C. received in daily airplane weather reports), a true PH represents 102 meters. Heights are customarily repre- curve may be dlrectly constructed from the data. This sented above sen level by drawing the PH curve so that it curve does not require the 2-percent correction referred to
is displaced horizontally with respect to the zero of the above. (Section IV of this paper may be consulted for
height scale by an amount (in O C.) equal to the height, further details regarding this type of PH curve.)
Porn lfo. ll97A-Aeh (WFH*)
U. S. DEPARTMENT OF AGRICULTURE, WEATHER BUREAU
FIQKLE 2.-Meteorogarn obtained in the sounding made at Omnhs, Nebr.. ?.I.~r:h 3, 1934. Arrows indicate points representing significaut levels.
(in meters) of the surface stat.ion above sea level divided
this manner, readings on t.he he,iglit scale, whe,n inc.re,ased
by 2 pe,rcent, give true heights above sea level directsly.
If the true heights above sea level corresponding bo
given barometric pressure.s are kno~vn for a number of
levels in the free air (RS for example wlien such data are
11. PROCEDURE FOLLOWED IN EVALUATING A SOUNDING,
Figure, 3 shows R nieteorogram on which are inscribed simu1t:tneous records of the temperature, relative hu-
midity, and baronietric pressure in the free air as ob-
taiuecl by means of an aerometeorograph which is illus-
by 102, and then regarding zero height as sea level. In EXCEPT FOR DETERMINATION OF ALTITUDES
1
FIGUBE 3.-Side view of the Friez type aerometeorograph, with protective cover removed, the instrument carried on United States Weather Bureau observation airplanes to measure and record automatically the temperature, relative humidity and barometric pressure of the free air. On the left is the cylindrical drum which rotates by clockwork and carries aruled sheet, the meteorogram, on which the three upper pens trace records of the three data in auestion. Near the lower center is the SvlDhon element. an evacuated box of thin metal which serves in the capacity of an aneroid barometer. Near the right center mounted 6n the upright support is the curved bilh6taIlic element for measuring the temperature; and beside it, to the left and extendin above and below, may be seen a strand'of human hairs whose change in dngth with change in relative humidity permits the measurement of the moisture in the atmosphere. %he lowest pen, actuated. by a small electro-magnet controlled by a button switch in the pilot's cockpit, is contacted with the meteorogram when the pilot wishes to note the time of any observation d u n g flght.
Monthly Weather Review, April 1935
APRIL 1935
SpeciBc humid-
ity
MONTHLY WEATHER REVIEW
Vlrtusl tempara- ture -273
127
~~
I A l b
998.3
877.3
132.4
i22.3
692.6
mli. 5
540.8
615.3
gin. z
835.2
trated in figure 3. Figure 4 illustrates the latter instru-
ment mounted on a biplane in position for a sounding. The following paragraph describes briejly the procedure employed by the Weather Bureau in emlitaling the continu- ous record sf free-air conditions made by the insti ument in jlight after the record i s placed at tlie disposal of the ground ohsewer following the completion of the airplane ascent. After the completion of a sounding, points are niarlced
off on the riieteorogram traces to indicate significant
levels, that is, levels where marked changes in the 1-erticnl temperature or relative huiiiidity gradients have occurred.
Synchronous points on the three traces are determined by taking equal distances along the horizontal tiine scale
measured from the arcs which mark the known time when
the pens are finally set to give a continuous record (called
second pens-down). The vertical displacements of these
sj-nclironous points are measured from the respective base
lines of the three traces. (The base lines are horizoutal
portions of the traces made just prior to the souncling
while the aerometeorograph is subjected to known con-
stant conditions of pressure, temperature, and relative
humidity in a well-ventilated highly humid box.) The displacements of the given base lines froni the base lines
employed during the calibration of the instrument are determined by referring the known base-line conditions
of p , t , and r to the calibration tables (or curves) for the
given instrument and finding the corresponding ordinate
displncenients. These respective base-line displacements froni the calibration positions are added to the appropriate
displacements of the synchronous points from tlie re-
spective base lines for the given meteorogram, and the total ordinate displacements thus obtained when referred
to the calibration tables (or curves) give the correspond-
TABLE 1.-Evaluation of ascent of Aeronieteorograph no
I
Qr./Kg.
1.7
1.3
1.4
2.2
1.7
2 .0
1.3
.7
.4
.4
in5 values of free-air pressure, temperature, and relative
humidity. [All pressure readings of the instrument are corrected
for the effect of temperature upon the pressure recording element in accordance with the well-known equation (5)
6p= -At(A+ap)
where 6p=error in the recorded pressure; At= (tenipera- ture of instrument a t the time error is to be determined)
minus (tempertitiire of the instrument at the time of the pressure calibration), algebrnicnlly; y=pressure indicated
l y the instrument at the time tlie error (6p) is to be de-
termined: d and a are constants. a=-0.00013, very closely, for the type of instrument pictured in figure 3.
In this t,ype of instrument, when the pressure element
is properly calibrated, the vnlue of d for a given instru-
ment should lie between 0.05 and 0.10, preferably a t
about 0.078. It is not desirable to use a pressure element which has a value of A greater than 0.30.1
The columns headed Observed values in table 1
exhibit tlie free-air barometric pressures, temperatures,
and relative huniidities obtnined from the rneteorogram
illustrated in figure 2, in the nianner outlined in t,he above
paragrnph. The vapor pressures corresponding to the observed values of temperature and relative humidity are next
computed by means of the equation
e = r.es(t)
which has already been discussed (I: 4).
these computatioiis are also shown in table 1.
The results of
. 24) Alar. 9, 1934, 90th meridian time, Omaha, Nebr.
I 1
C. -8.2
-13.9
-14.1
-9.4
-13.9
-12.6
-12.6
-20.3
-27.2
-27.6
I
Observed values
Tempera- ture
OC. -8. 5
-14. 1
-14.3
-9.7
-14.2
-12.8
-12.8
-20.4
-?7.3
-27.7
I I Graphically obtained values from chart
Relative j Vapor
1
Altitude
humid- pressure a h ~~~~e a ity
Percent 84
94
94
100
100
100
61
58
fQ 57
Nh 2.71
1.94
1.91
2.94
2.04
2.29
1.40
.70
* 39
.36
hider s
300
1,000
1,301
1, G57
2.6s3
2.790
3,096
4.111
4.957
5.304
Potential tempera-
ture
Oh-.
261.6
266.0
268.6
277.3
283. 1
295.8
289.2
291.7
293.3
296.9
PILOTS NOTES.-Maximum altitude of 17,400 feet reached at 8:12 a. m. Entered base of St Cu clouds at 5.000 feet at 7:20 a. m. (contact no. 2 on meteoroqram). Peak of St Cu clouds at 8.800 feet nt 7:28 a. m. (contact no. 3 on meteorogram). These were loll0 overcsst. There were 3/10 St clouds at time of t?ke-otl but was unable to get contact on base or peskofthese. They were estimated to be at 1,500 and 3,000 feet, respectively. Rard, clear ice of about 46 inch thickness was accumulated. Altitudes in feet above surface by alti- meter.
a. m. Times of beginnin nnd ending of precipitation encountered during Bight: Snow, 7:06 a. m. (take-off) to 720 a. m. Snowing from surface to base of St Cu clouds. snow observed at surlace tfwoughout flight by weather observer at the airport..
ADDITIONAL NOTE.-% inch hard, clear ice formed on all exposed parts of plnne. Times when plane entered and leIt clouds in which ice formed: Entered, 7 2 0 a. m.: left. 7%
Light
The steps necessary in plotting the results of the
sounding on the adiababic chart are as follows (see fig. 1):
(1) Isobaric Zin.es.-First, sets of points are locnted on
the vertical logarithmic pressure scale.s, one on each side of the cha.rt, to correspond with the respective baro- niet,ric pressures for the several significant levels. The
sets of points on each side of the chart are conne.c.ted by straight horizontal lines (called isobaric. lines).
(2) Tempera.ture-pressu.re cuwe. -On each of these hori- zontnl (isobaric) lines there is indicated by a dot t,lie
temperature prevniling at tlie corresponding level. These
points are connected by st8raight lines. The 1et)t)er T is
p1:tced a t the top and bottom of the broken line thus obtained to identify it as the teiiiper,zture-pressure curve.
(3) Hum itlity-pre.ssure curpe.-A convenient 10 inter- val to the left of tlie cliart is selected for the humidity
scale; then, below the isobaric line that represents the surf nce barometric pressure there are written the limiting
values 0 percent and 100 percent, respectively, at the left and riqlit extremities of tllis interval. On this hori- zontnl scale 1311 interval of 1 C. thus represents 10 percent
of relntive humidity. As in the case of teiiipernture, there is indicnted by n point on the isobaric line for each
significant level the corresponcliug relntive humidity,
128 MONTHLY WEATHER REVIEW APRIL 1935
And all the points thus plotted are connected by stmrnight8 lines. The top and bottom of the re,sulting broken line
is labeled witah the letters R H for identification purposes.
(4) Tupor pressure-pressure curw.-Using the hori-
zont,al vapor-pressure sc.ale Ilidicat,e,d near the upper right-hand corner of t.he chnrt,, the vapor pressure,s for the several levels are indicated by plncing a point on
each of the respec.tive isobaric lines. Straight lines are
drawn connect~ing the points thus olAned, and the
broken line so Iorniecl is hbeled with t,he 1et)ters T P at
top and bottoni.
(5) TTi.rtu.u.l temperntures.4n each of t,he isobaric lines drawn, t,here is indicated by a point) the virtual tempera-
t,ure for the level in question. This is accomplished
graphically by applying t,he proper virtual-t8emperature
c,orrection (Tu- Tj, which is always positive, to the tem- perature for the same level. To obtain (Tu- T ), one goes
vert,ically on the chart from the temperature a t the given
pressure level to the nearest horizont,d lines, above and below, representing pressures which are niultiples of
100 nib. Then the values (T,,-TT) there indicated by
the dist,ance,s between the short vertical dashes are sub- divided graphically t,o obtain t8he proportion hhereof
equal to the relative humidity at the given level. This ai-res values (Tu--) for t,he standard 100 mb isobaric lmes, and a slight interpolation is then necessary to
reduce to the correckion applicable a t t,he isobaric line, for the given le.vel. This correction is indicate.d by a
clot (fig. 1) placed at t,he proper distance to the right of the point representing the temperature for the significant
level unde.r consideration. The temperatures correspond-
ing to the positions of these dots conse.quently represent the virtual temperatures for the levels. The column in
t,able 1 headed Virtual temperature-273  indkates the values obtained from the result,s of t,he sounding
illustrated in figure 1.
(6) Potential te.mperatur.e.-The potential temperature
for any level, if desired, may be found from the adiabat,ic c.hart by noting t,he value assigned to the adiabat passing t,hrough t,he inter~ect~ioii of the temperature-pressure
curve (T) wit,li t,he isobaric line for t,he level in question.
The data in the column of table 1 headed Potential temperature  shows the values thus found from figure 1. (7) SpeciJic humidity.-The spcific. humidity for any
level, if desired, niay be found froin the adiabatic chart
by notsing the value assigned to the curve of constant specific. hunlidity which passes through the intersection
of the vapor pressure-pressure curve, 1, P, wit,h t,he iso- baric line for the given level. The column of table 1
headed Specific humidity, gflig indicates the ..dues
t,hus obtained from figure 1.
?
111. MATHEMATICAL THEORY UNDERLYING THE GRAPHICAL
DETERMINATION OF HEIGHTS OF LEVELS I N THE FREE
AIR BY MEANS OF THE ADIABATIC CHART
Symbols employed
p = density of moist air (grams/cm3). P= total barometric pressure of moist air (dynes/cmZ) *. e=vapor pressure, i. e. partial pressure of aqueous vapor of
T=absolute temperature of moist air (OK.). T=virtual temperature of moist air (OK.).
po=density of pure dry air under standard conditions of pressure and temperature (gramslcm3).
Po= standard pressure (dynes/cml). To=standard temperature (OK.).
R=gas constant for 1 gram of pure dry air,
moist air (dynes/cml) *.
dyne.cm.
gram. OK.
g=acceleration due to gravity at level of moist air (cm./sec.a). h=height of moist air (cm). J=mechanical equivalent of heat, ergs/cal. C,=specific heat of dry air at constant pressure, cal./gram. OK.
*Pressure units used are dynes/cmZ up lo equation (5), but mb. heginning with equation (5) and continuing.
From the hydrostatic equation, we have, in the free air
where the density of moist air is p and the gravity accelera-
tion g,
(1) d P = - p g dh,
in which dP denotes vertical change in barometric pressure
and dh change in height. From the characteristic equation for perfect gases, and
from the known ratio between the density of water vapor
and that of pure dry air, riz. 0.623 under average condi- tions, we may express the density of moist air in terms of
measurable quanti ties. Thus
dynes. c.111. from which R=2.86S6X1O8 grams.OK. , where we take the -
standard values
po=density of pure dry air under standard condi-
tions Po and To, viz. 0.001293 g m ./~m .~.
Po= 1 atmosphere (=760 nim. of me.rcury) =
1013250 dynes/cm2.
T,,=fre,ezing point of water=273.18 K. (6)
Subst,ituting (2 ) in (l), we obtain
which is the differential form of the hypsometric equation. We now proce.ed to show how the terms in equation 4
may be evaluated by means of the adiabatic chart, and
t,he equation thus solved graphically. Consider an adiabatic chart formed of a rectanyular
coordinate system with absolute temperatures, T, as
abscissas and log, ( o ~n ~b  ) as ordinates (where P from
hereon is understood to be measured in d l i b a r s J 6 i. e.,
1,000 dynes/cm2, instead of dynes/cm2), upon which coordinate system there is constructed the family of curves of constant potential temperature given by the
equation R
where J=mechanical equivalent of heat; C, =specific
heat of dry air at constant pressure; R=gas constant for
1 gram of pure dry air.
dP or e 8 This is permisslble since P hereafter occurs only in the form of ratloa 21 so that
if the same udts vlz. mb. ace used in both numerators and denominators, the values of the ratios rema& unchanged whether expressed in dynas/cms or mb.
APRIL 1935 MONTHLY WEATHER REVIEW 129
The esponent, of (y) is later denoted hg k .
(See page 131.)
Taking natural logarit8hms ' of both members of this equation, we get
whence
(7)
log 8=log T+ ($3 7 log (Y O ), -
log T+ (&p) log (9) 1
e=
where e = base of Napierian logarithms.
From (7) it is evident that e is n scalar point function of
the coordinates of the adiabatic chart, T and log (y ", -
that is, with each point on the chart specified by a vnlue
T and a value log ('yo),
~
there is associated a value of
the scalar quantity 0 that depends only upon the cn-
ordinates of the point. It is therefore permissible to
find the rate of chnn:t of 0 with distance along any direction in the field of the scalar point function e, viz,
the adiehatic chart. Let s represent, distance measured on the nditthatic
chart; then taking the directional derivative of 0 in the direction of the tangent to nily one n7enil)er of the family
of curves B=constant (adiabats), defined by either of
the last three equations, we get from (7), after simpli- fication. since e z o .
where the notatmion represents different,iation wit,li
respect to s in such a direction as to keep e constant,. We note that by taking the directional derivative along the tangent to an adiabat, we are choosing a direct,ion in which 0 remains constant, hence the left member of
equation (8) becomes zero as indicated. From (8) we get
Equation (9) obviously reduces to
where the expression
just introduced represents the slope of an adiabat in the
1000 coordinate system T, log [ 7)- Hence (10) states that
\L /
the reciprocal of the absolute temperature a t any point
on the adiabatic chart is proportional to the slope of the tangent to the adiabat through t,hat point. A decluc.tion
7 Logarithms hereafter designated by "log " represent Napierian logarithms. Log- arithms to the base 10 are designated by "loglo".
we may draw from (10) is that the slopes of the adiabats a t nll points along a given isotherm (vertical line on the adiabatic chart) are iclontical.
Introducing the following simplifying not,at.ion, let
IJ =log ( T)l 1000 and (1 1)
(12) r=T; then equttt,inn (10) obviously may be rewritten
e) represents t,he slope of the adiahat on the chart a t
the coordinates T, log ('To), - i. e., (r, y).
We have thus arrived at n graphical means of espress- ing the reciprocal of T, wliicli is necessary for the graphical
solution of equation (4). There remains the factor
This may be done as
follows: From the considerations in the paragraph im-
mediately following equation (10)) we note that the slope of an adiabat a t a point on the chart depends on the value
of the absolute temperature corresponding to that point,
but not on the value of log ( 'F0) __ . Hence for each tem-
pera ture, T, there is a corresponding slope of the adiabats
which is independent of the pressure. Now, consider a point on the adiabatic chart with coordinates T', log
('F), where, in general, T'f T. From (13) we get
1-0.377 - to take into account. ( P ">
where the prime associated with the symbol on the right denotes that the pertinent temperature is T' instead of T.
From (13) and (14)) we get
Let T' have the particular value
T (16) T'=
(1 -0.377 P '>
tual teniperature; then (15) becomes
, which is the expression for vir-
@):=($)8(1-0.377 P
Now multiplying bot,h members of (13) by the factor
(1-0.377 - 1 and subst8ituting (17) in the equation thus
obtained, we get P e )
We note that
1000 (19) d P -- -d log P=-d log (T)= -dy;
130 MONTHLY WEATHER REVIEW APRIL 1935
hence on subst,itutiiig equations (IS) and (19) in (4) and simplifying, we ohtnin the result,
From geometiical considerations, we see that (2) is
the trigonometric tangent, of the angle between the adiabat and the isobar (horizontal line) at the point where the coorclinates on the acliabatic chnrt are T, log
(F); and dy is the vertical distilnce between two
neighboring isobars a t the same coordinates. Therefore,
the ratio of the latter to the former represents the hori- zontal projection of that portion of the geometric tangent
to the acliabat which is bounded by the two neighboring
isobars. That is,
e
where (d,r)g= (dT)s is the projectmion, on the axis of
abscissas (T axis), of the line-segment tangent to the
adiabnt at T, log (y) and delimited by the isobars
[ log (y o ) - ->:hd log (lono)] p
Therefore, with the neglect of the sniall vertical varia-
tion in gravity, equation 21 tells us that on the adia-
batic chart, horizontal projections of small segments of adiabats give a relative (graphical) measure of the differ- ences in height that correspond to given differences in
pressure when the virtual temperatures for the intervals of pressure in question a!e known. Then integrating
equation (20) between the limits of height (I t 2 , h and the corresponding h i t s of pressure (Pz, Pl), we obtain
If the virtunl tenipernture, T, is constant throughout a given finit8e interval of pressure, then the slope of the
nclinbat corresponding to this value of T is constant,
that is (2); =constant; and the integration in equa-
tion (22) may be performed since dy=d log (yo) - is
integrable. Hence in the special case where Tl=con-
stant, we obtain from equat,ion (22) the result
where
y l =l o g (F ) 1000 and %=log() 1000
If T varie,s in value throughout 2, given finite interval
of pressure, the segments of the adiabnts corresponding to
the various values of T must vary in slope; and in general
we cannot rigorously perform the integration indicated
in equation (32). However, t,hrough small finite intervals
of pressure in the free atmosphere where T is variable,
we mtiy as an approsinintion adopt n constmt value of the
slope fg) w~r-llicli corresponds to tlie mean value of T for
the internl. That is, we may nssume that the integra- tion will be sufficiently emct if we employ a single value of
slope that represents the nwo!) slope of the segments of the aidinbats corresponding to tlir vnrinble vnlues of T throughout the interrnl of presure. Where the value of
T varies linearly withlog (T)), - - as is custoni:~ril~~ assumed
to he the case in nerologicnl TVGIEI for each successive layer
of air (usually) mnrkecl off 1)) discontinuities in vertical teniperature gradient at top tint1 hottom, the iiiean value
of T is obviously to be founcl :it the center of the virtunl
tempera ture-pressure curve in the i n t e r d of log -
for any given lnyer. Therefore the mean slope of the seg-
ments of the adinbnts tlint correspond to the variable
viilues of T in the interval is to be found by determining the center point of the virtual temperature-pressure curve
for the interval nncl taking the slope of the adiabat passing
throvgh, and at, this center point. Hence if e) =?nl e
denotes the slope of the adiabnt a t the teniperature wllich
corresponds to the meum virtual teniperature (i. e., center)
of the interval of log (To) - in question, me obtain from
e
(Yo)
 w e a n )
a-hich is more generzl than equation (23).
The variables y1 and y, obviously represent ordinates of isobars on the ndiabatic chnrt, m c l m, represents a slope.
Suppose we construct a straight line segment that has slope
?nl nnd terminates in the isobars y1 and y2; then since the
equation (~yI---yJ=ml (.rl-z?), where .rl 2nd .rz we abscissas corresponding to the ordinntes y, ant1 y2 respectively, is the
rectangular equation of n straight line, we recognize the
expression (yl-yz)/rnl on the right of equation (24) as the value (zl-.r2) of the projection, on the axis cf abscissas, of the straight line segment from the point (.rl, yl) to the point
(sz, y2). The abscissas xl, x2 are temperatures on the chart hence if TI = .rl nnd T2 = .cat ecluation (24) becomes
&-Al)= (T) JCD e (Tl-T2).
Therefore, if we construct a straight line segment on the
adiabatic chart, of slope nil= ($)e as defined above,
and terminate the segment in the isolms y, -log(T)
and y? -log(?) for the given lnyer, then the horizontal
projection of the line segment expressed in O K. (or O C.)
gives a value which, when multiplied by the constant factor
r*)is the difference in height between the top and
hottom of the layer. Since both t n l and (yl, y2) may be constructed grnpliically on the ncliabtttic chart, u e are
thus provided with a graphical means of computing thick-
I mean)
1000
APRIL 1935 MONTHLY WEATHER REVIEW 131
nesses of layers of air in the free atmosphere, provided the necessary observational data are a t hand.
If it is required to compute the elevation of the upper
limit of a layer superimposed upon several others, then equation (24) provides the mems; for if h l , h2, ha, - - - - --h ,
nre the eleviitions of the respective boundary surfaces separating the successive layers (hl being the bottom of the
lowerniost layer, and h, the top of the uppermost layer),
yl, ya, y,, - - - - - -yn ase the corresponding ordinates on the adiabatic chart as derived from the observed barometric
pressures, and 1711, m2, m3, - - - - - -n ~(,-~) are the appropri- ate slopes for the respective layers (subscripts indicating the identification numbers of the lower bounclaries), then
for each layer we obtain an equation similar to equation
(24) and by acldition of these (n-1) equations we obtain
the result
1
+ (?/,i-l-Y")
l k -1
If hl is the elevation of the surface of the ground above
sea level, then the elevation A, above sea level is obvioiisly
1
+ h -,-?/,z )
inrL-,
Equation (27) tells us that we may graphically cleteriiiine
the elevation above sea level of any point in tlie free air, in terms of tlie total horizontal projection on the acliabatic
chart of the appropriate straight line segments for the successive layers of air, provided we add an allowance
for the elevation of the ground equal to (2- h,);
(see the curve designated PH in fig. 1).
mien the value of C, refers to perfectly dry air we recog- nize this as the expression for the reciprocal of the dry
adiabatic lapse rate (see, for example, Humphreys' Physics of the Air, second edition, 1929, p. 2s). To compute this term on the basis of the latest accepted
values of the several constants involved, we shall adopt
the following values:
J=4.1S52 s lo7 ergs per 15O calorie (given by Birge (6)); C,=0.2405 cal.l,./gram. O IC. at one atmosphere pressure and 0' C. for dry air free of CO, (given by Hol- born, Scheel, and Henning (7)); this should be essentinlly
tlie same as C, for ntmosDheric drv air containinc. 3 Darts
JCP
(3-
It is now necessary to evaluate the term
of GO, per 10~000 parts of air, si;ice the value of C, for
C'O, is 0.197 a t 0' C . The value of C, given by Moody (8) (viz, 0.2412
cal.2,./g.oI<. a t one atmosphere pressure and 30' C.) when reduced to the standard conditions emnloved above.
become,s 0.24057, in essential agreement witl; t,hit adopted
he.rein . Tlie values of gravity acceleration (9) herein ndopted
are based on the United States Coast and Geodetic Sur- very forniula (9) for the variation of g wit,h latitude a t sea
level, and Helmert's formula (10) for the variation of g
with height a,bove sea level. Since most airplane weather
observation ascents atta.in an elevation of about 5,000 meters, we shall employ values of g pertinent to the mean height of 2,500 m. Thus we obtain:
g (at (p=3Oo, h=2500 m.)=97S.556 cm./sec z .
g (at (0=40', h=2500 m.)=979.400 " ('
902-35-2
g (at (0=50', h=2500 m.)=9S0.299 " "
g (standard gravity) =980.665 " ('
where cp=latitude, and h =height above sen level.
Let h. denote t8he exponent of ( p> in eq. (5) [also
the constant coefficient in eq. (IS)], then (compare equa- tions 4, ls, 19, and 20) from the above values of the
adopted constants we find
'1000
=lo386 c~li./'I<. for (p=30, h=2500 in. =io277 6 6 (' (p=4Oo, <i
=IO268 ' I (L p=5Oo, l (
= 10264 (' " standard gravity.
Converting these to round nurnhers esprersecl in meters
per ' I<., we finally obtain 103 meters per ' I<. for the value
of r?). Hence applying this to equations (23)-(27),
we arrive a t the conclusion that $ file rrdia6nf.s are con-
structed on the adiabatic chart i n accordatice wifh the con- stunts here adopted, a horizontal projection of 1' C. by a
straight line segment of proper slope corresponds to a difference of elevation of 103 meters.
If the adiabats are not constructed on the adiabatic
chart in accordance with the values of the constants adopted above, then the factor 103 just discussed is not
applicable in general. Thus if we evaluate the exponent
k = in equation (5) on the basis of the adopted
constants, we obtain k=0.2S.50 instead of 0.288 as usually employed; the latter value is commonly assumed
to be the one appropriate to average values of humidity.
Assuming that k=0.2SS, then for $=40, h=2500 m.,
[ (:,>I
and for stanclard gravity 10157 cm./OI<.
Therefore, when ihe acliabafs are constructed on the basis
of 0.288 as the exponent, which i s the case i i ~ -figure 1 , we must employ a factor of 102 meters per O K . in determining
heights graphically by means of the adiabatic chart. NOTE.-one may inquire why, if the actual adiabatic
chart (fig. 1) has its ordinate system constructed on the
basis of the relation y10=75 log,, ('To), __ is i t permissible
to consider the theoretical chart which we have discussed
in this section with an ordinate system constructed on the
basis of the relation ye = log, (y). Since log,,
t,he modulus of
common logarithms, 0.43429+, y,,= (75 M) log, (y o )- - -
(75 A 0 ye. Then dyl0= (75 ill) dy,, and if x = T as before,
(p )i =(i 5 AI) BY division, we have
..
but equation (20) is equivalent to dYl0 dye .
d~ e CIX e
(&,)
I----
(&)"
dx e
132 MONTHLY WEATHER REVIEW APRIL 1935
hence by virtue of the equality just previously found we
may rewrite this equation in the form
Therefore, all the discussion regarding the graphical
computation of height is as valid in the case of the actual adiabat,ic chart as in the case of the theoretical adiabat,ic
chart.
IV. VARIOUS METHODS FOR THE CONSTRUCTION OF THE
PRESSURE-HEIGHT (PH) CURVE ON THE ADIABATIC CHART,
AND GRAPHICAL COMPUTATION OF HEIGHTS OF FREE-AIR
LEVELS
From equation (27) it is obvious that we inay graphic- ally determine the elevation above sea level corre- sponding to any pressure in the free air by adding tlie
horizontal projections (on the axis of abscissas= tein-
perature) of several straight line segments of appro- priate slope, one for each successive stratum of air,
assuming an allowance is made for the elevation of the
ground above sea level. We recall that the straig.lit line segments in question must ternilllate in the isobars (ordinates) for the upper and lower limits of each respec-
tive stratum, and that they must each have an npyro-
priate slope corresponding to the slope of the tangent to
the adinbat at the center of the virtual temperature-
pressure curve for the stratum (which is assumed to be linear). Under these circumstances, a horizontal projec-
tion of any segment equal to 1 C. corresponds to a
difference in elevation of 102 meters.
To facilitate the addition of the horizontal projections
of the straight-line segments required by equation (2 7 ),
instead of constructing each segment so that its center
coincides with the center of the virtual temperature- pressure curve for its particular stratum, which requires
the addition of separate horizontal projections, we may
displace each one horizontally, maintaining the same
slope, until they join a t their extremities and form a continuous (broken-line) curve. The continuous curve
thus formed is called the pressure-height (PH) curve, shown in figure 1.
To facilitate the determination of the inagnitude of the
total horizontal projection of any portion of the PH curve froin its origin (lower terminus) to any point on the curve, it is most convenient to place the origin so that the
projections in degrees Centigrade may be read off at a
glance. Since each degree Centigrade of projection cor-
responds to 102 meters elevation, it is convenient to regard the temperature scale as a sort of height scale,
whereby the projection in degrees increased by 2 percent
gives the elevation in hundreds of meters. Thus a 10
horizontal projection corresponds to 1,020 meters, or
1 lun plus 20 meters. The following instructions briefly indicate a procedure
for the construction of the PH curve on the adiabatic
chart, and the deterinination of the elevations that cor-
respond to the pressures at the various levels attained in
an aerological ascent (see fig. 1):
(1) Immediately below the isobar that corresponds to
the barometric pressure a t the surface, write in the height-
scale values: 0 km (or s. L., representing sea level),
1 km, 2 km, 3 km, 4 km, 5 Ian, 6 km, etc., increasing values extending to the left, one value beneath successive
vertical lines for consecutive printed loo intervals of
temperature. The origin (i. e., S. L. position) of this
scale is placed somewhere to the right of the chart a t any
suitable arbitrary point located on a vertical line repre-
senting a 10 multiple of temperature. (This scale is in error by 2 percent as mentioned above, but serves to facilitate the exact computation of elevations.)
(2) Place a dot on the surface pressure isobar with
reference to the height scale (constructed as just indi-
cated) a t the apparent height equal to (h41.02) (see first term in brackets in equation (27)), where hl is the
height of the station above sea level, more particularly tlie lieiglit a t wliich the isobar in question represents the
e.\isting barometric pressure. (This should be the sume height ns that nt wliich the station temperature reading
was made-usually the height of the instrument shelter).
For esample, if both the surface barometric pressure and
temperature re er to the height 300 meters above sea
level, the dot should be placed a t the position on the
horizon t 1x1 height scale corresponding to the height
300/1.03 or 394 meters. This dot is the origin of the PH ciirve.
(3) To construct the segment of the PH curve for the
first stratum of air above the statidn, first refer to the
tein pera t ure-pressure, T, curve for this stratum and
mark by dots the virtual temperatures corresponding
to the top and bottoni, respectively, of the stratuin. Iiiiagine a line through these two dots, the virtual tein- pernturi-pressure  curve, and mnrk n dot at tlic cmter
of this iiiiaginary line. (This dot represents the mean virtzml-tempernture for the stratimi.) Find tlie ndinbst
passing through this (central) dot, or if an adiabat does
not happen to pass through the dot, place some identifying mark such as a sinal1 cross on a point located vertically
above the dot and lying on an adiabat. Place a drafts- mans triangle a t the dot or cross in question and orient it
until one edge thereof is tangent to tlie adiabat a t the
point inarked by the dot or cross. (This gives tlie slope
of the straight line segment corresponding to tlie niesn
virtual temperature for the stratum.) Place a straiglit-
edge, such as a ruler, against one of the edges of the
trinngle otlier than that which is tangent to the adiabat, and slide the triangle by parallel displacenient until the
tancentin1 edge passes througli the origin of the PH curve,
wliich is the dot marked in the manner indicated in
paragraph (2) above. Draw a straight line segment along
the tangential edge, starting it in the dot just referred to ani1 ending it in the isobar for the top of the stratum.
This is the first segment of the PH curve.
NoTE.-Instead of employing a draftsmans triangle and a
straightedge as indicated above, a set of parallel rules may be used
to advantage, especially in making the parallel displacement of slope.
(4) The segments of the PHcurve for the other succes-
sive strata of air may be obtained in a manner similar to
that just described for the fmt stratum; in each case the
initial point for any segment is the terminal point of the segment for the preceding lower stratum. Thus a succes-
sion of segments may be constructed to form a continu- ous curve, marked by the letters PH in figure 1.
(5) To determine the elevations corresponding to the several isobars on the chart which represent the signifi- cant levels attained in the sounding, from each point
where the PH curve intersects an isobar project an im- aginary vertical line down on the height scale and read
the value there indicated. Increase each reading by 2 per- cent of itsev, thus finally obtaining the actual elevations in meters above sea level. The results found in this
manner for the aerological sounding illustrated in figure
1 are entered in table 1 in the column headed Altitude
APRIL 1935 MONTHLY WEATHER REVIEW 133
above sea level, meters." These results are also entered in figure 1 in the left-hand margin just above each
respective isobar.
NOTE.-OnCe the PH curve is constructed, the pres- sure, temperature, or other data corresponding to any
particular elevation may be obtained. (6) To determine the pressure corresponding to any given elevation, the PH curve must be employed in a
manner the inverse of that just described. First divide
the given elevation by the factor 1.02, then imagine a point on the horizontal height scale at this diminished
elevation (i. e., h/1.02) and move vertically on the chart from this point until the PH curve is intersected. The
required pressure niay be determined by reading the pressure scale on the side of the chart at the isobar
passing through the point of intersection in question.
(7) The values of the temperature, virtual-temperntiire,
relative humidity, vapor pressure, potential temperatsure,
or specific humidity corresponding to any given elevation may be determined by finding the isobar corresponding
to that elevation as outlined in paragraph (6) and then
noting the values of the desired elements at the intersec- tions of the appropriate curves with the isobar in question.
I t may be seen from equation (27) that if the slope of
each segment of the PH curve constructed in the manner indicated in the preceding paragraphs be decreciacd i n the proportion (1/1 .Oi?), then the horizontal projections of the resulting straight line segments will fulfill the condition that 1' C. of projection corresponds to 100 meters dif-
ference in elevation instead of 102 meters. Hence, if we
can modify the slopes in this proportion, the horizontal height scale referred to in paragraph (1) will give actuctl
heights above sea level directly without a correction of 2 percent being necessary. This assumes, of course, that
the origin of the new PH curve is placed at the position
on the scale corresponding to height hl instead of (h,/1.03)
as before. The slopes of the segments may easily be decreased in the required proportion if the honzontal pro-
jection of each segnient of the original PH curve which
we have described be increased in the proportion (1.02/1). This is most easily accomplished by (1) constructing a
PH curve in the manner described in paragraphs (1)-(4)
or preferably by placing a dot on each isobar where t'he
original PH curve would intersect the isobar; (2) reading
the horizontal projections of these dots on the height scale and increasing these readings by 2 percent; (3) plac-
ing new dots on the respective isobars at positions verti- cally above the height scale corresponding to the readings
increased by 2 percent just referred to; and (4) construct-
ing a new PH curve formed of straight line segments connecting the new dots. This new PH curve has the
advantage that elevations corresponding to any pres-
sure attained in the sounding may be read directly on the height scale. Conversely, pressures corresponding to
any elevation may be determined directly by noting the
pressures on the side of the chart at the level of the
isobar that passes through the point in the PH curve
vertically above the point on the height scale where the given elevation is marked.
Instead of making the slope of each segment of the PH curve parallel to the tangent to the appropriate adiabat at
the point where it passes through the center of the virtual- temperature-pressure curve (or vertically above or below
that central point), one might employ the approximation
of making the slope of each segment respectively parallel
to the chord of the portion of the appropriate adiabat bounded by the isobars for the given stratum. This may
easily be accomplished with the aid of a draftsman's
Thus:
compass. Briefly, the instructions for tlus procedure are
as follows:
(a) Beginning with the shratum between the ground and the first significant level, set the point of a compass
on the intersection of the proper printed adiabat and the
surface pressure isobar. With the conipnss thus centered, mark a point on the surface pressure isobar at the place
on the horizontal height scale (see priragraphs nos. 1
and 2 above) that corresponds to the station elevation
divided by t'he factor 1.02 (i. e., hJl.02). With this same sett,ing of the compass, place one compass point on the intersection of the proper adiabat with the isobar for
t>he first level. Wit,li the compass t,lius set, niarlr a point
011 this isobar wit,li t>he 0the.r c.onipass point extended
toward the left of the chart. The point, thus fimlly ob- tained on the isobar fur the first level and tlie point desig-
nn.t,ccl by h,/l.O2 on tile liorizontixl height, scale nia.rkecl on the surface pressure isobar, represent the upper a.nd
lower termini respect,idy of t.he first segment of the originnl PH curve discussed in pai.ngr:xph number (3)
above. (b ) Nest,, adjust, the coiiipass t,o t,lie dist,ance between
the point h s t obt,n.i;ied and t,he int,ersec.t,ion of t.lie. proper
print,e.cl adia.bat for the second strtitum of air with the
isobar for the first level. With t'his setting, place one point of the coiiipa,ss on t,he iiit,ersect,ion of t,he adiitbnt
just 1.eferre.d to with tlie isobar for the second significant level, and inark a. point on t>his isobnr with 6lie ot,lier
compass point est,eiided toward the left of t,he chart.
This 1nt.ter point represent's the upper terminus of the segment of t,he original PH curve for the second st,ratuni,
while the point last obtained in t,he preceding paragraph
represent,s the lower terminus of this segment. In a simi1:w manner the point,s for a.11 the segments of
the PH curve may be obt'ainecl. If desired, these points may be displaced to the left in the increased proportion
of 2 percent on the horizontal height scale, thus giving a
new set of points through which a PH curve niny be
drawn that gives actual elevations above sea level di- rectly without correction. The object,ion t,o the method just outlined, wherein chords of xdiabats inst,end of tange.nts are used to obt,ain
the segments of the PH curve, is that it is tantamount to assuming that the vertical virtual-temperature gradient of
each stratum is always equal 60 the dry adiabatic gradient
of approximately -1 O C. per 100 nieters,whereas actually the verticnl virtual-temperature gradients niay have a wide range from positive vdues (inversions) to super-
adiabatic values (algebraically less than - 1 ' C./lOO m).
This may be seen from the following considerations. A section of an xdiabat subtended by 8 chord must have
ident,icnlly the same horizontal projection as the chord. Therefore, since the difference in elevation between two
levels is determined merely by the horizontal project,ion
of the portion of the PH curve lying betw-een the respec-
tive isobars for the two levels, the chord may be rephced
by the portion of the adiabat wluch i t subtends when constructing the portion of the PH curve lying between
the isobars in question. That is, the portion of the PH curve for the given stratum may be made curvilinear
instead of linear and yet give the same diffemnce in
elevat,ion between the top and bottom of the stratum
so long as the same horizontal projection obtains in
both cases. From the interpretation of the symbol
(d lOg;y))'
e
134 MONTHLY WEATHER REVIEW APRIL 1935
(see discussion following equations 10,14, and 20, sec. 111)) which controls the construction of segments of the PH curve, and from equation (18)) we note that a given slope
of 8 tangent to an adiabat at a point must correspond to an equivalent virtual temperature represented by an
isotherm (vertical line) passing through the point of
tangency. Also, the virtual temperature corresponding to a given point on the PH curve must be represented
as being a t the same pressure that is applicable at the
point in question, i. e., the isobar (horizontal line) pass-
ing through the point must intersect the virtual teni-
perature-pressure curve for the stratum at a point hav-
OI assuming that t,he vertical virtual-temperature gradient is equal to t,he dry adiabatic gradient. The only justifi-
cation for the procedure under consideration would
appear to be the practical one that tangents to ndiabats
cannot be obtained wit,h great accuracy in ordinary work, and that the errors involved through the use of chords
are of the same order of magnitude as those involved
through the determination of tangents.
We have shown in the preceding paragraph that the
PH curve for a stratum wherein the vertical virtual-
temperature gradient is equal to the dry adiabatic gradient should rigorously be curved and not straight,
+ 75mperafure
I/erf/i=lu/ "v/rfua/ - fernpermure grodenfs
/ = D r y ad/abuf/c yrudlenf ;
2 = /sofherma/ I# ; 2'1 I1 I f I?
3 = /nverfed B I - 3') efc. 4= Gradlent befween dry ab/bbaf//c and /sotherma/ 5 = Superad/ubahc (dry) qrud/enf
//= Correspond/ng 'kW"curve
u
FIQURE J.--Examples of appropriate PI9 curves (greatly exaggerated) for various vertical virtual-temperature gradients.
ing the appropriate virtual-temperature value. Thus, if we chose a point A on the portion of the adiabat
which fornis the PH curve for the stratum, there must in general, be a point, B on the virtual temperature-pressure
curve for the same stratum which lies both on a vertical line passing through A and on a horizontal line passing
through A also. The only way in which both these things may occur is for B to coincide with A. If we apply this argument to every point on bhe curvilinear section of the PH curve, we conclude that the virtual teniperatnre-
pressure curve must coincide with the portion of the
adiabat in question. Therefore, by const,ructing the
PH curve out of chords of adiabats, we do the equivalent
and should have a curvature and form identical with that of the portion of the adiabat along which the virtual
temperature-pressure curve for the stratum lies. How- ever, when the vertical virtual-temperature gradient is
zero, that is when the virtual temperature remains constant with decreasing pressure throughout a stratum,
equation (18) indicates that the slope of the adin,bats at every point along the virtual temperature-pressure curve
must reninin constant. Therefore, when conditions in a
stratum are isothermal with respect to virtual tempera-
ture, the PH curve for the stratum should rigorously be
a straight line. On the other hand, when the vertical virtual-temperature gradient is such that the virtual
APRIL 1935 MONTHLY WEATHER REVIEW 135
temperature increases with decreasing pressure, or in
other words when an inversion of virtual temperature
prevails in a stratum, the PH curve for the stratum should again be curved. Reference to the adiabatic
chart portrayed in figure 1 will show that the ndiabats
are concave upwards, and that tangents to the adiabats tend to approach verticality at lower temperntures,
and horizontality a t higher temperatures. Thereforeit fol-
lows that the appropriate PI curve for n stratum that
has an inversion of virtual temperature should rigorously be concave downwards, i. e., opposite in curvature to the
PH curve for a stratum with a dry adiabatic vertical virtual-tempera ture gradient.
By virtue of the facts presented in the preceding two
paragraphs, it is obvious that a linear PH curve rigorously has a sort of median position between the appropriate
PH curves for dry adiabatic and inverted vertical virtunl-
temperature gradients. T l ~s may be best seen froin figure 5 , which shows the three types of PH curves in question, with greatly exaggerated curimtiu-e to illustrate
their respective differences ; the mean t$?-tual temperature f o r the stratum is different for each different vertical vir-
tual-temperature gradient. Since the virtual temperature at the bottom of the given stratum shown in figure 5 is fised, tlie slope of the PH curve at tlie corresponding pres-
sure level is tlie same no matter what the vertical virtud- temperature gradient is in tlie stratum iininediittely above.
Therefore the three PH curves in question all start with tlie same slope, viz, that corresponding to the slope of the adiabats st the virtual temperature of the bottom of the
stratum. This niust be the slope of tlie linear PH curve which would obtain for isothernial conditions of virtual
temperature. Then as a consequence of the upward con- cave character of the curve appropriate for dry adiabatic
conditions, and of the downward concave character of the
curve appropriate for inverted conditions of virtual tern- perature, the two curves in question niust deviate froill
tlie straight line, the former to the right and the latter to the left. The differences between the horizontill projec-
tions of tlie respective PH curves shown in figure 5 are largely accounted for by the different mean virtual tem-
peratures for the stratum that correspond to the different virtual temperature-pressure curves shown in the lower part of the figure. If the mean virtual temperature
were identical for each given virtual temperature-pressure
curve, then the horizontal projections would be tlie sanie
in each case. The discussion in the last paragraph suggests that a
very nearly rigorous PH curve may be constructed by employing templates of various curvatures depending
upon the mean virtual temperatures and upon the vertical
virtual-temperature gradients of the particular strata.
For example, one can prepare sets of templates for say
three different mean virtual temperatures. Each set may
consist of templates for the following vertical virtual-
temperature gradients: (1) Dry adiabatic, (2) half-way
between dry adiabatic and isothermal, (3) isothermal
(use straight edge), (4) moderate inversion, and (5 )
strong inversion. Possibly (4) and (5 ) may be replaced by one suitable intermediate template. The templates
may be made of some transparent material, such as
celluloid, similar in form to that of a triangle with one side having a curvature appropriate to a given mean virtual
teinperature and a given vertical virtual-temperature
gradient. This triangle may conveniently be slid to and fro with one straight edge parallel to the isobars, while the curved edge retains the proper direction relative thereto.
The range of vertical virtual-temperature gradients to
which the given template p e r t :~s may be graphically indicated by two fine dark lines of proper slope inscribed
on the transparent material. These refinements are of course not justified in many cases; under such circum-
stances the methods previously outlined for the construc-
tion of tlie PH curve may be employed.
REFERENCES TO LITERATURE
(1) V. Bjerknes and Collaborators. Dynamic Meteorology and
Hydrography. Part I, Statics, pp. 6 1 4 8 . Washington, 1910.
(2) G. Stfive. Aerologische Untersuchungen ziim Zwecke der Wetterdiagnose. Arbeiten des Preussisclien Aeronautischen Observatoriums bei Lindenberg, Band S I V , 1932, Wiss. Abh., pp. 104-116.
(3) H. Hertz. Grsphische Methode zur Bestimmung der adia- txttischen Zustandsbnderungen feuchter Luft. Meteorolo- gische Zeitschrift, I. Jnlirgang, 1584, Berlin, pp. 421-431.
(4) 0. Neuhoff. Adiabatische Zustsndshderungen feuchter Luft uncl deren rerhnerische und graphische Bestimmung. Abhnndlungen des Koniglich Preussischen Meteorolo- gischen Instituts, Bd. I, No. 6, Berlin, 1900, pp. 271-305.
(Translation of this may be found i n Cleveland Abbes
 Mechanics of the Earths Atmosphere , third collection, Washington, D. C., 1910).
uber die Kompensation von Aneroidbarometern gegen Temperatureinwirkungen;
and Nachtrag zu der Arbeit ifber die Iiompensation von ,4iieroidbnrometern gegen Temperatureiiinirlruiigen . Bei-
trdge zur Physik der freien Atmosphare, Band 1, 1904-05,
pp. 108-119 and 20%210.
Probable V.tlues of the General Physical Con- stants. Phpical Review Supplement (now Reviews of Rfodern Physics) vol. 1, No. 1, July 1929, pp. 1-73 (see particularly pages 2W29 and 30-33).
(7) L. Holborn, I<. Scheel, F. Helining. Warmetabellen der Physikalischen Technischen ReiL*hsanstalt. 72 pp., Braun-
schweig, 1919 (see particularly pages 5G-58).
(8) H. W. Moody. A Determination of the Ratio of the Specific Heats and the Specific Heat a t Constant Pressure of Air and Carbon Dioside. Physical Review, vol. 34, 1st series,
1913, pp. 275-295 (see pp. 29@-294).
(9) W. Bowie. Investigations of Gravity and Isostasy. U. S. Cortst
and Geodetic Survey, Special Publication No. 40, Wash- ington, D. C., 1917, 196 pp. (see p. 134).
(10) F. R. Helmert. Uber die Reduction der auf der physischen Erdoberflache beobachteten Schwerebeschleunigungen auf ein gemeinsames Niveau. Sitzungsberichte der Iioniglich Preussischen Akademie der Wissenschaften, Berlin, 1902,
pp. 843-855 and 1903, pp. 65D-667 (see particularly 1903,
p. 651).
(5 ) H. Hergesell und E. Kleinschmidt.
(6) R. T. Birge.
FLOODS IN THE SACRAMENTO VALLEY DURING APRIL 1935
By E. H. FLETCHER
[Weather Bureau, Sacramento, Calif., May 18351
On ,4pril 6-7 a low-pressure area of marked intensity
and wide extent, whose center moved inland from the ocean along the California-Oregon boundary, caused
heavy rains generally throughout northern Californirt,
culminating in torrential downpours in several locnlit,ies of the Sacramento basin. This was the piim:try cnuse
of the flood in question. However, there were two other
contributine factors of imDortance.
First, there was an unusually heavy snow cover-in the mountain areit a t moderate elevations; and the rapid run-
off that occurred over the American, Feather-Yuba, and
upper Sacramento drainage areas WAS augmented by melting snow from rains that extended well up into the
mountain snow fields. The material source of snow
water was in a belt from about 4,000 to 5,000 feet, where the snow was less compact. The winter and earls spring