BRUNT-ADI,4B.4TJC L.\€'SE-RATE FOR DRY S. SATURATED AIR 351

T H E ADI\B..\TIC LAPSE-RATE FOR DRY AND SATURATED AIR

By D. BRUNT, M.A. , B.Sc.

talanusfript received February 1CRcad hIay 17, 19331

I . INTRODUCTIOX The ascent of saturated air has been discussed in detail by

Hertz,' Neuhoff,' and Fjeldstad.3 These writers have given graphical methods for evaluating the temperature of ascending air at any level. But so far a s I have been able to ascertain, no table of the actual lapse-rate has yet been given by any writer, in such a form as to make i t possible t o ascertain the value for any com- bination of temperature and pressure. I have therefore thought it worth while giving in the following paper a simple method of doing this. The ascent of dry a i r is discussed in a few lines, in order to indicate the outline of the method to be followed.

A fluid in equilibrium with a steady distribution of temperature may be unstable, if the lapse-rate exceeds a certain value. This value, for dry air, is slightly in excess of the lapse-rate corres- ponding t o uniform entropy at all heights. The latter demands a lapse-rate equal to the dry adiabatic lapse-rate. Jeffrey+ in a n interesting extension of the work of Rayleigh5 to a compressible fluid, has shown that in the atmosphere, over ranges of height such a s are of meteorological interest, the limiting lapse-rate at which instability sets in differs only very slightly from the dry adiabatic lapse-rate. I t is essentially for this reason that the adiabatic lapse-rate is of fundamental importance in meteorology. When the atmosphere is unsaturated, the effect of neglecting the effect of the water vapour in i t i s slight, as is shown in standard textbooks on the subject.

In the following paper an effort is made to derive the value of the adiabatic lapse-rate, first for dry air, and subsequently for saturated air, when displaced slightly from its position of .equiiibri um.

3. THE .ISCENT OF D R Y .4IR The lapse-rate of dry ascending air i s most readily deduced

from the thermal equation of energy. I f a quantity of heat is added to a mass of air, the heat is used partly in raising the tem- perature of the air and partly in expansion against the external pressure of the environment.

Let T, p , and 3 represent the temperature on the absolute scale, the pressure in millibars, and the specific volume, respectively, and let d T , d p and dv be the changes in the values of these variables when an amount of heat d Q is added to unit mass. In the following 1 Met . Zeit., Braunschweig, 1, 1884, a SmitLsonicrn Inst. Misc Coll.. Mechanics of the Earth's Atmosphere, Vol. 3. 3 Geof. Publ., Oslo, 3. 1925. No. 13. 4 P y a . Camb. Phil. SOC.. 26, 1930, pp 170-2. J Collected Papers, Vol. 6, p 432.

pp. 421-431.

352 BRUNT-ADIABATIC LAPSE-RATE FOR D R Y Sr SATURATED AIR

pages R is the gas constant, c, and C . the specific heats at constant pressure and temperature respectively, and p the density, all for dry air. W e have the relations

pv = R T , and by differentiation, pdv + v d p = RdT. C , - c,= A R

Here A is the reciprocal of the mechanical equivalent of heat. We then have

dQ=c,dT+ApdV=(c ,+AR) d T - A v d p

' (1)

a p = - g p a z . - (3)

dQ=c,dT+Agdz . * (3)

A P

=c,aT-Avap =c ,aT--ap . If the changes of pressure are due to vertical motion, then

Substituting for dp in equation ( I ) , we find

We could describe this result by saying that the heat given to the rising mass of air is in part used in changing the intrinsic energy of the air, and in part in increasing its potential energy.

If the motion is adiabatic, no heat being communicated to the moving- air from outside, by radiation or convection, then d Q = o , and the rate of change with height of the temperature of the moving air is given by

- (4) dT _ _ n.4 dz --z

3. TRE -\SCENT OF SATURATED AIR The state of a mass of air with regard to its content of water

vapour can be specified in a number of ways. W e might, for example, specify the state of the air by its temperature T and its vapour pressure e , or by T and the humidity mixing ratio x , which measures the number of grammes of water vapour associated with one gramme of dry air. Then

where e is the ratio of the densities of water vapour and dry air at the same temperature and pressure.

Consider the changes in unit mass of dry air with its associated water vapour, during a small element of its ascending path, it being supposed that any water drops which may be condensed are carried upward with the air. Let the initial value of t be E , and let the communication of a quantity of heat dQ produce changes of dT, d p , d v , and dx in T , p , v and x . The heat is used up in

(a) Increasing the internal energy of the dry air ; (b) Increasing the internal energy of the liquid water and

(c) Increasing the potential energy. water vapour ;

Equation (3) now has to be modified in two particulars. If p is the density of the dry air, then px is the density of the water vapour, and p ( I + x ) is the total density of the moist air. The term c,dT must be mcdified to take account of both (a) and (b)

BRUNT-ADIABATIC LAPSE-RATE FOR DRY 8: SATURATED AIR 353

above, and as we are now concerned with I +z grammes of moist air, we must take account of the change of potential energy of I + z grammes of moving air. The last term in equation (3) there- fore becomes Ag ( I + x ) d z , plus a term t o represent the change of potential energy of any liquid water drops in the mixture.

At any instant, the internal energy of z grammes of water vapour and of ((-5) grammes of liquid water, a11 at temperature T, is equal to the internal energy of ( grammes of liquid water + the latent heat of z gramrnes of water vapour, both estimated a t temperature T.

where c i s taken to be a mean value of the specific heat of water. The change in internal energy of the system consisting of I gramme of dry air, T grammes of water vapour, and ( E - x ) grammes of liquid water, may therefore be written,

(c,+(c) d T + d (Lz) and for adiabatic motion, with dQ=o, it follows that equation (3) now becomes

The total internal energy is therefore cT( -I- Lz

Here d / d z ( L r ) is to be interpreted a s the rate of change of Lx with height, on account of the change of temperature and pressure with height. Equation (6) may be used to compute the lapse-rate of temperature of a mass of saturated air displaced from its position of equilibrium. For this particular aspect of the problem, ( becomes effectively equal to x, since initially it is supposed that the air is saturated, and contains no liquid water. For z (and E ) we can therefore substitute the value of z given by equation (j).

p de dT -€- L e d p . d d L e se d L d T dz d z p - e - p - e dT dz ( p - e ) Z dT dz ( ~ - e ) ~ d z + € L - - - - -- ( L x ) = E - - - - - -

(7) L e d p } ( p - e ) ? d z

+ L P - - € - - .

But *= - g x total density= - g p ( I +() = -9- az R T Q - e * (8) = - - ( p - ( r - e ) e ) 9 .

RT Substituting in (6) from (7) and (8) for ( L x ) and d p l d z , we find

ee d L { cD+sc+ F d T (9)

p - ( x - e ) e , { p - ( 1 - s ) e } L e -eg RT ( p - e ) a = - g A

P - e So far, no approximations have been made, and equation (9)

is strictly accurate, but a number of approximations can now be made. In the factors p-e and p - ( x - e ) e we shall neglect e by comparison with p . The. error involved in this approximation is very small for values of e and p such a s occur in practice, the

354 BRUNT--ADI:\BATIC LAPSE-RATE FOR DRY 9r SATURATED AIR

maximum error in dTlda not exceeding one per cent at the most. (See p. 356 below.) Equation (9) then reads

All the variables in this equation except p are T. 'The equation may be written in the form

8 Le I

dT s A A R T P I + - - -

- ( 1 0 )

known functions of

Equation ( 1 1 ) is then in a form suitable for computation by the use of tables of the vapour pressure e. I t may also be written

. (12) X + p z+p - dT-dry adiabatic lapse-rate x z-

where X=- e Le - , and Z=- { e ( c + ~ ~ ) + ~ + } AR T c,

Thus .Y and Z are functions of T only, and a table of values of these functions can be computed for different temperatures, and the saturated lapse-rate for any temperature and pressure can then be computed by the use of equation (12 ) . For some purposes i t may be more convenient to write the last equation in the form

saturated lapse-rate 2-,Y = I - - . * (13) dry adiabatic lapse-rate Z + p

where p is mcasured in millibars. X table of the values of X, 2, and 2 - X has been constructed

for a range of temperatures from amo:\ to 32oO-A. covering rather more than the whole range of temperatures which occur in the atmosphere. The details of the computation have been shown, in order that the reader may see the relative magnitudes of the terms. I t is readily seen that the effect of the neglect of the terms in e in the expressions p - e and p - ( I -c ) e of equation ( g ) , is very small, and that the approximations made at the stage of equation (10) were justified.

The values adopted for the saturation vapour pressures are those given in the Smithsonian tables, and the values of the latent heat L are shown in column (2) of the table. At temperatures below freezing point, i t has been assumed that the latent heat of sub- limation of ice is constant, as stated by Fjeldstad. Above the freezing point c + d L / d T has been taken a s 1.5 , while below freezing point d L l d T has been taken as zero and c a s O'j.

The tabulated values of X and 2 can be used to compute the lapse-rate for any temperature and pressure, the values for inter- mediate degrees being interpolated where required. In Fig. I is shown the distribution of the values of the lapse-rate, lines of equal values of the lapse-rate being drawn. The numbers shown alongside the lines in Fig. I are the measures of the lapse-rates in ,degrees C. per IOO metres, This diagram is sufficiently accurate

BRUNT-ADIABATIC LAPSE-RATE FOR DRY & SATURATED AIR 355

57 1 574 577 579 582 584 587 590 592 594 595 677

I

I

1

I ,

9

I

1

TABLE OF THE FUNCTIONS 2, X. AND 2 - X

e

106.3 82'1 62.8 47'6 35'7 26.5 I94 14.0 10'0

7' 1 6 1 6 1 4'8 3.1

1.26 078 0'47 0.211 0.16 0.09 I 0.050 0'027 0014 0.0071 0.0035 0'00 15

2'0

5'4 4'3 3'1 2'7

I .6 1'3 0.9 069 0.51 0'44 0.50 0'40 0.27 0.18 0'12 0'077 0'049 0.030 0.018 0'01 I ow36 0004

2.1

0'002 0001 0'00 I 0001

-

-- (5) I (el

I- I59 3083 123 2468 94 1962 71 I563 54 I222 40 934 29 763 21 531 15 408 I1 29 1 9 262 3 338 2 270

I 81 0.4 52 0'24 38.2 0.14 203 0.08 12'2 0.046 7.4 0.025 4'1 0.014 2.7 0.007 1.4 00036 0.7 03018 0 7 O.O0O8 o 6

2 183 I I22

I

3243 2591 2056 1634 1276 974 792 552 423 302 271 341 272 185 123 82 52 38

20.4 12.3 7'4 4'' 2.7 1 '4 0 7 0.7 0.6

~

8560 IW 6760 1330 5360 1040 4260 816 3330 616 2540 466 2060 350 I440 254 1100 188 787 136 700 120 885 136 712 107 481 70 321 46 214 30 136 18.8 99 11.6 53 7'0 32 4.1 19 2'4 11 1.3 7 0.74

3'6 0 39 1.9 0 2 0 1'9 010 1.6 0'045

(10) -

2 - X

6870 5430 4320 3444 2714 2074 1710 1186 912 651 580 749 605 411 275 184 117 87 46 28

166 9'7 6.3 3'2 1 '7 I .n I *6

Pacssuar I N MILLIMRS Fro. l.-Ie.odeths of saturated adisbatio lapse-rate.

356 BRUNT-ADIABATIC LAPSE-RATE FOR DRY & SATURATED AIR

to permit of the interpolation of the values for temperatures and pressures intermediate to those shown on the lines.

The actual lapse-rates given by Neuhoff were compared with the diagram of Fig. I , and in all cases the agreement was as good as could be expected, in view of the differences in the adopted values of the various constants involved in the computation. A check on the accuracy of the approximation made in going from equation (9) t o the approximate equation ( 1 1 ) was also made. The order of the errors involved can be seen by a comparison of the values of the saturated adiabatic lapse-rate computed from the two equations, for p=100o mb. and T=3m0A. The full computation without approximation gave the value '376, whereas the approximate equation gave the value '374, showing that the method here adopted is sufficiently accurate for all practical purposes. The omission of e by comparison with p leads to an error of the same sign in both numerator and denominator of equation (11), and the errors very largely balance each other.

Attention is drawn to the fact that the saturated lapse-rate which we have discussed above is the rate of fall of temperature of an element of saturated air displaced from its position as part of the normal environment. If the displacement of the air is over a wide range of height, allowance must be made for the difference of temperature or density of the moving air and its environment. This is most conveniently done by the graphical m e t h d s of Hertz and Neuhoff.

4. THE DEPENDEBCE OF THE LAPSE-RATE ON "EMPEBATURE AND PRESSURE

The general nature of the way in which the saturated adiabatic lapse-rate depends on temperature and pressure can best be seen in the figure, which shows isoplcths of the ratio of the saturated adiabatic to the dry adiabatic lapse-rate for a wide range of con- ditions which is probably wider than we shall require in the earth's atmosphere.

For temperatures below freezing point, the latent heat of sub- limation of ice has been assumed to be constant, as suggested by Fjeldstad. The change in the latent heat at the freezing point is reflected in the break in the continuity of the isopleths a t 273'A. The points at which the isopleths run into the isotherm of 273' have been determined with some care, in order that the diagram may be sufficiently accurate for interpolation.

When the condensation a t temperatures below the freezing point takes the form of supercooled water, the saturated adiabatic lapse- rate should be determined by the continuation of the curves drawn for temperatures above 273' into the region below 273'. I have refrained from representing this continuation in the diagram, so as not to confuse it by too much complication.

Conditions a t the earth's surface are represented approximately by a pressure of 1000 mb. and temperatures between about 230'A. in the Siberian anticyclone in winter, to about 315'A. at the equator. The saturated adiabatic lapse-rate at the earth's surface therefore varies between -98 and -30 times the dry adiabatic. I t is perhaps not generally realised how close to the dry adiabatic is the saturated

BRUNT--ADIABATIC LAPSE-RATE FOR DRY 9r S.\TURATED AIR 357

adiabatic lapse-rate in the polar regions, so that convection cannot possibly occur as readily in high as in low latitudes.

The temperature is the more important factor in determining the saturated adiabatic lapse-rate, as is seen by an examination of the figure. The diagram is applicable t o any atmosphere, and is not restricted to the earth’s atmosphere. The isopleths are for selected values of the ratio of the saturated to the dry adiabatic, and i t happens that in the earth’s atmosphere the dry adiabatic has the value of almost exactly I O C . per 100 metres. Hence thk isopleths may be interpreted for the earth’s atmosphere as lapse- rates in degrees C. per 100 metres.

I t is not without interest t o consider the application of the results we have obtained to the atmospheres of other planets, notably Venus and Mars. If t h e planet Venus were unclouded, the equatorial temperature would be little, if any, short of 400°A., since Venus, if cloudless, would receive about three times the amount of radiation which actually reaches the earth’s surface after diminution by clouds. At this temperature the saturated adiabatic lapse-rate would be less than one-fifth of the dry adiabatic, and violent convection would be readily set up. The formation of a dense layer of cloud would of necessity follow, if there is water on the planet. On the planet Mars, on the other hand, the equatorial temperature, even with a clear sky, would not surpass about 240°A. a t most, and the saturated adiabatic lapse-rate would be nearly equal t o the dry adiabatic. Vertical convection currents would be set up only with difficulty, and even when set up could not produce thick clouds on account of the small water vapour con- tent of the atmosphere a t such low temperatures.

Returning t o Fig. I , we sec that for a given temperature the lapse-rate decreases with decreasing pressure, so that in the earth’s atmosphere its value at a given temperature will be the greater, the lower that temperature occurs in the atmosphere. This result is of importance in the consideration of a number of atmospheric problems. I t may be recalled that Golds showed that the observed lapse-rate in the atmosphere is very closely governed by the adiabatic lapse-rate for saturated air. A similar result appears t o follow from an examination of the tephigrams in Vol. 3 of the Manual of Meteorology, in a large number of which the plotted curve of obser- vations hugs closely the saturated adiabatics. We should therefore expect that on an isothermal surface in the earth’s atmosphere the observed lapse-rate should be greatest where the surface is lowest. This raises a question which can only be satisfactorily discussed by an examination of observations, and a discussim of all the factors which determine the lapse-rate in the atmosphere. A fuller treat- ment of this question is postponed to another occasion.

5 . AN ALTERNATIVE DERIVATION OF EQUATIO?; (6)

In view of the importance of equation (6), and in order to bring out the essential simplicity of the equation which is normally used in the discussion of the ascent of saturated air, an alternative derivation is added here. 6 London Meteor. Of., Geophys. Mem., No. 5 , p 101, and table XI.

358 BRUNT-ADI;\BATIC LAPSE-R;\TE FOR DRY & SATURATED AIR

The notation will be that used in the earlier paragraphs, one gramme of dry air having associated with it G grammes of water vapour and (E-z) grammes of liquid water, where [ is a constant. The entropy of z grammes of water vapour plus ((-3) grammes of liquid water, all a t temperature T I will be entirely independent of the manner in which the water vapour is formed. The simplest way to visualise it is to imagine ,$ grammes of water to be first brought up to the temperature T I and that then z grammes of the water are evaporated at that temperature. The evaporation involves the communication of L z units of heat, and therefore of Lz/T units of entropy. The total entropy of the water and water vapour is therefore +q+ $ The entropy of one gramme of dry air is

cp log T - AR log ( p - e). Hence the total entropy (p of I gramme of dry air, plus x grammes of water vapour, plus (E-z) grammes of liquid water is given by

Lx c p = ( c , + & ) l o g T + - - . A R l o g ( p - e ) T . - (14)

W e here assume that c, the specific heat of liquid water is constant. Equation (14) gives the relation between the temperature, vapour pressure, and the pressure. For a given mass of air ascending adiabatically p remains constant, and this constant value is readily evaluated when the initial conditions are known. If To, zo, eor and ( p - e ) , are the initial conditions, a t any level, then

(cp+[c)logT0 T +(La:-L,x,)-ARlog- P - e =o . (14a) ( P - 810

\Ve can represent the fact that the conditions are adiabatic by differentiating equation. (14) with respect to a.

- -+-. - ---- l h i s gives

( c , + dT T dz d", (?)- ( p - e ) da

or,

Now the Clausius-Clapeyron equation, stated in our notation, is

* (16) Lx de L x d T Av d e = - - d T o r Av- = - - T da T d a ' *

Substituting for d e l d z in (IS), and taking the substituted term t o the left-hand side, we find

dT da d T T dz

da ( L x ) L x dT - Av - dP da (cD+(c)-+!P- - + ---

which is again equation (6). I t appeared worth while giving this alternative proof in order

t o confirm equation (6), and also in order to draw attention to the

BRUNT-.4DI.\Ba\TIC LAPSE-R.4TE FOR DRY S: SATURATED AIR 350

ease with which equation (14) i s derived. The latter equation can in fact be written down without any preliminary argument. Most writers, including Fjeldstad, who has recently extended the work ot Neuhoff and Hertz t o lower pressures, give a derivation of equation ( 1 s ) which originated with Clausius, and which is far more involved than that given above ; they then obtain our equation (14) by integration. I t is possible that many readers of Fjeldstad's paper have been discouraged by the tedious nature of the derivation of the standard equation, and the statement of the equation given above is put forward a s a substitute for the proofs given elsewhere. A number of writers have given equation ( I j) without proof, but they appear to quote the equation a s one proved by Clausius. Humphreys' gives a less accurate equation for the lapse-rate of saturated air, but his form i s n o simpler, and no easier to derive, than the accurate form given above.

6. SUMMARY The equation g iv ing the saturated adiabatic lapse-rate is derived

as an energy equation. and a slight approximation makes it possible to reduce this t o a form suitable for direct computation. The results obtained are represented graphically, isopleths of different values of the saturated lapse-rate being shown Lvith pressure and tem- perature a s co-ordinates.

.An alternative derivation of the fundamental equation for rising saturated air is given, which, by assuming the principle of entropy from the beginning, reduces the derivation to very brief compass.

DISCUSSION

Dr. H. JEFPREYS \vrites : Mr. Brunt has given an elegant account of a rather tricky matter. I have tried to understand previous work on the saturated adiabatic lapse-rate, but have always failed; that may be my fault, but I did succeed in understanding this paper. I t is pleasant to see the question brought into relation with the problem of the stability af R layer of fluid heated below, and to see no mention of the mythical '' surrounding air " that plays so large a part in meteorological literature.

There is only one criticism I should like to offer. The equation c ~ - c , = A R

holds if cp and c, are heat capacities per gramme molecule: and then R has the same value for all gases. If, as stated in the paper, we are considering one g r a m m e of gas, the right side should be divided by the molecular weight, which of course differs for dry air and for water vapour. This slip does not affect later work, for the author works in terms of cp for air alone, and cunningly absorbs the specific heat of water vapour into that of liquid water and the latent heat of volatilisation.

Could the author tell us a little more explicitly what happens to the latent heat of fusion?

Mr. D. BRUNT, in reply, pointed o u t that he had used the equation

in the manner common among meteorologists, using cp and c, to denote heat capacities per gramme, so that R is not the universal gas constant,

7 Humphreys, W. J., Physics of the n i r , p. 33.

cp - C, = A R

360 BRUNT-ADIABATIC LAPSE-RATE FOR DRY SI SATURATED AIR

but the special value appropriate to dry air. He had allowed for this in the case of water vapour by using R / e as the g a s constant for water vapour. With regard to the query concerning the latent heat of fusion, he had assumed that this was at each stage conveyed t o the dry a i r and the remaining water vapour, and so could be treated in precisely the same manner as the latent heat of condensation.

A New Meteorological Society

Nacional de Meteorologia,” has been formed ar Santiago de Chile. We are informed that a new society, to be known as the I ‘ Sociedad

Note on the paper on “Meteorological Acoustics” in the July Number I t has been pointed out to me that Rayleigh’s equation for the path

of a sound ray, quoted in my paper on “ Meteorological Acoustics ’’ in the last issue, makes a n approsimation which can hardly be justified. The true equation for such rays in a n isentropic atmosphere can be es- pressed by y = D cos ?$ which is a cycloid f o r m d by the rolling. of a circle of diameter D on a horizontal line above the earth’s surface whose position is determined by the lapse-rate of temperature a n d corresponds to that level (hypothetical) a t which the absolute temperature is zero and y is measured from that line.

The value of D is further given by the relation

where I.’ is the velocity of sound a t the lowest point of the cycloidal path. This value works out to about 420,000 feet, and for points near the

ground it can be shown that the cycloid and a circle described with radius 2D are nearly coincident.

This approximation was made in determining the elevation of the acoustical horizon and range of audibility for a n aircraft a t IO,OOO feet quoted in the paper.

For heights small compared with the 420,000 feet quoted, Rayleigh’s catenary is also nearly coincident with the cycloid and the circle.

The order of error for the angle of inclination $ of, the sound ray a t 10,ooo feet, which ultimately just touches the ground, is shown by the following values :-

Cycloid (correct) 12’ 43’ Circle ... ... 12’ 39’ Catenary ... 12’ 30’

These errors a re the maximum as they refer to sound rays showing the maximum curvature. For heights of 1,000 feet above the earth’s surface the variations in the value of $ for these three curves is less than one minute of arc.

Allusions to the cycloidal form of path for sound transmission are to be found in the literature of abnormal audibility.

W. S. TUCKER.