ARCHIV Arch. Met. Geoph. Biokl., Ser. A, 26, 31-38 (1977)
FUR METEOROLOGIE GEOPHYSIK UND BIOKLIMATOLOGIE 9 by Springer -Vorlag 19 7 7 551.571:551.511.13
Carribean Meteorological Institute, Bridgetown, Barbados, West Indies
Effect of Moisture on the Potential Temperature M. L. Khandekar With 2 Figures Received August 2, 1976
Summary Three simple technique of incorporating the influence of moisture on potential temperature calculations are briefly discussed. These techniques lead to slightly differing values of potential temperature for a given sample of moist unsaturated air. It is shown that these small differences may be of importance in certain numerical modelling studies. Zusanmmnfassung Der Einflug der Feuchtigkeit auf die potentielle Temperatur Es werden drei einfache Methoden zur Berticksichtigung des Einflnsses der Feuchtigkeit bei Berechnungen der potentiellen Temperatur kurz besprochen. Diese Methoden fithren zu geringfiigig unterschiedlichen Werten der potentiellen Temperatur bei einer gegebenen unges~ittigten feuchten Luft. Es wird gezeigt, daft diese kleinen Unterschiede bei bestimmten numerischen Modellstudien yon Bedeutung sein k6nnen.
1. Introduction One o f the most widely used p a r a m e t e r s in a t m o s p h e r i c t h e r m o d y n a m i c s is the p o t e n t i a l t e m p e r a t u r e (3 which is defined by the standard e q u a t i o n 0 =
- -
(1)
Here T is t e m p e r a t u r e (~ p is pressure in millibars and n = R/cp, where R is the specific gas c o n s t a n t and Cp the specific heat capacity of dry air at c o n s t a n t pressure. In this definition, no a c c o u n t is t a k e n o f the moisture that m a y be present in the given sample of air. In general, the presence o f moisture will influence all the three parameters, n a m e l y , p, T and Ir on the right side o f (1). Thus the influence o f m o i s t u r e on the p o t e n t i a l t e m p e r a t u r e can be i n c o r p o r a t e d using m o d i f i e d values o f p, T and K.
32
M.L. Khandekar
The purpose of the present note is to consider some simple techniques of incorporating moisture in the potential temperature calculations. These techniques lead to modified and slightly differing values of potential temperature for a given sample of most air. These modifications, which are in general small in magnitude, may be of importance in certain atmospheric modelling studies.
2. Basic D e f i n i t i o n s and E q u a t i o n s
A very common technique by which ordinary temperature T of a given sample of moist air can be modified in the presence of moisture is by defining the virtual temperature Tv given by
Tv = T(1 + 0,61 q).
(2)
Here q is the specific humidity of the given sample of moist air. By definition the virtual temperature is the temperature that dry air would have if its pressure and specific volume were equal to those of a sample of moist air. Thus by introducing the concept of virtual temperature, the effect of moisture is taken into account without having to change the value of the specific gas constant. This concept of virtual temperature can be extended to define a "virtual" potential temperature 0v given by =
- -
= 0(1 + 0.61 q).
(3)
The interpretation of the virtual potential temperature is similar to that of virtual temperature as defined by (2). In the above definition of 0v, the exponent ~: is obtained using values of R and Cp pertaining to dry air. Since the introduction of virtual temperature Tv is meant to replace a given sample of moist air by that of dry air having same pressure and specific volume, the use of ~: pertaining to dry air in (3) appears consistent. However, it is possible to consider the influence of moisture directly on the values of R and Cp by taking a mixture of dry air and water vapour. Assuming no condensation, the first law of thermodynamics can be written for adiabatic condition as
dT Rm dp T Cprn p
(4)
Here R m refers to the specific gas constant for moist air and Cprn refers to the specific heat capacity of moist air at constant pressure. Using appropriate definitions we can write
R m = R (1 + 0 . 6 1 q )
and
Cpm =cp (1 + 0 . 8 7 q )
(5)
Effect of Moisture on the Potential Temperature
33
R and cp refer to dry air values. Substitution of (5) into (4)yields, after slight manipulation ~d T= ~ ( 1 - 0 . 2 6 q) dp P . Integration of this equation leads to the potential temperature which can be expressed as
(1_~)
K (1-0.26 q)
0m = T
(6)
0rn may be called as the potential temperature of unsaturated moist air (see [7]). Using (6), we may now redefine the virtual potential temperature as 0v* = 0m(1 +0.61 q).
(7)
Eq. (7) may be considered as the precise definition of the virtual potential temperature which may be interpreted as the temperature that dry air would have if its pressure and density were equal to that of moist air after having been brought adiabatically to 1000 mb. A slight manipulation allows us to write (7) as 0v* = ~)v (1000) -0'074q -~ 0v since the factor ( 1 ~00) -0'074 q -is very close to unity; accordingly in the following discussion ev as defined by (3) will be referred to as the virtual potential temperature. So far we have not considered the influence of moisture on the pressure exerted by the column of moist air. Let the pressure exerted by water vapour in a given sample of moist air be given by e mb so that p - e is the pressure exerted by dry air, p being the pressure exerted by the column of moist air. Using the partial pressure of dry air, we may define (~d = T
--
(8)
(3a may be defined as dry air (partial) potential temperature. This temperature was introduced and used in a thermodynamic diagram designed by Rossby [10]. We thus see that inclusion of moisture can lead to three different expressions for potential temperature as defined by (3), (6) and (8). All these expressions involve functional dependence on the specific humidity q (or partial vapour pressure e) and as would be expected these definitions reduce to the value of 0 defined by (1) when the given sample of air is completely dry. For non-zero values of q these three expressions lead to slightly differing values of potential temperature as will be seen by the following.
Arch. Met. Geoph. Biokl. A. ]Bd. 26, ]~. 1
3
34
M, L. Khandekar
3. Illustrative Examples Two sets of data over two widely different climatological regions are utilized to make some sample calculations. The first set of data are Jordan's [8] mean soundings for the West Indies area for the hurricane season, July to October. Using the values of q and the various expressions obtained earlier, values of ~v, 0m, and 9d are c o m p u t e d at various levels. The vertical profiles of these temperatures together with the profile of potential temperature ~ are shown in Fig. 1, The profiles are constructed only up to 500 mb level since there is
mb 500-
/
600 -
700
800
-
-
,y..-~ 900-
t000 295 ~
O
~'/
300 ~
-
-
8d .............
305 ~
310 ~
315"
320 ~
3]25 ~
Fig. 1. Vertical profiles of 9, era, 9d and % for Jordan's [8] data. The profiles are drawn only up to 500 mb level
very little moisture above that level. We see that the 0m profile is very close to the 6 profile everywhere with maximum departure o f up to half a degree in the lower layers. In contrast, the values of 0v (or 9d) differ from ~ by as much as two to three degrees from surface to 850 rob, while the difference is one degree or more above 800 mb tapering off to less than half a degree at 500 mb level. This relatively large difference between ~? and 0v (or Oa) especially in the lower layers cannot be neglected in general. As a second example, we have chosen the sounding at Baker Lake (64~ 96~ Northwest Territorties, Canada for 12 GMT August 2, 1973. The vertical distribution of q (Table 1) shows the airmass to be almost saturated up to 500 mb level. Also shown in Table 1 are values 8, 0v, 0m and 0d obtained using equations developed earlier. Once again, we find that 0m differs from ~ by at most one tenths o f a degree everywhere, while 9v differs from 0 by about two degrees below 900 mb and by about half a degree around 500 mb level.
Effect of Moisture on the Potential Temperature
35
Table 1. Baker Lake Sounding, 12 GMT August 2, 1973; Values of VariousPotential
Temperatures Pressure mb
Temperature ~
500 600 700 800 850 900 950 1000
14.9 - 5.4 1.0 7.2 10.1 13.8 14.9 15.4
Mixing Ratio gm/Kg
0
(3v
0m
0d
314.7 309.6 303.4 298.6 296.5 295.6 292.1 288.4
315.0 310.3 304.2 299.8 297,8 296.9 293.6 289.8
~
2.2 4.2 5.9 7.9 9.1 9.4 10.7 10.5
314.7 309.7 303.4 298.7 296.6 295.6 292.2 288.4
315.0 310.4 304.5 300.0 298.1 297.3 294.1 290.2
A simple analysis allows us to write an inequality between the various potential temperatures so defined. A comparison of (1) and (6) shows that em < (3 for non-zero values o f q; similarly it can be seen that both 0v and 0d are greater than (3 when q is not zero. To compare ev and Oct, eq. (8) can be manipulated to write ed = T ( I-~ -0 0 ) K approximated to obtain -~
- -
( 1- p)-K
1 +--
. This can be further
~ 0(1+.46q).
(9)
Comparing (9) with (3), (6) and (8) we have the inequality 0m ~ 0 ~ (3d~ 0v where the equality sign holds when the airmass is completely dry.
4. Discussion The small difference between ern and 0 at all levels suggests that moist air ascent or descent can essentially be treated as if the air were dry. However, the appreciable departure of 0v (or eg) from e, especially in the lower layers, suggests that these differences cannot be neglected in general. Since ev-0 represents change in the specific volume o f a sample of moist air from that o f a dry air (at same pressure), use o f 0v becomes desirable in modelling studies involving c o m p u t a t i o n of hydrostatic pressure. An excellent example is the study of atmospheric flow over a heated tropical Island as reported by Bhumralkar [2]. To consider the influence o f moisture the hydrostatic equation in potential temperature form is modified to write
~z
g('t
CpO
1---+~"
(10) 3*
36
M.L. Khandekar
Here the symbol; have standard meaning. The factor 1/(1 + A e) in (10) arises because of presence of moisture in vapour or liquid form and this factor togehter with 0 constitutes the virtual potential temperature. Bhumralkar has further shown [3] using data over Grand Bahama Island in the Caribbean Sea that integration of eq. (10) without the factor 1/( 1 + AB) would overestimate the surface pressure by 2.2 mb for "normal" moist conditions and by 1.1 mb when condensed water is also present. Virtual potential temperature thus provides a useful means of simulating real-atmosphere pressure distribution. Table 1 shows that the departures ev--B are appreciable even for a subartic region like the Northwest Territories in Canada. The sounding at Baker Lake shows an almost saturated airmass with a lapse rate close to saturated adiabatic above 950 mb level. Such an airmass is conducive to convective showers which are not uncommon in this region of Boreal climate during summer. A simple numerical integration using (10) over the lowest 150 mb layer for the Baker Lake sounding yields an overestimate of the surface pressure by about one millibar when e (instead of ev) is used. Thus, use of virtual potential temperature becomes desirable in certain modelling studies even over the midlatitude and higher regions. Recent numerical simulation studies on the atmospheric boundary layer have found it useful to incorporate By in defining the stability length the Monin-Obukhov length; according to Deardorff [4] use of ev allows the influence of water vapour upon buoyancy to be taken into account. In a more recent study, Mak [9] finds By a useful quantity for calculating vertical heat flux and its influence on the steady-state structure of the trade wind convective layer. As mentioned earlier, the dry air potential temperature Bd was introduced by Rossby and was utilized in his "Rossby" diagram. The use of ed allowed Rossby to obtain vertical and equally spaced saturation mixing ratio lines which are quite unequally spaced in a Tephigram, for example. Since Tephigram and other similar thermodynamic diagrams are extensively used at present, the advantage offered by Rossby diagram does not appear to be of particular significance now; however the introduction of ed becomes useful in obtaining an expression for the equivalent potential temperature Be as Be =
Bd exp ( L w s / c p T )
(see [6].
(1 1)
Here L is the latent heat of condensation and w s is the saturation mixing ratio. Quite often, ed is replaced by O in (11) to yield a simpler expression for the equivalent potential temperature; this approximation could lead to appreciable difference in Be values in the lower layers of tropical atmosphere. The equivalent potential temperature is an important parameter for classification of tropical atmosphere into various convective regimes [1, 5]. Using
Effect of Moisture on the Potential Temperature
37
such a classification Betts further computes a cloud work function suitable for parameterization of cumulus convection. Since these clouds work functions and the various convective classifications appear to be sensitive to changes in 0e, more precise determination of 0e via (11) would be desirable. As an illustration, two (?e profiles over Barbados, West Indies for a typical dry and a wet day are shown in Fig. 2. The profiles refer to 1200 GMT for 8 and 23 October 1973. Climatologically October is the rainiest month in Barbados. The wet day (October 8) was characterized by widespread preci-
200rnb 300
400-
I~
/111Ill
500-
600 -
700-
800 s,# f
f
t
I
I
900" 1000-
sis"
3 ~ o " 3es"
3 3 0 " 33s"
a ~ o " aks"
3~o'K
Fig. 2. Vertical profiles of Oe for a typical dry day (solid line) and a typical wet day (dashed line) in October 1973 over Barbados, West Indies
pitation over the island of 430 km 2 (about 50% of the total reporting stations, recorded rainfall amounts of over 5 cm each for the 24-hour period ending 1000 GMT); while the dry day (October 23) reported essentially no precipitation over most parts of the island. The markedly different profiles of 0e (in Fig. 2) are indicative of the suppressed and disturbed conditions over a tropical island as suggested by earlier workers. An attempt is being undertaken to relate the 0e profile with the spatial distribution of rainfall over Barbados,
38
M.L. Khandekar: Effect of Moisture on the Potential Temperature
Acknowledgements This work was initiated while I was associated with the University of Alberta at Edmonton, Canada. I wish to thank Dr. Chandrakant Bhumralkar of Stanford Research Institute, Menlo Park, California, USA for some helpful communications. Thanks are also due to Miss Donna P. Greene for typing the manuscript.
References 1, Betts, A. K.: Thermodynamic Classification of Tropical Convective Soundings. Monthly Weather Rev., 102, 760-764 (1974). 2. Bhumralkar, C. M.: An Observational and Theoretical Study of Atmospheric Flow Over a Heated Island: Part II. Monthly Weather Rev., 101,731-745 (1973). 3. Bhumralkar, C. M.: Effect of Moisture on the Hydrostatic Pressure. J. Appl. Meteor., 13, 174-175 (1974). 4. Deardorff, J. W.: Numerical Investigation of Neutral and Unstable Planetary Boundary Layers. J. Atmospheric Sci., 29, 91-115 (1972). 5. Garstang, M., N. E. Lesur and R. Hadlock: Results From a Comprehensive Tropical Field Experiment. Proc. of the Intern. Conference on Tropical Meteor., June 2-11, 1970, Honolulu, Hawaii, BI-1, BI-2 (1970). 6. Holton, 3. R.: An Introduction to Dynamic Meteorology. New York: Academic Press. 1972. 7. Iribarne, J. V., and W. L. Godson: Atmospheric Thermodynamics. Dordrecht: D. Reidel PuN. Comp. 1973. 8. Jordan, C. L.: Mean Soundings for the West Indies Area. J. Meteor., I5, 91-97 (1958). 9. Mak, Man-Kin: A Semi-Empirical Dynamical Model for the Trade-Wind Convective Layer. Monthly Weather Rev., 102,561-570 (1974). 10. Rossby, G. G.: Thermodynamics Applied to Air-Mass Analysis. Mass. Inst. Technology, Met. Papers 1, No. 3, 1932. Author's address: M. L. Khandekar, WMO Lecturer in Meteorology, Caribbean Meteorological Institute, P.O. Box 130, Bridgetown, Barbados, West Indies.