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Bulletin American Meteorological Society BULLETIN OF THE AMERICAN METEOROLOGICAL SOCIETY Entered as second class matter at the Post Office, Worcester, Massachusetts, under Act of Aug. 24, 1912. Issued monthly except July and August. Annual subscription, $3.00; single copies of this issue, 35c. Address all communications to ROBERT G. STONE, Editor Blue Hill Observatory, Harvard University Milton, Mass., U. S. A. Vol. 20 OCTOBER, 1939 No. 8 A Diagram for Obtaining in a Simple Manner Different Humidity Elements and its Use in Daily Synoptic Work DR. W. BLEEKER Subdirector, Royal Netherlands Meteorological Institute, De Bilt, Holland 1. DEFINITIONS AND SYMBOLS 2. CONSTRUCTION OF THE DIAGRAM T — Temperature. The relations between the tempera- Te — Equivalent temperature, the ture, T, the equivalent temperature, temperature which the air Te, and the wet-bulb temperature, Tw, would assume if all water va- are given by pour were condensed at con- (cp+tfcpw) (T-Tw)=Lw M), (1) stant pressure and the heat of condensation used to increase cP (Te-Tw) = Lw Xw, (2) the temperature of the air (von cP (T-Te)+xcPw (T-Tw) = -LwX. (3) Cpw Bezold, 1905). Since = about 2 and x seldom Tw = theoretical wet-bulb tempera- cP ture, the lowest temperature to exceeds 0.025 we may write: which the air can be cooled un- der constant pressure by evap- T-Tw - —— (x-xw), (la) cP orating water into the air -Lw (Normand, 1921; Brunt, 1934). Te-T w = ( Xe — X w ) , (2a) fw equals the temperature of a well-ventilated wet-bulb ther- T-Te = (X - Xe), (3a) mometer (Assmann-psychrome- Cp ter. where xe denoting the mixing ratio at cP = Specific heat at constant pres- the equivalent temperature Te, is, and sure of dry air. may be written for, zero. = specific heat at constant pres- This leads to the conclusion that on sure of water vapour. a T, x-diagram the points (Tw, xw), e = vater vapour pressure at tem- (T,x) and (Te,xe) lie on a straight perature T. line inclined to the T-axis at arc tan ew — max. water vapour pressure at (-oP) temperature Tw. V — pressure of the air. (-Lw) x = humidity mixing-ratio. We may draw two principal lines xw — humidity mixing-ratio of air on such a T, ^-diagram: saturated at the wet-bulb tem- I) The Tw, Xw - curve, giving the re- perature, TV lation between the saturation Lw = latent heat of water vapour at mixing ratios at various tempera- the wet-bulb temperature, Tw. tures, Unauthenticated | Downloaded 09/27/21 11:51 PM UTC • • h XJls P g hrj 4 1=O4 QTQ i_L gAS " S'11 || I fill gs-S-I: CD Torq 5cj CD' tr1 ^ S ™ C P g (D & P fD SjoB *-> Pas O£ m. B 2, S w ^ oO 3 C Oi-". CD /rs 3 2- 2.5 » Orq •• S^ ft) (D L g* 4 K w^OQ g1 ^ g * S3 II O CD 4 p II) the Te, Xe - line, which, since xe = 0, the T axis. These lines are lines of will be the T axis. constant wet-bulb- and constant equiv- We may now draw through all alent temperatures. They are called points of the 2V, xw - curve a straight by e - lines and they are nearly paral- / -cP \ lel (see Figure 1). line inclined at arc tan I I to V Lw/_ With the aid of these b, e- lines 2V Unauthenticated | Downloaded 09/27/21 11:51 PM UTC and Te can be very easily found for For a given pressure we can also any given value of T and x; Tw by mark along the x -ordinate a scale for following the b, e - line from the point water vapour pressure, e, from the x (T,x) to the Tw,Xw -curve (satu- relation: ration-curve), and Te by following the e same 6, e -line down to the T axis. x = 0.622 . p — e Again, it is possible to draw on the The diagram in figure 1 is computed diagram lines of constant relative for a pressure of 1000 mbar. humidity, The influence of pressure will be + 0.622 R = - X discussed in Tf8. x + 0.622 3. DETERMINATION OF HUMIDITY ELEMENTS FROM VENTILATED THERMOMETERS AT APPROXIMATELY MEAN SEA-LEVEL From the completed diagram we IV). Relative humidity, is inter- may, given the wet- and dry-bulb polated from the curves of equal temperatures obtain directly other relative humidity adjacent to the humidity elements as follows. point A. I) Mixing ratio, x. Intersection of V). Equivalent temperature is given the b, e -line with the given wet-bulb by following the b, e -line from A temperature and the dry-bulb tempe- down to intersection with the T-axis, rature-ordinate gives a point A (fig- or if no intersection occurs at the ure 1). The mixing ratio x is given scale at the right hand side of the by the ordinate of A measured on the diagram. sc-axis. VI). The differences (xm — x), or, II). Water vapour pressure, e. At at a given pressure, (em-e), which the pressure for which the e-scale has are expressions considered in the the- been calculated the value of e corres- ory of evaporation, are obtained by ponding to the point A may be read proceeding vertically from A to inter- off directly from that scale. section with the Tw, xw — curve (satu- III). Dew point, is given by inter- ration-curve) and reading off xm from section (B) of the horizontal through the x — scale, Or 6m from the e - scale, A and the Tw, xw - curve (saturation- and subtracting from them the values curve) read off on the !T-axis. originally found for the point A. 4. SOME THEORETICAL APPLICATIONS a. Determination of mean wet-bulb where Te is the equivalent tempera- temperatures after mixing at constant ture of the parcel of mass m', and Te" pressure. is the equivalent temperature of the It is easy to deduce the final wet- parcel of mass m' bulb temperature of two air parcels If we proceed from this tempera- of different wet-bulb temperatures ture Tem on the T-axis of our dia- and different masses, when these are gram along the b, e - line to intersec- mixed, by using their equivalent tem- tion with the Tw, xw m curve, the tem- peratures. For, since the mixing perature of the point of intersection, equation holds without exception for read along the T-axis, will give the equivalent temperatures, we may de- required wet-bulb temperature of the termine the final equivalent tempera- mixture. ture of the mixture from the relation Since particles of the same wet-bulb m' T\ + m'' TV ' temperature are represented on the Te = • m'+m'' same b, e -line and have the same Unauthenticated | Downloaded 09/27/21 11:51 PM UTC equivalent temperature, it follows that temperature in the case that conden- after mixing the wet-bulb tempera- sation does occur. The final mixing ture of the entire mass remains un- ratio, xm, after condensation can be changed. read off along the #-axis. The quan- b. Condensation after mixing. tity of condensed water vapour is Mixing of moist air parcels may then given by the expression: lead to condensation (Brunt, 1934), [ (m' x' + m" x" - but the mean temperature after con- (m' + m ') xm) ]. densation is not the weighted mean Brunt computed for the slope of of the original temperatures owing to the line going from the weighted mean the liberation of heat. of the initial temperatures of the two Brunt (1935) pointed out the pos- masses to the saturation water vapor sibility of calculating the final tem- pressure-temperature curve a value perature by the aid of the Clausius- / 1600 cCPp \ Clapeyron-equation. It is however arc tan f possible to simplify this calculation. V ) This is the same as our value, since For, if we first determine from the Brunt used the saturated water vapour diagram the two equivalent tempera- pressure-temperature-curve, while we tures, find their weighted mean and have used the saturated mixing ratio- then from this temperature follow the temperature curve at pressure 1000 b, e - line to intersection with the mbar and JTW, Xw — curve, we get the final tem- perature Tm, which is the mean wet- #=0.622 - or approximately- bulb temperature when condensation 1000-e 1600 does not occur, and is the mean real 5. APPLICATION OF THE DIAGRAM FOR NIGHTFROST- AND RADIATION-FOG FORECASTING For forecasting of nightfrost and The extracooling necessary for reach- radiation fog the diagram is very ing, after saturation, the "supersatu- helpful, because it is possible to de- ration" of 0.5, 1.5 and 3.0 gr/m3 water termine at a glance the difference be- is then given by the horizontal tem- tween dry-bulb and dew-point tem- perature-distances between the satu- peratures. ration-curve and the different "fog- Petterssen (1939) has introduced lines". It is easily seen from the dia- the term "mist-interval". He points gram that these distances increase out that at least 0.5 gr of liquid water with decreasing temperature. must be present in 1 m3 of air before The question: "given Tw and T, to the visibility falls below 1000 meters what temperature must the air be and, that such a "supersaturation" cooled for fog to occur" may be an- 0.5 gr/m3 water asks for much more swered without any computation or cooling at low than at high tempera- use of tables. The given temperatures tures, as may be seen from Petterssen Tw and T enable the point A to be fog-prediction-diagram.
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