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On Radiation Cooling of a Layer of Fog* On Radiation Cooling of A Layer of Fog* RAYMOND WEXLER Box 136, RFD No. 2, Neptune, N. J. NOTE IN THIS BULLETIN, Sept. 1941, in a cloud or fog. Fifty meters is then the A p. 288, briefly described and explained length of a path which contains 0.1 mm of observations of isothermal lapse rates liquid water per square centimeter of cross- in dissipating stratus at Spokane, Wash. section. This thickness of liquid water is In this article an effort will be made to explain within a few percent of being a black-body lapse rate changes within a fog during radia- radiator. tive conditions at night. The theory of Assume a fog of 50 meters or more thick- temperature changes in a fog with its upper ness. Let T0Ti represent the known initial portion remaining at a fixed level will be temperature distribution, assumed linear, of analyzed. In a subsequent article it is the top fifty meters. At the end of time t let planned to analyze lapse rate changes in a the temperature distribution be T0T2 in the dissipating fog. fog (see Fig. 1). In other words we assume 1. Lapse Rates in Fog.—At Leafield, Eng- land a superadiabatic lapse rate often de- veloped in the ground layer after the fog had become sufficiently deep.1 An inversion of temperature was most frequently observed just below the top of the fog and the steepest inversion existed just above the fog. The upper portion of a fog cools more rapidly due to radiation than the air imme- diately above or below. Hence a fog with its upper surface at a fixed level would develop superadiatic lapse rates within the fog and a steep inversion immediately above. Heywood cites one example of superadiabatic lapse rates observed within a fog and a steep inversion above. Apparently this is not a common condition. Presumably if the fog grows rapidly in depth, cooling at any one FIGURE 1. level may be insufficient to change the initial that the cooling of the fog due to radiation inversion conditions to a lapse of temperature. extends 50 meters down from the top of the Hence a rapidly deepening fog should be fog layer and that the temperature distribu- characterized throughout by a temperature tion in this layer is linear. Assume also that inversion. A slow rate of ascent of the top the cooling extends upward from the top of surface of the fog would cause a slight inver- the fog through eddy turbulence. sion in the upper portion and superadiabatic The total energy loss in cooling a column lapse rates a short distance below. of fog with unit cross-section and 50 meters 2. Theory.—According to Brunt2 a layer of T1 -f- T 0 1 fog 50 meters thick can be treated as a black in length from mean temperature 9—- to body radiator. This is on the assumption that one cubic meter of cloud contains 2 cc of liquid water, not unreasonable conditions (1) C/fog = cpP {Tl 2 Ti) • 5 • 103 cal. 1 Heywood, G. S. P., "Some Observations on Fogs and their Accompanying Lapse Rates," Q. Jn. Roy. where cp is the specific heat of air at constant Met. Soc., Jan. 1931, 97-101. pressure and p is the average density of the 2 Brunt, D., "Physical and Dynamic Meteorology," London, Cambridge University Press, 1939, p. 122. column. During the cooling latent heat of condensa- radiation). E decreases with decrease in tem- tion will be released. The energy that must perature but this is very gradual and no great be compensated by radiative cooling will be error will result if E is assumed constant. given by: Substituting (3) in (6) and eliminating 2 we {Ti - To) obtain: (2) Uu 5LP - (7) ^ + . ™ - * This is an approximate relationship computed dt 2(a + &V0V< a + b<t' on the basis that for a mean temperature loss of 3°C, one gram of water vapor per kilogram where b = 2cpP^K. This is a first order of air will be released. This is approximately linear differential equation easily solved5 with true between 265°A and 275°A. Here L is the result: the latent heat of condensation assumed con- Et (8) T = stant at 596 calories per gram, the value at + 6VR 0°C. Actually L decreases with increase in temperature. Since more water vapor con- = 2 Et (9) Ti - T2 denses at higher temperatures the assumption a + by/t of a constant L counteracts to a certain extent Equation (9) represents the cooling with time the assumption of a constant amount of of the top surface of the fog. condensation. As the fog cools due to radiation the air above the fog will be cooled by turbulent mixing. According to Taylor3 and Brunt4 the height z above some boundary to which turbulence is effective at time t is given by the equation: (3) z2 = 4 Kt, where K, in Brunt's notation, is the eddy diffusivity, assumed constant. The energy loss of the air above the fog will be: (Ti - 7Y) (4) Ua Cpp The total energy loss due to radiation will be the sum of (1), (2), and (4): (5) U = (bcpp • 103 + HLp) {Tl ~ Tl) FIGURE 2. Cooling of the Fog Surface. (T1 - Tt) + CppZ According to Elsasser6 the mean nocturnal Differentiate this with respect to time, letting heat loss of the ground for cloudless skies during the winter ranged from about 24 to 30 (Ti - T2) . T = calories per square centimeter per three hours. Since that article the radiation chart has been dU dT , dT , mdz (6) llt=a-dt+C»pZ-dt+C>pTli = E> corrected so that these values have decreased by about a third. Assuming a value of 7 where the constant a has been substituted for calories per square centimeter per hour for E 5cpp-103+^Lp. Here E is the effective we can plot the cooling of the top of the fog nocturnal radiation (difference between black for different values of K (see Fig. ,2). The body radiation from the fog and the return 5 See for example: Osgood, W. F., "Advanced Cal- ^Taylor, G. I., Phil. Trans. Roy. Soc. A, 215, p. 1, culus," Macmillan Co., New York, p. 312 (1932). 1915. 6 Elsasser, W., "Radiation Cooling in the Lower " Brunt, D., op. cit., pp. 227-228. Atmosphere," Mo. Wea. Rev., 68, 185-188, July 1940. .
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