Effect of Radiative Heat Transfer on Profiles of Wind, Temperature and Water Vapor in the Atmospheric Boundary Layer

Transcription

1 April 1971 Junsei Kondo 75 Effect of Radiative Heat Transfer on Profiles of Wind, Temperature and Water Vapor in the Atmospheric Boundary Layer By Junsei Kondo Institute of Coastal Oceanology, National Research Center for Disaster Prevention, Hiratsuka City, Japan (Manuscript received 6 October 1970, in revised from 2 February 1971) Abstract The diurnal changes of wind, air temperature, and water vapor profiles near the ground have been investigated by integrating numerically the differential equations of momentum, heat and vapor transfers. The equation for change of air temperature includes the long-wave radiative heat transfer. Equality of the diffusivities for momentum, heat and vapor transfers is assumed. Profiles obtained deviate from the theoretical profiles which exclude the long-wave radiative heat transfer, and assume the steady state. In the stable case, the deviation of estimated wind velocity is relatively small and that of air temperature is somewhat larger, and the largest discrepancy is seen in the profile of water vapor. In unstable cases, the so-called constant-flux layer is at the heights of several tens of meters, but in stable cases, it is about several meters high. In the daytime at the height of several centimeters, there is a layer of intense heating due to radiative flux divergence. It appears that such excess rate of heating might yield the heat energy of convective mixing. In the nighttime, radiation fog started to form in a layer about 0.1 to 1m high from the surface, and the layer, after spreading slowly upwards, finally disappeared after sunrise. 1. Introduction The problem of turbulent transfer near the atmospheric boundary layer has been studied by many workers. Most of the recent progresses in this field have been brought about by the similarity theory (Monin and Obukhov, 1954). According to this theory the principal variables, such as temperature and wind velocity, are expressed nondimensionally as fractions of a friction velocity u*, a stability length L, and a scaling temperature T*. Under impetus from the similarity theory, KEYPS equation has been derived (Kazanski and Monin, 1956; Ellison, 1957; Yamamoto, 1959; Panofsky, 1961; Sellers, 1962). It was found by Panofsky, Blackadar and Mc Vehil (1960) and Kondo (1962) that KEYPS equation and observations agree well for nearneutral and unstable air; in stable air, however, the theory does not fit all the points. Panofsky et al. (1960) suggest that the eddy structure in neutral and unstable stratifications is governed by the distance from the ground, whereas in stable air the eddies have too small a vertical extent to be governed by this factor. Takeuchi and Yokoyama (1963) introduced the scale function, which depends on the stability and represents a scale of eddy, and they obtained an improved profile formula. Assumption of the quadratic combination of the exchange coefficient, due to forced and free convections, leads to KEYPS equation (Sellers, 1962). In contrast with this assumption, assuming a linear combination of the exchange coefficients Yamamoto and Shimanuki (1966) give a modified profile equation. Other trials of improving the profile formula and comparison of profiles between the theory and the observation have been made by many workers, Taylor (1960), McVehil (1964), Pandolfo (1966), Chanock (1967), Dyer (1967), Rijkoort (1966), Swinbank (1964, 1968), Arya and Plate (1969), Ito (1969), etc. Yet, at the present time, any conclusive profile formulae have not been obtained. In the study of very unstable lapse case, a significant advance was made by Priestley (1955). He found that the vertical gradient of potential temperature comes to take a form of **/*z*z-4/3,

2 76 Journal of the Meteorological Society of Japan Vol. 49, No. 2 where z is the height from the ground. The humidity. Simultaneously, the expected profile same form can be derived also from KEYPS or for the nonsteady state in radiation plus turbulent modified KEYPS equations in the limited case of transfers, which is experienced in the field, will very unstable condition. Nevertheless, the discrepancy be compared with that of the theory hitherto between the theory and observation also made for the steady state in turbulent transfer. occurs in very unstable air. Webb (1958) found an unexpected feature of temperature profile. 2. Preliminary estimation of radiation effect Above the strong lapse near the ground, there is a transition to zero potential temperature gradient. Its height decreases with increasing negative Richardson We consider a case of steady state, that is, number, falling to below 8 m in lightest where R is the net radiative flux (see the next winds. This feature can not be expected from paragraph), and Q the sensible heat due to turbulent the theory hitherto made. On the other hand, transfer, however, Elliott (1966) analysed the daytime temperature profiles and obtained an equation. The lapse rate of temperature varies as z-3/2 in very where cp and * represent the specific heat and light winds. Qualitatively, this relation exceeds density of air, respectively, and K is the eddy 4/3-power law. diffusivity. If R increases with height, as in the It was also found by Kondo (1962) that in case of stable air where the temperature gradient very stable air, not only the theoretical temperature /*z has positive value, then negative value ** of profile deviating from the observed one, Q should increase with height. In other words, but also the discrepancy between the theoretical the temperature gradient takes larger values than and observed temperature profiles was larger than that in case of no effect of radiation. that between the theoretical and observed wind In order to estimate this factor, a simple model profiles. Several workers have pointed out that of K is considered. this kind of discrepancy between wind velocity and temperature profiles can be attributed to the different mechanisms between momentum and heat Eq. (1) is solved by the trial and error method, transfers. Another cause of this discrepancy was and the results are shown in Fig. 1. The cases suggested by Kondo. One of the causes may be of a=0.87cm/s and a= 2.6cm/s are shown by attributed to neglecting of the radiative heat the solid and the dot-and-dash lines, respectively, transfer in the theory. As is well-known, in the layers near the ground at night the radiative flux increases with height, which implies the radiational cooling there. Therefore, if we assume the steady state and take into account both the turbulent and the radiative effects in the stable condition, then the resulting turbulent heat flux should decrease with increasing height. There is one more point worthy of consideration in comparing the theoretical and the observational. That is, so far as the thermal condition in the lower atmosphere is concerned, the steady state as postulated by the theory is hardly attained. In order to understand better the influences of radiative transfer and nonsteady condition upon Fig. 1. Temperature profile in steady state the wind, temperature and humidity profiles in including the turbulent and radiative the atmospheric boundary layer, taking the radiative heat transfer into consideration, the author heat transfers. Thermal diffusivity K=az is assumed. Solid line shows of the present paper will investigate, by means a=0.8cm/s and dot-and-dash line of numerical experiments, the diurnal changes of a=2.6cm/s, and dotted line expresses wind velocity, air and soil temperatures, and the case of no radiation.

3 April 1971 Junsei Kondo 77 and the dotted line represents the case of R= 0. These orders of magnitude of K are often experienced in stable air (see Fig. 13). Considerable effects of radiation on temperature profile can be seen from the figure. Next, we consider the case of radiational cooling of air, without turbulent transfer, The initial temperature profile, and the boundary condition of the earth surface temperature, radiation on transfer problems in the surface layer must be taken into consideration, especially, in the case of light wind condition. 3. The basic equations We consider the atmosphere as a horizontally stratified medium where all the physical variables are functions of the height only. If we are concerned with the lowest shallow layers near the surface, which are of depths not exceeding 50m, it should be usually possible to neglect the Coriolis force and pressure gradient force, and then the local rates of changes of the wind velocity u, potential temperature *, and specific humidity q respectively obey the following equations: are taken. By integration of Eq. (4) with time, the temperature profiles of t=0.083, 0.5, 1, 2, 3, and 4h are obtained, as shown in Fig. 2. The arrows pointing upwards and downwards represent the earth surface temperature and the temperature of air sticked to earth surface, respectively. The difference between these two temperatures are seen in the figure. This gap is attributed to the radiative heat transfer (see the radiation equilibrium profile of temperature, for example, by Emden, 1913). The air temperature profiles of Fig. 2 are very similar to those expected in field observations. It is concluded from Figs. 1 and 2 that the effect of where RL is the net flux of long wave radiation, and The change of the ground temperature TG at the depth of y is expressed by In this paper, we assume Fig. 2. Radiational cooling without turbulence. The profiles at time=0, 0.083, 0.5, 1, 2, 3 and 4h are shown. The arrows pointing upwards and downwards represent, respectively, the earth surface temperature and the air temperature at the surface. where K is the eddy diffusivity, L the local value of Monin-Obukhov length or the stability length, which is usually taken for constant with height but, in this paper, we take L for a function of height as shown by Eq. (15). The other symbols have their usual meaning, and * is the nondimensional shear function, and in this paper * is assumed as (see Appendix 1)

4 78 Journal of the Meteorological Society of Japan Vol. 49, No. 2 Fig. 3. Comparison of nondimensional shears as a function of nondimensional height. The upper half and the lower half of this figure are stable and unstable cases, respectively. Comparisons between Eq. (16) are *-function, obtained by other several workers, are shown in Fig. 3. As stated in introductory remarks, scattered *-functions may be seen from Fig. 3 and we have not any recommended function. Calculation of the radiation flux was made with the aid of the values of *(w), the mean transmission function for slab of water vapor of thickness w, which were obtained by Yamamoto (1952). In the present paper the correction of the flux divergence due to carbon dioxide was neglected. According to Yamamoto (1952) the upward and downward fluxes at the height zw are given by Then, the difference between the net flux radiation at zw and that at the surface, where *B =*T4 is the black-body radiation. In actual numerical calculations the Radiation Table was used (see Appendix 2). Eqs. (7), (8), (9), and (13) are solved numerically with the initial and boundary conditions. At the ground, the equation of heat budget is used

5 April 1971 Junsei Kondo 79 where Is and IL are the incoming short wave and long wave radiations at the ground surface, TS the surface temperature, le the latent heat due to evaporation, Q the sensible heat, G the downward soil heat transfer, and other quantities represent usual meanings. The empirical formulae by Kondo (1967) are used to estimate IS and IL, that is, where J0 is the solar constant, * the latitude, the solar declination, h the hour angle (h=0 at noon), d and d are the mean and the instantaneous distances between the sun and the earth, respectively,* the solar zenith angle, e the daily mean value of vapor pressure near the ground (in mb), T the daily mean value of absolute air temperature near the ground. In this paper, the values at the height of 300m are used for e and T. CS and CL are the correction factors due to cloud for the short wave and the long wave radiations, respectively (see Kondo, 1967). *IL is the correction of the downward long wave radiation due to inversion (see Appendix 3). If we take z1 to be a very small height from the ground, z0<zl* L, then Q and E may be expressed as follows (see Appendix 1): in the layers considered. Vapor pressure at the ground surface is assumed to be the saturation vapor pressure at the ground temperature *s. It is expressed by empirical equation, In the present study, the author has made the above formula from the Physical Tables of saturation vapor pressure with respect to water. Error arised from Eq. (32) is only 0.02 to 0.05 mmhg in the temperature range of -12 to 35*. The soil layer is divided into 50 equal increments of 1 cm each, and soil temperature below the depth of 50 cm is assumed to be constant. Since the present study we are concerned with the effect of radiative heat transfer on vertical profiles of wind, air temperature and humidity at the lowest layer of the atmosphere, which are of depths not exceeding several tens of meters, no change of air temperature and humidity above the height of 300 m is assumed for simplification of the calculation. At 300-m height, the potential temperature *H= 16*, wind velocity uh=6 m/s, water vapor pressure eh=10mmhg. Conditions of the upper atmosphere, which are necessary for calculation of radiation flux, are shown in Table 1. Table 1. Atmospheric initial condition. 4. Model Air heating due to solar radiative flux divergence has been omitted from Eq. (8), since the magnitude of such heating during the whole day is of the order of less than 1 deg and is negligible in comparison with heating or cooling by the long wave radiation and by the turbulent heat fluxes

6 80 Journal of the Meteorological Society of Japan Vol. 49, No. 2 The atmospheric boundary layer is divided into equal logarithmic height increments of 1/8. Those are the heights of 7.5, 10, 13.3, 17.8, 23.7, 31.6, 42.2, 56.2, 75cm, 1m,..., 300m. Initial soil (y=0 to 50cm) and air (z= 0 to 300m) potential temperatures are uniform values of 16*. And initial wind and water vapor profiles at boundary layer were assumed to be logarithmic. Time intervals of and 18sec are respectively adopted for integration of Eqs. 7 to 9 and Eq. 13, respectively. And the following numerical values are used; z1=10cm, z0=0.1cm,*=20, cp*= 0.3*10-3 cal deg-1cm-3, -1=10-3 cal cm-1 sec-1 deg-1, C1*1=1 cal deg-1 cm-3, *=35*, =0 (equinoxes), CS=CL=1 *(fine weather), J0(d/d)2=2,740 ly day-1, and rs=rl=0. 5. Daily changes of wind and temperature The basic equations with the numerical model described above were integrated for 42 hours, starting from the sunset. After the 18 hours the wind, temperature and humidity attain a uniform diurnal pattern, with little or no difference from 18 hours (at noon) to 42 hours (at noon). It was therefore decided to adopt the results of the 18 to 42 hours as representing the equilibrium state of diurnal variation. The diurnal variations of the profiles of wind and air temperature are shown in Fig. 4A. And the diurnal variations of wind velocity at 31.6m, 10m, 3.16m, 1m and 0.1m, and those of air temperature at the heights of 10m, 1m and 0.1 m, and those of soil temperature at the depths of 5cm and 10cm are presented in Fig. 5. The normal features of the temperature wave and wind velocity wave propagating by conduction and by radiative transfer from the ground surface to the upper layer are clearly shown. These features are the same as the observed results hitherto obtained usually at the field. The ground surface temperature, TS, gradually drops down after noon, and the minimum value occurs at the time of sunrise (6 A. M.). But, at the 10-m height, the minimum air temperature takes place one hour after sunrise. Nearly uniform air temperature occurs at the time of one hour before sunset. An interesting feature of wind velocity can be seen one or two hours after sunrise. The reason of this speed-down in wind velocity at 7 or 8 A. M. can be explained as follows. The sun has risen and it heats the ground surface, and after the ground temperature has gradually risen up, the air temperature is also heated by the convective heat and by the long wave radiative heat transfers from the heated surface, and a strong unstable air is formed at the lowest layer. An intense convection, therefore, occurs in this layer. This means that a large quantity of momentum aloft is carried away downward by the convective Fig. 4A. Diurnal change of wind (the left) and air temperature (the right) profiles.

7 April 1971 Junsei Kondo 81 Fig, 4B. Observational evidences of the speed-down in wind velocity occurred at few hours after sunrise. This is obtained from the Project Prairie Grass, Field Program in Diffusion (Barad, 1958). The top and central figures show, respectively, time variations of wind speed at several heights from 0.25m to 16m and insolation, (24 and 25, July 1956, O'Neill, Nebraska). The bottom figure represents vertical profiles of wind velocity (the left) and air temperature 0605, 0705 and 0805 CST, 25 July (the right) at

8 82 Journal of the Meteorological Society of Japan Vol. 49, No. 2 Fig. 5. Diurnal variation of wind, air temperature and soil temperature (the dot-and-dash line). condition of the upper layers above the 300-m height, several times higher than the heights of the lowest layers, is assumed to be constant with time. The magnitude of amplitude of the second depression in the wind velocity curve at 9 or 10 A. M. is not sufficiently correct, at least quantitatively, because of the too small height of the upper boundary in consideration. On the other hand, however, as the first speeddown in wind velocity at 7 or 8 A. M. is not affected by the upper boundary condition, the occurrence of this phenomenon can be explained from the present calculation. Some preliminary calculations show that in the layers below several tens of meters the difference between the divergences of radiative heat and sensible heat under the unchanged condition at air parcels, and then the wind velocity lowers the 300-m height and those under somewhat different there. On the other hand, however, the air below condition is very small. Therefore, so far gains momentum and the wind becomes fast. By as the shape of the vertical profiles of wind, air such mechanism the depressed shape of wind temperature and humidity are concerned, the results velocity propagates upward. of the present calculation for the layers of Another depressed shape of wind can be seen the heights not exceeding several tens of meters at 9 or 10 A. M. Several hours after sunrise, are expected to be reliable. the gradual increase in wind velocity rises near the ground surface and this trend propagates upward. For reference, some observational evidences of the speed-down in wind velocity which occurred Now let us consider when this wave reaches few hours after sunrise are shown in Fig. 4B. to some level near the boundary. There is a very This is obtained from the Great Plains Turbulence stable air above this level and a convective air Program, (Barad ed., 1958: Geophysical Research below. Since less amount of momentum is carried Papers, No. 59). From the bottom figure on Fig. from the upper layer to this layer and large quantity 4B we can easily understand the phenomenon of of momentum is carried away downward, the wind velocity lowers there till the stable air is destroyed. speed-down in wind velocity at few hours sunrise. after Diurnal variation of various meteorological parameters in the planetary boundary layer has been 6. Heat budget Energy balance at the ground surface is given studied by Krishna (1968), Sasamori (1969) and in the lower part of Fig. 6. At night, the net Yasuda (1969). From the Sasamori's paper, the flux radiation RN is almost balanced with the soil same depressed shape of wind velocity can be heat transfer G. The absolute values of sensible seen at the time of 8 to 10A. M. in the analysed heat Q and the latent heat le are very small in results of the observation of the Great Plains comparison with those of G and RN. This means Turbulence Field Program. Yasuda's (1969) analysis also that the ground surface temperature at night of wind data at WKY-TV Tower, Oklahoma, scarcely depends on wind velocity under the con- shows also the same depressed shape of wind dition of a light wind (see Kondo and Naito, velocity which occurred about at the time of 1969). sunrise. In the daytime, however, the values of RN and Since the main purpose of the present paper is E become to one and the same order of magnitude. l connected with the vertical profiles of wind, air The difference between RN and le is com- temperature and humidity in the lowest layers pensated by both Q and G. In the present paper near the earth surface, which are of depths not the wet surface is assumed, and so le becomes exceeding several tens of meters, the atmospheric larger than the value of Q. As was presented

9 April 1971 Junsei Kondo 83 Fig. 6. Diurnal variation of heat budget at the ground surface, and the friction velocity u0*, the scaling temperature T0* and the inverse of stability length L0. by Kondo and Naito (1969), if we adopt the dry surface with less quantity of evaporation, then Q must take larger value than that for the wet surface. Fig. 7. An example of nondimensional wind For reference, the inverse of stability length L0, (dot-and-dash line), air temperature the friction velocity u0*, and the scaling temperature, T0* =-Q0/(c*pku0*), at the ground surface line) profiles at 9 P.M. The solid (dashed line) and water vapor (dotted are shown in the upper portion of Fig. 6. The line represents the case of steady state most stable air can be seen at sunrise, the instability being intense one hour after sunrise. without radiative transfer. without the radiative heat transfer, then a profile 7. Estimated shear functions of wind, temperature, in such a case could be obtained (see Appendix and vapor pressure 1). The solid line represents the profile of steady From the observational point of view, several state. Deviation of the calculated wind profile of workers have pointed out that an inequality of nonsteady state from the profile of steady state the profiles of wind velocity and temperature or can be seen at the layer several meters above. vapor pressure is caused by the different mechanisms between the momentum transfer and the and vapor profiles can be seen also in this figure. Further, large discrepancies of the temperature heat or vapor transfers. The present author considers that such inequality is attributed not only observation by Kondo (1962). Same trends of discrepancy were shown in the to the different transfer mechanism but also to Nondmmensional shear functions at every 30 the radiative heat transfer and nonsteadiness. minutes,*,in stable case are shown in Fig. 8. An example of the present calculation, in which The abscissa is the nondimensional height, z/l0, the radiative heat transfer and the nonsteadiness L0 being the stability length at the surface and are included, is shown in Fig.7. A quantity with being defined by Q0 and *0. The dashed line suffix "O" represents the value at the ground represents the steady state without radiative heat surface or at the aerodynamic roughness height transfer. (A), (B), and (C) of Fig. 8 are the shear z0. The dot-and-dash line, the dashed line, and functions of wind velocity, air temperature, and the dotted line express, respectively, the nondimensional profiles of wind ku/u0*, air temperature height range from the ground surface to the vapor pressure, respectively. The line within a c*pku0*(*0-*)/q0, and specific humidity *ku0* height of 17.8m is drawn in this graph, that is, (q0-q)/e0. Since KM=KH=KW was assumed the end of right-hand side of each line expresses (see Eq. 14), if we consider the steady state the value at 17.8-m height. Scattered relations

10 84 Journal of the Meteorological Society of Japan Vol. 49, No. 2 Fig. 8(A) Wind velocity Fig. 9(A) Wind velocity Fig. 8 (B) Air temperature Fig. 9(B) Air temperature Fig. 8(C) Water vapor Fig. 8. Nondimensional shear function against z/l0, L0 being the stability length at the ground surface. The dashed line represents the case of steady state without radiative transfer, (stable case). can be seen from the figure. The estimated lines, excepting a few lines (e. g. a line for 6.30A. M.), are shown below the theoretical line for steady state. The deviation of wind velocity from the dashed line is relatively small, but that of temperature is considerably large. The largest discrepancy can be seen in the curve of vapor pressure. Fig. 9(C) Water vapor Fig. 9. Same as Fig. 8, except in unstable case. From these figures we can say that only one line of universal *-function, without scattering, would not be obtained from observational data by any method of analysis which has hitherto been applied by the researchers of this scientific field. Fig. 9 shows estimated nondimensional shear functions of wind (A), temperature (B), and vapor pressure (C), same as Fig. 8, but in unstable conditions. In this case the curves are shown for

11 April 1971 Junsei Kondo 85 the height range of z=0 to 31.6m. Discrepancy between the line of steady state and the estimated curve, excepting the curves of 7A. M. and 7.30 A. M., is relatively smaller in comparison with the stable condition of Fig. 8. With careful inspection of Fig. 9 it can be seen that the estimated shear function of wind for a range of about z/l0=-0.1 to -2 is generally smaller than the value of steady state (the dashed line), but for a range of about z/l0=-2 to -10, the estimated one becomes larger than the value of steady state. On the other hand, however, it can be also seen that the estimated one of temperature for the range of about z/l0=-0.1 to -2 is generally larger than that of steady case, and with increasing absolute value of z/l0 the estimated curve becomes smaller than the value of the dashed line. In general, the estimated shear function of humidity, shown in C of Fig. 9, is slightly smaller than the value of steady state. These trends of discrepancy between the estimated *-functions and those in steady state are attributed to the flux divergence of heat in nonsteady state with the radiative heat transfer. Fig. 10. Height variation of the net flux radiation. The difference, RL(z)-RL(0), is shown. 8. Height variation of radiative heat flux In the present paper, the difference between -the net flux of long-wave radiative heat at the height of z, RL(z), and that at the ground surface, RL (0), was directly calculated. Fig. 10 shows some examples of the height variations of RL (z)-rl (0). The net flux radiation at the ground surface is shown at the lower part of the right-hand side in this figure. In the boundary layer of the atmosphere, the magnitude of the height variation of RL(z)-RL(0) is of the order of several percent of RL(0), It may be seen from the figure that at night the net flux decreases with height up to the height of about one or two centimeters, and thereafter it increases with height, however, in the daytime (9A. M., 0P. M. and 3P. M.) vice versa below the height of about 100m, and therefrom it turns to increase with height. Cooling or heating rate due to the radiative Fig. 11. Height-time variation of radiational flux divergence can be calculated. The heighttime variation of these rate is shown in Fig. 11. temperature change. Funk (1960) measured directly the nocturnal profiles in the daytime and nighttime, and estimated the temperature changes due to radiative radiative flux divergence profiles in the lowest few meters and found a rate of the temperature flux divergence. Their results are qualitatively in cooling there. And also, Yamamoto and Kondo accord with the present results. But some differences between the results by Yamamoto (1959) observed the temperature and humidity and

12 86 Journal of the Meteorological Society of Japan Vol. 49, No. 2 Kondo and the present one can be seen, because the former has estimated the flux divergence for the reflective surface but the latter has assumed the surface of black body. Just above the ground surface, heating rate during nighttime and cooling rate during daytime are seen from the figure. As stated in paragraph 2, these features are attributed to the nature of the radiative transfer. We should like to emphasise an important interesting factor in the atmospheric layers near the ground. A very large value of heating rate, beyond 5 deg/h, can be seen during the daytime in the low layer of about one-meter height. This heating rate exceeds the values ordinarily observed in the field. In compensation for this excess heating rate, therefore, a convective heat by the turbulent transfer has arisen. 9. Height variations of momentum, heat and water vapor fluxes, and eddy diffusivity Fig. 12 shows the height variation of the momentum (dot-and-dash line), the sensible heat (dashed line), and the water vapor (dotted line) fluxes. The ratio of each value at the height z and that at the ground surface are shown. The profiles at 9A. M., 3P. M., 9P. M., and 6A. M. are expressed in the figure, respectively, from the left to the right. Dividing the value of *RL=RL (z)-rl (0) with the absolute sensible heat at the Fig. 12. Height variation of momentum (dotand-dash line), sensible heat (dashed line) and water vapor (dotted line) fluxes. The solid line shows *RL/ Q, where *RL-RL(z)-RL(0) and Q0 is the sensible heat at the ground surface. Fig. 13. Height variation of eddy diffusivity. ground surface Q0, and *RL/ Q0 is shown by the solid line. It is concluded from the figure that in unstable case the fluxes of entities are almost constant at the heights less than several tens of meters, but in stable case, as the so-called constant-flux layer is very small, at the heights of about several meters from the ground surface. Fig. 13 expresses the height variation of the eddy diffusivities every three hours. In the daytime the diffusivity increases with height more rapidly than in the case of the so-called linear variation. On the other hand, however, the interesting feature at night is the maximum diffusivity at the heights of about several tens of meters. Elliott (1964) obtained the profiles of vertical heat flux by subtracting the component of temperature change due to radiative flux divergence from some observed temperature changes and by integrating the resultant temperature change. And he obtained the results that the resultant fluxes of sensible heat showed significant height variations near the ground by night but almost no variation by day, and that by night the coefficient of eddy conductivity increased with height up to the heights of about 50 meters and subsequently decreased. These observed results by Elliott are consistent with the present calculated results. 10. Formation of radiation fog Fleagle (1953) made a qualitative study on the occurrence of radiation fog from theoretical radiation consideration, and Fisher and Caplan (1963) studied a dynamic model for forecasting the development and dissipation of fog and stratus, based on numerical solutions of the diffusion equation. The present paper contains the equations of

13 April 1971 Junsei Kondo 87 Fig. 14. Height-time variation of relative humidity in percent. The dashed line expresses the height of maximum relative humidity. Fig. 15. Air (solid) and dew-point (dotted) temperature profiles at 0 and 3 radiation and diffusion. Both of these equations A.M. The arrow shows cooling rate act on temperature change, and the diffusion due to radiation. equation regulates the water vapor profile. Therefore, the mechanism of formation of radiation difference between the dot-and-dash line and fog will be examined actually from the present numerical calculations. the solid line of 3A. M.), and the air temperature becomes cooler than the dew-point temperature Time-height variation of relative humidity is (the dotted line), that is, the supersaturation shown in Fig. 14. The dashed line expresses the layer is formed. height of maximum relative humidity. Because the present paper has assumed a supersaturation 11. Conclusions humidity, no droplets of water have fallen off by As was stated earlier, this research had a major gravitation. objective, i, e., an examination of the effect of From the figure we can find in this model that radiative heat transfer and nonsteadiness on the supersaturation humidity does not occur in lower profiles of wind, air temperature and water vapor atmospheric layers which are directly in contact in the atmospheric boundary layer. It can be with a cold ground surface, but at heights of said from the calculated results that even if the several tens of centimeters the supersaturation eddy diffusivities for momentum, heat, and water layer spreads slowly upwards, and finally it disappears vapor transfers had the same order of magnitude in the late morning. These features are with each other, the resulting profiles of the wind, very consistent with the observational results of air temperature and water vapor come to be of Davis (1957) and Funk (1962). If the occurrence different shape from each other. Especially in of water droplets and their falling velocity due to calm nights, the radiation has yielded a very gravity were taken into this calculation, the layers important factor in lower atmospheric layers near of supersaturation would be formed at the heights the ground, and it changes the shape of temperature somewhat lower than that shown in Fig. 14. profile, simultaneously changing also the Fig. 15 shows an examples of the air temperature and the dew-point temperature changes. The solid lines on the right- and left-hand sides are eddy diffusivity. Accordingly, the transfer of each entities becomes to be not constant with height. In other words, expected profiles of wind, air the profiles of air temperature at 0A. M. and at temperature, and water vapor are deviated from 3A. M., respectively. During three hours from 0 those in steady state of turbulent transfer. to 3A. M., the air is cooled by radiation (the Fig. 16 shows qualitatively the expected profiles order of magnitude of this effect is shown by the of nondimensional shear as a function of nondimensional arrows), and by turbulent mixing (shown by the height, which is obtained from the

14 88 Journal of the Meteorological Society of Japan Vol. 49, No. 2 wishes also to thanks Dr. M. Sugawara for his very generous providing of the computer time, and Mr. K. Ohmura for his help in operating the computer of the National Research Center for Disaster Prevention, Tokyo, and also to express his thanks to Mrs. M. Koizumi for typing the manuscript. Fig. 16. Representative shear function against z/ L0, for wind velocity (dot-anddash line), air temperature (dashed line) and water vapor (dotted line). The solid line expresses the steady state without radiation. present calculation. Some discrepancies between the observed profile of wind velocity and that of air temperature, which are found from the observations (for example, Kondo, 1962; Webb, 1958, etc.), may be partly, though not perfectly, explained from the present research. It is concluded that in unstable case the turbulent fluxes of entities are almost constant at the heights less than several tens of meters, and in stable case at the heights of about several meters. In the daytime, we can find a layer of intense heating, due to radiative flux divergence, at the heights from several centimeters to one meter above the ground. It appears that such excess heating rate might yield the heat energy of convective mixing. Thereafter, a parcel of hot air is set up in these layers and penetrates into upper layer, and simultaneously in compensation for such hot air element, a cold air parcel is carried from the upper layer to the lower layer of intense heating. In such a way an equilibrium state seems to be formed by radiation and convective mixing in the boundary layer. Acknowledgments The author is paticularly indebted to Prof. G. Yamamoto of Tohoku University and Dr. K, Terada, Director of National Research Center for Disaster Prevention, for encouragement during the course of this study. Thanks are also extended to Mr. N. Yasuda of Tohoku University and Dr. T. Sasamori of National Center for Atmospheric Research for valuable comments. The author References Arya, S. P. S. and E. J. Plate, 1969: Modeling of the stably stratified atmospheric boundary layer. J. Atmos. Sci., 26, Charnock, H., 1967: Flux-gradient relations near the ground in unstable conditions. Quart. J. Roy. Meteor. Soc., 93, Davis, F. K., Jr., 1957: Study of time-height variations of micrometeorological factors during radiation fog. Publications in climatology. 10, No. 1, 1-37, Drexel Inst. Techn. Dyer, A. J., 1967: The turbulent transport of heat and water vapor in an unstable atmosphere. Quart. J. Roy. Meteor. Soc., 93, Elliott, W. P., 1964: The height variation of vertical heat flux near the ground. Quart. J. Roy. Meteor. Soc., 90, : Daytime temperature profiles. J. Atmos. Sci., 23, Ellison, T. H., 1957: Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech., 2, Emden, R., 1913: Uber Strahlungsgleichgewicht und atmospharische Strahlung: ein Beitrag zur Theorie der oberen Inversion. Sitz. Ber. Mi nchen. 55. Fisher, E. L, and P. Caplan, 1963: An experiment in numerical prediction of fog and stratus. J. Atmos. Sci., 20, Fleagle, R. G., 1953: A theory of fog formation. J. Mar. Res., 12, 43. Funk, J. P., 1960: Measured radiative flux divergence near the ground. Quart, J. Roy. Meteor. Soc., 86, : Radiative flux divergence in radiation fog. Quart. J. Roy. Meteor. Soc., 88, Ito, S., 1969: A mechanism of turbulent transfer in the atmospheric surface layer. J. Meteor. Soc. Japan, 47, Kazanski, A. B. and A. S. Monin, 1956: The turbulence in surface inversion. Izv. AN SSSR, Ser. Geofiz., 1, Kondo, J., 1962: Observations on wind and temperature profiles near the ground. Sci. Rep. Tohoku Univ., Ser. 5, Geophys., 14, : Analysis of,-solar radiation and downward long-wave radiation data in Japan. Sci. Rep. Tohoku Univ., Ser. 5, Geophys., 18,

15 April 1971 Junsei Kondo 89,,,- and G. Naito,1969: Numerical experiment on diurnal changes of the soil and air temperatures near the earth's surface. Rep. Nat. Res. Ctr. Disas. Prey., Tokyo, No. 2, (with English abstract). Krishna, K., 1968: A numerical study of the diurnal variation of meteorological parameters in the planetary boundary layer. Mon. Wea. Rev., 96, McVehil, G. E., 1964: Wind and temperature profiles near the ground in stable stratification. Quart. J. Roy. Meteor. Soc., 90, Monin, A. S. and A. M. Obukhov, 1954: Basic regularity in turbulent mixing in the surface layer of the atmosphere. Tr. Geofiz. In-ta AN SSSR, No. 24, 163. Pandolfo, J. P., 1966: Wind and temperature profiles for constant-flux boundary layers in lapse conditions with a variable eddy conductivity to eddy viscosity ratio. J. Atmos. Sci., 23, Panofsky, H. A., A. K. Blackadar and G. E. McVehil, 1960: The diabatic wind profile. Quart. J. Roy. Meteor. Soc., 86, :,- An alternative derivation of the diabatic wind profile. Quart. J. Roy. Meteor. Soc., 87, Priestley, C. H. B., 1955: Free and forced convection in the atmosphere near the ground. Quart. J. Roy. Meteor. Soc., 81, Rijkoort, P. J., 1968: The increase of mean wind speed with height in the surface friction layer. Med. en Verh. KNMI, No. 91. Sasamori, T., 1969: A numerical study of atmospheric and soil boundary layers. NCAR Manuscript No Sellers, W. D., 1962: A simplified derivation of the diabatic wind profile. J. Atmos. Sc., 19, Swinbank, W. C., 1964: The exponential wind profile. Quart. J. Roy. Meteor. Soc., 90, : A comparison,- between predictions of dimensional analysis for the constantflux layer and observations in unstable conditions. Quart. J. Roy. Meteor. Soc., 94, Takeuchi, K., 1961: On the structure of the turbulent field in the surface boundary layer. J. Meteor. Soc. Japan, 39, and 0.,- Yokoyama, 1963: The scale of turbulence and the wind profile in the surface boundary layer. J. Meteor. Soc. Japan, 41, Taylor, R. J., 1960: Similarity theory in the relation between fluxes and gradients in the lower atmosphere. Quart. J. Roy. Meteor. Soc., 86, Yasuda, N., 1969: Shearing stress and eddy diffusivity in the Ekman layer. Read at the National Conference of the Meteor. Soc. Japan, 1969, p Yamamoto, G., 1952: On a radiation chart. Sci. Rep. Tohoku Univ., Ser. 5, Geophys., 4, : Theory of turbulent,- transfer in non-neutral conditions. J. Meteor. Soc. Japan, 37, ,- and J. Kondo, 1959: Effect of surface reflectivity for long wave radiation on temperature profiles near the bare soil surface. Sci. Rep. Tohoku Univ., Ser. 5, Geophys., 11,1-9. and A. Shimanuki,,- 1966: Turbulent transfer in diabatic conditions. J. Meteor. Soc. Japan, 44, Yokoyama, O., 1962: On the contradiction and modification of the equation of diabatic wind profile. J. Meteor. Soc. Japan, 41, Appendix 1. Nondimensional shear function, *, is usually defined by Putting Eq. (16) into Eq. (A-1) and assuming the steady state with turbulent transfer, without radiative heat transfer, and integrating the equation with a boundary condition of u=0 at z=z0, we have the wind profile formula. By the same method we have also the air temperature specific humidity profiles. For unstable case, L<0, where A=**z+1, A0=**0+1 and *=*/ L. If z* L, Eqs. (A-4) and (A-5) yield that is, the so-called (log + linear) wind profile. Appendix 2. and A method of numerical calculation of net flux

16 90 Journal of the Meteorological Society of Japan Vol. 49, No. 2 radiation is shown below. We take 62 height levels according to the optical thickness of water vapor w, (zw1=0; zw2, zw3,*, zw62=zw*), from the ground to the top of the atmosphere. In a unit of ly/day (net flux radiation RL and the black-body radiation *T4) and * (temperature T), it is expressed that where T1 is the earth surface temperature and T2 the air temperature at w= g, T3 that at w= g,*, etc. (see Table A-1). And putting we have the difference of net flux radiation between the values at wi and at the surface, where {**(wi)}n is the difference between the mean transmission function for slab of water vapor of wi-wn and that of wn, and {** (wi)}n, n= 1 to 62, are shown in Table A-2. The value of *R(w), wi*w*wi+1, can be interpolated from *R(wi) and *R(wi±1). Appendix 3. The correction of long wave radiation due to inversion (last term of Eq. (27)) can be expressed Table A-1. Optical thickness w at n-th level. * If total optical thickness w*<2 g, w*= *p00(qp/ps)dp/g, it should be put T62=-273.2*.

17 April 1971 Junsei Kondo 91 Table A-2 Radiation table for lower atmosphere.

18 92 Journal of the Meteorological Society of Japan Vol. 49, No. 2 Table A-2. (Continue)

19 April 1971 Junsei Kondo 93 Table A-2. (continue)

20 94 Journal of the Meteorological Society of Japan Vol. 49, No. 2 by the approximate equation, where and *m is the instantaneous mean temperature at the boundary layer (in *), and w is the optical path of water vapor, g being the acceleration due to gravity, q the specific humidity, p the atmospheric pressure, ps the standard pressure, p0 the pressure at the ground surface.