Sensible Heat Flux from the Earth’s Surface under Natural Convective Conditions

Junsei Kondo Geophysical Institute, Tohoku University, Sendai, Japan

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Sachinobu Ishida Geophysical Institute, Tohoku University, Sendai, Japan

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Abstract

A value for the exchange speed of sensible heat CHU under natural convective conditions was determined by both indoor and field experiments. Regardless of the type of experiment, the relationships for the CHU were obtained as CHU = b(TST)1/3. For a wet surface, ΔTv should be substituted for (TST). Here, TS is the ground surface temperature, T the air temperature, and ΔTv the virtual temperature difference. In addition, b is a coefficient having a value of 0.0011 m s−1 K−1/3 for a smooth surface and 0.0038 m s−1 K−1/3 over a rough surface. From the field observation data, it was concluded that under strongly unstable conditions (−1 > ζ > −477) the best pair of stability profile functions was proposed.

Corresponding author address: Mr. Sachinobu Ishida, Geophysical Institute, Tohoku University, Sendai, 980-77, Japan.

Email: ishida@wind.geophys.tohoku.ac.jp

Abstract

A value for the exchange speed of sensible heat CHU under natural convective conditions was determined by both indoor and field experiments. Regardless of the type of experiment, the relationships for the CHU were obtained as CHU = b(TST)1/3. For a wet surface, ΔTv should be substituted for (TST). Here, TS is the ground surface temperature, T the air temperature, and ΔTv the virtual temperature difference. In addition, b is a coefficient having a value of 0.0011 m s−1 K−1/3 for a smooth surface and 0.0038 m s−1 K−1/3 over a rough surface. From the field observation data, it was concluded that under strongly unstable conditions (−1 > ζ > −477) the best pair of stability profile functions was proposed.

Corresponding author address: Mr. Sachinobu Ishida, Geophysical Institute, Tohoku University, Sendai, 980-77, Japan.

Email: ishida@wind.geophys.tohoku.ac.jp

1. Introduction

There have only been a few studies concerning the sensible heat flux under natural convective conditions, due to the difficulty in observing the necessary meteorological elements. The sensible heat flux under these conditions is not a negligible quantity, and a method for evaluating the flux has not been properly established.

Tropical atmospheric convection is a critical element in determining the earth’s climate, as can be demonstrated by a well-known example, the El Niño–Southern Oscillation (ENSO) phenomenon. As part of the ENSO phenomenon, a so-called warm pool region exists in the western equatorial Pacific, characterized by high SSTs (>27°C) and low wind speeds over the entire year. Seager et al. (1988) pointed out that an SST change of 1°C can be accomplished by a perturbation of only 12 W m−2 in the sensible heat flux from the warm pool. Yet an uncertainty on the order of 80 W m−2 is apparent in climatological estimates of the heat budget of the surface mixed layer in this region (Godfrey and Lindstorm 1989; Weare 1989).

Over arid areas, Garratt (1992) suggested that extreme maximum land surface temperatures were likely to occur under natural convective conditions. With 1000 W m−2 noontime solar radiation, a dry surface must divide this between net thermal emission, heat flux into soil, and sensible heat flux, all depending on surface temperature. The numerical factors are such that as much or more has to go into sensible heat as the other two terms, implying the stated 300–500 W m−2. This is a very large quantity for the heat balance of the land surface. The value of the temperature difference between the land surface and the air for such a heat flux would reach 20°–30°C.

In this paper, an attempt is made to parameterize the sensible heat flux under natural convective conditions by analyzing indoor experiments and field observations.

2. Review of natural convective heat transport

a. Over the ocean

The turbulent transports in the surface boundary layer can be expressed as
HcPρw
and
lEρlwq,
where H is the sensible heat, lE the latent heat, ρ the air density, and cP the isobaric specific heat of the air. Furthermore w′, T ′, and q′ are the fluctuating components of the vertical wind speed, temperature, and specific humidity, respectively. Time-averaged components are expressed by an overbar. In the eddy correlation method, H and lE are given by direct observations of w, T ′, and q′.
In practical applications, H and lE can also be expressed by bulk formulas as
HcPρCHUTST
and
lElρCEUqsq
Here, T and q are the air temperature and specific humidity at the reference height, respectively, while TS and qs are those at the ground surface. From Eqs. (1)–(4), the bulk transfer coefficients CH and CE can then be obtained.
But the individual value of CH or CE is meaningless in the range of small wind velocity such as U < 1 m s−1. Under natural convective conditions, the exchange speed CHU or CEU is more useful. Figure 1 displays the relationship between CEU10 and the wind velocity U10 (at a height of 10 m), made of the data of Bradley et al. (1991). In the figure, it is assumed that the exchange speed of sensible heat CHU is equal to that of latent heat CEU. The dotted line and dot-chain line in this figure are taken from Kondo (1975). Here, the virtual temperature difference is defined as
i1520-0469-54-4-498-e5
where e(hPa) is the water vapor pressure at the reference height and es the saturation value for the sea surface temperature TS. The observations in Bradley et al. (1991) were conducted under the conditions that TST = 1–3°C and ese = 10–12 hPa. In Fig. 1, it can be seen that CEU10 converges to a positive constant value as U10 → 0. Even with no wind, the virtual temperature difference creates a natural convection heat transfer.

b. Similarity theory

On the basis of Monin–Obukhov similarity theory, wind speed and virtual temperature profiles can be expressed as
i1520-0469-54-4-498-e6
and
i1520-0469-54-4-498-e7
Here, ϕM and ϕH are the nondimensional shear functions for momentum and heat, respectively; k(= 0.4) is the von Kármán constant; ζ = z/L; uT∗ = −H/cPρ; u∗ the friction velocity; and T∗ the temperature scale. The Obukhov length L can be defined by
i1520-0469-54-4-498-e8
where
HvHTcPE.
Integration of Eqs. (6) and (7) results in
i1520-0469-54-4-498-e11
where
i1520-0469-54-4-498-e13
Therefore, the relation for the exchange speed of sensible heat is obtained as
i1520-0469-54-4-498-e14
Under calm conditions, the value of u∗ is physically meaningless. [But it is necessary to obtain the value of z0 or zT of smooth surface as Eqs. (32) and (33) under these conditions.] Therefore, assuming the natural convection velocity scale w∗ = [Hvhg/(cPρT)]1/3, along with the definition given by Eq. (8), results in
i1520-0469-54-4-498-e15
where h (m) is the convectional scale length. As will be discussed in section 3 (the indoor experiment), assigning a value of h = 1.5 m results in w∗ = α × Hv1/3 (α = 0.034 W−1/3 m1/3 s−1), and in section 4 (the field observation), if z = 10 m, then ζ = −100. Beljaars (1995) used w∗ for the wind velocity as |U|2 = U2l + (βw∗)2, where Ul is the large-scale velocity components and β is the constant coefficient. But in this paper, it is taken the following traditional way.
Dyer and Hicks (1970) proposed ϕ functions for unstable condition (0 ≥ ζ ≥ −1) as
i1520-0469-54-4-498-e16
More recently, Kader and Yaglom (1990) expressed the shear function for heat when −0.01 > ζ > −20 as
i1520-0469-54-4-498-e18
Integration of Eqs. (16)–(18) leads to
i1520-0469-54-4-498-e19
and
i1520-0469-54-4-498-e21
Here, z0 and zT are the roughness lengths for the wind and temperature profiles, respectively.

Figure 2a displays the shape of ΨM according to Eq. (19), and Fig. 2b ΨH from Eqs. (20) and (21) for values of −ζ0 = −ζT = 10−4. These figures reveal how the profiles of u(z) and T(z) for unstable conditions differ from the respective logarithmic profiles (dotted lines) under neutral conditions.

There have been few reports on these profiles under strongly unstable conditions such as ζ = −100. In the present paper, clarification of this point will be discussed.

c. Engineering of heat transfer

On the basis of the “engineering of heat transfer,” natural convection on an isothermal horizontal plate is treated by following nondimensional parameters:
i1520-0469-54-4-498-e22
and
i1520-0469-54-4-498-e25

The parameter of Nu denotes the nondimensional exchange speed, while Gr indicates the influence of convection due to buoyancy. Here, h (m) is the scale length of the natural convection (e.g., the length of the surface or the size of the experimental convective cell), κ (m2 s−1) the thermometric conductivity, ν (m2 s−1) the kinematic viscosity, g (m s−2) the acceleration of gravity, and β (K−1) the coefficient of thermal expansion.

From many experimental studies, it is well known that these parameters are related in natural convection as
Aa
Table 1 lists several numerical values of A and a (also see Fig. 7). Engineering studies have pointed out that an inclination of the test plate results in differences in the sensible heat flux. Mikheyev (1968) showed that the sensible heat flux on a horizontal plate is 1.3 times greater than that on a vertical plate. The results of Takeyama et al. (1983) shown in Table 1 have been corrected by this factor.
Since a = 1/3 for large values of Ra (i.e., 8 × 106 < Ra < 1013), CHU can be described with use of the proportional constant b as
i1520-0469-54-4-498-e27
It should be noted in Eq. (27) that CHU does not depend on h. The value of b has been estimated at 0.0013–0.0018 m s−1 K−1/3. For wet surface conditions, the buoyancy of the water vapor pressure difference should be taken into consideration, that is, ΔTv [Eq. (5)] should be substituted for (TST).

Since engineering studies rely on indoor experiments, determination of CHU for large values of Ra or over a rough surface has not been researched in detail. In the present paper, the values of sensible heat flux for large Ra (>1012) will be determined by field observations and those for a rough surface by indoor experiments.

3. Indoor experiment

a. Smooth surface

The indoor experiments were carried out with the use of readily available materials. Figure 3 is a conceptual illustration of the indoor experiment. The nearly constant temperature surface constructed of styrene foam is realized by exposition to solar radiation coming through the glass window. Since the glass window cuts off longwave radiation from the outside atmosphere, and the wall temperatures in the experimental room have almost the same value as TW, the longwave radiation to the test surface is expressed by σT4W. Here, σ (= 5.67 × 10−8 W m−2 K−4) is the Stefan–Boltzmann constant, and TW is the effective infrared temperature of the walls. The value of TW is obtained from the average temperature of the walls and ceiling as observed by an infrared thermometer. The latent heat flux lE at the test surface is always zero, since the surface is dry. The heat capacity of styrene foam is so small that the thermal conduction G into the surface is small. The test surface is painted black, having an albedo ref = 0.074. An experiment for a weak sensible heat flux was also conducted, which was realized by covering the window with a semitransparent film to reduce the shortwave flux at the horizontal test surface.

The heat balance on the test surface is described by
i1520-0469-54-4-498-e28
where Rn is the net radiaton, S the solar radiation incident on the horizontal test surface, csρs (= 32.2 × 103 J K−1 m−3) the heat capacity of the styrene foam, TS the test surface temperature, Tg(z) the styrene foam temperature at depth z (m), and z the height from the test surface. The height of the experimental room is 2.65 m, the width 4 m, and the length 6.55 m.

Values of Tg(z) were measured at heights of z = −0.01, −0.02, −0.07, and −0.12 m by thermocouples; the representative air temperature in the room T is measured at z = 1.5 m by a ventilated thermometer; and S determined at a location next to the test surface by a pyranometer. The values of Rn, G, and H are averaged over every 30 min; observation with the value of CHU (U → 0) is then obtained through the bulk formula given by Eq. (3). The experiments were conducted on 29, 30, 31 January; 1 February; and 4, 9, 13, 15, 16, 22 March 1993. Table 2 lists the representative conditions of the indoor experiment.

Figure 4 shows an example of the horizontal distribution of TS during the experiment. When the test surface was heated on a clear day, the temperatures were measured by an infrared thermometer at z = 1.5 m, which transversed the surface 10 times in the horizontal direction, and were then averaged. It appears that the low temperatures at both ends are the result of the stagnation of convection. In the horizontal, opposite the window, the surface temperature was lower, but the temperature anomaly remained small near the center of the test surface. And horizontally averaged temperatures are almost the same as those measured by the thermocouple (these errors are less than ±1°C).

Figure 5 shows the vertical profile of air temperature, which is measured by the thermocouples covered with sunshades for z ≤ 0.15 m, above the center of the test surface. A large lapse rate gradient can be seen for 0 < z < 0.15 m, with a weak lapse rate for z > 0.15 m.

Figure 6 displays the relationship between H and TST calculated from the experimental data. The error is estimated at ±1°C for TST and ±15 W m−2 for H. Therefore, the data in this figure are limited to value of TST > 5°C. The number of data meeting this criterion is 98.

The straight line in Fig. 6 represents H = 1.4 × (TST)4/3; therefore, the coefficient b in CHU = b(TST)1/3 was determined by a least squares fit as
b−1−1/3
An attempt is made to obtain the value of CHU for the smooth surface from similarity theory [Eq. (14)] under calm conditions. Using values of H = 100 W m−2, L = −0.1 m, h = 100 m, and T = 300 K as representative of calm conditions and a strongly unstable surface layer results in w∗ = 0.65 m s−1 and u∗ = 0.05 m s−1. The Obukhov length L, defined by Eq. (8), is determined by the observed sensible heat flux H. The roughness length zT for the smooth surface is obtained by following Kondo (1975) as
i1520-0469-54-4-498-e32

By use of Eqs. (14), (21), (32), and (33), the coefficient b in CHU can be obtained as b = 0.0011 m s−1 K−1/3. This value is the same as that given by Eq. (31).

On the other hand, the average value of b in other studies (Table 1 and Fig. 7) is 0.0015 m s−1 K−1/3, which is 1.36 times larger than the present result. This difference can be explained by the following. In the experiments of Mikheyev (1968) and Lloyd and Moran (1974), the test surface was placed in a fluid, so the heat flux was directed downward from the downfacing surface. In the case of Fujii and Imura (1972) and the present experiment, the test surface was placed on the floor. The effect of convection from the surface may be less than in the former experiments.

b. Rough surface

Experiments on the rough surface are carried out in a manner similar to those on the smooth surface.

To create a black rough surface, small cubes constructed of black styrene foam were arranged on the black painted styrene foam surface. Figures 8 and 9 display a schematic representation and a photograph of how the cubes were arranged, respectively. The side dimension of the test surface is 1.8 m and the layer has a thickness of 0.135 m (see Fig. 3), while the length and width of each cube are 0.02 m. The cubes create regions of sun and shadow having different temperatures. The difference in surface temperatures between sun and shadow is measured by an infrared thermometer. The maximum temperature difference is 15°C, and the average value is 5.3°C. The representative temperature of sunny locations is measured on the surface of the cube, and that for shade on the vertical shadow surface by thermocouples. The surface temperature TS is the average value of these, which agrees with the average temperature measured by the infrared thermometer, within ±0.5°C.

The experiments were conducted on 22, 23, 26, 29 December and 6 January 1994. The representative conditions of the rough surface indoor experiment are shown in Table 2.

In this experiment, the surface temperature exhibits very high local values, but the average temperature of the test surface TS is less than the smooth case under the same solar conditions. It was concluded that the reason for this is that the alternate existence of sun and shade creates an effective sensible heat flux.

Figure 10 shows the results of the rough surface experiment. The data and results of the experiment are also tabulated in Table 3. The value for the sensible heat flux is obtained as H = 4.6 × (TST)4/3, with the coefficient b in CHU = b(TST)1/3 determined as
b−1−1/3
This value of b is about 3.5 times larger than that of the smooth surface cases. Compared with experiments by Sparrow and Vemuri (1986) as shown in Fig. 7, 3.5 may be the proper ratio of the coefficient b of the rough surface to the smooth surface. It should be noted that Sparrow and Vemuri (1986) conducted experiments for natural convection heat transfer on a horizontal plate having pin–fin arrays. The baseplate was square with side dimensions of 0.0762 m, and individual fins consisted of rods 0.00635 m in diameter and exposed length of 0.0254 m. The number of pins N was 27 and 68 for smooth and rough surfaces, respectively.

4. Field observations

The sensible heat flux in natural convection was observed above a smooth and horizontally homogeneous land surface by the eddy correlation method.

a. Observations

The observations were carried out at the Atmospheric Boundary-Layer Observatory of Tohoku University, located in Kitaura, in the town of Kogota, Miyagi Prefecture. The fetch of the prevailing wind direction for this area was about 2 km. The area was surrounded by horizontally homogeneous paddy fields. The observations were gathered on 18, 21, and 29 May, and 2 and 23 June 1994, which were calm days that occured just after the planting of rice. Table 4 lists the apparatuses used and the heights at which they were set for the observations. The surface temperature TS was measured by the infrared radiometer. Use was made of the ultrasonic anemometer–thermometer (KAIJO, DAT-100), having a sound pathlength of 0.2 m, a sampling frequency set at 10 Hz, and an observation run over a 10-min span. Under stationary conditions, the output fluctuations of w′ and T ′ were used to evaluate the sensible heat flux H = cPρw.

The roughness length for the wind profile, z0 = 1.5 × 10−2 m, and that for the temperature profile, zT = 10−5 m, were obtained by the profile method under windy neutral conditions. These are nearly the same values as those expected for the smooth surface of the indoor experiment. Since the surface is almost entirely covered with water during the observational period, it may be assumed that the value of emissivity is ϵ = 0.98 and qs is the saturation specific humidity at TS.

Figure 11 displays an example of the raw temperature and wind data under calm conditions with U11 = 0.98 m s−1 (the wind speed at z = 11 m). The fluctuations in w and T reveal on intermittent warm air plume.

The raw values of T measured by the sonic thermometer contain an error due to the specific humidity q. This error was corrected (see appendix).

Table 5 displays a tabulation of the data for conditions when U11 < 1.5 m s−1 and when H > 10 W m−2. Figure 12 shows the relationship between the corrected sensible heat flux H and the temperature difference TST. It can be seen that the values of the sensible heat flux are generally slightly larger than those of the indoor experiment over a smooth surface [Eq. (31), Fig. 6].

Taking into account the buoyancy due to the water vapor pressure difference, the relation between ΔTv and CHU is shown in Fig. 13 for the results from the indoor experiment with the smooth surface and the field observations.

Although the plotted points are widely scattered for small values of ΔTv, the following relation when ΔTv ≥ 10°C can be obtained as
CHUbTv1/3b−l−1/3

b. Ψ-functions comparisons

Several comparisons are made between the observed sensible heat flux H and those estimated Hp using the Ψ functions. Using the roughness lengths of z0 and zT measured under windy conditions, the observed values of U(z), T(z), and the Ψ functions [e.g., Eqs. (19) and (20)], the estimated sensible heat Hp can be evaluated by successive approximations by use of Eqs. (8), (9), (11), and (12). The initial values of u∗ and T∗ are determined by use of Eqs. (11) and (12) and Ψ functions under neutral conditions as ΨM = ln(z/z0) and ΨH = ln(z/zT). Making use of Eqs. (19) and (21) rather than (19) and (20) will also yield estimated values of Hp. Figure 14 shows comparisons between the observed values of H and the estimated Hp values. Plotted data are distinguished by the symbols (see the figure legend) according to the value of ζ = z/L. In Fig. 14, values of the sensible heat flux were estimated with the use of (a) Eqs. (19) and (20) and (b) Eqs. (19) and (21). Comparing Fig. 14a and 14b, it can be seen that the Ψ functions given by Eqs. (19) and (21) produce better estimated values of sensible heat under strongly unstable conditions. In Fig. 14b, the most unstable value of ζ is −477. (This value is obtained by the data in the second line from the bottom of Table 5.) So the above pair of Ψ functions is useful for the range 0 > ζ > −477.

5. Conclusions

An observational study of the sensible heat flux was conducted under natural convective conditions. Figure 15 displays the relationship between TST and H obtained from the present data and those found in other studies. The solid line and dotted-chain line in the figure indicate the results obtained from the indoor experiments over smooth and rough surfaces, respectively. The plus symbol represents the results of the observations in the paddy field, while the other symbols are quoted results from reference papers (see the figure legend). Along the bottom abscissa, the scale of water temperature difference TST at 20°C has been added for the experiment conducted in water (e.g., Townsend’s experiment). The ordinate on the right-hand side represents the scale of the temperature flux H/cPρ. The rightmost ordinate indicates value of Nu × Ra, while the top abscissa is the nondimensional parameter Ra for the convectional scale length h = 10 m.

The following are the clarified concrete results.

  1. The ϕ and Ψ functions defined in Eqs. (16) and (19) by Dyer and Hicks (1970), and Eqs. (18) and (21) by Kader and Yaglom (1990) are useful in unstable conditions and can be extended to the strongly unstable conditions such as ζ = −477.

  2. Over a smooth surface, the relationship between TST and H obtained from field observations is nearly the same as that found in the indoor experiment.

The exchange speed for the sensible heat flux under natural convection is expressed as CHU = b(TST)1/3. When the surface is wet, ΔTv should be substituted for (TST). For a smooth surface, the coefficient b is about 0.0011 m s−1 K−1/3 for U10 ≤ 1 m s−1. Over the tropical ocean, when TST = 5°C, the sensible heat flux H is estimated at about 12 W m−2. When T = 300 K and ese = 10 hPa, the total heat flux H + lE amounts to about 47 W m−2.

When applying these results to more complicated land surfaces, further investigations should be conducted to obtain the heat flux from various rough surfaces under natural convective conditions.

Acknowledgments

The authors would like to thank Dr. T. Kuwagata of the Tohoku National Agricultural Experimental Station and Dr. I. Tamagawa of the University of Nagoya for their helpful advice and valuable comments concerning the ultrasonic anemometer–thermometer.

REFERENCES

  • Beljaars, A. C. M., 1995: The parametrization of surface fluxes in large-scale models under free convection. Quart. J. Roy. Meteor. Soc.,121, 255–270.

  • Bradley, E. F., P. A. Coppin, and J. S. Godfrey, 1991: Measurements of sensible and latent heat flux in the western equatorial Pacific Ocean. J. Geophys. Res.,96 (Suppl.), 3375–3389.

  • Deardorff, J. W., and G. E. Willis, 1985: Further results from a laboratory model of the convective planetary boundary layer. Bound.-Layer Meteor.,32, 205–236.

  • Dyer, A. J., and B. B. Hicks, 1970: Flux-gradient relationships in the constant flux layer. Quart. J. Roy. Meteor. Soc.,96, 715–721.

  • Fujii, T., and H. Imura, 1972: Natural convection heat transfer from a plate with arbitrary inclination. Int. J. Heat Mass Transfer,15, 755–766.

  • Garratt, J. R., 1992: Extreme maximum land surface temperature. J. Appl. Meteor.,31, 1096–1105.

  • Godfrey, J. S., and E. J. Lindstorm, 1989: The heat budget of the western equatorial Pacific surface mixed layer. J. Geophys. Res.,94, 8007–8017.

  • Kader, B. A., and A. M. Yaglom, 1990: Mean fields and fluctuation moments in unstably stratified turbulent boundary layers. J. Fluid Mech.,212, 637–662.

  • Kaimal, J. C., and J. A. Businger, 1963: A continuous wave sonic anemometer–thermometer. J. Appl. Meteor.,2, 156–164.

  • ———, and J. E. Gaynor, 1991: Another look at sonic thermometry. Bound.-Layer Meteor.,56, 401–410.

  • Kondo, J., 1975: Air–sea bulk transfer coefficients in diabatic conditions. Bound.-Layer Meteor.,9, 91–112.

  • Lloyd, J. R., and W. R. Moran, 1974: Natural convection adjacent to horizontal surface of various planforms. J. Heat Transfer,96, 443–447.

  • Mikheyev, M., 1968: Fundamentals of Heat Transfer. Peace Publishers, 376 pp.

  • Mitsuta, Y., 1966: Sonic anemometer–thermometer for general use. J. Meteor. Soc. Japan,44, 12–24.

  • Monin, A. S., and A. M. Yaglom, 1971: Statistical Fluid Mechanics. The MIT Press, 769 pp.

  • Priestley, C. H. B., 1955: Free and forced convection in the atmosphere near the ground. Quart. J. Roy. Meteor. Soc.,81, 139–143.

  • Schotanus, P., F. T. M. Nieuwstadt, and H. A. R. DeBruin, 1983: Temperature measurement with a sonic anemometer and its application to heat and moisture fluxes. Bound.-Layer Meteor.,26, 81–93.

  • Schumann, U., 1988: Minimum friction velocity and heat transfer in the rough surface layer of a convective boundary layer. Bound.-Layer Meteor.,44, 311–326.

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  • Sparrow, E. M., and S. B. Vemuri, 1986: Orientation effects on natural convection/radiation heat transfer from pin-fin arrays. Int. J. Heat Mass Transfer,29, 359–368.

  • Takeyama, T., S. Ooya, and T. Aihara, 1983: Heat Transfer (in Japanese). Maruzen, 254 pp.

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APPENDIX

Correction to the Ultrasonic Thermometer

The temperature measured by the ultrasonic thermometer Tobs contains an error induced by water vapor as
TobsTq
where q is the specific humidity. The components of Eq. (A1) can be separated into the average and fluctuation as Tobs = Tobs + T ′obs. Subtracting the average from the total results in
T ′obsTqTqTqq
Multiplying T ′obs by w, taking average, and then ignoring the third-order fluctuation and small component (wT ′q) yields
i1520-0469-54-4-498-ea3
Therefore, the evaluated sensible heat Hobs = cPρwT ′obs equals
i1520-0469-54-4-498-ea4
and contains the error in approximate proportion to the latent heat lE (Kaimal and Businger 1963; Mitsuta 1966; Schotanus et al. 1983; Kaimal and Gaynor 1991). Here, H is the true sensible heat. Assuming the air temperature T is 300 K and substituting this value in Eq. (A4) results in H = Hobs − 0.06lE.

Fig. 1.
Fig. 1.

Relationship between the exchange speed CEU10 and the wind speed at the height of 10 m U10 under calm conditions over the ocean, from various studies according to the figure legend.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 2.
Fig. 2.

The stability profile functions ΨM and ΨH vs the nondimensional height ζ. (a) The solid line represents Eq. (19) according to Dyer and Hicks (1970); the thin solid line denotes ζ < −1, while the dotted line indicates ΨM = ln(z/z0) (neutral conditions). (b) The solid and broken lines represent Eqs. (20) and (21), respectively, while the dotted line indicates ΨH = ln(z/zT) (neutral conditions).

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 3.
Fig. 3.

Illustration of the setup for the indoor experiments.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 4.
Fig. 4.

An example of the horizontal surface temperature distribution over the smooth test surface of the indoor experiment.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 5.
Fig. 5.

An example of the vertical air temperature profile at the center of the smooth test surface.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 6.
Fig. 6.

Relationship between TST and H over the smooth surface of the indoor experiment.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 7.
Fig. 7.

Relationship between Ra and Nu from various studies, including the present experiments (see the figure legend). The letters SBL indicate the surface boundary layer.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 8.
Fig. 8.

Partial overhead view of the array of cubes used to create the rough surface of the indoor experiment.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 9.
Fig. 9.

A photograph of the rough test surface showing the placement of the small cubes.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 10.
Fig. 10.

Relationship between TST and H over the rough surface of the indoor experiment.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 11.
Fig. 11.

An example of the raw data of w′ and observed between 0805 and 0815 LST 2 June 1994. Here, U11 = 0.98 m s−1, TST = 8.82°C, and H = 47.4 W m−2.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 12.
Fig. 12.

Relationship between TST and H from the field observation data. The plus symbols represent values of H corrected for specific humidity. The dashed line denotes the relationship determined from the indoor experiment over a smooth surface.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 13.
Fig. 13.

Relationship between ΔTv and CHU over smooth surfaces. The dots represent results from the present indoor experiments, the triangles the results from the field observations, and the solid line according to Eq. (31).

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 14.
Fig. 14.

Comparisons between the estimated sensible heat flux Hp by the profile method and the observed H corrected for specific humidity. Comparisons are shown for (a) using Eqs. (19) and (20), and (b) using Eqs. (19) and (21).

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Fig. 15.
Fig. 15.

Relationship between TST and H for the present results and those of previous studies (see legend). Included on the abscissa (TST) is a scale for water temperature difference at 20°C (Townsend 1964). The left-hand ordinate represents H (W m−2), while the right-hand denotes values of H/cPρ (m s−1 K), the rightmost ordinate values of Nu × Ra (for h = 10 m). The top abscissa represents the nondimensional parameter Ra for a convectional scale length of h = 10 m.

Citation: Journal of the Atmospheric Sciences 54, 4; 10.1175/1520-0469(1997)054<0498:SHFFTE>2.0.CO;2

Table 1.

The coefficients A and a in Nu = A × Raa, and b in CHU = b(TST);d1 over the respective range ofRa in several engineering heat transfer experiments.

Table 1.
Table 2.

Representative conditions during the indoor experiments (smooth and rough surfaces). Here, rh is the relative humidity in the room, ref is the albedo of the test surface, and TWINDOW and TTOP are the surface temperatures of windows and ceiling, respectively.

Table 2.
Table 3.

Data from the rough surface indoor experiment.

Table 3.
Table 4.

Measuring instruments and A/D converters used in the field observations. Thermodac-E: Digital DataLogger (ETO-Denki Co., Ltd.); ADn1400: A/D converting board for Note-PC (Canopus Co., Ltd.).

Table 4.
Table 5.

Data from the field observations. Here, S is the solar radiation, U11 the wind velocity at z = 11 m, T10 the air temperature at z = 10 m, q10 the specific humidity of the air at z = 10 m, Hobs the observed sensible heat, H the corrected sensible heat, lE the latent heat, u&lowast; the friction velocity, and T∗ the friction temperature.

Table 5.
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  • Fig. 1.

    Relationship between the exchange speed CEU10 and the wind speed at the height of 10 m U10 under calm conditions over the ocean, from various studies according to the figure legend.

  • Fig. 2.

    The stability profile functions ΨM and ΨH vs the nondimensional height ζ. (a) The solid line represents Eq. (19) according to Dyer and Hicks (1970); the thin solid line denotes ζ < −1, while the dotted line indicates ΨM = ln(z/z0) (neutral conditions). (b) The solid and broken lines represent Eqs. (20) and (21), respectively, while the dotted line indicates ΨH = ln(z/zT) (neutral conditions).

  • Fig. 3.

    Illustration of the setup for the indoor experiments.

  • Fig. 4.

    An example of the horizontal surface temperature distribution over the smooth test surface of the indoor experiment.

  • Fig. 5.

    An example of the vertical air temperature profile at the center of the smooth test surface.

  • Fig. 6.

    Relationship between TST and H over the smooth surface of the indoor experiment.

  • Fig. 7.

    Relationship between Ra and Nu from various studies, including the present experiments (see the figure legend). The letters SBL indicate the surface boundary layer.

  • Fig. 8.

    Partial overhead view of the array of cubes used to create the rough surface of the indoor experiment.

  • Fig. 9.

    A photograph of the rough test surface showing the placement of the small cubes.

  • Fig. 10.

    Relationship between TST and H over the rough surface of the indoor experiment.

  • Fig. 11.

    An example of the raw data of w′ and observed between 0805 and 0815 LST 2 June 1994. Here, U11 = 0.98 m s−1, TST = 8.82°C, and H = 47.4 W m−2.

  • Fig. 12.

    Relationship between TST and H from the field observation data. The plus symbols represent values of H corrected for specific humidity. The dashed line denotes the relationship determined from the indoor experiment over a smooth surface.

  • Fig. 13.

    Relationship between ΔTv and CHU over smooth surfaces. The dots represent results from the present indoor experiments, the triangles the results from the field observations, and the solid line according to Eq. (31).

  • Fig. 14.

    Comparisons between the estimated sensible heat flux Hp by the profile method and the observed H corrected for specific humidity. Comparisons are shown for (a) using Eqs. (19) and (20), and (b) using Eqs. (19) and (21).

  • Fig. 15.

    Relationship between TST and H for the present results and those of previous studies (see legend). Included on the abscissa (TST) is a scale for water temperature difference at 20°C (Townsend 1964). The left-hand ordinate represents H (W m−2), while the right-hand denotes values of H/cPρ (m s−1 K), the rightmost ordinate values of Nu × Ra (for h = 10 m). The top abscissa represents the nondimensional parameter Ra for a convectional scale length of h = 10 m.

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