## 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

*H*

*c*

_{P}

*ρ*

*w*′

*lE*

*ρ*

*l*

*w*′

*q*′

*,*

*H*is the sensible heat,

*lE*the latent heat,

*ρ*the air density, and

*c*

_{P}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*′.

*H*and

*lE*can also be expressed by bulk formulas as

*H*

*c*

_{P}

*ρ*

*C*

_{H}

*U*

*T*

_{S}

*T*

*lE*

*l*

*ρ*

*C*

_{E}

*U*

*q*

_{s}

*q*

*T*and

*q*are the air temperature and specific humidity at the reference height, respectively, while

*T*

_{S}and

*q*

_{s}are those at the ground surface. From Eqs. (1)–(4), the bulk transfer coefficients

*C*

_{H}and

*C*

_{E}can then be obtained.

*C*

_{H}or

*C*

_{E}is meaningless in the range of small wind velocity such as

*U*< 1 m s

^{−1}. Under natural convective conditions, the exchange speed

*C*

_{H}

*U*or

*C*

_{E}

*U*is more useful. Figure 1 displays the relationship between

*C*

_{E}

*U*

_{10}and the wind velocity

*U*

_{10}(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

*C*

_{H}

*U*is equal to that of latent heat

*C*

_{E}

*U.*The dotted line and dot-chain line in this figure are taken from Kondo (1975). Here, the virtual temperature difference is defined as

*e*(hPa) is the water vapor pressure at the reference height and

*e*

_{s}the saturation value for the sea surface temperature

*T*

_{S}. The observations in Bradley et al. (1991) were conducted under the conditions that

*T*

_{S}−

*T*= 1–3°C and

*e*

_{s}−

*e*= 10–12 hPa. In Fig. 1, it can be seen that

*C*

_{E}

*U*

_{10}converges to a positive constant value as

*U*

_{10}→ 0. Even with no wind, the virtual temperature difference creates a natural convection heat transfer.

### b. Similarity theory

*ϕ*

_{M}and

*ϕ*

_{H}are the nondimensional shear functions for momentum and heat, respectively;

*k*(= 0.4) is the von Kármán constant;

*ζ*=

*z*/

*L*;

*u*∗

*T*∗ = −

*H*/

*c*

_{P}

*ρ*;

*u*∗ the friction velocity; and

*T*∗ the temperature scale. The Obukhov length

*L*can be defined by

*H*

_{v}

*H*

*Tc*

_{P}

*E.*

*u*∗ is physically meaningless. [But it is necessary to obtain the value of

*z*

_{0}or

*z*

_{T}of smooth surface as Eqs. (32) and (33) under these conditions.] Therefore, assuming the natural convection velocity scale

*w*∗ = [

*H*

_{v}

*hg*/(

*c*

_{P}

*ρ*

*T*)]

^{1/3}, along with the definition given by Eq. (8), results in

*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*×

*H*

_{v}

^{1/3}(

*α*= 0.034 W

^{−1/3}m

^{1/3}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}=

*U*

^{2}

_{l}

*β*

*w*∗)

^{2}, where

*U*

_{l}is the large-scale velocity components and

*β*is the constant coefficient. But in this paper, it is taken the following traditional way.

*ϕ*functions for unstable condition (0 ≥

*ζ*≥ −1) as

*ζ*> −20 as

*z*

_{0}and

*z*

_{T}are the roughness lengths for the wind and temperature profiles, respectively.

Figure 2a displays the shape of Ψ_{M} according to Eq. (19), and Fig. 2b Ψ_{H} from Eqs. (20) and (21) for values of −*ζ*_{0} = −*ζ*_{T} = 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

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* (m^{2} s^{−1}) the thermometric conductivity, *ν* (m^{2} s^{−1}) the kinematic viscosity, *g* (m s^{−2}) the acceleration of gravity, and *β* (K^{−1}) the coefficient of thermal expansion.

*A*

^{a}

*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.

*a*= 1/3 for large values of Ra (i.e., 8 × 10

^{6}< Ra < 10

^{13}),

*C*

_{H}

*U*can be described with use of the proportional constant

*b*as

*C*

_{H}

*U*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, Δ

*T*

_{v}[Eq. (5)] should be substituted for (

*T*

_{S}−

*T*).

Since engineering studies rely on indoor experiments, determination of *C*_{H}*U* 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 (>10^{12}) 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 *T*_{W}, the longwave radiation to the test surface is expressed by *σT* ^{4}_{W}*σ* (= 5.67 × 10^{−8} W m^{−2} K^{−4}) is the Stefan–Boltzmann constant, and *T*_{W} is the effective infrared temperature of the walls. The value of *T*_{W} 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.

*R*

_{n}is the net radiaton,

*S*the solar radiation incident on the horizontal test surface,

*c*

_{s}

*ρ*

_{s}(= 32.2 × 10

^{3}J K

^{−1}m

^{−3}) the heat capacity of the styrene foam,

*T*

_{S}the test surface temperature,

*T*

_{g}(

*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 *T*_{g}(*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 *R*_{n}, *G,* and *H* are averaged over every 30 min; observation with the value of *C*_{H}*U* (*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 *T*_{S} 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 *T*_{S} − *T* calculated from the experimental data. The error is estimated at ±1°C for *T*_{S} − *T* and ±15 W m^{−2} for *H.* Therefore, the data in this figure are limited to value of *T*_{S} − *T* > 5°C. The number of data meeting this criterion is 98.

*H*= 1.4 × (

*T*

_{S}−

*T*)

^{4/3}; therefore, the coefficient

*b*in

*C*

_{H}

*U*=

*b*(

*T*

_{S}−

*T*)

^{1/3}was determined by a least squares fit as

*b*

^{−1}

^{−1/3}

*C*

_{H}

*U*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

*z*

_{T}for the smooth surface is obtained by following Kondo (1975) as

By use of Eqs. (14), (21), (32), and (33), the coefficient *b* in *C*_{H}*U* 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 *T*_{S} 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 *T*_{S} 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.

*H*= 4.6 × (

*T*

_{S}−

*T*)

^{4/3}, with the coefficient

*b*in

*C*

_{H}

*U*=

*b*(

*T*

_{S}−

*T*)

^{1/3}determined as

*b*

^{−1}

^{−1/3}

*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 *T*_{S} 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* = *c*_{P}*ρ**w*′

The roughness length for the wind profile, *z*_{0} = 1.5 × 10^{−2} m, and that for the temperature profile, *z*_{T} = 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 *q*_{s} is the saturation specific humidity at *T*_{S}.

Figure 11 displays an example of the raw temperature and wind data under calm conditions with *U*_{11} = 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 *U*_{11} < 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 *T*_{S} − *T.* 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 Δ*T*_{v} and *C*_{H}*U* is shown in Fig. 13 for the results from the indoor experiment with the smooth surface and the field observations.

*T*

_{v}, the following relation when Δ

*T*

_{v}≥ 10°C can be obtained as

*C*

_{H}

*U*

*b*

*T*

_{v}

^{1/3}

*b*

^{−l}

^{−1/3}

### b. Ψ-functions comparisons

Several comparisons are made between the observed sensible heat flux *H* and those estimated *H*_{p} using the Ψ functions. Using the roughness lengths of *z*_{0} and *z*_{T} 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 *H*_{p} 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 Ψ_{M} = ln(*z*/*z*_{0}) and Ψ_{H} = ln(*z*/*z*_{T}). Making use of Eqs. (19) and (21) rather than (19) and (20) will also yield estimated values of *H*_{p}. Figure 14 shows comparisons between the observed values of *H* and the estimated *H*_{p} 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 *T*_{S} − *T* 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 *T*_{S} − *T* 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*/*c*_{P}*ρ*. 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.

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.Over a smooth surface, the relationship between

*T*_{S}−*T*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 *C*_{H}*U* = *b*(*T*_{S} − *T*)^{1/3}. When the surface is wet, Δ*T*_{v} should be substituted for (*T*_{S} − *T*). For a smooth surface, the coefficient *b* is about 0.0011 m s^{−1} K^{−1/3} for *U*_{10} ≤ 1 m s^{−1}. Over the tropical ocean, when *T*_{S} − *T* = 5°C, the sensible heat flux *H* is estimated at about 12 W m^{−2}. When *T* = 300 K and *e*_{s} − *e* = 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.

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## APPENDIX

### Correction to the Ultrasonic Thermometer

*T*

_{obs}contains an error induced by water vapor as

*T*

_{obs}

*T*

*q*

*q*is the specific humidity. The components of Eq. (A1) can be separated into the average and fluctuation as

*T*

_{obs}=

*T*

_{obs}

*T*

^{ ′}

_{obs}.

*T*′

_{obs}

*T*

*q*

*T*

*q*

*T*

*q*

*q*′

*T*

^{ ′}

_{obs}

*w*′

*,*taking average, and then ignoring the third-order fluctuation and small component (

*w*′

*T*′

*q*

*H*

_{obs}=

*c*

_{P}

*ρ*

*w*′

*T*′

_{obs}

*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*

*H*=

*H*

_{obs}− 0.06

*lE.*

The coefficients *A* and *a* in Nu = *A* × Ra^{a}, and *b* in *C*_{H}*U* = *b*(*T*_{S} − *T*)^{;d1} over the respective range ofRa in several engineering heat transfer experiments.

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 *T*_{WINDOW} and *T*_{TOP} are the surface temperatures of windows and ceiling, respectively.

Data from the rough surface indoor experiment.

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.).

Data from the field observations. Here, *S* is the solar radiation, *U*_{11} the wind velocity at *z* = 11 m, *T*_{10} the air temperature at *z* = 10 m, *q*_{10} the specific humidity of the air at *z* = 10 m, *H*_{obs} the observed sensible heat, *H* the corrected sensible heat, *lE* the latent heat, *u*∗ the friction velocity, and *T*∗ the friction temperature.