NOTES AND CORRESPONDENCE. Toward a Robust Phenomenological Expression of Evaporation Efficiency for Unsaturated Soil Surfaces

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1 1330 JOURNAL OF APPLIED METEOROLOGY NOTES AND CORRESPONDENCE Toward a Robust Phenomenological Expression of Evaporation Efficiency for Unsaturated Soil Surfaces TERUHISA S. KOMATSU Department of Environmental Sciences, Japan Atomic Energy Research Institute, Tokai, Naka, Ibaraki, Japan 24 May 2002 and 23 December 2002 ABSTRACT The evaporation rates of water from several soil types were measured under controlled conditions. When the layer of soil is sufficiently thin, the evaporation efficiency, the ratio of the evaporation rate from the soil surface relative to that from the watered surface, is described well by a function of the mean volume fraction of water. The function has a common form for several types of soil and can be parameterized by a single parameter, the characteristic volume fraction c of water, as 1 exp( / c ). The value of c depends on the soil type and also on the wind speed. The latter dependence can be robustly described in terms of the evaporation resistance for the watered surface. These findings are expected to become a basis for calculating the evaporation efficiency for the soil with an arbitrary depth in a systematic way. 1. Introduction Evaporation of water is a key process of the hydrologic cycle on the earth. In order to simulate water circulation, it is important to estimate the evaporation rate from the surface of the land. Until now, a standard description for the evaporation rate of water from a water surface has been established (e.g., Brutsaert 1982), and the effects of environmental temperature, humidity, and wind speed can be well described. In contrast, the evaporation from soil surfaces is much harder to describe because of the vast variety of soil types. Although specific descriptions have been proposed for various applications (Mahfouf and Noilhan 1991, and references contained therein), their applicability seems to be limited because they are mostly based on measurements under limited conditions of soils, for example, the soils of a specific thickness or type. For example, Kondo et al. s (1990) results for 0.02-m-deep soil are well known and widely used in many studies. However, it is not clear if one can apply the results to soils of varying depths. It is, thus, important to develop a unified description and a robust parameterization of evaporation rate from soil surfaces, which can be applied to soil of arbitrary depth in a systematic way. In our previous study (Komatsu 2001), the evapo- Corresponding author address: Teruhisa S. Komatsu, Department of Physics, Gakushuin University, Mejiro 1-5-1, Tokyo , Japan. teruhisa.komatsu@gakushuin.ac.jp ration rate of water from several types of soil of variable thickness was systematically measured under a calm state in indoor experiments. From these experiments, it was concluded that the evaporation rate of water can be described by a unique function of the mean volume fraction of water contained in the soil, within an adequate range of thickness. The function characterizing evaporation rate is common for several types of soils and is parameterized by a single parameter, characteristic volume fraction c of water. In the present study, the experiments were conducted under the conditions in which there was wind with constant speed. As in the cases without wind, the evaporation rates for several types of materials were found to be parameterized by a unique common function for sufficiently thin layers of soils. These findings are expected to become a basis for calculating the evaporation efficiency for a soil with an arbitrary depth in a systematic way. 2. Experimental method The two different experimental conditions, as in Fig. 1, were examined. In the first case (case I), experiments were performed in the wind tunnel (whose measurement section was a 1 m 1 m cross section and was 3 m in length in the transverse direction), where wind speed and air conditions (temperature and humidity) were kept at constant values (25 C, 50% with various wind speeds). The profile of the wind velocity in the tunnel 2003 American Meteorological Society

2 SEPTEMBER 2003 NOTES AND CORRESPONDENCE 1331 FIG. 1. Experimental configurations. Pans were filled with soil and put on the weighing machine so that their surfaces were located at the same height as the bottom plate. (a) In the first case, experiments were performed in the wind tunnel. (b) In the second case, evaporation under different wind conditions was examined. measured at points above the pan by anemometer is shown in Fig. 2. Note that the velocities at a sufficiently high vertical position were constant with height. These constant values of U can be used to characterize the wind speed in the tunnel. It should be noted that the specific value of U can characterize wind speed only in the present wind tunnel. In order to study robust properties of evaporation, experiments were performed in a less controlled environment in the second case (case II). Here, the wind stream was created by a small fan placed beside the pan, and the air conditions, such as temperature and humidity, were uncontrolled. In order to keep constancy of the wind condition, the system was placed in the center of a 2 m 2 m area surrounded by vertical side walls with 1.5 m height, where only a gradual exchange of air with air outside of this area was allowed through the top of the system. A calm state in the same setting was also prepared by stopping the fan. In both cases, the soils were poured in cylindrical pans (150-mm diameter with various depths) made of acrylic plastic (whose surface is hydrophobic) until they just filled the pans, which were then placed on the base (which insulates thermal conduction from the bottom) on the electric weighing machine. An acrylic plastic plate, which had a hole of the pan size, was placed at the bottom of the system. The vertical position of the pan was adjusted so that the soil surface and the bottom plate have equal heights (see Fig. 1). In this alignment, where the boundary condition for the wind at the bottom was nearly identical with that formed by a single flat plane, we can study properties of evaporation under steady wind above a single flat plane. The weight of the pan was measured every 60 s with a resolution of g. We then calculated the evaporation rate E of water from the change in the weight and the average volumetric water content in the soil from the weight. We thus obtained the function E( ) from the measured data of the weight. At the same time, the temperature T s of the soil surface, air temperature T a, and specific humidity q a (mass density of water vapor per mass density of the air) at the height of the pans FIG. 2. Velocity profiles in the wind tunnel. The horizontal axis is the wind velocity, and the vertical axis is the height from the bottom plate. Each symbol corresponds to different settings of the wind. Rotational speeds of the controlling fan were 30 (plus), 45 (cross), 140 (circle), 230 (open square), and 340 rpm (solid square). were measured using thermocouples (T-CC), platinum resistance, and the humidity sensor (OMRON s ES2), respectively. In this experiment, we examined several kinds of media soils, including an agricultural soil summarized in Table 1. The agricultural soil was a kind of andisol sampled at a field in Tokai, Ibaraki, Japan. The two kinds of sand were sampled at a sand beach in Tokai, were well rinsed in water, and were sifted by several sieves. All media soils were fully oven dried before the start of the experiment. In addition, although it was not a real soil, cornstarch was used as a test material, which was the finest media studied here. For the initial conditions, soil particles were spread uniformly in the pans and made fully wet so that the surface of the soil was covered by water. In the present study, we focused only on the drying processes of wet soil. Because the experiments were performed indoors, there was no precipitation or radiation fluxes from the sun. 3. Parameterization of evaporation rate Let us first recall the standard formula for the evaporation rate E w from the watered surfaces, Ew a q/r a, (1) where E w is the weight of water evaporating from the unit area within the unit time, and the evaporation resistance r a is a function of wind speed; a is the mass density of the air; and q is the difference of the spe- TABLE 1. Characteristics of the soils and the corresponding experimental results. Each column represents void fraction, bulk density b (g cm 3 ), solid density s (g cm 3 ), characteristic volume fraction c0 of water, and characteristic evaporation resistance (s m 1 ) (see the text). Soil type b s c0 Sand A( m) Sand B( m) Agricultural soil Cornstarch

3 1332 JOURNAL OF APPLIED METEOROLOGY FIG. 3. (a) Temperature ( C) and humidity (%) of environmental air, (b) evaporation rate E w (g cm 2 h 1 ) from water pan in a control experiment, and (c) evaporation resistance r a (s m 1 ) (see text). cific humidity between environmental air and the water surface; that is, q q s q a, where q s is assumed to be equal to the saturated value at the temperature of the surface, that is, q s q sat (T s ). {Here we use the formula q sat (T) [0.622e sat (T)/p]/[ e sat (T)/p], where p is the atmospheric pressure and e sat (T) T/(237.3 T) (hpa).} In our study, the evaporation rate E w, the temperatures, and the humidities were measured, and the evaporation resistance r a was computed by inverting Eq. (1). In Fig. 3, we present results for Eq. (1). Figure 3a is the time series of the temperature and the humidity of environmental air in case II with no wind. Because the environment was uncontrolled in case II, the data in Fig. 3a show daily fluctuation, which results in the fluctuation of the evaporation rate E w, as seen in Fig. 3b. On the other hand, the evaporation resistance r a plotted in Fig. 3c shows relatively small fluctuation. This suggests that r a can robustly parameterize the evaporation rate even in a fluctuating environment. Now let us examine the evaporation from the soil surfaces. We introduce an empirical parameter called evaporation efficiency, which is the ratio between the evaporation rate from the soil surface and that from the watered surface. More precisely, it is defined by the relation E a q/r a, (2) where E is the evaporation rate of water from the soil surface, and q is, as before, the difference of the specific humidity ( q q s q a ), but between environmental air and the soil surface (q s is assumed to be equal to the saturated value at the temperature of the soil surface), and r a is the evaporation resistance from the watered surface (the saturated soil evaporation resistance). Because r a is obtained from the results of control experiments with the water pans and from Eq. (1), FIG. 4. Evaporation efficiency vs volumetric water content for various soil thicknesses. The type of soil, speed of wind U, and thicknesses of the samples are displayed in each figure. The guide lines in each figure is 1 exp( / c ) with (a) c 0.07, (b) c 0.08, (c) c 0.028, (d) c 0.035, (e) c 0.028, and (f) c 0.1. can be obtained from the experimental runs. At the primary stage of evaporation where soils were sufficiently wet, the evaporation rate from the soil pan was almost equal to that from the water pan, and we obtained 1. As evaporation proceeds and the soil becomes increasingly dry, evaporation was strongly restricted by the presence of the soils, and, hence, decreased. Because the evaporation rate from the fully wet soil surfaces was almost equal to that from water surfaces, the values of r a can also be calculated from the data at the primary stage within measurement errors. Using the obtained r a, the values of ( ) were calculated from E( ). 4. Experimental results In Fig. 4, the evaporation efficiency for the soil samples (agricultural soil, sand, and cornstarch) with various thicknesses under the equal wind condition in case I is shown as a function of average volumetric water content. Here, one finds that the form of ( ) changes with thicknesses. For the thinner samples, ( ) has a monotonically increasing convex form and seems to converge to a unique convex function of in the limit of the thin layer. For the thicker samples, on the other

4 SEPTEMBER 2003 NOTES AND CORRESPONDENCE 1333 FIG. 5. The dependence of the characteristic volumetric water content c on the strength of the wind. The horizontal axis represents the inverse of evaporation resistance from the water surface. Symbols represent soil types: sand A (circle), sand B (triangle), agricultural soil (square), and cornstarch (inverted triangle). Data from case I (filled symbols) and II (open symbols) are shown. Each guide line is c c0 (1 /r a ). hand, ( ) has a monotonically increasing S-shaped form and is not described by a simple common function. This S-shaped form of ( ) is qualitatively similar to results from Kondo (1990). Nevertheless, our results indicate that the expression for the evaporation efficiency of the samples with a definite thickness cannot be applied to samples with different thicknesses. The nonuniversal behavior of ( ) found in thicker layers may be caused by nonuniformity in the vertical distribution of water in the soil. For a thicker soil layer, the local water content near the surface and that near the bottom would be totally different; for example, the surface may be dry when the bottom is wet. For a thinner soil layer, on the other hand, the nonuniformity would be relatively smaller, and, thus, the local water content near the surface could be approximated by the average water content. This explains the convergence of ( ). From Fig. 4, it is found that the values of are smaller in the thicker case for the same values of. This is reasonable because the surface of thicker samples could be more dry for the same value of and, thus, the values of the evaporation efficiency for thicker samples could be lower. The crossover length for the convergence of is found to depend on the wind speed and also to be smaller under higher wind speeds (comparing Figs. 4a and 4b). This is reasonable because the nonuniformity of vertical distribution of water is larger when the wind speed is higher. It is reasonable to conclude that the converged forms of evaporation efficiency ( ) in the limit of the thin layer reflects the property of the soil itself. Furthermore, it is found that the limiting ( ) for various soil types and various wind speeds can be well approximated by a single common function as 1 exp( / ). c (3) FIG. 6. Scaled characteristic volumetric water content c / c0 vs 1/r a. This figure is the replot of the same data in Fig. 5. Symbols are used in the same manner as in Fig. 5. Here, c, which we call characteristic volumetric water content, is a single parameter that characterizes the soil type and the wind speed. We note that the convergence of ( ) in the thin limit and the common form Eq. (3) in the calm state were already noted by the present author (Komatsu 2001). In order to estimate c reliably, sufficiently thin samples where ( ) converges to the limit should be used. The convergence was checked by testing the linearity of the plot of log [1 ( )] versus, and experiments with thinner layers were conducted until the convergence was confirmed. In most cases, data in the range were used for these analysis. The characteristic volumetric water content c depends not only on the soil type but also on the wind speed. In order to investigate the latter dependence systematically, data must be plotted versus the wind speed. As noted before, U can characterize the wind speed only in the present wind tunnel, and so it is more suitable to plot the data versus another more robust variable that robustly characterizes wind speed. For such a variable, r a was selected here. It is natural because r a is almost solely determined by wind speed, and it can robustly parameterize the evaporation rate, as shown in Fig. 3. In Fig. 5, the values of c for four soil samples (agricultural soil, two kinds of sand, and corn starch) are shown versus 1/r a, where data in case I and II are plotted at the same time. It is found that data in case I and II for the same soil type agree well, and that c linearly depends on 1/r a. The relation is summarized as c c0(1 /r a ), (4) where c0 and are given in Table 1. Figure 6 is the replot of same data in Fig. 5, where the vertical axis is scaled as c / c0. From this figure, it is observed that differences of among different soil types are small when compared with the measurement errors, and is estimated around 100 s m 1. Because has less dependence on the soil type, c0 is the essential parameter classifying each soil type. Thus, Eq. (4) includes the

5 1334 JOURNAL OF APPLIED METEOROLOGY effect of the wind through /r a and that of the soil type through c0. 5. Summary and discussion The evaporation efficiency [see Eq. (2) for the definition], as a function of the average volumetric water content, was measured for various soil types under various constant wind speeds. Surprisingly, ( ) in the limit of the thin layer was well described by a single common function as Eq. (3). The only arbitrary parameter c in Eq. (3) was further reduced to the simple form of Eq. (4), where c0 and /r a characterized the soil type and the wind speed, respectively. This finding is expected to become a basis of systematic calculation of evaporation from various types of soil surfaces under various environments. We conjecture that the common result found for thin layers of several soil types, which was that the evaporation efficiency can be expressed as in Eq. (3), holds for the homogeneous system composed of granular substances. Because materials used in our experiments are limited to several kinds, exhaustive works for other soil materials should be done in the future. Considerations about inhomogeneous systems and systems with different pan sizes are also issues in the future. In our system, temperature variation was about 10 and temperature dependence of our results was not fully investigated. Solar radiation effects, which are a major factor of evaporation in a real field, were not considered here, but it might be treated as another problem in predicting the surface temperature. The assumption that the humidity at the soil surface was equal to the saturated humidity at the temperature of the surface might be modified. Detailed studies for temperature dependence of the parameterization will clarify these points. Here we employed the description [Eq. (2)] of the evaporation rate for unsaturated soil surfaces, which was based on evaporation efficiency and the so-called method. There is also another description, called the method, where parameter is introduced by the relation, E a ( q sat q a )/r a, instead of Eq. (2). Comparing this form and Eq. (2), the relation 1 q/q sat (1 ) is obtained. Then, has the similar form to Eq. (3) with some offset values of due to the factor q/q sat. Because this offset value depends on the difference of the specific humidity, we can select the better method of parameterization by comparing the dependence on humidity. To clarify the relative merits between two methods, more detailed study under various humidity conditions must be done; these are future issues. Theoretical estimation of c0 and (those parameters characterize soil types and dependence on the wind), and simple theoretical interpretation of the exponential forms of Eqs. (3) and (4) will lead to a better understanding of evaporation processes. As a first step toward a robust phenomenological description of evaporation, the expressions in the limit of the thin layer, Eqs. (3) and (4), are obtained here. The next step will be to deduce the evaporation efficiency for the samples with arbitrary depth based on these results. As noted before, the S-shaped form of ( ), for thicker samples, originates from the nonuniformity in the vertical distribution of water in the soil. Thus, if we can estimate the distribution of water and water content s at the surface for a given average water content, we can obtain evaporation efficiency as ( s ) using Eq. (3). It would be interesting to investigate how correctly this procedure can reproduce evaporation efficiency for thicker samples by some model calculations. Acknowledgments. The author thanks H. Yamazawa and M. Hayano for their helpful advice and discussions. He also thanks H. Tasaki for his critical reading of this manuscript and acknowledges T. Kishii, Y. Kuzuha, and all members at the Atmospheric and Hydrospheric Science Division, National Research Institute of Earth Science and Disaster Prevention (NIED). This research were performed in part as joint research between JAERI and NIED, and the experiments in the wind tunnel were performed at NIED. REFERENCES Brutsaert, W., 1982: Evaporation into the Atmosphere. Kluwer Academic, 309 pp. Komatsu, T. S., 2001: Evaporation speed of water from various soil surfaces under calm state. J. Phys. Soc. Japan, 70, Kondo, J., N. Saigusa, and T. Sato, 1990: A parameterization of evaporation from bare soil surfaces. J. Appl. Meteor., 29, Mahfouf, J. F., and J. Noilhan, 1991: Comparative study of various formulations of evaporation from bare soil using in situ data. J. Appl. Meteor., 30,