Duplicate study sites “A” and “B” were set in a meadow field of 2.8 [ha] (350 × 80 [m]), called the ‘3-1 field’ (36˚29'23"N, 139˚59'14"E), located in the Utsunomiya University Farm in Moka city, Tochigi, Japan. The study period was from May 2018 to April 2019. The grasses and forbs grown in the meadow in the study period were orchard grass (Dactylis glomerata L. cv. Natsumidori), tall fescue (Festuca arundinacea Schreb. cv. Hokuryo), hybrid ryegrass (Lolium hybridum Hausskn. cv. Tetrelite II), and red clover (Trifolium pratense L. cv. Makimidori).

The air temperatures were measured at each site at the heights of 0.7 [m] and 1.8 [m] with 30-minutes intervals by temperature loggers (HOBO U23 Pro v2 External temperature/relative humidity data logger “U23-002A”; Onset Computer Corp; Bourne, MA, USA) enveloped in radiation shields (Solar radiation shield “RS1”; Onset Computer Corp; Bourne, MA, USA). A data set of wind speed was utilized, which had been measured at 10 m in height at Moka weather station (36˚28'36"N, 139˚59'12"E; 1.4 kilometers south of the study site) ( Japan Meteorological Agency, 2025 ).

The aerodynamic conductance K [m s⁻¹] was evaluated by the air temperatures T_a1 [K] and T_a2 [K] measured at the two heights z_t1 = 0.7 [m] and z_t2 = 1.8 [m], respectively, with the wind speed u_m [m s⁻¹] measured at z_u = 10 [m]. Since the combinations of T_a1 [K], T_a2 [K], and u_m [m s⁻¹] were obtained every 30 minutes, values of K [m s⁻¹] were evaluated with 30-minute intervals by solving a system of nonlinear equations that includes the models of the wind- and temperature-profiles in vertical direction. The wind and temperature profiles were formulated by applying the Monin-Obukhov similarity theory ( Monin & Obukhov, 1954 ) as stated below.

According to Monin-Obukhov’s theory, the vertical profile of wind speed u [m s⁻¹] above a land surface can be expressed as follows:

$u=\frac{u_{\ast }}{k}\left(\ln \left(\frac{z-d+z_m}{z_m}\right)-{\psi }_m\right)$ (1)

where u_* [m s⁻¹] is the friction velocity, k [−] = 0.4 is the von Karman constant, z [m] is a vertical location at which u [m s⁻¹] is given, d [m] and z_m [m] are the height of zero plane displacement and the roughness length of momentum of the surface. And ψ_m [−] is the correction factor of atmospheric stability for momentum transfer.

Linear relations between d [m] and z_m [m] had been found on many types of air-surface interactions, while the height of plants h [m] covering the land had also been related to d [m] and z_m [m]. Maki (1975) thoroughly reviewed these relations in 16 papers and represented them with the following expressions:

$d=a_{d}h$ (2)

$z_m=a_{zm}h$ (3)

where the slopes a_d and a_zm averaged 0.717 and 0.106 among the reviewed values with their standard deviations of 0.080 and 0.045, respectively. This study adopted a_d = 0.717 and a_zm = 0.106, since these values allowed to approximate general trends observed for the dense canopies with all the variety of conditions ( Maki, 1975 ). The time series of h [m] was approximated as a sequence of linear functions of time based on such observations that the first, second, and third cuttings of the crops in the study field were done on 5/24, 7/25, and 9/24 in 2018, and the height of plants returned to about 0.1 [m] on each cutting date, while it reached around 0.5 [m] during each growing period. On these settings, d [m] and z_m [m] were regarded as known variables that were specified for a given plant height h [m].

The ψ_m [−] in Equation (1) is defined by using the universal similarity function for momentum transfer φ_m [−] as follows:

${\psi }_m\left(x\right)={\psi }_m\left(x_0\right)+{\displaystyle {\int }_{x_0}^x\frac{1-{\phi }_m\left(x\right)}{x}\text{d}x}$ (4)

where x [−] is an auxiliary variable defined as the ratio of z − d [m] to the Monin-Obukhov length L [m] ( Monin & Obukhov, 1954 ), and the integrand φ_m [−] was of one of the existing models as below ( Paulson, 1970 ; Dyer, 1974 ):

${\phi }_m\left(x\right)=\{\begin{array}{l}1+5x\left(L\ge 0\right)\\ {\left(1-16x\right)}^{-0.25}\left(L<0\right)\end{array}$ (5)

And, by assuming that x₀ [−] in Equation (4) is negligibly small, ψ_m [−] is formulated as below:

${\psi }_m\left(x\right)\approx \{\begin{array}{l}-5x\left(L\ge 0\right)\\ \ln f_m\left(y\right)-2{\tan }^{-1}y+\frac{\pi }{2}\left(L<0\right)\end{array}$ (6)

where $y={\left(1-16x\right)}^{0.25}$ and $f_m\left(y\right)={\left(\left(1+y\right)/2\right)}^2\left(1+y^2\right)/2$.

Equations (4) through (6) show that a value of ψ_m [−] is identified when values of d [m], L [m], and z [m] are given. As mentioned above, d [m] and z_m [m] in this study were known variables. Thus, when a pair of u = u_m [m s⁻¹] and z = z_u [m] are given as measured values, unknown variables in Equation (1) are u_* [m s⁻¹] and L [m]. And, as explained below, L [m] can be expressed as a function of the sensible heat flux from the surface to the atmosphere H [W m⁻²], the friction velocity u_* [m s⁻¹], the air temperature T_a [K], and the atmospheric pressure p_a [Pa]. This means that H [W m⁻²] and u_* [m s⁻¹] are unknown variables for L [m] under a given weather condition and, in turn, that the unknown variables in Equation (1) are H [W m⁻²] and u_* [m s⁻¹] under the given weather condition.

The Monin-Obukhov length L [m] is formulated as below ( Monin & Obukhov, 1954 ):

$L=-{\left(\frac{\tau }{{\rho }_a}\right)}^{\frac{3}{2}}\frac{C_h{T}_{ave}}{kgH}$ (7)

where τ [N s m⁻² s⁻¹] is the momentum flux, ρ_a [kg m⁻³] is the density of air, g [m s⁻²] is the acceleration of the earth’s gravity, H [W m⁻²] is the sensible heat flux from the surface to the atmosphere, T_ave [K] is the average temperature in the air layer of interest, and C_h [J m⁻³ K⁻¹] is the volumetric heat capacity of air. Equation (7) was derived with the assumption that spatial variation in temperature in the air layer is small compared with T_ave [K]. Because the u_* [m s⁻¹] in Equation (1) is defined as the square root of the ratio of τ [N s m⁻² s⁻¹] to ρ_a [kg m⁻³] ( Monin & Obukhov, 1954 ), Equation (7) can be rearranged as below:

$L=-u_{\ast }^3\frac{C_h{T}_{ave}}{kgH}$ (8)

The C_h [J m⁻³ K⁻¹] in Equation (8) is defined as the density of air ρ_a [kg m⁻³] multiplied by the specific heat of air c_p [J kg⁻¹ K⁻¹]:

$C_h={\rho }_a{c}_p$ (9)

The ρ_a [kg m⁻³] was approximately quantified by using the ideal gas law as follows:

${\rho }_a=\frac{p_a{M}_a}{R{T}_a}$ (10)

where T_a [K] and p_a [Pa] are air temperature and atmospheric pressure, M_a [kg mol⁻¹] is the molar weight of the air (≒ 0.028 966 [kg mol⁻¹]), R [J mol⁻¹ K⁻¹] is the ideal gas constant (= 8.314 [J mol⁻¹ K⁻¹]). The c_p [J kg⁻¹ K⁻¹], too, was described as a function of T_a [K] and p_a [Pa] by applying the multivariate regression to the data set tabulated in Lemmon (2010) :

$c_p=a_0+a_1{T}_a+a_2{p}_a$ (11)

where a₀, a₁, and a₂ are 1120.42 [J kg⁻¹ K⁻¹], −0.369 05 [J kg⁻¹ K⁻²], and 15.5494 × 10⁻⁶ [J kg⁻¹ K⁻¹ Pa⁻¹], respectively. The coefficient of determination R² of Equation (11) was 0.909 53. As shown in Equations (8) through (11), the L [m] is a function of H [W m⁻²], u_* [m s⁻¹], T_a [K], and p_a [Pa], meaning that H [W m⁻²] and u_* [m s⁻¹] are unknown variables for L [m] under a given weather condition.

In summation, Equation (1) should hold true when a proper pair of H [W m⁻²] and u_* [m s⁻¹] is found when a set of relevant measured values is given.

As Equation (1) is formulated, the vertical profile of temperature T_a [K] is expressed as follows:

$T_a=T_s-\frac{H}{k{C}_h{u}_{\ast }}\left(\ln \frac{z-d+z_h}{z_h}-{\psi }_h\right)$ (12)

where T_s [K] is the temperature at z = d [m], the z_h [m] is the roughness length of heat on the land surface. Some linear relations between z_h [m] and z_m [m] had been proposed ( Garratt & Francey, 1978 ; Shaw & Pereira, 1982 ), and the ratio of z_h [m] to z_m [m] is likely to take a value between 0.1 and 0.2. Among the linear relations, Garratt & Francy (1978) found such a relation as:

$\ln \frac{z_m}{z_h}\approx 2$ (13)

by plotting ln(z_m/z_h) against the roughness Reynolds number Re_* (= u_*z_m/ν, where ν [m² s⁻¹] is the kinematic viscosity of air) in the range 10 < Re_* < 10⁵ for various types of surfaces, including bare soil, grass and crops, shrubs, and forested area.

The ψ_h [−], the stability correction factor for heat transfer, was also introduced. The derivation process of ψ_h [−] is like that of ψ_m [−] as follows:

${\psi }_h\left(x\right)={\psi }_h\left(x_0\right)+{\displaystyle {\int }_{x_0}^x\frac{1-{\phi }_h\left(x\right)}{x}\text{d}x}$ (14)

The definition of the auxiliary parameter x [−] in Equation (14) is the same as that in Equation (4). The universal similarity function for heat transfer φ_h [−] is expressed in the same manner as φ_m [−] as follows ( Dyer, 1974 ):

${\phi }_h\left(x\right)=\{\begin{array}{l}1+5x\left(L\ge 0\right)\\ {\left(1-16x\right)}^{-0.5}\left(L<0\right)\end{array}$ (15)

And, for a very small x₀ [−] in Equation (14), the resultant ψ_h [−] can be calculated as below:

${\psi }_h\left(x\right)\approx \{\begin{array}{l}-5x\left(L\ge 0\right)\\ \ln f_h\left(y\right)\left(L<0\right)\end{array}$ (16)

where $f_h\left(y\right)={\left(\left(1+y^2\right)/2\right)}^2$, and $y={\left(1-16x\right)}^{0.25}$ as defined in Equation (6).

When the variables in Equation (12) are classified into variables with given values and unknown variables in the same way as those in Equation (1), it is noticed that Equation (12) should hold true when a proper set of H [W m⁻²], T_s [K] and u_* [m s⁻¹] is identified for a series of measured values.

As formulated above, Equations (1) and (12) can coincidentally hold true when an adequate set of H [W m⁻²], T_s [K], and u_* [m s⁻¹] are input for a given set of relevant measured values. This means that three equations are required for identifying the unknowns H [W m⁻²], T_s [K], and u_* [m s⁻¹]. Therefore, this study used the following three equations:

$\{\begin{array}{l}{u}_m=\frac{u_{*}}{k}\left(\ln \left(\frac{z_u-d+z_m}{z_m}\right)-{\psi }_m\right)\\ T_{a1}=T_s-\frac{H}{k{C}_h{u}_{*}}\left(\ln \frac{z_{t1}-d+z_h}{z_h}-{\psi }_{h1}\right)\\ T_{a2}=T_s-\frac{H}{k{C}_h{u}_{*}}\left(\ln \frac{z_{t2}-d+z_h}{z_h}-{\psi }_{h2}\right)\end{array}$ (17)

Because Equation (17) is a system of nonlinear equations, the Newton-Raphson iterative scheme was adopted to solve it.

Finally, every time a set of H [W m⁻²] and T_s [K] was determined, the aerodynamic conductance K [m s⁻¹] was obtained by using the following expression:

$K=\frac{H}{C_h\left(T_s-T_a\right)}$ (18)

Figure 1 shows the time series of air temperatures observed every 30 minutes in the two sites. T_a1 [K] and T_a2 [K] denote the air temperatures monitored at 0.7 [m] and 1.8 [m] in height at each site, respectively. The sensor at 0.7 [m] in height in site A had failed to record the data due to a battery trouble from 2018/8/19 19:00 to 2018/10/15 12:00 and from 2019/4/24 5:00 to 2019/4/30 23:30. The portion of the data obtained at 0.7 [m] in height in site B had also been lost from 2018/10/28 4:30 to 2018/12/28 13:00.

The curves together showed annual peaks around 300 [K] in July and found bottoms around 275 [K] in January. And the daily ranges were roughly 10 to 20 degrees, being larger in winter than in summer. These four temperature records showed almost the same patterns, and the features of the four data sets had no clear distinctions among them. In fact, the diurnal trends of the four data sets were almost the same as each other ( Figure 2 ), in which the daily maxima and minima appeared around 15:00 and the sunrise, respectively, regardless of season, and the daily ranges narrowed in summer and widened in winter.

However, the temperature difference T_a2 − T_a1 [K] clearly showed that the two temperatures had literally differed from each other ( Figure 3(i) ), fluctuating between −1 and +2 degrees. It took smaller values in warmer seasons, while large upward temperature gradients were found mainly in winter. And, their monthly-averaged diurnal patterns ( Figure 3(ii) ) showed that T_a2 − T_a1 [K] became positive mainly at night, causing the air on the land surface to be stable, suggesting that the size of the aerodynamic conductance reduced during the nights. On the other hand, the negative values were observed during the daytime for every month, with a peak one hour before the daily maximum temperature. These implied that upward sensible heat flux appeared solely during daytime, while almost no sensible heat flux was found during nighttime due to a very stable atmospheric condition.

The wind speed fluctuated primarily between 0 and 5 [m s⁻¹] regardless of season, and occasionally took more than 5 [m s⁻¹] during the study period ( Figure 4(i) ). However, it became strong from spring to early summer and weakened from late autumn to winter, according to its monthly-averaged diurnal patterns ( Figure 4(ii) ). In addition, unlike the trends found on the air temperature measurements, the peak time of the wind speed had moved forward from around 16:00 in spring to around 13:00 in winter.

Figure 5 shows the time series of the sensible heat flux H [W m⁻²] upward positive, the temperature at the height of zero plane displacement T_s [K], the friction velocity u_* [m s⁻¹], and the aerodynamic conductance K [m s⁻¹], which were evaluated by solving Equations (17) and (18) for the measured data sets on Figure 1 and Figure 4(i) .

The values of H [W m⁻²] ( Figure 5(i) ) ranged from −9.8 to 262.6 [W m⁻²] for the site A, while those for the site B fell between −9.7 and 243.7 [W m⁻²]. The temporal averages of H [W m⁻²] were 9.1 and 11.8 [W m⁻²] with the standard deviations of 24.5 and 26.2 [W m⁻²] in sites A and B, respectively. The median of H [W m⁻²] for each site was 0.0 [W m⁻²], meaning that the number of the data points with positive H [W m⁻²] was almost equivalent to that with negative H [W m⁻²] and, thus, large positive values in H [W m⁻²] occurred very occasionally, though many data points with H > 50 [W m⁻²] were found.

The time series of T_s [K] ( Figure 5(ii) ) showed almost the same patterns of annual cycle as the measured air temperatures T_a1 [K] and T_a2 [K] in Figure 1 . However, the daily ranges of T_s [K] were larger than those of T_a1 [K] and T_a2 [K] throughout the study period, certainly driving the sensible heat flux H [W m⁻²] in either upward or downward direction by making the temperature difference in Equation (17) significant.

The friction velocity u_* [m s⁻¹] ( Figure 5(iii) ) fluctuated between 0.00 and 0.52 [m s⁻¹] in site A and 0.00 and 0.69 [m s⁻¹] in site B in the study period. The temporal averages of u_* [m s⁻¹] were 0.08 and 0.09 [m s⁻¹] with the standard deviations of 0.09 and 0.09 [m s⁻¹] in sites A and B, respectively. The medians of 0.04 and 0.07 [m s⁻¹] for sites A and B indicated that the small number of large values pulled up the averages.

The aerodynamic conductance K [m s⁻¹] ( Figure 5(iv) ) fluctuated mainly between 0.00 and 0.02 [m s⁻¹] and rarely exceeded 0.03 [m s⁻¹] during the study period, with the annual averages of 0.0051 and 0.0064 [m s⁻¹] in sites A and B, respectively. These values were one-digit smaller than those found on forests of juniper ( Duman et al., 2021 ) or of spruce ( Hurtalova et al., 2005 ), and about half of those found on tall aquatic plants ( Hemes et al., 2018 ), suggesting the positive relation between h [m] and K [m s⁻¹].

The diurnal patterns of H [W m⁻²], T_s [K], u_* [m s⁻¹], and K [m s⁻¹] were displayed in Figure 6 . All the parameters rose after sunrise and fell toward sunset every month, and the diurnal cycle changed gradually in height and length from month to month. As a result, the diurnal patterns of the four parameters showed some seasonal cycles.

Particularly, the aerodynamic conductance K [m s⁻¹] ( Figure 6(iv) ) rose only in daytime and stayed below a daily average with almost a constant value during night. Every month, the value of K [m s⁻¹] sharply increased just after sunrise, and then, the rate of increase in K [m s⁻¹] became moderate until K [m s⁻¹] reached its daily peak. After that, K [m s⁻¹] lowered steeply by the time it got dark. Thus, it can be said that the times of sunrise and sunset define the seasonal contraction and expansion of the period when K [m s⁻¹] effectively promotes the heat exchange between the land and the atmosphere.

The daytime K [m s⁻¹] showed such a seasonality that it grew in spring, slightly declined in summer, and steeply reduced through autumn to winter, while the nighttime K [m s⁻¹] was larger in warmer seasons and got close to zero in winter. In this annual cycle, the daily maximum of K [m s⁻¹] peaked in April, reaching beyond 0.015 [m s⁻¹], and started dropping in the warmest season, reduced through autumn, and bottomed out amid winter. The time of the daily maximum K [m s⁻¹] also shifted from season to season, which was found around 16:00 in summer and moved forward in winter.

The diurnal and seasonal features of K [m s⁻¹] resembled those of u_* [m s⁻¹] as closely as the shapes of the curves on Figure 6(iii) were like those on Figure 6(iv) , while they were different from the diurnal and seasonal patterns of T_a [K] ( Figure 2 ) and T_s [K] ( Figure 6(ii) ). These evaluations suggested that u_* [m s⁻¹] or the wind speed u [m s⁻¹] is the principal environmental variable for defining K [m s⁻¹], though the air temperatures surely affect the size of K [m s⁻¹] as the theory behind Equations (17) and (18) tells.

In fact, the comparisons of the values of u_* [m s⁻¹] on Figure 6(iii) with those of K [m s⁻¹] on Figure 6(iv) gave a linear relationship expressed as below:

$K=b{u}_{\ast }$ (19)

The value of the proportionality coefficient b [−] obtained by fitting Equation (19) to the whole-year data set of u_*-K relation was 0.075 for the site A, while that for the site B was 0.077, with the coefficient of determination R² = 0.989 for any of the two sites. When b [−] was determined for an every-month data set, it varied with such an annual cycle that it took its minimum of 0.065 in January and its maximum of 0.088 in May, with R² being consistently more than 0.999. These features implied that once a pair of u_* [m s⁻¹] and K [m s⁻¹] is obtained, a value of K [m s⁻¹] at any other time can be estimated solely by evaluating u_* [m s⁻¹] through wind profile measurements, and monitoring air temperatures can no longer be required. Of course, a pair of u_* [m s⁻¹] and K [m s⁻¹] obtained once a month can give more precise estimations of K [m s⁻¹] in each month.

Since K [m s⁻¹] and u_* [m s⁻¹] seemed to be cognate parameters derived from the wind speed u [m s⁻¹], the relation between K [m s⁻¹] and u [m s⁻¹] was also probed ( Figure 7 ). Every curve of the u-K relation was made for every one-month data set by matching the values of K [m s⁻¹] on Figure 6(iv) against the values of u [m s⁻¹] on Figure 4(ii) in order of the time of day. Any of the u-K relations formed a diurnal loop with a clockwise trajectory with time on the graph. And the loop size varied seasonally, reflecting the seasonality of the sizes in u [m s⁻¹] and K [m s⁻¹]. In many of the loops, a value of K [m s⁻¹] in the morning often became two times larger or more than that in the evening for a given u [m s⁻¹].

The theory that gives the wind- and temperature-profiles of Equations (1) and (12) indicates that K [m s⁻¹] can be expressed as a multiple of u [m s⁻¹] mediated by u_* [m s⁻¹]. However, as Equations (6) and (16) express, the relation between u [m s⁻¹] and K [m s⁻¹] is not unique and, instead, is defined by two different sets of expressions depending on whether the atmosphere is stable or not and on the extent of the atmospheric stability measured by the Monin-Obukhov length L [m]. For instance, the times of day at which u [m s⁻¹] took 1.5 [m s⁻¹] in July were around 8:00 and around 21:00 ( Figure 4(ii) ), while the vertical gradient of air temperature in July was negative at 8:00 and reversed its direction before 21:00 ( Figure 3(ii) ), resulting in two different values in K [m s⁻¹] for the single u [m s⁻¹]. In addition, it can be found on Figure 3(ii) and Figure 4(ii) that the time of daily maximum u [m s⁻¹] did not coincide with the time of daily maximum T_a2 − T_a1 [K], meaning that the diurnal pattern in T_a2 − T_a1 [K] slightly differed in temporal phase from that in u [m s⁻¹] in any month. This feature can also explain the cause of the loop-like behaviors in the u-K relations. Consequently, it can be said that the loop-like behavior of the u-K relation owes to the temporal phase lag between the diurnal peak of u [m s⁻¹] and the vertical temperature gradient dT_a/dz [K m⁻¹]. The loop-like behavior of the u-K relation may be called a hysteretic behavior, because the u-K relations differed between the processes with dK/du > 0 and with dK/du < 0. However, since there was such a true cause for the two different u-K paths as the change in the sign of dT_a/dz, the expression “a hysteretic behavior” can be qualified with the adjective “apparent”. In any case, it holds true that to know the diurnal patterns of aerodynamic conductance requires the consideration of the effects of the atmospheric stability, instead of simply assuming the atmospheric condition is neutral.