Sensible Heat and Latent Heat Flux Estimates in a Tall and Dense Forest Canopy under Unstable Conditions

Abstract

A method to estimate the sensible heat flux (H) for unstable atmospheric condition requiring measurements taken in half-hourly basis as input and involving the land surface temperature (LST), H_LST, was tested over a tall and dense aspen stand. The method avoids the need to estimate the zero-plane displacement and the roughness length for momentum. The net radiation (Rn) and the latent heat flux (λE) dominated the surface energy balance (SEB). Therefore, λE was estimated applying the residual method using H_LST as input, λE_R-LST. The sum of H and λE determined with the eddy covariance (EC) method led to a surface energy imbalance of 20% Rn. Thus, the reference taken for the comparisons were determined forcing the SEB using the EC Bowen ratio (BREB method). For clear sky days, H_LST performed close to H_BREB. Therefore, it showed potential in the framework of remote sensing because the input requirements are similar to current methods widely used. For cloudy days, H_LST scattered H_BREB and nearly matched the accumulated sensible hear flux. Regardless of the time basis and cloudiness, λE_R-LST was close to λE_BREB. For all the data, both H_LST and λE_R-LST were not biased and showed, respectively, a mean absolute relative error of 24.5% and 12.5% and an index of agreement of 68.5% and 80%.

1. Introduction

In non-dry climates, the evapotranspiration or latent heat flux (λE) is one of the dominant components of both the land-surface water and energy budgets. In dry and semi-arid climates λE is a critical component of living things. Therefore, among other issues, monitoring evapotranspiration rates is of interest to understand ecosystem sustainability/vulnerability regardless of the climate [1,2,3,4,5]. Different methodologies are available to determine λE and micrometeorological methods often involve the surface energy balance and similarity theory [6,7,8,9,10]. The latter methods require the identification of the dominant terms in the surface energy balance. On a half-hourly basis, for instance, the energy storage in tall and dense canopies is significant [11,12,13]. When the dominant surface fluxes can be determined using measurements taken at low frequency, the method becomes friendlier because the instrumentation involved is more robust and affordable and requires minor power supply. In addition, most similarity-based formulations allow involving remote sensing products. Hence, provided they are used on clear-sky days, they can be applied at non-local scales [14,15,16,17,18,19,20].

During the day, the sensible heat flux (H) on a half-hourly basis is a dominant term in the surface energy balance. Thus, the aim of this paper was to continue research on testing a recent method to estimate the H under unstable cases which involves measurements taken at low frequency including the land surface temperature (LST), H_LST [21]. Currently, H_LST has been tested over agricultural landscapes and mountain meadows, showing potential in the framework of remote sensing [22,23,24]. The good performance obtained in different scenarios aimed here to adapt H_LST for a tall and dense canopy. Here, the scenario selected to test H_LST was a Boreal Forest at a site which allowed estimation of the latent heat flux (LE) as a residual of the surface energy balance. In this study the data analysis was mainly based on the unstable condition and the criterion for the selection of unstable data was through the sign of sensible heat flux, because under stable condition the measured surface fluxes are expected to have major errors using the eddy covariance method [25,26]. Therefore, it was omitted with the aim of excluding errors in the estimated H_SR. The testing included clear and cloudy days.

2. Materials

The campaign was carried out in a forest located in Prince Albert National Park (Saskatchewan, Canada; 53.629° N, 106.200° W, 600.63 m asl) in 1996. A brief description of the campaign accessed on (28 December 2021) (http:///boreas/TF/tf01tflx/comp/TF01_Tower_Flux.txt; Black, 2000) is the following. The forest consisted in an even-aged 70-year-old aspen stand with a mean height of 21.5 m. Hazelnut (2.0 m tall) dominated the understory. The forest had a minimum fetch of 3 km extending on a gentle rolling topography and the aspen canopy closure averaged 89% from June to August. For July and August, the mean leaf area index was 5.4 m²m⁻² and 5.25 m²m⁻², respectively [27] and the following measurements on a half-hourly basis were obtained from the dataset tf01_tower_flux.dat (freely downloadable). The soil heat flux was sampled at 3 cm below the ground at seven points. The mean soil temperature and the volumetric fraction of water were placed in a sublayer (3 cm thick) above the soil heat flux plates. The temperature and pressure of the air, the wind speed, the friction velocity and the sensible heat and latent heat fluxes (determined using the eddy covariance (EC) system) were measured at 39 m above the ground. The wind speed was measured at the canopy top and the downwelling and upwelling shortwave and longwave radiation were measured at 33 m above the ground.

3. Theory and Methods

3.1. Monin–Obukhov Similarity Theory (MOST)

The atmospheric surface layer is also known as the constant flux layer because fluxes are nearly constant with height, with variations of less than 10 % under steady-state and horizontal homogeneous conditions [28]. Monin and Obukhov (1954) introduced two scaling parameters, which are independent of height in the surface layer for the structure of turbulence [29]. The friction velocity $(u_{*})$ expressed as Equation (1) is [30]:

$$u_{*}={\left[{\left(\overline{u^{\prime }{w}^{\prime }}\right)}^2+{\left(\overline{v^{\prime }{w}^{\prime }}\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$$

(1)

where u′, v′ and w′ are the fluctuations from the mean three-dimensional windspeed of u, v and w respectively. The second scaling parameter, which is a function of momentum and sensible heat fluxes, is the Obukhov length, (L_o) expressed as Equation (2):

$$L_o=-\frac{\rho c_p{T}_a{u}_{*}^3}{kgH}$$

(2)

where ρ is the density of air, c_p the specific heat capacity of air at constant pressure, T_a the absolute air temperature, k the von Karman constant and g the acceleration due to gravity. The negative sign is the sign of the sensible heat flux corresponding to the vertical air temperature difference between two heights. The Obukhov length can be interpreted as the height, above (d_o + z), for which free convection dominates [31,32]. Another, similar key scaling parameter which determines the structure of turbulence is the turbulent temperature scale (T_a) expressed as Equation (3) [30,31]:

$$T_a=\frac{H}{\rho c_p{u}_{*}}$$

(3)

On the other hand, turbulent characteristics depend upon a dimensionless stability parameter (ζ) expressed as Equation (4):

$$\zeta =\frac{z}{L}=\left(z-d\right)/L_o$$

(4)

where z is the height above the surface and d is the zero-plane displacement height. The dimensionless stability parameter ζ is a theoretical indicator of the atmospheric stability.

3.2. Sensible Heat Flux

The summary of the method, fully described in Castellví and González-Dugo (2021) [24], to estimate the sensible heat flux (half-hourly basis) is as follows. It combines similarity and surface renewal formulation through Equations (5)–(7):

$$H=\rho C_p\sqrt{\frac{kz{u}_{*}{\Phi }_h^{-1}}{\pi \tau }}A$$

(5)

$$\frac{1}{\tau u_{*}}=\frac{1}{2h_c}$$

(6)

$$LST-\left(T+b\right)=s A$$

where

$$s=\frac{{\lambda }_h }{2k}\frac{ Z}{h_c} \left(\ln \left(\frac{z}{z_{0m}}\right)+2\right)\hspace{1em}$$

(7)

In Equation (5), z is the measurement height above the zero-plane displacement (d), thus z = Z − d where Z is measurement height above the ground, Φ_h is the stability correction function for heat transfer, λ and A, namely ramp period and amplitude, are two parameters that identify a coherent motion in the surface sublayer above the canopy top. By defining a macro parcel of air as a parcel of air which volume, per unit area, may cover all the sources of scalar at the surface, a coherent motion (regular eddy motion of macro parcels of air) is carried out by a coherent structure which is defined as an eddy capable to impose some organization in the turbulence in the surface layer. Thus, the ramp period accounts for the time that a macro parcel of air remained in contact with the surface and the ramp amplitude for the net temperature increase (for an unstable case) while it remained in contact with the surface [33,34,35].

Coherent structures near the canopy are mostly driven by shear rather than by buoyancy [36]. Thus, Chen et al. (1997b) proposed to relate the ramp frequency and u_∗/z [34]. Approaching the shear at the canopy top by u_∗/h_c (h_c is the canopy height), the semi-empirical Equation (6) was proven reliable for a variety of canopies. Assuming that the maximum temperature to be reached by the parcel of air is LST, Equation (7) proposes a linear relationship between quantities (LST–T) and A where T is the potential temperature of the air measured at a reference height above the canopy [21,22]. In Equation (7), b is a coefficient (hereafter referred to as offset) that corrects for the fact that (LST–T) may not necessarily follow the adiabatic lapse rate at neutral conditions (i.e., case where A = 0 K) and z_0m is the roughness length for momentum. Because the factor [8k/π]^1/2 approaches to one, combining Equations (5)–(7) the sensible heat flux can be estimated above the canopy as Equation (8):

$$H_{LST}=\rho C_p\frac{{\left[z{h}_c{\mathsf{\Phi }}_h^{-1}\right]}^{1/2}}{Z}\frac{k{u}_{*}\left(LST-T-b\right)}{\left(\ln \left(\frac{z}{z_{0m}}\right)+2\right)}$$

(8)

3.3. Latent Heat Flux

The latent heat flux can be estimated as a residual of the surface energy balance (the residual method) as Equation (9):

λE_R = Rn − G − S − H

(9)

where Rn is the net radiation, G is the soil heat flux and S is the energy storage (per unit time and surface) below the measurement height. The net radiation (Rn) was calculated using four radiative components. The soil heat flux (Wm⁻²) and the total storage in a column were estimated using Equations (10) and (11), respectively [37]:

$$G=\overline{G{z}_s}+Cszs\raisebox{1ex}{$\Delta Ts$}\!\left/ \!\raisebox{-1ex}{$\Delta t$}\right.$$

(10)

where $C_s=0.3+4.18{\theta }_w$.

S = 44.5∆ + 1.66

(11)

In Equation (10), overbar denotes spatial average, z_s is the depth at which the soil heat flux plates were collocated and the second term in the right hand denotes heat storage in the soil above z_s where Cs is the soil heat capacity (MJm⁻³K⁻¹) and θ_w is the soil water content. In Equation (10), ΔT is the change of temperature of the air at the reference height on a half-hourly basis. Hereafter, λE_R determined using as input H_LST and H_EC is denoted as λE_R-LST and λE_R-EC, respectively.

3.4. Solving the Sensible and Latent Heat Fluxes

For tall canopies, the zero-plane displacement and the roughness length for momentum are related to each other and depend on the stability parameter [15,38,39]. It was accounted here and Appendix A describes an iterative procedure to simultaneously estimate d, z_0m, u_∗, H_LST and λE_R-LST.

The land surface temperature was retrieved from the Stefan–Boltzman law and the effective emissivity of the surface (ε₀) was estimated as [40]; ε₀ = ε_v(1 − f_s) + ε_sf_s(1.74f_s − 0.74) + 1.7372f_s(1 − f_s) where f_s is the fraction cover and ε_v and ε_s are representative emissivities for the canopy and understories. During the experiment, the fraction cover remained fairly constant (f_s = 0.89) and the emissivities for the aspen canopy and understories (hazelnut and soil) were set to 0.97 and 0.96, respectively [41]. Thus, ε₀ was set to 0.97. The offset in Equation (7) was determined as in Castellví et al. (2016) [22]. Thus, the coefficient b at sunrise and sunset was determined setting A = 0 K, at noon the offset was set to zero and for the rest of the day it was calculated using a linear relationship from sunrise to noon and from noon to sunset.

3.5. The Reference, Datasets and Performance Evaluation

The sum of the EC sensible heat flux and latent heat flux was significantly smaller that the available net surface energy (shown in Section 4.1) which has implications on how H_EC and λE_EC should be interpreted. Kidstone et al. (2010) conducted experiments over two different land surfaces and implications for CO₂ fluxes measurement using eddy covariance method. Results suggested that spatio-temporal distributions of total surface fluxes were in good agreement and difference between the relative magnitudes of the fluxes for several investigated energy balance closure classes was observed [42]. Noman et al. (2019) estimated surface fluxes including sensible and latent heat fluxes along variable fetch using flux variance technique, linear regression was performed between the sum of turbulent fluxes and the available energy fluxes and energy balance closure was well behaved with linear regression of R² = 0.83 [43]. J. L. Chavez et al. (2009) evaluated the performance of eddy covariance over cotton with large weighing lysimeters; results suggested the estimated surface fluxes were underestimated with average error of about 30%. Energy balance closure was 73.2–78.0% for daytime fluxes [44]. R. Ding et al. (2019) conducted experiments in maize field of northwest China for evaluating performance of eddy covariance using lysimeters. Energy balance ratio was 0.84 for stable conditions, indicating that lack of energy balance closure occurred and estimated ET was adjusted by Bowen ratio forced closure method [45]. Noman et al. investigated performance of flux variance method at different heights and results suggested relatively better agreement between the energy balance closure with slope of regression 0.55 and RMSE of 33.95 Wm⁻² [46]. X. Wang et al. (2010) performed experiments in maize fields for the estimation of energy fluxes and evapotranspiration in arid area with shallow groundwater. Estimated latent heat flux was adjusted using Bowen ratio with linear regression of R² = 0.94 and slope of regression was 0.732 and 0.634 respectively for half-hourly data in 2017 and 2018. The water balance results showed average water table depth of 1.52 and 1.76 m for the growing season of 2017 and 2018 [47]. Here, the EC Bowen ratio (β_EC) was used to refine the EC sensible heat and latent heat fluxes, Equation (12), which were taken as a reference [27]:

$$H_{BREB}=\frac{\left(Rn-G-S\right)}{\left(1+{\beta }_{EC}\right)}{\beta }_{EC}$$

(12)

A dataset was formed excluding rainy events [25]. July only included cloudy days. August had 11 clear sky days and the rest were cloudy. Datasets were formed with samples having different friction velocity thresholds (up to 0.6 m/s).

The performance of the flux estimates, Fest, was analyzed determining the sum of the estimates over the sum of the reference (D), the mean absolute difference or error (MAE) and the index of agreement (IA) expressed as below [48]:

$$D=\frac{{\displaystyle \sum _{i=1}^N{F}_{est}{}_i}}{{\displaystyle \sum _{i=1}^N{F}_i}}MAE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|F_{est}{}_i-F_i\right|} \mathrm{and} IA=1-\frac{1}{2}\frac{{\displaystyle \sum _{i=1}^N\left|F_{est}{}_i-F_i\right|}}{{\displaystyle \sum _{i=1}^N\left|F_i-\overline{F}\right|}}$$

(13)

where F is the flux taken as reference, N is the total number of samples and overbar denotes average. The expression given in Equation (12) is valid when ${\displaystyle \sum }_{i=1}^N\left|F_{esti}F\right|\le 2{\displaystyle \sum }_{i=1}^N\left|F_i-\overline{F}\right|$ [48] and, here, the full expression to calculate IA was omitted because the datasets formed accomplished this constrain. The performance of H_LST (determined as proposed in the Appendix A) was compared with the estimates obtained by assuming a fixed value for the zero-plane displacement (selected by rule of thumb in a range of 0.6–0.85 times the canopy height) and, in addition, by assuming that the offset may be neglected.

4. Results

4.1. Surface Energy Balance Using H_EC and λE_EC

Figure 1 shows the accumulated H_EC, λE_EC and (H_EC + λE_EC + G + S) standardized by the net radiation for different friction velocity thresholds. Regardless of the month, it is shown that the weight of the latent heat flux in the surface energy balance was much higher than of the sensible heat flux. The energy partitioning in Figure 1 showed the need to refine the EC fluxes. For all the data, about 20% of R_n was unexplained by (H_EC + λE_EC + G + S). The surface energy balance improved as the friction velocity threshold increased. A good closure (of about 10%) was obtained for a short dataset, for friction velocities higher than 0.5 m/s.

Table 1. EC and LST sensible heat and latent heat flux estimates.

Dataset	July (N = 282)			August (N = 521)
Statistics	D	MAE a	IA	D	MAE a	IA
Method		(Wm−2%)	(%)		(Wm−2%)	(%)
HEC	0.87	28  22	74	0.82	32  23	69
λEEC	0.77	79  58	57	0.75	80  57	48
λER-EC	1.07	28  11	84	1.09	32  12	78
HLST	1.00	36  27	67	1.04	30  22	70
λER-LST	1.00	36  14	80	0.98	30  11	80

^a MAE expressed in % indicates relative mean absolute error (MAE over the mean reference flux). N is the number of samples.

shows the coefficients D, MAE and IA for the sensible heat fluxes H_EC and H_LST and the latent heat fluxes determined using the EC method, λE_EC, and the residual method, λE_R-EC and λE_R-LST. Regardless of the month, Table 1 shows that λE_R-EC was closer to λE_BREB than λE_EC. The reasons of the surface energy imbalance are unknown (i.e., yet an unresolved problem), however, (on the basis of forcing the closure using the Bowen ratio) Table 1 shows that the residual method was suitable to estimate the latent heat flux. This is of interest because estimation of the Bowen ratio in the present context appears difficult given that over very tall vegetation, measurements are expected in the roughness sublayer (i.e., a layer where gradients are weak).

4.2. Sensible Heat Flux

Figure 2 compares H_EC and H_LST against H_BREB and Figure 3 shows the coefficients D, MAE and IA for H_EC and H_LST determined for different friction velocity thresholds and cloudiness. Regardless of the month, Figure 2 shows that H_ECt ended to underestimate H_BREB while H_LST scattered around H_BREB. Therefore, the coefficients D in Figure 3 show that the accumulated H_LST was consistently closer to the reference than using H_EC. On a half-hourly basis, the MAE and IA obtained for clear sky days showed that H_LST and H_EC performed similarly. For cloudy days H_EC was closer than H_LST to H_BREB, though the MAE coefficients differed about 10 Wm⁻² (regardless of the month and friction velocity threshold). It is difficult to explain why the accumulated H_LST was closer to H_BREB than H_EC. The lack of closure of the EC surface energy balance was, partly, attributed to lack of stationarity [49]. Though a steady flow on a half-hourly basis is also assumed in H_LST, perhaps the use of measurements taken at low frequency as input tended to balance such source of error. On the other hand, the higher IA coefficients obtained using H_EC than H_LST can be attributed to the use of measurements taken at high frequency. That is, the EC method requires as input direct measurements of the turbulence and, therefore, it is expected to be highly correlated with the actual eddy flux.

4.3. Latent Heat Flux

Figure 4 shows the coefficients D and IA obtained for λE_R-EC and λE_R-LST for different friction velocity thresholds and cloudiness. Here, Figure 4 was included for completeness (i.e., the results shown can qualitatively be inferred from Figure 3), however, the MAE values were not shown (i.e., they are identical to the values shown in Figure 3). Regardless of the cloudiness and friction velocity threshold and for all the data (Table 1), it is shown that λE_R-LST nearly matched λE_BREB. λE_R-EC tended to overestimate (7% and 9% in July and August, respectively) λE_BREB and the worst performance was obtained using λE_EC. For clear sky days, λE_R-EC and λE_R-LST performed close to λE_BREB. For cloudy days, λE_R-EC was closer to λE_BREB than λE_R-LST but in August (which includes cloudy and clear sky days), Figure 5, which compared λE_EC, λE_R-EC and λE_R-LST against λE_BREB, shows that λE_R-LST scattered the 1:1 line and λE_R-EC tended to overestimate.

The results of the linear regression analysis obtained for the samples collected during clear sky days (August) showed the λE_R-LST performed close to λE_BREB, λE_R-LST = 0.98 λE_BREB + 4 Wm⁻², with a coefficient of determination (R²) of 0.89. The bias was negligible (1 Wm⁻²) and the mean absolute relative error was 9%. For cloudy days in July, it was found: λE_R-LST = 0.96 λE_BREB + 13 Wm⁻², R² = 0.85; bias = 0 Wm⁻², and the mean relative absolute error was 14% (Table 1). For cloudy days in August: λE_R-LST = 0.98 λE_BREB + 3 Wm⁻², R² = 0.82, bias = 8 Wm⁻² and the mean relative absolute error was 12%. Thus, −λE_R-LST performed excellently.

4.4. The Zero-Plane Displacement and the Offset

Table 2. LST sensible heat and latent heat flux estimates determined using different zero-plane displacements at neutral conditions and neglecting the offset.

Dataset			July (N = 282)				August (N = 521)
Statistics		D	MAE a		IA	D	MAE a		IA
Case:			(Wm−2	%)	(%)		(Wm−2	%)	(%)
dN = 0.60hc	HLST	1.24	49	36	55	1.30	43	31	53
	λER-LST	0.89	49	20	55	1.30	43	16	70
dN = 0.65hc	HLST	1.17	42	31	62	1.21	38	27	61
	λER-LST	0.82	42	17	77	0.91	38	14	75
dN = 0.70hc	HLST	1.07	38	28	66	1.12	30	22	68
	λER-LST	0.97	38	15	79	0.95	30	11	78
dN = 0.75hc	HLST	1.02	36	27	67	1.05	35	22	70
	λER-LST	0.99	36	14	80	1.01	30	11	80
dN = 0.80hc	HLST	0.88	36	27	67	0.97	30	22	70
	λER-LST	1.05	36	14	80	1.01	30	11	80
dN = 0.85hc	HLST	0.85	39	14	63	0.89	30	11	70
	λER-LST	1.06	39	16	79	1.05	30	11	78
b = 0	HLST	1.21	45	33	60	1.13	35	25	64
	λER-LST	0.91	45	18	75	0.94	35	13	78

^a The MAE in % indicates relative mean absolute error (MAE over the mean reference flux). N is the number of samples and d_N, h_c and b denote zero-plane displacement at neutral conditions, canopy height and offset, respectively. In bold is the input modified.

shows the coefficients D, MAE and IA for H_LST obtained for each month, setting a given value for the zero-plane displacement and setting the offset to zero. For completeness, the comparisons between λE_R-LST and λE_BREB were also included.

4.4.1. Setting a Fixed Value for the Zero-Plane Displacement

The half-hourly difference, E, in closing Equation (A6) for a fixed d_N/h_c ratio, E = z_0m − (2.05 d_N/d − 1.05)z_0mN), was averaged for samples collected in cloudy days, clear sky days and for all the data, hereafter referred to as Error. Figure 6 shows the Error obtained for d_N/h_c ratios in the range 0.6–0.85. The smallest Errors were obtained for d_N values close to 0.75 h_c . Comparing the results obtained in Table 1 for H_EC with those obtained in Table 2, it is inferred that H_LST would be neither close to H_EC nor H_BREB using approaches close to d = 0.65 h_c , which are often taken as a rule of thumb [50,51]. Thus, the procedure to objectively select the zero-plane displacement for each sample is recommended.

4.4.2. The Offset

The accumulated frequency of the offset (Figure 7) shows that, in general, the offset ranged between −1 K and 1.25 K. The probability to observe an offset equal and smaller than 0 K in August doubled July because for clear sky days the offset ranged between 0.18 K and 0.15 K. Thus, in practice, for clear sky days the offset could be neglected. While Table 1 shows that the coefficient D was, in practice, one, Table 2 shows that when the offset was neglected H_LST overestimated H_BREB by 21% and 13% in July and August, respectively. The MAE and IA coefficients were worse, especially in July because August included clear sky days. Therefore, it is not recommended to neglect the offset for cloudy days.

5. Concluding Remarks

A method based on similarity and surface renewal formulation to estimate the sensible heat flux for unstable cases was adapted to operate over tall canopies avoiding the need to estimate the zero-plane displacement and roughness length for momentum. The input requirements are the potential temperature of the air and wind speed at a reference height above the canopy, the land surface temperature, the canopy height, the leaf area index and the wind speed at the canopy top, though the latter is not required as input when the reference height is in the inertial sublayer. The method includes an offset to adjust the adiabatic temperature lapse rate which can be neglected for clear sky days. The latter implies that, except for bare soils (research is pending), the offset can be neglected regardless of the canopy type implying that (for clear sky days) the method shows potential of application in the framework of remote sensing over forests. Given that the net radiation and the latent heat flux dominated the surface energy balance, the latent heat flux was determined using the residual method and, regardless of the cloudiness, the role of the offset played a minor role. Thus, for cloudy days H_LST scattered the reference and the accumulated H_LST closely matched the reference. Further research is required to compare it with other models requiring the same or similar input. However, here it is concluded that the method proposed to estimate the sensible heat flux is an alternative to consider for modelling surface energy budgets in tall and dense canopies. In particular, it showed potential in remote sensing applications.