Evaluation of Large Eddy Effects on Land Surface Modeling Based on the FLUXNET Dataset

Abstract

Surface fluxes are vital to understanding land–atmosphere interactions, with similarity theory forming the basis for their parameterization. However, this theory has limitations, particularly due to large eddy effects, which have not been widely considered in Earth system models. A novel scheme was proposed to address this, considering large eddy effects under unstable atmospheric conditions. This study systematically evaluates the proposed scheme using the CoLM2014 model, FLUXNET2015 data, and ERA5 data. Based on the analysis of flux parameterization mechanisms, it proposes specific improvements aimed at enhancing the scheme’s performance. Our findings indicate that the proposed and classical schemes yield similar results, partly because they employ the same dimensionless wind speed gradient under near-neutral conditions. Furthermore, the results revealed that friction velocity responded more strongly to large eddies than did heat flux, as friction velocity influenced atmospheric stability and thereby mitigates the large eddy effects on heat flux. Additionally, our analysis reveals that bare soil exhibits the most pronounced changes in surface fluxes and energy partitioning, while grassland-type and forest-type sites display more complex responses. These findings indicate that different land cover types respond distinctly to the influence of large eddies. Overall, this research deepens our understanding of large eddy impacts and improves Earth system modeling by enhancing land–atmosphere interaction parameterization.

1. Introduction

The exchange of energy, mass, and momentum between the surface and the atmosphere, commonly referred to as the surface fluxes, plays a critical role in understanding and quantifying land–atmosphere interactions. For instance, both latent fluxes and friction at the land surface play a key role in sustaining the longevity of convective systems [1,2]. Accurately parameterizing these fluxes is therefore crucial for improving the reliability of numerical models in meteorology and climate science. Since its proposal, the Monin–Obukhov similarity theory (MOST) [3] has been widely recognized as a key theoretical framework for surface flux parameterization.

The classical similarity theory asserts that in a horizontally homogeneous and statistically steady atmospheric surface layer, the key statistical properties are determined by a single dimensionless stability parameter, which depends on the height above the ground and the Obukhov length $L$. Over the years, the Monin–Obukhov similarity theory has been rigorously tested in numerous field studies, which has contributed to the development of several universal functions at quantifying wind and temperature gradients under a variety of atmospheric conditions, encompassing both stable and unstable stratification [4,5,6,7]. These universal functions are critical for the accurate calculation of surface fluxes in atmospheric models. However, despite their practical significance, numerous studies have identified inconsistencies in these functions [8,9,10,11,12]. Such discrepancies are typically attributed to omitted physical processes in the MOST and inherent random errors [13].

Recent studies have increasingly highlighted the role of large-scale eddies in the surface layer and their implications for the similarity theory [12,14,15,16,17,18,19,20,21,22,23]. The neglect of large eddies in the MOST has led to inaccuracies in flux predictions, particularly in environments with high instability or complex terrain [16]. Large eddy simulations (LES) and observational data have been used to systematically quantify the influence of large eddies on surface-layer processes, confirming that these eddies play a crucial role in flux variability, often underrepresented in traditional flux parameterizations [18,19,21]. The role of large-scale eddies, particularly under convective conditions, has been investigated, with findings showing that their presence can significantly modify surface heat and moisture fluxes [12]. Despite recognizing these shortcomings, these studies do not propose an explicit form for the similarity function that accounts for large eddies.

A novel surface flux estimation scheme for unstable atmospheric conditions, which considers the influence of large eddies in land surface modeling, was proposed more recently [24]. They demonstrated that the boundary layer depth is linearly associated with large eddy effects. In their study, it was shown that the dimensionless wind gradient is a function of both height above ground $z$ and the boundary layer depth $z_i$, implying the relationship ${\varphi }_m={\varphi }_m(z/L,z_i/L)$, while the normalized temperature gradient consistently satisfies the similarity hypothesis.

In this work, we examine the modified similarity function presented by [24], hereafter referred to as LZD2022, and evaluate its differences compared to traditional similarity schemes. Additionally, we investigate the physical mechanisms and identify key factors affecting surface flux parameterization through model simulations. Furthermore, we assess the effects of large eddies on different land cover types, including bare soil, grassland-type, and forest-type sites, and explore their energy distribution patterns. The data utilized in this study were obtained from the FLUXNET2015 and ERA5 datasets, and the CoLM2014 model was employed for simulation purposes. The paper is structured as follows: Section 2 introduces the Common Land Model, the classical and modified universal functions, along with an overview of the FLUXNET2015 and ERA5 datasets. Section 3 analyzes the simulation results, focusing on the flux parameterization mechanisms and the effects of large eddies across various land surface types. Section 4 offers a discussion and concludes with final remarks.

2. Theory, Model, and Data

2.1. Common Land Model (CoLM) and Classical Flux-Gradient Relations Under Unstable Conditions

The Common Land Model (CoLM) is based on the Land Surface Model (LSM) [25], the Biosphere-Atmosphere Transfer Scheme (BATS) developed by [26], and the Land Surface Model developed by the Institute of Atmospheric Physics, Chinese Academy of Sciences (IAP94) as described by [27]. By combining the strengths of these three models, CoLM integrates multiple component processes within the land surface system. Key land surface processes, including soil water dynamics, lake interactions, and surface runoff, have been enhanced, and comprehensive global land surface datasets, such as vegetation and soil information, have been developed [28,29]. In this study, we utilized the 2014 version of CoLM (CoLM2014), which is characterized by the introduction of a Two-Big-Leaf Model [30] that separately considers sunlit and shaded leaves. Compared to earlier versions, CoLM2014 significantly improves the accuracy of radiation calculations, leaf temperature estimations, and stomatal conductance for photosynthesis, while employing a simplified two-stream approximation scheme to calculate vegetation albedo.

According to the MOST, the universal function are used to derive surface fluxes (e.g., momentum flux and heat flux). The universal functions in [31] (hereafter ZENG) are widely used and can be expressed as follows:

$${\varphi }_m(\zeta )=\left\{\begin{array}{ll}{\varphi }_{m1}(\zeta )={\left(1-16\zeta \right)}^{-1/4},& \mathrm{for} -1.574\le \zeta <0\\ {\varphi }_{m2}(\zeta )=0.7{\kappa }^{2/3}{\left(-\zeta \right)}^{1/3},& \mathrm{for} \zeta <-1.574\end{array}\right.$$

(1)

$${\varphi }_h(\zeta )=\left\{\begin{array}{ll}{\varphi }_{h1}(\zeta )={(1-16\zeta )}^{-1/2},& \mathrm{for} -0.465\le \zeta <0\\ {\varphi }_{h2}(\zeta )=0.9{\kappa }^{4/3}{(-\zeta )}^{-1/3},& \mathrm{for} \zeta <-0.465\end{array}\right.$$

(2)

Specifically, ZENG adopts the universal functions from [5] (hereafter Dyer1974) under slightly unstable conditions (i.e., $-1.574\le \zeta <0$ for ${\varphi }_m$ and $-0.465\le \zeta <0$ for ${\varphi }_h$). Under very unstable conditions (i.e., $\zeta <-1.574$ for ${\varphi }_m$ and $\zeta <-0.465$ for ${\varphi }_h$), universal functions from [32] are used.

The exact forms of Fm in flux-gradient relations (Equation (A6)) under unstable conditions are

$$\mathrm{Fm}=\left\{\begin{array}{ll}\ln {\displaystyle \frac{-1.574L}{z_{0m}}}-{\psi }_m(-1.574)+1.14\left[{(-\zeta )}^{1/3}-{(1.574)}^{1/3}\right]+{\psi }_m({\displaystyle \frac{z_{0m}}{L}})& \mathrm{for} \zeta <-1.574\\ \ln {\displaystyle \frac{z}{z_{0m}}}-{\psi }_m(\zeta )+{\psi }_m({\displaystyle \frac{z_{0m}}{L}})& \mathrm{for} -1.574<\zeta <0\end{array}\right.$$

(3)

The exact forms of Fh (Equation (A7)) under unstable conditions are

$$\mathrm{Fh}=\left\{\begin{array}{ll}\ln {\displaystyle \frac{-0.465L}{z_{0h}}}-{\psi }_h(-0.465)+0.8\left[{(0.465)}^{-1/3}-{(-\zeta )}^{-1/3}\right]+{\psi }_h({\displaystyle \frac{z_{0h}}{L}})& \mathrm{for} \zeta <-0.465\\ \ln {\displaystyle \frac{z}{z_{0h}}}-{\psi }_h(\zeta )+{\psi }_h({\displaystyle \frac{z_{0h}}{L}})& \mathrm{for} -0.465<\zeta <0\end{array}\right.$$

(4)

The specific expressions of ${\psi }_m$ and ${\psi }_h$ can be found in [31]. For a comprehensive understanding of the derivation and application of this formula, refer to the detailed discussion provided in that article.

2.2. Modified Flux-Gradient Relations Considering the Large Eddy Effects

A modified universal function for wind gradient was developed to account for the contribution of large eddy impact (hereafter LZD2022). The universal function ${\varphi }_m$ in LZD2022 under unstable conditions is given by:

$${\varphi }_m(\zeta )=\left\{\begin{array}{ll}{\varphi }_{m1}(\zeta )={\left(1-16\zeta \right)}^{-1/4},& \mathrm{for} {\zeta }_{m2}\le \zeta <0\\ {\varphi }_{m3}(\zeta )=B_{m2}{\left(-\zeta \right)}^{-1/2},& \mathrm{for} \zeta <{\zeta }_{m2}\end{array}\right.$$

(5)

where $B_{m2}=\max \left(B_m,0.2722\right)$, ${\zeta }_{m2}=\min \left({\zeta }_m,-0.13\right)$, $B_m=0.0047\left(-z_i/L\right)+0.185$, ${\zeta }_m={\displaystyle \frac{-16-\sqrt{256+4{\left(B_m\right)}^{-4}}}{2{\left(B_m\right)}^{-4}}}$, and $z_i$ is the boundary layer depth.

Specifically, in LZD2022, ${\varphi }_m$ takes the form of Dyer1974 from $\zeta =0$ to $\zeta ={\zeta }_{m2}$, which is the same form as ZENG. For $\zeta <{\zeta }_{m2}$, the new formula accounting for large eddy impact is used. The exact forms of Fm under unstable conditions are

$$\mathrm{Fm}=\left\{\begin{array}{ll}\ln {\displaystyle \frac{-1.574L}{z_{0m}}}-{\psi }_m(-1.574)+1.14\left[{(-\zeta )}^{1/3}-{(1.574)}^{1/3}\right]+{\psi }_m({\displaystyle \frac{z_{0m}}{L}})& \mathrm{for} {\zeta }_{m2}\le \zeta <0\\ \ln {\displaystyle \frac{{\zeta }_{m2}L}{z_{0m}}}-{\psi }_m({\zeta }_{m2})-2B_{m2}\left[{(-\zeta )}^{-1/2}-{(-{\zeta }_{m2})}^{-1/2}\right]+{\psi }_m({\displaystyle \frac{z_{0m}}{L}})& \mathrm{for} \zeta <{\zeta }_{m2}\end{array}\right.$$

(6)

Note that LZD2022 takes the same form as ZENG for ${\zeta }_{m2}\le \zeta <0$. For $\zeta <{\zeta }_{m2}$, LZD2022 has a different form.

2.3. FLUXNET2015 Dataset and Planetary Boundary Layer Height Data

FLUXNET aims to provide a high-quality, shared-site dataset for the validation and development of land surface models. The earlier versions of FLUXNET include the FLUXNET Marconi Dataset (2000) and the FLUXNET La Thuile Dataset (2007). These have evolved into FLUXNET2015 [33]. FLUXNET2015 provides the most comprehensive dataset of terrestrial ecosystems, encompassing a wide range of biome types and climate zones [34]. Compared to previous versions, FLUXNET2015 incorporates more sites, broader time scales, and novel protocols for data quality control, such as standardized measurement procedures and validation methods. The shared flux data undergo uniform quality control, interpolation, and disaggregation, and are available for download directly from the FLUXNET website (https://fluxnet.org/data/fluxnet2015-dataset/, accessed 6 March 2021).

ERA5 is the fifth version of the ECMWF (European Centre for Medium-Range Weather Forecasts) atmospheric reanalysis dataset, covering global climate data from January 1950 to the present [35]. ERA5 became publicly available online in 2017 (https://www.ecmwf.int/en/forecasts/, accessed on 12 March 2025), with a spatial resolution of 0.25° and a temporal resolution of 1-h intervals. Due to the absence of boundary layer heights in the FLUXNET2015 data, the planetary boundary layer height for each site was extracted from the ERA5 dataset.

In this study, nine land cover types were selected for analysis.

Table 1. Characteristics of FLUXNET Sites and Data Information.

Site Name	Longitude	Latitude	Year	Land Cover Type (USGS)	Height (m)
AU-Cpr	140.58	−34.00	2011–2017	Savanna	20
AU-DaP	131.32	−14.06	2009–2012	Grassland	15
AU-DaS	131.38	−14.15	2010–2017	Savanna	23
AU-Dry	132.37	−15.25	2011–2015	Savanna	15
AU-How	131.15	−12.49	2003–2017	Savanna	23
AU-Lit	130.79	−13.1	2016–2017	Savanna	31
AU-Stp	133.35	−17.15	2010–2017	Grassland	4.8
BE-Bra	4.52	51.30	2004–2014	Mixed Forest	41.0
BE-Lon	4.74	50.55	2005–2014	Irrigated Cropland and Pasture	2.7
BE-Vie	5.99	50.30	1997–2014	Mixed Forest	40
CA-Qfo	−74.34	49.69	2004–2010	Evergreen Needleleaf Forest	24
CA-SF3	−106.00	54.09	2003–2005	Mixed Shrubland/Grassland	18.29
CH-Cha	8.41	47.21	2006–2014	Grassland	2
CH-Dav	9.85	46.81	1997–2014	Evergreen Needleleaf Forest	35
CN-Cng	123.50	44.59	2008–2009	Grassland	6
CN-Du2	116.28	42.04	2007–2008	Grassland	4
CN-HaM	101.18	37.37	2002–2003	Grassland	2.2
CZ-wet	14.77	49.02	2007–2014	Herbaceous Wetland	2.7
DE-Bay	11.86	50.14	1997–1999	Evergreen Needleleaf Forest	32
DE-Geb	10.91	51.10	2001–2014	Irrigated Cropland and Pasture	6
DE-Gri	13.51	50.94	2004–2014	Grassland	3
DE-Hai	10.45	51.07	2000–2012	Deciduous Broadleaf Forest	42
DE-Kli	13.52	50.89	2005–2014	Irrigated Cropland and Pasture	3.5
DE-Obe	13.72	50.78	2008–2014	Evergreen Needleleaf Forest	30
DE-SfN	11.32	47.80	2013–2014	Herbaceous Wetland	6
DE-Tha	13.56	50.96	1998–2014	Evergreen Needleleaf Forest	42
DE-Wet	11.45	50.45	2002–2006	Evergreen Needleleaf Forest	30
DK-Lva	12.08	55.68	2005–2006	Grassland	2.5
DK-Sor	11.64	55.48	1997–2014	Deciduous Broadleaf Forest	57
ES-LMa	−5.77	39.94	2004–2006	Savanna	15.5
FI-Hyy	24.295	61.84	1996–2014	Evergreen Needleleaf Forest	23
FI-Sod	26.63783	67.36	2008–2014	Evergreen Needleleaf Forest	23
HU-Bug	19.60	46.69	2003–2006	Grassland	4
IE-Dri	−8.75	51.98	2003–2005	Grassland	10
IT-Amp	13.60	41.90	2003–2006	Grassland	4
IT-Isp	8.63	45.81	2013–2014	Deciduous Broadleaf Forest	38
IT-Noe	8.15	40.60	2004–2014	Shrubland	2
IT-SR2	10.29	43.73	2013–2014	Evergreen Needleleaf Forest	22.5
NL-Ca1	4.92	51.97	2003–2006	Grassland	20
NL-Loo	5.74	52.16	1997–2013	Evergreen Needleleaf Forest	24.4
RU-Fyo	32.92	56.46	2003–2014	Evergreen Needleleaf Forest	29
SD-Dem	30.47	13.28	2005–2009	Savanna	9
US-AR1	−99.42	36.42	2010–2012	Grassland	2.84
US-Aud	−110.50	31.59	2003–2005	Grassland	4
US-Bkg	−96.83	44.34	2005–2006	Grassland	4
US-Bo1	−88.29	40.00	1997–2006	Irrigated Cropland and Pasture	10
US-FPe	−105.10	48.30	2000–2006	Grassland	3.5
US-Goo	−89.87	34.25	2004–2006	Grassland	4
US-Ho1	−68.74	45.20	1996–2004	Evergreen Needleleaf Forest	30
US-KS2	−80.6715	28.60858	2003–2006	Shrubland	3.5
US-Me2	−121.557	44.4523	2002–2014	Evergreeen Needleleaf Forest	32
US-Me4	−121.62	44.49	1996–2000	Evergreeen Needleleaf Forest	47
US-SRG	−110.82	31.78	2009–2014	Grassland	3.25
US-SRM	−110.86	31.82	2004–2014	Savanna	7.8
US-Ton	−120.96	38.43	2001–2014	Savanna	23.5
US-Var	−120.95	38.41	2001–2014	Grassland	2.2
US-Whs	−110.05	31.74	2008–2014	Mixed Shrubland/Grassland	6.5
US-Wkg	−109.94	31.73	2005–2014	Grassland	6.4

presents the locations of the stations, observation years, the United States Geological Survey (USGS) classifications, and observation heights. As shown in Table 1, the stations correspond to different data years and fulfill the criterion of having at least two consecutive years of data. The USGS classification scheme with a spatial resolution of 30 arc-seconds served as the basis for surface cover classification, and all 58 sites were assigned to the following surface types: Irrigated Cropland (CRP), Grassland (GRA), Shrubland (SHR), Mixed Shrubland/Grassland (MSG), Savanna (SAV), Deciduous Broadleaf Forest (DBF), Evergreen Needleleaf Forest (ENF), Mixed Forest (MF), and Wetland (WET). In this study, the grassland-type sites included six land cover types: CRP, GRA, SHR, MSG, SAV, and WET; while the forest-type sites included three categories: DBF, ENF, and MF. The distribution of sites according to the USGS classification is shown in Figure 1. The sites were predominantly located in North America and Western Europe, as well as a few in South Africa, Asia, and Australia.

3. Results

3.1. Comparison of Modified and Classical Flux-Gradient Relations

This study first compares the simulation results of four variables from the LZD2022 and the ZENG schemes: the integral momentum function (Fm), friction velocity ($u_{*}$), sensible heat flux (SH), and latent heat flux (LE) across various land cover types. This paper focuses on periods of atmospheric instability during the day (0800 LST–1600 LST).

Figure 2, Figure 3, Figure 4 and Figure 5 present the multi-year averages and relative biases of Fm, $u_{*}$, SH, and LE for the LZD2022 and ZENG schemes across various land cover types. For most land cover types (e.g., DBF, ENF, MF, MSG, SHR, and WET), the simulation results of the LZD2022 and ZENG schemes were almost identical, with relative biases near 0%. For some land cover types (e.g., GRA and SAV), the simulation results showed minor variations. The integral momentum function (Fm) decreased by a relative bias between 2% and 4%. The friction velocity ($u_{*}$) increased by a relative bias between 3% and 8%. Additionally, the variations in sensible heat flux (SH) and latent heat flux (LE) were smaller compared to friction velocity, with relative biases of less than 3% for SH and less than 1.5% for LE.

Overall, the differences between LZD2022 and ZENG were little, particularly in heat flux. Further analysis will investigate the causes of these minor differences. By examining the mechanisms of surface flux parameterization, we aimed to identify the key factors that influence the performance of these schemes.

3.2. Factors Contributing to Minor Differences

The integral momentum function Fm (Equation (A6)) can also be written as:

$$\mathrm{Fm}={\displaystyle {\int }_0^{\zeta }\frac{{\varphi }_m(\zeta )}{\zeta }}d\zeta$$

(7)

The differences between the LZD2022 scheme and the ZENG scheme arise from ${\displaystyle {\int }_0^{\zeta }\frac{{\varphi }_m(\zeta )}{\zeta }}d\zeta$ in Equation (7), which represents the integral of ${\displaystyle \frac{{\varphi }_m(\zeta )}{\zeta }}$ with respect to $\zeta$ over the range from 0 to $\zeta$.

To facilitate analysis, we define the integral of ${\displaystyle \frac{{\varphi }_m(\zeta )}{\zeta }}$ over the range from 0 to $\zeta$ in the ZENG scheme as ${\mathrm{Fm}}_{\mathrm{ZENG}}$, and the corresponding integral of ${\displaystyle \frac{{\varphi }_m(\zeta )}{\zeta }}$ in the same interval for LZD2022 as ${\mathrm{Fm}}_{\mathrm{LZD}2022}$. In Equation (5), the dimensionless wind speed gradient in LZD2022 consists of two terms. Specifically, ${\varphi }_{m1}(\zeta )$ has the same form as in ZENG, while ${\varphi }_{m3}(\zeta )$ represents a new term. Figure A1 in Appendix A shows a schematic diagram of ${\mathrm{Fm}}_{\mathrm{A}1}$, ${\mathrm{Fm}}_{\mathrm{A}2}$, and ${\mathrm{Fm}}_{\mathrm{A}3}$. In this study, we define ${\mathrm{Fm}}_{\mathrm{A}1}={\displaystyle {\int }_0^{\zeta }\frac{{\varphi }_{m1}(\zeta )}{\zeta }}d\zeta$ as the component that overlaps with ZENG in LZD2022, with $\zeta$ ranging from $\zeta =0$ to a transition value of $\zeta$ (denoted as ${\zeta }_{m2}$). Similarly, we define ${\mathrm{Fm}}_{\mathrm{A}2}={\displaystyle {\int }_{{\zeta }_{m2}}^{\zeta }\frac{{\varphi }_{m3}(\zeta )}{\zeta }}d\zeta$ as a new term that accounts for the effects of large eddies for $\zeta <{\zeta }_{m2}$. Additionally, ${\mathrm{Fm}}_{\mathrm{A}3}$ represents the difference between the integrals of ZENG and LZD2022, which allows us to assess the discrepancies between the two schemes. The integral in LZD2022, denoted as ${\mathrm{Fm}}_{\mathrm{LZD}2022}$, is the sum of ${\mathrm{Fm}}_{\mathrm{A}1}$ and ${\mathrm{Fm}}_{\mathrm{A}2}$. The integral in ZENG (denoted ${\mathrm{Fm}}_{\mathrm{ZENG}}$) is the sum of ${\mathrm{Fm}}_{\mathrm{A}1}$, ${\mathrm{Fm}}_{\mathrm{A}2}$, and ${\mathrm{Fm}}_{\mathrm{A}3}$. The exact forms of ${\mathrm{Fm}}_{\mathrm{A}1}$ and ${\mathrm{Fm}}_{\mathrm{A}2}$ under unstable conditions are

$$\left\{\begin{array}{ll}{\mathrm{Fm}}_{\mathrm{A}1}=\ln {\displaystyle \frac{z}{z_{0m}}}-{\psi }_{m1}(\zeta )+{\psi }_m({\displaystyle \frac{z_{0m}}{L}}),{\mathrm{Fm}}_{\mathrm{A}2}=0& \mathrm{for} {\zeta }_{m2}<\zeta <0\\ {\mathrm{Fm}}_{\mathrm{A}1}=\ln {\displaystyle \frac{{\zeta }_{m2}L}{z_{0m}}}-{\psi }_m({\zeta }_{m2})+{\psi }_m({\displaystyle \frac{z_{0m}}{L}}),{\mathrm{Fm}}_{\mathrm{A}2}=-2B_{m2}\left[{(-\zeta )}^{-1/2}-{(-{\zeta }_{m2})}^{-1/2}\right]& \mathrm{for} \zeta <{\zeta }_{m2}\end{array}\right.$$

(8)

In this study, ${\mathrm{Fm}}_{\mathrm{A}1}{/(\mathrm{Fm}}_{\mathrm{A}1}{+\mathrm{Fm}}_{\mathrm{A}2})$ denotes the ratio of the overlapping component in the LZD2022 scheme with ZENG. Figure 6 shows the probability of overlapping distribution for different land cover types in LZD2022. As shown in Figure 6, for all land cover types, the ratio ${\mathrm{Fm}}_{\mathrm{A}1}{/(\mathrm{Fm}}_{\mathrm{A}1}{+\mathrm{Fm}}_{\mathrm{A}2})$ mostly fell within the range of 0.7 to 1.0, with the highest proportion in the 0.9–1.0 range. For CRP, GRA, SHR, and WET, the proportion in the 0.9–1.0 range exceeded 90%. For MSG and SAV, the proportion in the 0.8–1.0 range exceeded 80%. For forest-type sites, such as DBF, ENF, and MF, the proportion in the 0.7–1.0 range exceeded 80%.

Table 2. Mean proportions of ${\mathrm{Fm}}_{\mathrm{A}1}{/(\mathrm{Fm}}_{\mathrm{A}1}{+\mathrm{Fm}}_{\mathrm{A}2})$ across nine land cover types.

Land Cover Type (USGS)	${\mathbf{Fm}}_{\mathbf{A}1}/({\mathbf{Fm}}_{\mathbf{A}1}+{\mathbf{Fm}}_{\mathbf{A}2})$
CRP	0.98
DBF	0.88
ENF	0.91
GRA	0.98
MF	0.87
MSG	0.95
SAV	0.92
SHR	0.98
WET	0.97

presents the average proportion of ${\mathrm{Fm}}_{\mathrm{A}1}$ relative to ${(\mathrm{Fm}}_{\mathrm{A}1}{+\mathrm{Fm}}_{\mathrm{A}2})$. For all land cover types, the ratio of ${\mathrm{Fm}}_{\mathrm{A}1}$ to ${(\mathrm{Fm}}_{\mathrm{A}1}{+\mathrm{Fm}}_{\mathrm{A}2})$ exceeded 85%, with seven land cover types (i.e., CRP, ENF, GRA, MSG, SAV, SHR, and WET) showing ${\mathrm{Fm}}_{\mathrm{A}1}$ proportions greater than 90%. This suggests a significant overlap between LZD2022 and ZENG, resulting in minor differences between the two schemes.

3.3. Overlapping Contributions and Indirect Influences

Section 3.2 shows that a rigorous comparison between LZD2022 and ZENG requires removing the overlapping contributions in LZD2022, quantified as ${\mathrm{Fm}}_{\mathrm{A}1}$. By excluding ${\mathrm{Fm}}_{\mathrm{A}1}$, ${\mathrm{Fm}}_{\mathrm{LZD}2022}$ consists solely of ${\mathrm{Fm}}_{\mathrm{A}2}$, while ${\mathrm{Fm}}_{\mathrm{ZENG}}$ includes both ${\mathrm{Fm}}_{\mathrm{A}2}$ and ${\mathrm{Fm}}_{\mathrm{A}3}$. Thus, ${\mathrm{Fm}}_{\mathrm{A}3}$ corresponds to the difference between ${\mathrm{Fm}}_{\mathrm{LZD}2022}$ and ${\mathrm{Fm}}_{\mathrm{ZENG}}$. In this study, we introduced a parameter $\mathsf{\alpha }$, defined as ${\mathsf{\alpha }=\mathrm{Fm}}_{\mathrm{A}3}{/(\mathrm{Fm}}_{\mathrm{A}2}{+\mathrm{Fm}}_{\mathrm{A}3})$, to quantify the proportion of differences between the LZD2022 and ZENG schemes relative to the ZENG scheme, after excluding overlapping components. A larger $\mathsf{\alpha }$ value indicates that the differences between the two schemes are significant. Subsequently, $\mathrm{Fm}=(1-\mathsf{\alpha }){\mathrm{Fm}}_{\mathrm{ZENG}}$ represents ${\mathrm{Fm}}_{\mathrm{LZD}2022}$ after removing the overlapping contributions.

Figure 7 presents the distribution of $\mathsf{\alpha }$, denoted by ${\mathrm{Fm}}_{\mathrm{A}3}{/(\mathrm{Fm}}_{\mathrm{A}2}{+\mathrm{Fm}}_{\mathrm{A}3})$ across nine land cover types. The $\mathsf{\alpha }$ values did not exhibit any clear pattern among the different land cover types, ranging from 0.01 to 0.2. This indicates that, even after removing the overlapping contributions, the differences between the LZD2022 and ZENG schemes constituted a relatively small ratio within the ZENG scheme, with a maximum value of 0.2. Notably, the highest ratio was observed at the US-Ton site with the SAV land cover type.

In this study, we set the Fm value to $(1-\mathsf{\alpha }){\mathrm{Fm}}_{\mathrm{ZENG}}$ to represent the LZD2022 scheme after removing the overlapping contributions (hereafter referred to as LZD2022_R1). The $\mathsf{\alpha }$ value used here was 0.2, taken from the US-Ton site as a reference. Figure 8 compares the relative biases of the LZD2022 and LZD2022_R1 relative to ZENG across bare soil, grassland-type, and forest-type sites. Due to the absence of bare soil observation sites in the FLUXNET2015 database, we set the US-Ton site as a bare soil site in this study. Furthermore, we selected eight grassland-type sites (i.e., AU-DaS, AU-Dry, AU-Lit, CA-SF3, CN-Du2, DE-Geb, IT-Amp, and US-Ton) and six forest-type sites (i.e., CH-Dav, DE-Tha, IT-Isp, RU-Fyo, US-Me2, and US-Me4) for detailed analysis. As shown in Figure 8, the relative bias in the LZD2022_R1 scheme was greater than that in the LZD2022 scheme for each land cover type. Taking the friction velocity $u_{*}$ as an example (Figure 8b,f,j), the relative bias between the LZD2022 and ZENG schemes was generally less than 8%, whereas the relative bias between the LZD2022_R1 and ZENG schemes ranged from 15% to 22%. This suggests that after eliminating the overlapping contributions, the LZD2022_R1 scheme exhibited more significant changes compared to the LZD2022 scheme. This result emphasizes the importance of considering overlapping effects in model comparisons, suggesting that researchers should fully recognize the potential impact of overlapping components when conducting scheme evaluations. Furthermore, across all land cover types, the relative bias of sensible heat flux (SH) and latent heat flux (LE) remained consistently smaller than that of friction velocity ($u_{*}$), even after removing the effects of overlapping components. In the LZD2022_R1 scheme, specifically, the relative bias of friction velocity was approximately 22%, while the relative biases of both sensible heat flux and latent heat flux were less than 5%. This indicates that the response of friction velocity to large eddies was more pronounced compared to that of heat flux. Additionally, the simulation performance of each variable varied across different land cover types. For the integral momentum function (Fm) and friction velocity ($u_{*}$) (as shown in Figure 8a,b,e,f,i,j), the simulation differences between the LZD2022 and LZD2022_R1 scheme were similar across the three land cover types. This suggests that the response of momentum flux to large eddies was relatively consistent across different land cover types. In contrast, for the sensible heat flux (SH) (Figure 8c,g,k), the differences between the LZD2022 and LZD2022_R1 scheme were most pronounced at bare soil sites, followed by grassland-type sites, with the smallest differences observed at forest-type sites. This indicates that, for sensible heat flux, the response to large eddies was most significant at bare soil sites, followed by grassland-type sites, and least at forest-type sites. Similarly, for latent heat flux (LE) (Figure 8d,h,l), the relative bias between the LZD2022 and LZD2022_R1 scheme was most prominent at bare soil sites, moderately pronounced at forest-type sites, and minimal at grassland-type sites.

Based on the above analysis, after removing the overlapping effects of the ZENG scheme, the variations in heat flux were minimal, particularly at grassland-type and forest-type sites. In practical applications, surface flux parameterization is an iterative process. As discussed in Section 3.1, due to the influence of large eddies, the decreases in integral momentum function (Fm) leads to an increase in the surface friction velocity ($u_{*}$), which in turn causes an increase in the Obukhov length $L$ (Equation (A5)) and a decrease in stability parameter($\zeta =z/L$), ultimately approaching a neutral condition. According to the empirical expression for the dimensionless temperature gradient (Equation (2)), ${\varphi }_h(\zeta )$ increases, which ultimately leads to a decrease in the dimensionless temperature scale ${\theta }_{*}$ (Equation (A4)). Since the changes in $u_{*}$ and ${\theta }_{*}$ are opposite, the variations in sensible heat flux remain insignificant (Equation (A2)). Overall, under the influence of large eddies, friction velocity ($u_{*}$) influences the stability, which, in turn, affects the dimensionless temperature scale (${\theta }_{*}$). Consequently, this indirect influence mitigates the impact of large eddies on heat flux.

To mitigate the indirect influence of large eddies on heat flux, we developed the LZD2022_R2 scheme. Specifically, LZD2022_R2 builds on LZD2022_R1 by further blocking the indirect effects of large eddies on heat flux. Figure 9 shows the relative biases of integral momentum function (Fm), friction velocity ($u_{*}$), sensible heat flux (SH), and latent heat flux (LE) under the LZD2022, LZD2022_R1, and LZD2022_R2 schemes across three land cover types. The results show that, after blocking the indirect influence on heat flux, the relative change in sensible heat flux (SH) increased across all three surface types (Figure 9c,g,k), with the most pronounced change occurring at bare soil sites, followed by grassland-type sites, and the least at forest-type sites. Specifically, comparing the changes in sensible heat flux (SH) between the LZD2022_R1 and LZD2022_R2 schemes relative to ZENG, the relative bias at bare soil sites increased from 3% to approximately 8%, at grassland-type sites from 2% to around 4%, and at forest-type sites from little to about 2%. This finding indicates that different land cover types modulate the impact of large eddies in distinct ways. The response to large eddies was most pronounced at bare soil sites, followed by grassland sites, with the least response observed at forest sites. In contrast, for latent heat flux (LE), after removing the indirect influence on heat flux, the relative bias at bare soil sites increased (Figure 9d), while no clear pattern emerged at grassland and forest sites (Figure 9h,l). This suggests that the effect of large eddies on latent heat flux is more complex than its impact on sensible heat flux.

3.4. Surface Energy Balance

In this section, we explore the energy partitioning characteristics associated with different land surface types. Figure 10 illustrates the daily patterns of sensible heat flux (SH), latent heat flux (LE), and ground heat flux (G) relative to net radiation (Rnet) for three surface types: bare soil, grassland-type, and forest-type sites. It is evident that, on bare soil, the SH/Rnet ratio rose sharply during the afternoon, which implies a constrained evaporative process that favors sensible heat release. Meanwhile, grassland-type and forest-type sites exhibited more uniform SH/Rnet profiles, suggesting enhanced transpiration and moisture control. Moreover, the latent heat contribution (LE/Rnet) was lower in bare soil compared to the other two surface types, suggesting that vegetation facilitated greater conversion of net radiation into latent heat. The G/Rnet values were uniformly low, especially for grassland and forest. The figure also shows the difference in energy partitioning between the LZD2022_R2 and ZENG schemes. Notably, differences in SH/Rnet and G/Rnet were most significant for bare soil, followed by grassland-type sites, and were least apparent in forest-type sites. For LE/Rnet, the differences remained largely uniform across all surfaces. Collectively, these results indicate that the large eddy effects induced more substantial energy partitioning changes on bare soil, moderate changes on grassland-type, and minimal effects on forest-type sites, likely due to varying vegetation cover, soil moisture conditions, and transpiration dynamics.

Figure 11 presents the diurnal variations in the ratios of sensible heat flux (SH), latent heat flux (LE), and soil heat flux (G) to net radiation (Rnet) for different land types (bare soil, grassland-type, forest-type sites) during the summer and winter periods. Firstly, Figure 11 presents the seasonal differences in energy partitioning for the ZENG scheme, with black circles and star symbols marking the data for summer and winter. The results indicate a significant seasonal contrast, especially in bare soil where the variation was greatest, followed by grassland and then forest. Furthermore, Figure 11 illustrates the seasonal differences in energy partitioning between the ZENG and LZD2022_R2 schemes, as indicated by blue circles and star symbols. On bare soil, the differences in energy partitioning between the ZENG and LZD2022_R2 schemes were more pronounced for SH/Rnet in summer and for LE/Rnet in winter. In contrast, on grassland-type surfaces, the differences between the two schemes were relatively consistent between summer and winter. For forest-type surfaces, the difference in LE/Rnet in winter was particularly significant.

4. Conclusions and Discussion

This study employed the Common Land Model (CoLM2014) along with FLUXNET2015 and ERA5 data to evaluate the modified flux-gradient relations, which account for large eddy effects under unstable conditions, as outlined in the LZD2022 scheme introduced by [24]. The aim was to test its applicability in improving the accuracy of surface flux parameterization compared to classical schemes. First, we conducted a comparative analysis between the LZD2022 scheme and the widely used surface flux parameterization scheme [31]. Subsequently, we investigated the underlying physical mechanisms governing the flux-gradient relations and identified the key factors influencing surface flux parameterization through model simulations. Finally, we assessed the effects of large eddies on different land cover types, including bare soil, grassland-type, and forest-type sites, and explored their energy distribution patterns.

While preliminary comparisons indicated minimal differences between the LZD2022 scheme and the ZENG scheme, subsequent analysis demonstrated that two distinct factors contributed significantly to this result. The first significant factor is attributed to the LZD2022 scheme’s implementation of a Dyer-form dimensionless wind speed gradient [5], which ensures continuity of the wind profile under near-neutral conditions and overlaps with the gradient employed in the ZENG scheme. As a result, more than 85% of the dimensionless wind speed gradient is shared between the two schemes, leading to little differences. Therefore, to accurately evaluate the potential effects of large eddies on surface fluxes, it is crucial to remove the overlapping contribution in the LZD2022 scheme. After eliminating the overlap between the LZD2022 and ZENG schemes, significant differences in friction velocity ($u_{*}$) were observed, with these differences becoming more pronounced. However, the changes in sensible heat flux (SH) and latent heat flux (LE) were relatively small. Further analysis indicates that a crucial second factor is the role of friction velocity in affecting atmospheric stability, which subsequently governs the dimensionless temperature scale (${\theta }_{*}$) and reduces the effect of large-scale eddies on heat flux. After blocking the indirect influence on heat flux, the variations in sensible heat flux (SH) became significantly more pronounced, highlighting the stronger response of heat flux to large eddies. Considering the two factors discussed above, our research advises that, under near-neutral conditions, an alternative form of the dimensionless gradient formulation be adopted in the LZD2022 scheme to account for large eddy effects. In addition, we recommend that the model application of LZD2022 remove the indirect impact on stability, which mitigates the influence of large eddies on heat flux.

This study compared the impact of large eddies across three different surface types: bare soil, grassland-type, and forest-type sites. The results show that the response of momentum flux to large eddies is relatively consistent across different land cover types. In contrast, the response of sensible heat flux to large eddies was most pronounced at bare soil sites, followed by grassland-type sites, with the least effect at forest-type sites. For latent heat flux (LE), the influence of large eddies was most pronounced at bare soil sites, showing a response similar to that of sensible heat flux. However, the latent heat flux response to large eddies at grassland-type and forest-type sites was inconsistent, indicating that the mechanisms governing latent heat flux are more complex than those influencing sensible heat flux. Furthermore, our analysis of energy partitioning revealed that large eddy effects induce more pronounced changes on bare soil, moderate changes on grassland-type surfaces, and minimal effects on forest, likely due to differences in vegetation cover, soil moisture conditions, and transpiration dynamics. These findings indicate that different land cover types respond differently to the influence of large eddies.

This study employed an offline land surface model to evaluate a scheme that considers the effects of large eddies. Through an in-depth analysis of the surface flux parameterization mechanisms, this study provides recommendations for improving the scheme that accounts for large eddy effects. Additionally, by comparing the response of surface fluxes and energy partitioning across different land cover types, this work has contributed to enhancing our understanding of land–atmosphere interactions.