Eddy Covariance Theory: A Review

ABSTRACT

Eddy covariance (EC), the gold standard for measuring ecosystem scale gas and heat exchanges, has transformed our understanding of the breathing of the biosphere, and thus global change biology. Despite numerous methodological improvements and insights gained from the technique, the community faces persistent challenges that have been present since the first EC measurements. Here, we review the theoretical developments underpinning EC. We present theoretical developments in four important areas that have relevance to EC measurements of the net ecosystem exchanges (NEE) of gases and heat from a single tower: (i) measuring the total vertical flux density, (ii) flux attenuation, (iii) coordinate rotations, and (iv) energy balance closure. Persistent problems with EC measurements, such as the inability to close the energy budget, led us to identify two priorities for revisiting the theory underlying: (i) sensible heat flux calculations, and (ii) constraining the mean vertical wind velocity. We present a framework for improved calculation of sensible heat flux derived from first principles of fluid mechanics and thermodynamics that considers coupled heat and mass transfer so that conservation of both is obeyed. These refinements are motivated by the need for unbiased measurements of energy and mass transfer between the land surface and atmosphere for ecosystem research and to validate satellite observations and land surface models.

Eddy covariance is often applied using a single tower to measure net ecosystem exchanges of gases and heat‐in other words, to infer the integrated source/sink strengths.

1. Introduction

Eddy covariance (EC) is considered the gold standard for measuring ecosystem scale gas and heat exchanges (D. Baldocchi ²⁰¹⁴; Sabbatini et al. ²⁰¹⁸) and its proliferation at sites around the world (Baldocchi et al. 2001) has transformed understanding of the breathing of the biosphere from ecosystem to global scales (Baldocchi et al. 2024; D. D. Baldocchi ²⁰²⁰). Originally, EC was the domain of meteorologists concerned with the heat budget of the lower atmosphere, and answering questions like “What happens to the sunlight?” (Suomi 1953) by measuring sensible and latent heat fluxes (Dyer et al. 1967; Dyer ¹⁹⁶¹; Dyer and Maher ¹⁹⁶⁵; Swinbank ^{1951, 1955}; Taylor and Dyer ¹⁹⁵⁸) in short “golden days” experiments with fair weather and favorable winds (Haugen et al. 1971; Kaimal and Wyngaard ¹⁹⁹⁰). Note that EC measures the flux densities of matter or energy (i.e., mol m⁻² s⁻¹ or J m⁻² s⁻¹, respectively). For convenience, we use the term flux consistently throughout the main text as a synonym for flux density.

A convergence of remarkable advances in instrumentation and computing, along with the seminal theory of Webb et al. (1980), hereafter WPL, enabled continuous flux measurements and led to the application of EC by a broader community of scientists interested in ecosystem carbon balances (Lee and Massman 2011), and ultimately the founding of regional and global flux networks (Aubinet et al. 2000; Baldocchi et al. ²⁰⁰¹; Gu and Baldocchi ²⁰⁰²). The founding of flux networks spurred tremendous growth in science applications of EC (Baldocchi et al. 2024; D. D. Baldocchi ²⁰²⁰). Additionally, EC moved from strictly tower‐based deployments to mobile platforms including aircraft (Desjardins et al. 1994) and ships (McGillis et al. ²⁰⁰¹).

In the 1990s and early 21st century, methodological advances emphasized data filtering (Barr et al. 2013; Gu et al. ²⁰⁰⁵), gap‐filling (Falge et al. 2001; Moffat et al. ²⁰⁰⁷), uncertainty characterization (Finklestein and Sims 2001; Richardson and Hollinger ²⁰⁰⁷), and the partitioning of net fluxes of CO₂ (Griffis et al. 2006; Lasslop et al. ²⁰¹⁰; Reichstein et al. ²⁰⁰⁵) or water vapor (Berkelhammer et al. 2016; Lee et al. ²⁰⁰⁷; Nelson et al. ²⁰¹⁸; Perez‐Priego et al. ²⁰¹⁸; Zhou et al. ²⁰¹⁶) or both simultaneously (Scanlon and Sahu 2008; Wohlfahrt et al. ²⁰¹²). With the emergence of a plethora of processing algorithms, there have also been significant efforts towards standardizing data quality assurance and data pipelines (Foken and Wichura 1996; Papale et al. ²⁰⁰⁶; Pastorello et al. ²⁰²⁰; Sabbatini et al. ²⁰¹⁸; Vickers and Mahrt ¹⁹⁹⁷; Wutzler et al. ²⁰¹⁸).

Despite challenges like measuring implausible carbon dioxide flux dynamics (Goulden et al. 1996) and our inability to close the ecosystem energy balance (Leuning et al. 2012; Mauder et al. ^{2020, 2024}; Wilson et al. ²⁰⁰²), critical re‐examinations of the EC theory guiding ecosystem flux measurements received comparatively less attention in the literature, with some notable exceptions (Finnigan 2004; Finnigan et al. ²⁰⁰³; Fuehrer and Friehe ²⁰⁰²; Gu ²⁰¹³; Gu et al. ²⁰¹²; Lee ¹⁹⁹⁸; Lee and Massman ²⁰¹¹; Leuning ²⁰⁰⁷; Paw U et al. ²⁰⁰⁰; Sun et al. ²⁰²¹).

We are now ~75 years since the first EC measurements of Swinbank (1951) and have yet to resolve issues like the energy balance closure problem, suggesting a fresh look at EC theory is warranted. The overall goal of this paper is to review historical developments of EC theory and point to areas warranting further attention. For the purposes of this review, we focus on theoretical advances that guide the application of tower‐based EC for measuring net ecosystem exchanges (NEE) of gases and heat (Box 1). The review is structured into four sections:

1. An overview of theoretical developments.

2. What is an eddy flux?

3. How do we apply eddy covariance to measure net ecosystem exchanges of gases and heat?

4. Novel theories: Where does our theory go from here?

BOX 1. The most common eddy covariance (EC) application is measuring net ecosystem exchanges (NEE) of gases and heat using a single tower.

2. An Overview of Theoretical Developments

A chronology of major theoretical developments influencing EC (Table 1) can be broadly separated into three thematic periods. The earliest advances concerned the application of turbulence theory (Reynolds 1895) to the atmosphere in ways amenable to interpreting available data (Richardson 1920; Schmidt ¹⁹¹⁷; G. I. Taylor ¹⁹¹⁵), and a thorough review of these developments is provided in Priestley and Sheppard (1952).

TABLE 1.

Chronology of theoretical developments of eddy covariance theory.

	Year	Contribution
First period
O. Reynolds	1895	Decomposition of fluid flow into bulk flow and turbulent components
G. I. Taylor	1915	Application of turbulence theory to eddy diffusion of momentum, mass, and heat in the atmosphere; introduced theories of eddy diffusivities and mixing length a
W. Schmidt	1917	Austausch theory of turbulent mass exchange
L. F. Richardson	1920	Criteria for turbulence in the atmosphere; the energy cascade
G. I. Taylor	1938	Taylor's frozen turbulence hypothesis
A. Kolmogorov	1941	Structure and decay of isotropic turbulence
R. Montgomery	1948	Rigorous definition of eddy flux of heat based on thermodynamics of an open system
Second period
W. Swinbank	1951, 1955	Measuring the total vertical flux must address the mean and eddy flux components; energy balance considerations reveal EC flux underestimation
Monin & Obukhov	1954	Monin–Obukhov similarity theory
Webb, Pearman & Leuning	1980	Equations to account for the influence of sensible and latent heat fluxes on CO2 flux measurements (i.e., so‐called, “density corrections”)
J. Gash	1986	Defining the modern interpretation of flux footprints
C. Moore	1986	Developed the theoretical transfer function approach to spectral corrections
Third period
Leuning	2004, 2007	WPL theory for non‐steady state and non‐homogeneous flows
Gu et al.	2012	Fundamental equation of eddy covariance
Metzger	2018	Environmental response functions and virtual control volumes

Note: This chronology emphasizes major advances rather than incremental ones.

^a Called eddy viscosity and eddy conductivity at the time.

The second period, beginning in the early 1950s, saw the first applications of EC for measuring sensible and latent heat fluxes (Dyer 1961; Dyer and Maher ¹⁹⁶⁵; Swinbank ^{1951, 1955}; Taylor and Dyer ¹⁹⁵⁸) at which point technical advances and theory advanced in an iterative fashion.

In the earliest applications of EC (McIlroy ¹⁹⁵⁵; Swinbank ^{1951, 1955}) the major technical limitation was not sensor frequency response, but rather the data recording rate and limits on the length of “runs” (i.e., averaging periods). The laborious nature of hand analyzing EC data motivated the development of integrated systems with electric circuits automatically multiplying signals from different sensors and integrating to obtain the mean fluxes (Dyer 1961; Taylor and Dyer ¹⁹⁵⁸).

The most influential theoretical advances in this period derived from the work of WPL (others at the time were working on the “density corrections” problem and a thorough review of these developments is available in Fuehrer and Friehe (2002)) who proposed the equations for calculating fluxes of mass and heat that have largely guided EC measurements through the present day (Lee and Massman 2011). Several major field campaigns pertinent to advancing EC were conducted during this time‐period (Table 2), with the Kansas experiment being particularly relevant in the theoretical context because the published cospectral models (Kaimal et al. 1972) are still used when processing EC data to correct for flux attenuation; for example, TK3, and EddyUH softwares (Mammarella et al. 2016; Mauder and Foken ²⁰¹⁵).

TABLE 2.

Important micrometeorological field campaigns in the development of the eddy covariance technique through 2000.

Objectives	Year | location	Citation
Obtain experimental data on diffusion of a gas over ~800 m; characterize wind velocity fluctuations (Project Prairie Grass)	1956 | USA	Haugen (1959)
Intercomparison of sonic anemometers	1965 | USA	Businger et al. (1969)
Intercomparison of EC systems for measuring sensible the heat flux	1966 | Australia	Businger et al. (1967)
Intercomparison of sonic anemometers over water (ITCE‐1968)	1968 | Canada	Miyake et al. (1971)
Investigate spatial structure of the surface layer; test for constancy of flux profiles (“The Kansas Experiment”)	1968 | USA	Kaimal and Wyngaard (1990)
Intercomparison of EC systems; investigate (horizontal) spatial structure of atmospheric turbulence (ITCE‐1970)	1970 | USSR	Tsvang et al. (1973)
Resolve low‐frequency behavior of u and v spectra in the surface layer; determine if atmospheric boundary layer depth is the scaling length in upper boundary layer (“The Minnesota Experiment”)	1973 | USA	Kaimal and Wyngaard (1990)
Intercomparison of EC systems; determine flux‐profile relationships, and von Karman's and Kolmogorov's constants (ITCE‐76)	1976 | Australia	Dyer et al. (1982)
Intercomparison of EC systems; evaluate models of flux‐gradient relationships (ITCE‐81)	1981 | USSR	Tsvang et al. (1985)
Determine role of boreal forest in global change; intercomparison of commercially available instrumentation (BOREAS)	1994, 1996 | Canada	Sellers et al. (1997)
Determine causes of lack of energy balance closure (EBEX‐2000)	2000 | USA	Oncley et al. (2007)

There were also intensifying efforts towards predicting the source area contributing to EC observations and the first definition of the flux footprint in the modern connotation (Gash 1986). Early work developed from one‐dimensional analytical solutions to the advection–diffusion equation (Gash 1986; Horst and Weil ¹⁹⁹²; Schuepp et al. ¹⁹⁹⁰). Footprint modeling was also tackled as an inverse problem using backward Lagrangian stochastic approaches (Flesch 1996; Leclerc and Thurtell ¹⁹⁹⁰). Notably, this line of work showed that the 100:1 fetch‐height‐ratio rule of thumb was highly inaccurate over smooth surfaces, under stable atmospheric stratification, and with high measurement heights (Leclerc and Thurtell 1990).

During the third period, which began in the 1990s, rugged instrumentation suitable for long‐term deployments was introduced and the first estimates of integrated annual fluxes were reported (Wofsy et al. 1993). Advances in tunable diode laser spectroscopy enabled the application of EC for measuring the fluxes of methane (Edwards et al. 1994; Verma et al. ¹⁹⁹²) and nitrous oxide (Wienhold et al. 1994). With the first long‐term EC deployments, implausible measured carbon dioxide flux dynamics (Goulden et al. 1996) and an inability to close the ecosystem energy balance (Wilson et al. 2002) motivated a revisiting of EC theory (Gu et al. 2012; Lee ¹⁹⁹⁸; Lee and Massman ²⁰¹¹; Leuning ²⁰⁰⁷; Massman and Tuovinen ²⁰⁰⁶; Paw U et al. ²⁰⁰⁰). Later, Taylor's Hypothesis was also revisited in light of advances in distributed temperature sensing (Cheng et al. 2017; Hilland and Christen ²⁰²⁴). Theories of flux footprints advanced with the application of more sophisticated approaches like large eddy simulation to focus on unstable conditions (Leclerc et al. 1997). From the perspective of practical application to long‐term datasets, the most important developments in flux footprint modeling were the extension to two dimensions and improved performance under a broader range of stabilities (Horst and Weil 1994, 1995; Hsieh et al. ²⁰⁰⁰; Kljun et al. ²⁰¹⁵; Kormann and Meixner ²⁰⁰¹).

Amiro (1998) advanced operational applications through developing weighted flux footprint climatologies for evaporation, noting the importance of this information for interpreting flux measurements in heterogenous landscapes. Footprint awareness was underscored more recently in Chu et al. (2021), who noted few sites were in homogenous landscapes and that the percentage of sites with high footprint‐to‐target‐area representativeness around towers decreased from ~62% to ~42% to ~30% as the radius increased from 250 to 500 to 1000 m. A logical extension of footprint applications is for developing two‐dimensional flux maps, which was first applied to airborne EC observations (Schuepp et al. 1992). Different mapping applications to tower‐based EC followed (Chen et al. 2009; Lewicki et al. ²⁰⁰⁹). Further efforts to address the issue of surface heterogeneity led to a new theory on environmental response functions and virtual control volumes (Metzger 2018).

It is important to consider that EC developed in conjunction with efforts aimed at conceptual and quantitative treatments of atmospheric turbulence. Furthermore, technical challenges such as flux attenuation and recalcitrant problems like energy balance closure have been treated in detail elsewhere. There is thus a huge body of work that is beyond the scope of this review, or alternatively, treats content in more detail than is possible here. We therefore point the reader to reviews that document different aspects of the development of theoretical and practical knowledge of atmospheric turbulence as well as the application of EC (Table 3).

TABLE 3.

Reviews or papers on topics with some relevance to the theory of eddy covariance and atmospheric turbulence.

Topic(s)	Citations
Progress in the statistical theory of turbulence	von Kármán (1948)
Turbulence and transfer processes in the atmosphere	Priestley and Sheppard (1952)
Work on the microstructure of atmospheric turbulence in the U.S.S.R.	Obukhov and Yaglom (1959)
The Lettau–Schwerdtfeger balloon experiment: measurement of turbulence via Austausch theory	Lewis (1997)
The Kansas and Minnesota experiments	Kaimal and Wyngaard (1990)
Backwards Lagrangian stochastic models for estimating gas emissions	Flesch et al. (1995)
A century of turbulence	Lumley and Yaglom (2001)
“Density corrections”	Fuehrer and Friehe (2002)
Flux footprints	Schmid (2002) and Vesala et al. (2008)
Eddy covariance for measuring CO2 exchanges	D. Baldocchi (2003)
50 years of Monin–Obukhov similarity theory	Foken (2006)
Perspectives on WPL theory	Lee and Massman (2011)
Energy balance closure	Mauder et al. (2020)
Reynolds averaging and decomposition	Kowalski and Abril‐Gago (2025)

3. What Is an Eddy Flux?

The foundation of EC theory derives from the notion that fluid flow can be decomposed into mean and fluctuating parts (Reynolds 1895). Theoretically, the averaging should be applied to space because eddies are spatial structures within the fluid; however, the EC community is more familiar with the Reynolds decomposition applied in the temporal domain, which for the case of the vertical transport of a scalar, s, is:

$$\overline{w{\rho }_s}=\overline{w}\overline{{\rho }_s}+\overline{w\prime {\rho }_s^{\prime }}$$ (1)

where w is the vertical wind velocity (m s⁻¹). Here, ρ _s is molar density (mol m⁻³) for mass transfer; or, though not as commonly written, energy density (J m⁻³) for energy transfer. Overbars and primes denote mean and departures from the mean, respectively. The key theoretical insight that the total flux ($\overline{w{\rho }_s}$) is the sum of contributions from the mean convective ($\overline{w}\overline{{\rho }_s}$) and turbulent ($\overline{w\prime {\rho }_s^{\prime }}$) components, the latter becoming known as the “eddy flux.” We use the term convective here for the $\overline{w}\overline{{\rho }_s}$ term following (Finnigan 2009). In the context of NEE measurements, Equation (1) gives the expression for the total vertical flux across the top of the control volume (Box 1), of which, the eddy flux is only one component.

Applications of Reynolds's ideas to the atmosphere began appearing around two decades later, beginning with G. I. Taylor (1914, 1915), who introduced the notion of what we now call eddy diffusivities and the mixing length concept. At nearly the same time, a similar idea called “Austausch theory” was developed independently by Schmidt (1917). Through the 1940s, theories conceptualizing the eddy fluxes of heat and mass in analogy to thermal and molecular diffusion through the respective gradients and diffusivities were refined conceptually and mathematically (Brunt 1929; Priestley and Swinbank ¹⁹⁴⁷; Richardson ¹⁹²⁰; Schmidt ^{1917, 1921}; Sutton ¹⁹³²).

Mixing length theory (i.e., K‐theory), where eddy fluxes were related to their respective eddy diffusivities and mean scalar gradients, were desirable for parameterizing governing equations of turbulent flow and importantly, were measurable given the available instrumentation (Priestley and Sheppard 1952). Another development in the parameterization of turbulent transport in a surface layer where the temperature distribution influences turbulence was what became known as Monin Obukhov Similarity Theory, MOST (Monin 1950; Monin and Obukhov ¹⁹⁵⁴; Obukhov ¹⁹⁴⁸). MOST is relevant to modern EC because spectral corrections following the theoretical transfer function approach require cospectral models that depend on stability that is parameterized using the Obukhov length (Massman and Clement 2005; Moore ¹⁹⁸⁶).

The first access to fine‐scale knowledge of the characteristics of turbulent flows was through high frequency wind velocity measurements. The turbulence spectrum was first characterized in wind tunnel experiments (Simmons and Salter 1938) that led to Taylor's frozen turbulence hypothesis (G. Taylor ¹⁹³⁸). Taylor's hypothesis states that eddies of all sizes maintain their characteristics and are transported on average, at the mean horizontal wind speed. Taylor's hypothesis is of practical importance for single‐tower EC applications to enable the inference of spatial properties of eddies from repeated measurements over time at a fixed point. Taylor's hypothesis has been revisited, and the validity of assuming all eddies advect at the mean horizontal wind speed is questioned (Cheng et al. 2017; Hilland and Christen ²⁰²⁴; Lumley ¹⁹⁶⁵; Wyngaard and Clifford ¹⁹⁷⁷), though a generalizable replacement to the theory has yet to emerge. Finally, the theory of the spectral properties of isotropic turbulence was further advanced by Kolmogorov (1941) (see Kolmogorov et al. (1997) for an English translation), with scaling properties of (co)spectra in the inertial subrange still used today for quality assurance and control of EC observations.

In our view, a particularly important but perhaps underappreciated line of work was the consideration of eddy fluxes of heat when treating the atmosphere as an open thermodynamic system (Montgomery 1948; van Mieghem ^{1935, 1951}). Montgomery's work was more directly relevant to EC because it conceptualized a control volume with a lower boundary at the surface. However, his definition of the system was not strictly compatible with EC deployments since the upper boundary was extensive and allowed to vary locally with the vertical wind velocity. The implication being that the mass of the system was constant, which simplified the problem considerably compared to EC deployments where the upper boundary is defined by the fixed measurement plane (Box 1). Nevertheless, the consideration of open system thermodynamics is crucially important and something that has been forgotten for many years.

From the perspective of contemporary EC applications there are several salient points to be made regarding the question of “what is an eddy flux?” and how theoretical developments influenced contemporary EC applications. Notably, early work was not strictly motivated by researchers aiming to measure ecosystem‐atmosphere mass and heat exchanges. Rather, interests were focused on understanding atmospheric fluid mechanics and thermodynamics, including scales that were up to the general circulation (Priestley and Sheppard 1952; van Mieghem ¹⁹⁵¹). Indeed, the framing of K‐theory was often a thin layer of the atmosphere and a relatively large horizontal extent, what Priestley and Sheppard (1952) refer to as a “differential problem”; they further pointed out that the more important question is that of a flux at a boundary—an “integral problem.” It is an integral problem that is most relevant for EC applications for measuring ecosystem fluxes.

In applying EC for single‐tower NEE measurements, the Reynolds decomposition (Equation 1) is a pivotal step for the practical application of the technique because offset errors (i.e., bias) in measurements needed for computing fluxes. In applying Reynolds decomposition and averaging, the eddy flux, $\overline{w\prime {\rho }_s^{\prime }}$, is determined by the fluctuating quantities, which can be measured with high precision and sufficient accuracy. However, this still leaves $\overline{w} \overline{{\rho }_s}$, which is sensitive to offset errors in both the vertical wind and scalar measurements (and the former in particular), to be estimated. In fact, it was precisely this problem that was stated as motivation for the classic work of Webb et al. (1980), and which formed the basis of all other early solutions to the so‐called “density corrections” (Fuehrer and Friehe 2002). Fuehrer and Friehe (2002) provide an excellent review of the work preceding WPL, all of which focused on parameterizing $\overline{w}$, which cannot be measured accurately using current sonic anemometers. Reynolds decomposition also works together with Taylor's hypothesis to enable the derivation of spatial information about the turbulence field from time series measured at a fixed point.

Finally, from a practical perspective, since the 1970s the development of EC as a technique has largely been motivated by desires for a way to measure the exchanges of heat and mass between ecosystems and the atmosphere. There has been some ambiguity owing to imprecise and interchangeable terminology; for example, “eddy fluxes,” “turbulent fluxes,” or just “flux” being used in reference to the total flux across a measurement plane (Kowalski et al. 2021). We thus wish to emphasize that in the context of what is of interest to most EC practitioners, the eddy flux is only a part of the total vertical flux, and thus only a part of the target variable of interest, which is NEE, which must account for changes in storage of the atmospheric constituent below the measurement height. Indeed, the challenge of measuring the total vertical flux was identified in the first paper on EC measurements (Swinbank 1951) and driven home by Taylor and Dyer (1958) in stating some assumptions of EC for measuring evaporation, “The validity of this equation $\left[\overline{\left(\mathit{\rho w}\right)\prime q\prime }\right]$ demands that there be no vertical transport of water vapor by mean vertical motion nor accumulation of water vapor between the ground, and the level of the instrument;” here, ρ is moist air density (kg m⁻³), q is specific humidity (kg kg⁻¹).

4. How Do We Apply Eddy Covariance to Measure Net Ecosystem Exchanges of Gases and Heat?

In the 1990s, commercially produced, rugged instrumentation became available, and EC transitioned from use in “golden days” experiments to permanent deployments where the goal was obtaining annual integrated flux estimates. It was at this time that the term “net ecosystem exchange” was introduced in the first paper reporting on annual integrated net CO₂ exchange (Wofsy et al. 1993). The NEE (of CO₂) was first defined in terms of its relation to the gross biological carbon fluxes (gross primary productivity and ecosystem respiration) before formulating as a general EC observable that considers conditions of non‐steady state for which Equation (1) would not yield the true ecosystem gas exchange when there are changes in storage below the measurement height.

We wish to emphasize that the term NEE should be adopted in the general sense (Finnigan et al. 2003; Gu et al. ²⁰¹²; Lee ¹⁹⁹⁸) and not restricted to the case of carbon dioxide. It is unfortunate that NEE came to be associated exclusively with CO₂. In general, it is NEE that most practitioners of EC wish to measure—the exchanges of mass or heat between all surfaces in the control volume and the atmosphere.

Eddy covariance deployment considerations are often stated as having requirements of horizontal homogeneity of surface conditions, relatively flat terrain, and stationarity of scalar fields (Meyers and Baldocchi 2005). In practice, most flux towers are in heterogeneous landscapes (Chu et al. 2021), and requirements are unlikely to be met in a strict sense. In situations with more complex terrain, the associated multi‐dimensional flows require more involved theory and measurements to apply EC (Belcher et al. 2008; Gu et al. ²⁰¹²; Leuning ²⁰⁰⁴; Paw U et al. ²⁰⁰⁰; Raupach and Finnigan ¹⁹⁹⁷).

The main theoretical requirement limiting single‐tower EC in practice from a measurement perspective is the need for sensors that can resolve the high frequency turbulent fluctuations of the atmospheric properties of interest (Garratt 1975).

The first application of EC was to measure momentum flux (Scrase 1930). As technology advanced EC was then applied to measure sensible and latent heat fluxes (Cramer and Record 1953; Dyer ¹⁹⁶¹; Dyer and Maher ¹⁹⁶⁵; Panofsky ¹⁹⁵³; Swinbank ^{1951, 1955}; Taylor and Dyer ¹⁹⁵⁸) and later net carbon dioxide (CO₂) fluxes (Bakan 1978; Desjardins and Lemon ¹⁹⁷⁴; Jones and Smith ¹⁹⁷⁷). With further technical advances, EC has been applied to other trace gases (Asaf et al. 2013; Aurela et al. ¹⁹⁹⁶; Edwards et al. ¹⁹⁹⁴; Karl et al. ²⁰⁰¹; Sun et al. ²⁰¹⁵; Verma et al. ¹⁹⁹²; Wienhold et al. ¹⁹⁹⁴) and aerosols (Buzorius et al. 2003).

It is instructive to note the very earliest forays into EC applications encountered challenges that have persisted through time. First, while acknowledging that the total vertical flux is composed of mean convective and eddy flux components, expressions of the heat fluxes were derived to eliminate the former because of limitations of being able to directly measure the total flux (Swinbank 1951, 1955). The problem of constraining the mean vertical wind velocity and/or the mean convective flux remains a yet‐to‐be fully resolved problem that became particularly noteworthy as EC was applied to trace gas flux measurements (Finnigan 2009; Fuehrer and Friehe ²⁰⁰²; Gu ²⁰¹³; Gu et al. ²⁰¹²; Lee and Massman ²⁰¹¹; Massman and Tuovinen ²⁰⁰⁶; Rannik et al. ²⁰⁰⁹; Webb et al. ¹⁹⁸⁰).

Additionally, the first formal checks on the EC‐derived sensible heat fluxes through assessing the energy balance suggested an underestimation of up 20% (Swinbank 1955). Dyer (1961) implicated the “limited speed of response of recording equipment” (i.e., galvanometers) as the cause of high frequency flux attenuation. Recording to magnetic tapes allowed for higher sampling rates and decreased high frequency flux losses, however, a single tape‐recorder had a channel‐to‐channel standard deviation of 7% when recording the signal from one sonic anemometer (Tsvang et al. 1973). Despite the best efforts of the EC community, the energy balance closure problem persists (Mauder et al. 2020, 2024). Furthermore, notwithstanding the significant advances in instrumentation, the problem of high frequency flux attenuation has remained a concern (Fratini et al. 2012; Ibrom et al. ²⁰⁰⁷; Smidt et al. ²⁰²⁵).

We next present theoretical developments in four important areas that have relevance to EC measurements of NEE: (i) measuring the total vertical flux, (ii) flux attenuation, (iii) coordinate rotations, and (iv) energy balance closure.

4.1. Measuring the Total Vertical Flux

From a physical perspective, when heat fluxes are directed upward, away from the ecosystem, rising parcels of air must be, on average, warmer and less dense than descending ones, and there must therefore be a nonzero, positive $\overline{w}$ (Webb et al. 1980). The problem is that in practice, $\overline{w}$ is impossible to directly measure with the accuracy needed for EC applications.

The problem was addressed by WPL, considering a trace gas flux across a horizontal plane at the measurement height (i.e., the total vertical flux, not considering changes in storage below the EC system). Unfortunately, the WPL theory became known as the so‐called “density corrections”—but is more accurately conceived as a constraint on $\overline{w}$ (Finnigan 2009; Fuehrer and Friehe ²⁰⁰²). The key to the WPL approach was the constraint on $\overline{w}$ of no source/sink of dry air at the ground and thus no vertical flux of dry air (Lee and Massman 2011). The WPL derivation, valid only for conditions of steady state (i.e., there is no change in the storage of s below the measurement height), led to the familiar form of the equation, written here for gas molar densities:

$$\overline{F_s}=\overline{w\prime {\rho }_s^{\prime }}+{\overline{\chi }}_s\left[\overline{w\prime {\rho }_v^{\prime }}+\overline{\rho }\overline{w\prime T\prime }/\overline{T}\right]$$ (2)

where $\overline{\rho }$ is the moist air molar density, χ_s is the mole mixing ratio of trace constituent s (mol mol⁻¹‐dry air), ρ_v is the molar density of water vapor (mol m⁻³) and T is thermodynamic air temperature (K). The WPL work has guided EC measurements since its publication and a pivotal development for enabling the establishment of flux networks (Lee and Massman 2011).

Owing to new and old challenges like implausible observations of CO₂ flux dynamics and the energy imbalance problem, the WPL theory was revisited in the late 1990s and early 2000s (Finnigan 2009; Fuehrer and Friehe ²⁰⁰²; Gu et al. ²⁰¹²; Lee ¹⁹⁹⁸; Leuning ^{2004, 2007}; Paw U et al. ²⁰⁰⁰). In broad terms, this work led to more complete and generalizations of the original WPL theory (Lee and Massman 2011). Fuehrer and Friehe (2002) provided a more complete treatment of the dry air constraint, including higher order terms in the expansion of the ideal gas law than did WPL. Leuning (2007), deriving from mass conservation, extended the theory to non‐steady state conditions with the more general constraint of no net ecosystem source/sink of dry air. Meanwhile, other treatments extended the theory for heterogeneous flows (Leuning 2004; Massman and Lee ²⁰⁰²; Paw U et al. ²⁰⁰⁰). Finally, Gu et al. (2012) derived a generalized non‐steady state theory for homogeneous and heterogeneous flows, that can be constrained by conservation of any other atmospheric constituent; with the constraint of no net ecosystem source/sink of dry air and for homogeneous flow, the approximate equation collapses to that of Leuning (2007), that is, Equation (B1.2).

The convergent solutions of the extended WPL theory are all subject to the constraint of no net ecosystem source/sink of dry air. Fundamentally, we cannot measure dry air density directly; we must derive by applying the ideal gas law and infer it from other measurements. Whereas WPL makes (reasonable) approximations to eliminate some terms in the derivation, Fuehrer and Friehe (2002) include a more complete expression. Gu et al. (2012) in generalizing the EC theory of gas exchange measurements, suggested that using a single gas species such as dinitrogen or argon as constraining constituents would have advantages if fast response analyzers were available. Additionally, Gu (2013) proposed using the ratio of the exchanges of molecular oxygen (O₂) to CO₂ as an improved constraint over dry air, though it does require O₂ mixing ratios to be measured. Taken together, constraining $\overline{w}$ through assuming no net ecosystem source/sink of dry air, while both appealing in theory and practice, is imperfect because the total vertical flux of trace gases, which have similar magnitudes of eddy and convective fluxes, can be sensitive to $\overline{w}$.

Another consequence of WPL's dry air constraint is that if the concentration of a target gas is expressed as a mixing ratio, a simplified expression for the total vertical flux is obtained for steady state conditions:

$$\overline{F_s}=\overline{{\rho }_d}\overline{w\prime {\chi }_s^{\prime }}$$ (3)

where ρ _d is the dry air molar density (mol m⁻³). According to Paw U et al. (2000), it became common for micrometeorologists to favor using Equation (3) and to apply coordinate rotations to force $\overline{w}$ to zero to allow the mean convective term to be eliminated. This was not the reason coordinate rotations were developed sensu stricto, and moreover, is problematic from a theoretical perspective because gas analyzers do not directly measure χ _c . Furthermore, regardless of whether fluxes are computed using Equations (2) or (3), if not at steady state, storage below the EC measurement height must be taken into account to enable calculation of NEE.

Finally, it is important to note that the above discussion was restricted to mass fluxes (as well as latent heat fluxes which are determined by multiplying the measured water vapor fluxes by the latent heat of vaporization) and did not strictly apply to sensible heat fluxes. WPL derived the commonly used expression for sensible heat flux (H), which neglects mean transport associated with mass flow, invoking the concept of an ill‐defined base temperature, which eventually gets waved away to arrive at:

$$H\cong \left(c_{p,d}\overline{{\rho }_d}+c_{p,v}\overline{{\rho }_v}\right)\overline{w\prime T\prime }\cong c_p\overline{\rho }\overline{w\prime T\prime }$$ (4)

where c _p is specific heat capacity at constant pressure (J mol⁻¹ K⁻¹) and d and v subscripts denote dry air and water vapor, respectively. Equation (4) was presented by WPL as representing the total vertical sensible heat flux across the measurement plane.

After the WPL theory was revisited in the early 2000s and the advancement of mass conservation‐based derivations, there was an implicit (e.g., Leuning et al. ²⁰¹²) or explicit (e.g., Finnigan et al. ²⁰⁰³) extension to sensible heat fluxes by considering the EC mass conservation equations to be identical and applicable to energy conservation, with temperature being the scalar. The energy conservation problem is, however, more complicated because energy can change forms and the EC control volume is an open thermodynamic system (van Mieghem ^{1935, 1951}). We contend that the theory underlying conventional sensible heat flux calculations requires revision and must be derived from total energy conservation and consider coupled heat and mass transfer.

In summary, we believe that further consideration of the theory of sensible heat flux measurements and constraints on $\overline{w}\overline{{\rho }_s}$ are needed.

4.2. Spectral Corrections

The first (co)spectral analyses of turbulence of the form familiar today were reported in the 1950s, for example, Panofsky and McCormick (1954). There was much interest in developing fundamental knowledge of the spectral nature atmospheric turbulence (Griffith et al. 1955; MacCready ¹⁹⁵³; Panofsky ¹⁹⁵³; Panofsky and van der Hoven ¹⁹⁵⁵). It was also recognized that cospectral analysis was a valuable tool for assessing the limitations of EC imposed by frequency responses of the instrumentation (Businger et al. 1967; Garratt ¹⁹⁷⁵; McBean ¹⁹⁷²; Panofsky and Mares ¹⁹⁶⁸; Tsvang et al. ¹⁹⁷³). Early EC applications (McIlroy ¹⁹⁵⁵; Swinbank ^{1951, 1955}) were constrained by technology. Interestingly, it was the response time of recording data that limited the high frequency response of the earliest systems, while the laborious method of flux calculations limited the length of averaging period (Dyer 1961). Immediate concerns about the high frequency response of EC systems motivated various forms of cospectral analysis (Deacon 1959; Swinbank ¹⁹⁵¹).

Since no EC system is perfect, the need for spectral corrections persists (Fratini et al. 2012; Ibrom et al. ²⁰⁰⁷; Smidt et al. ²⁰²⁵). The concept underpinning spectral corrections is that the EC system must resolve all turbulent motions contributing to the flux, which is expressed mathematically as:

$$\overline{w\prime {\rho }_s\prime }={\int }_0^{\infty }{\mathit{Co}}_{w{\rho }_s}\left(\omega \right)\mathit{d\omega }$$ (5)

where Co _wρs is the cospectral density between w and ρ_s and ω is the angular frequency, which is equal to $2\mathit{\pi f}$, where f is ordinary frequency (D. Baldocchi ²⁰⁰³). The covariance, $\overline{w\prime {\rho }_s\prime }$, represents the contributions of all “flux‐carrying eddies,” and so the integration on the right‐hand side of Equation (5) must be done over all relevant frequencies (Garratt 1975). In principle, direct measurement of the eddy flux, $\overline{w\prime {\rho }_s\prime }$, requires the measurement of atmospheric properties at the same instant and location, which is in practice, impossible because no EC system is perfect (Box 2). In practice, sensors are limited by the frequency response, also called bandwidth. Moreover, any physical separation of sensors introduces a lag between signals and phase distortions that decreases the magnitude of the measured covariance.

BOX 2. The challenge of measuring the eddy flux.

There has been much work on the problem of flux attenuation, the systematic underestimation of eddy fluxes because of imperfect sensor (and system) responses (Massman and Clement 2005). The two most common approaches to spectral corrections are the theoretical transfer function method and empirical, in situ techniques. The theoretical transfer function method, initially developed by Moore (1986), has been updated through time as instrumentation advanced (Horst 1997, 2000; Massman ²⁰⁰¹) and to address averaging procedures (Kaimal et al. 1989; Rannik ²⁰⁰¹; Rannik and Vesala ¹⁹⁹⁹). Regardless of the approach, in simple terms, a multiplier is determined and applied to boost the uncorrected covariance magnitude.

The transfer function approach involves convolving a series of transfer functions based on theoretical knowledge of sensor responses that are applied to model cospectra and compared to measured cospectra to estimate flux attenuation. The strength of the method is a grounding in solid theory; however, there are difficulties in practice (Massman and Clement 2005; Nordbo et al. ²⁰¹⁴). One is the requirement of spectral models like those derived from the Kansas experiments over a flat wheat field (Kaimal et al. 1972). Additionally, the measured cospectra for any individual averaging period will be noisy.

The empirical, in situ approach to spectral corrections is a family of methods (Hicks and McMillen ¹⁹⁸⁸; Koprov and Sokolov ¹⁹⁷³; Massman and Clement ²⁰⁰⁵). The general idea is that, assuming similarity, attenuated cospectra are compared to a measured reference cospectrum that is assumed to be unattenuated. In this way, the reliance on model cospectra is eliminated. However, there is the assumption that the reference cospectrum suffers from no attenuation, which in theory is impossible. Furthermore, similarity among cospectra must be assumed, and this is not always true (Katul and Hsieh 1999). The problem of noisy measured cospectra is also an issue with the in situ technSince the mid‐2000s, work on spectral corrections has favored empirical approaches (Aslan et al. 2021; Fratini et al. ²⁰¹²; Ibrom et al. ²⁰⁰⁷; Smidt et al. ²⁰²⁵).

There have also been concerns regarding the attenuation of low frequency fluxes because of having to use finite averaging periods (Businger et al. 1967; Kaimal et al. ¹⁹⁸⁹; Malhi et al. ²⁰⁰⁵; Sakai et al. ²⁰⁰¹). However, compared to the problem of high frequency flux attenuation, the picture is less clear for low frequency losses. Furthermore, disentangling wide‐scale advection and locally relevant fluxes is challenging (Malhi et al. 2005). We would also point out that in theory, for conservation‐based derivations of net ecosystem exchange (Equation B1.2) (Gu et al. 2012; Leuning ²⁰⁰⁷) measuring a meaningful total vertical flux is constrained by sampling for long enough to derive a robust covariance statistic and does not necessarily have to sample all low frequency atmospheric transport in that averaging period.

4.3. Coordinate Rotations

Thorough reviews of coordinate systems and rotations for interpreting EC measurements have been provided elsewhere, and we refer the reader to these substantive treatments (Finnigan 2004; Lee et al. ²⁰⁰⁵; Paw U et al. ²⁰⁰⁰; Wilczak et al. ²⁰⁰¹). Though commonly referred to as coordinate systems or frames, it is the associated basis vector that is defined by rotations in most practical situations (Finnigan et al. 2003; Lee et al. ²⁰⁰⁵). Here, we highlight some salient points.

Coordinate rotations have become a common step in the EC data processing pipeline (Finnigan et al. 2003; Lee et al. ²⁰⁰⁵; Paw U et al. ²⁰⁰⁰). The history of coordinate rotations traces back to concerns of sonic anemometer tilt biasing Reynolds stress (i.e., $\overline{u_i^{\prime }{u}_j^{\prime }}$, where the subscripts denote different velocity components) measurements because of the horizontal wind components contaminating w (Deacon 1968; Kraus ¹⁹⁶⁸; Pond ¹⁹⁶⁸). Kaimal and Haugen (1969) suggested that accurate Reynolds stress measurements required sonic anemometers to be leveled to within ±0.1° of the horizontal plane defined by a uniform horizontal surface (N.B. scalar fluxes are less sensitive to tilt errors than are the Reynolds stresses; Lee et al. ²⁰⁰⁵). If the tilt angles are known, measurements made in the instrument coordinate can be rotated so that horizontal and vertical wind velocity components are parallel and perpendicular to the surface, respectively (Kaimal and Haugen 1969; Wilczak et al. ²⁰⁰¹). However, this is not usually practical, and motivated further consideration of how rotations might solve the “tilt problem.”

The natural wind system developed by Tanner and Thurtell (1969) for homogeneous flow (i.e., a one‐dimensional surface layer), aligns the x‐axis with the streamline (i.e., the mean flow) at the anemometer to force the mean crosswind and vertical wind velocities, and the crosswind Reynolds stress ($\overline{v\prime w\prime }$) to zero; though forcing $\overline{v\prime w\prime }$ to zero often yielded physically unrealistic results and thus ceased to be implemented (Finnigan 2004; Finnigan et al. ²⁰⁰³; Lee et al. ²⁰⁰⁵). The natural wind system is fairly simple and can be implemented with online flux calculations and became the most used coordinate rotation. Another, perhaps unintended, reason for adoption of the natural wind rotation was the desire to force $\overline{w}$ to zero to ease the application of Equation (1) when the gas concentration was expressed as a mixing ratio (Paw U et al. ²⁰⁰⁰). Forcing $\overline{w}$ to zero for each averaging period is, however, problematic because real flows will have non‐zero mean vertical velocities varying over time and is likely to introduce bias that is particularly important when integrating fluxes (Finnigan et al. 2003; Lee et al. ²⁰⁰⁵; Massman and Lee ²⁰⁰²). To overcome these challenges, physical streamline coordinate systems were developed (Finnigan 2004; Paw U et al. ²⁰⁰⁰; Wilczak et al. ²⁰⁰¹).

The family of physical coordinate systems are defined by the flow field (Finnigan 2004; Lee et al. ²⁰⁰⁵). In this approach, one set of rotation angles are computed from analyzing data collected over several weeks (or more) such that the long‐term mean $\overline{w}$ is zero, but for individual averaging periods it is allowed to be non‐zero, which is a more realistic representation of the actual flow. Additionally, with physical coordinate systems, the alignment of the anemometer relative to the ground need not be known nor are the relative orientations (Finnigan 2004). However, it is critical that the orientation of anemometers does not change during the long‐term period used to compute the rotation angles (Wilczak et al. 2001). For these reasons, physical streamline coordinates are attractive for applications where deployments are relatively static over time and the problem of complex flows acute (e.g., over forests and complex terrain). Thus, in general, physical streamline coordinates should be preferred. However, this may not always be practical, like over rapidly growing crops where the tower configuration is changed frequently.

4.4. Energy Balance Closure

The lack of energy balance closure has remained a recalcitrant challenge to the EC community (Aubinet et al. 2000; Baldocchi et al. ²⁰⁰¹; Foken ²⁰⁰⁸; Mauder et al. ^{2020, 2024}; Swinbank ¹⁹⁵⁵; Wilson et al. ²⁰⁰³) and motivated intense efforts to find a solution. Given the technical requirements of EC, it should come as no surprise that instrumentation and data post‐processing have been suspected as causes of the energy imbalance problem (Dyer 1961), a notion that remained a theme through time (Foken 2008; Foken and Oncley ¹⁹⁹⁵; Mauder et al. ²⁰⁰⁷; Oncley et al. ²⁰⁰⁷). However, given the large body of work, instrumentation and data processing have now been ruled out as the likely cause (Leuning et al. 2012; Mauder et al. ²⁰²⁰). It is important to consider that as EC matured and applied over productive, dense, and/or tall canopies, it has become clear that biomass and biochemical energy storages are collectively non‐trivial if attempting to achieve energy balance closure (Blanken et al. 2001; Gu et al. ²⁰⁰⁷; Meyers and Hollinger ²⁰⁰⁴); though these terms are often neglected (Mauder et al. 2020), thereby magnifying non‐closure.

We use the general term, biochemical heat storage to represent the net storage of chemical energy, which reflects the balance of ecosystem photosynthesis and respiration. Sometimes called “photosynthetic energy storage,” the often‐used definition is $-0.48\times {\mathrm{NEE}}_{{\mathrm{CO}}_2}$, where the multiplier (J/μmol) is the standard free energy change of the balanced photosynthesis reaction, assuming glucose is the product: ${\mathrm{H}}_2\mathrm{O}+\mathrm{C}{\mathrm{O}}_2\to \frac{1}{6}\left(\text{glucose}\right)+{\mathrm{O}}_2$. Note that this definition represents a simplified view of the problem of biological energy storage. An important point to note is that if estimating biochemical energy storage from EC‐derived CO₂ fluxes, it is the NEE of CO₂ that should be used and not gross primary productivity.

The favored hypothesis to explain EC energy balance closure is horizontal advection that cannot be resolved from single tower measurements (Butterworth et al. 2024; Mauder et al. ²⁰²⁰; Wang et al. ²⁰²⁴). The energy balance closure problem is global in that it affects EC sites all around the world (Mauder et al. 2024; Wilson et al. ²⁰⁰²), therefore, a hypothesized explanation should pass a universality test. In other words, does it apply everywhere? Although we acknowledge advection is certainly a possible source of error in the ecosystem energy balance, we wonder how universal it would be. In other words, is advection more site‐specific than universal? We suspect that advection cannot universally explain the underestimation of available energy at all flux sites as it requires the presence of flux divergence and no flux convergence at all sites and at all times.

Given that over the ~75 years since the first EC measurements, there was much emphasis on technical advances and algorithms, we ask the question, does our theory of EC measurements of net ecosystem heat exchanges need revising? Contemporary EC heat flux measurements are guided by the WPL theory that has not changed—but if it does need revision, this would represent a universal source of bias.

5. Novel Theories: Where Do We Go From Here?

In the preceding sections, we identified two key areas where we believe further consideration is warranted: revisiting the theory (i) underlying sensible heat flux calculations, and (ii) of constraints on $\overline{w}$. We focus our attention here on the first point, the theory of sensible heat flux measurements.

The expression for the net ecosystem exchange of sensible heat (NEE _H ) is conventionally stated as follows:

$$\overline{{\mathrm{NEE}}_H}\cong {\left.c_p\overline{\rho }\overline{w^{\prime }{T}^{\prime }}\right|}_{z_m}+{\int }_0^{z_m}\rho c_p\frac{\overline{\mathrm{d}T}}{\mathrm{d}t}\mathrm{d}z.$$ (6)

Interestingly, Equation (6) was not explicitly derived from first principles in a strict sense. Rather, the mass conservation‐based derivation (Gu et al. 2012; Leuning ²⁰⁰⁷) is presented and temperature is treated as a scalar that is equivalent to mass (Finnigan et al. 2003; Leuning et al. ²⁰¹²), with the first term on the right‐hand side of Equation (6) defined as the eddy flux of sensible heat according to WPL, and the second being the sensible heat storage flux. The situation is, however, considerably more complicated because energy changes forms (Sun et al. 2021; Sun and Tribbia ²⁰²⁵). Furthermore, the atmosphere is an open thermodynamic system (Montgomery 1948; van Mieghem ^{1935, 1951}), and thus coupled heat and mass transfer must be considered.

The conventional sensible heat flux formulation is essentially an attempt to approximate the exchange of enthalpy. It uses the specific heat capacities at constant pressure (i.e., c _p,d and c _p,v ) to calculate both the storage term and the eddy flux term. From an energy balance point of view, this calculation is problematic because for an open thermodynamic system, which simultaneously exchanges mass and mechanical (kinetic and potential) energies and does work on its environment, the heat energy exchange does not equal its enthalpy change. Moreover, simultaneous heat and mass exchange is not a constant pressure process.

To reinforce these points, consider that the total energy is the sum of internal, kinetic, and potential energies. When solar radiation heats Earth's surface, the overlying air is warmed creating thermals that have internal energy defined by mass and temperature, as well as kinetic and potential energies that are determined by their mass and velocity, and their height, respectively. As thermals rise, energy can change forms, and they do work on the surrounding air. Furthermore, mass is added and removed from the atmosphere by evaporation and condensation, respectively. We have developed a theory capable of resolving these issues that will appear elsewhere. Here, we explain the key concepts of our new theory.

The correct formulation of NEE _H must be derived from first principles of fluid mechanics and thermodynamics and consider coupled heat and mass transfer so that conservation of both is obeyed. Because energy changes forms, the derivation must begin from conservation of total energy to arrive at the correct expression for NEE _H . We derived the equations for non‐homogeneous flows, and with the typical assumptions to simplify to a 1‐D problem for non‐steady state conditions (Gu et al. 2012), with the key concept encapsulated by:

$${\overline{\mathrm{NEE}}}_H=\underset{{\overline{\mathrm{NEE}}}_{\mathrm{TE}}}{\underbrace{{\overline{\mathrm{NEE}}}_{\mathrm{IE}}+{\overline{\mathrm{NEE}}}_{\mathrm{KE}}+{\overline{\mathrm{NEE}}}_{\mathrm{PE}}}}-{\overline{\mathrm{NEE}}}_{\mathrm{TEd}}-{\overline{\mathrm{NEE}}}_{\mathrm{TEv}}$$ (7)

where all terms are mean net ecosystem exchanges, and the subscripts are defined as follows: IE, KE and PE are the internal, kinetic and potential energies, respectively; TEd and TEv represent the energy exchanges associated with mass exchanges of dry air and water vapor, respectively.

Equation (7) reveals two important aspects of the new theory. First, it allows for energy interconversions that alter the internal and thus thermal energy transfer as sensible heat. Second, it accounts for changes in energy caused by mass transfer. Changes of mass and thus the internal energy of the control volume are caused by gains or losses of water vapor from evaporation and condensation, respectively.

To render Equation (7) more tangible for EC applications, we can express ${\overline{\mathrm{NEE}}}_H$ for homogeneous flows (the complete derivation for non‐homogeneous flows includes horizontal advection terms, but is not presented here) as:

$${\overline{\mathrm{NEE}}}_H={\overline{S}}_H+{\overline{F}}_{H,E}+{\overline{F}}_{H,C}.$$ (8)

The three terms on the right‐hand side of Equation (8) represent the mean storage, mean eddy flux and mean convective flux, respectively. The last two terms on the right‐hand side of Equation (8) are together, the mean total flux across the top of the control volume. The storage term, ${\overline{S}}_H$, is given by:

$$\begin{array}{c}{\overline{S}}_H=\underset{0}{\overset{z_m}{\int }}\overline{\frac{\partial {\rho }_{\mathrm{IE}}}{\partial t}}\mathrm{d}z+\underset{0}{\overset{z_m}{\int }}\overline{\frac{\partial {\rho }_{KE,A}}{\partial t}}\mathrm{d}z+\underset{0}{\overset{z_m}{\int }}\overline{\frac{\partial {\rho }_{KE,P}}{\partial t}}\mathrm{d}z\hfill \\ +\underset{0}{\overset{z_m}{\int }}\overline{\frac{\partial {\rho }_{\mathrm{PE}}}{\partial t}}\mathrm{d}z-\underset{0}{\overset{z_m}{\int }}{c}_{v\mathit{,}d}{\overline{T}}_s\overline{\frac{\partial {\rho }_d}{\partial t}}\mathrm{d}z-\underset{0}{\overset{z_m}{\int }}{c}_{v\mathit{,}v}{\overline{T}}_s\overline{\frac{\partial {\rho }_v}{\partial t}}\mathrm{d}z.\hfill \end{array}$$ (9)

Here, we express energy densities (J m⁻³), ρ _IE, ρ _KE,A, and ρ _PE are energy densities of internal, kinetic and potential energies of air in the control volume, respectively. ρ _KE,P, is kinetic energy density of the swinging plant canopies and other non‐gas elements, for example, aerosols and particulates (Table 4). The last two terms on the right‐hand side of Equation (9) are storages of energy associated with changes of dry air and water vapor in the control volume. c _v,d and c _v,v are the constant volume specific capacities of dry air and water vapor, respectively, and T _s is surface temperature (K). The eddy flux term from Equation (8), ${\overline{F}}_{H,E}$, can be decomposed into:

$${\overline{F}}_{H,E}=\overline{w^{\prime }{\rho }_{\mathrm{EN}}^{\prime }}+\overline{w^{\prime }{\rho }_{KE,A}^{\prime }}+\overline{w^{\prime }{\rho }_{\mathrm{PE}}^{\prime }}-c_{v\mathit{,}d}{\overline{T}}_s\overline{w^{\prime }{\rho }_d^{\prime }}-c_{v\mathit{,}v}{\overline{T}}_s\overline{w^{\prime }{\rho }_v^{\prime }},$$ (10)

and similarly for the mean motion convective flux:

$${\overline{F}}_{H,C}=\overline{w} \overline{{\rho }_{\mathrm{EN}}}+\overline{w} \overline{{\rho }_{KE,A}}+\overline{w} \overline{{\rho }_{\mathrm{PE}}}-c_{v\mathit{,}d}{\overline{T}}_s\overline{w} \overline{{\rho }_d}-c_{v\mathit{,}d}{\overline{T}}_s\overline{w} \overline{{\rho }_v}.$$ (11)

TABLE 4.

Definitions of energy densities required for deriving the sensible heat flux from total energy conservation.

Energy density	Definition
Internal energy, ${\rho }_{\mathrm{IE}}$	${\rho }_{\mathrm{IE}}=\left({\rho }_d{c}_{v\mathit{,}d}+{\rho }_v{c}_{v\mathit{,}v}\right)T$
Enthalpy, ${\rho }_{\mathrm{EN}}$	${\rho }_{\mathrm{EN}}={\rho }_{\mathrm{IE}}+p$
Kinetic energy in air, ${\rho }_{\mathrm{KE},\mathrm{A}}$	${\rho }_{\mathrm{KE},\mathrm{A}}=\left({\rho }_d{m}_d+{\rho }_v{m}_v\right)\frac{\mathit{u}·\mathit{u}}{2}$
Kinetic energy of non‐air objects, ${\rho }_{\mathrm{KE},\mathrm{P}}$	${\rho }_{\mathrm{KE},\mathrm{P}}=\frac{{\varrho }_P}{2}{V}^{\mathbf{2}}$
Potential energy, ${\rho }_{\mathrm{PE}}$	${\rho }_{\mathrm{PE}}=\left({\rho }_d{m}_d+{\rho }_v{m}_v\right)\mathit{gz}$
Total energy, ${\rho }_{TE}$	${\rho }_{\mathrm{TE}}={\rho }_{\mathrm{IE}}+{\rho }_{\mathrm{KE},\mathrm{A}}+{\rho }_{\mathrm{KE},\mathrm{P}}+{\rho }_{\mathrm{PE}}$

Note: ρ _d , dry air density (mol/m³); ρ _v , water vapor density (mol/m³); c _v,d , specific heat capacity of dry air at constant volume (J K⁻¹ mol⁻¹); c _v,v , specific heat capacity of water vapor at constant volume (J K⁻¹ mol⁻¹); T, thermodynamic temperature (K); p, pressure (Pa); m _d , molar mass of dry air (kg/mol); m _v , molar mass of water vapor (kg/mol); $\mathit{u}$, 3‐component wind velocity vector; ϱ _p , density of non‐air objects (kg/m³); V, mean velocity of non‐air objects (m/s); g, gravitational acceleration (9.81 m s⁻²); z, height (m).

Note that the first terms on the right‐hand sides of Equations (10) and (11) are enthalpy fluxes, with ρ _EN representing the enthalpy density. The need to introduce enthalpy for the fluxes is because heat transfer involves air parcels displacing surrounding air to move across the measurement plane, and thus work is done. This contrasts with the storage term, which represents a change of energy in a constant volume, and thus it can be conceptualized in terms of internal energy.

Here, we have not presented the full equation set with approximations needed for practical application, which will be reported elsewhere. Rather, we wish to emphasize the major conceptual aspects of the theory including considering total energy conservation, coupled heat and mass transfer, and constraining the mean convective flux. It must be stressed that although temperature is a scalar, it cannot be substituted into governing equations derived for EC measurements of mass exchanges and applied for sensible heat flux measurements. The governing equations must allow for energy transformations and how mass exchanges influence energy conservation of the control volume.

6. Conclusions

Eddy covariance is considered the gold standard for measuring ecosystem scale gas and heat exchanges and has transformed our understanding of the breathing of the biosphere. Despite numerous methodological improvements and insights gained from the technique, challenges remain. Here, we reviewed the theoretical developments underpinning EC in four important areas that have relevance to EC measurements of the net ecosystem exchange (NEE) of gases and heat. Persistent problems, such as the inability to close the energy budget, led us to identify two priorities for revisiting the theory underlying: (i) sensible heat flux calculations, and (ii) constraining the mean vertical wind velocity. We present a framework for improved calculation of sensible heat flux derived from first principles of fluid mechanics and thermodynamics that considers coupled heat and mass transfer so that conservation of both is obeyed. These refinements are motivated by the need for unbiased measurements of energy and mass transfer between the land surface and atmosphere for ecosystem research and to validate satellite observations and land surface models.

Author Contributions

Jeffrey D. Wood: conceptualization, funding acquisition, investigation, project administration, writing – original draft, writing – review and editing. Lianhong Gu: conceptualization, funding acquisition, project administration, writing – review and editing. Adam P. Schreiner‐McGraw: writing – review and editing.

Conflicts of Interest

The authors declare no conflicts of interest.

Wood, J. D. , Gu L., and Schreiner‐McGraw A. P.. 2026. “Eddy Covariance Theory: A Review.” Global Change Biology 32, no. 7: e70947. 10.1111/gcb.70947.

Data Availability Statement

No data sets were generated or analyzed in preparing this paper.