Agricultural and Forest Meteorology 113 (2002) 121–144

Eddy covariance flux corrections and uncertainties in long-term studies of carbon and energy exchanges W.J. Massman a,∗,X.Leeb a USDA/Forest Service, Rocky Mountain Research Station, 240 West Prospect, Fort Collins, CO 80526, USA b School of Forestry and Environmental Studies, Yale University, New Haven, CT 06511, USA Accepted 3 April 2002

Abstract This study derives from and extends the discussions of a US DOE sponsored workshop held on 30 and 31 May, 2000 in Boulder, CO concerning issues and uncertainties related to long-term eddy covariance measurements of carbon and energy exchanges. The workshop was organized in response to concerns raised at the 1999 annual AmeriFlux meeting about the lack of uniformity among sites when making spectral corrections to eddy covariance flux estimates and when correcting the eddy covariance CO2 fluxes for lack of energy balance closure. Ultimately, this lack of uniformity makes cross-site comparisons and global synthesis difficult and uncertain. The workshop had two primary goals: first, to highlight issues involved in the accuracy of long-term eddy covariance flux records; and second, to identify research areas and actions of high priority for addressing these issues. Topics covered at the workshop include different methods for making spectral corrections, the influence of 3D effects such as drainage and advection, underestimation of eddy covariance fluxes due to inability to measure low frequency contributions, coordinate systems, and nighttime flux measurements. In addition, this study also covers some new and potentially important issues, not raised at the workshop, involving density terms to trace gas eddy covariance fluxes (Webb et al., 1980). Wherever possible, this paper synthesizes these discussions and make recommendations concerning methodologies and research priorities. Published by Elsevier Science B.V.

Keywords: Eddy covariance; Long-term flux records; Carbon balance

1. Introduction climate, air pollution, and CO2 concentrations (Wofsy and Hollinger, 1998). Because the eddy covariance The main scientific goals of the AmeriFlux network method directly measures the net flux of CO2,itis are to: (1) understand the factors and processes regu- the logical choice for attempting measurements of the lating CO2 exchange, including soil processes, vege- net CO2 exchange to and from terrestrial ecosystems. tation structure, physiology, and stage succession, and However, implementing the eddy covariance method (2) determine principal feedbacks that may affect the can vary significantly between sites. This is partic- future of the biosphere, such as responses to changes in ularly true for CO2 flux measurements which can be measured by either open- or closed-path systems ∗ Corresponding author. Fax: +1-970-498-1314. (e.g., Leuning and King, 1992; Suyker nad Verma, E-mail address: [email protected] (W.J. Massman). 1993). Although the greatest difference in eddy

0168-1923/02/$ – see front matter. Published by Elsevier Science B.V. PII: S0168-1923(02)00105-3 122 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 covariance instrumentation is likely to be between 2. Fundamental equations of eddy covariance open- and closed-path systems, there are also differ- ences between sonic anemometer designs, sampling 2.1. Summary frequencies, processing algorithms, the relative geo- metries of the instruments, and the degree of aerody- In this section, we present the fundamental equa- namic interference by the measurement platform. To tions of eddy covariance. However, because we wish further complicate the issue of cross-site, long-term to be as general as possible, all fluxes are expressed as comparisons of net CO2 exchange is the nearly uni- 3D vectors and the gradient operator, ∇, should be un- form inability to close the surface energy balance. derstood as independent of coordinate system. Wher- At most, if not nearly all, sites the energy available ever necessary and appropriate, a coordinate system to drive evaporation, sensible heat, photosynthesis, will be specified. The five fundamental equations, de- and canopy storage almost always exceeds sum of rived in Appendix B, detail the relationships between these other processes by 10–20%. Because sensible the various fluxes. Each equation is derived in a fully and latent heat fluxes are measured by eddy covari- consistent manner with the minimum number of as- ance, the concern naturally arises about whether the sumptions and wherever appropriate include heat and net CO2 flux is also underestimated and how or moisture effects. Here we present the results primarily if to correct for this. Without some understanding as a summary and as background for later discussions. of and ability to compensate for these differences, Eq. (1) shows the relationship between the turbulent cross-site comparisons and global scale synthesis are T  3D temperature flux, v a , and the measured 3D sonic difficult and uncertain at best. In an effort to address T  virtual temperature flux, v s , the measured turbulent these site-to-site differences in flux systems and data p 3D pressure flux, v a, and the 3D vapor covariance, processing, the National Institute for Global Environ-   v ρ . [Note here throughout this paper, we use the mental Change (NIGEC) sponsored an AmeriFlux v term covariance to mean that part of the turbulent flux workshop on 30 and 31 May, 2000 in Boulder, CO exclusive of the WPL term (Webb et al., 1989 and to address eddy covariance flux corrections and un- Appendix B). The complete fluxes (or those turbulent certainties in long-term studies of carbon and energy fluxes that include the WPL term) are denoted with a exchanges. The purpose of this paper is to synthe- ρ F size, and where necessary extend, the discussions and superscript F, e.g., v v .]

 conclusions of the workshop. Wherever possible, this T  T  α¯ ( +¯χ ) ρ paper also provides recommendations on methodolo- v a = 1 v s − v 1 v v v T¯ 1 + δ λ¯ T¯ 1 + δ λ¯ ρ¯ gies and priorities for future research. a
oc v s oc v d The remainder of this paper is divided into five β¯ ( +¯χ ) p + v 2 v v a sections. The next section discusses the fundamental ¯ p¯ (1) 1 + δocλv a equations of eddy covariance. Section 3 discusses the flux loss due to physical limitations of instrumentation, where v is the 3D turbulent (fluctuating) velocity; ¯ such as line averaging effects, sensor separation, data α¯ v = 0.32µv/(1 + 1.32χ¯v); βv = 0.32χ¯v/(1 + ¯ ¯ processing, and related issues that cause spectral atten- 1.32χ¯v); λv = βv(1 +¯χv); χ¯v the volume mixing ra- uation of the flux. 2D and 3D effects, such as drainage tio or mole fraction for water vapor (=¯pv/p¯d); p¯v the and advection, are examined in Section 4. Section 5 mean vapor pressure; p¯d the mean partial pressure of discusses coordinate systems and Section 6 focuses dry air (i.e., ambient air devoid of water vapor); p¯a the specifically on night time flux issues. The paper closes mean ambient pressure (=¯pd +¯pv); µv (=md/mv) with two appendices. Appendix A lists the workshop is the ratio the molecular mass of dry air, md,to participants, speakers, and organizing committee. the molecular mass of water vapor, mv; ρ¯d the mean ¯ Appendix B provides a detailed discussion and deriva- ambient dry air density; Ts the mean temperature ¯ tion of the fundamental equations of eddy covariance. measured by sonic thermometry; Ta the mean ambi- These equations are developed in three dimensions ent temperature and δoc = 1 for an open-path sensor and include the WPL terms associated with fluxes of and δoc = 0 for a closed-path sensor. We use the δoc temperature and water vapor (Webb et al., 1980). notation to unify the mathematical development for W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 123

ρ −¯ω ρ both the open- and closed-path systems. Note here v c cv d (Webb et al., 1980; Paw U et al., 2000, that Eq. (1) assumes that the cross-wind correction Appendix B). [We note here that for Eq. (6),we ¯ to Ts (Kaimal and Gaynor, 1991) is included in the have dropped a small correction term to Sc related to sonic signal processing software and as such it does the stoichiometry of photosynthesis and respiration not explicitly appear in Eq. (1) (see Appendix B). ρ (Appendix B).] The turbulent dry air flux, v d is Eqs. (2) and (3) are the turbulent water vapor and given as   CO2 fluxes including the WPL terms as developed in vT  vp Appendix B and adapted from Paw U et al. (1989) and vρ =−ρ ¯ (1 +¯χ ) δ a − a −µ vρ (7) d d v oc ¯ p¯ v v Webb et al. (1980): Ta a

  F   Although not all issues raised by these equations were v ρ = (1 +¯χv)v ρ v v   discussed at the workshop, it is important for the pur- T  p +¯ρ ( +¯χ ) δ v a − v a poses of the workshop and this paper to discuss some v 1 v oc ¯ p¯ (2) Ta a of the implications of these equations to the practice of eddy covariance.   T  p ρ F = ρ +¯ρ ( +¯χ ) δ v a − v a 2.2. Some implications v c v c c 1 v oc ¯ p¯ Ta a +¯ω µ ρ The equation of mass conservation, Eq. (6),isthe c vv v (3) basis for long-term studies of the CO2 budget. The where ρ¯c is the mean ambient CO2 density, ω¯ c traditional method of obtaining (an approximate) CO2 (=¯ρc/ρ¯d) the mean mass mixing ratio for CO2 and ρ¯v budget over a 24 h period, usually involves the vertical the mean ambient water vapor density. As discussed component of Eq. (6) integrated over the vertical depth in Appendix B, these two equations are generaliza- extending from the soil surface to height of the flux tions of the original Webb et al. (1980) formulations. measurement. The storage (integral of the time rate of The major quantitative difference between Eqs. (2) change term) and flux terms (integral of the flux diver- and (3) and the corresponding formulations in Webb gence term) are each measured and summed over 24 h et al. (1980) is the 3D formulation and the inclusion (e.g., Moncrieff et al., 1996; Lee, 1998). However, to p of the pressure flux term, v a. date none of the CO2 budget studies have included the ρ · The total 3D water vapor (Fv) and CO2 (Fc) fluxes second term of the left-hand side of Eq. (6),[v d are presented by Eqs. (4) and (5). These equations ∇¯ωc−Vω¯ c·∇ρ ¯d], here called the quasi-advective term. differ from Eqs. (2) and (3) only by the inclusion of Because of the component involving the dry air flux,   the mean flow terms, Vρ¯v and Vρ¯c: ρ ·∇¯ω v d c, in the quasi-advective term, Eqs. (6) and F (7) suggest that the vertical profiles of the water vapor, F = Vρ¯ + vρ (4) v v v temperature, and pressure fluxes may also need to be = ρ¯ + ρ F measured. The potential importance of the dry air gra- Fc V c v c (5) dient term, Vω¯ c ·∇ρ ¯d, is less clear. Under most condi- where V is the mean 3D velocity vector. tions, this term should be negligibly small. We expect The general equation for CO2 mass conservation this because dry air is likely to be well mixed so that for application to long-term ecosystem studies of the horizontal components of ∇¯ρd are probably insignif- CO2 budget is given as icant in most situations. Over the depth of the profile measurements hydrostatic conditions do not appear to ∂ω¯ c   contribute significantly to ∇¯ρ and mean velocities, ρ¯d + [v ρ ·∇¯ωc − Vω¯ c ·∇¯ρd] d ∂t d V, are generally quite low within a canopy. +∇·( ρ¯ + ρ −¯ω ρ ) = S¯ V c v c cv d c (6) Concerning the attenuation of temperature fluctu- ations for closed-path systems, we have found only ¯ where t is the time, Sc the mean source/sink term for one study that measures the attenuation of temper- −¯ω ρ ρ F = CO2, cv d the WPL term for CO2 and v c ature fluctuations within a cylindrical tube. Frost 124 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144

(1981) found that turbulent temperature fluctuations of) addressing the behavior of pressure fluctuations in wp ≤ were reduced to the level of instrument noise beyond turbulent tube flow. However, because a 0, the a downstream distance greater that about 11 tube possibility exists that any long-term CO2 studies may diameters. This observation should be useful in en- have a bias in NEE resulting from ignoring this term T  → suring the validity of a 0 for current and future during turbulent high wind speed conditions. For ex- wp ≈− −1 closed-path eddy covariance systems. ample, assuming that a 10Pams (Fig. 1), 5 −3 Throughout this study, we have included the pres- p¯a ≈ 10 Pa, and ρ¯c ≈ 675 mg m , then the verti- p −¯ρ wp /p¯ sure flux term, v a, because there are special circum- cal pressure flux term, c a a,ofEq. (3) is ap- −2 −1 stances under which it may be important. Fig. 1 is proximately +0.06 mg CO2 m s which can be a wp a time course of the vertical pressure flux, a, for significant fraction of either the daytime or nighttime 2 days in January 2000. Also included on this fig- CO2 flux. Over the course of a year this term would −wp ρ¯ u3 u −1 ure is the ratio of a to a ∗, where ∗ is the yield an additional 5.1tCha to the annual carbon friction velocity. These are half hourly eddy covari- balance of a (perpetually turbulent and windy) site. |wp | ance data obtained at a high elevation (3200 m) alpine But, because the magnitude of a is usually less site in southern Wyoming USA at a height of 27.1 m than 10 Pa m s−1, this additional 5.1tCha−1 is likely above the ground over a forest of approximately 18 m to be the maximum possible amount. in height. Mean wind speeds during this period were In closing, the purpose of this section (and between 5 and 15 m s−1 and exceeded 10 m s−1 for Appendix B as well) is to provide a framework that − several hours at a time and u∗ exceeded 1 m s 1 at all unifies the elements and discussions of the work- |(wp /p¯ )/(wT /T¯ )|≥ times. During this period a a a a shop. Although the workshop did not specifically 20%. In other words, during periods of high winds focus on these fundamental equations, their presen- and significant turbulence the pressure flux can con- tation here allows each subject covered at the work- −¯ω ρ shop to be referenced to a process or an equation, tribute to the WPL term, cv d, for CO2 or any other trace gas. Therefore, for an open-path system thereby allowing them to be more precisely defined the pressure flux can be relatively significant. But, the and quantified. The next section discusses the spectral implications to a closed-path systems are less obvi- corrections associated with each of the covariance ous because there have been no studies (we are aware terms (given on the right-hand side) of Eqs. (1)–(7).

wp −1 Fig. 1. Time course of half hourly eddy covariance data. The bottom curve is the vertical pressure flux, a (Pams ), the top curve is −wp /ρ¯ u3 u the non-dimensionalized pressure flux, a a ∗ (where ∗ is the friction velocity), and the zero line if highlighted. Data taken at a high elevation site in southern Wyoming USA between 14 and 16 January, 2000 under neutral, windy, and very turbulent atmospheric conditions − (u∗ ≥ 1ms 1). Fluxes include spectral corrections using spectra developed from data obtained at the site. The non-dimensionalized flux −wp /ρ u3 ≈ z/L ≈ indicates that a a ∗ 2 for 0 in agreement with the observations of Wilczak et al. (1999). W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 125 √ 3. Flux loss due to physical limitations where j = −1, ω = 2πf , f the input forcing of instrumentation frequency (Hz) and AI the amplitude of the input forc- ing. Note that throughout this study, we use complex All eddy covariance systems attenuate the true tur- notation because it simplifies the analysis. bulent signals at sufficiently high and low frequencies The transfer function of a linear first-order sensor (e.g., Moore, 1986). This loss of information results (system), h1(ω), is the ratio of the output signal to the from limitations imposed by the physical size of the input signal, XO(t)/XI(t),or instruments, their separation distances, their inherent φ (ω) 1 1 + jωτ ej 1 time response, and any signal processing associated h (ω) = = 1 =  (10) 1 − ωτ 2 2 1 j 1 1 + ω τ + ω2τ 2 with detrending or mean removal (Moore, 1986; 1 1 1 Horst, 1997; Massman, 2000, 2001; Rannik, 2001). There are a variety of ways to assess and correct the where φ1(ω) is the phase of the filter and is defined −1 raw covariances for this loss of information. However, as tan (Im[h1(ω)]/Real[h1(ω)]). the workshop focused primarily on two methods. One Although not specifically derived by Horst (1997) method, proposed by Goulden et al. (1997), is termed or Massman (2000), Eq. (10) is the same function the low-pass filtering method, and the other, proposed they use in their analyses. The major advantage of this by Massman (2000, 2001), is termed the analytical general methodology of describing dynamic charac- approach. While neither is perfect or the ultimate so- teristics of sensors is that it allows the use of Fourier lution to the problem of flux loss, comparison of the analysis to describe complex input and output signals two methods showed the strengths and weaknesses of in terms of an amplitude and phase characteristics. both. But, before discussing these two methods and Because the system is linear the superposition prin- detailing the differences between them, we must first ciple applies and the input and output signals can define the concept of a transfer function, a first-order be decomposed into their individual spectral compo- filter, and a low-pass recursive filter. nents. Another advantage of this general approach is that it has a direct analog in electrical circuit design. 3.1. Preliminaries For example, Eq. (10) is the same equation that de- scribes an RL-circuit (e.g., Eugster and Senn, 1995) The basic premise for describing the physical or an RC-circuit (or RC-filter). In the case of an characteristics and behavior of a sensor or measur- RC-filter the time constant, τ1, is specifically iden- ing instrument is that its dynamic performance can tified as RC, the product of the circuit’s resistance, be modeled by an appropriate differential equation. R, and capacitance, C. Consequently the terminology The behavior of an ideal first-order instrument (or used in circuit analysis and filtering can be applied to system) is defined by the following linear first-order sensor input and response. non-homogeneous ordinary differential equation: The first-order transfer function, Eq. (10), shows ω → h (ω) → dXO that for low frequencies (i.e., 0) that 1 1 τ + X = X (t) (8) ω →∞ h (ω) → 1 dt O I and that for high frequencies ( ) that 1 0. Therefore, the filter defined by Eq. (10) passes the where X is the input or forcing function, X the I O low frequencies relatively unaffected and attenuates output or response function, t the time and τ the 1 the high frequencies, thereby, defining a low-pass instrument’s time constant. Eq. (8) can be used to filter. The corresponding first-order high-pass filter, assess the system’s response to any type of forcing, hHP(ω), is the complement of h (ω), i.e., hHP(ω) = but the response to sinusoidal forcing is of great- 1 1 1 1 − h (ω). est interest because it is the basis for describing the 1 To this point, we have assumed that the input and response to much more complicated forcing. The output signals are continuous functions. In addition, steady-state solution to Eq. (8) assuming sinusoidal we can also define the low-pass recursive filter in terms input, X (t) = A exp(−jωt),is I I of a discretely sampled time series, noting that for any −jωt AI e XI(t) given filter applicable for continuous input and output, XO(t) = = (9) 1 − jωτ1 1 − jωτ1 there is always an analog for discretely sampled input 126 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 and output. Consider a discrete equally spaced time Although Eqs. (10) and (12) may appear different their series, xi, where i = 1, 2, 3,... indicates the time ti moduli are functionally similar. This is the basis of at which the data are sampled. The difference equation Massman’s (2000, Table 1) claims that the equivalent for a first-order low-pass recursive filter is defined as first-order time constant for hr(ω) is equal to τr and follows: for many references to Eq. (11) as an RC-filter as well. However, there is a relatively significant difference be- yi = Ayi− + (1 − A)xi (11) 1 tween the phases of these two filters. Fig. 2 compares where xi is the ith input datum, yi the ith output the phases of a first-order filter and a recursive filter. datum, A = exp(−1/(fsτr)) with fs as the sampling The consequence of this phase difference will be dis- frequency and τr as the filter time constant. Eq. (11) is cussed in the following section. Also, see Shaw et al. the basis for the low-pass filtering procedure employed (1998) for an example of a first-order filter phase anal- in some present eddy covariance systems. However, ysis, Berger et al. (2001) for an example of a phase it is possible to develop filtering procedures using analysis of the recursive filter, and Massman (2000) higher order recursive filters, i.e., ones with more for an example of the potential importance of phases recursive terms (yi−2,yi−3,...) or non-recursive and phase shifts for flux attenuation. filters, i.e., ones with more input terms (... ,xi−2, xi−1,xi+1,xi+2,...). But these more complicated 3.2. Strengths and weaknesses of the two methods filters are beyond the intent of the present study. Eq. (11) is the low-pass complement of the When applying either of the two spectral correction high-pass recursive filter discussed in McMillen methods some basis for estimating the specific correc- (1988), Moore (1986), Massman (2000, 2001), except tions must first be defined. In the case of the low-pass that the time constant used in the present study does filtering method, which is applied to closed-path not have the same value as that used in Massman systems, the sonic temperature flux and the univer- (2000, 2001). Its transfer function is the complement sality of scalar spectra are the bases. For the analyt- of Eq. (4) in Massman (2000) and is given as ical approach, the basis is defined by how well an instrument’s transfer function can be approximated by − A − A (ω/f ) − A (ω/f ) h (ω) = [1 ][1 cos s j sin s ] a first-order filter and by how well the true cospectra, r 2   1 − 2A cos (ω/fs) + A Cowβ (f ), for any given flux measurement, w β , can (12) be approximated by the following simple model of a

Fig. 2. Comparison of phase shifts associated with a first-order filter, Eq. (10), and a low-pass recursive filter, Eq. (12). Two cases are presented: one assuming that τ1fs = τrfs = 5 and the other τ1fs = τrfs = 20, where τ1 is the time constant for the first-order filter, τr the time constant of the recursive filter, and fs the sampling frequency. The method of calibrating the recursive filter results in τr = τ1. Phase differences beyond the Nyquist frequency (ω/fs = π) are not included. W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 127 frequency-weighted normalized cospectrum: then estimate the time constant associated with the f Cowβ (f ) 2 f/fx time decay or rise of the signal (Munger et al., 1996). = (13) This time constant is the effective first-order time wβ π 1 + (f/fx)2 constant, τ1. Another method is implemented as fol- where fx is the frequency at which f Cowβ (f ) reaches lows. First, spectra of the sonic temperature and CO2 its maximum, or, in the case of the model cospectrum (or any other trace gas) are calculated and compared. given by Eq. (13), fx is also the ‘mid-point’ of the Next, the low-pass filter, Eq. (11), is applied to the cospectral power, i.e., the cospectral power contained sonic temperature data stream and the time constant in the frequency bands [0,fx] and [fx, ∞] are both τr, is adjusted until the filtered temperature spectra equal to 50% of the total cospectral power. One ob- resemble the CO2 spectra (see Hollinger et al., 1999). vious approximation that results from Eq. (13) is that A third method uses the frequency dependent phase −2 the high frequency cospectral power decays as f , characteristics of the CO2 signal relative to, e.g., the unlike true cospectra which typically decay as f −7/3. sonic temperature signal to infer both the time lag be- The consequences of this and other approximations tween the sonic and the intake tube and the first-order associated with the analytical method can vary with time constant, τ1, of the CO2 system (e.g., Lenschow wind speed and atmospheric stability, but tend to be et al., 1982; Shaw et al., 1998). Fundamentally, all small during unstable atmospheric conditions for eddy procedures accomplish the same thing—they calibrate covariance systems that have a little instrument related the recursive filter so that the filter time constant ac- filtering (Massman 2000, 2001). counts for the effects of the signal attenuation associ- To use the analytical approach, fx must be provided ated with tube flow and the analyzer, i.e., τr = τ1 and externally or developed by generalizing results from all methods should produce approximately the same observed cospectra. For example, Moore (1986), Horst value for the effective time constant. But, regard- (1997), and Massman (2000) all used models adapted less of the method for estimating τ1 or τr the same from Kaimal et al. (1972) to parameterize fx as a func- algorithm is employed for correcting the CO2 flux. tion of stability; whereas, Wyngaard and Coté (1972) To examine this algorithm mathematically, let developed models of fx using inertial subrange ar- hT(ω)ZT be the Fourier transform of the measured guments. Nevertheless, the analytical approach is not temperature time series and hc(ω)Zc be the Fourier necessarily limited to previous models of fx. More transform of the measured CO2 signal. Here the precise site-specific models of fx could also be used Fourier transform of the true (unfiltered) atmospheric with the analytical approach. fluctuations in temperature is ZT and for CO2 it A significant advantage that the low-pass filtering is Zc. The transfer function hT(ω) includes the fil- method has over the analytical method is that the tering effects associated with sonic line averaging accuracy of the former is not dependent upon any (see Moore, 1986 or Massman, 2000 and references specific cospectral shape, whereas for the analytical therein) or with the intrinsic properties of any sep- method it is (Massman, 2000). This distinction is arate fast response temperature sensor. The transfer likely to be most important for situations or sites with function hc(ω) is associated with tube flow and the highly variable cospectra. However, to date no quan- trace gas analyzer attenuation. The Fourier trans- titative comparison of the two methods has actually form of the recursively filtered temperature time been performed and we recommend that such a com- series is hr(ω)hT(ω)ZT. Calibrating the recursive fil- parison be conducted. Other important differences ter matches the spectrum of the filtered temperature and further strengths and weaknesses are detailed in signal with the spectrum of CO2, which yields the the following discussions. following approximation: To implement the low-pass filter method requires ∗ determining the effective first-order time constant for [hr(ω)hT(ω)ZT][hr(ω)hT(ω)ZT] the filter. This can be accomplished by several meth- ∗ ≈ [hc(ω)Zc][hc(ω)Zc] (14) ods. One method is to supply a step change in CO2 concentration at the mouth of the intake tube (e.g., where ∗ denotes complex conjugation. Next assuming from zero to ambient concentration or vice versa) and similarity between the true (unfiltered) temperature 128 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144

Z Z∗ Z Z∗ spectrum, T T, and the true scalar spectrum, c c , the right-hand side of this equation yields yields h (ω)h∗(ω) C w c −jφw (ω) Cw = e c [Co − jQa] (16b) 2 2 2 c h∗(ω) |hr(ω) ||hT(ω)| ≈|hc(ω)| (15) r ∗ where the complex cross spectrum, ZwZ , has been where |h(ω)|2 = h(ω)h∗(ω). The filtering effect as- c replaced by [Co − jQa] with Co as the true cospec- sociated with sonic line averaging can be expected trum and Qa as the quadrature spectrum (Kaimal and to be much smaller than that associated with the Finnigan, 1994). Finally recognizing (a) that hw(ω) is recursive filter provided that the height of the sen- real, i.e., sonic line averaging does not cause a phase sors above the surface greatly exceeds the sonic path   shift or time delay between w and ρ (Kristensen and length. As this is typically the case, it follows im- c Fitzjarrald, 1984) and (b) that the real part of CC , mediately that for a properly tuned recursive filter, wc denoted CoM , is the measured cospectrum after cor- |h (ω)2|≈|h (ω)|2. Therefore, tuning the recursive wc r c rection by the recursive filter yields low-pass filter matches the modulus of the filters.    ∗ − φ (ω) Next we investigate the consequences of applying the h (ω) j wc M =h (ω) c e − calibrated low-pass filter technique to the cospectrum Cowc w Real h∗(ω) [Co jQa] (fluxes). For this, we apply the same Fourier analysis r used in Eqs. (14) and (15) to the flux case. (16c) The simplest flux correction factor based on the An examination of the right-hand side of Eq. (16c) low-pass filter technique is the ratio of the measured clarifies some of the compromises associated with sonic temperature flux, wT , to the temperature s the low-pass filter method. First, it does not correct flux calculated with the filtered temperature, wT  . sr for sonic path or line averaging effects [hw(ω)]or T  [Note that before filtering s , it is shifted by the for possible phase (time) shifts inherent in the rel- corresponding CO2 lag time.] Therefore, the CO2 ative placement of the sensors or residual lag times covariance corrected for high frequency flux loss [exp(−jφw (ω))]. Second, the phase difference be- wT /wT  wρ c is [ s sr] c. Using the Fourier transform tween h (ω) and h (ω) is not accounted for in this wT /wT  wρ c r method on [ s sr] c yields the following approach. This second issue can be significant in expression for the corrected complex cospectrum, some situations. For example, Fig. 2 shows the dif- CC wc: ference between the phases of a first-order system, ∗ which hc(ω) is assumed to be, and the recursive filter, [hw(ω)Zw][h (ω)Z ] CC = T T h (ω). For relatively low frequencies, ω/f < 0.1, the wc h (ω)Z h (ω)h (ω)Z ∗ r s [ w w][ r T T] phase difference is small, but it does increase rapidly ∗ −jφwc(ω) × [hw(ω)Zw][hc(ω)Zc] e (16a) as ω increases. Therefore, for scenarios where most of the cospectral power is well sampled and located h (ω)Z h (ω)h (ω)Z ∗ where the denominator, [ w w][ r T T] , in relatively low frequencies (e.g., fx/f s < 0.01), the is the transform of the recursively filtered covariance phase difference is of little consequence because the wT  h (ω)Z h (ω)Z ∗ sr; the numerator, [ w w][ T T] , the associated effect (i.e., uncorrected flux loss) is con- wT  h (ω)Z h (ω) transform of the covariance s ;[ w w][ c fined to frequencies that carry very little cospectral Z ∗ wρ h (ω) f /f ≥ . c] the transform of the covariance c; w power. But, for other cases where, e.g., x s 0 1 the filter associated with line averaging of the sonic the measured (but low-pass corrected) flux could be  vertical velocity signal w and φwc(ω) has been in- significantly underestimated if the effects of the phase troduced to account for the possibility of a shift in difference are not accounted for. [Note here we use phase (or time) between the sonic and CO2 signals a value of 0.1 as a cutoff for ω/fs because it sum- caused by any longitudinal separation between the marizes the results of Fig. 2 relatively well. But, as sonic sensing path and the closed-path intake tube Fig. 2 also shows, the cutoff value is in fact more or any unresolved lag time after performing digital precisely determined by the values of τrfs and τ1fs.] time shifts to resynchronize the sonic and closed-path An analysis similar to that provided by Eqs. (16a)– sensor time series (e.g., Massman, 2000). Simplifying (16c) shows that attempting to eliminate the phase W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 129 difference between hc(ω) and hr(ω) by low-pass fil- frequencies near fx than can be well approximated by tering w is not necessary. In this formulation of the Eq. (13). Nevertheless, Massman (2000, 2001) shows wρ C that the analytical method can be adapted to account low-pass filtering method, the corrected flux c wT /w T  wρ w for this possible broadening of true cospectra. is estimated by [ s r sr] r c, where r is the recursively filtered sonic vertical velocity time se- In addition to providing estimates of the eddy co- ries. This analysis results in an expression similar to variance correction factors, the analytical method is Eqs. (16b) or (16c) and all the compromises associated also useful for planning and design of eddy covari- with the low-pass filter method remain. The funda- ance systems. Following the notation of Massman mental concern here with the low-pass filter method is (2000, 2001), the general criteria for minimizing that calibrating the recursive filter constrains only the errors due to the relative placement of sensors and magnitude (modulus) of the filter, Eq. (15), without time response characteristics is summarized by the constraining the phase difference between hc(ω) and following expressions: h (ω) or accounting for hw(ω) or exp(−jφw (ω)). r c 2πfxτ  1 (17a) In theory, the analytical method includes correc- h tions for the phase differences (time shifts) between 2πfxτb  1 (17b) the various sensors (Massman, 2000). However, this method does assume that for a given period (of 2πfxτe  1 (17c) approximately one-half hour) of flux data the observed τ cospectra can be well approximated by a relatively where h is the equivalent time constant associated smooth function (i.e., Eq. (13) above). Unfortunately, with trend removal (McMillen, 1988; Gash and Culf, τ most observed (half hourly) cospectra show sig- 1996), b the equivalent time constant associated with nificant variability from one cospectral estimate to block averaging and mean removal (Kaimal et al., τ another. Consequently, they are not necessarily very 1989; Kristensen, 1998; Massman, 2001) and e is the smooth and the analytical approach may produce cor- equivalent first-order time constant for the entire set rection factors that can be in error (Massman, 2000; of low-pass filters associated with sonic line averag- Laubach and McNaughton, 1999). The low-pass filter ing, sensor separation, finite response times, etc. [see method does not suffer from this problem because it Table 1 and Eq. (9) of Massman, 2000]. If these three is applied directly to the eddy covariance time series criteria are met then the analytical method suggests (Eq. (11) above) rather than to the flux. little need for spectral correction. On the other hand, the analytical method includes Finally, the low-pass filter method is questionable corrections for low frequency losses due to any re- during conditions when the magnitude of the heat −2 cursive high-pass filtering (McMillen, 1988), linear flux is less than about 10 W m . When this occurs |wT |≈ |wT  |≈ detrending of the eddy covariance time series (Gash s 0 and sr 0 and the low-pass filter and Culf, 1996), or mean removal (Kaimal et al., correction term becomes undefined. Similarly, the 1989; Kristensen, 1998; Massman, 2000), whereas analytical method is suspect for very stable atmo- the low-pass filter method does not. These low fre- spheric conditions because correction factors for CO2 quency losses may be of greater importance than has fluxes can exceed 1.5 or even 2.0 (Massman, 2001). been attributed to them in the past because (a) the Ultimately, neither correction method is likely to be frequency-weighted spectra (and by implication the useful for conditions where the turbulent transfer is cospectra) of (Kaimal et al., 1972) actually result dominated by intermittent events because all eddy from too much high-pass filtering (Högström, 2000), covariance measurements become less reliable under (b) the surface layer may be disturbed during flux such conditions. measurement periods (McNaughton and Laubach, 2000), or (c) significant flux-bearing low frequencies have been inadvertently removed from the data dur- 4. Flux error due to 2D and 3D effects ing processing (Finnigan et al., 2002). All three of these possibilities imply that true cospectral power A major goal of many micrometeorological studies may actually be distributed more uniformly across is to quantify the net exchange of a trace gas of interest 130 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 between the atmosphere and the surface. This is usu- analyses. 2D changes in scalar fluxes caused by step ally achieved by approximating the net exchange with changes in surface roughness have also been studied the measured vertical eddy flux corrected for storage (e.g., Mulhearn, 1977; Lee et al., 1999) and previ- below the level of measurement and thereby ignoring ously reviewed by Garratt (1990). Nevertheless, any all the other terms of the mass conservation equation guidelines developed from these previous studies, because they are difficult to measure. This approxima- while helpful, cannot be used with assurance of elim- tion works if the flow and scalar fields are nearly hor- inating either 2D and 3D effects or the concomitant izontally homogeneous. However, under 2D and 3D possibility of biases in the measured fluxes. There are influences the vertical eddy flux may systematically several reasons for this. First, no previous study has deviate from the true net exchange. Mathematically considered the full complexity of Eqs. (6) and (18). this is expressed by integrating Eq. (6) from the soil Second, almost all studies reviewed by Garratt (1990) surface (z = 0) to some height (z = zm) at which the have assumed near-neutral atmospheric stability, con- flux measurements are made, yielding: sequently their results may not be accurate under  z  z m ∂ω¯ m extreme conditions, such as very stable air or free ρ¯ c dz + [vρ ·∇¯ω − Vω¯ ·∇¯ρ ]dz convection. Third, mesoscale motions, which are not d ∂t d c c d 0  0  included in these studies and which can bias vertical zm zm + ∇ · ρ¯ z + ∇ · ρ F z flux measurements, occur on 2D and 3D scales much H V c d H v c d 0 0 larger than the scale of micrometeorology measure- + W(z )ρ¯ (z ) + wρ F(z ) ments, which can be characterized by site fetch and  m c m c m zm height scales. Thermally driven circulations, such as a = S¯ z + W( )ρ¯ ( ) + wρ F( ) c d 0 c 0 c 0 (18) land-lake breeze and stationary convective cells, and 0 drainage flows are examples of 3D motions that may where the first term on the left is the storage term, the subject observations to advective influences. Fourth, second is the integrated form of the quasi-advective these earlier micrometeorological studies assume no term (and has never been previously included in the variation in background topography. budget equation before), the third term is related to Clearly, a proper understanding of 2D and 3D flows the mean horizontal advective term with ∇H as the and their role in micrometeorological flux observation horizontal gradient operator, the fourth term is the is of importance to any site, but the problem of 2D and vertically integrated horizontal flux divergence, and 3D flows is most difficult to treat at sites on non-flat the fifth term on the left is the measured (mean plus topography. At least four topographic effects are rele- turbulent) flux with W as the mean vertical velocity vant to the surface layer flux observations: and w as the fluctuating component of the vertical velocity. The term on the right side of Eq. (18) is the (1) Terrain can generate its own nighttime gravita- net ecosystem exchange. We include the mean ve- tional or drainage flows. A good example of this locity term, W(0)ρ¯c(0), as part of the net ecosystem is the Walker Branch forest (Baldocchi, 2000). exchange primarily for mathematical completeness. This forest is situated on a ridge top and night- In many situations, it is reasonable to assume that time wind speed tends to be low (mean nighttime W(0) = 0. However, there may be scenarios, possi- friction velocity 0.15 m s−1; KB Wilson, personal bly related to pressure pumping, where W(0) during communication). These two characteristics favor a flux-averaging period, although small, may not the occurrence of drainage flow. At other sites on be 0. more even terrain, drainage flow is more likely Simpler forms of Eq. (18) or Eq. (6) have been to be driven by background topography larger used in many previous studies of 2D and 3D effects. than the tower footprint/fetch scale. Models of For example, local 2D advection in which there is a drainage flow have been developed for simple step change in the surface source strength of a passive topography without vegetation (e.g., Brost and scalar has been studied by Philip (1959) and further Wyngaard, 1978). However, at present, we lack developed by Dyer (1963) to estimate the so-called models for the air layer within the height of the fetch-height ratio and by Schmid (1994) for footprint tower. W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 131

(2) Terrain obstacles can modify the ambient flow ral coordinate system. But, it can introduce additional via a bluff body effect. Because the streamline in noise or uncertainty into the flux estimates (Wilczak the tower air layer can depart significantly from et al., 2001) and is often ignored in many flux studies. the local terrain surface, persistent mean vertical Furthermore, there may be dynamical and diagnostic motion may be expected. The severity of vertical reasons why wv should not be minimized (Weber, advection will depend on vertical concentration 1999; Wilczak et al., 2001). gradient of the scalar of interest. Change of turbu- The main application of the natural coordinate sys- lent stress in response to the change in wind field tem is for the calculation of fluxes in sloping terrain may produce spatial variation of the scalar flux and and there are several valid reasons for working in this hence horizontal advection (Finnigan, 1999).An particular coordinate system in non-uniform terrain analytical solution for advective flow over isolated (Wilczak et al., 2001). However, a major disadvan- low hills under neutral stability was first proposed tage to long-term studies is the possibility that W = 0 by Jackson and Hunt (1975). This theory was later during the flux-averaging periods. Setting W = 0 for extended to canopy flow on hills (Finnigan and every half an hour (a) eliminates the mean flow com- Brunet, 1995; Wilson et al., 1998) and to scalar ponent of the flux, thereby causing either a significant concentration fields (Raupach et al., 1992).How- bias or a systematic underestimation of the individual ever, the utility of solutions of the Jackson and fluxes and in the long-term balance (Lee, 1998) and Hunt type in elucidating the advection problem is (b) filters (attenuates) the low frequency components subject to debate (Finnigan, 1999; Lee, 1999). of the turbulent flux (Finnigan et al., 2002). Wilczak (3) Surface source strength may not be uniform in et al. (2001) and Paw U et al. (2000) outline a method, the streamwise direction. For example, Raupach termed planar fit method, that can be used to estimate et al. (1992) showed that significant horizontal W. In fact, the planar fit method defines the preferred (along slope) advection of energy can result from coordinate system for single point (single tower) flux variations in the incident solar radiation along a measurements (Finnigan and Clement, in preparation). curved slope. However, unlike the natural coordinate system, the (4) Gravity waves generated by terrain obstacles are planar fit method cannot be used in real time for each beyond the scope of traditional micrometeorology flux-averaging period. Rather it must be used over a because of the extent of their horizontal spatial set of many flux-averaging periods. Nevertheless, the scales and their 3D nature. This motion type is planar fit method has been shown to reduce sampling common in stratified air with moderately strong errors (or the variability from one flux-averaging pe- winds (Smith, 1979). Their role in the surface-air riod to another) for flux data sets obtained over water fluxes is yet to be understood. (Wilczak et al., 2001). This method has yet to be tested over land in complex terrain and we recommend that it be evaluated for its impact on long-term CO2 fluxes 5. Issues arising from choice of coordinate and carbon balances. systems and data processing In addition to the two coordinate systems just des- cribed, there is another coordinate system that can To date, the most common coordinate system used be used for estimating fluxes in complex terrain for flux measurements is a rectangular coordinate sys- (Finnigan, 1983). For studies of the vertical flux tem sometimes called the ‘natural’ coordinate system divergence, ∂wc/∂z, this particular streamline coor- (Tanner and Thurtell, 1969; Kaimal and Finnigan, dinate system is recommended because it should give 1994) or the ‘streamline’ coordinate system (Wilczak the most reliable estimate of the flux divergence in et al., 2001). In this coordinate system, the x-axis is curved flows than with Cartesian coordinate systems. parallel to the local mean horizontal wind (U) and A second data processing issue concerns the the z-axis is perpendicular to the x-axis, thus the possible loss of the low frequency portion of mea- mean cross-wind (V ) and the mean vertical wind sured fluxes. For example, choosing a flux-averaging (W) are zero. A third rotation, which minimizes the period that is too short will attenuate the low frequen- cross-stream stress term wv, is also part of the natu- cies components of the flux (Lenschow et al., 1994; 132 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144

Mann and Lenschow, 1994; Kristensen, 1998),as Large footprints. It is known that the eddy covari- will overfiltering with any high-pass (e.g., recursive) ance footprint expands rapidly as air becomes increas- filter (Högström, 2000). Loss of these low frequency ingly stratified (Leclerc and Thurtell, 1990; Schmid, components has been implicated in the lack of en- 1994) and can extend beyond the vegetation type ergy balance closure (Sakai et al., 2001; Finnigan et under investigation. Footprint correction is however al., 2002) and in a 10–40% underestimation of the not straightforward as the existing footprint models daytime CO2 fluxes over forests (Sakai et al., 2001). are built on principles of eddy diffusion established Coordinate rotation can also act as a complicated for conditions of near-neutral stability. For example, non-linear high-pass filter (Finnigan et al., 2002). air stability over a forest often exceeds the range One possible solution to this problem involves using over which the empirical Monin–Obukhov similarity the raw (fully sampled) high frequency data without functions are valid. filtering and evaluating the fluxes with the planar fit Gravity waves. Shear-generated gravity waves are a method. Potentially this approach could circumvent common motion type in the canopy at night (Lee and many of the concerns about low frequency losses. Barr, 1998; Fitzjarrald and Moore, 1990; Paw U et al., Nevertheless and subject to the constraints outlined in 1989, 1990). The wave motion manifests itself in the Section 3 above, the analytical method should be able form of periodic time series of velocities, temperature to correct the fluxes regardless of whether the data are and scalar concentrations. Strictly speaking, the sta- recursively high-pass filtered or not (Massman, 2001). tionarity condition is not satisfied during gravity wave However, cospectra that describe the appropriate flux events because the coefficient of auto-correlation does energy distribution is still required for the analytical not vanish at a finite lag time and consequently no approach. Such cospectra need not be the same as the integral time scale can be defined. Numerical simu- cospectra of Kaimal et al. (1972) (e.g., Sakai et al., lations show that a constant flux layer does not ex- 2001). ist in the presence of the wave motion (Hu et al., 2002). Instead, fluxes of momentum and scalars can very greatly with height over the canopy with the flux 6. Nighttime flux measurements: a co-occurrence peaking at the so-called critical level, i.e., the height at of all eddy covariance limitations which the wave propagation speed matches the mean wind speed. For these reasons, eddy fluxes appear very Almost all eddy covariance limitations occur at noisy during a gravity wave event. However, when night when the air becomes stably stratified. Some averaged over a long enough time period CO2 fluxes of these are instrumental, others are meteorological. collected at the Borden forest during a gravity wave The instrumental limitations ultimately result from event show the same dependence on soil temperature the fact that eddy covariance instruments are best established for other periods (Lee, unpublished data). suited for daytime convective conditions when the This suggests that although the raw data may appear dominant turbulent motions are frequent and large noisy, the wave motion does not introduce a detectable enough that sensor limitations are not significant. systematic bias into the ensemble averaged fluxes. At night or during stable atmospheric conditions, Advection. Under very stable conditions, the verti- when turbulent motions shift toward relatively higher cal gradient of the Reynolds stress is small within the frequencies and become more intermittent, the lack vegetation and therefore the horizontal pressure gra- of instrument response due to finite time constant, dient, associated with baroclinic forcing (Wyngaard sensor separation, path-length averaging, and tube and Kosovic, 1994), synoptic weather systems, or the attenuation becomes a severe limitation. Corrections gravitational force on a slope (Mahrt, 1982), is rel- developed with either the analytical method or the atively large. Simultaneously, large vertical gradients low-pass filtering method are suspect under very stable in scalar quantities exist near the ground due to the conditions. lack of vigorous turbulent mixing. Under these condi- Some of the meteorological limitations include tions, air motions within the canopy and surface layer large footprints, gravity waves, advection, and aerody- are inherently 2D or 3D and the resulting drainage namic or low turbulence issues. or (vertical and horizontal) advection that occurs is W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 133 likely to be of a magnitude much larger than that oc- to the same level as observed at high wind conditions curring in the daytime (e.g., Sun et al., 1998; Mahrt (Fig. 4). In some cases, one can identify a critical or et al., 2001). This diel asymmetry could introduce a threshold friction velocity, u∗c, beyond which the flux large bias into the estimates of annual net ecosystem seems to level off, while in other cases no threshold production (Lee, 1998). exists (e.g., Fig. 3 and windy sites reported by Aubinet Aerodynamic issues. A common phenomenon at et al. (2000)). Similarly, energy balance closure is gen- long-term flux sites is that turbulent CO2 flux ap- erally poor at low u∗ conditions and improves as u∗ proaches zero as the level of turbulence, measured by increases (Black et al., 2000; Aubinet et al., 2000). the friction velocity, drops to zero (Goulden et al., A common practice is to replace the flux during pe- 1996). This should be expected on the basis of aero- riods with u∗ u∗c). general turbulent scalar fluxes are proportional to (Here Q10 is the relative increase in respiration result- ◦ u∗(∂c/∂z). In other words, as the turbulence decreases ing from a 10 C increase in temperature.) Lavigne so also must the turbulent fluxes. Fig. 3 is an exam- et al. (1997) use a single u∗c across all sites in a com- ple of the observed dependence of nighttime vertical parative study of nighttime eddy flux and chamber F wρ u flux of CO . However, it is now recognized that u∗ CO2 flux, c , on friction velocity, ∗. However, 2 c several issues remain unsettled during conditions of and Q10 are site-specific parameters (Table 1). An- low turbulence. other concern with the u∗c–Q10 approach is the risk of Wofsy et al. (1993) and Goulden et al. (1996) sug- double counting due to morning flush of CO2 (Grace gest that biological source strength of CO2 is not et al., 1995; Aubinet et al., 2000). Studies of the sen- a function of air movement, implying that the stor- sitivity of annual NEP to u∗c suggest that imposing age corrected eddy flux should be independent of u∗ a u∗ threshold will increase the annual estimate of − if the 1D approach accurately approximates the sur- NEPby0.5tCha 1 per year or more (Grelle, 1997; face layer mass balance. Numerous observations show Aubinet et al., 2000; Goulden et al., 1996; Barr et al., however that storage correction does not bring the flux 2002).

wρ F u Fig. 3. Dependence of nighttime air storage and vertical eddy CO2 flux, c , on friction velocity, ∗, at the Great Mountain Forest during May–September 1999. Data are averaged over 0.05 m s−1 bins. Shaded area is one standard deviation about the mean flux. Dots are flux predictions based on Monin–Obukhov similarity theory for very stable air, where a =−(0.25θ/g)1/2, θ the potential temperature, g the gravitational acceleration, and ρc the CO2 concentration. The F superscript indicates that the fluxes include the WPL terms. 134 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144

Fig. 4. Schematic diagram showing nighttime CO2 eddy flux and air storage as a function of friction velocity. The F superscript indicates that the fluxes include the WPL terms.

The assumption that biological source strength is tant for a variety of soils (Hillel, 1980; Nilson et al., invariant with turbulence intensity is reasonable, ex- 1991). Quasi-stationary pressure fields induced by cept for the possibility of pressure pumping effects. wind blowing over rough topography could further Variations in barometric pressure at the ground sur- enhance diffusional fluxes significantly more than face are correlated with turbulence intensity (Shaw ground level turbulent pressure fluctuations (Farrell and Zhang, 1992). Such variations will introduce ad- et al., 1966; Colbeck, 1989). Although pressure fluc- vective air movement into and out of the soil, thus tuations are not a standard parameter called for by the enhancing the soil efflux of gases with concentrations AmeriFlux science plan (Wofsy and Hollinger, 1998), exceeding ambient concentrations (Hillel, 1980). The further investigation of this phenomenon is warranted. pressure pumping effect has been proposed as the pos- Causes of nighttime flux underestimation remain the sible cause of episodic emissions of CO2 from soils subject of debate. Poor instrument response at high fre- (Baldocchi and Meyers, 1991) and snowpacks (Yang, quencies contributes to the flux loss, but is unlikely the 1998, Y. Horazano, personal communication). Model root of the problem because the flux will be still be too simulations by Massman et al. (1997) suggest that low even using the large correction factors predicted the turbulent pressure pumping effect can increase for the stable conditions (Massman 2000, 2001).We or suppress the diffusive flux through a snowpack by conclude that the problem is meteorological in nature 25% and the effect may be significantly more impor- and recommend an experiment that simultaneously measures all the terms of the mass conservation equa- Table 1 −1 tions [Eqs. (6) and (8)]—an admittedly difficult task. Summary of friction velocity thresholds, u∗c (m s ), and the corresponding rates of increase of whole-ecosystem respiration Drainage flow is one possible reason why fluxes mea- over a 10 K increase in temperature, Q10 sured under very stable stratification always seem bi- ased toward underestimation (Grace et al., 1996; Lee, Forest type u∗c Q10 Reference 1998). This raises the possibility that CO2 fluxes from Aspen 0.6 5.5 Black et al. (1996) Pine 0.5 2.6 Lindroth et al. (1998) low areas that accumulate CO2 from drainage must be Maple-tulip poplar 0.5 1.9 Schmid et al. (2000) relatively high to compensate. Such high fluxes have Black spruce 0.4 2.0 Jarvis et al. (1997) yet to be observed. Studies of the influence of drainage Douglas-fir 0.4 4.5 Jork et al. (1998) flows on trace gas movement are strongly encouraged. Beech 0.25 3.0 Pilgaard et al. (2001) Black spruce 0.2 2.0 Goulden et al. (1997) Oak-maple 0.17 2.1 Goulden et al. (1996) Spruce-hemlock 0.15 2.4 Hollinger et al. (1999) 7. Summary and recommendations Maple-aspen 0.15 2.9 Lee et al. (1999) Tropical 0.0 NAa Malhi et al. (1998) The findings of this study and areas needing further a Not available. research are: W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 135

p (1) The pressure covariance term: v a. Although (4) 2D and 3D effects. Advective effects are a major usually ignored, the pressure covariance com- source of uncertainty, particularly in complex ter- ponent of the WPL term (Webb et al., 1980) is rain, and they may not be fully quantified without likely to be important during windy turbulent the aid of 2D and 3D models. However, drainage conditions. Ignoring this term could lead to a flows are likely to be amenable to observational significant bias at sites that have frequent high studies and more studies of CO2 drainage should winds and strong turbulence. be performed. ρ ·∇ω ¯ − ω¯ ·∇ρ ¯ (2) The quasi-advective term: v d c V c d. (5) Coordinate systems. Choice of a coordinate sys- We have identified a new (or previously uniden- tem is quite important. For example, processing tified) term in the CO2 budget equations, flux data with the ‘natural’ (Tanner and Thurtell, Eqs. (6) and (18), which we have termed the 1969) or ‘streamline’ (Wilczak et al., 2001) coor- quasi-advective term. This term originates from dinate system may cause the loss of the vertical the 3D dry air density correction (Paw U et al., advective component of the flux (because W = 0 2000, Appendix B). The significance of this term in the natural coordinate system) and remove to long-term CO2 eddy covariance studies is un- some of the low frequency contribution to the known. But, it is suggested that this term is likely fluxes (Finnigan et al., 2002). Consequently, other to be important anywhere that within-canopy coordinate systems, most notably the planar fit gradients of CO2 mass mixing ratio, ∇¯ωc, and method of Wilczak et al. (2001) and PawUetal. T /T¯ the profiles of temperature covariance, v a a, (2000), should be investigated and their impact on ρ /ρ¯ long-term flux and energy should be quantified. water vapor covariance, v v v, and pressure p /p¯ We also note that when writing a budget equation covariance, v a a, are important. (3) Methods for correcting frequency attenuation. in any particular coordinate system it is important Two methods of correcting eddy covariances for to use the 3D form of the coordinate system first spectral attenuation are reviewed and analyzed. and then to simplify to the 1D case as necessary, The low-pass filter method (Goulden et al., 1997; and finally, where possible, measure or account Hollinger et al., 1999) has a potentially significant for all terms in the budget equation. advantage over the analytical method (Massman, (6) Nighttime and low flux issues. Many of the previ- 2000, 2001) for high frequency cospectral atten- ously discussed shortcomings of eddy covariance uation because it is independent of cospectral technology coincide when attempting flux mea- shape. The analytical method, on the other hand, surements at night. Spectral correction methods does include some high frequency attenuation are unreliable or questionable, drainage is more factors related to phase shifts that are not part of likely to occur during relatively stable nighttime the low-pass filter method. The analytical method conditions, and turbulent transfer may become also incorporates low frequency attenuation which intermittent in time and space. There are other is not included in the low-pass filter method. This issues or phenomena that further confound flux last difference between the two spectral correction measurements made at night. The presence of methods further highlights the importance of the shear-generated traveling gravity waves trapped loss of low frequency cospectral power as a po- in near-surface atmospheric stable layers inval- tentially significant source of error for long-term idate the constant flux layer assumption. One flux and energy balances (Högström, 2000; Sakai practical method for estimating nighttime fluxes et al., 2001; Finnigan et al., 2002). Neither spec- employs data filling during periods of low turbu- tral correction method is entirely satisfactory for lence using a u∗ threshold. However, it is now very stable conditions. Nevertheless, a detailed recognized that this approach is relatively site quantitative comparison of the two methods and specific. Further complicating both nighttime and their impacts on long-term fluxes has yet to be daytime issues is the possibility that atmospheric performed and further investigations into low pressure pumping may augment or reduce soil frequency issues and very stable conditions are diffusive CO2 efflux particularly over rough to- necessary. pography. Such an effect raises some uncertainty 136 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144

in Q10-based algorithms developed for nighttime Michael Goulden∗† data filling because these algorithms assume that Lianhong Gu biological source strength is independent of tur- Jeffrey Hare bulence and pressure pumping. We conclude that Dave Hollinger∗ the difficulties of making nighttime flux measure- Thomas Horst† ments are largely meteorological in nature, not Larry Jacobsen instrumental. To insure further progress on these Gabriel Katul nighttime flux issues and the other previously Xuhui Lee∗ discussed issues more research is needed into Don Lenschow how gravity waves, intermittency, drainage, and Ray Leuning pressure pumping affect flux measurements. Yadvinder Mahli† Larry Mahr Bill Massman∗† Acknowledgements Russel Monson Steve Oncley This workshop was supported by grant 901214- Kyaw Tha Paw U HAR from the US DOE NIGEC Northeast Regional Üllar Rannik Center. The second author also acknowledges sup- Ruth Reck† port by the US National Science Foundation through Luis Ribeiro grants ATM-9629497 and ATM-0072864. Both au- Scott Saleska thors wish to thank Jielun Sun, Don Lenschow, and an HaPe Schmidt anonymous reviewer for their comments and insights Jielun Sun on the subjects covered by this study. Andy Suyker Bert Tanner Andy Turnipseed Appendix A. The UFO committee and Sashi Verma∗ Participants of the UFO workshop Marv Wesely Eric Williams This appendix is an alphabetical list of the par- Steve Wofsy∗† ticipants of the workshop for unaccounted flux in long-term studies of carbon and energy exchanges (UFO). Those participants whose names are marked with ∗ are members of the UFO committee. The in- Appendix B. Derivation of the fundamental vited speakers are denoted by †. The workshop was equations of eddy covariance co-chaired by Bill Massman and Xuhui Lee. This appendix derives and discusses the fundamen- Dean Anderson tal eddy covariance equations for the measurement of Peter Anthoni the fluxes of water vapor, heat and CO2, Eqs. (2)–(6) Marc Aubinet of the main text. These equations are derived in a Dennis Baldocchi∗ fully consistent manner with the minimum number of Brad Berger assumptions and include temperature, pressure, and Constance Brown-Mitic moisture effects and generalize the results of Webb George Burba et al. (1980) (WPL) and Paw U et al. (2000). Ex- Robert Clement pansion of all requisite equations with respect to the Ken Davis perturbation fields follows WPL, but also include pres- John Finnigan† sure effects. However, unlike WPL, we keep only the David Fitzjarrald† first-order (linear) terms in the perturbation fields in John Frank accordance with the findings of Fuehrer and Friehe W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 137 ∂ρ (2002). To allow for the possibility of horizontal ad- c +∇·(vρc) = Sc (B.7) vection, we employ the general 3D mass conservation ∂t for dry air (Paw U et al., 2000) when deriving the ∂ρ d +∇·( ρ ) = S WPL term, rather than assume the net mean vertical ∂t v d d (B.8) dry air mass flux is zero, as do WPL. Furthermore, we expand on previous studies by deriving an explicit where vectors are denoted in bold type, ∇ is the spatial relationship between the source terms for dry air and gradient operator, v the velocity and the subscripted S CO2. Finally, we note that, although we focus on CO2, the corresponding source or sink term. It should also the method outlined here is generalizable to all other be emphasized that equations of continuity, Eqs. (B.7) trace gases as well. and (B.8), are expressed in 3D vector form and are, therefore, independent of any assumptions regarding B.1. Trace gas fluxes horizontal gradients or any particular coordinate sys- tem. This discussion begins by listing the key equations For the purposes of the present discussion, which on which the derivation for trace gas fluxes is based. we limit to photosynthesis and respiration, the source S S The total density of the atmosphere, ρa, is the sum term for dry air, d, can be expressed in terms of c. of dry and vapor components, i.e., The coupling between O2 and CO2 is such that for ev- ery mole of one gas used during photosynthesis or res- ρa = ρd + ρv (B.1) S /m = piration a mole of the other is created, i.e., O2 O2 −S /m , where S is the source strength of O and where, henceforth, ρ denotes density, the subscript c c O2 2 m is the molecular mass of O . As long as these pro- ‘a’ denotes ambient or total, the subscript ‘d’ denotes O2 2 the dry air component, the subscript ‘v’ denotes the cesses do not significantly alter the basic composition of dry air, we m ay also assume that S + S = S . vapor component, and where necessary, the subscript c O2 d ‘c’ denotes the trace gas component, which in this case Therefore we can make the following substitution for Sd in Eq. (B.8): Sd = (1 − (mO /mc))Sc. will be taken to be CO2. 2 Dalton’s law of partial pressure is Before formally manipulating this set of equations, we need to define two more terms. The CO2 mass p = p + p a d v (B.2) mixing ratio (or CO2 mass fraction), ωc, and the CO2 volume mixing ratio (or CO mole fraction or CO where p denotes pressure. 2 2 volume fraction), χ , are given as follows: The ideal gas laws for the three constituents and the c ρ ambient air are ω = c c ρ (B.9) ρdRTa d pd = (B.3) md pc χc = (B.10) ρ pd p = vRTa v m (B.4) v Assuming the CO2 and dry air components are isother- ω χ ω = ρ RT mal, the relationship between c and c is c p = c a (B.5) (m /m )χ c m c d c. Similar relationships can be defined for c water vapor. ρaRTa Combining Eqs. (B.1)–(B.6), yields pa = (B.6) ma ρ ρ p d + v = a (B.11) where Ta is the ambient temperature, R the universal md mv RTa gas constant and m the molecular mass of the gas as Performing the Reynolds’s decomposition on indicated by the subscript. Eqs. (B.7)–(B.11), yields the following four equations By ignoring molecular diffusion, the conservation of mass, or the equation of continuity, for CO2 and ρ¯c mc 1 p¯c ω¯ c = = χ¯c = (B.12) dry air are ρ¯d md µc p¯d 138 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144
     T p  ρ¯ + ρ −¯ω ρ ρ =−¯ρ ( +¯χ ) δ a − a − µ ρ of the total flux, V c v c cv d. This, in turn, d d 1 v oc ¯ p¯ v v (B.13) Ta a emphasizes that the dry air or density effects have a 3D ρ ·∇ω ¯ − ω¯ ·∇ρ ¯ ∂ρ¯ aspect, expressed by the term [v d c V c d], c +∇·( ρ¯ + ρ ) = S¯ ∂t V c v c c (B.14) that Webb et al. (1980) did not include. In other words, WPL did not specifically include the within-surface ∂ρ¯ m d +∇·( ρ¯ + ρ ) = − O2 S¯ layer effects associated with vertical and horizontal V d v d 1 c (B.15) ∂t mc structure of the fluxes and mean density of dry air. In a 1D setting, we could state that the dry air den- µ = m /m µ = m /m where c d c, v d v, the mean wind is sity fluctuations influence, not only the vertical trace denoted by V rather than denoting it with the overbar gas fluxes, but that they extend throughout the surface notation, and all deviation quantities (here and hence-  layer and can influence exchanges below the level of forth) are denoted by . Note that all products in the flux measurement. A second difference between the deviation quantities were dropped from Eqs. (B.12) present approach and WPL is the use of the continuity and (B.13) and that Eq. (B.13) has been linearized in equation for dry air, Eq. (B.15). WPL assume that the the deviation quantities similar to Webb et al. (1980). Wρ¯ +wρ = W δ T  1D dry air flux, d d 0. In doing so, their Finally oc is introduced in the a term of Eq. (B.13) to distinguish between open- and closed-path sensors. becomes a drift velocity and it loses the interpretation For an open-path sensor δ = 1 and for a closed-path of a mean flow velocity appropriate to atmospheric oc flows. This is an important distinction for applications sensor δoc = 0. For closed-path systems, δoc = 0 because by the time the gas sample has reached the where the fluxes are rotated into a coordinate system W = W analyzer the temperature fluctuations have been at- that allows for 0. In this case, is a mean tenuated so strongly by the sampling tube that they vertical velocity associated with atmospheric or topo- can probably be ignored (Frost, 1981; Leuning and graphic forcing independent of any dry air flux. Nev- Moncrieff, 1990; Rannik et al., 1997). This distinc- ertheless, it is important to recognize that if the point tion between open- and closed-path sensors relative measurement of CO2 fluxes is the only concern, then ρ the present results are the same as WPL. But, if an ac- to the fluctuations in density, d, and its implications to eddy covariance measurements are discussed in curate accounting of the aerodynamic budget for CO2 more detail in the main text. is the main goal, then Eq. (B.16), is more appropriate. The only situation where Eq. (B.16) and the original Next multiplying Eq. (B.15) by ω¯ c, subtracting the result from Eq. (B.14), and then manipulating the WPL tend to correspond with one another is when ρ ·∇ω ¯ ≡ ω¯ ·∇ρ ¯ terms algebraically yields: v d c V c d, which seems unlikely at best. Several other points need to be noted here regarding ∂ω¯ c   this equation. First, the incompressibility assumption ρ¯d + [v ρ ·∇¯ωc − Vω¯ c ·∇¯ρd] ∂t d for the mean flow (∇·V = 0) has been employed in the     +∇·(Vρ¯ + v ρ −¯ω v ρ ) derivation of Eq. (B.16), otherwise the term −Vω¯ c ·
c c  c d m ∇¯ρd should be replaced by ω¯ c∇·Vρ¯d. Second, Paw U = + O2 − ω¯ S¯ 1 m 1 c c (B.16) et al. (2000) derive a related 1D version of this equa- c tion. Third (discussed below), Eq. (B.16) is not neces- This is the fundamental equation of continuity for sarily correct for in situ flux measurements based on ω χ  in situ measurements of CO2 fluxes and background the mixing ratio fluctuations c or c. Fourth, greater concentrations using one or more eddy covariance mathematical precision is possible, particularly con- sensors that directly measure fluctuations in density. cerning issues involving horizontal variability, when Mathematically Eq. (B.16) is not unique, i.e., it can developing a budget equation like Eq. (B.16) by first be written in other ways. But, expressing Eq. (B.16) defining a control volume of some horizontal extent as we have aids in the interpretation of the WPL term. and then beginning that development by integrating In traditional applications, the WPL term is applied Eq. (B.16) over that control volume (e.g., Finnigan, solely to fluxes measured at a single level. Therefore, personal communication). But such precision is not −¯ω ρ we include the dry air flux term, cv d, as part necessary for the present study, because the insights W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 139 offered by this more complete approach are fully cov- Hygrometer (Tanner et al., 1993). These corrections ered by Finnigan (personal communication). Finally, are summarized below in two sets of equations. The ((m /m ) − )ω¯  we note that because O2 c 1 c 1, it is not first set, Eqs. (B.18) and (B.19), is for the complete be included in the main text, but, for the purposes of (or symmetric, but rather unlikely) case of measuring completeness, it is kept in this appendix. the CO2 flux with one sensor and the water vapor flux The interpretation of the three terms on the left-hand with a Krypton Hygrometer: side of this equation is fairly standard, even if the
     T p explicit form is not. The first term is the time rate of ρ {corrected}=ρ {KH O}+γ ρ¯ a − a  v v 2 O2 ¯ ρ ·∇¯ω − ω¯ · T p¯a change term. The second term, [v d c V c a ∇¯ρd], is a quasi-advective term and the third is the flux (B.18) divergence term. From this third term, the appropriate = ρ¯ + ρ − α trace gas flux, Fc, can be identified; Fc V c v c ρ { }=ρ { }− ρ {K } ω¯ ρ c corrected c raw β v H2O cv d, which when combined with Eqs. (B.1), (B.12)
 and (B.13) yields: α T  p − γ ρ¯ a − a (B.19)   β O2 T¯ p¯ T  p a a = ρ¯ + ρ +¯ρ ( +¯χ ) δ v a − v a Fc V c v c c 1 v oc ¯ p¯ The more likely scenario, measuring both water vapor Ta a and CO2 fluctuations with a single instrument, is given +¯ω µ ρ c vv v (B.17) by the following equations: ρ { }=ρ { } Note that Eq. (B.17) explicitly includes the pressure v corrected v raw (B.20) p /p¯ covariance term, v a a, which WPL did not. Under α ρ { }=ρ { }− ρ { } most circumstances involving water vapor and CO2 c corrected c raw β v raw (B.21) fluxes, this term is probably small enough to ignore, however, as discussed in the main text, there are sit- where α/β ≤ 10−3, γ ≈ 0.05 (Tanner et al., 1993), uations where the pressure covariance term could be ρ¯ −3 and O2 is the ambient concentration of O2 [kg m ]. significant. ρ¯ = . ρ¯ (Note O2 0 23 d.) These corrections apply to both While the interpretation of Eq. (B.16) may be rou- the concentration flux, vρ , and the mass flux term, tine, the implications are potentially significant and c vρ ,ofEqs. (B.16) and (B.17), where the ‘corrected’ result from the quasi-advective term. This can be d fluctuations replace the ‘raw’ or ‘uncorrected’ quanti- seen by noting that budget equations for trace gases ties in Eqs. (B.13)–(B.17). developed by vertically integrating equations similar We end this section of this appendix by citing (with- to Eq. (B.16) (e.g., Moncrieff et al., 1996; Lee, 1998, out proof) the fundamental equation of eddy covari- and others) usually do not include the quasi-advective ance for in situ flux measurements based on mixing term. Eq. (B.16) indicates that applications of mass ratio, ω or χ .Itis flux term requires not only augmentation of the mea- c c     sured CO2 covariance v ρ with −¯ωcv ρ , but knowl- ∂ω¯ c d ρ¯ c + [vρ ·∇¯ω − Vω¯ ·∇¯ρ ] ρ d ∂t d c c d edge of the vertical profiles of components of v d as
 well. Consequently, there is the potential for some in- m +∇·( ρ¯ ω¯ +¯ρ ω )= + O2 − ω¯ S¯ V d c dv c 1 1 c c accuracies in the current calculations of the vertically mc integrated CO2 budgets and the inferred NEE. (B.22) One benefit of the present formalism for develop- ing the WPL term for trace gas exchange is that it The derivation of this equation follows from substi- readily adapts to the inclusion of two instrument re- tuting vρdωc for vρc in the divergence term of lated corrections: the correction to CO2 fluctuations Eq. (B.7) and then employing the same assumptions due to sensor sensitivity to water vapor (Leuning and and general approach used to derive Eq. (B.16). Moncrieff, 1990) and the oxygen (or O2) correction Eq. (B.22) is in agreement with the results of to water vapor fluctuations measured with a Krypton Webb et al. (1980) in that no additional covariance 140 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144

ρ F p term is required when estimating the total flux, be- In addition to the vapor correction (v v and v a ρ¯ ω cause it is measured directly as dv c. However, it terms), it may also be necessary to correct the sonic T  should also be noted that the quasi-advective term, virtual heat flux, v s , for cross-wind effects (e.g., ρ ·∇¯ω − ω¯ ·∇¯ρ [v d c V c d], remains part of this form Kaimal and Gaynor, 1991; Hignett, 1992). Including of the continuity equation. this correction term in Eq. (B.26) yields

F B.2. Turbulent temperature flux: vT  T  T  ρ p U u  a v a = v s −¯α v v + β¯ v a + δ 2 n nv ¯ ¯ v ρ¯ v p¯ un ¯ Ta Ts d a γdRdTs Eqs. (B.13), (B.16) and (B.17) clearly indicate the (B.27) T  importance of the ambient temperature flux, v a ,to γ R = 2 −2 −1 U u the WPL term of the water vapor and CO2 fluxes. where d d 402 m s K , n and n are the However, modern sonic thermometry does not di- mean wind speed (Un) and the wind speed fluc- T u rectly measure the ambient temperature, a,orthe tuations ( n) normal to the axis of the sonic that T  w δ turbulent fluctuations a (Kaimal and Gaynor, 1991). measures temperature (usually the axis), and un Rather, modern sonics measure the sonic virtual tem- is 0 if the sonic’s internal signal processing software perature, Ts, defined by Kaimal and Gaynor (1991) includes this correction and 1 if it does not. We do as Ta(1 + 0.32pv/pa). This section derives the rela- not consider this correction any further in this study T  T  tionship between v s and v a . because some sonics already include this correction Assuming the definition of Ts just given, we first internally in their signal processing software. But, be- ¯ decompose Ts into a mean, Ts, and a fluctuating com- cause we are uncertain whether all sonics include this T  ponent, s and then perform the Reynolds averaging correction or not, we feel it important to point out the on the resulting equation. This yields the following existence of these cross-wind effects. For example, two equations: for most applications the vertical component of this ¯ ¯ term is proportional to Uuw, which implies that it Ts = Ta(1 +¯σv) (B.23)
 should not be ignored during windy or very turbulent ρ p T  = T ( +¯σ ) +¯σ T¯ v − a conditions. s a 1 v v a (B.24) ρ¯v p¯a B.3. The combined turbulent fluxes where σ¯v = (0.32p¯v/p¯a). For this derivation we have (i) taken advantage of Eq. (B.4) to simplify p , (ii) ig- v T  nored the small cross-correlation terms in Eq. (B.23), Eq. (B.26) indicates that v a measured using sonic p /p¯ thermometry is a function of the water vapor flux. and (iii) linearized Eq. (B.24) in the a a term. Multiplying Eq. (B.24) by v and taking the Reynolds However, as indicated in Section B.1, the vapor flux, ρ F average of the resulting equation yields the following v v , must include the WPL term, which in turn is a T  T  equation for the turbulent temperature flux, v a . function of v a . Therefore, the temperature and va-   por flux expressions form coupled equations. We com- T  σ¯ ρ F p   v s v ¯ v v v a v T = − Ta − (B.25) plete this section by combining the results of the two a 1 +¯σ 1 +¯σ ρ¯ p¯ T  v v v a previous sections stating the solution for v a and the F F resulting expressions for vρ and CO , vρ .For Dividing both sides of this equation by T¯ and sim- v 2 c a the present purposes, none of these fluxes include the plifying the ratio σ¯ /ρ¯ for the vapor flux term and v v mean flow component. σ¯v/(1 +¯σv) for the pressure flux term yields the

 following (more computationally useful) equation: T  T  α¯ ( +¯χ ) ρ v a = 1 v s − v 1 v v v T  T  ρ F p T¯ + δ λ¯ T¯ + δ λ¯ ρ¯ v a = v s −¯α v v + β¯ v a a 1 oc v s 1 oc v d ¯ ¯ v ρ¯ v p¯ (B.26) Ta Ts d a
 β¯ ( +¯χ ) p α¯ = . µ /( + . χ¯ ) β¯ = . χ¯ / + v 2 v v a where v 0 32 v 1 1 32 v and v 0 32 v ¯ p¯ (B.28) 1 + δocλv a (1 + 1.32χ¯v). W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144 141

F ρ = ( +¯χ ) ρ +¯ρ ( +¯χ ) Baldocchi, D.D., Meyers, T.P., 1991. Trace gas exchange above v v 1 v v v v 1 v   the forest floor of a deciduous forest. 1. Evaporation and CO     2 v T v p efflux. J. Geophys. Res. 96, 7271–7285. × δ a − a (B.29) oc ¯ p¯ Baldocchi, D., Finnigan, J., Wilson, K., Paw U, K.T., Falge, E., Ta a 2000. On measuring net ecosystem carbon exchange over tall vegetation on complex terrain. Bound.-Lay. Meteorol. 96, 257–   291.     F v T v p vρ = vρ +¯ρ (1 +¯χ ) δ a − a Barr, A.G., Griffis, T., Black, T.A., Lee, X., Staebler, R.M., c c c v oc ¯ p¯ Fuentes, J.D., Chen, Z., Morgenstern, K., 2002. Comparing Ta a the carbon budgets of mature boreal and temperate deciduous +¯ω µ ρ forest stands. Can. J. For. Res. 32, 813–822. c vv v (B.30) Berger, B.W., Davis, K.J., Yi, C., Bakwin, P.S., Zhao, C.L., 2001. ¯ ¯ ¯ where λv = βv(1 +¯χv) and α¯ vω¯ v = βv. By con- Long-term carbon dioxide fluxes from a very tall tower in vention, Eq. (B.28) expresses vT  in terms of the a northern forest: flux measurement methodology. J. Atmos. a Ocean. Technol. 18, 529–542. covariances between the sonic anemometer and the Black, T.A., den Hartog, G., Neumann, H.H., Blanken, P.D., Yang, instruments used to measure vapor and pressure P.C., Russell, C., Nesic, Z., Lee, X., Chen, S.G., Staebler, R., T  fluctuations. Once v a has been determined from Novak, M.D., 1996. Annual cycles of water vapor and carbon Eq. (B.28), it then can be used in Eqs. (B.29) and dioxide fluxes in and above a boreal aspen forest. Glob. Change (B.30) to estimate the fluxes of water vapor and other Biol. 2, 219–229. Black, T.A., Chen, W.J., Barr, A.G., Arain, M.A., Chen, Z., Nesic, trace gases. Z., Hogg, E.H., Neumann, H.H., Yang, P.C., 2000. Increased In summary, in the main text we cite Eqs. (B.16), carbon sequestration by a boreal deciduous forest in years with (B.17), (B.28)–(B.30) and its equivalent for water va- a warm spring. Geophys. Res. Lett. 27, 1271–1274. por, and the WPL or dry air flux equation, the equation Brost, R.A., Wyngaard, J.C., 1978. A model study of the stably for vρ estimated from Eq. (B.13), as the fundamen- stratified planetary boundary layer. J. Atmos. Sci. 35, 1427– d 1440. tal equations of eddy covariance. Colbeck, S.C., 1989. Air movement in snow due to wind pumping. J. Glaciol. 35, 209–213. B.4. The turbulent heat flux Dyer, A.J., 1963. The adjustment of profiles and eddy fluxes. Quart. J. R. Meteorol. Soc. 89, 276–280. Eugster, W., Senn, W., 1995. A cospectral correction model for For the sake of completeness, Eq. (B.31) below measurement of turbulent NO2 flux. Bound.-Lay. Meteorol. 74, gives the turbulent 3D heat flux, H, in terms of the 321–340. T  Farrell, D.A., Graecen, E.L., Gurr, C.G., 1966. Vapor transfer in temperature flux, v a , and is adapted from Sun et al. (1995): soil due to air turbulence. Soil Sci. 102, 305–313. Finnigan, J., 1983. A streamline coordinate system for distorted = ρ¯ C +¯ρ C T  turbulent shear flows. J. Fluid Mech. 130, 241–258. H [ d pd v pv]v a (B.31) Finnigan, J., 1999. A comment on the paper by Lee (1998): “On micrometeorological observations of surface air-exchange over where Cpd is the specific heat capacity for dry air − − tall vegetation”. Agric. For. Meteorol. 97, 55–64. = 1 1 Cp ( 1005 J kg K ) and v is the specific heat ca- Finnigan, J., Brunet, Y., 1995. Turbulent airflow in forests on flat = −1 −1 pacity for water vapor ( 1846 J kg K ). Other rel- and hilly terrain. In: Coutts, M.P., Grace, J. (Eds.), Wind and atively small terms (Sun et al., 1995) are negligible Trees. Cambridge University Press, Cambridge, UK, pp. 3–40. for the present purposes. Finnigan, J.J., Clement, R., 2002. A re-evaluation of long-term flux measurement techniques. Part 2. Coordinate systems. Finnigan, J.J., Clement, R., Mahli, Y., Leuning, R., Cleugh, H., 2002. A re-evaluation of long-term flux measurement References techniques. Part 1. Averaging and coordinate rotation. Bound.-Lay. Meteorol., (in press). Aubinet, M., Grelle, A., Ibrom, A., Rannik, Ü., Moncrieff, Fitzjarrald, D.R., Moore, K.E., 1990. Mechanisms of nocturnal J., Foken, T., Kowalski, A.S., Martin, P.H., Berbigier, P., exchange between the rain forest and the atmosphere. J. Bernhofer, C., Clement, R., Elbers, J., Granier, A., Grunwald, Geophys. Res. 95, 16839–16850. T., Morgenstern, K., Pilegaard, K., Rebmann, C., Snijders, Frost, S.R., 1981. Temperature dispersion in turbulent pipe flow. W., Valentini, R., Vesala, T., 2000. Estimates of the annual Ph.D. Thesis. University of Cambridge, Cambridge. net carbon and water exchange of forests: the EUROFLUX Fuehrer, P.L., Friehe, C.A., 2002. Flux corrections revisited. methodology. Adv. Ecol. Res. 30, 113–175. Bound.-Lay. Meteorol. 102, 415–457. 142 W.J. Massman, X. Lee / Agricultural and Forest Meteorology 113 (2002) 121–144

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