Journal of Geophysical Research: Atmospheres

Modeling nighttime ecosystem respiration from measured CO2 concentration and air temperature profiles using inverse methods



[1] A major challenge for quantifying ecosystem carbon budgets from micrometeorological methods remains nighttime ecosystem respiration. An earlier study utilized a constrained source optimization (CSO) method using inverse Lagrangian dispersion theory to infer the two components of ecosystem respiration (aboveground and forest floor) from measured mean CO2 concentration profiles within the canopy. This method required measurements of within-canopy mean velocity statistics and did not consider local thermal stratification. We propose a Eulerian version of the CSO method (CSOE) to account for local thermal stratification within the canopy for momentum and scalars using higher-order closure principles. This method uses simultaneous mean CO2 concentration and air temperature profiles within the canopy and velocity statistics above the canopy as inputs. The CSOE was tested at a maturing loblolly pine plantation over a 3-year period with a mild drought (2001), a severe drought (2002), and a wet year (2003). Annual forest floor efflux modeled with CSOE averaged 111 g C m−2 less than that estimated using chambers during these years (2001: 1224 versus 1328 gCm−2; 2002: 1127 versus 1230 gCm−2; 2003: 1473 versus 1599 gCm−2). The modeled ecosystem respiration exceeded estimates from eddy covariance measurements (uncorrected for storage fluxes) by at least 25%, even at high friction velocities. Finally, we showed that the CSOE annual nighttime respiration values agree well with independent estimates derived from the intercept of the ecosystem light-response curve from daytime eddy covariance CO2 flux measurements.

1. Introduction

[2] Long-term measurements of net ecosystem exchange (NEE) are now routinely employed to estimate ecosystem carbon budgets using eddy covariance (EC), yet the large error in the measurement of ecosystem respiration (RE) under nighttime conditions remains an unresolved problem that must be confronted [Baldocchi et al., 1996; Goulden et al., 1996; Law et al., 1999a, 1999b; Lindroth et al., 1998; Moncrieff et al., 1996; Schmid et al., 2000; Valentini et al., 2000; Wofsy et al., 1993]. Often, nocturnal conditions are dominated by vertical subsidence, lack of steadiness in mean atmospheric conditions, and intermittent turbulent transport often initiated by transients such as passage of clouds [Cava et al., 2004]. When viewed from the one-dimensional vertically integrated scalar continuity equation, these factors contribute to increased “decoupling” between the desired RE quantity and the CO2 flux above the canopy, the latter being the observed quantity by EC methods. Furthermore, these nocturnal conditions tend to amplify the limitations of the EC instrument configurations for measuring the turbulent flux. For example, separation distance between gas analyzers and anemometers, volume averaging by anemometers across some path length, and finite sampling periods that may be too short to resolve intermittency (and other low-frequency contributions) contribute to a reduction in the measured turbulent flux by EC systems [De Bruin et al., 1993; Kaimal and Finnigan, 1994; Kaimal and Gaynor, 1991; Leuning and Judd, 1996; Massman, 2000; Moncrieff et al., 1996].

[3] In this study, we argue that these theoretical and sampling reasons necessitate exploring other micrometeorological methods that are sensitive to different set of assumptions and approximations to constrain or independently verify nighttime RE estimates derived from EC.

[4] An independent approach to estimating RE is to utilize a functional relationship between aboveground mean CO2 sources, equation image, or turbulent fluxes, equation image, and a relatively simpler quantity to measure such as mean CO2 concentration profiles, equation image, within the canopy volume, where t is time, z is the height from the forest floor and the overbar denotes the temporal and spatial averaging operator. This framework is not new and dates back to Woodwell and Dykeman [1966]. The basic premise is that equation image and equation image can be related to equation image using the temporally and horizontally averaged one-dimensional continuity equation for a planar homogeneous flow, given by

equation image

which, upon vertical integration, yields

equation image

where h is the mean canopy height, and RE is defined as

equation image

where equation image is the forest floor efflux. In equation (2), equation image (which can be measured by EC) represents RE when equation image = 0. A primitive approach to compute equation image in equation (1) can be formulated on the basis of equation image measurement using first-order closure principles (or K-theory) by assuming that

equation image

where Kt is the eddy diffusivity.

[5] Over the past 30 years, however, theoretical developments and many laboratory and field experiments have demonstrated that scalar and momentum fluxes within many canopies do not always obey K-Theory [Corrsin, 1974; Deardorff, 1972, 1978; Denmead and Bradley, 1985; Finnigan, 1985; Raupach, 1988; Shaw, 1977; Sreenivasan et al., 1982; Wilson, 1989]. To alleviate K-Theory limitations, other theoretical and practical methods were developed without resorting to a local eddy diffusivity approximation [Katul and Albertson, 1999; Raupach, 1988, 1989a, 1989b; Siqueira and Katul, 2002].

[6] For example, Lai et al. [2002a] used the Localized Near Field (LNF) theory to relate equation image to equation image and demonstrated some success in estimating the two components of RE (i.e., equation image and equation image) over a 1-year period for near neutral and mildly stable flows. However, they pointed out a drawback of their method, titled Constrained Source Optimization (CSO), in that it was incapable of resolving the effects of local thermal stratification at a particular z within the canopy except through a Lagrangian integral timescale. Previous Lagrangian methods attempted to correct the Lagrangian timescale via a uniform multiplier derived from Monin-Obukhov similarity theory [Hsieh et al., 2003; Leuning, 2000]. Several basic issues within Lagrangian models remain subject to debate outside the stability effects – most notable is that almost all Lagrangian models assume a vertically uniform timescale [Lai et al., 2002a]. This assumption cannot be reconciled with a uniform mixing length scale inside the canopy [Katul et al., 2004].

[7] On the other hand, Siqueira et al. [2002, 2003], and Siqueira and Katul [2002] developed Eulerian closure models that are capable of accounting for local thermal stratification within the canopy volume if mean air temperature profile measurements are available.

[8] This study combines the two approaches by revising the CSO method of Lai et al. [2002a] to include local thermal stratification within the canopy volume using higher-order closure principles. We tested this modified CSO method over a 3-year period at a maturing southeastern pine forest using independent measurements of equation image and equation image. The study period includes a mild drought (2001), a severe drought (2002), and a very wet year (2003) so that widest ranges of hydrologic and climatic conditions at this site are sampled. Improvements over RE estimation from EC measurements using standard friction velocity u* thresholds are discussed within the context of annual carbon balances.

2. Theory

2.1. Governing Equations and Turbulent Transport

[9] Rather than using K-theory, we consider the steady state one-dimensional budget equation for the temporally and horizontally averaged carbon flux for high Reynolds and Peclet numbers flows (i.e., the molecular diffusion and conductive heat transfer are neglected), given by:

equation image

where w is the vertical velocity, equation image is the mean air temperature, p is the turbulent static pressure normalized by air density ρ, g is the gravitational acceleration constant, and equation imageC is the molecular dissipation term. The symbol prime denotes the departures from averaging operator. To solve equation (4) from measured mean CO2 concentration profiles, further parameterizations are needed to quantify the vertical velocity variance equation image, the covariance between temperature and CO2 turbulent fluctuations equation image (i.e., the local atmospheric stability effects), the concentration-pressure interaction term equation image, the flux dissipation term equation imageC, and the triple moment equation image. For equation image, we employ a second-order closure model [Katul and Albertson, 1998; Wilson and Shaw, 1977] that solves for the mean velocity equation image and Reynolds stresses equation image, as discussed in Appendix A.

[10] In equation (4), the air temperature equation image and equation image need to be determined. The corresponding steady state one-dimensional temporal and horizontally averaged budget equations of the mean air temperature equation image and the vertical kinematic turbulent flux of sensible heat equation image can be derived as:

equation image
equation image

where equation imageT is the molecular dissipation term for sensible heat, and equation image is the corresponding heat source/sink term at level z.

[11] Unlike the LNF approach utilized by Lai et al. [2002a], the covariance equation image and variance equation image explicitly characterize the local (i.e., at given level z within the canopy) buoyant production/destruction effects. To compute the budgets of these two variables, additional steady state one-dimensional prognostic equations are needed and given by Meyers and Paw U [1987]:

equation image


equation image

where ɛTC and ɛTT are the corresponding molecular dissipation terms.

[12] In equations (4), (6), (7), and (8), the pressure-gradient diffusion terms equation image, equation image, molecular dissipation terms equation imageT, equation imageC, equation imageTC, equation imageTT, and triple correlation terms equation image, equation image, equation image, equation image are unknowns that need parameterizations. To solve these additional variables, standard second-order closure approximations are employed [Donaldson, 1973; Katul and Albertson, 1998; Mellor, 1973; Mellor and Yamada, 1974; Meyers and Paw U, 1986, 1987; Wilson and Shaw, 1977]. After utilizing these closure parameterizations, equations (4), (6), (7), and (8) can be rewritten as

equation image
equation image
equation image


equation image

where Q is the characteristic turbulent velocity (square root of the mean turbulent kinetic energy equation image); λ1, λ2, λ3 are length scales for the various terms as in Wilson and Shaw [1977] and Katul and Albertson [1998], and physically represent the characteristic length scales for the triple velocity correlations, the pressure-velocity gradient correlations, and the viscous dissipations, respectively. The parameterizations for these length scales are discussed in Appendix A.

[13] Coupling equations (1), (5), (9), (10), (11), and (12) with the set of equations (A1) for momentum in Appendix A (mainly to solve for Q and equation image) results in 6 equations with 8 unknowns (equation image, equation image, equation image, equation image, equation image, equation image, equation image, equation image). If equation image and equation image measurements are available, the system reduces to 6 equations with 6 unknowns thereby permitting one to numerically determine equation image, equation image and, in turn, RE.

2.2. Eulerian Inverse Model for Heat

[14] We used the Eulerian inverse model proposed by Siqueira and Katul [2002] to determine equation image and equation image from mean air temperature profile measurement. These variables are needed to solve equations (5), (10), and (12). The boundary conditions for equation image and equation image are as proposed by Meyers and Paw U [1987] and are applied to equations (10) and (12), respectively. After estimating equation image and equation image from measured temperature profiles, the heat source term equation image can be directly determined from equation (5).

[15] The advantage of this inverse model is that the effects of atmospheric stability within the canopy volume can be explicitly considered. As discussed by Siqueira and Katul [2002], the impact of atmospheric stability is most pronounced in the scalar-temperature covariance equations (equation image and equation image). These terms are now directly considered via their budget equations.

2.3. Source Calculation

[16] We estimate the CO2 turbulent fluxes and source terms differently from temperature for several reasons: (1) The aboveground plant area density is indicative of the relative “distribution” of aboveground respiring biomass thereby providing an additional constraint on equation image; (2) small errors in measured mean CO2 concentration profile can dramatically impact the inference of equation image from measured equation image because of the absence of any redundancy [Siqueira et al., 2003]; and (3) the temperature sensitivity of equation image, while not precisely known, can be constrained from leaf measurements.

[17] The estimation of equation image at each level from measured C(t, z) can be reformulated as an optimization problem [Lai et al., 2002a; Styles et al., 2002] in which the relative strength of equation image and its temperature sensitivity is a priori defined. Thus, rather than solving equations (1), (5), (9), (10), (11), and (12) for equation image forced by mean CO2 concentration profile measurements, the system can be forced by an estimate of equation image and predict the mean CO2 concentration distribution, which can in turn be compared to measurements (e.g., every half hour).

[18] To formulate a model for equation image, we note that the woody and leaf foliage tissue respiration have different physiological properties and hence their contribution to the total aboveground respiration is different. However, Lai et al. [2002a] argued that in a first-order analysis, the respiration of woody tissue is less important than the contribution from foliage because the total woody surface area is less than the total leaf surface area (at least for this pine site), and the woody parts have smaller tissue-specific respiration rates than the foliage [Hamilton et al., 2002]. With this simplification, Lai et al. [2002a] estimated the carbon source vertical distribution equation image by assuming that the entire plant surface area was only foliage leading to:

equation image

where a(t, z) is the plant area density (PAD, in m2 m−3), and Rd(t, z) is the dark respiration rate per unit plant tissue surface area (in μmol m−2 s−1). Rd(t, z) can be estimated from the Farquhar et al. [1980] model:

equation image

where α(t) is a constant that needs to be determined at a given time t, and Vcmax(t, z) is the maximum catalytic capacity of Rubisco per unit leaf area. The temperature dependency of Vcmax(t, z) can be expressed as:

equation image

where a1 and a2 are the species-specific adjustment coefficients, which are obtained experimentally (e.g., via porometry) and Vcmax,25(t, z) is the value of Vcmax,25(t, z) at 25°C [Campbell and Norman, 1998; Collatz et al., 1991; Farquhar et al., 1980]. From previous studies conducted at the site [Ellsworth, 1999; Naumburg and Ellsworth, 2000; Naumburg et al., 2001], Vcmax,25, a1 and a2 are 59 μmol m−2 s−1, 0.051 and 0.205, respectively for the upper canopy pine foliage, and are 30 μmol m−2 s−1, 0.088 and 0.290, respectively for the subcanopy broadleaved plants [Lai et al., 2002a]. With the exception of the product {α(tVcmax,25(t, z)} these physiological parameters were considered temporally constant for the model calculations.

2.4. Modified Constrained Source Optimization

[19] With this formulation for equation image, the problem reduces to a two-parameter estimation ({α(tVcmax,25(t, z)} and equation image) from the measured nighttime CO2 concentration profiles equation image. Thus the question is what is the optimum combination of {α(tVcmax,25(t, z)} and equation image so that that the solution to equations (1), (5), (9), (10), (11), and (12) best matches the measured equation image? Because only two parameters describe the entire source and they can be constrained to vary within a limited range, a global search for the optimum {α(tVcmax,25(t, z)} and equation image can be conducted until the root mean square error (RMSE) between the calculated and measured CO2 concentrations at different levels is minimized on a 30-min timescale. For example, if foliage respiration is the only dominant aboveground respiration component, we anticipate α(t) to be near 0.015, which is a commonly used value in many studies [Collatz et al., 1991; Farquhar et al., 1980; Lai et al., 2002a]. Furthermore, nighttime forest floor respiration should not exceed the maximum daytime photosynthesis in magnitude. An upper limit on the maximum daytime canopy photosynthesis An,c can be determined from daytime water vapor flux measurements (i.e., latent heat flux) using

equation image

where LE is determined as the maximum latent heat flux measured for each day throughout the experiment, VPD is the mean daytime vapor pressure deficit and Ci/Ca is the ratio of intercellular to ambient atmospheric CO2 concentration, estimated at 0.66 for sunlit foliage [Katul et al., 2000]. This leads to a maximum conservative estimate of mean photosynthesis and sets an upper limit to a priori constrain nighttime respiration. Because of these constraints and its Eulerian formulation to account for thermal stratification within the canopy, we refer to this method as the Eulerian constrained source optimization (CSOE).

3. Study Site and Measurements

3.1. Study Site

[20] The measurements were made at the Blackwood Division of the Duke Forest near Durham in North Carolina (site location: 35°58′N, 79°05′W, 163 m above sea level) as part of the AmeriFlux long-term CO2 flux monitoring initiative [Baldocchi et al., 2001]. This study site is a uniformly planted loblolly pine (Pinus taeda L.) forest (planted in 1983 at 2 m × 2.4 m spacing) that extends 300 to 600 m in the east-west direction and 1000 m in the south-north direction. The subcanopy also contains about 40 woody species, of which Liquidambar styraciflua L., Acer rubrum L., Ulmus alata Michx., and Cornus florida L. are the most prevalent [Palmroth et al., 2005]. The local topographic variations are small (slope < 5%) enough to ignore the effect of the complex terrain on the flow statistics [Siqueira et al., 2002]. The study period extends from year 2001 to 2003. Table 1 describes the variations in ecological, hydrologic, and climatic conditions for these 3 years at this study site.

Table 1. Overall Variability Ranges in Ecophysiological, Hydrological, and Climatic Factors From 2001 to 2003 at the Duke Forest Pine Sitea
  • a

    The soil moisture content, soil temperature, and the air temperature are half-hourly averages from the profile measurement.

  • b

    In December of 2002, an ice storm reduced the PAI of the pine stand.

  • c

    The growing season is from 1 April to 30 September (Julian day: 152 to 273).

  • d

    The bracketed numbers represents the range during the growing season.

Average canopy height, m17.017.518.0
Total PAI range, m2m−22.20 ∼ 6.532.22 ∼ 5.162.09 ∼ 4.16b
Pine PAI range, m2m−21.85 ∼ 3.261.91 ∼ 2.961.74 ∼ 3.23b
Growing seasonc precipitation, mm529.2371.4789.8
Soil moisture content (θ) range, m3m−30.13 ∼ 0.54 (0.13 ∼ 0.51)d0.13 ∼ 0.47 (0.13 ∼ 0.37)0.20 ∼ 0.54 (0.20 ∼ 0.54)
Soil temperature (Ts) range, °C3.1 ∼ 23.5 (8.8 ∼ 23.5)5.4 ∼ 23.8 (9.6 ∼ 23.8)3.7 ∼ 23.6 (9.0 ∼ 23.6)
Air temperature (Ta) range, °C−11.2 ∼ 35.5 (−2.8 ∼ 35.5)−10.7 ∼ 38.9 (−2.6 ∼ 38.9)−12.2 ∼ 35.2 (0.3 ∼ 35.2)

3.2. Eddy Covariance Flux Measurements

[21] The momentum components, Reynolds stresses, sensible heat, latent heat and CO2 fluxes above the canopy were measured by a conventional eddy covariance system comprising a Li-Cor 7500 CO2/H2O open-path infrared gas analyzer (Li-Cor Inc., Lincoln, Nebraska, USA) and a triaxial sonic anemometer (CSAT3, Campbell Scientific Inc., Logan, Utah, USA). Both the gas analyzer and the triaxial sonic anemometer were positioned at z = 20.23 m, which is above the canopy top (from 17 to 18 m during the study period).

[22] The flux measurements were sampled using a Campbell Scientific 23X data micrologger with all digitized signals transferred to a portable computer via an optically isolated RS232 interface for future processing. All the variables in this eddy covariance measuring system were sampled at 10 Hz and averaged every 30 min. The correction for the effects of air density on flux measurements after Webb et al. [1980] was applied.

3.3. Mean CO2 Concentration and Air Temperature Profiles Within the Canopy

[23] A multilayer concentration monitoring system was installed to sample the mean water vapor pressure and CO2 concentration at 10 different levels throughout the canopy volume (z = 0.1 m, 0.75 m, 1.5 m, 3.5 m, 5.5 m, 7.5 m, 9.5 m, 11.5 m, 13.5 m and 15.5 m) using a Li-Cor 6262 CO2/H2O infrared gas analyzer. This profiling system includes a multiport gas-sampling manifold to sample each level for 1 min (45 s sampling and 15 s purging) with a repeating cycle of 10 min for the 10 sampled levels. Data were averaged every 30 min. In addition, a mean air temperature profiling system was installed to measure the mean air temperature every 30 min at eight different levels (z = 1.5 m, 3.5 m, 5.5 m, 7.5 m, 9.5 m, 11.5 m, 13.5 m, and 15.5 m) throughout the canopy volume using copper-constantan shielded thermocouple sensors (see Siqueira and Katul [2002] for details).

3.4. Volumetric Soil Moisture Content and Soil Temperature Measurement

[24] Long-term volumetric soil moisture content θ (m3 m−3) was sampled using 4 Campbell Scientific CS615 reflectometers placed in the top 30 cm of the mineral soil, and the soil temperature was measured at 10–12 cm via nonlinear thermistor probes (M 841/S1, Siemens, Germany). All signals were sampled every 30 s using a CR23X data logger and averaged every 30 min. The mean soil moisture content was obtained by averaging over all 4 CS615 probes.

3.5. Forest Floor CO2 Efflux Measurements

[25] The forest floor CO2 efflux was measured with the automated carbon efflux system (ACES, US Patent 6692970) developed by the USDA Forest Service, Southern Research Station Laboratory in Research Triangle Park, North Carolina [Butnor and Johnsen, 2004; Butnor et al., 2003; Palmroth et al., 2005]. The ACES is an open system with an infrared gas analyzer connected to several soil chambers equipped with soil and air thermocouples, pressure equilibration ports, and reflective covers. The ACES system was installed at the site in February of 2001. The details of the ACES configuration, quality checks, and spatial sampling area, are described by Palmroth et al. [2005].

[26] To quantify the variation of forest floor efflux with varying volumetric soil moisture content equation image (m3 m−3) and soil temperature Ts (°C), Palmroth et al. [2005] derived a equation modified from Fang and Moncrieff [1999], of the form

equation image

where Rb is the base respiration (μmol m−2 s−1), which is defined as the intercept at 0°C, a is the temperature sensitivity (Q10 = ea×10) when soil moisture content is not limiting, and the constants b and c are fitted parameters of the soil moisture reduction function. All the constants in equation (16) were determined via nonlinear regression methods using the ACES measured respiration, the mean 10–12 cm soil temperature, and the CS615 soil moisture and are summarized in Table 2 [Palmroth et al., 2005]. Equation (16) constitutes the spatially averaged chamber measurements of forest floor efflux and will be used to independently test the CSOE estimates of equation image.

Table 2. Regression Parameters for the Chamber-Based Forest Floor CO2 Efflux Equation F(t, 0) = imagee(−bθ+c)⌋ Given by Palmroth et al. [2005] for 2001 to 2003 at the Duke Forest Sitea
001 ∼ 1790.5700.12325.2282.301
180 ∼ 3650.6480.11827.1202.882
001 ∼ 0880.6480.11827.1202.882
089 ∼ 2380.9080.09227.9523.099
239 ∼ 3650.7530.10632.3023.987
001 ∼ 3650.8140.10432.3023.987

3.6. Plant Area Index and Plant Area Density

[27] The plant area index (PAI, m2 m−2) is routinely measured several times a year using a pair of Li-Cor LAI 2000 optical sensors. The plant area density (PAD, m2 m−3) measurements were conducted at 1 m intervals from the bottom to the top of canopy. Calibration of PAI was done using allometric relationships derived from different individual species within the canopy volume [Lai et al., 2000b; Pataki et al., 1998; Schafer et al., 2003].

[28] LAI-2000 measurements, coupled with the abovementioned allometric functions, were used to estimate the vertical distribution of PAD at daily time steps [Schafer et al., 2003]. The range of total PAI and pine PAI in 2001, 2002, and 2003 are given in Table 1.

4. Results and Discussion

[29] To address the study objective, the results and discussion are organized as follows: (1) We use the measured equation image to estimate the nighttime storage fluxes in equation (1) and compare their magnitude to the EC measured nighttime NEE noting that almost all sites that utilize a large u* threshold for the EC data neglect storage fluxes in equation (1). (2) We use the CSOE model to individually estimate the two components of nighttime ecosystem respiration and compare them to the results from chamber measurements and to independent estimates of aboveground respiration. (3) We discuss the sensitivity of the modeled RE to local thermal stratification by comparing model calculations with and without the consideration of buoyant production/destruction terms for thermally stratified condition and neutral flows condition, respectively. (4) Finally, we discuss the CSOE respiration components within the context of the annual carbon balance at the site, and explore other methods to constrain annual nighttime ecosystem respiration (e.g., intercept of the NEE light response curve).

[30] To ensure that nighttime conditions are not “contaminated” by photosynthesis, we define nighttime hereafter from 2000 to 0500 LT throughout the 3-year study period.

4.1. Storage Flux

[31] Gap-filled nighttime EC measured flux (hereafter referred to as equation image) is often derived from high u* runs, in which the canopy is likely to be ventilated (except for a region close to the ground). Under such conditions, it is reasonable to assume that equation image. Hence, when determining time series of RE, equation image is often neglected when using gap-filled equation image collected for high u*. On a daily basis, the mean value of equation image is often close to zero, but can be significantly large during sunrise, sunset and during nighttime conditions of low u* [Lai et al., 2000a, 2002a].

[32] Because equation image is the key determinant of equation image, we show the 2-month ensemble averaged equation image (in ppm) measured at different times of the day during 2003 for illustration (Figure 1). It is clear that the mean CO2 concentration is unsteady during nighttime conditions and this buildup trend is even stronger during high leaf area season (i.e., May to October). This finding is not surprising because the canopy respires more during the summer months, because of both higher leaf mass and higher temperature, and because the turbulence is dampened during the high leaf area season (see Appendix A). More subtle is the observation that the temporal variation of the measured ensemble averaged u* (red solid line) is also different across seasons with lower values measured in the summer. This finding is important when using a global u* threshold for gap filling equation image measurements because such a threshold may disproportionately eliminate summertime runs.

Figure 1.

Normalized depth/time of day variations of nighttime (2000 to 0500 LT, bounded by two dashed lines in each subplot) ensemble averaged CO2 concentration profiles (ppm) for 2-month periods in 2003. The 2-month ensemble averaged friction velocity u* (m s−1, solid red line) is shown for reference.

[33] To quantify the effect of nighttime variations in u* on EC and storage flux, we compared equation image at different u* thresholds (Figure 2, showing 14-day averages during nighttime from the entire 3-year study period). The value of equation image is derived from the numerical integration of the measured mean CO2 concentration profile every 30 min run and ensemble averaged every 14 days.

Figure 2.

Ensemble variation of the ratio of storage flux to EC measured flux (equation image) with friction velocity u* during nighttime runs (2000 to 0500 LT) for the entire measurement period (2001–2003). The flux ratio equation image is expressed as 14-day ensemble averages, and vertical bars represent one standard deviation. The solid line is the regression curve.

[34] Figure 2 indicates that the ensemble averaged equation image is almost always greater than 0.27 at this experimental site. The mean equation image ratio increases from 0.27 to about 0.44 when u* drops from 0.45 to 0.15 m s−1, but significantly increases when u* drops below 0.15 m s−1. Note also that measured u* < 0.15 m s−1 is a common occurrence for summertime runs, especially in 2003 (Figure 1). Thus this analysis indicates that RE may be larger than equation image by at least 27% at this experimental site if storage is neglected. However, we emphasize that determining equation image from a single tower is subject to several theoretical and practical limitations, and the need to ensemble average concentration data (e.g., 14-day) beyond averaging random noise are further discussed in Appendix B.

4.2. Optimized Forest Floor Carbon Efflux

[35] The optimized {α(tVcmax,25(t, z)} and equation image were determined over a 14-day ensemble average period on the basis of the root mean squared error (RMSE ≤ 10 ppm) of 30-min comparisons between CSOE modeled and measured equation image. The available numbers of 30-min runs used in the model calculations for 2001 to 2003 are 396, 432, and 450 runs, respectively. The resulting CSOE optimized equation image is then regressed with measured 10–12 cm soil temperature Ts (Figure 3a). The relationship is expressed as

equation image

where A and B are regression parameters (Figure 3a; with an individual regression fit for each year) presented in Table 3. From Table 3, it is clear that these fitted parameters change from year to year. For example, forest floor carbon efflux values modeled with CSOE for the severe drought year of 2002 were different than the other 2 years, especially when soil temperature was high (Figure 3a). Calculated from parameter B, the Q10 values for 2001, 2002 and 2003 are 2.34, 1.82 and 2.32, respectively, consistent with the values reported by Palmroth et al. [2005]. Using a Student's t-test, the reduction in Q10 for 2002 is statistically significant at the 95% confidence level.

Figure 3.

(a) Variation of the CSOE optimized forest floor efflux (open circle) with soil temperature for each of the 3 years. The solid lines are obtained by regressing soil temperature to the CSOE optimized values of equation image. They demonstrate the nonstationarity in forest floor respiration-soil temperature curve parameters. (b) Soil temperature effect (left plot) for θ ≥ 0.2 m3 m−3 and soil moisture reduction curve (solid line on the right plot) for all 3 years. All equations and regression statistics are shown in Table 3.

Table 3. Regression Curves for F(t, 0) Shown in Figure 3a
YearFitted CurveR2RMSEQ10
  • a

    Unit is μmol m−2 s−1. The coefficient of determination R2 and the root mean squared error RMSE (in μmol m−2 s−1) are also shown. For reference, we also show the equivalent Q10 values.

Ts-Dependent Only
2001F(t, 0) = 0.846·exp(0.085·Ts)0.661.082.34
2002F(t, 0) = 1.191·exp(0.060·Ts)0.551.221.82
2003F(t, 0) = 1.036·exp(0.084·Ts)0.761.032.32
Unique Fitted Curve Couple With θCorrection
2001–2003F(t, 0) = FTs·[1−exp(−37.829θ + 3.948)0.521.122.09
Ts dependenceFTs = 0.974·exp(0.085·Ts)0.731.052.34
θ correction1−exp(−37.829θ + 3.948)0.15N/AN/A

[36] To investigate whether the variability in parameters are driven by soil moisture effects, we fit equation (16) [Palmroth et al., 2005] to the entire 3-year record. We separate the model results into two different equation image regions (equation image ≥ 0.2 m3 m−3 and equation image < 0.2 m3 m−3), where the value of equation image < 0.2 m3 m−3 is the critical point at which θ significantly affects equation image [Palmroth et al., 2005]. For the non-soil-moisture-limiting region, it is clear from Figure 3b that one temperature curve suffices to explain the entire optimized forest floor flux variability (hereafter, the estimate of equation image from this curve is referred to as FTs). For the soil-moisture-limiting region, we plot relative equation image (expressed as equation image) against θ and show that resulting reduction is consistent with the chamber data. When combining these two findings, a unique multivariate curve for the entire 3-year record can be derived (Table 3).

4.3. Optimized Aboveground C Source

[37] There are no explicit measurements for the aboveground respiration during this period and hence the evaluation of the CSOE model is not direct. Nonetheless, we can assess whether the CSOE model is sensitive to well-documented variability in Vcmax,25(t, z). Toward this end, we compare the seasonal dynamics in {0.015·Vcmax,25(t, z)} as derived from porometry [Ellsworth, 2000] with the optimized {α·Vcmax,25(t, z)} from the CSOE.

[38] If we set α = 0.015 and compute Vcmax,25(t, z) via equations (14) and (15), the normalized seasonal variation of Vcmax,25 (expressed as Vcmax,25/mean Vcmax,25) derived from the CSOE model calculations can be compared to the porometry data shown in Figure 4. The comparison with the porometry data cannot be direct because the published porometry measurements given by Ellsworth [2000] were conducted from 1998 to 2000. Nonetheless, the qualitative agreement in Figure 4 suggests that the mean CO2 concentration profile data, when combined with the CSOE model, can resolve seasonal shifts in aboveground physiological properties due to leaf acclimation. This agreement also lends indirect support to the CSOE above ground respiration estimates.

Figure 4.

(top) Relative variation of optimized Vcmax,25 from CSOE model from 2001 to 2003 and (bottom) the reported relative changes in Vcmax,25 from 1998 to 2000 after Ellsworth [2000].

4.4. Ecosystem Respiration

[39] From the optimized {α·Vcmax,25(t, z)} and equation image described in the previous two sections, we proceed to estimate the ecosystem respiration. Figure 5 shows separately the modeled monthly variation of equation image, aboveground respiration, and RE (in gC m−2 month−1) from 2001 to 2003, along with measured monthly averaged air and soil temperature. On the basis of the CSOE calculations, the contribution of the forest floor efflux is larger than the contribution of the aboveground biomass to total ecosystem respiration. In the winter, modeled equation image can be as much as 85% of modeled RE, while in the summer, it drops to about 70%. This finding is consistent with a recent study at the site based on stable isotope measurements and analysis [Mortazavi et al., 2005].

Figure 5.

(top) CSOE model results for monthly forest floor efflux, aboveground respiration, and ecosystem respiration and (bottom) monthly mean air temperature and soil temperature from 2001 to 2003. The error bars and shaded area in Figure 5 (bottom) represent the standard deviation of air temperature and soil temperature, respectively.

4.5. CSOE Model Testing

[40] To check the performance of the CSOE model, we compare the monthly modeled forest floor carbon efflux with monthly equation image determined from chambers [Palmroth et al., 2005] (Figure 6). When comparing the monthly data, the largest divergence between the chamber estimates and the CSOE model is during the severe drought in 2002. It appears that the CSOE model predictions of equation image are lower than estimates by the chambers suggesting oversensitivity to drought. Furthermore, the CSOE model underpredicts the chamber-based high respiration rate. Despite these differences, there is a good agreement between these two independent estimates on annual timescales (Table 4). These differences result in CSOE modeled efflux that is about 111 gC m−2 year−1 smaller than the chamber-based estimates. The difference might be attributed to several factors that are difficult to deconvolve: (1) The footprints of the chambers and CSOE model are very different, and it is possible that the average of the patches sampled with the chambers consistently respired more than the area sampled by the mean concentration used in the CSOE and (2) the CSOE modeled turbulent diffusivity near the ground (highly sensitive to how equation image and Q decay near the forest floor) may be consistently low (because of both model formulation of the mixing length and plant area distribution near the ground) thereby biasing the CSOE model inversion to lower values. Regardless of the reason, relative to the annual rate of forest floor efflux (>1000 gC m−2), the difference in annual estimates based on these very different approaches is surprisingly small (about 10%), especially considering the large differences (23%) obtained using different approaches [Law et al., 1999a].

Figure 6.

Comparisons between the monthly forest floor effluxes from CSOE and the chamber data generated from the regression equation given by Palmroth et al. [2005] for all 3 years.

Table 4. Annual Carbon Budgets at the Duke Forest Pine Site From 2001 to 2003a
Carbon Budget Components200120022003Notes
Annual ecosystem respiration, RE176716232022RE = F(t, 0) + RAB
Forest floor carbon efflux, F(t, 0)122411271473CSOE model
(1328)(1230)(1599)ACES chamber exp.
(1344)(1180)(1565)ACES model equation (16)
Aboveground respiration, RAB543498549CSOE model = equation image
(Dark respiration)(391)(387)(412)equation image
Total root respiration, RR673620810RR = 0.55·F(t, 0)b
Autotrophic respiration, RA121611181359RA = RR + RAB
Heterotrophic respiration, RH551507663RH = 0.45·F(t, 0)b
Modeled GPP221120332471equation imagec
Modeled NPP9959151112 
Nighttime values    
RE from CSOE837791987 
RE from F(t,h)-PPFD curve9877961033 

[41] Finally, we compared equation image with CSOE modeled equation image for different u* thresholds and for the entire 3-year period (Figure 7). equation image is consistently lower than modeled equation image by almost 30% for small u* and almost 8% for high u*. Note that this comparison is a direct flux comparison between measured and modeled fluxes above the canopy and not a respiration comparison, which is dependent on storage flux estimates. To explore whether high-frequency corrections to equation image (not applied to the EC data here) alone may explain this underestimation, we used the analytical model by Massman [2000]. For the model calculations (also shown as dashed line in Figure 7) we employed the following configuration: the CSAT3 sonic anemometer has collocated vertical and horizontal paths of length 0.15 m; the sampling period is 30 min; the sampling frequency is 10 Hz; the measurement height above the zero plane displacement is 9.95 m; no anti noise band pass filtering or detrending is used; block averaging is conducted every 30 min; planar separation distance between the CSAT3 and the LI7500 gas analyzer is 0.15 m with no vertical separation; and the LI7500 sensor path length is 0.20 m with a time constant determined by assuming line averaging only. The ensemble ratio of corrected to uncorrected fluxes predicted by this analytical model only explains about half of the differences of equation image (i.e., 15%–4% with increasing u*).

Figure 7.

Ratio of modeled CSOE flux above the canopy (equation image) to that from eddy covariance measurements (equation image) in relation to the u* threshold employed for data collected during the 3-year period. The circles are ensemble-averaged equation image, and the vertical lines are one standard deviation around the average. The dot-dashed line is the high-frequency spectral corrections to the equation image predicted by the Massman [2000] model.

4.6. Nighttime Net Ecosystem Respiration Comparison: Effects of Atmospheric Stability

[42] In Figure 8, we compare the CSOE model calculations assuming neutral atmospheric stability conditions with the density-stratified CSOE model results for RE. By setting g = 0 (i.e., the contribution from terms equation image and equation image = 0) and not correcting the upper boundary conditions for atmospheric stability guarantee neutral stratification within the general CSOE model. We found that by ignoring local atmospheric stability, the modeled RE is about 10% lower for the entire study period. For reference, Figure 9 also shows the nighttime ecosystem respiration comparison between the CSOE model (solid lines) and equation image (dot-dashed line) and equation image (dotted line). This comparison demonstrates that resolving the storage flux and correcting for local thermal stratification tends to increase RE over its eddy covariance estimate (without storage). Interestingly, correcting for monthly storage fluxes may be comparable to correcting for the stability effects (∼20 gC m−2 month−1 in summer of 2003).

Figure 8.

Canopy-scale light response curve determined from eddy covariance flux measurements and PPFD measurements at the top of canopy for each year. The open circle and error bar show the statistics (mean and standard deviation) of the flux measurement against different PPFD levels, and the solid line shows the fitted light response curve for each year. The values of Ro are shown for convenience.

Figure 9.

Comparison between nighttime (monthly ensemble average from 2000 to 0500 LT) ecosystem respirations obtained from eddy covariance measurements and CSOE model results with and without the consideration of atmospheric stability.

4.7. Ecosystem Carbon Budget at the Duke Forest Pine Site

[43] From the CSOE model results, we summarize the carbon budget for the site from 2001 to 2003 (Table 4). Summing up the modeled forest floor carbon efflux and modeled aboveground respiration leads to total ecosystem respiration of 1767, 1623, and 2022 gC m−2, respectively, for 2001, 2002, and 2003. These values are consistent with independent estimates made earlier at the site (see Table 4). To further assess whether the modeled RE is also consistent with the expected overall carbon balance at the site, we estimated root respiration from RR = 0.55·equation image [Andrews et al., 1999] using CSOE modeled equation image. The autotrophic respiration RA can be determined from the RR and the CSOE modeled aboveground respiration (RAB). To determine gross primary production (GPP) and net primary production (NPP) from RA, we used the following relationship:

equation image

Lai et al. [2002b] quantified the NPP/GPP ratio using aboveground biomass for a young (6 year old) pine stand. In this study, we used the averaged aboveground biomass of 5128 gC m−2 estimated by Hamilton et al. [2002] to determine the NPP/GPP ratio as about 0.45 for this study site. Using this estimate, the modeled GPP computed from modeled RA varied from 2033 to 2471 gC m−2 for these 3 years. This range is comparable to other estimates [Hamilton et al., 2002; Lai et al., 2002a; Schafer et al., 2003] conducted earlier at the site (2371 to 2486 gC m−2 from 1998 to 2000). As for NPP, the modeled values here ranged from 915 to 1112 gC m−2 during the 3-year study period. This range is higher by about 200 gC m−2 when compared to biometric estimates [Hamilton et al., 2002; Schafer et al., 2003] conducted for an earlier period from 1998 to 2000 (705 to 1060 gC m−2).

[44] Up to this point, we showed how the CSOE model is used to constrain annual nighttime respiration from CO2 concentration data. Here, we compare these CSOE results to other proposed methods that attempt to constrain nighttime respiration. In particular, we used the so-called light response curve method [Lee et al., 1999], which is based on determining the intercept of the equation image and photosynthetically active photon flux density (PPFD) [Clark et al., 1999; Lai et al., 2002a; Law et al., 1999a]. The curve is expressed as:

equation image

where ωp is the mean apparent quantum yield, Fsat is the net CO2 flux at light saturation, and intercept Ro is the mean net CO2 flux when PPFD = 0. The Ro can provide estimates of mean nighttime ecosystem respiration independent of the nocturnal CO2 concentration or equation image data.

[45] Figure 9 shows the light response curves for 2001 to 2003, respectively. The lower daytime fluxes in 2002 are due to the severe drought event. The Ro estimated for each year resulted in nighttime ecosystem respiration of 987, 796, and 1033 gC m−2 yr−1, which are slightly higher than the estimates from the CSOE model (higher by about 0.5% to 16%).

5. Conclusion

[46] We developed a Eulerian version of the constrained source optimization (CSOE) model that considers local atmospheric stability and storage fluxes. The model uses simultaneous mean air temperature and mean CO2 concentration profiles in the inversion for forest floor efflux and above ground source distribution. On the basis of model calculations and measurements at a maturing pine forest in the southeastern United States, we demonstrated the following:

[47] 1. At this study site, the contribution of the storage flux during nighttime conditions is at least 27% of the EC measured flux, even under high friction velocity u* conditions.

[48] 2. Considering local atmospheric stability in the CSOE model increases the modeled ecosystem annual respiration by about 10%, and can be comparable to storage fluxes.

[49] 3. The CSOE model captures well forest floor carbon efflux during both wet and dry years. Also, the variation of the optimized aboveground source parameter is consistent with seasonal variation in Vcmax,25.

[50] 4. The CSOE modeled CO2 flux above the canopy was systematically higher than the eddy covariance measurements by about 30% for low u* and about 10% for high u*. A separate analysis using the Massman [2000] analytical model revealed that high-frequency corrections to the eddy covariance measurements can explain only 50% of this difference.

[51] 5. The CSOE modeled ecosystem respiration, when evaluated within the overall carbon balance at the site, appears consistent with various independent component estimates.

[52] 6. The CSOE modeled ecosystem respiration agreed well with independent respiration estimates derived from the intercept of the annual equation image-PPFD light-response curves. This agreement lends support to a symbiotic use of both methods to further constrain nighttime ecosystem respiration.

[53] The broader implications of this work are twofold. Given the large uncertainties in RE, a logical starting point is to derive multiple estimates of RE, with each estimate sensitive to different assumptions. Chamber estimates provide bottom-up values with limited spatial extent; EC methods provide top-down estimates that can be linked to RE using numerous assumptions and simplifications (but independent from the chamber data). Agreement between these estimates hints at a robust value for RE, while disagreement flags uncertainties. The proposed CSOE provides an additional, independent estimate of RE at the EC spatial scale but has the decisive advantage over EC based estimates because of its ability to separate forest floor effluxes from aboveground fluxes. Therefore the model can serve as a link between EC based measurements and chamber measurements of RE, helping to isolate uncertainties in RE originating from forest floor estimates from those generated by above ground estimates.

[54] The CSOE model can be readily linked to stable isotope measurements. Information from stable isotope measurements can be combined into the CSOE optimization by providing further constraints on the ratio of floor efflux and above ground CO2 production at multiple levels within the canopy. The optimization solutions above ground can also be qualitatively assessed against expected shifts in physiological properties (e.g., Vcmax,25).

[55] Although the CSOE model is a useful step for constraining nighttime RE, certain difficulties remain. For example, the Eulerian formulation provided is one-dimensional and neglects topography-induced drainage flows. The closure formulations are derived assuming fully developed turbulence; an assumption that may frequently be violated at night. Last, the CSOE formulation has several inconsistent “internal” approximations. For example, the assumption of nonsteady state means continuity equation (to account for mean storage fluxes) versus that of steady state flux budget equations (for simplicity).

Appendix A:: Second-Order Closure Approximation

A1. Momentum and Reynolds Stress Budget Equations

[56] Wilson and Shaw [1977] proposed a set of higher-order closure approximations to parameterize each term in the momentum equation image equations, and utilized a method similar to Mellor [1973] for the closure the Reynolds stress equation image equations. For higher-order closure approximation schemes, the gradient diffusion approximation introduced by Mellor [1973] and Donaldson [1973] is employed to close each term in the governing equations of fluxes.

[57] Katul and Albertson [1998] simplified Wilson and Shaw [1977] model by assuming horizontal homogeneity and steady state conditions and obtained the following closure approximation equations to describe the full budget of the longitudinal wind velocity component and corresponding Reynolds stresses.

equation image
equation image
equation image
equation image
equation image

where Q is the characteristic turbulent velocity (square root of the mean turbulent kinetic energy equation image), and Cd is the drag coefficient. Parameters λ1, λ2, and λ3 are the characteristic length scales as discussed in section 2.1. These three length scales are determined from the mixing length, L(z) (λi = ai × L, where i = 1, 2, 3), where ai and Cw are the constants to be determined in Appendix A.2.

[58] In equations (A1a), the closure approximations of all the triple correlation terms are described in Appendix A.3.

A2. Determination of Closure Constants and the Stability Dependency

[59] Applying the linear relationship between σui and u* above the canopy in the neutral surface layer results in:

equation image

where Au, Av, and Aw could be obtained from the eddy covariance measurement [Katul and Albertson, 1998; Shaw, 1977].

[60] To derive the relationship between Aw and atmospheric stability, we used the EC data from 2001 to 2003 and plotted Aw against the stability (equation image, where d is the zero-plane displacement derived from equation image profiles [Katul and Albertson, 1998]), as shown in Figure A1. The fitted curve is

equation image

and is employed to quantify Aw for different atmospheric stability conditions. The number 1.15 in equation (A3a) is the mean value of Aw at neutral stability. The variations of Au and Av for different stability conditions were obtained using the same procedure:

equation image
equation image
Figure A1.

Relationship between Aw and atmospheric stability parameter ζ. All the data points are from EC measurements above the canopy from 2001 to 2003. Neutral atmospheric stability conditions are defined when the absolute value of ζ is less than 0.05 (the dotted vertical line) [Siqueira and Katul, 2002]. The dashed line represents the mean value (1.15) of Aw under neutral atmospheric stability, and the solid line is the fitted curve.

[61] The corresponding value of AQ then can be derived from Au, Av and Aw.

[62] To quantify the effect of canopy structure and atmospheric stability on the momentum component profiles, we present the vertical distributions of modeled σw/u* and Q/u* for two distinct plant area density distributions and for both neutral (∣ζ∣ ≤ 0.05) and stable (ζ = 1.2) stability conditions in Figure A2. From Figure A2, we found that atmospheric stability is much more important than the variations of PAD.

Figure A2.

Sensitivity of plant area density (PAD) and atmospheric stability to closure model predictions. (left) Two end-members of the measured PAD profiles (solid line for day 124 and dashed line for day 329). The bold lines and thin lines represent neutral (∣ζ∣ ≤ 0.05) and stable (ζ = 1.2) atmospheric stability conditions, respectively, for (middle) σw/u* and (right) Q/u*. The solid lines and dashed lines in the middle and right plots correspond to the PAD profiles shown in the left plot.

[63] By using the parameters derived above, Katul and Albertson [1998] summarize the following equations,

equation image

to determine the value of closure constant a2, a3 and Cw. The value of a1 is determined by the equation equation image [Katul and Albertson, 1998; Shaw et al., 1974]

A3. Second-Order Closure Parameterization for Triple Correlations

[64] For the triple velocity correlation components, the general form of the second-order closure model parameterizations are [Katul and Albertson, 1998; Mellor, 1973]:

equation image

[65] The one-dimensional closure approximation for the transport of scalar fluxes and buoyant terms can be express as follows:

equation image


equation image

Appendix B:: Theoretical and Practical Issues for Estimating Storage Flux at a Single Tower

[66] The u* thresholds are often employed for multiple reasons:

[67] 1. The first reason is to eliminate nonturbulent conditions. For example, Cava et al. [2004] showed how canopy waves are generated and how they can transport significant CO2 inside to outside and outside to inside the canopy and over periods that if not properly captured by the averaging interval can lead to negative CO2 fluxes (i.e., photosynthesis like) at night. Employing a u* threshold is primarily to ensure that runs collected under such nonturbulent conditions are removed.

[68] 2. The second reason is to reduce high-frequency losses. Eddy covariance fluxes themselves suffer from high-frequency cospectral losses under moderately stable and very stable atmospheric conditions due to instrument separation distances, limited sampling frequency, and path length averaging by instruments. So, filtering by u* eliminates those runs. There are ways to correct for some of those losses using temperature time series data, but the similarity between CO2 and temperature breaks down even at high frequency for very stable conditions [see Katul and Parlange, 1994].

[69] 3. The third reason is to minimize the effect of storage flux on the estimate of ecosystem respiration from eddy covariance data. The difficulty in estimating the storage flux that is most consistent with the scalar continuity equation can be demonstrated as follows: the depth-integrated 1-D continuity equation:

equation image

and this equation is a spatially averaged equation. The term in parentheses is the desired ecosystem respiration (RE) (spatially averaged). The origin of the term equation image arises because of spatially averaging (after temporally averaging say over half an hour) the point equations. Now, to correctly estimate the spatially averaged RE, one must determine the spatially averaged equation image. Estimating this quantity from a single tower at half hourly time step assumes that we have sampled all the volume within the averaging domain numerous times (to ensure sufficient sample size for the purposes of statistical averaging stability). For low winds inside the canopy, this convergence is problematic. For stronger winds (or higher u* above the canopy), one is likely to sample, at the tower, fluid parcels originating from further distances, thereby ensuring that a bigger volume has been sampled over a 30 min period at a given point. This is primarily the ergodic hypothesis (in space), likely to be more accurate at higher u* than lower u*. Alternatively, one can construct an ensemble of equation image for similar u* and temperature and average those ensemble values assuming that the ensemble average better represents the spatial at the tower when compared to an individual half hour run. In all cases, a major uncertainty remains in the determination of equation image from a single tower, exasperated by low u* conditions. This was the main reason for choosing a 14-day ensemble average: to ensure that the spatial average is estimated from ensemble averages rather than half hour rums. The 14-day period was chosen because the respiring biomass did not drastically change. There is another practical advantage to using ensemble averages vis à vis single runs, which is a reduction in the random error known to contaminate CO2 concentration samples.


[70] This study was supported, in part, by the United States Department of Energy (DOE) through both the Office of Biological and Environmental Research (BER) and the National Institute of Global Environmental Change (NIGEC) Southeastern Regional Center at the University of Alabama (Cooperative Agreement DE-FC02-03ER63613) and by the National Science Foundation (NSF-EAR and NSF-DMS).