Complexity in water and carbon dioxide fluxes following rain pulses in an African savanna
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The idea that many processes in arid and semi-arid ecosystems are dormant until activated by a pulse of rainfall, and then decay from a maximum rate as the soil dries, is widely used as a conceptual and mathematical model, but has rarely been evaluated with data. This paper examines soil water, evapotranspiration (ET), and net ecosystem CO2 exchange measured for 5 years at an eddy covariance tower sited in an Acacia–Combretum savanna near Skukuza in the Kruger National Park, South Africa. The analysis characterizes ecosystem flux responses to discrete rain events and evaluates the skill of increasingly complex “pulse models”. Rainfall pulses exert strong control over ecosystem-scale water and CO2 fluxes at this site, but the simplest pulse models do a poor job of characterizing the dynamics of the response. Successful models need to include the time lag between the wetting event and the process peak, which differ for evaporation, photosynthesis and respiration. Adding further complexity, the time lag depends on the prior duration and degree of water stress. ET response is well characterized by a linear function of potential ET and a logistic function of profile-total soil water content, with remaining seasonal variation correlating with vegetation phenological dynamics (leaf area). A 1- to 3-day lag to maximal ET following wetting is a source of hysteresis in the ET response to soil water. Respiration responds to wetting within days, while photosynthesis takes a week or longer to reach its peak if the rainfall was preceded by a long dry spell. Both processes exhibit nonlinear functional responses that vary seasonally. We conclude that a more mechanistic approach than simple pulse modeling is needed to represent daily ecosystem C processes in semiarid savannas.
KeywordsEvapotranspiration Net ecosystem carbon exchange Pulse response Soil water dynamics Ecosystem modeling
About half of global lands experience rainfall in deficit of evaporative demand at timescales from weeks to years (e.g., Slaymaker and Spencer 1998). In these water-limited systems, pulses of rain excite ecosystem dynamics such as microbial decomposition, germination and growth of plants, predator outbreaks and community succession (Schwinning and Sala 2004), with intervening periods of relative inactivity. Ecosystem water use, productivity, and respiration all respond to the episodic availability of water (Austin et al. 2004; Baldocchi et al. 2006; Huxman et al. 2004; Jenerette et al. 2008; Porporato et al. 2001; Rodriguez-Iturbe et al. 1999; Xu et al. 2004). This has led dryland ecologists to propose various conceptual and mathematical models to describe ecosystem functional responses to pulses of rainfall or soil water (Schwinning et al. 2004). The pulse-reserve paradigm of Noy-Meir (1973), recently modified by Reynolds et al. (2004), and threshold-delay model of Ogle and Reynolds (2004) elaborated by Collins et al. (2008), span the earliest and most recent conceptual models proposed to represent ecological pulse responses. These conceptual descriptions of general system behavior tend not to be tested with data. It therefore remains to be seen if the pulse modeling paradigm is useful in the prediction of ecosystem water and C fluxes.
The pulse paradigm has promoted a focus on the roles of thresholds and lags in ecosystem dynamics (Schwinning et al. 2004). For example, evapotranspiration (ET) and net ecosystem CO2 exchange are both stimulated by rainfall pulses (Gutierrez and Whitford 1978; Huxman et al. 2004; Knapp et al. 2002; Knapp and Smith 2001; Webb et al. 1978; Williams and Albertson 2004), but the degree, duration and timing of stimulation may be highly variable. While rapid responses to even small wetting events can occur (Fravolini et al. 2005; Sala and Lauenroth 1982), in other cases water and C fluxes exhibit a delayed response of several days associated with upregulation (physiological recovery from the preceding dry period), and this may reduce the impact of small pulses or reduce sensitivity during certain times of year (Baldocchi et al. 2006; BassiriRad et al. 1999; Schwinning and Sala 2004; Schwinning et al. 2003). Such nonlinear behavior has typically been evaluated using only a few wetting events and would benefit from more extensive quantitative assessment.
If ecosystem responses to rainfall are strongly dependent on when in the year they occur (seasonality) or antecedent vegetation and soil conditions, the simplest versions of the pulse paradigm may not be adequate for predictive modeling. “Path dependency” (i.e., a sensitivity to history), typical of highly seasonal environments (e.g., Scholes and Walker 1993) or those undergoing functional or structural transition (e.g., Briggs et al. 2005), adds complication to the model. Part of the within-pulse and seasonal pattern of water and CO2 fluxes can be clearly attributed to direct effects of time-variation in environmental conditions (Baldocchi et al. 2004; e.g., Falge et al. 2002; Hui et al. 2003; Reichstein et al. 2002). Additional variability can be introduced by transience in functional responses to environmental conditions (Hui et al. 2003). For example, seasonal variation in temperature dependence of ecosystem respiration is often partially explained by coincident soil water status (Reichstein et al. 2002; Tang and Baldocchi 2005; Tang et al. 2005; Xu and Baldocchi 2004) but even this interaction fails to explain the particularly large CO2 releases sometimes observed after even small wetting events following prolonged dry periods (e.g., Jarvis et al. 2007; Jenerette et al. 2008; Xu and Baldocchi 2004). The work of Ogle and Reynolds (2002) illustrates another time-varying functional response with Larrea, whose success in the face of drought may be enhanced by seasonal acclimation of stomatal response to plant water status with growing season temperature. Such functional adjustments are a challenge to model using simplified pulse-response approaches. This leads us to investigate in this paper the degree of variation of water and CO2 flux responses to soil water pulses in a semi-arid savanna, and how much of that variation is captured by models of increasing complexity.
We address the following hypotheses:
Soil water pulses stimulate ET, productivity, and respiration processes, with a general decline in activity during the ensuing dry-down (hypothesis 1).
The time lag between the increase in soil water and the peak of activity varies between these three (i.e., ET, productivity, and respiration) processes (hypothesis 2).
The magnitude of the peak, and lag time to peak, both increase with the intensity of ecosystem dryness prior to re-wetting (hypothesis 3). This stems from a general expectation of the need for upregulation of ecosystem physiological activity following dryness-induced dormancy, akin to the “Birch effect” discussed below.
As a consequence of the above factors, a time-invariant pulse-response model with a fixed characteristic shape misses modulation by seasonal phenology and antecedent wetness, and thus represents daily fluxes poorly (hypothesis 4).
To address these hypotheses we analyze time series of soil water, ET, and net ecosystem CO2 exchange (partitioned into production and respiration) measured over 5 years in an Acacia–Combretum savanna in South Africa. Our analysis assesses the degree to which coincident weather and surface states control ecosystem flux responses to discrete rainfall events. It begins by documenting the average response of C and water fluxes to soil water pulses. Then, using a 5-month case study which contains three main pulse events, we show such responses to be time-variant and nonlinear. We quantify lags in the response of water and CO2 fluxes to wetting, and then test the ability of alternate “pulse models” of increasing complexity to capture lagged ecosystem responses. The discussion explores minimum parameterization of ecosystem flux responses to soil water pulses, comments on the implications of our findings for the pulse paradigm, and outlines specific challenges that remain in representing complex, transient ecosystem responses to pulses.
Materials and methods
The shallow coarse sand to sandy loam soils (about 65% sand, 30% clay, and 5% silt) overlay granite, which outcrops in places (Biggs et al. 2003; Sankaran et al. 2004). The landscape is gently undulating, with broad-leaved savanna on the crests dominated by the small tree Combretum apiculatum, and fine-leaved savanna in the valleys dominated by Acacia nigrescens (Scholes et al. 1999). Within the 300 × 300-m square centered on the tower, tree cover is about 28% and evenly mixed between Combretum and Acacia. Woody plant basal area is 7 m2 ha−1 (Scholes et al. 1999). Tree canopies average 5–8 m high with occasional trees (mostly Sclerocarya birrea) reaching 10 m. The grassy and herbaceous understory contains Panicum maximum, Digitaria eriantha, Eragrostis rigidor, and Pogonarthria squarrosa among others. The tree and herbaceous vegetation both root throughout the ~0.6-m-deep soils.
Leaf area index (LAI; m2 leaf m−2 ground) from 2000 to 2004 was obtained from MODIS LAI cutouts centered over the Skukuza site, made available through the Oak Ridge National Laboratory Distributed Active Archive Center and recalibrated (LAIfield = 0.5 LAIMODIS) against in situ observations made on multiple occasions with a Decagon Accupar ceptometer (Pullman, Wash.) at 49 systematically sampled locations within 150 m of the tower. Growing season LAI averaged about 1.1 (Fig. 1) during the period of flux tower observations reported here, but has reached as high as 2.0 for the more extended MODIS LAI record.
Measurements and data processing
A 22-m tower was instrumented at 16 m with a sonic anemometer (Gill Instruments Solent R3, Hampshire, England) measuring three-dimensional, orthogonal components of velocity (u, v, w; m s−1) as well as the “sonic” air temperature (T a; °C), and a closed-path infrared gas analyzer (IRGA; LiCor 6262; LiCor, Lincoln, Neb.) measuring concentrations of water vapor (q; mmol H2O mol−1 moist air), and CO2 (μmol CO2 mol−1 air). Manual IRGA calibrations were performed approximately monthly. Calibrations were done in the field using Level-5 (99.999% pure) N for CO2 and H2O zero (this was also the reference gas running continuously through the reference cell in the 6262). Spans were carried out using National Oceanographic and Atmospheric Administration (NOAA) secondary standard calibration gases calibrated to <0.2 p.p.m. for CO2 and a LiCor dew point generator for H2O. Analytical flux footprint estimation (Gash 1986; Hsieh et al. 2000) indicates that a source area within 130 m upwind of the tower typically contributes 90% of the measured flux and a source area within 1.3 km contributes 99% of the flux.
Post-processing of the raw high frequency (10 Hz) data for calculation of above-canopy turbulent fluxes of sensible heat (H; W m−2), water vapor (LE; W m−2), and CO2 (F c; g CO2 m−2 time−1) calculated for half-hour periods is consistent with methods advised in Lee et al. (2004), and involved standard spike filtering, planar rotation of velocities applied to monthly data for 60° sectors (Wilczak et al. 2001), as well as lag correction to CO2 and q from analysis of monthly peaks in lagged cross-correlations of vertical velocity and scalars. Heat and mass fluxes were calculated with conventional equations (see e.g., Aubinet et al. 2000; Moncrieff et al. 1997). All fluxes are reported as positive upward from the land to the atmosphere. Frequency response correction of some of the energy lost due to instrument separation, tube attenuation, and gas analyzer response for LE and F c was performed with empirical cospectral adjustment to match the H cospectrum (Eugster and Senn 1995; Su et al. 2004).
Canopy storage flux was estimated from the half-hourly time derivative of a 16-m-column integral based on CO2 concentrations measured with PP Systems (Amesbury, Mass.) CIRAS-SC infrared gas analyzers in two separate profiles 40 m on either side of the main tower, with air intake at heights of 0.75, 2.0, 3.5, and 5.25 m, plus the 16-m concentration on the main tower, and added to the above-canopy turbulent flux for data analysis. Average half-hourly incoming and outgoing longwave and shortwave radiation was measured with a Kipp and Zonen (Delft, The Netherlands) CNR1 radiometer mounted at 22 m, the balance of which provides net radiation (R n ; W m−2).
Average half-hourly volumetric soil water content (θ; m3 H2O m−3 soil) was estimated with 15-cm-long Campbell Scientific (Logan, Utah) frequency domain reflectometry probes (CS615) installed horizontally in four separate profiles, two in the Acacia-dominated [probes at soil depths (z, cm) of 3, 7, 16, 30, and 50 cm] and two in a Combretum-dominated area (probes at 5, 13, 29, and 61 cm). The probes have not been locally calibrated, and while manufacturer notes suggest absolute accuracy to within 2%, our estimates of soil water content are approximate, still providing a robust measure of the relative moisture dynamics. Half-hourly averaged soil heat flux (G, W m−2) was obtained with HFT3 plates (Campbell Scientific) installed 5 cm below the ground both under and between tree canopies. Rainfall per half-hour was measured with a tipping bucket rain gage (TE525-L; Campbell Scientific) located on the tower top.
C fluxes are also represented as temporal sums over daily (24 h), daytime (12 h from 0800 to 2000 hours), and nighttime (8 h from 2200 to 0600 hours) of the day. Data were only included in the summation if friction velocity exceeded 0.1 m s−1, a widely used threshold for valid eddy covariance measurements (Baldocchi 2003). In addition, a 15-day mean diurnal replacement as in Falge et al. (2001) was applied to fill missing values for daytime, nighttime and daily summations. Summations of gap-filled data were included in analyses only if fewer than 10% of the data had been filled by this procedure.
The average temporal dynamics of fluxes responding to significant soil wetting events (≥ + 5 mm soil water) are calculated for dry and wet season populations during the period April 2000 to July 2004. Here, observations were only included for a lag of one up to n days if a second soil wetting (S ≥ 5 mm) did not occur during the period. Neither the mean depth of a rainfall event nor the post-pulse accumulated rainfall differs between the dry and wet populations; the key difference is the gap between rainfall events. Functional control of fluxes by root-zone soil water was also examined with averages for data during the growing season. In both the temporal and functional analyses we investigate the presence of lags to peak response.
To more directly assess dynamics of the main C exchange processes we apply a simple approach to separating the net flux (F c) into respiration and production components. Respiration is estimated by scaling mean nighttime CO2 flux to a 24-h total, then subtracting this from total daily CO2 flux to obtain an estimate of daily photosynthetic productivity. Temperature variations during the summer growing seasons at this site do not include extreme low or high temperatures that severely limit plant or microbial physiology. Lacking clear relationships between nighttime CO2 flux and air or soil temperature, it was not appropriate to include temperature as an independent variable in the estimation of daily respiration based on nighttime CO2 flux despite its conventional appeal.
The final analyses involve a “model intercomparison”, evaluating data support from the full measurement record for empirical models of daily water and C processes that vary in complexity. We used the simplest possible representation of a pulse response—a “square wave” in which ET/PET, respiration, and productivity are assigned a constant high value where S is greater than 5 cm, and a low value if not—as a reference point, and then added progressive refinements. Rather than imposing a particular functional form to these more complicated models a priori, base models were first selected from regression fits of a wide range of linear and nonlinear models using the CurveFinder function of the program CurveExpert 1.3. From this analysis we found that linear and logistic models tended to provide the best functional descriptions without an excessive number of parameters. We then explored linear and logistic dependence of ET/PET, respiration, and productivity each on: (1) day of year as a purely seasonal model, (2) 0- to 30-day average rainfall, (3) 0- to 20-day average S, and (4) 0- to 100-day WDD histories (Eq. 3). As a third step in model testing, we performed a linear regression of the residuals on LAI (linearly interpolated to a daily series). Full details of the model exercise can be found in the Electronic Supplementary Material.
The rapid response of respiration to soil wetting dominates the daily total net C exchange (F c) during the first 3 days after rainfall, suggesting that upregulation of productivity in response to the moisture pulse is delayed. This interpretation is supported by the continued net release of CO2 for the dry season population, in contrast to the switch to net ecosystem C uptake (negative F c) in the wet season, peaking on average about 9 days after wetting. A wide spread about the average for each functional response indicates that a single characteristic response irrespective of the degree of wetting, prevailing environment, or antecedent conditions provides a poor system description even when seasonally stratified. The fit is poorest for daily F c.
In contrast, responses of both daytime (12 h) and daily (24 h) net ecosystem exchange are more complex, since they include the longer lag inferred for photosynthetic upregulation. CO2 exchange during daylight hours shows an apparent minimum (i.e., maximum uptake given that downward fluxes are negative) at intermediate rather than high soil water content, because the rapid decline in soil moisture in the days after wetting, coupled with a photosynthetic response lagged by up to a week, tends to obscure the expected monotonic relation between soil moisture and net ecosystem exchange. Similarly, net 24-h CO2 flux is apparently near zero at high soil water content, reaches a minimum at intermediate soil water content, and is near zero again when the soil is dry.
Comparing the second and third major wetting events (DOY 63 and 117), peak total root zone soil water is nearly equal (Fig. 4). However, soil water declines more rapidly in the second compared to the third wetting event, corresponding with a peak ET that is 1.7 times higher in the former. This is largely explained by the relatively low PET at the time of the third compared to second wetting event: note that the magnitude of ET/PET is similar in both cases (Fig. 4). However, this cannot explain the large difference in net C flux between the second and third events. Net C flux for each day after wetting is about 10 g CO2 m−2 day−1 higher (more positive) for the third than second event. The difference is likely due to the respiration term remaining high but photosynthesis being reduced due to leaf senescence and abscission. Die-off of the herbaceous vegetation typically starts at the end of April in this region, while tree leaf fall begins during May if the soil dries out.
For ET/PET, predictive skill is greatly improved from the square wave to a model that includes a linear dependence on soil water content (S), and was further modestly improved by a logistic dependence on S. AIC only weakly supported LAI as an additional predictor, accounting for longer term (8 days to several months) variations, explaining only an additional 2% of the variability in ET/PET. Remaining variation is partly due to the hysteresis resulting from the lag to maximal ET following wetting. However, a logistic function of soil water history (S x ) decreases the fit relative to simply using the soil water content on the day. Taken together, an equation of the form ET = PET × [logistic f (soil water) + linear f (leaf area)] explains 72% of the variation in daily ET at the savanna site studied in this work.
Respiration is also better described by a logistic function of soil water than a square wave or linear model, but only 36% of the variance is explained and LAI adds little to no explanatory power. Models of productivity benefited most from added complexity. Including the effect of water stress history (the 15-day average soil moisture, or accumulated water stress days) generally improved the fit to productivity by capturing a short-term “memory” of ecosystem water status and the associated lag in productivity response. Residual dependence of productivity on LAI was poorly supported by information criteria and provides only a marginal increase to the coefficient of determination or reduction to SE of regression.
Results confirm that rainfall events stimulate pulse responses of ET, respiration and productivity at the daily time scale in the savanna under study (hypothesis 1). Only a subtle lag to peak stimulation for ET and respiration, is contrasted by a significant lag to peak for photosynthesis, taking a week or longer to reach its peak and supporting hypothesis 2. Correspondingly, prior dryness was not a predictor of lags in ET or respiration responses; however, intensity and duration of pre-pulse dryness is a valuable predictor of lags in productivity, supporting hypothesis 3. It is still worth mentioning that a short, 1–3 day, lag to maximal ET following wetting is a mild source of hysteresis in the ET response to soil water even after controlling for PET variation.
The temporal dynamics of NEE reported here, with faster upregulation and ensuing decay for respiration compared to productivity, are consistent with those documented for irrigated grass communities of Arizona (Huxman et al. 2004), as well as a number of conceptual models (e.g., Schwinning et al. 2004). Rapid, punctuated increase in respiration following wetting as reported here is common in water-limited systems (Jenerette et al. 2008; Xu and Baldocchi 2004). However, soil wetness accounts for only 36% of the total variation in respiration observed at the Kruger savanna. Again, pre-pulse dryness is not a significant predictor of respiration dynamics. Temperature dependence of respiration is unclear at the site though results from Kutsch et al. (2008) suggest that it may still account for some of the remaining variation. Perhaps more importantly, some measure of the availability of C substrate is a sensible candidate, possibly capturing a “Birch effect” (see Jarvis et al. 2007), meaning a burst of microbial decomposition and N mineralization excited by the rewetting of soils that have been dry for a long period. However, it is not possible to separate autotrophic and heterotrophic respiration from the ecosystem total fluxes alone and thus attribution of pulse respiration to a Birch effect cannot be evaluated here.
Results suggest that an integrated measure of plant water status such as stress history would improve model predictions of productivity. We note, however, that the peak magnitude does not depend on pre-pulse dryness. Thinking about mechanisms, it is not surprising that ecosystem physiological responses to wetting are delayed by prolonged dry spells, particularly in a water-limited system in which vegetation dehydrates and becomes dormant at monthly to seasonal time scales. One might then expect part of the variation in lag to peak productivity to be captured by LAI, so the general lack of residual dependence on LAI may seem surprising. However, this can be explained by the temporal correlation of LAI with S x and WDD x (the correlations were 0.79 and −0.79 for x equal to 20 days). Thus most of the effect of LAI is already expressed in the water stress index.
Implications for the pulse-reserve paradigm
The pulse-reserve paradigm, introduced by Noy-Meir (1973), modified by Reynolds et al. (2004), and reconceived as the threshold-delay model by Ogle and Reynolds (2004), purports to describe ecological dynamics based on a very simple principle: there is a characteristic pulse of activity following a wetting event. It is attractive because it reduces the system to a few simple inputs, such as initial resource condition and the time since a pulse. But this study demonstrates that in order to realistically capture the observed behavior of a pulsed system, pulses need detailed parameterization to adjust the generalized unit response function for weather, vegetation, and the recent history of the system (hypothesis 4), and likely for soils and climate. Such parameterizations may be site specific, and thus not suitable for extrapolation in general predictive models.
While the pulse-response approach lends itself to synthetic parameterization of system lags and thresholds such as those documented here for the Skukuza site, our observations suggest that a reduced-form pulse-response model is probably not an appropriate choice for capturing the daily dynamics of net ecosystem CO2 exchange (hypothesis 4). Models that represent the differentially lagged temporal responses of ET, respiration and productivity appear necessary. Thus, while the pulse-response paradigm has value for conceptualizing time-domain variation of ecosystem processes, the approach has few advantages for predicting how water and C dynamics relate to a changing environment.
Nonetheless, suggestions for minimum parameterizations emerge. For example, findings from the Skukuza tower site lend confidence to widely applied space-time models describing the co-evolution of ET and soil water dynamics (e.g., Rodriguez-Iturbe et al. 1999; Williams and Albertson 2005). In such minimum parameterizations, the core system behavior that needs to be carefully prescribed is the daily rate of ET as a function of soil water, with additional adjustment by vegetation fractional cover or leaf area. The daily canopy-scale ET of the heterogeneous savanna studied here is well-characterized by PET, total root zone soil water, and seasonal vegetation dynamics, consistent with Williams and Albertson (2005). While a sub-daily resolution may be important for some applications (i.e., land surface coupling to the atmospheric boundary layer) we find that 72% of the interstorm to seasonal dynamics are captured at a daily resolution.
Our findings also provide general support for the use of simple (“bucket”) soil water models, indicating that the added complexity of multi-layer models that parameterize wetting fronts, root distributions and soil water potential may not always be needed for describing daily to monthly time-scale behavior of the coupled ET and soil water dynamics. However, we note that sites with deeper soils might still require a soil water model and root distribution model with vertical detail, especially where there exists a vertical root niche separation between trees and grasses influencing their relative activity.
Previous work has explored the use of a constant canopy-scale water use efficiency to characterize daily net CO2 flux as a function of ET during the growing season (Scanlon and Albertson 2004; Verhoef et al. 1996; Williams et al. 2004) motivated by functional coupling of leaf-scale water and CO2 exchanges combined with similar sensitivity to soil water for productivity and respiration. In this study, the different time scales of the respiration and productivity responses to wetting call that idea into question. We note that the ratio of average monthly F c to ET is surprisingly conservative during the growing season but different in the dry season (not shown but can be loosely inferred from Fig. 1). While the application of seasonally varying parameters could improve fit, a more mechanistic approach would be preferred, possibly by representing temporal dynamics of likely drivers of ecosystem process peak rates, such as labile soil C, or lags, such as vegetation water stress history.
Space-time variation in the critical parameters that control how soil water status influences bare soil evaporation and transpiration is still limited by scarcity of detailed observational data. The few datasets that do exist tend to describe canopy-scale aggregate ET, field-scale ET from lysimetry, or leaf-scale transpiration on individual plants (Baldocchi et al. 2004; Black 1979; Dunin and Greenwood 1986; Federer 1979; Kurc and Small 2004; Mahfouf et al. 1996; Teuling et al. 2006; Williams and Albertson 2004). Aggregating or disaggregating these measures to the scale of functional units such as whole plants, plant functional types, or landscape patches continues to be a challenge, though stable isotopes (e.g., Ferretti et al. 2003; Williams et al. 2004; Yepez et al. 2003) could be very helpful.
The biological mechanisms that cause lagged responses of respiration and productivity are still speculative. While recovery from a desiccated, water-stressed state is qualitatively understood and may even be parameterized experimentally, biochemical and biophysical models have not advanced to the level of refinement needed to characterize such up- or downregulation in a mechanistic way. Similarly, models that contain the detailed soil and plant biogeochemistry needed for characterizing C pool dynamics (e.g., Century, CASA, or Biome-BGC) are only beginning to be implemented at the daily time scale, and likely still miss the pulse-like responses that characterize water and C fluxes in intermittently wetted, semi-arid biomes.
The authors would like to thank Ian McHugh and Walter Kubheka of Kruger National Park, South Africa for field technical support. The fieldwork supporting these findings has been conducted in full compliance with current laws in South Africa. The Skukuza flux tower was supported by grants from the National Aeronautic and Space Administration, the United States National Science Foundation and NOAA, as well as funds from the South African National Research Foundation and Council for Scientific and Industrial Research and the Deutsche Forschungsgemeinschaft (grant Ku 1099/2-1).
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