Revisiting the parameterization of potential evaporation as a driver of long-term water balance trends



[1] We examine the effects of two different parameterizations of potential evaporation on long-term trends in soil moisture, evaporative flux and runoff simulated by the water balance model underlying the Palmer Drought Severity Index. The first, traditional parameterization is based on air temperature alone. The second parameterization is derived from observations of evaporation from class-A pans. Trends in potential evaporation from the two parameterizations are opposite in sign (±) at almost half the stations tested over Australia and New Zealand. The sign of trends in the modelled soil moisture, evaporative flux and runoff depends on the parameterization used and on the prevailing climatic regime: trends in water-limited regions are driven by precipitation trends, but the choice of parameterization for potential evaporation is shown to be critical in energy-limited regions.

1. Introduction

1.1. Background

[2] Simple water balance models are often used to assess land surface moisture conditions. A typical example is the water balance model underlying calculations of the Palmer Drought Severity Index (PDSI) [Palmer, 1965]. Routinely used in the US to assess developing drought conditions, this model was recently extended on a worldwide, long-term basis leading to the conclusion that many regions have become drier and more drought-affected during recent decades [Dai et al., 2004]. In the PDSI, the water-balance is calculated using a bucket model, yielding a moisture anomaly from which the non-dimensional, monthly drought index is derived. In the bucket model, soil fills with precipitation (Prcp) and empties through actual evapotranspiration (ETa) and Runoff. At each model time step, the maximum possible ETa is the minimum of the evaporative demand, herein denoted potential evaporation (Ep), and the available water.

Table 1. Magnitude, Range, Direction, and Significance of Temporal Trends for the Inputs and Both Parameterizations' Outputs Across the Two Sets of Stationsa
ParameterizationVariableTrend StatisticsTrend Direction (# p ≤ 0.05)
Mean ± StdevMinMax+ve−ve
  • a

    Units are mm/year2 for trends in Prcp, Epan, Ep, ETa, and Runoff; °C/year for trends in Tair; and mm/year for trends in SM. Means and standard deviations are across the 27 stations in Australia and eight stations in New Zealand. Numbers in parentheses count trends significant at 95%.

Australia, 27 stations, 19752004
ObservedPrcp0.10 ± 3.92−10.316.8414 (1)13 (1)
 Tair0.01 ± 0.01−0.010.0322 (6)5 (0)
 Epan−2.67 ± 7.62−17.4810.9510 (3)17 (7)
EpPDSIEp1.14 ± 1.25−0.593.8320 (6)7 (0)
 ETa0.45 ± 3.02−9.376.7116 (2)11 (1)
 SM−0.27 ± 0.99−3.750.9913 (0)14 (2)
 Runoff−0.50 ± 1.72−7.521.2517 (0)10 (0)
EpPanEp−2.00 ± 5.72−13.118.2210 (3)17 (7)
 ETa0.15 ± 3.20−9.666.7614 (1)13 (1)
 SM−0.05 ± 0.94−3.941.1714 (0)13 (2)
 Runoff−0.14 ± 2.02−6.686.2418 (0)9 (1)
New Zealand, 8 stations, 19742003
ObservedPrcp−3.24 ± 6.34−12.164.883 (0)5 (4)
 Tair0.02 ± 0.01−0.010.047 (3)1 (0)
 Epan−1.77 ± 2.21−4.740.893 (0)5 (2)
EpPDSIEp1.03 ± 0.98−0.492.547 (3)1 (0)
 ETa0.51 ± 1.55−2.142.516 (2)2 (0)
 SM0.19 ± 0.64−1.550.395 (1)3 (1)
 Runoff−3.58 ± 5.80−12.904.753 (0)5 (4)
EpPanEp−1.24 ± 1.55−3.320.623 (0)5 (2)
 ETa−0.89 ± 2.77−5.582.364 (1)4 (2)
 SM−0.05 ± 0.61−0.940.885 (1)3 (2)
 Runoff−2.21 ± 4.83−8.834.833 (0)5 (3)

1.2. Water- and Energy-Limited Environments

[3] Ep and ETa can be considered in the classical framework of the limitations on ETa. In environments with a limited supply of soil-water to evaporate, ETa is less than Ep. In these “water-limited” environments, it is changes in the availability of water (through Prcp), and not of energy, that dominate changes in ETa. As water availability increases, ETa converges towards Ep until they are equal in wet environments. Here only the availability of energy limits ETa, and in such “energy-limited” environments, increasing Prcp will not change ETa but will increase soil moisture and/or Runoff. Increasing the availability of energy alone at energy-limited sites will raise ETa and consequently depress soil moisture and/or Runoff. Most regions in the world lie in the continuum between the water and energy limits, reaching either only seasonally. The method used to estimate Ep (and hence constrain ETa) influences the water balance calculation and the resulting conclusions on long-term trends, hence it is the effects of different Ep formulations on long-term water balance variables that motivate this paper.

1.3. Ep Trends

[4] In the traditional formulation of the PDSI bucket model, evaporative demand is derived from an approach that can be traced back to Thornthwaite [1948] and is solely a function of air temperature (Tair). Consequently, as Tair steadily increases with global warming, the calculated value of Ep in the model also steadily increases. In contrast, the measurements of pan evaporation (Epan) that are better physical representations of Ep show widespread declines over the last 30-50 years [Peterson et al., 1995; Chattopadhyay and Hulme, 1997; Golubev et al., 2001; Hobbins et al., 2004; Liu et al., 2004; Roderick and Farquhar, 2004, 2005; Tebakari et al., 2005; Wu et al., 2006]. Similarly, calculations of reference ET [Allen et al., 1998] (also see auxiliary material) using observations of solar radiation, Tair, humidity, and wind speed also show declines [Thomas, 2000; Chen et al., 2005; Shenbin et al., 2006], in general agreement with the Epan record in China. The poor performance of Tair-based Ep in predicting observed evaporative demand was further highlighted in China, where rising Tair but declining Epan were found across eight of the 10 river basins [Chen et al., 2005].

[5] The key point here is that Ep is less affected by changes in Tair than by changes in surface radiation, wind speed, and humidity deficit [Roderick et al., 2007]. Indeed, when formulating the PDSI, Palmer [1965] recognized the importance of all of these dynamics driving the more physically complete Penman [1948] Ep-formulation, but justified his choice of an equation for Ep based solely on Tair by an appeal to expediency, having previously noted, “…Thornthwaite's empirical formula can be used for any location at which daily maximum and minimum temperatures are recorded. It is this simple universal applicability rather than any claim to outstanding accuracy which has led to the widespread use of this method” [Palmer and Havens, 1958]. In fact, Thornthwaite [1948], too, expected his Tair-based approach to be replaced by a more physically based method as the necessary theory was developed and supporting data became more widely available. As evaporation from a water surface inside a pan integrates all these factors [Rotstayn et al., 2006], its measurements have long been widely used as a physical measure of Ep in agricultural and engineering applications. While other workers have examined the sensitivity of long-term PDSI to various model features [e.g., Karl, 1986], no-one has yet examined the effects of using such a physically based Ep.

[6] Clearly, the declining trend in physically based measures of Ep compared to the increasing trend implied by calculations based solely on Tair might lead to different conclusions about trends in drying [Moonen et al., 2002]. In this paper, we examine that proposition by running the PDSI water balance model at 35 sites across Australia and New Zealand. At each site, the model is run twice. In the first run, we calculate Ep per the standard PDSI implementation using Tair measurements. In the second run, Ep is based on pan evaporation observations. No other forcings (e.g., Prcp) and model parameters are changed, thereby isolating the effect of different formulations of evaporative demand on the outputs of the water balance model (ETa, soil moisture, and Runoff) and hence on the modelling of drought.

2. Methods and Materials

2.1. Description of the Water Balance Model

[7] The PDSI bucket model uses a two-layer soil column, with the water content at field capacity set as 25.4 mm (1 inch) for the upper layer, where it is assumed freely available for evapotranspiration, and that for the lower layer (or root zone) prescribed by the analyst. Evaporative demand in the PDSI is traditionally set as a time-series of Tair-based Ep. Prcp is partitioned into recharge, filling first the upper soil layer and then the lower layer, until the soil column fills; Runoff occurs when whole-column soil moisture (SM) is at field capacity and evaporative demand (i.e., Ep) from both layers has been met. When Prcp and upper-layer soil moisture cannot meet Ep, the soil dries in two stages: moisture in the upper soil-layer completely evaporates at the potential rate (i.e., Ep), then that in the root-zone declines at a rate depending on root-zone soil moisture and unmet Ep. Note that the physiologic response of vegetation to water limitation is modelled in this two-layer soil column, not in the parameterization of evaporative demand. The water-equivalent depth of the bucket (AWC) was set using the same 2.5° × 2.5° global grid used by Dai et al. [2004] (see Figure S1a).

2.2. Data and Study Area

[8] We chose Australia and New Zealand as our study area. This represents a wide climatologic range because Australia is primarily water-limited and New Zealand energy-limited (see section SI.3 and Figure S3). Australian stations were restricted to the 27 sites with monthly Tair, Prcp, and Epan data used by Roderick et al. [2007] that were also part of the High Quality Annual Tair data network [Della-Marta et al., 2004]. The period of analysis for Australian data was 1975-2004. We selected the eight New Zealand sites for which long-term monthly Prcp, Tair, and Epan records were available from the National Institute of Water and Atmospheric Research (NIWA). Analysis periods varied across the New Zealand sites but were constrained to within 1974–2003 (see Table S2).

[9] As the soil moisture-accounting in the PDSI bucket model requires continuous monthly input time-series, missing data were filled as follows: (i) when only one or two months' data were missing from a year, these monthly data were scaled from their climatological monthly means according to the proportion in the rest of the year; or (ii) when over two months' data were missing from a given year, all months of that year were replaced by their climatological means. In the worst case (Epan for New Zealand), less than 3% of data were infilled. For further details on data sources, see section SI.2.

2.3. Two Estimates of Ep

[10] Our first estimate of Ep is that used operationally in the PDSI by NOAA [Dai et al., 2004], and here denoted EpPDSI (see equations (S1)-(S3)). The constant heat factors (B and H, see equation (S2)) required for EpPDSI were prescribed using the 2.5° × 2.5° global grids from Dai et al. [2004] (see Figures S1b and S1c). Our second Ep estimate uses observed monthly Epan multiplied by the traditional pan coefficient k, i.e., EpPan = k.Epan. While k has been shown to vary seasonally [Allen et al., 1998], the widely adopted value of 0.7 [Stanhill, 1976] was used for the unscreened New Zealand pans. For the Australian pans, k was set to 0.75 to account for the 7% reduction in Epan due to the bird-guards [van Dijk, 1985].

2.4. Model Calculations

[11] In running the PDSI bucket model, initial SM was specified equal to AWC, for the following reasons: first, it is standard PDSI procedure to initiate the model run with maximum moisture in both soil layers; second, a sensitivity analysis (not shown) indicated that the effects of initial conditions disappeared within two years. Nonetheless, to eliminate bias and minimize the effects of high initial SM conditions, the models were run for two years prior to the analysis period. Trends were defined as the slope of an Ordinary Least Squares regression through annual time-series, with the significance of annual trends determined by t-tests. As the model conserves mass, the trends in the component fluxes (Prcp, ETa, and Runoff) sum to the second time-differential of SM. We report the first time-differential of SM as it is of greatest climatological interest.

3. Results

[12] The observed trends in the input time-series—Tair (for the EpPDSI run), Prcp (for both runs), and k.Epan (for the EpPan run)—are shown in Figures 1a, 1b, and 1d, respectively, and closely match previous results for Australia and New Zealand [Roderick and Farquhar, 2004, 2005]. The difference between the two Ep measures is apparent in Figures 1c and 1d. Note that of the 35 sites, most (27) show increases in EpPDSI, in line with the general increase in Tair shown in Figure 1a. In contrast, only 13 sites showed increases in EpPan.

Figure 1.

Annual trends at 27 stations in Australia and eight in New Zealand over the periods 1975–2004 (Australia) and 1974–2002 (New Zealand). Maps show direction and scale of trends for (a) Tair, (b) Prcp, (c) EpPDSI, (d) EpPan, (e) ETaPDSI, (f) ETaPan, (g) SMPDSI, and (h) SMPan. Grey circles indicate positive trends, black circles negative, with areas proportional to trend magnitudes. In (i) the station-trends in SMPDSI (Figure 1g) are plotted against those in SMPan (Figure 1h).

[13] ETa-trends in Australia (Figures 1e and 1f) generally follow the Prcp-trends (Figure 1b). This result is expected because Australia is largely water-limited (section SI.3 and Figure S3), so trends in moisture supply, not in evaporative demand, largely determine the ETa-trend. However, in New Zealand, whether ETa-trends increased or decreased depended on the Ep parameterization. Again, this result is expected because at energy-limited sites, ETa is sensitive to changes in Ep. Clearly, ETaPan in New Zealand responds to dynamics in evaporative demand missing from the EpPDSI parameterization. While differences exist between the scale and direction of SM-trends, both parameterizations generate generally declining SM in eastern Australia and central New Zealand, with increases in western Australia and northern and southern New Zealand (Figures 1g and 1h). Again, in Australia, both models generate spatial patterns of SM-trends more closely resembling trends in Prcp than in Ep. Figure 1i shows the relations between SM-trends between the two parameterizations at all stations mapped in Figures 1g and 1h. Clearly, there is little relationship between the SM-trends predicted using EpPDSI and EpPan: at seven of the 35 stations, they are of different signs, and five of these stations are located in important agricultural and/or populated regions of southwestern Western Australia, New South Wales, and Victoria.

[14] The potential for mismatch between the hydrologic trends estimated by the two parameterizations is further demonstrated at Darwin (the northernmost station in the Australian set), which is energy-limited on an annual basis. Here moisture supply is increasing (dPrcp/dt = +4.4 mm/year2), but the Ep-parameterizations' respective drivers are in conflict: warming leads to increasing EpPDSI (dTair/dt = +0.02°C/year, dEpPDSI/dt = +2.8 mm/year2), but Epan is decreasing (dEpPan/dt = -10.4 mm/year2) due to declining wind speeds and solar radiation [Rayner, 2007; Roderick et al., 2007]. The decline in the energy available for evaporation is captured by the EpPan observations, but not by the EpPDSI calculations because the latter do not respond to solar radiation or wind speed. Thus, the two parameterizations drive the PDSI bucket model to indicate opposite trends in ETa (dETaPDSI/dt = +1.1 mm/year2, dETaPan/dt = -4.2 mm/year2), which leads to the soil drying in the case of EpPDSI (dSMPDSI/dt = -1.1 mm/year) and wetting in the case of EpPan (dSMPan/dt = +1.2 mm/year).

4. Discussion

[15] In assessing the likely ecohydrologic impacts of climate change, projections of continental drying must be reconciled with observations of declining evaporative demand. The PDSI, upon which much support for many of these conflicting projections is based, is an index based on a hydrologic model that predicts evaporative demand from the direct effect of surface warming only. However, the effects of other accompanying changes, such as those in surface radiation, wind speed and humidity, likely play an important role over many land areas [Roderick et al., 2007].

[16] Here we have compared long-term trends in ETa and SM from the PDSI bucket model forced by two parameterizations of Ep: one based on Tair alone; the other derived from observations of a physical metric of evaporative demand (i.e., Epan) that responds to the appropriate dynamics—Tair, humidity, wind speed, and net available energy [Rotstayn et al., 2006]. The primary attraction of the Tair-based EpPDSI is simplicity: only Tair data are required, and such data are widely available in time and space, including historical global grids at many spatio-temporal resolutions. That historical Tair-trends are more certain than trends in radiation, humidity, and wind speed, demonstrates the value of the Epan record: these variables are physically integrated into Epan measurements. Here we have shown that Tair-based Ep cannot predict the directions of observed trends in evaporative demand: 46% of the long-term trend directions predicted by EpPDSI are opposite to those observed in Epan across Australia and New Zealand.

[17] It is important to note that using Tair-based parameterizations in climate change analyses—particularly in deriving trends in drought dynamics—invokes retrograde assumptions about the relationships of Tair to all other evaporative drivers. In fact, using Tair-based Ep produces an apparent paradox with declining Epan in Australia, which has been resolved by Roderick et al. [2007] as caused by reductions in wind speed and some regional changes in solar radiation—both drivers unaccounted for in a Tair-based Ep. It can be no surprise then that Tair-based predictions of drying under warming contradict a gathering wealth of reports of decreasing evaporative demand in the face of global warming.

[18] More subtle interactions between trends in the surface water balance arise in energy-limited regions than in water-limited regions, so the answer to the question “does warming mean drying?” depends critically on accompanying changes in Prcp and also on the climate regime in question. Our Budyko-based climatological analysis (see Figure S3) indicates that, over water-limited regions such as most of Australia, trends in ETa are largely determined by trends in Prcp, and not in Ep. However, this conclusion does not assist in the essential regional analyses: it is in the southern and eastern regions of Australia—areas that are wetter (i.e., more energy-limited) than the continental mean and, crucially, where most people live and most agricultural production occurs—that the differences in direction and scale of SM-trends between parameterizations are most apparent. Here, the Ep parameterization choice is crucial.

[19] Our conclusions regarding the disjunction between drying predicted by Tair-based parameterizations and observations of wetting are also implicit in the results of Robock et al. [2005], who, working with a large-scale set of observed soil moisture in the Ukraine from 1958 to 2002, found that decreasing summer Prcp under a warming trend had still resulted in increased summer soil moisture.

5. Conclusions

[20] Our comparison of different Ep formulations in the PDSI bucket model has highlighted the problems with the Tair-based parameterization of evaporative demand classically used in a well-established water balance model (i.e., EpPDSI). We find that, in energy-limited regions, what constitutes drying under EpPDSI frequently shows as wetting under EpPan. While the analysis here is limited to Australia and New Zealand, our station selection covers a broad climatic range, so we expect our findings—wetting often projected by a physically based parameterization where drying had previously been projected by one that is Tair-based—to apply elsewhere, as over the last few decades declining Epan has been widely observed in the face of globally rising Tair.

[21] The PDSI model could be parameterized with potential evapotranspiration, such as that from the Penman-Monteith approach. However, the limited availability of data on humidity, solar radiation, wind speed, and surface resistance currently preclude such an observational study on a global or even continental basis, although work progresses on developing such datasets [Dai et al., 2005]. In the meantime, however, results from a multi-model ensemble-mean (see Figure 10.12 in the IPCC AR4 WG1 report [Meehl et al., 2007]) indicate patterns of projected trends in ETa broadly similar to those in Prcp over much of the global land surface.

[22] Our continent-wide results warn against an over-simplistic treatment of evaporative demand in water balance models and highlight the urgent need for a more rigorous approach to assessing long-term changes in the terrestrial water balance. The effects of this over-simplicity may be obscured in water-limited regions as evapotranspiration there is strictly constrained by water availability. However, more rigorous analyses are particularly wanting over regions where evapotranspiration is limited frequently or consistently by the availability of energy, and where this empirical study shows important dependencies of ETa and SM on Ep. Meeting this need over more than select regions presents the hydroclimatological community with the same challenge originally laid down by Penman [1948], Thornthwaite [1948], and Palmer [1965]: to develop reliable global datasets of radiation, humidity and wind as well as air temperature and precipitation to permit physically sound estimations of water balance trends.


[23] We acknowledge funding support from a Gary Comer Award. We acknowledge the BoM and NIWA and especially the numerous observers whose work formed the ultimate basis of this study. The National Center for Atmospheric Research is sponsored by the U.S. National Science Foundation. Dai is also partly supported by NCAR's Water Cycle Program.