Budyko s Framework and Climate Elasticity Concept in the Estimation of Climate Change Impacts on the Long-Term Mean Annual Streamflow D. S. Reis Junior 1, C. M. Cerqueira 2, R. F. Vieira 3, E. S. Martins 4 1 University of Brasília, Department of Civil and Environmental Engineering, Campus Darcy Ribeiro, Brasília-DF, Brazil, 7.91-9, email: dirceureis@unb.br 2 University of Brasília, Department of Civil and Environmental Engineering, Campus Darcy Ribeiro, Brasília-DF, Brazil, 7.91-9, email: carolinacerq@gmail.com. 3 Research Institute for Meteorology and Water Resources FUNCEME, Av. Rui Barbosa, 1246, Aldeota, Fortaleza/CE, Brazil, 6115-221, email: robson@funceme.br 4 Research Institute for Meteorology and Water Resources FUNCEME, Av, Rui Barbosa, 1246, Aldeota, Fortaleza/CE, Brazil, 6115-221, email: espr.martins@gmail.com. ABSTRACT Understanding climate change impacts on hydrologic regime is important to evaluate water resources systems vulnerability and develop adaptation strategies. Traditional studies are complex and expensive. A recent and simple methodology, based upon Budyko s hypothesis and climate elasticity concept, allows estimation of sensitivity of long-term streamflow to changes in climate variables. This methodology was applied to the Upper Jaguaribe River basin, Brazil, wherein a traditional climate change impact study has been carried out. Results for the sensitivity analysis suggest that a 1% increase in precipitation results in a 2.75% increase on runoff, while a 1% increase in potential evapotranspiration results in a decrease of 17.3% in runoff. Comparison of theses results with those obtained from a more traditional approach (statistical downscaling followed by hydrologic modeling) showed that estimates of relative changes in runoff are very similar, with errors within.6 and correlation of.99, suggesting that the methodology could be used in preliminary studies to identify most sensitive basins that need more detailed studies.
INTRODUCTION Many approaches have been developed by the scientific community to evaluate the impacts of climate change in the water availability in a given region with different levels of complexity. However, there is no consensus on which methodology should be used and when, and it continues to be a topic of intense research. This paper employs an elegant approach developed by Budyko (1974), based on the long-term temporal scales of water and energy conservation principles, which allows to explain spatial patterns of evapotranspiration (E) and streamflow (Q) by a single equation that depends only on precipitation (P) and potential evapotranspiration (E ). This approach is referred to in the literature as Budyko s hypotheses, Budyko s model or even Budyko s equation. More recently, many other studies revisited the Budyko s approach, trying to better understand how other factors, besides P and E, should affect the way precipitation is divided between E and Q, resulting in different equations that relate E/P to E /P. These equations are usually called Budiko-like equations. Zhang et al. (24) developed an analytic equation, denoted here Fu s equation, with only one parameter, that shows how basin and climate characteristics should affect the partition of P between E and Q. Later, Yang et al. (28), employing the same hypotheses, and based on dimensional analysis and mathematical reasoning, also derived a one-parameter analytic equation to relate E/P with E /P. Both parameters are linearly related, meaning that both equations are almost equivalent. Other Budyko-like equations were proposed in the literature. But what is interesting about them is that they can be used along with the climate-elasticity concept to understand how the hydrologic cycle would be changed due to possible changes in climate variables. This possibility has been investigated in many recent studies, where one can appreciate the potentials of this relatively simple approach (Nemec e Schaake, 1982; Dooge, 1999; Sankarasubramanian et al., 21; Arora, 22; Niemann, 25; Fu et al., 27; Wang et al., 28; Fu et al., 29; Reneuer et al., 211; Roderick e Farquar, 211; Yang e Yang, 211; Donohue et al., 212). The goal of this paper is to show how the estimated impacts on the long-term streamflow due to climate change, obtained by the simple methodology based on Budyko framework and climate-elasticity concept, compare to those obtained by traditional approaches, based on downscaling techniques of global climate models projections of climate variables followed by some sort of hydrologic modeling. The study makes use of a recent evaluation of climate impacts on hydrology in several basins located in the semiarid region of Northeastern Brazil, where key reservoirs had been built during the twentieth century to reduce vulnerability to drought. The hope is that this simple approach could be used in a large scale to identify the most sensitive basins that need more detailed and complex studies.
METHODOLOGY Budyko Framework There is a clear link between water and energy balance within a river basin. In a long-term analysis, in which temporal variations of the water storage in the basin can be neglected, long-term mean streamflow (Q) equals the difference between precipitation (P) and evapotranspiration (E). The latter is one of the main components of the energy balance equation along with the sensible heat (H). In a long-term analysis, it is reasonable to say that net radiation near the surface (R n ) is transformed into latent heat (L e E) and sensible heat fluxes, R n = L e E + H, where L e is the latent heat of vaporization of water. Budyko (1974) presented an interesting analysis relating Q with components of the energy balance equation in large temporal scale, which became the basis for the development of more sophisticated models that appeared later in the literature. Budyko s analysis is based upon two extreme climate conditions: one that is very arid, in which the net radiation near the surface, in units of water evaporated, is much larger than P, R n /(L e E) >> 1, and another in which there is plenty of water available, R n /(L e E) << 1. Considering that the average soil water content diminishes as R n /L e increases and P is reduced, the soil tends increase its storage capacity resulting in a reduction of surface flow. So, when the aridity index (R n /(L e P)) tends to infinity, evapotranspiration approaches total precipitation. In this conditon, the evapotranspiration is limited by the amount of water avaiable. Mathematically, it is described by, E lim Rn L e P P =1 (1) When the aridity index (R n /(L e P)) diminishes, the average soil water content increases and the ratio E/P reduces, resulting in an increase of the surface flow. Therefore, as the net radiation near the surface diminishes and P increases, there will be an increase in the average soil water content near the surface so that in the limit, when the aridity index tends to zero, one will observe plenty of water in the soil surface layer. In this case, evapotranspiration is solely related to the amount of energy available. Mathematically, this extremely humid condition is described by, lim Rn L e P E P = R n L e P (2) Based on these two extreme conditions, Budyko (1974) proposed the following, E P = Φ R n (3) L e P
wherein the function Φ needs to be defined. Previous studies (Schreiber, 194; Ol dekop, 1911), based upon empirical analyses, had proposed similar equations, as point out by Budyko himself. The final equation proposed by Budyko is based on eqs. (2) and (3), along with data from several european river basins, E P = E P 1 exp E tgh P P E 1/ 2 (4) wherein tgh is the hyperbolic tangent and E is the potential evapotranspiration, which represents the maximum net radiation near surface that would be used to evaporate water. One can check that eq. (4) obeys the conditions imposed by eqs. (2) and (3). Fu s Equation Zhang et al. (24), based on Fu (1981), analytically derived an equation that relates E/P with E /P. Fu s equation, as it is referred to herein, has one parameter that is supposedly dependent on factors and characteristics of the basin that affects the way precipitation is transformed into evapotranspiration and surface flow. The whole analysis is based on large temporal scales such that variations of water storage whitin the basin can be neglected. The analysis is based on the following observations about E / P and E / E. For a fixed value of E, E / P increases with (E E) and diminishes with P, whereas for a fixed value of P, E / E increases with (P E) and diminishes with E. Based on these observations and dimensional analysis, Zhang et al. (24) obtained the following equations, E P = φ E E 1 and E P E = φ P E 2 (5) E wherein φ 1 and φ 2 are functions of the dimensionless variables (E E)/P e (P E)/E, respectively. Selection of functions φ 1 and φ 2 were made based on the boundary conditions of the problem. In extremely humid conditions, E is limited by E and doesn t depend on P, therefore when the dimensionless variable (E E)/P tends to zero, the term E / P also tends to zero. Function φ 1 should satisfy this boundary condition. In an extremely arid condition, E = P, and the evapotranspiration doesn t depend on E, thus when the dimensionless variable (P E)/E tends to zero, E / E also tends to zero. This is the boundary condition that function φ 2 must satisfy. Fu s equation derived by Zhang et al. (24) is based on eq. (5) and boundary conditions identified previously, E P =1+ E P 1+ E P 1/ (6)
wherein is the only parameter of Fu s equation, supposedely related to basin characteristics. More recently, Yang et al. (28) developed a new analytic equation based on dimensional analysis and mathematical reasoning, E P = E P n n + E ( ) 1/ n (7) which has also only one parameter, n. However, it has been observed that parameter n is linearly related to parameter, which means that eq. (7) is basically equivalent to Fu s equation. Sensitivity of Long Term Streamflow The concept of elasticity, very popular in the field of economics, can be employed to describe the sensitivity of the river streamflow to changes in the regional climate. This paper adopts the concept of streamflow climate-elasticity, which represents the percentage change in the long-term streamflow of a given river basin due to a percentage change on climate variables, such as precipitation and temperature. Sankarasubramanian et al. (21) identify different ways to estimate streamflow climate-elasticity. It can be accomplished by employing calibrated conceptual or physically based hydrologic models, adjusting a multivariate statistical model, or employing analytical equations, such as Fu s equation, based on climate and hydrologic principles, as is done in this paper. The study followed the methodology presented by Yang et al. (211), who obtained analytical equations to estimate streamflow climate-elasticity. Two climate factors are employed, precipitation and potential evapotranspiration. The latter depends on many climate variables, such as temperature, radiation, wind velocity and air humidity. Based on Fu s equation, the evapotranspiration (E) is a function of potential evapotranspiation (E ), precipitation (P) and parameter, E = f(e, P, ), where f( ) represents Fu s equation, although any other Budyko-like equation could have been used in the analysis. Total variation in evapotranspiration, de, can be approximated by employing a firstorder Taylor s series expansion, de = f f dp + de P E + f d (8) As Q = P E, then dq = dp de. Substituing this into eq. (8) leads to eq. (9), dq = 1 f dp f de P E f d (9)
Assuming that characteristics of the basin that affects the way precipitation is transformed into evapotranspiration and streamflow won t change over time, which may not be true (this discussion is beyond the scope of this paper), d = and the relative change in the long term streamflow due to relative changes in precipitation and potential evapotranspiration is given by eq. (1), dq Q = 1 f P P P E ε 1 dp P f E E P E ε 2 de E (1) wherein ε 1 and ε 2 are respectively the streamflow precipitation-elasticity and streamflow potential evapotranspiration-elasticity coefficients. Although it hasn t been explored in this paper, one may be interested in estimating the sensitivity of the long-term streamflow due to changes in specific climate variables that affect the potential evapotranspiration. The analysis can be easily carried out. For example, if potential evapotranspiration is given by the Hargreaves formula, the total change of evapotranspiration is given by de = E ds S + E dδ δ T + E dt (11) T T wherein S is the extraterrestrial solar radiation in units of water evaporated, δ T the difference between maximum and minimum temperatures, and T the average temperature. If only changes in average temperature is taken into account, the relative change in potentail evapotranspiration is given by, de = 1 E E E T dt = 1.23S E δ T ( )T (12) Therefore, the relative change in long-term streamflow due to changes in precipitation and temperature can be estimated by, wherein dq Q = ε dp 1 P +ε 2ε 3 dt (13)
ε 1 = 1 f P P P E ε 2 = f E E P E ( ) ε 3 = 1 E.23S δ T f P =1 1+ E P f =1 1+ E E P 1/ 1 + 1+ E P E P 1 1 E P (14) CASE STUDY The methodology was applied to the upper Jaguaribe river basin, located in the semiarid region of the State of Ceará, Brazil (Figure 1). The sensitivity of the longterm streamflow due to changes in climate was estimated for the Iguatu streamflow gauge, located upstream from the Orós reservoir, the second largest reservoir in the State of Ceará, part of a system of reservoirs and other water infrastructures built to reduce vulnerability to drought. Figure 1: Hydrosystems of Jaguaribe and Piranhas-Açu river basins. (Source: FUNCEME) Besides its strategic role in the water management system, this basin was chosen because of a recent climate change impact study (NLTA), organized by World Bank, and supported by the National Water Agency (ANA) and water state agencies in the
region. The NLTA study (Martins, 211), which includes the possible changes in the long-term streamflow into the Orós reservoir, was based on traditional methodologies (downscaling of global climate models and hydrologic modeling), is used to understand how the methodology presented herein, based on the Budyko framework and climate-elasticity concept, agrees with more traditional approaches. RESULTS The methodology was applied to the Iguatu streamflow gauge, with 69 years of record length, and an area of 2,68.9 km 2. Otsuki and Reis (212) provide the longterm averages for P = 661.4 mm, E = 624.2 mm, and Q = 37.2 mm, whereas Barros (212) reports the long-term average for E = 1,864.5 mm. The aridity index is large, E /P = 2.8, resulting in a relatively small streamflow of 37.2 mm, only 5.6% of total average precipitation. Based on these values, the parameter of Fu s equation was estimated to be = 2.77. Based on eq. (14), the climate-elasticity coefficients were estimated to be ε 1 = + 2.74, ε 2 = - 1.73, and ε 3 = +.22. Therefore, the sensitivity of the long-term streamflow (dq/q) for the site of interest due to changes in P and E cab be computed by, dq Q = 2.74 dp P 1.73 de E (15) This first-order result indicates that a 1% reduction in P, results in a 27.4% reduction in the long-term streamflow, whereas a 1% increase in E results in a 17.3% decrease in Q. This means that a reduction of 66.1 mm in P represents a reduction of 1.2 mm in Q, whereas an increase of 186.5 mm in E would result in a decrease of only 6.4 mm in Q. The results obtained based upon the Budyko framework and climate-elasticity concept were compared to those obtained during the NLTA study (Martins, 211) which employed statistical downscaling of three global climate models to provide monthly series of precipitation and potential evapotranspiration for the 241-27 period that were used to feed a calibrated conceptual hydrological model, which provided the streamflow series. Series of potential evapotraspiration were estimated based on the Hargreaves equation. It should be noted that the NLTA study was applied to the Orós reservoir, located downstream from the Iguatu streamflow gauge, with an incremental contributing area of 5, km 2. Values of dp/p and de /E, obtained from the NLTA study for the Orós reservoir, based on three different models and two climate change scenarios, are presented in Table 1, along with the estimated values of dq/q. For the sake of comparison, values of dq/q obtained by the methodology applied herein are also presented. One can see the estimates of dq/q obtained by both methodologies are relatively close, with absolute errors within.6, and R 2 equal to.985. This comparison is also presented in Figure 2, which shows graphically the degree of agreement of the estimates of dq/q. A linear regression analysis showed that the bias is small, less than.1, with angular coefficient equal to.87.
Table 1: Results of relative change in Q obtained from the NLTA study and those based on the Budyko framework and climate-elasticity concept. Model Scenario NLTA Budyko/Elasticity concept dp/p de /E dq/q dq/q BCM2 A2.1.2.3. BCM2 B1.2.2..2 INCM3 A2.5.5.7.5 INCM3 B1.1.3 -.5 -.4 MIMR A2 -.2.15 -.37 -.31 MIMR B1..13 -.21 -.24 Figure 2: Comparison of the results on the sensistivity of long-term streamflow (dq/q) obtained by the methodology based on the Budyko framework and those obtained by traditional analysis reported in the NLTA report. The dashed line represents the 45 degree line. Correlation between results obtained by both methodologies is equal to.99. Bias is less than 1% while the angular coefficient of the fitted line between results is equal to.87.
CONCLUSIONS This paper provides preliminary results of an on-going study that aims at evaluating a simple methodology, based on Budyko s hypothesis and climate-elasticity concept, to estimate the sensitivity of the long-term streamflow due to changes in climate. The methodology was applied to the Upper Jaguaribe river basin, located in the semiarid region of state of Ceará, northeast of Brazil. Results for the Iguatu streamflow gauge, based on the methodology presented herein suggest that a 1% reduction in precipitation would result in a 27.3% reduction in the streamflow, whereas a 1% increase in potential evapotranspiration would result in a decrease of 17.3% in streamflow. Results obtained from the NLTA study, coordinated by the World Bank, were used to compare the estimates of dq/q obtained by the methodology presented herein and those obtained by more traditional and complex methodologies. The NLTA study employed a statistical downscaling technique to generate monthly time series of precipitation and potential evapotranspiration for the 241-27 period that were used to feed a hydrologic model that generated monthly streamflow series. The whole analysis was carried out to the Orós reservoir, also located in the Jaguaribe river basin, but a few kilometers downstream the Iguatu gauge station (~2, km 2 ), representing an incremental area of almost 5, km 2. Despite this difference in location, the results of dq/q obtained by both methodologies are very similar, with absolute differences within.6 and correlation of.99. These preliminary results suggest that the methodology based on Budyko s hypothesis and climate-elasticity concept may be reliable and useful in identifying most sensitive basins that need more detailed and complex studies. REFERENCES Arora, V.K. (22). The Use of the Aridity Index to Assess Climate Change Effect on Annual Runoff. Journal of Hydrology, 265 (1): 164 177. Budyko, M.I. (1974). Climate and Life. Academic Press, Nova Iorque. Donohue, R.J., Roderick, M.L., McVicar, T.M. (211). Assessing the Differences in Sensitivities of Runoff to Changes in Climatic Conditions Across a Large Basin. Journal of Hydrology, 46 (3-4) (September 6): 234 244. doi:1.116/j.jhydrol.211.7.3. Dooge, J.C.I., Bruen, M., Parmentier, B. (1999). A Simple Model for Estimating the Sensitivity of Runoff to Long-Term Changes in Precipitation Without a Change in Vegetation. Advances in Water Resources, 23 (2): 153 163. Fu, G., Charles, S.P., Chiew, F.H.S. (27). A Two-Parameter Climate Elasticity of Streamflow Index to Assess Climate Change Effects on Annual Streamflow. Water Resources Research, 43 (11) (November 24): 1 12. doi:1.129/27wr589.
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