[1] The coupled water-energy balance on long-term time and catchment scales can be expressed as a set of partial differential equations, and these are proven to have a general solution as E/P = F(E 0 /P, c), where c is a parameter. The state-space of (P, E 0 , E) is a set of curved faces in P À E 0 À E three-dimensional space, whose projection into E/P À E 0 /P two-dimensional space is a Budyko-type curve. The analytical solution to the partial differential equations has been obtained as E = E 0 P/(P n + E 0 n ) 1/n (parameter n representing catchment characteristics) using dimensional analysis and mathematic reasoning, which is different from that found in a previous study. This analytical solution is a useful theoretical tool to evaluate the effect of climate and land use changes on the hydrologic cycle. Mathematical comparisons between the two analytical equations showed that they were approximately equivalent, and their parameters had a perfectly significant linear correlation relationship, while the small difference may be a result of the assumption about derivatives in the previous study.
[1] On the basis of long time series of climate and discharge in 108 nonhumid catchments in China this study analyzes the spatial and temporal variability of annual water-energy balance using the Budyko hypothesis. For both long-term means and annual values of the water balances in the 108 catchments, Fu's formula derived from the Budyko hypothesis is confirmed. A high correlation and relatively small systematic error between the values of parameter v in Fu's equation optimized from the water balance of individual year and calibrated from the long-term mean water balance show that Fu's equation can be used for predicting the interannual variability of regional water balances. It has been found that besides the annual climate conditions the regional pattern of annual water-energy balance is also closely correlated with the relative infiltration capacity (K s /i r ), relative soil water storage (S max /E 0 ), and the average slope (tan b). This enables one to estimate the parameter v from catchment characteristics without calibration from the long time series of water balances. An empirical formula for the parameter v in terms of the dimensionless landscape parameters is proposed. Applications of Fu's equation together with the parameter v estimated by this empirical formula have shown that Fu's equation can predict both long-term mean and annual value of actual evapotranspiration accurately and predict both long-term mean and interannual variability of runoff reasonably. This implies that the Fu's equation can be used for predicting the annual water balance in ungauged basins.
[1] The river discharges have decreased continuously during the last half century in the Yellow River, the second-largest river basin in China. In particular, a drying up of the main river along the lower reach has occurred since 1972, and the situation has become more and more serious during the 1990s. Using 50 years of meteorological data from 108 stations together with a collection of irrigation data, the long-term changes in the river discharge have been investigated with a view to identifying the reason for the drying up of the Yellow River. It was found that the annual precipitation generally decreased (À45.3 mm/50 yr) while the air temperature generally increased (+1.28°C/50 yr). From the 1960s to the 1970s the precipitation decreased by 29.6 mm/10 yr, the evaporation increased by 7 mm/10 yr (for pan evaporation), and the irrigation water usage increased by 10.5 mm/10 yr. As a consequence the drying up of the Yellow River has occurred since 1972. Irrigation was developed continuously in the 1980s, but the drying-up situation maintained at the same level as during the 1970s. The reason for this was the increase in precipitation (by 10.3 mm/10 yr) and the sharp decrease in the evaporation (by 133 mm/10 yr for pan evaporation). During the 1990s the irrigation was maintained at a level similar to that during the 1980s, but the drying-up situation was greatly aggravated. The reason for this was found to be a result of the decrease in precipitation (by 38.2 mm/10 yr) and the increase in evaporation (by 52 mm/10 yr for pan evaporation).
[1] The Budyko, Bouchet, and Penman hypotheses together describe, using different formulae, the tight connections and feedbacks between water -energy balances and the landscape. The discrepancy between the Penman and Bouchet hypotheses is usually highlighted in non-humid regions. In this paper, using Fu's equation for annual water-energy balances at the catchment scale, which was derived phenomenologically and mathematically on the basis of the Budyko hypothesis, consistency among the three hypotheses is explained. In non-humid regions, change of actual evaporation is dominated by change in precipitation rather than potential evaporation, and the Bouchet complementary relationship between actual evaporation and potential evaporation comes about because actual evaporation and potential evaporation are correlated via precipitation. The annual water balances in 108 non-humid catchments of China have been examined as part of this study, and the results supported the complementary relationship. In humid regions, change in actual evaporation is controlled by change in potential evaporation rather than precipitation, and this is identical to the Penman hypothesis.
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