[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.

Abstract:A GIS-based RUSLE model is employed to study the global soil erosion potential for viewing the present situation, analysing changes over the past century, and projecting future trends with reference to global changes in land use and climate. Scenarios considered in the study include historical, present and future conditions of cropland and climate. This research gives the first overview of the global situation of soil erosion potential considering the previous century as well as the present and future. Present soil erosion potential is estimated to be about 0Ð38 mm year 1 for the globe, with Southeast Asia found to be the most seriously affected region in the world. It is estimated that nearly 60% of present soil erosions are induced by human activity. With development of cropland in the last century, soil erosion potential is estimated to have increased by about 17%. Global warming might significantly increase the potential for soil erosion, and the regions with the same increasing trend of precipitation and population might face much more serious problems related to soil erosion in the future.

[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.

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[1] The growth of vegetation is affected by water availability, while vegetation growth also feeds back to influence regional water balance. A better understanding of the relationship between vegetation state and water balance would help explain the complicated interactions between climate change, vegetation dynamics, and the water cycle. In the present study, the impact of vegetation coverage on regional water balance was analyzed under the framework of the Budyko hypothesis by using data from 99 catchments in the nonhumid regions of China, including the Inland River basin, the Hai River basin, and the Yellow River basin. The distribution of vegetation coverage on the Budyko curve was analyzed, and it was found that a wetter environment (higher P/E 0 ) had a higher vegetation coverage (M) and was associated with a higher evapotranspiration efficiency (E/E 0 ). Moreover, vegetation coverage was related not only to climate conditions (measured by the dryness index DI = E 0 /P) but also to landscape conditions (measured by the parameter n in the coupled water-energy balance equation). This suggests that the regional long-term water balance should not vary along a single Budyko curve; instead, it should form a group of Budyko curves owing to the interactions between vegetation, climate, and water cycle. A positive correlation was found between water balance component (E/P) and vegetation coverage (M) for most of the Yellow River basin and for the Inland River basin, while a negative correlation of M $ E/P was found in the Hai River basin. Vegetation coverage was successfully incorporated into an empirical equation for estimating the catchment landscape parameter n in the coupled water-energy balance equation. It was found that interannual variability in vegetation coverage could improve the estimation of the interannual variability in regional water balance.