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Accurate detection and quantification of vegetation dynamics and drivers of observed climatic and anthropogenic change in space and time is fundamental for our understanding of the atmosphere-biosphere interactions at local and global scales. This case study examined the coupled spatial patterns of vegetation dynamics and climatic variabilities during the past three decades in the Upper Heihe River Basin (UHRB), a complex multiple use watershed in arid northwestern China. We apply empirical orthogonal function (EOF) and singular value decomposition (SVD) analysis to isolate and identify the spatial patterns of satellite-derived leaf area index (LAI) and their close relationship with the variability of an aridity index (AI = Precipitation/Potential Evapotranspiration). Results show that UHRB has become increasingly warm and wet during the past three decades. In general, the rise of air temperature and precipitation had a positive impact on mean LAI at the annual scale. At the monthly scale, LAI variations had a lagged response to climate. Two major coupled spatial change patterns explained 29% and 41% of the LAI dynamics during 1983-2000 and 2001-2010, respectively. The strongest connections between climate and LAI were found in the southwest part of the basin prior to 2000, but they shifted towards the north central area afterwards, suggesting that the sensitivity of LAI to climate varied over time, and that human disturbances might play an important role in altering LAI patterns. At the basin level, the positive effects of regional climate warming and precipitation increase as well as local ecological restoration efforts overwhelmed the negative effects of overgrazing. The study results offer insights about the coupled effects of climatic variability and grazing on ecosystem structure and functions at a watershed scale. Findings from this study are useful for land managers and policy makers to make better decisions in response to climate change in the study region.

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Abstract. Aridity indices have been widely used in climate classification. However, there is not enough evidence for their ability in identifying the multiple climate types in areas with complex topography and landscape, especially in those areas with a transition climate. This study compares a traditional meteorological aridity index (AI), defined as the ratio of precipitation (P) to potential evapotranspiration (PET), with a hydrological aridity index, the evaporative stress index (ESI) defined as the ratio of actual evapotranspiration (AET) to PET in the Heihe River Basin (HRB) of arid northwestern China. PET was estimated using the Penman–Monteith and Hamon methods. The aridity indices were calculated using the high-resolution climate data simulated with a regional climate model for the period of 1980–2010. The climate classified by AI shows a climate type for the upper basin and a second type for the middle and lower basin, while three different climate types are found using ESI, each for one river basin, indicating that only ESI is able to identify a transition climate zone in the middle basin. The difference between the two indices is also seen in the interannual variability and extreme dry/wet events. The magnitude of variability in the middle basin is close to that in the lower basin for AI, but different for ESI. AI had a larger magnitude of the relative interannual variability and a greater decreasing rate from 1980 to 2010 than ESI, suggesting the role of local hydrological processes in moderating extreme climate events. Thus, the hydrological aridity index is better than the meteorological aridity index for climate classification in the arid Heihe River Basin.

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In this paper, we consider the entire solutions to the parabolic 2-Hessian equations of the form −utσ2(D 2 u) = 1 in R n × (−∞, 0]. We prove some rigidity theorems for the parabolic 2-Hessian equations in R n × (−∞, 0] by establishing Pogorelov type estimates for 2-convex-monotone solutions of the parabolic 2-Hessian equations.

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