Empirical nonlinear dependence of variability on mean runoff 

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Patterns of variability and long-term erosion rates

The band of low rainfall variability may help to explain the inconsistency between mea-sured long-term erosion rates and rainfall rates in the region. Burbank et al., (2003) and Thiede and Ehlers, (2013) showed that in central Nepal, erosion rates increase north of the band of high magnitude rainfall. It can be seen that mean erosion rate increases even as the mean rainfall rate decreases. Erosion rates peak close to the topographic divide, well north of the highest mean rainfall rates. If the region is in dynamic equilib-rium, we expect the erosion rate to be set by the uplift rate rather than climate. How-ever, the specific stream power should still reflect the erosion rate. In this region the specific stream power still decreases northward (albeit more slowly that rainfall) (Bur-bank et al., 2003). This implies that steeper slopes and narrower channels can make up only partially for the decreasing rainfall. Increasing rainfall variability may explain the rest of the discrepancy between erosion rates and rainfall rates. Figure 7a shows a representative cross section over the topographic divide in this region. The rainfall variability increases rapidly as the mean rainfall rate decreases across the topographic divide, as demonstrated by the observed decrease in the shape parameter and mean wet day frequency. The observed increase in rainfall variability, in conjuction with nar-rowing channels and steepening slopes, is consistent with a peak erosion rate on the topographic divide despite significantly decreasing mean rainfall rate. As further sup-port for this, both steeper slopes and higher rainfall variability are expected to enhance other important erosion processes, such as landsliding (Gabet et al., 2004).
Increasing rainfall variability means that while the total amount of rainfall decreases sharply across the topographic divide, the size of large storms will decreases more slowly. The solid yellow line in figure 7a and b show the average maximum annual storm size, and it can be seen that it does not track with the mean daily rainfall rate, and does decrease more slowly across the divide. More than that, the east-west striking band of very low variability associated with the northern band of high rainfall results in the large storms in this region being exceptionally small given the high rainfall rate. The ratio of the mean maximum annual storm to the mean daily rainfall rate can be used as another measure of rainfall variability. Shown as the solid red line in figure 7, we see that it is consistent with the measured changes in the shape parameter and mean wet day frequency. It is very low over the northern band of high magnitude rainfall, and highest over the Tibetan plateau. Figure 7c and d show another representative cross section further to the west near the end of the arc of low variability. The same pattern holds for this cross section as well, with increasing variability across the topographic divide, reflected both in the ratio of the maximum annual storm to the daily mean rainfall as well as the shape parameter and wet day frequency. In this cross section, the rise in topography from south to north is gentler, as are the gradients in the erosion rate, mean daily rainfall, mean wet day frequency and the shape parameter.
We put forth the idea of rainfall variability resolving the inconsistency between long-term erosion rate and rainfall rate only as a possibility. The theoretical arguments provide a basis for this theory, and the APHRODITE data support it. However, the way that rainfall drives erosion rates across the landscape is not perfectly understood, so we must restrict ourselves to a qualitative assessment of the patterns of rainfall variability and mean. It is not clear how to asses whether the magnitude of increase in variability is sufficient to offset the magnitude of decrease in mean rainfall. Additionally, the TMPA data (7b and d) only broadly match the pattern observed in the APHRODITE data set (7a and c). The rise in variability observed in the TMPA data occurs further north than in the APHRODITE data, and is not coincident with the peak in the erosion rate. Further, the TMPA data do not resolve the northern rainfall maximum, making it more difficult to compare the peak in erosion to rainfall. Due to this, in the Nepal swath (7b) the TMPA data do not support the theory that increasing rainfall variability offsets decreasing rainfall mean (though they do in swath 7d). Additionally the elevation is an important parameter in the interpolation scheme of the APHRODITE data, making swath profiles with the APHRODITE data suspect.
One point in support of the theory comes from a data set derived from the TMPA 2B31 data that has much higher spatial resolution that the ones we used here Bookha-gen and Burbank, 2006; Bookhagen, 2010; Bookhagen and Burbank, 2010; Olen, Bookha-gen, and Strecker, 2016. While we do not have access to the data directly and cannot compute the shape parameter, mean frequency and mean intensity, the data agrees well with our results here. The TMPA derived data resolve the northern peak in rain-fall magnitude and, as far as we can tell, low in variability in a very similar location to the APHRODITE data. This lends support to the APHRODITE data. So, while we find the data presented here is suggestive that rainfall variability may be a key parameter influencing the erosion efficiency of rainfall in the Himalayan orogen, we conclude that better data, which may become available in the future, is necessary to confirm or refute this hypothesis.

Patterns of variability and short-term erosion rates

Olen, Bookhagen, and Strecker, (2016) conducted an analysis of the empirical relation-ships between vegetation density, precipitation rates and short-term denudation rates in the Himalayan orogen. They find a strikingly clear negative correlation between veg-etation density and the variability of measured denudation rates within a single basin. The lower the vegetation density, the higher the variation in measured denudation rates. They point out the logic in this; vegetation tends to stabilize soils, increasing the resistance to erosion. Basins with low vegetation density should be more vulnerable to substantial surface erosion during large rainstorms than those with high vegetation density. The erosion caused by large rainstorms will likely not be evenly distributed across the basin for a variety of reasons including localized high intensity rainfall, non-uniformly distributed vegetation, and different antecedent conditions on different hillslopes. This will lead to more episodic, localized erosion events, and consequently, more variation in denudation rates measured within a single basin or region.
It is also logical that rainfall variability would influence this trend by increasing the size and frequency of large storms in regions with high rainfall variability rela-tive to those with low variability. Olen, Bookhagen, and Strecker, (2016) therefore also compare rainfall variability to denudation variability and vegetation density. However, while they observe vegetation density to increase and denudation variability to de-crease from west to east along the strike of the orogen, they find rainfall variability trends in the opposite sense, increasing from west to east. This doesn’t match the de-crease in denudation variability from west to east, so they conclude that the influence of increasingly dense vegetation towards the east is so strong as to erase any effects of increasing rainfall variability on denudation variability.
Figure 7: Representative cross sections orthogonal to the strike of the orogen showing the pat-terns of daily rainfall mean and variability across the topographic divide. Cross sec-tions are aligned relative to the location of the northern rainfall peak. The lines show the average value across the section, and the shaded regions show the range. The mean daily rainfall rate is shown in blue, the mean maximum annual storm is shown in solid yellow, the estimated maximum annual storm is shown in dashed yellow, the ratio of the daily mean to the annual max. storm is shown in red, the gamma shape parameter is shown in purple and the mean wet day frequency is shown in green. Estimated erosion rates over the last 2 million years obtained from a 1D inversion of thermochronological data taken from Thiede and Ehlers, (2013) are are shown with the dashed black line. Average maximum topography shown as solid black line.
While we agree completely with their explanation of how vegetation density and rainfall variability are likely to influence the variation in measured denudation rates, we find that rainfall variability increases along strike from east to west rather than west to east. Olen, Bookhagen, and Strecker, (2016) use the number of times per year that extreme events occur as a measure of variability. Extreme events are defined by them as events above the 90th percentile. Since, by definition, extreme events consist of 10% of observed events, the number of extreme events is more a reflection of the mean wet day frequency than anything else. Although the data sets are not directly comparable because they analyze annual rainfall, and we only consider monsoon rainfall, the spa-tial pattern of mean wet day frequencies during the monsoon match the number of extreme events per year quite well.
The issue with using the number of events above the 90th percentile as a measure of daily rainfall variability is that there is not a standard relationship between the magnitude of the 90th percentile event and the mean rainfall intensity. This is in fact determined by the shape parameter of the distribution and varies from region to region. Figure 8 shows the mean magnitude of events above the 90th percentile as a function of the shape parameter. As gamma falls below one, the average magnitude of extreme events approaches ten times the mean rainfall intensity, but above one, it is only about 2-3 times the mean. Therefore, right where they find the most frequent extreme events, in central Nepal, is where those extreme events will be the smallest. This is an impor-tant point, because from the perspective of geomorphology it is not just the frequency of big storms that is important, but also their magnitude. The shape parameter and mean wet day frequency can be used to measure the magnitude of these big storms relative to the mean rainfall intensity in an unbiased way that allows for comparison between regions with significantly different mean rainfall intensities (such as central Nepal and the Tibetan plateau).
As figure 6 shows, these measures of variability point to an increase in varability from the east to the west and from the south to the north of the orogen. This is more consistent with the patterns of denudation variability and vegetation density observed by Olen, Bookhagen, and Strecker, (2016). Moving along the orogen from east to west, the mean annual rainfall drops, as does the mean rainfall intensity and the vegetation density. At the same time the variability of rainfall and the variability in denudation increase. As figure 6c shows, the maximum annual storm magnitude does not change along strike for most elevations, because of increasing rainfall variability. As a result, large storms become more extreme relative the mean rainfall intensity. It is unsur-prising that the most episodic erosion is observed in the west. However, whether the denudation variability increases to the west because of increasingly significant large storms, or because of decreasing vegetation density or both is less clear.

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conclusions

In this study we have made a careful characterization of the distribution of rainfall in the Himalayan orogen. We find a consistent pattern appearing with the gamma shape parameter, mean wet day frequency and mean rainfall intensity in many places along the Himalayan arc, particularly in Nepal. We observe moderate rainfall variability in the foreland basin up to and including the southern band of high magnitude rainfall. This is where the biggest storms are taking place during the monsoon as the moist air coming from the bay of Bengal collides with the first rapid rise in topography and relief (Bookhagen and Burbank, 2006). Although a significant amount of moisture makes it past the initial mountain front to collide with the second steep rise in topography and relief near the topographic divide and form the northern band of high magnitude rainfall, the storms there are not as intense. Instead a more frequent, more moderate rainfall regime is observed. This is reflected in the rainfall variability which begins to drop rapidly starting at the southern band moving north, and reaches a low right at or directly north of the northern band. Further into the orogen from the northern band, mean rainfall amount decreases rapidly while variability increases due to both the mean wet day frequency and the shape parameter. In general the plateau directly behind the mountains possesses moderate to high rainfall variability. Similarly, while the mean monsoonal rainfall amount and intensity decrease along strike from east to west, the rainfall variability increases. As a result, the magnitude of moderate to large storms remains constant along the strike of the orogen above 500 m elevation. These two trends point to monsoonal rainfall having a larger geomorphic impact in the north and the west of the orogen than the mean rainfall intensity and amount suggest. This demonstrates the potential importance of rainfall variability in understanding the relationship between erosion and climate.
As a result of the postulated influence of climate on landscape morphology and tec-tonic deformation rates, the erosional response of mountainous bedrock rivers to changes in climatic forcing has been the subject of intense research. However, due to the chal-lenges in upscaling daily climatic forcing to geological time, physically realistic models describing how rainfall drives fluvial erosion are lacking. We derive a theoretical frame-work for long-term fluvial erosion rates driven by realistic climate by integrating an established stochastic-mechanistic model of hydrology into a threshold-stochastic for-mulation of stream power. The hydrological theory provides equations for the daily streamflow distribution and variability as a function of climatic boundary conditions that are applicable across most of the observed range of streamflow regimes on Earth. The new parameters introduced are rooted firmly in established climatic and hydrolog-ical theory and are easily measured. We predict how fluvial erosion rates respond to changes in realistic climatic forcing, observing an anti-correlation between streamflow mean and variability, peak erosion rates for moderate climatic conditions, and an in-sensitivity to increasing mean streamflow above a certain point in many cases. We find that hydrological processes can have a significant influence on how erosive a particu-lar climatic forcing will be. Based on the generally unaccounted for role of hydrology, we conclude that the failure to find a clear dependence of long-term erosion rates on rainfall rates or steepness indices does not exclude the possibility that climate is an important control on erosion rates and landscape evolution.

introduction

Rivers play an important role in shaping Earth’s surface, especially in high relief, moun-tain environments; they have become a major focus of research in quantitative geo-morphology (e.g. Whipple, Kirby, and Brocklehurst, 1999; Whipple, 2009; Tucker and Hancock, 2010b), because they are one of the main links between climate and erosion. (e.g. Whipple and Tucker, 1999; Tucker and Bras, 2000; Crave and Davy, 2001; Burbank et al., 2003; DiBiase et al., 2010; DiBiase and Whipple, 2011; Lague, 2014). Mountainous bedrock rivers control landscape evolution in unglaciated mountain environments by steepening hill slopes through bedrock incision and transporting the resulting debris away. In doing so, they not only set the relief structure of a mountain, but also com-municate tectonic signals throughout the landscape (Whipple, Kirby, and Brocklehurst, 1999; Whipple, 2009). All of this is accomplished ultimately by the ability of flowing water to transport sediment. Therefore it is natural to conclude that climate, or more specifically, rainfall plays an important role in landscape evolution as the main source of river water. Gilbert, (1877) pointed this out more than 140 years ago. In the interven-ing time there have been a myriad of theoretical models postulating, and sometimes demonstrating, that climate should play an integral part in determining the form and rate of change of Earth’s surface. Some consider climate specifically in the context of a landscape dominated by fluvial erosion (e.g. Beaumont, Fullsack, and Hamilton, 1992; Whipple, Kirby, and Brocklehurst, 1999; Roe, Montgomery, and Hallet, 2002), and oth-ers in a more general sense (e.g. Dahlen and Suppe, 1988; Willett, 1999; Whipple and Meade, 2006).

Table of contents :

1 introduction 
1.1 Climate and Erosion
1.2 Short-term climate variability and long-term erosion rates
1.3 State of the art
1.3.1 Nonlinearity
1.3.2 General conclusions of nonlinear models
1.3.3 State dependency
1.4 Summary
1.4.1 Measurement techniques
1.4.2 Realistic models
1.4.3 General theory
2 rainfallvariability inthehimalayanorogen 
2.1 Abstract
2.2 Introduction
2.3 Data and Methods
2.3.1 Data
2.3.2 Statistical model
2.3.3 The applicability of the statistical model
2.3.4 Data fitting techniques
2.4 Results
2.4.1 Spatial distribution of the gamma shape parameter
2.4.2 Spatial distribution of mean wet day frequencies
2.4.3 Spatial distribution of mean storm intensities
2.4.4 Stationarity
2.4.5 Correlations between intensity, frequency and shape parameter
2.4.6 General along and across strike trends in variability
2.5 Discussion
2.5.1 Rainfall variability and erosion rates
2.5.2 Relevance to the Himalayan orogen
2.6 Conclusions
2.7 Acknowledgements
3 rainfall, hydrology and fluvial erosion efficiency 
3.1 Abstract
3.2 Introduction
3.3 The (eco)hydrological model
3.3.1 Rainfall
3.3.2 Soil moisture
3.3.3 Catchment-scale water balance and the streamflow ratio
3.3.4 Daily streamflow
3.4 Integration of hydrology into the stream power incision model
3.4.1 Channel hydraulics and distributions of daily erosion
3.4.2 Mean long-term erosion rates as a function of climate and ecohydrology
3.5 Results and Discussion
3.5.1 Daily streamflow variability and longterm erosion rate
3.5.2 Hydrologic and climatic controls on streamflow
3.5.3 Ecohydrology, climate and erosion
3.5.4 Outstanding issues and limitations
3.5.5 Significance and Future work
3.6 Conclusions
3.7 Appendix A: Distribution of daily streamflow when b = 2
3.8 Appendix B: Analytical approximations for C
3.9 Appendix C: Channel hydraulics
3.10 Appendix D: long-term erosion rate
3.10.1 pdf of daily erosion
3.10.2 long-term erosion rate
3.11 Appendix E: Importance of large floods
4 negative correlation between mean and variability of dis charge
4.1 Abstract
4.2 Introduction
4.3 Section 1: Estimating distributions of discharge from hydrographs
4.3.1 Theory
4.3.2 Results and discussion
4.4 Section 2: Negative correlation of mean and variability
4.4.1 Coefficient of variability
4.4.2 Covariation of daily runoff variability and mean
4.5 Results and Discussion
4.5.1 Monte Carlo simulations
4.5.2 Empirical nonlinear dependence of variability on mean runoff
4.5.3 The effect of the basin response time
4.5.4 Understanding controls on discharge variability
4.5.5 Comparison to data
4.6 Conclusion
5 process independent influence of climatic variability 
5.1 Abstract
5.2 Introduction
5.3 Results and Discussion
5.4 Appendix A: Threshold stream power model
5.5 Appendix B: Derivation of tool and cover transport law
5.6 Appendix C: Monte Carlo Simulation Setup

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