Global potential to increase crop production through water management in rainfed agriculture

Stefanie Rost¹, Dieter Gerten^1,4, Holger Hoff^1,2, Wolfgang Lucht¹, Malin Falkenmark³ and Johan Rockström^2,3

¹ Potsdam Institute for Climate Impact Research, Research Domain of Climate Impacts and Vulnerabilities, Telegraphenberg A62, 14473 Potsdam, Germany
² Stockholm Environment Institute (SEI), Kräftriket 2B, 106 91 Stockholm, Sweden
³ Stockholm Resilience Centre, Stockholm University, 106 91 Stockholm, Sweden

⁴ Author to whom any correspondence should be addressed

E-mail: gerten@pik-potsdam.de

Received 23 February 2009
Accepted 25 September 2009
Published 9 October 2009

Keywords:  food security, vapor shift, water harvesting, climate change, agriculture,  water resources

Contents

• 1. Introduction
• 2. Methods
  • 2.1. Model and data
  • 2.2. Simulation of water fluxes and water limitation to crop production
  • 2.3. Water management scenarios
  • 2.4. Computation of water requirements for crop production
  • 2.5. Model setup
• 3. Results and discussion
  • 3.1. Present water limitation of crop production
  • 3.2. Potential of water management strategies under present climate
  • 3.3. Future potential of water management under climate change only
  • 3.4. Future potential of water management under climate and CO₂ change
  • 3.5. Implications for food self-sufficiency
• 4. Conclusion
• Acknowledgments
• References

1. Introduction

On average, 1300 cubic meters per capita per year (m³ cap^–1 yr^–1) of fresh water are required to produce the food for a healthy diet of 3000 kcal cap^–1 day^–1 with a 20% meat share (Rockström et al 2007). This means that  > 8000 km³ yr^–1 of water are consumed (i.e. evapotranspired on rainfed and irrigated land) to feed the current world population (Rost et al 2008), and that an additional c.  5000 km³ yr^–1 will be required if the population rises to 10 billion in 2050 as suggested by the IPCC's SRES A2r scenario (Grübler et al 2007). A further increase in agricultural water requirements can be expected due to rising incomes and dietary changes towards more meat consumption (Liu et al 2008). Global climate change, elevated atmospheric CO₂ concentration, and altered incidence of pests, diseases and weeds will also noticeably impact future global food production (Easterling et al 2007). Whereas rising temperatures and regional soil moisture declines tend to negatively affect crop production (Parry et al 2004, Lobell and Field 2007, Battisti and Naylor 2009), direct effects of elevated CO₂ upon plants will probably increase production (Tubiello and Ewert 2002, Challinor and Wheeler 2008), albeit the magnitude of this effect is debatable (Long et al 2006, Tubiello et al 2007).

The availability of fresh water and arable land are approaching their limits. Already about 15 million km² of the land surface is covered by cropland (Ramankutty et al 2008), and about 16% of this area is equipped for irrigation (Siebert et al 2005). During the last century, the rate of increase in `blue' water withdrawals (from rivers, lakes, and aquifers) for irrigation and other purposes was higher than the growth rate of the world population (Shiklomanov 1998). A crucial question is to what extent cropland will have to be expanded in the future to guarantee sufficient food production for the growing world population, and to what extent the additional land and water requirements can be minimized through better management of existing cropland.

In an optimistic scenario, cropped area will increase by 9% and agricultural water withdrawals by 13% by the year 2050 (Molden 2007), which due to assumed increases in water use efficiency is notably lower than the highest expected increase in population by 67% by that time (SRES A2r). Nonetheless, a higher efficiency of blue water use or an expansion of irrigated areas is most likely insufficient to fulfill future food demand and to alleviate the still existing malnutrition of at least 850 million people (FAO 2006). In this context, it has to be noted that 60–70% of the world's current crop production is rainfed, i.e. it relies on the evapotranspiration of `green' water (precipitation stored in the soil; Falkenmark and Rockström 2004). Altogether more than 85% of water consumed in global agriculture is green (Rost et al 2008). Thus, increases in crop productivity are ultimately needed not only in irrigated but also in rainfed agriculture.

To meet the water and food needs of local communities, a new agricultural revolution has been proposed (IAASTD 2008). The plea is for fundamental changes in farming away from industrial, energy-intensive agriculture that strongly depends on synthetic fertilizers and pesticides toward small-scale and agro-ecological farming including low-cost water management options and indigenous methods. From a hydrological perspective there may indeed be enough rainfall to significantly increase yields even in semiarid regions without large-scale irrigation (Rockström et al 2007). Many of the low yields can be explained by the processes of unproductive soil evaporation, interception losses, deep percolation and surface runoff, which amount to losses of up to 70–85% of rainfall (Rockström 2000). Despite the success of a variety of soil and water management strategies to optimize the use of rainwater for crops in demonstration plots, there is still no widespread adoption of these methods (Pandey et al 2003, Faures and Santini 2008). We note the high number of irrigation tanks in regions such as southern India (Gunnell and Krishnamurthy 2003), but here we focus on rainfed cropland while assuming such tanks to be at least partially installed in irrigated areas and their contribution to be covered in our irrigation simulations. Using an optimistic scenario of water availability and management in which a theoretical maximum of 85% of total evapotranspiration from cropland and pasture was assumed to be available for plant transpiration and thus biomass production, Rockström et al (2009) suggested that without such improvements in water productivity, a horizontal cropland expansion by about 1000 Mha would be required to produce the food for  > 10 billion people. This renders water management in rainfed agriculture a very important strategy to increase food production globally (Molden 2007), but the large-scale potential of these methods has not yet been quantified systematically.

While only a few field experiments have investigated local successes with these techniques, ecohydrological models can contribute significantly to understand their potential at larger scales (Kahinda et al 2007) and help identify suitable management options (Prinz et al 1998, Mbilinyi et al 2007). The present study is the first to quantify at a global scale yet geographically explicitly the present water limitation of crop production and the potentials of the above water management strategies for improving crop production. Analysis is based on simulations with the LPJmL dynamic global vegetation and water balance model (Bondeau et al 2007, Rost et al 2008), in which we represented several management strategies (reducing soil evaporation to achieve a vapor shift, and harvesting runoff for use during dry spells; see section 2).

2. Methods

2.1. Model and data

The LPJmL model computes the growth and productivity of natural and agricultural vegetation coupled with biogeochemical and water fluxes on 0.5° spatial and daily temporal resolution (Bondeau et al 2007). Key ecosystem processes such as photosynthesis, respiration, carbon allocation, evapotranspiration and its individual components, soil moisture, and drought stress are simulated in a process-based way so that they respond dynamically to climatic variations. The model considers 9 natural plant functional types and 12 crop functional types (CFTs, either irrigated or rainfed, corresponding to the world's most important field crops: temperate/tropical cereals, temperate/tropical roots, rice, maize, pulses, sunflower, soybean, groundnuts, rapeseed, and pasture).

The annual fractions of cropland and pasture per grid cell were prescribed for the years 1901–1992 using the datasets by Ramankutty and Foley (1999) and Klein Goldewijk and Battjes (1997); for the period up to the year 2000 the coverage was assumed to follow the trend of the preceding 20 years. Future land use and irrigation areas were held constant at the values of 2000. The distribution of CFTs within a grid cell was taken for the year 1990 from Leff et al (2004) and interpolated backwards and forwards assuming constant relative proportions. Irrigation was assumed to occur on the fractions of a grid cell equipped for irrigation (Siebert et al 2007). Irrigation water requirements were derived from the soil water deficit and from country-specific irrigation efficiencies that result from differences in dominant irrigation types among countries. We assumed that these gross requirements can always be met (Rost et al 2008). The growing seasons of the CFTs were simulated to be initiated by a sowing date dependent on temperature or precipitation and to end with harvest when maturity is reached. The intensity of present agricultural management is represented by assuming the CFT's maximum leaf area index to be higher in intensively managed regions, as derived from data on fertilizer input (Bondeau et al 2007). We assume that management intensity will not change in the future. LPJmL is validated against biogeochemical and hydrological observations including crop yields, river discharge, soil moisture, and rainfed and irrigated water consumption (Gerten et al 2004, Bondeau et al 2007, Rost et al 2008, Biemans et al 2009).

2.2. Simulation of water fluxes and water limitation to crop production

Soil water content W is updated daily for each CFT within each grid cell, following the balance between the amount of infiltrated water (precipitation P, snowmelt M, and irrigation water `Irr' minus interception loss from leaves E_I) and that removed from the soil through surface and subsurface runoff R (generated from excess water above field capacity), soil evaporation E_S, and plant transpiration E_T (all in mm day^–1) (Gerten et al 2004):

E_S depends on potential evapotranspiration determined by the Priestley–Taylor method (E_pot times the Priestley–Taylor coefficient α = 1.32) and the relative moisture in the top 20 cm of the soil W_r20. It is computed for the fractions of bare soil of cropland as given by the coverage with vegetation f_v and the daily status of crop-specific leaf phenology `phen':

Crop transpiration E_T is determined for each CFT as the minimum of atmospheric demand D and water supply S:

S is regulated by soil moisture and plant hydraulic traits; when the soil is saturated, E_T reaches a maximum E_max (5–8 mm day^–1) and declines linearly with W weighted by the CFT-specific fraction of roots in each soil layer. D (mm day^–1) is a function of E_pot times a maximum Priestley–Taylor coefficient (α_m = 1.391) and the potential, water-unlimited canopy conductance g_pot (mm s^–1):

where f_wet is the fraction of the day that the canopy is wet, and g_m is a scaling coefficient (3.26 mm s^–1). g_pot is calculated from the photosynthesis rate and ambient CO₂ concentration (details in Gerten et al 2007). E_I is determined as:

where LAI is the leaf area index and i a factor accounting for vegetation type and rainfall regime.

Total evapotranspiration E, i.e. crop water consumption, is computed as the sum of E_T, E_S and E_I. Net primary production NPP (Gt dry matter) is defined as gross primary production less autotrophic respiration. The former is estimated daily from the fractional coverage of a CFT in a grid cell, leaf phenology and weather conditions. The latter is the compound of maintenance respiration (calculated based on the size of the living tissue pools, their assigned C:N ratios and weather data) and growth respiration (the cost of producing new tissues, calculated as a fraction of NPP) (Sitch et al 2003). The degree of water limitation of crop production was computed as the ratio between NPP and the production that could theoretically be reached if E_T was always at its maximum (NPP^*). The CO₂ effect upon E_S and NPP includes both physiological and structural vegetation responses (Gerten et al 2007).

2.3. Water management scenarios

Several simulations were performed to estimate crop NPP under different conditions (table 1):

We analyzed the differences between these model runs to estimate the effect on NPP of irrigation (INO minus BAS), and of the different management options (VS, RH, and VS + RH minus BAS) for the present, under SRES-A2 climate change (CC), and under climate change with CO₂ increase (CC + CO₂), respectively. Estimates for the future were represented as averages of the results under three climate scenarios (see below). We also used the spatial distribution of the revised high population growth scenario A2r for the year 2050 (Grübler et al 2007) to relate the effects of the water management options to global food demand. Since the A2r population scenario is at the upper end of the SRES range (10.2 billion in 2050) and because food and water demand are strongly affected by the population number, we also analyzed results for medium (B2) and low fertility (B1) population scenarios that show, respectively, a world population of 9.4 and 8.7 billion in 2050. To reduce the number of simulations, we did not compute the climate change impacts under B1 and B2 emissions. We expect the adverse effects of climate change (NPP reductions) reported below to be less severe in these scenarios, but at the same time the beneficial effects of rising CO₂ concentration would also be less pronounced.

2.4. Computation of water requirements for crop production

To estimate the effect of the water management options on securing global crop production, we followed the approach of Rockström et al (2007), assuming that as a global average 1300 m³ cap^–1 yr^–1 of blue and green water are required under present conditions to produce 3000 kcal cap^–1 day^–1 for a healthy diet (20% meat share). Actually, however, this water requirement differs among regions. To roughly account for these differences, we distributed the global average among countries according to their water productivity (i.e. the water consumed per unit of NPP), such that regions with low water productivity got values higher than 1300 m³ cap^–1 yr^–1 (e.g. around 2000 m³ cap^–1 yr^–1 in many countries of sub-Saharan Africa), and vice versa. (Note that these values refer to the normative goal of a diet of 3000 kcal; for producing today's actual diets, water requirements are significantly lower in many regions of the South.) Countries in which the water availability per capita falls below the thus computed value are regarded as water-stressed. Analogously, the water stress threshold changes under each of the scenarios computed here, following the individual changes in NPP and E and, thus, water productivity.

2.5. Model setup

LPJmL was driven by an updated version of the CRU TS2.1 climate dataset (0.5° spatial resolution) of monthly air temperatures, precipitation, and cloud cover (Österle et al 2003) for the period 1901–2000, preceded by a 1000 yr spinup during which the climate of 1901–1930 was repeated. CO₂ concentrations were as in Rost et al (2008). For the simulations under future climate, starting from the same equilibrium, the model was driven by the output from three different climate models (HadCM3, ECHAM5/MPI-OM, and CCSM3 (Randall et al 2007) disaggregated to 0.5° resolution and normalized to observed climate for 1961–1990) under the A2 emissions trajectory for the period 1901–2070. Sub-monthly weather variability was derived by statistical procedures (Gerten et al 2004). All results were computed at daily time steps for each CFT separately, but are presented as annual averages over all CFTs for 1971–2000 and 2041–2070, respectively.

3. Results and discussion

3.1. Present water limitation of crop production

We estimated a maximum achievable global crop NPP^* of 23.5 Gt in the absence of water limitation (OPT simulation). Only about half of this theoretical potential would be realized without any irrigation (11.4 Gt INO simulation; table 2). This reflects the fact that NPP is strongly water-limited in many regions of the world (figures 1 and 2), and suggests a high potential for increasing NPP through proper management of blue and green water in water-scarce regions. In contrast, in tropical, marine west coast, humid subtropical, and subarctic climate zones NPP (INO) is already close to NPP^* (OPT; figure 1), which implies low potentials for increasing NPP. Thus, as confirmed by earlier studies (Gregory et al 2000, Hatfield et al 2001, Pandey et al 2003), the likely success of soil and/or water management differs among regions, mainly due to differences in climate.

Through present irrigation, an increase in global crop NPP by 17% up to 13.3 Gt is already achieved (INO versus BAS; table 2), mainly in regions with large areas equipped for irrigation (figure 3(a)). On irrigated land, crop NPP would be lower globally by 69% without irrigation, i.e. green water contributes about a third to crop production there. Comparable estimates were found by Siebert and Döll (2009) who suggested a decrease in crop production by 17.8% and on irrigated land by 54% in the absence of irrigation. Previous estimates of global crop NPP (rainfed and irrigated) were of the order of 12 Gt (Rojstaczer et al 2001, Monfreda et al 2008). Our slightly higher estimate of 13.3 Gt might be explained by the larger area of agricultural land according to Ramankutty and Foley (1999) used in our study, and by our assumption of optimum irrigation on equipped areas.

3.2. Potential of water management strategies under present climate

The potential to increase global crop NPP by reducing soil evaporation is estimated to be 0.31 Gt (2.3%) under low (VS10) and 3.3 Gt (24.7%) under intense (VS85) vapor shift management (figure 2, table 2). The VS25 simulation suggests a high potential ( > 20%) mainly in semiarid regions such as the midwestern United States, parts of South America, the Sahel, southern Africa, Central Asia, and south-eastern Australia (figure 3(b)). Even on irrigated areas, vapor shift management can reduce water requirements by 5% (147 km³ yr^–1) under low (VS10) and by 83% (1499 km³ yr^–1) under intense management (VS85) (data not shown). Together with improvements in irrigation efficiency not studied here—which may, together with expansions of irrigated areas, amount to about 800 km³ yr^–1 globally (Falkenmark and Rockström 2004)—the thus saved blue water could be used for purposes other than agriculture, or for expanding irrigated areas to currently rainfed cropland.

Runoff harvesting was simulated to increase global NPP by 0.57 Gt (4.2%) under low (RH10) and by 4.1 Gt (30.7%) under high storage (RH85) (figure 2, table 2). In the RH25 simulation tropical wet and dry regions like Central America, parts of Brazil, eastern and southern Africa in particular show large potentials ( > 20%) for increasing NPP (figure 3(c)). These magnitudes are comparable to those in Wisser et al (2009), who found that overall cereal production in low-yield regions could be increased by about 15% for a medium variant of small reservoir construction, with highest potentials in Africa and Asia.

Regional differences in the potential to increase NPP between the VS and RH scenario can be explained primarily by climatic differences. Whereas vapor shift management seems to be more efficient in dry regions with low R, runoff harvesting was found to be more effective if substantial amounts of R can be stored outside of the growing period. In addition, crop management practices depend strongly on soil texture, with improvements in crop production being most promising on soils with high water holding capacity (Gregory et al 2000, Ngigi 2003).

The combination of both management strategies (VS + RH) results in an increase in NPP by 0.92 Gt (6.9%) in VS10 + RH10 and by 7.05 Gt (52.8%) in VS85 + RH85 (figure 2, table 2). Pronounced increases in NPP can be achieved mainly in regions where present NPP reaches less than 10% of NPP^* (figures 1 and 3(d)), which agrees well with recent findings by Faures and Santini (2008) for sub-Saharan Africa.

The aim of this study is to demonstrate the range of NPP increases given different levels of water management intensity without deeming a particular scenario to be most realistic. Nonetheless, we think that a global implementation of the 85% scenario is clearly unrealistic (though physically possible), and even the 50% scenario is very ambitious, while the 25% scenarios seem to be a possible option: Falkenmark and Rockström (2004) suggested an improvement of green water productivity by 1530 km³ yr^–1 through a combination of various techniques, which is comparable to our estimates of a water saving of about 1650 km³ yr^–1 and an associated NPP increase of 2.5 Gt in the V 25 + RH25 scenario. The actual potentials and mixtures of VS and RH methods will certainly differ between regions and require investigation in dedicated regional studies. Note that we did not account for evaporation losses of harvested water nor for the possibility that water harvesting reduces blue water availability in downstream locations, which may lead to lower overall water savings within river basins and globally, especially in the scenarios of high-level management (see also Wisser et al (2009)).

3.3. Future potential of water management under climate change only

The effect of climate change only (CC) is a decrease in NPP^*, and in NPP under all water management simulations, compared to the present situation (decrease by 9% in BAS by 2050, figure 2). This is due primarily to regional precipitation declines and generally higher temperatures that lead to higher crop water limitation and more direct crop damage. Similar results were found by Parry et al (2004), who estimated a climate change-driven decrease in global crop production of 10% by 2080. Under climate change only, irrigation on areas currently equipped for irrigation would increase the otherwise rather low crop production by 19.5% from 10.2 to 12.2 Gt (figure 2, table 2), i.e. future irrigation will continue to play an important role on present land equipped for irrigation.

The potential to increase NPP by the other water management strategies ranges from 0.26 Gt (2%) in VS10 to 3.9 Gt (32%) in RH85, and from 0.82 Gt (7%) in VS10 + RH10 to 6.5 Gt (54%) in VS85 + RS85 (figure 2, table 2). The absolute rise in crop production that can be attained was found to be slightly lower under future climate than at present, due to globally lower amounts of water stored in the soil and/or harvestable (data not shown). The relative potential to upgrade NPP by water management, however, is of a similar magnitude under future climate change as at present.

Note that either a vapor shift of about 50% or a water harvesting level of 25% would have to be achieved if the adverse consequences of global climate change were to be counterbalanced, i.e. if the current crop production levels were to be held. However, crop production will have to be increased significantly in the future, with the actual amount of this increase depending on population development (figure 2).

3.4. Future potential of water management under climate and CO₂ change

Other than under CC, the global joint effect of climate and CO₂ change (CC + CO₂) is an increase in NPP^* and NPP, respectively (figure 2). The isolated effect of rising CO₂ concentration is an increase in global crop NPP by 28% by the 2050s, which is somewhat higher than suggested by forest FACE experiments (about 23% at a CO₂ level of about 500 ppm; Norby et al 2005). Irrigation was simulated to increase NPP by 16% to 17 Gt. The potential to increase NPP by the diverse water management options ranges from 0.26 Gt (1.5%) in VS10 to 3.7 Gt (22%) in RH85, and from 0.78 Gt (5%) in VS10 + RH10 to 6.3 Gt (37%) in VS85 + RH85 (figure 2, table 2).

It is noteworthy that the CO₂ effect more than offsets the global NPP decrease induced by climate change (figure 2), as was also shown elsewhere (Tubiello and Ewert 2002, Gerten et al 2007, Challinor and Wheeler 2008). At any rate, the beneficial CO₂ effect should be interpreted as a maximum effect, the realization of which represents a major challenge due to the complex interactions between photosynthesis and yields, and limitations to crop growth through low nutrient availability, soil degradation, pests, weeds, and diseases (Ainsworth et al 2008). These co-limitations, and especially N and P limitations, are not explicitly considered in our model, but, as Hickler et al (2008) concludes, current observational evidence suggests that the projected strong impacts of increasing CO₂ on NPP are realistic if other limitations are absent. Please also note that we assumed the present management intensity to remain constant in the future, meaning that absolute NPP may be higher (or lower) in the non- CO₂ case. In this context we recognize that the computed E_T:E ratio—which is influenced by management and which, in turn, influences the outcome of the VS simulations in particular—depends on the daily phenological status of the CFTs. Preliminary studies based on more recent land use data and a refined parameterization of agricultural management show, however, that the present results are still valid (Fader et al 2009).

3.5. Implications for food self-sufficiency

As a measure of a country's food self-sufficiency, we calculated the number of people that will be water-stressed by 2050 (see section 2.4 for computation of this water stress), exemplary for the A2 scenario and under consideration of a successfully implemented VS25 + RH25 scenario (figure 4). Under these conditions, water availability will still be sufficient to fulfill the food demand in most developed and water-rich countries. In contrast, future food self-sufficiency remains unachievable for many countries especially in North Africa, the Middle East and South Asia, such that the global number of people living in countries without enough blue and green water (on present cropland) for producing a healthy diet increases from the present 2.3 billion to about 6 billion by the 2050s (data not shown). The associated additional water demand would be about 4500 km³ yr^–1 (Gerten and Rost 2009).

Import of agricultural products and the embedded virtual water eases countries' water shortages (Chapagain et al 2006). However, even if the above assumption of self-sufficiency was discarded by assuming that virtual water trade fully balances the regional differences in blue and green water consumption on present irrigated and rainfed cropland, a water gap of about 2800 km³ yr^–1 would remain at a population size of 10.2 billion (data not shown). Since water savings in irrigation probably will not be able to fill this gap (see above), continuing global expansion of cropland at a rate similar to the present rate appears to be inevitable (Rockström et al 2009). It is questionable, however, whether this land is actually available and what tradeoffs with other land uses will have to be made. We aim to explore these land limitations and tradeoffs in future studies, but a qualitative comparison with recent results based on our model already indicates that tough choices on land and water use will have to be made if the global demand for sustainable bioenergy production is taken into account (WBGU 2009).

The above estimates are based on the assumption that present diet composition will remain stable in the future, which is unlikely, however, given the recent trends toward increased meat consumption (which imply higher water consumption) and other prospective lifestyle changes. Also, our related assumption that 1300 m³ cap^–1 yr^–1 of water are required for producing a healthy diet of 3000 kcal cap^–1 day^–1 represents a global average, while the actual water requirements differ among regions depending, for example, on the virtual water content of individual crops. In addition, water demand crucially depends not only on lifestyle choices but also on population size, and we find that there will be future water gap even under the low fertility scenarios (figure 2; see also Brichieri-Colombi (2008)). Forthcoming analyses will have to account comprehensively for a wider range of climate, population, and lifestyle scenarios as well as for the concrete potential of virtual water trade.

4. Conclusion

This study represents the first spatially explicit quantification of water limitations in global crop production and the potential of different water management strategies to upgrade crop growth, under both present and projected future climate conditions. A key finding is that an area-wide combination of water harvesting and vapor shift techniques could increase global crop production by almost 20%. This result underscores quantitatively and scales up to a global scale previous local observations of high potentials for rising crop production and reducing risk of crop failure by proper soil and water management in rainfed agriculture (Rockström 2003, Ali and Talukder 2008, Faures and Santini 2008). But the study also indicates that even the most ambitious and large-scale water management efforts on present cropland will not be sufficient to guarantee the food demands of a growing world population. This evidence poses crucial questions about tradeoffs between future land and water use for irrigated and rainfed agriculture, natural ecosystems and bioenergy. It furthermore highlights the need for exploring and combining all options of more efficient irrigation and/or expansion of irrigated agriculture, of plant breeding and genetic development, and of more effective virtual water trade.

Acknowledgments

This study was funded by BMBF, DFG, and EC (ENSEMBLES, GOCE-CT-2003-505539), and it contributes to the scientific framework of the GWSP. We thank Alberte Bondeau, Marianela Fader, Jens Heinke, Ursula Heyder, Christoph Müller, Sibyll Schaphoff, and Stefan Siebert for discussions, Werner von Bloh for technical support, and an anonymous reviewer for very constructive comments.

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