Here we present a newly developed ice core gas-phase proxy that directly
samples a component of the large-scale atmospheric circulation:
synoptic-scale pressure variability. Surface pressure changes weakly disrupt gravitational isotopic settling in the firn layer, which is recorded in krypton-86 excess (
We present a
We emphasize that
Proxy records from around the globe show strong evidence for past changes in Earth's atmospheric circulation and hydrological cycle that often far exceed those seen in the relatively short instrumental period.
For example, low-latitude records of riverine discharge captured in ocean sediments (Peterson et al., 2000) and isotopic composition of meteoric water captured in dripstone calcite (Cheng et al., 2016) suggest large variations in tropical hydrology and monsoon strength, commonly interpreted as meridional migrations of the Intertropical Convergence Zone or ITCZ (Chiang and Friedman, 2012; Schneider et al., 2014). Such ITCZ movement is seen in response to insolation changes linked to planetary orbit (Cruz et al., 2005) as well as in response to the abrupt millennial-scale Dansgaard–Oeschger (D–O) and Heinrich cycles of the North Atlantic (Kanner et al., 2012; Wang et al., 2001); the organizing principle is that the ITCZ follows the thermal Equator and therefore migrates towards the warmer (or warming) hemisphere (Broccoli et al., 2006; Chiang and Bitz, 2005).
As a second example, the intensity of the El Niño–Southern Oscillation (ENSO), the dominant mode of global interannual climate variability, has changed through time. A variety of proxy data suggest that ENSO activity in the 20th century was much stronger than in preceding centuries (Emile-Geay et al., 2015; Fowler et al., 2012; Gergis and Fowler, 2009; Thompson et al., 2013). The vast majority of data and model studies suggest weakened ENSO strength in the middle and early Holocene, likely in response to stronger orbitally driven NH summer insolation at that time (Braconnot et al., 2012; Cane, 2005; Clement et al., 2000; Driscoll et al., 2014; Koutavas et al., 2006; Liu et al., 2000, 2014; Moy et al., 2002; Rein et al., 2005; Tudhope et al., 2001; Zheng et al., 2008); yet other studies suggest there may not be such a clear trend and simply more variability (Cobb et al., 2013). Intensification of ENSO (or perhaps a more El Niño-like mean state) may have occurred during the North Atlantic cold phases of the abrupt D–O and Heinrich cycles (Braconnot et al., 2012; Merkel et al., 2010; Stott et al., 2002; Timmermann et al., 2007). Overall, understanding past and future ENSO variability remains extremely challenging (Cai et al., 2015).
As a last example, the strength and meridional position of the Southern Hemisphere westerlies (SHWs) are thought to have changed in the past, which, via Southern Ocean wind-driven upwelling, has potential implications for the global overturning circulation (Marshall and Speer, 2012) and for carbon storage in the abyssal ocean (Anderson et al., 2009; Russell et al., 2006; Toggweiler et al., 2006). The SHWs are thought to have been shifted equatorward (Kohfeld et al., 2013) during the Last Glacial Maximum (LGM), a shift on which climate models disagree (Rojas et al., 2009; Sime et al., 2013). During the abrupt D—O and Heinrich cycles, the SHWs move in parallel with the aforementioned migrations of the ITCZ in both data (Buizert et al., 2018; Marino et al., 2013; Markle et al., 2017) and models (Lee et al., 2011; Pedro et al., 2018; Rind et al., 2001).
As these examples clearly illustrate, evidence of past changes to the large-scale atmospheric circulation is widespread. However, proxy evidence
of such past changes is typically indirect – for example via isotopes in
precipitation, sea surface temperature, ocean frontal positions, windblown
dust, or ocean upwelling – complicating their interpretation. Here we
present a newly developed noble-gas-based ice core proxy, Kr-86 excess (
Here we provide the first complete description of this new proxy. We validate and calibrate
The upper 50–100 m of the ice sheet accumulation zone consists of firn, the
unconsolidated intermediate stage between snow and ice. An interconnected pore network exists within the firn, in which gas transport is dominated by
molecular diffusion (Schwander et al., 1993). Diffusion in this stagnant air
column results in gravitational enrichment in heavy gas isotopic ratios such
as
Besides molecular diffusion, firn air is mixed and transported via three other processes: downward advection with the sinking ice matrix, convective
mixing (used in the firn air literature as an umbrella term to denote
vigorous air exchange with the atmosphere via, e.g., wind pumping and seasonal
convection), and dispersive mixing. These last three transport processes are
all driven by large-scale air movement that does not distinguish between
isotopologues, and we refer to them collectively as macroscopic air movement. Of particular interest for our proxy is dispersive mixing, which is driven by surface pressure variations. When a low-pressure (high-pressure) system moves into the site, firn air at all depth levels is forced upwards (downwards) to reach hydrostatic equilibrium with the atmosphere – a process called barometric pumping. One can think of the firn layer “breathing” in and out in response to a rising and falling barometer, respectively. Because firn has a finite dispersivity (Schwander et al., 1988), this air movement mixes the interstitial firn air. Note that an upward air movement also exists in the firn column relative to the overall downward advection of the ice, which is caused by the slow reduction of porosity with depth (Rommelaere et al., 1997). This upward airflow due to gradual pore closure (around 10
Any type of macroscopic air movement disturbs the gravitational settling,
reducing isotopic enrichment below
In the (theoretical) case of full gravitational equilibrium (and no gas loss
or thermal fractionation),
Kawamura et al. (2013) first describe this gravitational disequilibrium (or
kinetic) fractionation effect at the Megadunes site (Severinghaus et al.,
2010), where deep firn cracking leads to a 23 m thick convective zone. They
suggest that the isotopic disequilibrium can be used to estimate past convective zone thickness. We show here that sites with small convective
zones can nevertheless have very negative
The principle behind
Idealized firn air transport model experiments of
The ratio of macroscopic over diffusive transport is expressed via the
dimensionless Péclet number, given here for advection and dispersion:
In a last idealized experiment, we keep the fraction of dispersion fixed at
Note that these highly idealized experiments assume dispersive mixing to be a fixed fraction of total transport throughout the firn column, equivalent to a constant Péclet number in the diffusive zone (a convective zone is absent in these simulations). In reality, the Péclet number varies greatly on all spatial scales. On the macroscopic scale (
In this study we use ice samples from 11 ice cores drilled in Antarctica and 1 in Greenland. The Antarctic sites are the West Antarctic Ice Sheet (WAIS) Divide core (WDC06A, or WDC), Siple Dome (SDM), James Ross Island (JRI), Bruce Plateau (BRP), Law Dome DE08, Law Dome DE08-OH, Law Dome DSSW20K, Roosevelt Island Climate Evolution (RICE), Dome Fuji (DF), EPICA (European Project for Ice Coring in Antarctica) Dome C (EDC), and South Pole Ice Core (SPC14, or SP). Ice core locations in Antarctica are shown in Fig. 2a. In Greenland, we use samples from the Greenland Ice Sheet Project 2 (GISP2).
Calibrating Kr-86 excess.
We shall refer to late Holocene data from these sites as the calibration dataset, analogous to a core-top dataset in the sediment coring literature.
Site characteristics, coordinates, and number of samples included in the
calibration dataset are given in Table 1. The DE08-OH site is a recent revisit of the Law Dome DE08 site. The DE08-OH core was measured at
sub-annual resolution to understand centimeter-scale
Ice core sites used in this study, with
We broadly follow analytical procedures described elsewhere (Bereiter et al., 2018a, b; Headly and Severinghaus, 2007; Severinghaus et al., 2003). In short, an 800 g ice sample, its edges trimmed with a band saw to expose fresh surfaces, is placed in a chilled vacuum flask that is then evacuated for 20 min using a turbomolecular pump. Air is extracted from the ice by melting the sample while stirring vigorously with a magnetic stir bar, led through a water trap, and cryogenically trapped in a dip tube immersed in liquid He. Next, the sample is split into two unequal fractions. The smaller fraction (about 2 % of total air) is analyzed for
All calibration (core-top) data were measured using Method 2 as described by Bereiter et al. (2018a), with a longer equilibration time during the splitting step than used in that study to improve isotopic equilibration between the fractions. The exception is the DE08-OH site, where the ice sample (rather than the extracted gas sample) was split into two fractions – the advantage of this approach is that it does not require a gas splitting step that is time-consuming and may fractionate the isotopes; the downside is that the samples may have slightly different isotopic composition due to the stochastic nature of bubble trapping and the different gas loss histories of the ice pieces.
Measurements of the WDC downcore dataset were performed over five separate
measurement campaigns that occurred in February–April 2014, February–April 2015, August 2015, August 2020, and August 2021. The first three campaigns are described by Bereiter et al. (2018b), in which the
All samples were analyzed at the Scripps Institution of Oceanography, USA, with
the exception of the EDC samples, which were analyzed at University of Bern,
Switzerland (Baggenstos et al., 2019). Some of the EDC samples analyzed had
clear evidence of drill liquid contamination, which acts to artifactually
lower
The
Our study includes two ice cores from the Antarctic Peninsula: BRP (two ice
samples) and JRI (five ice samples). Measured
The
Kr-86 excess is sensitive to air movement (both upward and downward), which
in turn is controlled by the magnitude of relative air pressure change. Let
Present-day Antarctica has a wide range of
Calibrating Kr-86 excess.
The
Notably, there is a large spread in
Both theoretical considerations and observations thus suggest
Our interpretation of
Currently, 1-D and 2-D firn air transport model simulations underestimate the magnitude of the
Another likely explanation for the model–data mismatch is that certain critical sub-grid processes (such as the aforementioned pore size dependence
of the Péclet number) are not adequately represented in these models.
Barometric pumping may further actively shape the pore network through the
movement of water vapor, thereby keeping certain preferred pathways connected and open below the density at which percolation theory would predict their closure (Schaller et al., 2017). The fate of a pore restriction is determined by the balance between the hydrostatic pressure (that acts to close it) and vapor movement away from its convex surfaces (that acts to keep it open); we speculate that barometric Darcy airflow keeps high-flow channels connected longer by eroding convex surfaces. This enhances the complexity (and therefore dispersivity) of the deep firn pore network and possibly creates a nonlinear
Firn models predict that, after correcting for thermal fractionation, the
deviation from gravitational equilibrium for the elemental ratios (such as
Measurements on firn air samples, where available, suggest a smaller
Gas loss and thermal corrections are critical to the interpretation of
Besides the actual thermal gradients in the firn, the isotopic composition
may also be impacted by seasonal rectifier effects. If the firn air transport properties differ between the seasons (for example due to thermal contraction cracks, convective instabilities, or seasonality in wind pumping), this can result in a thermal fractionation of isotopic ratios in the absence of a thermal gradient
For the WDC, DSS, and GISP2 sites we obtain
For the Law Dome DE08-OH site we observe large (5-fold) sub-annual variations in
Another puzzling observation is the positive
The
The calibration of the
The observations presented in this section clearly highlight the fundamental
shortcomings of our current understanding of firn air transport, hinting at the existence of complex interactions, presumably at the pore scale, that are not being represented. Percolation theory finds that near the critical point (presumably the lock-in depth) a network becomes fractal in its nature; we suggest that this fractal nature of the pore network likely contributes to nonlinear pore-scale interactions that give rise to the
In this section we investigate the large-scale patterns of climate variability in the Southern Hemisphere that could affect
We use ERA-Interim reanalysis data for the 1979–2017 period (Dee et al., 2011) to evaluate the present-day controls on synoptic-scale pressure
variability in Antarctica. Kr-86 excess in an ice core sample averages over
several years of pressure variability, and therefore we focus on annual mean
correlation in our analysis. The annual mean
At all Antarctic sites investigated, a similar pattern exists; four representative locations are shown in Fig. 4, where we regress the zonal
wind in the lower (850 hPa, color shading) and upper troposphere (200 hPa,
contours) onto our surface pressure variability parameter
Zonal wind speed at 850 hPa (color shading, see scale bar) and 200 hPa (2 m s
Pressure variability at WDC is furthermore correlated with the strength of the Pacific subtropical jet (STJ) aloft (solid contour lines centered around
30
Next, we investigate how the well-known patterns of large-scale atmospheric
variability, such as SAM and ENSO, impact pressure variability in Antarctica. Figure 5 shows the correlation of
Modes of SH extratropical atmospheric variability and their link to synoptic-scale surface pressure variability in Antarctica.
Pearson correlation between
Globally, annual mean
The dominant mode of atmospheric variability in the SH extratropics is the
southern annular mode, representing the vacillation of atmospheric mass
between the middle and high latitudes (Thompson and Wallace, 2000). Figure 5b
shows 500 hPa geopotential height (
The second mode of SH extratropical variability is the Pacific–South American Mode 1 (PSA1), which reflects a Rossby wave response to sea surface temperature (SST) anomalies over the central and eastern equatorial Pacific
(Mo and Paegle, 2001) and is therefore closely linked to ENSO on interannual timescales (we find a correlation of
Next, we consider anomalies in sea ice area and extent (Parkinson and Cavalieri, 2012). We focus on the Ross and Amundsen–Bellingshausen seas where impacts on the WAIS Divide may be expected. At the 90 % confidence level we do not find significant correlations with sea ice area or extent at most core locations (Table 2). Correlations with sea ice extent are (even) weaker than those for sea ice area and consequently not shown. We performed a lead–lag study of the correlations between
Overall, we find that synoptic activity at WAIS Divide, the site of most
interest here, is controlled by the position and/or strength of the storm tracks at the southern edge of the SPJ in the Pacific sector of the
Southern Ocean (Ross, Amundsen, and Bellingshausen seas), with little
sensitivity to the SPJ behavior in the other sectors. Owing to its remote
southern location, WDC is only weakly impacted by the commonly defined
large-scale modes of atmospheric variability. Most notably, WDC has a modest
influence from the tropical Pacific climate, as shown by a correlation
around
The WAIS Divide downcore
Figure 6 compares
WAIS Divide Kr-86 excess records through the last deglaciation.
Visual inspection suggests that campaigns 1 and 2 agree well for both
We combine data from the first two campaigns and evaluate their offset to
data from the other three campaigns again using the linear interpolation
method. For campaigns 3, 4, and 5 we find an offset of
In the remainder of this paper we will interpret the combined data from
campaigns 1 and 2, but with the caveat that there is a persistent offset
with later campaigns. However, the features we interpret are corroborated by
the later campaigns if one takes the offset into account. To aid interpretation of the data, we apply a Gaussian smoothing spline with a
smoothing filter width that varies depending on the data density (from
250-year width in the deglaciation itself for which the data density is high to 1750 years in the Holocene and LGM for which data density is low). To estimate the uncertainty in the smoothing spline we use a Monte Carlo approach that considers uncertainty in following: (1) the gas loss correction by randomly sampling The Holocene shows a trend towards increasingly negative The most pronounced change occurs at the Younger Dryas (YD)–Holocene transition, where During the Last Glacial Maximum (LGM), WDC synoptic activity was perhaps slightly weaker than at present, but not significantly so ( The deglaciation itself has enhanced synoptic activity, in particular during the two North Atlantic cold stages Heinrich Stadial 1 (HS1) and the YD as highlighted with yellow bars in Figs. 6 and 7. Synoptic activity during these periods is enhanced relative to the adjacent LGM and early Holocene yet comparable to today. This feature is seen in campaigns 1 and 2 as well as in 4 and 5 for the transition into the Holocene.
Below we will interpret the deglacial WDC
Climate records through the last deglaciation with the Heinrich Stadial 1 (HS1) and Younger Dryas (YD) North Atlantic cold periods marked in yellow.
Local summer solstice insolation in Antarctica increases through the Holocene, with the highest values in the late Holocene. This may impact
The scatter in the late Holocene WDC
In the present day, synoptic-scale pressure variability at WAIS Divide is
correlated with zonal wind strength along the southern margin of the SPJ
(Sect. 4). In our interpretation, a more negative
The main features of the deglacial WDC
Changes in mean ITCZ position have a strong influence on the structure and strength of the SH jets. During periods when the NH is relatively cold (such as D–O stadials or periods with negative orbital precession index) the ITCZ is displaced southward and the SH Hadley cell is weakened, thereby also weakening the SH upper-tropospheric subtropical jet (Ceppi et al., 2013; Chiang et al., 2014). The reverse is also true, with the ITCZ shifted northward during NH warmth, associated with a strengthening of the SH Hadley cell and STJ. In a range of model simulations (Ceppi et al., 2013; Lee and Kim, 2003; Lee et al., 2011; Pedro et al., 2018) the weakening of the SH STJ (as during NH cold) is furthermore accompanied by a strengthening and/or southward shift of the SPJ as well as the eddy-driven jet and SH westerly winds. Recently, ice core observations have confirmed that in-phase shifts in the position of the SHWs occur during the D–O cycle in parallel to those of the ITCZ (Buizert et al., 2018; Markle et al., 2017). Marine records of fluvial sediment runoff off the Chilean coast suggest precession-phased movement of the South Pacific SPJ, again in parallel to the ITCZ movement (Lamy et al., 2019).
The SAM index reflects the meridional position of the SHWs and eddy-driven jet. During positive SAM phases the SHWs are displaced poleward and during negative phases equatorward. Present-day month-to-month changes in the SAM index represent a mode of internal variability, with anomalies persisting for only weeks to months – the timescale is longest in late spring and early summer, reflecting a stronger planetary wave–mean flow interaction (Simpson et al., 2011; Thompson and Wallace, 2000). By contrast, shifts in the ITCZ and SH jet structure on millennial and orbital timescales have a much longer lifetime and different dynamics, being driven from the tropics via hemispherically asymmetric changes in Hadley cell and STJ strength. Therefore, present-day SAM internal variability is not expected to be a good analog for past changes in SHW position. We find that the present-day SAM month-to-month internal variability mainly impacts synoptic variability over the Southern Ocean and does not have a statistically significant impact at WDC (Table 2). Such variability is likely to have also occurred during other climatic regimes, possibly just centered around a mean SHW position that is displaced meridionally relative to today. At first glance it may appear contradictory to state, as we do, that synoptic activity at WDC is not sensitive to the SAM while also suggesting that during the last deglaciation synoptic activity at WDC is linked to changes in the position of the SH eddy-driven jet and westerlies. Based on the considerations above, both claims may be true without contradiction.
Besides secular changes to the SPJ position and strength linked to meridional
ITCZ movement, WDC
The key features of the WDC
Going from the early Holocene to the Younger Dryas (YD), we observe a large increase in WDC synoptic activity. Enhanced ENSO activity during Heinrich stadials is generally supported by climate model simulations (Braconnot et al., 2012; Merkel et al., 2010; Timmermann et al., 2007) and by limited proxy evidence for stadial periods more broadly (Stott et al., 2002). Enhanced ENSO variability during the deglaciation is also found by Sadekov et al. (2013), although their record lacks the temporal resolution to resolve the individual stages. The zonal SST gradient in the equatorial Pacific further reaches a minimum during HS1, also consistent with higher El Niño intensity (Sadekov et al., 2013).
The observed variations in
The position of the SHWs during the LGM has been a topic of much scientific
inquiry. Proxy data have been interpreted to show a northward LGM shift of
the SHWs – with other scenarios, including no change at all, not excluded by
the data (Kohfeld et al., 2013). Such a shift is not supported by most
climate models (Rojas et al., 2009; Sime et al., 2013). Our
Changes to the SPJ and its associated westerly surface winds have implications for ocean circulation and marine productivity in the Southern Ocean via wind-driven upwelling. Opal flux records from the Antarctic zone
(Fig. 7g), reflecting diatom productivity, are commonly interpreted as a proxy for such upwelling – with enhanced upwelling during southward
displacement of the SHWs (Anderson et al., 2009). Here we only show records
from the Pacific sector, given that we find WDC
Here we present a new gas-phase ice core climate proxy, Kr-86 excess, that
reflects time-averaged surface pressure variability at the site driven by
synoptic activity. Surface pressure variability weakly disturbs the gravitational settling and enrichment of the noble gas isotope ratios
This interpretation is supported by a calibration study in which we measure
Current limitations of the new
Using atmospheric reanalysis data, we show that synoptic-scale barometric
variability in Antarctica is primarily linked to the position and/or strength of the southern edge of the eddy-driven subpolar jet (SPJ, also called the polar front jet) with a southward SPJ displacement enhancing synoptic-scale surface pressure variability in Antarctica. The commonly defined modes of large-scale atmospheric variability, such as the southern annular mode and the Pacific–South American Pattern, impact Antarctic only weakly as they are weighted towards the midlatitudes; the exception is the Antarctic Peninsula, where synoptic activity is well-correlated with the southern annular mode
(
We present a new record of
Kr-86 excess is a new and potentially useful ice core proxy with the ability to enhance our understanding of past atmospheric circulation. More work to better understand this proxy is warranted, and presently the conclusions of this paper should be considered tentative. In particular, replication of the deglacial Kr-86 excess record presented here in nearby cores is needed before these results can be interpreted with confidence. A full list of suggested follow-up studies is given in Sect. 3.3. Despite the many challenges of Kr-86 excess, its further development is worthwhile owing to the dearth of available proxies for reconstructing SH extratropical atmospheric circulation.
Gas loss processes artificially enrich the
Elemental ratios in the 11-site calibration study of late Holocene samples.
Severinghaus et al. (2009) hypothesize that the apparent
Gas loss is well-known to enrich ice samples in
Following Severinghaus et al. (2009), we assume that the
Argon isotopic enrichment due to gas loss in the Byrd core used to determine the
The value of
Given the uncertainty in the gas loss parameter, we verify that our results
are valid for a wide range of
The downcore WDC
In order to provide a consistent gas loss correction to the five measurement
campaigns, including campaigns 4 and 5 for which no
In the presence of a temperature gradient, thermal diffusion causes isotopic
enrichment towards the colder location. The thermal diffusion sensitivity
Influence of gas loss and thermal correction on the
For the downcore WDC record through the deglaciation we can no longer assume
a stationary
To correct the deglacial WAIS Divide record for elevation changes, here we
estimate the
Same as Fig. A3, but for
Gas loss and thermal corrections for the WDC time series.
Kr-86 excess dependence on site elevation. The vertical axis is the
The Law Dome DE08-OH site is a revisit of the DE08 site, drilled in the
2018–2019 austral summer Antarctic field season. We have samples from two
separate cores: (1) 13 24 cm long samples from a 10 cm diameter core
going from 97 to 193 m depth at
Both cores were dry-drilled (i.e., no drill liquid was used). The
10 cm diameter core used was drilled at the beginning of the field season and
the 24 cm diameter core at the end of the field season. Prior to shipment
off the continent, both cores were stored in a chest freezer at Casey Station; due to a miscommunication this freezer was set to
Both DE08-OH cores experienced more gas loss than the original DE08 core
that we also sampled (Fig. A1b). In particular, the samples from the
10 cm diameter core were strongly depleted in
High-resolution sub-annual sampling of
Figure B1 shows the high-resolution sub-annual DE08-OH sampling. The data were corrected for gas loss and thermal fractionation using a site mean
temperature gradient of
We refrain from interpreting the long-term variations in
Data are available at
CB, JS, AJO, and EJB designed research; SS, AS, BB, KK, DB, AJO, JDM, and IO contributed measurements; KK, DME, NB, RLP, RM, EMT, PDN, DT, and VVP contributed ice core samples; CB and WHGR analyzed reanalysis data; CB, AJO, and BB performed firn modeling; CB drafted the paper with input from all authors.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the special issue “Ice core science at the three poles (CP/TC inter-journal SI)”. It is not associated with a conference.
The idea for the Kr-86 excess proxy came out of discussions at the 2014 WAIS Divide Ice Core Science Meeting held at the Scripps Institution of Oceanography in La Jolla, CA. The authors want to thank John Chiang, Justin Wettstein, Zanna Chase, Bob Anderson, Tyler Jones, and Eric Steig for useful discussions, data sharing, and paper feedback, the NSF ice core facility (NSF-ICF, formerly the National Ice Core Laboratory) for curating and distributing ice core samples, the European Centre for Medium-Range Weather Forecasts (ECMWF) for making ERA-Interim reanalysis datasets publicly available, and the US ice drilling program for coordinating ice core drilling in Antarctica. Ice drilling and field support at Law Dome and sample handling in Hobart were provided by the Australian Antarctic Science Program, the Australian Antarctic Division, and (at DE08-OH) the US National Science Foundation.
This research has been supported by the National Science Foundation (grant nos. ANT-0944343, ANT-1543267, ANT-1543229, ANT-1643716, and ANT-1643669), the New Zealand Ministry of Business, Innovation and Employment (grant nos. RDF-VUW-1103, 15-VUW-131, and 540GCT32), and the Japan Society for the Promotion of Science KAKENHI grants (grant nos. 17H06320 and 15KK0027).
This paper was edited by Hubertus Fischer and reviewed by two anonymous referees.