Sediment mass conservation is a key factor that constrains river morphodynamic processes. In most models of river morphodynamics, sediment mass conservation is described by the Exner equation, which may take various forms depending on the problem in question. One of the most widely used forms of the Exner equation is the flux-based formulation, in which the conservation of bed material is related to the stream-wise gradient of the sediment transport rate. An alternative form of the Exner equation, however, is the entrainment-based formulation, in which the conservation of bed material is related to the difference between the entrainment rate of bed sediment into suspension and the deposition rate of suspended sediment onto the bed. Here we represent the flux form in terms of the local capacity sediment transport rate and the entrainment form in terms of the local capacity entrainment rate. In the flux form, sediment transport is a function of local hydraulic conditions. However, the entrainment form does not require this constraint: only the rate of entrainment into suspension is in local equilibrium with hydraulic conditions, and the sediment transport rate itself may lag in space and time behind the changing flow conditions. In modeling the fine-grained lower Yellow River, it is usual to treat sediment conservation in terms of an entrainment (nonequilibrium) form rather than a flux (equilibrium) form, in consideration of the condition that fine-grained sediment may be entrained at one place but deposited only at some distant location downstream. However, the differences in prediction between the two formulations have not been comprehensively studied to date. Here we study this problem by comparing the results predicted by both the flux form and the entrainment form of the Exner equation under conditions simplified from the lower Yellow River (i.e., a significant reduction of sediment supply after the closure of the Xiaolangdi Dam). We use a one-dimensional morphodynamic model and sediment transport equations specifically adapted for the lower Yellow River. We find that in a treatment of a 200 km reach using a single characteristic bed sediment size, there is little difference between the two forms since the corresponding adaptation length is relatively small. However, a consideration of sediment mixtures shows that the two forms give very different patterns of grain sorting: clear kinematic waves occur in the flux form but are diffused out in the entrainment form. Both numerical simulation and mathematical analysis show that the morphodynamic processes predicted by the entrainment form are sensitive to sediment fall velocity. We suggest that the entrainment form of the Exner equation might be required when the sorting process of fine-grained sediment is studied, especially when considering relatively short timescales.

Models of river morphodynamics often consist of three elements: (1) a
treatment of flow hydraulics; (2) a formulation relating sediment transport
to flow hydraulics; and (3) a description of sediment conservation. In the
case of unidirectional river flow, the Exner equation of sediment
conservation has usually been described in terms of a flux-based form in
which temporal bed elevation change is related to the stream-wise gradient of
the sediment transport rate. That is, bed elevation change is related to

An alternative formulation, however, is available in terms of an entrainment-based form of the Exner equation, in which bed elevation variation is related to the difference between the entrainment rate of bed sediment into the flow and the deposition rate of sediment on the bed (Parker, 2004). The basic idea of the entrainment formulation can be traced back to Einstein's (1937) pioneering work on bed load transport and has been developed since then by numerous researchers so as to treat either bed load or suspended load (Tsujimoto, 1978; Armanini and Di Silvio, 1988; Parker et al., 2000; Wu and Wang, 2008; Guan et al., 2015). Such a formulation differs from the flux formulation in that the flux formulation is based on the local capacity sediment transport rate, whereas the entrainment formulation is based on the local capacity entrainment rate into suspension. In the entrainment form, the difference between the local entrainment rate from the bed and the local deposition rate onto the bed determines the rate of bed aggradation–degradation and concomitantly the rate of loss–gain of sediment in motion in the water column. Therefore, the sediment transport rate is no longer assumed to be in an equilibrium transport state, but may exhibit lags in space and time after changing flow conditions. The entrainment formulation is also referred to as the nonequilibrium formulation (Armanini and Di Silvio, 1988; Wu and Wang, 2008; Zhang et al., 2013).

To describe the lag effects between sediment transport and flow conditions, the concept of an adaptation length–time is widely applied. This length–time characterizes the distance–time for sediment transport to reach its equilibrium state (i.e., transport capacity). Using the concept of the adaptation length, the entrainment form of the Exner equation can be recast into a first-order “reaction” equation, in which the deformation term is related to the difference between the actual and equilibrium sediment transport rates, as mediated by an adaptation length (which can also be recast as an adaptation time) (Bell and Sutherland, 1983; Armanini and Di Silvio, 1988; Wu and Wang, 2008; Minh Duc and Rodi, 2008; El kadi Abderrezzak and Paquier, 2009). The adaptation length is thus an important parameter for bed evolution under nonequilibrium sediment transport conditions, and various estimates have been proposed. For suspended load, the adaptation length is typically calculated as a function of flow depth, flow velocity, and sediment fall velocity (Armanini and Di Silvio, 1988; Wu et al., 2004; Wu and Wang, 2008; Dorrell and Hogg, 2012; Zhang et al., 2013). The adaptation length of bed load, on the other hand, has been related to a wide range of parameters, including the sediment grain size (Armanini and Di Silvio, 1988), the saltation step length (Phillips and Sutherland, 1989), the dimensions of particle diffusivity (Bohorquez and Ancey, 2016), the length of dunes (Wu et al., 2004), and the magnitude of a scour hole formed downstream of an inerodible reach (Bell and Sutherland, 1983). For simplicity, the adaptation length can also be specified as a calibration parameter in river morphodynamic models (El kadi Abderrezzak and Paquier, 2009; Zhang and Duan, 2011). Nonetheless, no comprehensive definition of adaptation length exists.

In this paper we apply the two forms of the Exner equation mentioned above
to the lower Yellow River (LYR) in China. The LYR describes the river
section between Tiexie and the river mouth and has a total length of about
800 km. Figure 1a shows a sketch of the LYR along with six major gauging
stations and the Xiaolangdi Dam, which is 26 km upstream of Tiexie. The LYR
has an exceptionally high sediment concentration (Ma et al., 2017),
historically exporting more than 1 Gt of sediment per year with only 49 billion tons of water, leading to a sediment concentration an order of
magnitude higher than most other large lowland rivers worldwide (Milliman
and Meade, 1983; Ma et al., 2017; Naito et al., 2018). However, the LYR has seen a substantial reduction in its sediment
load in recent decades, especially since the operation of the Xiaolangdi Dam beginning in
1999 (Fig. 1b), because most of its sediment load is derived from the
Loess Plateau, which is upstream of the reservoir (Wang et al., 2016; Naito
et al., 2018). Finally, the bed surface material of
the LYR is very fine, as low as 15

When modeling the high-concentration and fine-grained LYR, it is common to treat sediment conservation in terms of an entrainment-based rather than a flux-based formulation. This is because many Chinese researchers view the entrainment formulation as more physically based, as it is capable of describing the behavior of fine-grained sediment, which when entrained at one place may be deposited at some distant location downstream (Zhang et al., 2001; Ni et al., 2004; Cao et al., 2006; He et al., 2012; Guo et al., 2008). However, the entrainment formulation is more computationally expensive and more complex to implement. Because the differences in prediction between the two formulations do not appear to have been studied in a systematic way, here we pose our central questions. Under what conditions is it valid to use the entrainment form of the Exner equation, and under what conditions may the flux form be used? Or more specifically, which form of the Exner equation is most suitable for the LYR?

Here we study this problem by comparing the results of flux-based and entrainment-based morphodynamics under conditions typical of the LYR. The organization of this paper is as follows. The numerical model is described in Sect. 2. In Sect. 3, the model is implemented to predict the morphodynamics of the LYR with a sudden reduction of sediment supply, which serves to mimic the effect of the Xiaolangdi Dam. We find that the two forms of the Exner equation give similar predictions in the case of uniform sediment, but show different sorting patterns in the case of sediment mixtures. In Sect. 4, we conduct a mathematical analysis to explain the results in Sect. 3; more specifically, we quantify the effects of varied sediment fall velocity in the simulations. Finally, we summarize our conclusions in Sect. 5.

In this paper, we present a one-dimensional morphodynamic model for the lower Yellow River. The fully unsteady Saint–Venant equations are implemented for the hydraulic calculation. Both the flux form and the entrainment form of the Exner equation are implemented in the model for sediment mass conservation. For each form of the Exner equation, we consider both the cases of uniform sediment (bed material characterized by a single grain size) and sediment mixtures. Since the sediment is very fine in the LYR, the component of the load that is bed load is likely negligible (e.g., Ma et al., 2017), so we consider only the transport of suspended load. Considering the fact that most accepted sediment transport relations (e.g., the Engelund and Hansen, 1967, relation) underpredict the sediment transport rate of the LYR by an order of magnitude or more (Ma et al., 2017), in our model we implement two recently developed generalized versions of the Engelund–Hansen relation, which are based on data from the LYR. These are the version of Ma et al. (2017) for uniform sediment and the version of Naito et al. (2018) for sediment mixtures. In cases considering sediment mixtures, we also implement the method of Viparelli et al. (2010) to store and access bed stratigraphy as the bed aggrades and degrades.

Since the aim of this paper is to compare the two formulations of the Exner equation in the context of the LYR rather than reproduce site-specific morphodynamic processes of the LYR, some additional simplifications are introduced to the model to facilitate comparison. The channel is simplified to be a constant-width rectangular channel, and bank (sidewall) effects and floodplain interactions are not considered. The channel bed is assumed to be an infinitely deep supplier of erodible sediment with no exposed bedrock, which is justifiable because the LYR is fully alluvial and has been aggrading for thousands of years, as copiously documented in Chinese history. Finally, water and sediment (of each grain size range) are fed into the upstream boundary at a specified rate, and at the downstream end of the channel we specify a fixed bed elevation along with a normal flow depth. These restrictions could be easily relaxed so as to incorporate site-specific complexities of the Yellow River. Because of the severe aggradation of the LYR developed before the Xiaolangdi Dam operation, the LYR is famous for its hanging bed (i.e., bed elevated well above the floodplain) and no major tributaries need be considered in the simulation.

Flow hydraulics in a rectangular channel are described by the following
1-D
Saint–Venant equations, which consider fluid mass and momentum conservation,

In this paper, the full flood hydrograph of the LYR is replaced by a flood
intermittency factor

When dealing with uniform sediment, the flux form of the Exner equation can
be written as

When considering sediment mixtures, an active layer formulation (Hirano,
1971; Parker, 2004) is incorporated in the flux-based Exner equation so
that the evolution of both bed elevation and surface grain size distribution
can be considered. In this formulation, the riverbed is divided into a
well-mixed upper active layer and a lower substrate with vertical
stratigraphic variations. The upper active layer therefore represents the
volume of sediment that interacts directly with suspended load transport
and also exchanges with the substrate as the bed aggrades and degrades.
Discretizing the grain size distribution into

Summing Eq. (5) over all grain size ranges, one can find that the governing
equation for bed elevation in the case of sediment mixtures is the same as Eq. (4) upon replacing

The method of Viparelli et al. (2010) is applied in our model to store
substrate stratigraphy and provide information for

The entrainment-based Exner equation for uniform sediment is

For the sediment fall velocity

In the entrainment formulation the sediment transport rate

Summing Eq. (12) over all grain size ranges, we get the governing equation
for bed elevation.

To close the Exner equations described in Sect. 2.2 and 2.3, equations
for equilibrium sediment transport rate

In the relation of Ma et al. (2017), the dimensionless coefficient

We implement the relation of Naito et al. (2018) to
calculate the equilibrium sediment transport rate of size mixtures. Using
field data from the LYR, Naito et al. (2018)
extended the Engelund and Hansen (1967) relation to a surface-based
grain-size-specific form, in which the suspended load transport rate of the

We note that a form of the Engelund–Hansen equation for mixtures was introduced by Van der Scheer et al. (2002) and implemented by Blom et al. (2016). Blom et al. (2017) further extended this relation to a more general framework capable of including hiding effects. These forms, however, have not been calibrated to the LYR data and are thus not suitable for the LYR.

In this section, we conduct numerical simulations using both the flux form
and the entrainment form of the Exner equation, with the aim of studying under
what circumstances the two forms give different predictions. Numerical
simulations are conducted in the setting of the LYR. We specify a 200 km
long channel reach for our simulations, along with a constant channel width
of 300 m and an initial longitudinal slope of 0.0001. Bed porosity

A constant flow discharge of 2000 m

Two cases are considered here. In the first case, the sediment grain size
distribution of the LYR is simplified to a uniform grain size of 65

The 200 km channel reach is discretized into 401 cells, with cell size

Grain size distributions of both the initial bed and the sediment supply in the case of sediment mixtures. For the initial bed, the surface and substrate grain size distributions are the same. The grain size distribution of the initial bed is renormalized based on the field data at the Lijin gauging station. The grain size distribution of the sediment supply equals the grain size distribution of bed material load at equilibrium. Grain sizes in the range of wash load have been removed from both distributions.

Summary of computational conditions for numerical modeling of the LYR.

In this case, we implement a uniform grain size of 65

Figure 3 shows the modeling results using the flux form of the Exner equation. As we can see in the figure, the bed degrades and the sediment load decreases in response to the cutoff of sediment supply. Such adjustments start from the upstream end of the channel and gradually migrate downstream. Figure 4 shows the modeling results using the entrainment form of the Exner equation. A comparison between Figs. 4 and 3 shows that the entrainment form and the flux form give very similar predictions in this case. The entrainment form provides a somewhat slower degradation (at the upstream end the flux form predicts a 3 m degradation, whereas the entrainment form predicts a 2.3 m degradation) and a more diffusive sediment load reduction. Such more diffusive predictions of sediment load variation can be ascribed to the condition of nonequilibrium transport that is embedded in the entrainment form. This issue will be studied analytically in Sect. 4. Here we present the results for only 0.2 years after the cutoff of sediment supply, since the differences between the predictions of the two forms tend to be the most evident shortly after the disruption but gradually diminish as the river approaches the new equilibrium (El kadi Abderrezzak and Paquier, 2009). Modeling results over a longer timescale will be discussed in Sect. 4.3.

The 0.2-year results for the case of uniform sediment using the flux
form of the Exner equation: time variation of

The 0.2-year results for the case of uniform sediment using the
entrainment form of the Exner equation: time variation of

To further quantify the differences between the predictions of the two
forms, we propose the following normalized parameter:

Table 2 gives a summary of the maximum values of

Quantification of the difference between predictions of the flux
form and the entrainment form in the case of uniform sediment. The maximum
values of

The above results show that the flux form and the entrainment form can
provide similar predictions of the LYR when the bed sediment grain size
distribution is simplified to a uniform value of 65

The 0.2-year results for the case of uniform sediment using the
entrainment form of the Exner equation: time variation of

In Sect. S2, we also conduct numerical simulations with
hydrographs. Results indicate that our conclusions based on constant flow
discharge also hold when hydrographs are considered: the flux form and the
entrainment form (with the sediment fall velocity not adjusted) of the Exner
equation give very similar predictions using a characteristic grain size of
65

In this section we consider the morphodynamics of sediment mixtures rather
than the case of a uniform bed grain size implemented in Sect. 3.1. The
grain size distribution of the initial bed is based on field data at the
Lijin gauging station and is shown in Fig. 2. Using the sediment transport
relation of Naito et al. (2018) for mixtures, such a
grain size distribution combined with the given bed slope and flow discharge
leads to a total equilibrium sediment transport rate per unit width

Figure 6 shows the simulation results using the flux form of the Exner
equation. As a result of the reduced sediment supply at the inlet, bed
degradation occurs first at the upstream end and then gradually migrates
downstream. The total sediment transport rate per unit width

The 0.2-year results for the case of sediment mixtures using the flux
form of the Exner equation: time variation of

Figure 7 shows the simulation results obtained using the entrainment form of
the Exner equation. In general, the patterns of variation predicted by the
entrainment form have similar trends and magnitudes to those predicted by
the flux form: the bed degrades near the upstream end, the suspended load
transport rate is reduced in time, and both the bed surface and the suspended
load coarsen as a result of the cutoff of sediment supply. But the results
based on the two forms exhibit very evident differences when multiple grain
sizes are included. That is, the results predicted by the entrainment form
are sufficiently diffusive so that the variations of

The 0.2-year results for the case of sediment mixtures using the
entrainment form of the Exner equation: time variation of

Quantification of the difference between predictions of the flux
form and the entrainment form in the case of sediment mixtures. The maximum
values of

The results shown in Fig. 8 have also been calculated using the entrainment
form of the Exner equation, but here the sediment fall velocities

Table 3 summarizes the

The 0.2-year results for the case of sediment mixtures using the
entrainment form of the Exner equation: time variation of

In Sect. S3, we present additional numerical cases that are similar to the cases in this section, except that hydrographs are implemented instead of constant discharge. Results indicate that our conclusions based on constant flow discharge also hold when hydrographs are considered. The flux form and the entrainment form (with the sediment fall velocity not adjusted) of the Exner equation predict quite different patterns of grain sorting, with the flux form exhibiting a more advective character than the entrainment form.

In Sect. 3.1, our simulation shows that in the case of uniform sediment,
the flux form and the entrainment form of the Exner equation give very
similar predictions for a given sediment size of 65

In the entrainment form, the equation governing suspended sediment
concentration is

If we consider the spatial adjustment of sediment load shortly after the
cutoff of sediment supply, we can further neglect the nonuniformity of the
capacity (equilibrium) transport rate

For the case of uniform sediment in Sect. 3.1,

The evolution of bed elevation

The above analysis also holds for sediment mixtures, except that each grain size range will have its own adaptation length. Here we neglect the temporal derivative in Eq. (29) and analyze only the spatial adjustment of sediment load. If we neglect the spatial derivative in Eq. (29) and conduct a similar analysis for sediment concentration, we would find that the temporal adjustment of sediment concentration is also described by an exponential function of time, in analogy to Eq. (33).

In Sect. 3.2 we find that the flux form and entrainment form of the Exner equation provide very different patterns of grain sorting for sediment mixtures: kinematic sorting waves are evident in the flux form but are diffused out in the entrainment form. The diffusivity of grain sorting becomes smaller and the kinematic waves appear, however, if we arbitrarily increase the sediment fall velocity by a factor of 20. In this section, we explain this behavior by analyzing the governing equations.

First we rewrite the sediment transport relation of Naito et al. (2018) in the following form.

Relation between adaptation length

Now we turn to the entrainment form of the Exner equation. Combined with the
sediment transport rate per unit width

Substituting Eqs. (43) and (34) into Eq. (41), we find that

From Eq. (45) we can see that the governing equation for

From Eq. (47), we can also see that the diffusivity coefficient

Moreover, if we compare the celerity of kinematic waves in both the flux form
and the entrainment form, we have

In Sect. 3, two numerical cases are presented to compare the flux form and
the entrainment form of the Exner equation, but only within 0.2 years after
the cutoff of sediment supply. Here we run both numerical cases for a longer
time (5 years). Table 4 shows the results of the case of uniform sediment
(as described in Sect. 3.1) within 5 years, and Table 5 shows the results
of the case of sediment mixtures (as described in Sect. 3.2) within 5 years. For both cases, the

Quantification of the difference between predictions of the flux
form and the entrainment form in the case of uniform sediment. The maximum

Quantification of the difference between predictions of the flux
form and the entrainment form in the case of sediment mixtures. The maximum

Based on the numerical modeling and mathematical analysis in this paper, we
suggest that the entrainment form of the Exner equation be used when
studying the river morphodynamics of fine-grained sediment (or, more
specifically, sediment with small fall velocity). This is because the
adaptation length

It should be noted that in the morphodynamic models of this paper, we
implement the mass and momentum conservation equations for clear water
(i.e., Eqs. 1 and 2) to calculate flow hydraulics instead of the
mass and momentum equations for water–sediment mixture as suggested by Cao
et al. (2004, 2006). More specifically, Cui et al. (2005)
have pointed out that when sediment concentration in the water is
sufficiently small, bed elevation can be taken to be unchanging over
characteristic hydraulic timescales, and the effects of flow–bed exchange
on flow hydraulics can be neglected. For the two simulation cases in this
paper, the volume of sediment concentration

Considering the fact that in our numerical simulations a constant inflow discharge (along with a flood intermittency factor) is implemented, and also considering that the morphodynamic timescale is much larger than the hydraulic timescale in our case, the quasi-steady approximation or even the normal flow approximation can be introduced to further save computational efforts (Parker, 2004). But one thing that should be noted is that in our simulation results in Sect. 3, the bed exhibits an inverse slope near the upstream end. The normal flow assumption becomes invalid under such circumstances, thus requiring a full unsteady shallow water model.

By definition, the recovery coefficient

In this paper, a one-dimensional morphodynamic model with several simplifications is implemented to compare the flux-based Exner equation and the entrainment-based Exner equation in the context of the LYR. However, a site-specific model of the morphodynamics of the LYR without these simplifications would be much more complex. For example, in our 1-D simulation we observe bed degradation after the closure of the Xiaolangdi Dam, but we cannot resolve its structure in the lateral direction. In natural rivers, bed degradation is generally not uniform across the channel width, but may be concentrated in the thalweg. Moreover, the spatial variation of channel width and initial slope, which are not considered in this paper, are also important when considering applied problems. The abovementioned issues, even though not the aim of this paper, merit future research (e.g., He et al., 2012). Chavarrias et al. (2018) have reported that morphodynamic models considering mixed grain sizes may be subject to instabilities that result from complex eigenvalues of the system of equations. No such instabilities were encountered in the present work.

In this paper, we compare two formulations for sediment mass conservation in the context of the lower Yellow River, i.e., the flux form of the Exner equation and the entrainment form of the Exner equation. We represent the flux form in terms of the local capacity sediment transport rate and the entrainment form in terms of the local capacity entrainment rate. In the flux form of the Exner equation, the conservation of bed material is related to the stream-wise gradient of sediment transport rate, which is in turn computed based on the quasi-equilibrium assumption according to which the local sediment transport rate equals the capacity rate. In the entrainment form of the Exner equation, on the other hand, the conservation of bed material is related to the difference between the entrainment rate of sediment from the bed into the flow and the deposition rate of sediment from the flow onto the bed. A nonequilibrium sediment transport formulation is applied here so that the sediment transport rate can lag in space and time behind changing flow conditions. Despite the fact that the entrainment form is usually recommended for the morphodynamic modeling of the LYR due to its fine-grained sediment, there has been little discussion of the differences in predictions between the two forms.

Here we implement a 1-D morphodynamic model for this problem. The fully unsteady Saint–Venant equations are implemented for the hydraulic calculation. Both the flux form and the entrainment form of the Exner equation are implemented for sediment conservation. For each formulation, we include the options of both uniform sediment and sediment mixtures. Two generalized versions of the Engelund–Hansen relation specifically designed for the LYR are implemented to calculate the quasi-equilibrium sediment transport rate (i.e., sediment transport capacity). They are the version of Ma et al. (2017) for uniform sediment and the version of Naito et al. (2018) for sediment mixtures. The method of Viparelli et al. (2010) is implemented to store and access bed stratigraphy as the bed aggrades and degrades. We apply the morphodynamic model to two cases with conditions typical of the LYR.

In the first case, a uniform bed material grain size of 65

The results for the case of uniform sediment can be explained by analyzing
the governing equation of sediment load

In the second case the bed material consists of mixtures ranging from 15 to 500

The different sorting patterns exhibited in the case of sediment mixtures
can be explained by analyzing the governing equation for bed surface
fractions

Overall, our results indicate that the more complex entrainment form of the Exner equation might be required when the sorting processes of fine-grained sediment (or sediment with small fall velocity) is studied, especially at relatively short timescales. Under such circumstances, the flux form of the Exner equation might overestimate advection in sorting processes and the aggradation–degradation rate due to the fact that it cannot account for the relatively large adaptation length or diffusivity of fine particles.

All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplement. Additional data related to this paper may be requested from the authors.

CA, XF, and GP designed the study. CA performed the simulation with help from HM and KN. CA wrote the paper. AJM, XF, and YZ provided substantial editorial feedback. CA, AJM, and GP performed the analysis. HM, YZ, and KN collected data from the LYR for the simulation.

The authors declare that they have no conflict of interest.

The participation of Chenge An and Xudong Fu was made possible in part by grants from the National Natural Science Foundation of China (grants 51525901 and 91747207) and the Ministry of Science and Technology of China (grant 2016YFC0402406). The participation of Andrew J. Moodie, Hongbo Ma, Kensuke Naito, and Gary Parker was made possible in part by grants from the National Science Foundation (grant EAR-1427262). The participation of Yuanfeng Zhang was made possible in part by a grant from the National Natural Science Foundation of China (grant 51379087). Part of this research was accomplished during Chenge An's visit to the University of Illinois at Urbana-Champaign, which was supported by the China Scholarship Council (file no. 201506210320). The participation of Andrew J. Moodie was also supported by a National Science Foundation Graduate Research Fellowship (grant 145068). We thank the morphodynamics class of 2016 at the University of Illinois at Urbana-Champaign for their participation in preliminary modeling efforts. We thank Astrid Blom and two other anonymous reviewers for their constructive comments, which helped us greatly improve the paper. Edited by: Jens Turowski Reviewed by: Astrid Blom and two anonymous referees