channel network morphology and sediment dynamics under ...€¦ · 2. the periglacial environment...

Channel network morphology and sediment dynamics under

alternating periglacial and temperate regimes: a numerical

simulation study

Patrick W. Bogaarta,b,*, Gregory E. Tuckerc, J.J. de Vriesb

aNetherlands Centre for Geo-ecology ICG, The NetherlandsbFaculty of Earth and Life Sciences, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

cSchool of Geography and the Environment, Oxford University, Oxford, UK

Received 20 December 2001; received in revised form 12 November 2002; accepted 3 December 2002

Abstract

The occurrence of permafrost in a highly permeable catchment has a profound effect on runoff generation. The presence of

permafrost effectively makes the subsoil impermeable. Therefore, overland flow can be the dominant runoff-generating process

during periglacial conditions. The absence of permafrost will promote subsurface drainage and, therefore, saturation excess

overland flow can become the dominant runoff-generating process during temperate conditions. In this paper, we present a

numerical modelling study in which the effect of alternating climate-related phases of permafrost and nonpermafrost on

catchment hydrology and geomorphology is investigated. Special attention is given to the characteristics of the channel network

being formed, and the sediment yield from these catchments. We find that channel networks expand under permafrost

conditions and contract under nonpermafrost conditions. A change from permafrost to nonpermafrost conditions is

characterised by a decrease in sediment yield, while a change towards permafrost conditions is marked by a peak in sediment

yield. This peak is explained by the build-up of a reservoir of erodible sediment during the nonpermafrost phase. The driving

force behind this reservoir build-up may be local base-level change due to tectonic uplift or eustacy. We present a number of

experiments, which show the details of this process. The results are in line with existing reconstructions of climate and fluvial

dynamics during the Pleistocene in Europe and offer a new explanation to these observations.

D 2003 Elsevier Science B.V. All rights reserved.

Keywords: Quaternary; Permafrost; Geohydrology; Drainage networks; Landform evolution; Numerical models

1. Introduction

Unconsolidated Quaternary deposits of fluvial or

aeolian origin cover a significant portion of NW

Europe (Kasse, 1997). In these areas, the current

channel network appears to be in disequilibrium with

the valley network, which has a much larger extent

(Fig. 1). Numerous dry valleys occur and indicate that

0169-555X/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0169-555X(02)00360-4

* Corresponding author. Present address: Hydrology and

Quantitative Water Management Group, Department of Environ-

mental Sciences, Wageningen University, Nieuwe Kanaal 11,

Wageningen, 6709 PA, The Netherlands.

E-mail addresses: [email protected] (P.W. Bogaart),

[email protected] (G.E. Tucker), [email protected]

(J.J. de Vries).

www.elsevier.com/locate/geomorph

Geomorphology 54 (2003) 257–277

other hydrogeomorphological conditions existed dur-

ing the past. A number of hypotheses can be formu-

lated which are able to explain this former situation:

First, annual effective precipitation could have been

much higher, and fluvial geomorphological processes

would have been, thus, more intense. This explan-

ation can, however, be eliminated, since all palaeo-

climate reconstructions and modelling studies indicate

drier or similar conditions during the Last Glacial

(e.g. Renssen and Isarin, 2001). Second, runoff could

have been more peaked (fewer, but more intense

events) due to either a more peaked precipitation

regime or the distribution-moderating effects of snow

storage and melt dominance. As a result of this,

fluvial processes would have been more effective in

the past than they are now (Tucker and Slingerland,

1997; Tucker and Bras, 2000). Third, the hillslope

hydrological regime could have been different in the

past, e.g. switching between Hortonian (infiltration

excess) and saturation excess overland flow. In this

paper, we investigate the latter two hypotheses,

which deal with the way runoff is generated in low-

gradient, highly permeable catchments, under peri-

glacial conditions.

During interglacials or relative warm periods, the

subsoil is highly permeable and most runoff will occur

subsurface. Only in the lower parts of the catchment

will saturation excess overland flow occur (Dunne,

1978). This is also the current situation in most

temperate areas.

Fig. 1. Valley system in the southern Netherlands, as interpreted from the geomorphological map of The Netherlands. Map units indicating

fluvial activity during the Holocene, because of either an active floodplain or peat occurrences, are classified as ‘wet’ valleys. Map units

indicating no fluvial activity during the Holocene are classified as ‘dry’ valleys (after the 1:50,000 geomorphological map of The Netherlands,

Sheet 50: Breda, Stichting voor Bodemkartering, Wageningen, 1981).

P.W. Bogaart et al. / Geomorphology 54 (2003) 257–277258

During glacial or relative cold periods, the subsoil

is frozen for at least part of the year, and is no longer

capable of massive subsurface flow. Most runoff

would be taking place as overland flow and a more

Hortonian style of runoff generation prevails (e.g.

Dunne et al., 1976; FitzGibbon and Dunne, 1981).

A much more extended channel network is likely to

develop under these conditions. These changes in

runoff production are expected to have had a major

impact on the amount of sediment, which was deliv-

ered by catchments subject to these processes. This

sediment delivery is of special relevance to the study

of the Quaternary evolution of alluvial rivers since

these are, to a large extent, determined by water and

sediment input from the surrounding catchment.

The objective of this paper is to assess the impact

of alternating permafrost–nonpermafrost conditions

on channel network morphodynamics and the delivery

of sediment from upstream catchments to the main

river network, on glacial (stadial)–interglacial (inter-

stadial) time scales in lowland areas consisting of un-

consolidated, highly permeable sediments, by means

of numerical model experiments.

2. The periglacial environment

Periglacial and temperate geomorphology and

hydrology differ significantly in a number of ways.

Most, if not all, differences can be directly or indi-

rectly attributed to freezing and thawing of water on

or below the land surface. A brief overview of the

relevant hydrological and fluvial geomorphological

processes will be given in the next section.

2.1. Periglacial processes

In most temperate environments, saturation excess

is the dominant mode of storm runoff generation in

the vicinity of channels and channel heads (Dunne,

1978). The main factors that are responsible for this

are high-frequency, low-magnitude rainfall distribu-

tion, in combination with large values of infiltration

capacity and soil hydraulic conductivity, due to abun-

dant vegetation and soil fauna.

Under periglacial conditions, however, the situa-

tion is completely different. Vegetation and soil

fauna—if any—are scarce; soil hydraulic conductivity

is, therefore, more likely to be small, and infiltration

capacity may become a limiting factor. Hydraulic

conductivity and/or infiltration capacity is further

decreased by the presence of permafrost or seasonally

frozen soils. For example, steady-state infiltration rate

in a Vermont (USA) sandy-loam soil ranges from 8.0

cm/h for unfrozen conditions to 0.02 cm/h under

concrete frost (Dunne, 1978). Although there is some

evidence that permafrost systems are almost never

completely closed (even in ‘continuous’ permafrost

situations) (Woo, 1993), effective values for hydraulic

conductivity are much lower than under temperate

conditions. A more Hortonion-type of overland flow

can, therefore, be expected under periglacial condi-

tions. Fig. 2 shows the general hillslope hydrological

system under temperate and periglacial conditions.

Periglacial river flow regimes are extremely highly

peaked due to the springtime melting of snow (nival

regime; Woo, 1993). It is clear that climate-induced

changes in flow regime have a distinct impact on total

sediment transport, since (instantaneous) sediment

transport rate is a nonlinear function of (instantaneous)

river flow.

A large number of other hydrological processes are

active in the periglacial environment, such as inter-

hummock flow in the arctic tundra (Quinton and

Marsh, 1999) or slush flows and torrents (Gude and

Scherer, 1999). However, these processes are less

relevant for the issues dealt with in this paper, and

we will, therefore, ignore them for now.

Soil erodibility (defined here as the resistance

against erosion) is, in general, a function of soil

texture, vegetation cover, soil organic matter, etc.

Under temperate conditions, the vegetation will cover

most of surface, and soil organic matter content will

be high. Soils will, therefore, not be easily erodible.

(Palaeo) periglacial environments, on the other hand,

are generally characterised by much sparser vegeta-

tion. For example, the end of the Pleniglacial in NW

Europe was characterised by (relatively) dry condi-

tions with a sparse vegetation (Bohncke and Vanden-

berghe, 1991; Kasse, 1997). The response of

vegetation to climate change is also nonlinear and

lagged (e.g. because of nitrogen-deficiency induced

slow (primary) succession (van Geel, 1996)). In

addition, specific frost-related processes influence

erodibility. Freeze–thaw cycles weaken the soil and

riverbank strength (Gatto, 2000).

P.W. Bogaart et al. / Geomorphology 54 (2003) 257–277 259

Soil erodibility can, thus, be expected to be large

under temperate conditions, and low under periglacial

conditions. Whether or not soil erodibility has a major

impact on sediment production within the (unconso-

lidated, sandy) type of catchments that we are discus-

sing here is not certain. It can be argued that

erodibility during periglacial conditions is that of

loose sand because melt water derived from snow is

able to thaw enough soil to erode, and that it might be

higher during temperate conditions. However, under

temperate conditions, most overland flow will be

confined to small, incised channels, where vegetation

is more likely to be absent.

2.2. Reconstructed periglacial environments

Numerous workers have described the existence of

now dry valleys in formerly periglacial catchments.

Analysis of Polish fluvial valleys (Klatkowa, 1967)

suggest that the morphological evolution of fluvial

valleys is dependent on both scale and climate: the

larger, higher-order valleys were eroded during the

last interglacial (Eemian), and were filled by sediment

during the last glacial, while the smaller, lower-order

valleys (‘dells’) are the result of sheetwash erosion

under periglacial conditions.

Dry, incised valleys in ice-pushed ridges of the

Veluwe region (The Netherlands) are attributed to the

combined action of impervious soils and high snow-

melt discharges during the Weichselian (Maarleveld,

1949).

A reconstruction of the evolution of the Mark

(southern Netherlands) catchment (Vandenberghe et

al., 1987; Bohncke and Vandenberghe, 1991) suggests

a significant hydrological and geomorphological

response to climatic change. The interpreted river

response can be explained well by combining evi-

dence for soil frost and precipitation, as shown in

Table 1. The most prominent phase of river incision

occurred during the Weichselian Lateglacial, when

precipitation was high; a nival hydrological regime

prevailed and snowmelt peaked during springtime. In

Fig. 2. Cartoon illustrating the hillslope hydrological regime on a sandy subsoil under temperate conditions (top left and right) and under

periglacial conditions (bottom left and right).


combination with a (seasonally) frozen soil, high

overland flow discharges were generated.

A number of sections near the head of the present

drainage network show a relatively deep Lateglacial

incision and, thus, provides indirect evidence for a

more extended network during this period, compared

to the Holocene. The little activity during periods of

strong soil freezing, such as the Late Pleniglacial,

indicates that fluvial morphodynamics is forced by

both soil hydrological condition and total precipitation

and can, therefore, only be explained by taking both

hydrologic and precipitation regimes into account.

Analysis of drainage density and streamflow re-

gime properties in a number of catchments in the NE

United States (Carlston, 1963) showed that drainage

density (squared) is log-linearly related to both base

flow and mean annual flood, suggesting that drainage

density is controlled by transmissivity, and controls

flood discharge.

De Vries (1994) presented a conceptual model that

explains the relations between catchment morphology,

channel network structure and groundwater flow. This

model is valid for low-relief catchments in (thick)

unconsolidated, highly permeable substrate. Under

these conditions, groundwater and surface water are

tightly coupled, and it can be assumed that the

surficial channel network is adapted in such a way,

that precipitation surplus can be released through both

surface and subsurface drainages. Drainage density

can, therefore, be explained from geological, soil

physical and climatic parameters. The calculated rela-

tionships between stream density, channel morphol-

ogy, topography and groundwater dynamics compare

well with actual catchments from The Netherlands. In

a later study, it was shown that the seasonal expansion

and contraction of stream networks can be explained

by the seasonal changes in precipitation surplus and

the associated interplay of groundwater and stream

flow dynamics (de Vries, 1995). The mathematical

model describing this relation was shown to be in

agreement with the actual drainage density in the

Dutch Pleistocene area.

Kasse (1997) described the evolution of the Dutch

coversand areas during the Late Pleniglacial (OIS 2).

He attributed increased coversand deposition and

Table 1

Relation of climate, soil frost state, hydrology and morphodynamics for the Mark catchment, based on palynological, sedimentological and

geomorphological evidence (Bohncke and Vandenberghe, 1991)

Period Winter temperature (jC) Summer temperature (jC) Soil frost Total runoff Surface runoff Channel morphology

Early Holocene � 2 + 19 � + 0 very little activity

Younger Dryas � 20 + 10 + + � + minor activity

Lateglacial 2 � 13 + 12 + + ++ continuing incision

Lateglacial 1 � 6 + 15 + + + + ++ + major incision

Late Pleniglacial 2 � 12 + 12 + + � + filling

Late Pleniglacial 1 � 20 + 8 + + + + + ++ + incision

The number of (+) and (� ) symbols indicates the intensity of the process.

Table 2

Summary of changes in climate, hydrology and sedimentary

processes during the period of 20–12 ka, based on multiproxy

palaeo-climate (pollen, insects, periglacial features, etc.), sedimen-

tological and geomorphological evidence

Conditions Last Glacial

maximum

(20–15 14C kyr BP)

Late Pleniglacial

(14–12.4 14C kyr BP)

Climate/geology

Temperature (jC) <� 8 <� 1

Vegetation Tundra/absent Absent

Precipitation Low Very low

Permafrost Continuous No deep seasonal frost

Subsoil Frozen sand Unconsolidated sand

Hydrology

Infiltration Minimal Increased

Drainage density High Decreased

Overland flow Strong Absent

Surface moisture Wet Dry

Depositional processes

Aeolian activity Modest Strong

Preservation of

Aeolian sediments

Low High

Interfluves Erosion Aeolian accumulation

River valleys Aggradation Aggradation

Modified after Kasse (1997) and Huijzer and Vandenberghe (1998).


preservation during the latter phase of this period to

the lower soil moisture content due to improved

drainage caused by permafrost degradation and chan-

nel network retreat. The reconstructed environmental

parameters are shown in Table 2.

3. Methodology

The Channel-Hillslope Integrated Landscape De-

velopment (CHILD) numerical landscape evolution

model (Tucker et al., 2001a,b) was used for the

following purposes. Firstly, to assess whether the

hypotheses put forward above are physically plausi-

ble, given a simple quantitative model. Secondly, to

examine the kind of dynamics that might be expected

to emerge, given reasonable variations in (climate-

induced) soil hydrology and rainfall or runoff varia-

bility, if the dynamics are expressed as changes in

drainage density and amounts of sediment yielded

from the catchment.

3.1. The CHILD landscape evolution model

CHILD (Tucker et al., 2001a,b) is a comprehensive

landscape evolution model in which the temporal

evolution of catchment topography is simulated by a

feedback between surface hydrology, regolith detach-

ment and sediment transport. In this, CHILD is com-

parable to earlier efforts (Willgoose et al., 1991;

Howard, 1994; Tucker and Slingerland, 1997; Tucker

and Bras, 1998).

As will be discussed in the next section, the CHILD

model will be applied to a geological setting charac-

terised by low-gradient, highly permeable, unconsoli-

dated substratum, such that the processes like bedrock

incision and slope failure do not apply, and only a

subset of the CHILD model will be outlined here, i.e.

transport-limited fluvial processes; slow, continuous

creep processes; and slow, continuous local base-level

changes. See Tucker et al. (2001b,a) for a full discus-

sion of the model.

The CHILD model calculates the spatially variable

rate of erosion/deposition Bz/Bt (m/year) as a sum of

contributions by these three different processes:

Bz

Bt¼ Bz

Bt

��fluvial

þ Bz

Bt

��mass wasting

þ Bz

Bt

��base�level

ð1Þ

which is numerically solved on a regular hexagonal

grid. The three terms in Eq. (1) are discussed below.

Erosion/deposition due to fluvial processes is mod-

elled as the spatial divergence of sediment transport

Qs (m3/year)

Bz

Bt

��fluvial

¼ �jqs ¼ � BqsxBx

þBqsy

By

� �ð2Þ

where qs is the sediment flux per unit width (m3/m/

year).

Eq. (2) is solved numerically by averaging local

erosion or deposition over the area of one grid cell, so

that the equation for elevation changes in a given cell

due to fluvial sediment transport is

Bz

Bt

��fluvial

¼ 1

Av

XðQsinÞ � Qsout

� �ð3Þ

where Av (m2) is the area of the computational cells,

and Qsinand Qsout

are total volumetric fluxes of sedi-

ment entering and leaving the grid cell, respectively.

Volumetric sediment transport per unit flow width,

qs (m2/year) is modelled by a generic power law

(Tucker et al., 2001b):

qs ¼1

qsð1� gÞ kf ðktqmf Snf � scÞp ð4Þ

where qs (kg/m3) is sediment mass density; g (m3/m3)

is sediment porosity; q (m2/year) is overland flow or

channel discharge per unit flow width; S (m/m) is

topographic gradient; sc (N/m2) is critical shear stress;

kf, kt, mf, nf and p are model parameters. As discussed

by Howard (1980), many common sediment transport

formulas can be cast into the general form of Eq. (4).

Overland flow discharge Q (m3/year) is modelled

using a simple saturation excess hillslope hydrological

scheme, assuming that subsurface flow capacity is a

function of topographic gradient S (m/m) and sub-

strate transmissivity (horizontal soil hydraulic con-

ductivity, vertically integrated to the surface) T (m2/

year) (O’Loughlin, 1986):

Q ¼AP � STx; if APzSTx;

0; otherwise

8<: ð5Þ


where A (m2) is upstream catchment area, P (m/year)

is runoff (precipitation or snowmelt) rate and x (m) is

the width of the interface between adjacent grid cells.

Because of the threshold values in this equation, it

effectively models the seepage of water within well-

defined channel heads.

Erosion/sedimentation due to slow, semicontinuous

mass wasting processes like soli- or gelifluction and

creep are modelled by a simple diffusion model:

Bz

Bt

��diffusion

¼ �Kdj2z ¼ �Kd

B2z

Bx2þ B

2z

By2

� �ð6Þ

where Kd (m2/year) is a diffusivity parameter. Fast

mass wasting processes like slope failure are not

considered here.

Finally, absolute elevation change due to direct

relief changing process like tectonic uplift or local

base-level change is modelled by the simple model

Bz

Bt

��base�level

¼ U ð7Þ

where U (m/year) is the rate of tectonic uplift or base-

level change. Within all the simulations described

here, U will be a constant boundary condition, which

is applied to all grid nodes, except those which are

outlet locations for the channel network (see also

Section 4.2). These outlet nodes are kept at an

elevation of z = 0. Effects of base-level change, thus,

propagate upstream through the catchment. It should

be noted that base level here means local base level,

i.e. for the modelled domain and, thus, not global (e.g.

sea level) base level.

4. Model application

4.1. Parameterisation

In this section, we derive parameter values, which

are reasonable estimates of the process magnitudes for

a typical locality under typical time scales, e.g. the

Pleistocene areas of The Netherlands, during the Late

Quaternary (Weichselian and Holocene).

It should be noted that even the best model

parameterisation is only a rough approximation of

reality, since some potentially relevant processes are

omitted from the model; most model assumptions

simplify physical reality, scale problems occur, and

initial conditions are unknown. See Haff (1996) for a

full discussion. In this case, these problems are of

second-order significance because our aim is not

precise reconstruction, but rather hypothesis evalua-

tion and insight into the relevant morphodynamical

processes.

4.1.1. Transmissivity

Thickness of the upper part of the Pleistocene

aquifer within The Netherlands is in the range of

25–250 m, and has an average horizontal hydraulic

conductivity of 20 m/day, leading to a transmissivity T

ranging from 500 to 5000 m2/day, or 182,500–

1,825,000 m2/year (de Vries, 1974, see Fig. 3).

Transmissivity in the coversand areas in the south-

ern Netherlands has an average value of 1000 m2/

day = 365,000 m2/year. This value is taken to repre-

sent average ‘warm’ conditions. Under periglacial

conditions, we assume that permafrost is continuous,

and that subsurface flow is limited to a summer time

active layer of c 1 m and, thus, T= 7300 m2/year.

4.1.2. Fluvial sediment transport

Parameter values for the generic transport model

(4) can be estimated using, e.g. the Einstein–Brown

transport equation (Brown, 1950), which relates non-

dimensional sediment transport:

U ¼ qs;m

qsFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðs� 1Þd350

p ð8Þ

where qs,m (kg/s) is sediment transport rate per unit

flow width, qs (kg/m3) is sediment density, F

(dimensionless) is a sediment grain fall velocity

parameter, g (m/s2) is gravity, s (dimensionless) is

specific sediment grain density qs/q and d50 (m) is

median sediment grain size, to nondimensional flow

intensity

1

W¼ s

qgðs� 1Þd50ð9Þ

where s (N/m2) is shear stress, and q (kg/m3) density

of water, by

U ¼ 401

W

� �3

: ð10Þ


Sediment grain falling velocity parameter F in Eq.

(8) is defined as

F ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3þ 36m2

gd350ðs� 1Þ

s�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi36m2

gd350ðs� 1Þ

sð11Þ

where m (m2/s) is the kinematic viscosity of water.

Using s = qgSR, where R (m) is hydraulic radius,

Eqs. (8)–(10) can be rewritten as

qs;m ¼ 40qsF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðs� 1Þd350

q1

ðs� 1Þd50

� �3

ðRSÞ3:

ð12Þ

Following Willgoose et al. (1991), by combining

the Manning’s equation V=R2/3 S1/2/n, where V (m/s)

Fig. 3. Map of The Netherlands showing average horizontal transmissivity for those parts of the Pleistocene aquifer that contribute to runoff

(after de Vries, 1974).


is depth-averaged flow velocity and n is Manning’s

resistance parameter, with the conservation of water

Q =WDV and the assumption that alluvial channels

are much wider than deep, such that RcD, one

yields

R ¼ Qn

WffiffiffiS

p� �3=5

¼ q0:6n0:6S�0:3 ð13Þ

which, after insertion in Eq. (12), gives

qs;m ¼ 40qsF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðs� 1Þd350

q1

ðs� 1Þd50

� �3

� q1:8n1:8S2:1: ð14Þ

Setting sediment density qs = 2650 kg/m3, water

density q = 1000 kg/m3, gravity g = 9.8 m/s2, kine-

matic viscosity m = 1.3� 10� 6 m2/s, sediment median

grain size d50 = 0.3 mm (typical sand), gives

qs;m ¼ 1:7381� 107q1:8S2:1 ð15Þ

if qs,m and q are measured in (kg/s/m) and (m3/s/m),

respectively, and

qs;m ¼ 17:4q1:8S2:1 ð16Þ

if qs,m and q are measured in (kg/year/m) and (m3/

year/m), respectively. A final expression gives volu-

metric unit sediment transport qs (m2/year) as (assum-

ing porosity g = 0.4)

qs ¼1

qsð1� gÞ qs;m ¼ 0:0109q1:8S2:1: ð17Þ

Thus, Eq. (4) can be parameterised as

kf ¼ 1 ð18Þ

kt ¼ 0:011 ð19Þ

mf ¼ 1:8 ð20Þ

nf ¼ 2:1 ð21Þ

sc ¼ 0 ð22Þ

pf ¼ 1: ð23Þ

4.1.3. Hydraulic geometry

Hydraulic geometry (Leopold and Maddock, 1953)

relates channel dimensions (width, depth) to flow

discharge. It is used in the CHILD model to dimen-

sionalise fluvial channels and to calculate flow char-

acteristics such as flow depth.

Downstream hydraulic geometry relates bankfull

channel width Wb (m) to bankfull discharge Qb (m3/s):

Wb ¼ kwbQmwb

b : ð24Þ

Analysis of published data for >300 rivers world-

wide (van den Berg, 1995) showed that good approx-

imations are given by,

Wb ¼ 3:65Q0:50b for meandering rivers ðmÞ ð25Þ

Wb ¼ 3:81Q0:69b for braided rivers ðbÞ ð26Þ

which gives

kwb ¼ 3:65 ðmÞ or 3:81 ðbÞ ð27Þ

mwb ¼ 0:50 ðmÞ or 0:69 ðbÞ: ð28Þ

Here, only the meandering (m) cases are used.

At-a-station hydraulic geometry relates instantane-

ous discharge Q to water-body width Wh:

Wh ¼ kwhQmwh : ð29Þ

Recognising that for Q =Qb and Wh =Wb, param-

eter kwh can be expressed as a function of the other

parameters and Qb, leading to

Wh ¼ kwbQmwb�mwh

b Qmwh ð30Þ

where mwh can be interpreted as a channel shape

parameter which essentially determines how fast the

water surface width Wh approaches bankfull (channel)

width Wb when discharge increases towards Qb. In

these simulations, we assume that channels are non-

cohesive, and that

mwh ¼ 0:25 ð31Þ

(Knighton, 1998). It should be further noted that

(bankfull) channel width W is independent from grid

cell interface width x.


4.1.4. Bankfull discharge

Current mean precipitation within the southern

Netherlands is about 750 mm/year, about half of

which is lost by evapotranspiration. Mean discharge

hQi (m3/s) at a location within a catchment with

upstream area A (m2) is then

hQi ¼ 0:375A

31 536 000ð32Þ

where the factor 31,536,000 accounts for the conver-

sion between (year) and (s) time units.

The above assumptions were tested by taking the

Maas (Meuse, The Netherlands) catchment as an

example. For this catchment, with a size of 21,300

km2 upstream of station Borgharen, hQi, as calculatedfrom Eq. (32), is 253 m3/s, which compares well with

a measured value of hQi= 230 m3/s.

A general regression of bankfull discharge against

mean annual discharge, based on data presented by

van den Berg (1995) (see Fig. 4), reveals that the

model

Qb ¼ 25hQi0:75 ð33Þ

is a reasonable approximation of the underlying

hydrological regime processes.

Applied to the case of the river Maas, this yields

Qb = 1586 m3/s, which is also in good agreement with

present-day values of Q2.33 = 1616 m3/s (Passchier,

1995).

CHILD parameterises bankfull discharge as Pb (m/

year), which is the effective precipitation rate that

leads to bankfull discharge.

For a catchment with total area Ac, mean discharge

near the outlet is directly related to average precip-

itation rate P (m/year) as

hQi ¼ AcP=31; 536; 000 ð34Þ

and, thus, by definition

Qb ¼ AcPb: ð35Þ

The generic discharge scaling law Eq. (33) with Qb

and hQi in (m3/s), can be expressed for Qb and hQi in(m3/year) as

Qb ¼ 25NhQiN

� �0:75

ð36Þ

where N is the number of seconds within a year

(c 31,536,000). The bankfull event Pb is then

Pb ¼ Qb=Ac ð37Þ

Pb ¼25N

Ac

PAc

N

� �0:75

ð38Þ

Pb ¼ 25P0:75N 0:25

A0:25c

: ð39Þ

Applied to the case P= 0.375 m/year and for a

small catchment of Ac = 1205� 1250 m, this leads to

Pb ¼ 25:4 m=year: ð40Þ

Bankfull discharge events near the outlet are, thus,

generated by rainfall events that are 34 times the mean

rainfall rate. It should be noted that the units in which

Pb is expressed (m/year) do not imply that this effective

precipitation rate is valid on a yearly time scale, but

only for rare (once in 2.33 years) short-duration events.

4.1.5. Diffusional erosion/sedimentation

A number of studies have calculated hillslope dif-

fusivity values from the evolution of escarpments of

known age. For example, in a summary by Martin andFig. 4. Power law regression of bankfull discharge on mean annual

discharge, based on data in van den Berg (1995).


Church (1997), diffusivity due to slow, continuous

processes is estimated to be in the range 0.0001–0.01

m2/year. Rosenbloom and Anderson (1994) found a

rate of 0.01 m2/year for old marine terraces in Califor-

nia, while McKean et al. (1993) found 0.036 m2/year

for clay soils in California. Lacking better data, a value

of 0.01 m2/year is assigned to warm-phase diffusivity.

Diffusivity values for periglacial environments are

not explicitly listed in the literature. However, rea-

sonable values can be calculated from solifluction rate

measurements, as for example presented by French

(1996), using the following assumptions:

1. The mean movement rate can be approximated as

one quarter of the surface rate.

2. The active layer has a thickness of 50 cm.

Using the diffusional transport equation

qs ¼ KdS ð41Þ

where qs (m2/year) volumetric transport per unit slope

width, and S (m/m) slope gradient, the diffusion

coefficient Kd can then be calculated as

Kd ¼Dzv

Sð42Þ

where Dz (m) is the moving layer thickness and v (m/

year) is the mean downslope movement rate.

Table 3 presents the calculated diffusivity values.

The total range is about 0.01–0.06 m2/year. This

indicates that periglacial solifluction, as compared

with diffusional processes in more temperate environ-

ments, is less effective than one might expect. Given

the fact that solifluction acts in a spatially discontin-

uous manner (e.g. in ‘lobes’), areal average diffusivity

is even smaller. A value of 0.01 m2/year is, therefore,

assigned to cold-phase diffusivity also.

Using a more detailed, physically based temper-

ature- and topography-driven model for gelifluction,

Kirkby (1995) analysed the evolution of gelifluction

rate for Britain during the last 16 kyr. Model results

showed that total diffusivity peaked at c 0.3 m2/year

during the deglaciation, when mean annual temper-

ature passed the 0 jC isotherm. These temporary

effects during cold–warm transitions are not taken

into account here.

4.1.6. Base level changes

Changes in local base level of small rivers within

the Dutch Pleistocene areas depend on the dynamics

of the large river systems as the Maas and the Rhine.

The vertical movements of these rivers (incision and

aggradation) are controlled mainly by eustatic sea-

level change (leading to aggradation in the lower

reaches), climate-induced sediment supply (leading

to aggradation), tectonic uplift of the hinterland (lead-

ing to incision in the upper reaches) and overall

neotectonic activity in the area itself leading to a

complex variation of incising and aggrading stretches.

Local base-level lowering, however, is the driving

ultimate force behind all surface processes, since it

introduces relief and, thus, gradients within the mod-

elled catchment. Obtaining reliable estimates of base-

level change is often hindered by the coarse temporal

resolution of base-level indicators. On the glacial

cycle time scale, dated river terraces can be used to

derive an average rate on this time scale. For example,

the separation of Eemian and Holocene terraces in a

Maas tributary (Vandenberghe et al., 1987) leads to an

average base-level lowering rate of

U ¼ 20 mm=kyr ð43Þ

which is being used in our model experiments.

4.2. Transient evolution of an evolving drainage

network

In this numerical simulation, we investigate the

development of a drainage network in a highly per-

Table 3

Diffusivity values for periglacial areas, as calculated from

solifluction rates presented by French (1996)

Locality Gradient

(j)Rate

(cm/year)

Diffusivity

(m2/year)

Spitsbergen 3–4 1.0–3.0 0.024–0.054

Spitsbergen 7–15 5.0–12.0 0.051–0.056

Karkevagga, Sweden 15 4 0.019

Tarna area, Sweden 5 0.9–1.8 0.013–0.026

Norra Storfjell, Sweden 5 0.9–3.8 0.013–0.054

Okstindan, Norway 5–17 1.0–6.0 0.014–0.025

Ruby Range, YT, Canada 14–18 0.6–3.5 0.003–0.014

Sachs Harbour, Banks Island,

NWT, Canada

3 1.5–2.0 0.036–0.048

Garry Island, NWT, Canada 1–7 0.4–1.0 0.029–0.01


meable catchment, under the constraints of alternating

phases of periglacial and temperate climates, repre-

sented as an alternation of high and low values for

transmissivity T (Fig. 5).

All other model parameters are as described in

Section 4.1. The catchment is modelled as a square

1250� 1250 m regular grid, with a grid node

spacing of 25 m. This grid resolution is chosen as

a compromise between the fine resolution required

to resolve for small-scale hillslope processes, and a

coarse grid required for numerical efficiency. Neither

sediment nor water is allowed to leave the grid,

except through the outlet node. Topography is

initialised as ‘random’ (uniformly distributed), with

a very small relief ( < 1 cm). Time step length is not

prescribed to the model. Instead, optimal time step

length is calculated for each time step to as large

possible without causing numerical instability prob-

lems.

Resulting characteristic channel networks during

‘cold’ and ‘warm’ phases, as predicted by CHILD are

shown in Figs. 6 and 7. The temporal evolution of the

drainage density is shown in Fig. 8. It can be clearly

seen that drainage density under cold conditions is

much higher than under warm conditions.

Fig. 9 shows the sediment yield as produced by

the evolving catchment. As can be seen from this

figure, the sediment yield is highly dynamic and

cannot be fully correlated to climate directly. The

course of events during this numerical experiment,

leading to the response in Fig. 9, can be explained as

follows:

0 year. The experiment starts within a cold phase.

Transmissivity is set very low. The landscape is an

initial peneplain with very low relief. Drainage

Fig. 5. Temporal changes in transmissivity, T. Low and high values

indicate ‘cold’ and ‘warm’ conditions, respectively.

Fig. 6. Channel network under ‘cold’ conditions (low trans-

missivity) at time t = 69 kyr. The black lines indicate the flow

network. Grey lines have subsurface discharge only. Line width is

proportional to discharge. The ‘loose’ ends are artefacts of the

runoff scheme and have no physical meaning.

Fig. 7. Channel network under ‘warm’ conditions (high trans-

missivity) at time t = 59 kyr.


density is very high (essentially l at time t = 0)

because of the swamp-like conditions associated with

this low relief.

0–10 kyr. The system is slowly starting to evolve

towards a dynamic equilibrium state in which all

uplift is compensated for by hillslope erosion. Total

sediment yield is, therefore, slowly rising towards an

equilibrium value (uplift rate times catchment area).

Note that equilibrium is not to be reached within this

simulation’s time span (70 kyr).

10–20 kyr. Onset of the first warm phase. Trans-

missivity increases suddenly, and —in theory—a high

portion of discharge could now be subsurface.

However, most (water table) gradients are too small

to support a large amount of subsurface flow. Surface

runoff, therefore, is only slightly lower than during the

previous phase and, consequently, sediment yield is

still rising, although with a somewhat slower rate.

This small change is in sharp contrast with the sudden

drop in drainage density. The explanation for this

contrast is that during this stage, most sediment is

derived from the lower branches of the channel

network, which do not experience much change in

discharge.

20–30 kyr. The second cold phase. Frozen soils and

low transmissivity cause the channel network to

extend over the whole catchment and sediment yield

is rising faster.

30–40 kyr. The second warm phase. Transmissivity

increases and a high fraction of discharge is now

drained subsurface. The upper parts of the channel

network, as established during the previous phase, no

longer have any surface runoff and, therefore, do not

contribute to fluvial erosion. Sediment yield, there-

fore, decreases significantly. This decrease in drainage

density may be associated with an infilling of (former)

channel heads, leading to a physical or morphological

reduction in drainage density. Alternatively, it may be

associated with a drying up of the former channel

heads, leading to a hydrological reduction in drainage

density. Here, we do not distinguish between the two

types. Uplift (or base-level lowering) is continuing

and the local relief in the catchment is increasing, due

to decoupling of the low-order hillslopes without any

overland flow, and the channels, which are linked to

the local base level.

40–50 kyr. The third cold phase. Again, the channel

network extends over the whole catchment and

sediment yield is rising. The upper parts of the

catchments, which were not affected by fluvial erosion

during the previous phase, are now subject to

temporary intensified erosion. This is due to the larger

gradients here, which were established during the

phase of decoupling of channels and hillslopes. This

causes a (small) peak in sediment yield during the first

part of this phase (indicated by an arrow in Fig. 9).

50–70 kyr. The same cycle as during the previous two

phases. Note that the peak in sediment yield during

Fig. 8. Temporal evolution of drainage density, expressed as the

fraction of computational nodes that have surface runoff. Grey

panels indicate warm phases.

Fig. 9. Sediment yield as modelled (solid line) and under eventual

dynamic equilibrium conditions (dashed line) for an evolving

drainage basin.


the first part of the cold phase is larger than that

during interval 40–50 kyr.

A general trend worth noticing is that drainage

density (Fig. 8) is gradually decreasing. Over time,

catchment relief and overall gradients increase, more

water can be discharged subsurface, and fewer chan-

nels are needed to drain surface runoff. As noted

above, the system evolves towards a dynamic equili-

brium between hillslope erosion (or sediment yield)

and uplift (at tH70 kyr). Under these conditions, a

characteristic equilibrium hillslope form emerges,

leading to a characteristic equilibrium subsurface

discharge capacity. Therefore, the eventual equili-

brium drainage density will emerge as that drainage

density that is required to drain the runoff that cannot

be discharged subsurface. A further consequence of

these processes is that while sediment yield is increas-

ing over time, and drainage density is decreasing,

fluvial erosion processes are concentrating in a region

near the channels.

The most important feature of the catchment

response is the high peak in sediment yield, which

is produced during the onset of a cold phase. The

origin of this material is in fact the sediment, which

was not eroded during the preceding warm phase (a

more general analysis of the amount of this material

will be made in the next section).

The building up of this reservoir of erodible

material during the warm phase will be referred to

as the formation of a sediment source space. This

‘space’ should be thought of as an inverse analogue of

the term ‘accommodation space’, as used in basin

sedimentology. Where ‘accommodation space’

denotes (virtual) space where sediment can be stored

(and will, if there is supply), ‘sediment source space’

will denote (virtual) space where sediment can be

derived from (and will, if there is detachment/trans-

port capacity). It should be noted that the removal

of sediment from the ‘sediment source space’ is

not hindered by a too low erodibility, but by a lack

of sufficient overland flow, because of subsurface

drainage.

The ‘sediment source space’ is, thus, created or

‘filled’ by increasing relief near the channels, caused

by uplift-driven channel incision during warm peri-

ods, and ‘emptied’ by channel network expansion

during cold periods.

4.3. Analysis of the volume of the ‘sediment source

space’

As outlined in the previous section, a temporary

peak in sediment yield at the beginning of a cold period

is caused by the introduction of extra disequilibrium

during the preceding warm phase: the formation of the

‘sediment source space’ by gradual base-level low-

ering, which is emptied during the cold phase.

The volume of the ‘sediment source space’, or the

subsequent peak in sediment yield, is determined by

the depth and the areal extent of those areas within the

catchment that are subject to (fluvial) erosion during

cold phases, but not during warm phases. Adjacent

hillslopes (i.e. those parts of the catchment which are

not subject to fluvial erosion during both cold and

warm phases) do not contribute directly to the for-

mation of ‘sediment source space’.

During dynamic equilibrium conditions (not

reached during the model experiment discussed so

far), the depth of this volume is equal to the increase

in relief between hillslopes and channels during the

warm phase. Alluvial channels in dynamic equilibrium

are coupled to the catchment base level, and the

increase of relief is, thus, equal to the total rate of

base-level change or uplift U, times the length of the

warm phase Dtw. Depth of the ‘sediment source space’

is then UDtw.

Figs. 10 and 11 show the slope–area plots of the

catchment at the end of cold and warm phases,

respectively. It can be clearly seen which parts of

Fig. 10. Slope–area diagram of a typical ‘cold’ situation (t = 69

kyr).


the catchment (as expressed by contributing area) are

affected mostly by changes in transmissivity.

From Eq. (5), it follows that saturation occurs

where

A

SzTxP

ð44Þ

which is indicated in Figs. 10 and 11.

The channels, which are constrained to this satu-

ration domain, are approximated by (Tucker and Bras,

1998):

Seq ¼UA1�m

kfPm

� �1=n

ð45Þ

under dynamic equilibrium conditions. As can be seen

in Figs. 10 and 11, this equilibrium gradient is reached

during the simulations because (alluvial) channels have

a much shorter relaxation time than the hillslopes do.

The intersection of Eqs. (44) and (45) indicates the

position of the channel head (arrow). As shown by

Tucker and Bras (1998), this position is given by

As ¼T

nmþn�1ð ÞU 1

mþn�1ð Þ

k1

mþn1�ð ÞP mþnmþn�1ð Þ : ð46Þ

A comparison of Figs. 10 and 11 shows that (all

else being equal) the channel head position shifts in

time, as function of changes in transmissivity:

As;cold ¼ 2:8� 103 m2 ð47Þ

As;warm ¼ 2:9� 104 m2: ð48Þ

The area which is affected by alternating fluvial/

nonfluvial conditions is, thus, the range of locations

where

As;cold < A < Aswarm : ð49Þ

The total area for which Eq. (49) is valid is a

function of network structure, and can be determined

by analysing the cumulative distribution of contribu-

ting area. Fig. 12 shows this distribution as a cumu-

lative areal fraction P(A>a) plotted on logarithmic

axes.

The fraction A* of total catchment area which is

upstream of the channel heads then equals

1�P(A>As) and, thus, for cold and warm phase

channel heads:

Acold* ¼ 0:76 ð50Þ

Awarm* ¼ 0:89: ð51Þ

This indicates that 0.89–0.76 = 13% of the total

catchment contributes to the formation of a ‘sedi-

Fig. 11. Slope–area diagram of a typical ‘warm’ situation (t = 59

kyr).

Fig. 12. Cumulative distribution of contributing area, plotted as

cumulative areal fraction. Dotted lines indicate channel head

position during cold and warm phases.


ment source space’, whose total volume is then

given by

Vs ¼ 0:13AcUDtw: ð52Þ

4.4. Impact of hydrological regime

As discussed above, the hydrological regime under

temperate conditions is significantly different from

that under periglacial conditions, the latter being more

‘flashy’ or ‘peaked’ than the former.

To investigate the role of hydrological regime, a

series of simple numerical model experiments were

designed. In reality, processes like snow pack for-

mation and melting are responsible for the highly

peaked nival regimes of periglacial climates. How-

ever, explicit modelling of these processes would

require additional model parameters and model input,

especially high-resolution temperature time series.

Here, we assume that (changes in) the resulting

probability distribution of water application to the

land surface is much more relevant than the details

on how it originates. Therefore, we directly change

this distribution. Four scenarios have been investi-

gated, spanning the continuum from a ‘flat’ regime,

where precipitation and/or channel flow occurs at an

equal rate throughout time, towards a highly peaked

regime, where channel flow occurs only 12% of the

time. Intermediate scenarios are where channel flow

occurs at 50% and 25%, respectively. Total annual

effective precipitation is kept constant at 375 mm/

year.

Fig. 13 shows the catchment response, in terms of

sediment yield, to these four scenarios. The impact of

hydrological regime is twofold:

Sediment yield is higher for more peaked

regimes. This can be explained by the more-

than-linear relation of overland flow discharge

and sediment detachment and transport. Since the

long-term average sediment yield is constrained

by uplift or local base-level lowering rate, this

also implies that equilibrium will be reached

much faster under peaked regimes than under a

flat regime. The peak in sediment yield during the first part of

cold phases is higher for more peaked regimes.

This can also be attributed to the greater effective-

ness of the concentrated flow events associated

with this regime.

Using a comparable (but stochastic, and for a

different runoff-generating mechanism) numerical

experimental setup, Tucker and Bras (2000) obtained

similar conclusions: they found that increased climatic

variability results in higher erosion rates, a higher

drainage density and reduced relief.

Fig. 14 and Table 4 show a comparison of the

area–slope diagrams of the catchment under typical

‘warm’ and ‘cold’ climates, for two extreme flow

regimes, i.e. the most flat and the most peaked of the

four regimes above. The immediate difference

between the two regimes is that the contributing area

for the channel head is, for the warm as well for the

cold climate, a factor of 5 smaller for the most peaked

regime, than for the flat regime. More importantly, if

we make the assumption that a flat hydrological

regime is typical for a warm climate, and a peaked

regime for a cold climate, the difference in warm/cold

channel head contributing area increases from a factor

of 15 to a factor of 85. The changes in hydrological

regime that can be expected under alternation between

temperate and periglacial climate, thus, has a large

impact on the total area affected by the ‘sediment

source space’ (Table 4).

Fig. 13. Sediment yield for different hydrological regimes. (a) ‘Flat’

regime with even discharge throughout the year. (b) All discharge

within 50% of time. (c) All discharge within 25% of time. (d) All

discharge within 12.5% of time. Grey bars indicate warm periods

(high transmissivity).


4.5. Long-term evolution

The modelling examples presented so far did not

drive the system towards a state of dynamic equi-

librium. Therefore, an additional experiment has

been set up, in which the model was run for 400

kyr, using alternating warm and cold periods of 10

kyr each. The results, shown in Fig. 15, indicate

that equilibrium is approached only after c 0.5

Myr.

This experiment further shows that the peak in

sediment yield at the beginning of cold periods is

increasing in magnitude during the first c 250 kyr.

The real-world relevance of this experiment as a case

study, however, is limited. Most unconsolidated sandy

deposits experiencing alternations of temperate and

periglacial temperature regimes also experience aeo-

lian disturbance during the cold, dry periods of the

Quaternary (e.g. Late Pleniglacial; Kasse, 1997). It

can, thus, be expected that local relief is smoothed out

by aeolian processes.

Fig. 14. Comparison of area-slope diagrams for a catchment subjected to ‘flat’ or ‘peaked’ discharge regimes, under ‘cold’ and ‘warm’ climatic

conditions. Arrows indicate the position of the channel head. See also Table 4.

Table 4

Statistics for channel head locations in Fig. 14

Regime, climate As P (A>As)

Flat, warm 5�104 0.09

Flat, cold 4�103 0.22

Peaked, warm 1�104 0.16

Peaked, cold 6�102 0.40

Fig. 15. Long-term evolution of a channel network system under

alternating temperate periglacial climates. The dashed line indicates

sediment yield under dynamic equilibrium conditions.


5. Concluding remarks

The numerical model experiments presented show

clearly the response of (alluvial) channel networks and

associated sediment yield to climate induced changes

in subsurface hydrological parameters. Channel net-

works expand under periglacial conditions, and con-

tract under temperate conditions. A distinctive peak in

sediment yield is expected during a warm–cold tran-

sition, and is explained by the formation of a ‘sediment

source space’ in the upslope parts of the catchment. The

magnitude of this sediment yield peak is shown to be

controlled directly by the size of this ‘sediment source

space’, which itself is a function of relative uplift rate

and areal extent of those regions within the catchment

which have no surface runoff during the warm climatic

phase. A predictive equation for this sediment yield

peak is derived.

These results offer a good explanation for the

occurrence of dry valleys in many west and central

European catchments, and are in line with previous

field-based reconstructions of channel network loca-

tion and fluvial activity within these areas (Vanden-

berghe et al., 1987; Bohncke and Vandenberghe,

1991; Kasse, 1997). The presence of a thick uncon-

solidated Quaternary substratum, and the alternating

cold and warm phases of the Pleistocene provide

favourable conditions for these processes. The pro-

cesses described in this paper offer a good explanation

for the reconstructed changes in fluvial activity.

We have also shown that sediment yield from these

catchments is highly sensitive to the runoff regime. A

more peaked regime is associated with more effective

detachment and transport, resulting in higher sediment

yields and a shorter relaxation time of the system.

Although this paper is concerned primarily with

lower order catchments, it must be stressed that the

processes described here have a large impact beyond

these catchments. The dynamics of alluvial rivers is

determined by the amount of water and sediment

entering these rivers. Apart from within-catchment

sediment storage, the sediment yield from small catch-

ments will equal the sediment supply for a large-scale

river system. The periglacial extension of low order

networks, combined with peaked discharges is, there-

fore, a plausible explanation for river aggradation

under periglacial conditions, and incision under tem-

perate conditions. This type of behaviour is confirmed

in numerous studies (e.g. Budel, 1977; Huisink, 1999;

Mol, 1997).

These results further imply that the relationship

between climate and sediment yield/supply can be

highly nonlinear, and that the most pronounced sedi-

ment supply and fluvial activity can occur during a

climatic change, which is in line with previous model-

ling results (Tucker and Slingerland, 1997). This type

of response was previously recognised by, for example

Vandenberghe (1993), who attributed this enhanced

activity to time lags between climate and vegetation.

The results described here offer an alternative explan-

ation, although both factors may play a role.

A potentially important process, excluded from the

numerical experiments described here, is the impact of

aeolian processes. Aeolian activity is especially strong

in catchments consisting of sandy substrate during

Glacial periods because of the lack of vegetation and

the dry soil conditions (e.g. Kasse, 1997). One of the

central processes addressed in this paper is the alter-

nating active/inactive state of the first order segments

of the stream network. A possible mechanism is that

channel erosion within these segments during cold

periods (when they are active) is compensated by

aeolian input: these channel segments form local

depressions in the topography, and thus stimulate

aeolian deposition (because of air flowline diver-

gence) and prevent detachment by wind (because of

the wetness-induced aeolian erodibility). In fact, aeo-

lian infilling of fluvial channels has been found

(Kasse, 1997). A proper quantitative model-based

analysis of this aeolian–fluvial competition would

require the inclusion of process-based aeolian pro-

cesses within the model framework described.

In the model experiment described here, no consid-

eration is given to the form in which precipitation falls.

This can be as rain, as snow or as rain on snow. Instead,

a more simple approach has been adopted, in which it is

assumed that the exact source of water is less relevant

than the probability distribution of water supply to the

surface. Improvements of this approach can bemade by

using detailed process-based models for snowmelt and

storage and the water balance for the upper soil layers.

Such an enhanced model may better predict the prob-

ability distribution of water supply directly from cli-

mate and surface and soil properties data. However,

such a detailed modelling approach would require

additional data and model applications on a time scale


of days, which is beyond the scope of the present paper,

but may be the subject of further study. Furthermore,

(climate related) changes in evapotranspiration rates

are not taken into account yet.

Finally, it must be emphasised that although a

quantitative numerical model has been used in this

study, a precise quantitative relationship between soil

physical parameters, groundwater hydrology and (flu-

vial) erosion/deposition is not yet feasible. A number of

factors that were not accounted for in these numerical

experiments, for instance includes the impact of vege-

tation, soil erodibility, noncontinuous mass movement

processes and explicit groundwater tables. de Vries

(1994), for instance, showed that stream density in the

present warm phase is strongly related to average depth

of the groundwater table; the latter being a measure for

the groundwater storage capacity in this shallow aqui-

fer area. In addition, little is yet known of palaeo-

precipitation and -permafrost occurrence in a more than

indicative sense. These additional factors and a more

detailed model-data comparison will be the subject of

further study.

Notation

A Contributing area (m2)

Ac Total catchment area (m2)

As Threshold contributing area for saturation

(m2)

Av Grid cell area (m2)

D Channel depth (m)

Db Bankfull channel depth (m)

Dh Hydraulic channel depth (m)

d50 Median sediment grain size (m)

g Gravitational acceleration (m/s2)

Kd Hillslope diffusivity (m2/year)

kdb Downstream hydraulic geometry depth co-

efficient

kdh At-a-station hydraulic geometry depth coef-

ficient

kf Sediment transport coefficient

kt Sediment transport coefficient

kwb Downstream hydraulic geometry width co-

efficient

kwh At-a-station hydraulic geometry width coef-

ficient

mf Discharge exponent in sediment transport

equation

mdb Downstream hydraulic geometry depth ex-

ponent

mdh At-a-station hydraulic geometry depth ex-

ponent

mwb Downstream hydraulic geometry width ex-

ponent

mwh At-a-station hydraulic geometry width expo-

nent

N Number of days within 1 year

n Manning’s flow resistance factor

nf Slope exponent in sediment transport equa-

tion

P Precipitation rate (m/year)

Pb ‘Bankfull’ precipitation rate (m/year)

p Sediment transport exponent

Q Discharge (m3/s)

Qb Bankfull discharge (m3/s)

q Discharge per unit channel width (m2/s)

hQi Mean discharge (m3/s)

Qs Sediment transport rate (m3/year)

qs Sediment transport per unit channel width

(m2/year)

qs,m Mass unit sediment transport rate (kg/m/year)

R Hydraulic radius (m)

S Topographic gradient along flow path (m/m)

Seq Equilibrium gradient (m/m)

s Specific sediment density (qs/q)T Transmissivity (m2/year)

t Time (year)

U Uplift or base-level change rate (m/year)

V Depth averaged flow velocity (m/s)

Vs Volume of ‘sediment source space’ (m3)

v Average downslope gelifluction rate (m/year)

W Channel width (m)

Wb Bankfull channel width (m)

z Elevation above base level (m)

Dtw length of warm period (year)

Dz Thickness of active layer (m)

g Sediment porosity

q Mass density of water (kg/m3)

qs Sediment mass density (kg/m3)

s Shear stress (N/m2)

sc Critical shear stress (N/m2)

U Sediment transport number

W Flow number

m Kinematic viscosity of water (m2/s)

x Width of interface between adjacent grid

cells (m)


Acknowledgements

The Netherlands Organisation for Scientific Re-

search (NWO) is thanked for providing a travel grant

to the first author. Prof. Dr. J. Vandenberghe, Dr. C.

Kasse, Dr. R.T. van Balen, Dr. T. Dunne and an

anonymous reviewer are thanked for constructive

comments on an earlier draft of this paper.

This is a contribution to the Netherlands Environ-

mental Earth System Dynamics Initiative (NEESDI)

programme, partly funded by the Netherlands

Organization for Scientific Research (NWO grant

750.296.01).

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channel network morphology and sediment dynamics under ...€¦ · 2. the periglacial environment...

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