The hydraulic scaling of step- pool systems Paul A Carling, Wlodek Tych and Keith Richardson.

The hydraulic scaling of The hydraulic scaling of step-pool systemsstep-pool systems

Paul A Carling, Wlodek Tych Paul A Carling, Wlodek Tych and Keith Richardsonand Keith Richardson

RationaleRationale• In heterogeneous-gravel step-pools, free In heterogeneous-gravel step-pools, free

variable is the step spacing measured variable is the step spacing measured between clast accumulations between clast accumulations

• The latter are ‘mobile’ being destroyed The latter are ‘mobile’ being destroyed in big floods . Adjust through time and in big floods . Adjust through time and reformreform

• Uniform to irregular spacings resultUniform to irregular spacings result• Boulder size/accumulation/roughness Boulder size/accumulation/roughness

dictates pool depth/spacings: dictates pool depth/spacings: Curran and Curran and Wilcock, 2005 Wilcock, 2005

Step-pools with alternating scour. The standing figure is Hiroshi Ikeda. Byōbuga Ura, Chiba Prefecture, Japan. Flow from top to bottom. Unconsolidated Plio-Pleistocene sediments.

Step-pools. Formed in relatively homogeneous soil from uniform furrow after two days.Georgia

Step-pools in ice. Day 1

Step-pools in ice. Day 3

Homogeneous media and rapid evolution, without depositional sediment component

PositPosit• Inherent hydraulic character of flow Inherent hydraulic character of flow

that dictates step-pool geometrythat dictates step-pool geometry• Sediment accumulations are a Sediment accumulations are a

secondary considerationsecondary consideration• Posit derived from rapid evolution step-Posit derived from rapid evolution step-

pools in ice or soil: pools in ice or soil: Izumi & Parker, 2000Izumi & Parker, 2000• Posit derivedPosit derived from slow evolution in from slow evolution in

homogeneous bedrock: homogeneous bedrock: c. 5 publicationsc. 5 publications

Hypothesis:Hypothesis:• Step-pool systems of a defined geometry Step-pool systems of a defined geometry

develop in streams running across develop in streams running across homogeneous bedrock wherein the (bed) homogeneous bedrock wherein the (bed) sediment load is negligible sediment load is negligible

Test: Test: • Qualitative observations of homogeneous Qualitative observations of homogeneous

bedrock systemsbedrock systems• Quantitative observations of two Quantitative observations of two

homogeneous bedrock systems (Thailand)homogeneous bedrock systems (Thailand)

Qualitative: Homogeneous media and slow evolution(sediment free or sediment

poor)

Granite, Japan

Arkose, a course-grained sandstone, Uluru, Australia

Chute-over-fall: Nishi-zawa, Japan. Granite. The channel is approximately 6 m wide.

Free-over-fall: Woolshed Creek, Australia. Granite and microgranite. The channel is approx. 12m wide.

Qualitative: Homogeneous media and slow evolution(two over-fall styles)

S

Breached pool. A headcut has eroded from the lower pool (in which the photographer is standing) into the pool above (immediately behind the standing figure), through the intervening step (S). Than Rattana, Thailand, looking upstream. Andesite.

Breached long profiles

Longitudinal Profile of Nang Rong

-40

-20

0

20

0 100 200 300 400 500 600

Longitudinal distance (m)

Elev

atio

n (m

) (ar

bitr

ary

datu

m)

0 50 100 150 200 250 300 350 400 450-30

-20

-10

0

10Original data

0 50 100 150 200 250 300 350 400 450-30

-20

-10

0

10Fixed Interval interpolated data

Thalweg Oversteps

Longitudinal profile of Than Rattana

Rhyolitic agglomerate

Andesite

Data ProcessingData Processing• Fourier analysis averages through all Fourier analysis averages through all

space – any spatial variation (trend) is space – any spatial variation (trend) is lostlost

• Data-based mechanistic modelling Data-based mechanistic modelling (DBM)(DBM)

• Dynamic Harmonic Regression (DHR)Dynamic Harmonic Regression (DHR)• These latter methods are preferred for These latter methods are preferred for

non-stationary spatial seriesnon-stationary spatial series

0 50 100 150 200 250 300 350 400 450-1.5

-1

-0.5

0

0.5

1

1.5

2Harmonic components

period 55.784period 24.571period 12.077period 12.286period 5.147period 4.039

Resampled data for Than Rattana

0 50 100 150 200 250 300 350 400 450-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Model fit

data model

20 10 6.67 5 4 5

10

15

20

25

30

35

40

45

spatial periodicity [m]

Am

plitu

de s

pect

rum

dB

20 10 6.67 5 4 3.33 2.86 2.5 2.22 2

-0.5

0

0.5

1

1.5

2

2.5

Constant parameters AR(24) spectrum of oversteps data

PERIOD

POW

ER log10(am

p*pi)

Nang Rong

Than RattanaStatistically strong preferred component of periodicity – 23m for NR with harmonics and 55.8m for TR (but shift downstream to 24m)

Initial conclusionsInitial conclusions• At a ‘lumped’ reach-scale the DBM At a ‘lumped’ reach-scale the DBM

spectral periodicity represents a spectral periodicity represents a genuine periodic tendency for the genuine periodic tendency for the spacing of step-poolsspacing of step-pools

• Given homogeneous bedrock, the Given homogeneous bedrock, the periodic bed structure represents periodic bed structure represents large-scale periodic flow structureslarge-scale periodic flow structures

• Structural control (where present) and Structural control (where present) and boulder deposits are secondary effectsboulder deposits are secondary effects

Spatial varianceSpatial variance• Noted prior, a shift for TR from 50m to Noted prior, a shift for TR from 50m to

24m downstream24m downstream• DBM model isolates shifts in dominant DBM model isolates shifts in dominant

periodicity down channelperiodicity down channel• DBM model provides a sensitive DBM model provides a sensitive

means of investigating spatial means of investigating spatial variation in periodic linear topographyvariation in periodic linear topography

100 150 200 250 300 350 400

100

50

33.3

25

20

16.7

Spatially varying spectrum estimate

Perio

d [m

]

Distance

Than Than RattanRattanaa

• Spatially varying spectral signature with overprint of Spatially varying spectral signature with overprint of noise noise

• No evident control by channel slope or channel widthNo evident control by channel slope or channel width• Downstream limit is waterfall so energy slope might Downstream limit is waterfall so energy slope might

steepen – inverse relationship of spacing and slopesteepen – inverse relationship of spacing and slope

Nang Nang RongRong

• Spatially varying spectrum with little noiseSpatially varying spectrum with little noise• Overall strong signal at 23m, but strength varies Overall strong signal at 23m, but strength varies

down channel with four strong peaks down channel with four strong peaks

Step-pool pairs as hydraulic Step-pool pairs as hydraulic energy dissipatorsenergy dissipators• Initial transcritical flow on plane bed Initial transcritical flow on plane bed

develops defined periodicity and spatial develops defined periodicity and spatial variation in energy expenditure at the bedvariation in energy expenditure at the bed

• Evolving topography by maximized Evolving topography by maximized evorsion and abrasion in troughs is evorsion and abrasion in troughs is reinforcingreinforcing

• Two different models result: Two different models result: free-over-fall;free-over-fall; chute-over-fallchute-over-fall

Concentrated jet inflow dissipates through hydraulic jump and flow expansion

0 100 200 300 400 500 600-6

-4

-2

0

2

4

6

8

metres

1

h, metres

Step height data is plus/minus 2m about zero baseline: vertical drop is < 4m;Hydraulic jump length (Lj) may be 80% of pool length (Lp)

Nang Rong

E = 1E = 0.2

Morphological evolution occurs when flows are high: E = 0.4 to 0.2

λr= L/h Range is 6 to 7Indicates likely F1values

ConclusionsConclusions• DBM provide powerful tool to analyse DBM provide powerful tool to analyse

longitudinal variation in periodic longitudinal variation in periodic topography such as step-poolstopography such as step-pools

• Periodic step-pool spacings can be related Periodic step-pool spacings can be related in homogeneous-bedrock systems to in homogeneous-bedrock systems to inherent flow periodicityinherent flow periodicity

• Step-pools may be considered as efficient Step-pools may be considered as efficient hydraulic energy dissipatorshydraulic energy dissipators

• Step height:length relationship indicate Step height:length relationship indicate likely range of Froude numbers at likely range of Froude numbers at formative dischargeformative discharge