On modeling the variability of bedform dimensions

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1 On modeling the variability of bedform dimensions Rolien van der Mark A. Blom S.J.M.H. Hulscher S.F. Leclair D. Mohrig RCEM4 October 7, 2005

2 Bedforms Bedforms considered as regular features mean bedform length λ mean bedform height Δ mean angle of downstream face α mean migration speed c flow h α Δ λ

3 Irregular bedforms In reality bedforms are 3-dimensional and highly irregular in size, shape and spacing, even under steady flow conditions

Bed elevation (mm) 190 180 170 160 150 140 0 500 1000 1500 2000

4 Exp SAFL34, Leclair 02 Introduction Results Discussion Conclusions Examples Exp T10, Blom & Kleinhans 99 Exp BU21, Leclair Bed elevation (mm) X co-ordinate (mm) 3D, irregular in height, length, shape

5 Relevance Irregularity of bedforms needs to be taken into account: Shipping & burying of pipelines and cables Thickness of cross-strata sets Bed roughness Vertical sorting

6 Thickness of cross-strata sets Cross-strata sets: Deposited layers within bed Used to interpret ancient flow conditions flow direction Run 34 Leclair 02

7 Thickness of cross-strata sets Cross-strata sets: Result from migrating bedforms Thickness depends on varying bedform height Only deepest troughs leave a record Necessary to incorporate the stochastics of bedform dimensions Flow s From Paola & Borgman (1991)

8 Bed roughness Form roughness depends on size, shape and spacing of bedforms (e.g. Nelson et al, 1993) We expect that variability in bedform dimensions affects the form roughness and thus the total bed roughness Necessary to incorporate the stochastics of bedform dimensions in roughness models

9 Blom et al (2003) Vertical sorting Vertical sorting and its adaptation time scale are strongly related to variability in trough elevations Vertical sorting-model of Blom & Parker (2004) requires sub-model describing (time evolution of) PDF of the trough elevation

10 Objective To develop a model for the stochastics of bedform dimensions Approach The variability of bedform height, trough elevation and crest elevation are examined by analyzing three sets of flume experiments

Flume experiments 3 sets of experiments: Blom 00, Blom & Kleinhans 99, Leclair 02 11 experiments,

11 Flume experiments 3 sets of experiments: Blom 00, Blom & Kleinhans 99, Leclair experiments, bedforms occurred Steady + uniform flow Non-uniform sediment Measurements taken in equilibrium state

12 Flume experiments Individual crests and troughs are gathered from bed elevation profiles Bedform height is defined as vertical distance between crest and subsequent trough crest elevation η b trough elevation Δ bedform height η t mean bed elevation

13 PDFs For all 11 experiments: PDFs of: - bedform height, - crest elevation - trough elevation μ and σ Coefficient of Variation For this experiment: CV = σ/μ = 0.38 Δ (m) Δ (m) PDF of dune height T9 PDF of dune height T9 σ μ probability density (1/m) probability density (1/m)

14 0.05 σ versus μ for all experiments Bedform height σ σ (m) σ = 0.41 μ R 2 = 0.65 Variation σ scales with mean μ μ μ (m) For bedform height: CV = σ/μ 0.4 Confirmed by others: e.g. Gabel 93

15 0.03 Crest elevation σ σ (m) σ = 0.57 μ R 2 = 0.98 For crest elevation: σ/μ μ (m) μ (m) Trough elevation σ (m) σ = 0.64 μ R 2 = 0.69 For trough elevation: σ/μ μ (m) Introduction μ (m) Results Discussion Conclusions

16 Deviations Deviations from this linear trend can be seen, especially for trough elevations. Possible explanations: Turbulent wake in troughs Sediment composition within the bed

17 Deviations Coarse bed layer underneath migrating bedforms reduces variability in trough elevations (e.g. Wilcock & Southard, 1989) Trough elevation 0.03 σ (m) 0.02 Blom et al. (2003) σ = 0.64 μ R 2 = μ (m)

18 Conclusions The coefficients of variation for bedform height, trough elevation and crest elevation are within a narrow range As a first approximation, a constant CV can be used when describing the variability of bedforms Variability in bedform height can be modeled by only predicting the mean bedform height

19 Future work To develop a more generic model for the stochastics of bedform dimensions: 2D 3D Dunes Other types of bedforms Flume Field data Steady Non steady flow Normal Other distributions

21 Definition How do we define bedform dimensions? Lots of definitions suggested No consensus in literature Choice for definition is made subjectively bed elevation trough? mean bed level Δ? Δ? distance crest?

Quantification of variability in bedform geometry

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi:10.1029/2007jf000940, 2008 Quantification of variability in bedform geometry C. F. van der Mark, 1 A. Blom, 2 and S. J. M. H. Hulscher 1 Received 2 November