1. 1 A unified description of ripples and dunes in rivers 5 m Douglas Jerolmack, Geophysics, MIT; douglasj@mit.edu With David Mohrig and Brandon McElroy

2. 2 Scaling of bedforms 0.01 0.1 1 10 100 1000 0.001 0.01 0.1 1 10 100 Wavelength, λ [m] Height, H [m] Nikora et al., J. Fluid Mech., 1997 Flemming, Proc. Marine Sandwave Dynamics, 2000 Hino, J. Fluid Mech., 1968 Morphometric distinction between ripples and dunes based partly on this plot [Ashley, J. Sed. Petr., 1990] → λ = 0.6 m

3. 3 Extraction of topographic data from images 2 m Plan view image of N. Loup River Depth map from brightness [m] Time [min] 20 40 60 80 100 120 Profiles of bed evolution: ∆t = 6 min Space-time plot of bed evolution: ∆t = 2 min 1/c

4. 4 Roughness and Statistical Steady State l N. Loup River topo. data Scaling regime Rollover regime l x = 1.5 m 0.1 0.01 Window size, l [m] 0.001 1 10 100 0.01 0.1 Interface width, w [m] α = 0.64 f Wavenumber = 2π/l, k [m -1 ] 1 0.1 10 100 1000 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 Spectral density [m 3 ] w x = 0.02 m l x = 1.5 m Saturation regime w ~ l α -> α is the roughness exponent, characterizing scaling of η fluctuations Power Spectra: f = -(2α + 1)

5. 5 Interactions and feedback at sediment-fluid interface What are the necessary ingredients for realistic evolution of a train of bedforms? Data suggest that the same organizing processes act across all scales 1. Conservation of mass: 2. Meyer-Peter Müller: 3. Parameterization: Fluid Sediment flux Topography Modeling approach 1 32 x z y ‹h› η =0 - η +η+η

6. 6 A new “local growth model” Bed stress depends on elevation AND slope: (e.g., Smith, 1970) What about lateral transport? - Slope dependent, e.g. Murray and Paola, Earth Surf. Proc. Landforms, 1997. → Solve as 1D slices in the transport direction, and couple laterally via diffusion (Hersen, Phys. Rev. E, 2004). What about turbulence? – treat as stochastic variability in sediment flux, as observed by Gomez and Phillips, J. Hydr. Eng., 1999. Noise Lateral Diffusion Constant Nonlinear → n > 1! Topography Add an avalanching term for angles greater than repose

7. 7 Deterministic evolution Time x y a b Self-organizing bedforms grow from flat surface with small perturbations. Bedform merging occurs due to varying migration speeds. Sinuous crested bedforms develop whose crests occupy entire domain width. At long time, bed evolves toward a uniform, periodic train of straight-crested bedforms.

8. 8 Noisy evolution: dynamic steady-state Time [-] y [m] x [-] a b Plan view of bed in dynamic steady-state Profiles of bed in dynamic steady-state Addition of noise has profound influence on morphology and dynamics Bedform splitting and merging an ongoing process Bed roughness achieves statistical steady-state, but individual bedforms have a short life time New bedforms are created because noise continuously creates perturbations that grow

9. 9 Space-time plots of N. Loup and model N. Loup River profiles Model profiles Large dunes advance by spontaneous emergence of bedforms in troughs, which migrate and grow across dune and disappear in following trough As seen by Jain and Kennedy, J. Fluid Mech., 1974; Nikora et al., J. Fluid Mech., 1997; Harbor, J. Sed. Res., 1998

10. 10 t/t eq w/w eq Run 2 Run 3 Growth scaling of model – comparison to laboratory-derived relations Deterministic growth Noisy growth Exponential Growth – low transport stage → Ripples? Power Law Growth – high transport stage → Dunes? Scaling is insensitive to parameter values and noise amplitude. Most important thing is n>1

11. 11 Spatial roughness scaling – noisy model Again, normalized scaling not sensitive to coefficients 1 0 2 3 4 5 0 0.4 0.8 1.2 1.6 2.0 Normalized interface width, w/w x Normalized window size, l/l x lxlx wxwx Note Linear axes – inset is log-log

12. 12 Conclusions In many natural rivers, scale invariance exists below wavelength of largest bedforms → Scale of largest bedforms determined by mean channel properties → No morphometric basis to distinguish “ripples” from “dunes” Parameterization of shear stress in terms of local topography and noise reproduces temporal and spatial statistics of bed evolution → Bedforms at all scales arise from same transport processes → Non-local nature of fluid flow may be neglected for some problems Does deterministic model correspond to the ‘ripple instability’, while noisy model is the ‘dune instability’? Work supported by the National Center for Earth-surface Dynamics. Motivated by discussions at the “Novel methods for modeling the surface evolution of geomorphic interfaces” workshop (NCED) Contact: douglasj@mit.edu