A unified description of ripples and dunes in rivers
5m
Douglas Jerolmack, Geophysics, MIT; douglasj@mit.edu
With David Mohrig and Brandon McElroy
1
Scaling of bedforms
100
Height, H [m]
10
H  0.06770.81
H max  0.160.84
Nikora et al., J. Fluid Mech., 1997
1
0.1
0.01
0.001
0.01
0.1
1
10
100
1000
Wavelength, λ [m]
Flemming, Proc. Marine Sandwave Dynamics, 2000
Morphometric distinction between ripples and
dunes based partly on this plot [Ashley, J.
Sed. Petr., 1990] → λ = 0.6 m
2
Hino, J. Fluid Mech., 1968
Extraction of topographic data from images
Plan view image of N. Loup River
Depth map from brightness [m]
2m
Space-time plot of bed evolution: ∆t = 2 min
120
1/c
100
80
60
40
20
3
Time [min]
Profiles of bed evolution: ∆t = 6 min
Roughness and Statistical Steady State
l
w ~ lα
1 N

RMS ( )  w    (i   ) 2 
 N i 1

-> α
1/ 2
is the roughness exponent, characterizing scaling of η fluctuations
N. Loup River
topo. data
100
0.1
Power Spectra: f = -(2α + 1)
10-2
wx = 0.02 m
0.01
Spectral density [m3]
Interface width, w [m]
w  0.0186l 0.64
Saturation regime
α = 0.64
Rollover
regime
lx = 1.5 m
0.01
0.1
f
10-6
10-8
Scaling regime
0.001
10-4
1
Window size, l [m]
10
100
10-10
0.1
lx = 1.5 m
1
10
100
Wavenumber = 2π/l, k [m-1]
4
1000
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
z
‹h›
+η
Fluid
y
x
η =0
-η
2
Sediment flux
3
Topography
1
Modeling approach
• 1. Conservation of mass:    1
t
qs
(1  p) x
• 2. Meyer-Peter Müller:
qs  m n
• 3. Parameterization:
   b  f (topography)
5
A new “local growth model”
Bed stress depends on elevation AND slope:
(e.g., Smith, 1970)

 

 ( x )   b 1  A  B 

x 
h
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.
Nonlinear → n > 1!
Constant

n  A 
 2 

 
  qs
 B 2 1  A
B

t
(1  p)  h x
x 
h
x 
n 1
 2
 D 2   ( x, y , t )
y
Lateral
Diffusion
Noise
Topography
6
Add an avalanching term for angles greater than repose
Deterministic evolution
a
y
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.
b
Time
At long time, bed evolves toward a uniform,
periodic train of straight-crested bedforms.
x
7
Noisy evolution: dynamic steady-state
Plan view of bed in dynamic steady-state
a
Addition of noise has profound influence
on morphology and dynamics
y [m]
Bedform splitting and merging an
ongoing process
Bed roughness achieves statistical
steady-state, but individual bedforms
have a short life time
Profiles of bed in dynamic steady-state
Time [-]
b
New bedforms are created because
noise continuously creates perturbations
that grow
8
x [-]
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;
9
Harbor, J. Sed. Res., 1998
Growth scaling of model –
comparison to laboratory-derived relations
w/weq
Deterministic growth
Run 2
Exponential Growth – low transport stage
→ Ripples?
w
 1 e
weq

t
t eq
,  6
Noisy growth
w/weq
Power Law Growth – high transport stage
→ Dunes?

w  t 

,   0.28


weq  teq 
Run 3
t/teq
Scaling is insensitive to parameter values
and noise amplitude. Most important 10
thing is n>1
Spatial roughness scaling – noisy
model
2.02
Series1
N. Loup
1.6
1.6
w / w(lx)
Normalized interface width, w/wx
lx
model
1.2
1.2
wx
10
0.8
0.8
1
0.4
0.4
0.1
0.01
00
00
11
22
0.1
33
1
44
10
55
Normalized window
size,
l/lx
w indox
/ lx
Note Linear axes – inset is log-log
Again, normalized scaling not sensitive to coefficients
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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
12
evolution of geomorphic interfaces” workshop (NCED)
Contact: douglasj@mit.edu