SELF-ACCELERATION OF TURBIDITY CURRENTS

1
CEE 598, GEOL 593 TURBIDITY CURRENTS
MORPHODYNAMICS AND DEPOSITS
LECTURE 10 SELF-ACCELERATION OF TURBIDITY CURRENTS
Lets compare the momentum equation for a river
and a turbidity current River Turbidity
current. Consider a turbidity current and a
river, both flowing with the same layer thickness
H (depth in the case of a river) over the same
slope S. The ratio of the gravitational term of
the turbidity current to that of the river
is Now lets consider a turbidity current that
has a value of C of, say, 0.03, which is toward
the high end of what can be considered dilute.
Since R 1.65, the ratio becomes
2
THE INHERENTLY WEAK DRIVING OF TURBIDITY CURRENTS
COMPARED TO RIVERS
For the example considered, then, Now we know
that rivers can excavate deep submarine canyons,
such as the Grand Canyon to the left. If
turbidity currents have such weak driving, how
can they excavate canyons?
3
THE INHERENTLY WEAK DRIVING OF TURBIDITY CURRENTS
COMPARED TO RIVERS contd.
How do they do it?
4
THE SLOPE EFFECT
One way turbidity currents can be made stronger
is by jacking up the bed slope. Let Sf denote
the fluvial bed slope and St denote the slope of
the turbidity current channel. Again considering
the case of the same flow thickness H for both
cases, the ratio of the driving forces is
now Thus if the following relation
holds a turbidity current has the same
driving force as a river. For example, for the
example of the previous slides, And indeed,
the slopes of both canyons and fan channels
associated with turbidity currents tend to be
steeper than rivers of otherwise similar
dimensions.
5
LETS LOOK AT THE AMAZON CANYON/FAN SYSTEM
Pirmez (1994)
6
LONG PROFILE OF THE AMAZON SUBAERIAL/SUBMARINE
SYSTEM
Note that the submarine slopes are much higher
than the subaerial slopes.
Pirmez and Imran, (2003)
7
MORE DETAILS OF THE LONG PROFILE OF THE AMAZON
SUBMARINE CANYON-FAN SYSTEM
Amazondata.xls
8
ELEVATION AND SLOPE PROFILES OF THE AMAZON
SUBMARINE CANYON-FAN SYSTEM
9
DEPTH VERSUS SLOPE AMAZON SUBMARINE CHANNEL
VERSUS SOME OF THE LARGEST RIVERS IN THE WORLD
The river depths are the depths at bankfull flow.
The turbidity current channel depths are thalweg
(deepest point) to levee crest depths, which
exaggerates the depth somewhat (by a factor of
about 1.5?) Yes, the slopes for the turbidity
current channel are much steeper!
10
BUT THERE IS ANOTHER WAY TO MAKE STRONG TURBIDITY
CURRENTS SELF-ACCELERATION
Start them small
And grow them large!
11
CONSIDER THE FORMATION OF SNOW DONUTS
According to Mike Stanford from our WSDOT
avalanche team, snow doughnuts are a natural
occurrence in nature. We do not build them. They
form when there is a hard layer in the snow and
is then covered by several inches of dense snow.
Then you add a steep slope and a trigger, such as
a clump of snow falling out of a tree or off of a
rock face, and voila you have snow doughnuts.
http//www.wsdot.wa.gov/traffic/passes/northcascad
es/2007/pictures
12
SNOW DONUTS AS A CONCEPTUAL EXAMPLE OF
SELF-ACCELERATION
Snow doughnuts seemingly could grow very big if
conditions permitted. The one seen in the
photograph is about 24" in diameter.
http//www.wsdot.wa.gov/traffic/passes/northcascad
es/2007/pictures

• For some reason, a small ball (or donut) of snow
  starts rolling down a hill. If the hill is steep
  enough (and the snow is of the right stickiness),
• gravity accelerates the ball, which then
  gathers more snow as it roll,
• which makes it heavier,
• which makes it go even faster in a
  self-reinforcing cycle.
• The snowball eventually decelerates and comes to
  rest when it reaches the base of the slope.
• This is a way to make a big snowball out of an
  initially small snowball!
• Start it small, grow it big!

See also Fukushima and Parker (1990)
13
THE POSSIBILITY OF OF SELF-ACCELERATION
(IGNITION) AND SELF-DECELERATION OF A TURBIDITY
CURRENT
Consider the 3-equation model. We write Es
Es(U) to indicate that it is a function of
velocity.
Pantin (1979) Parker (1982) Fukushima et al.
(1985) Parker et al. (1986)

• Start off with a small (but not too small)
  current.
• The combination of initial (or upstream) U and
  C is just right so Es lt roC.
• The concentration of sediment C thus increases.
• As a result, the flow gets heavier.
• Because it is heavier, it accelerates and
  increases U
• Increased U leads to increased Es.
• Go back to step 2 and repeat.
• The same process can lead to self-deceleration if
  the current is too small.

14
A SIMPLIFIED CONCEPTUAL MODEL
Here m is a free parameter. Note that the
equations have an equilibrium (fixed point) at
(U, C) (1, 1).
15
A SIMPLIFIED CONCEPTUAL MODEL contd.
We can solve these equations subject to initial
conditions. For example, define t1 0, ti1
ti ?t where ?t is a time step, and Ui and Ci
denote the values of U and C at ti. Using the
simple Euler step method,
U, C
This gives us curves like
t
16
PHASE DIAGRAM
The solution can be represented in terms of a
phase diagram in which C is plotted against U.
In the case conceptually illustrated below, the
current eventually goes into self-acceleration,
with U and C both increasing.
17
CASE m lt 2
In this case the equation set is stable, and the
flow converges to the fixed point (U, C) (1, 1)
for any initial condition. Here the case (Uinit
, Cinit) (1.1, 0.2) is illustrated for m 1.7.
The calculation was performed with
IgnitPlotTurbCurr.xls
18
CASE m gt 2 SELF-ACCELERATION
In this case the equation set is unstable, and
the flow diverges away from the fixed point
(self-acceleration or self-deceleration) for any
initial condition for which (Uinit, Cinit) ? (1,
1). The case below, which illustrates
self-acceleration is for the case (Uinit, Cinit)
(1.3, 0.75) and m 2.5.
19
CASE m gt 2 SELF-DECELERATION
The case below illustrates self-deceleration for
the case (Uinit, Cinit) (1.3, 0.6) and m 2.5.
20
REGIME DIAGRAM
For any case m gt 2, it is possible to determine a
regime diagram that a) divides the
self-accelerating regimes from the
self-decelerating regime, and b) shows a line to
which the solution eventually converges for any
initial condition.
21
TURBIDITY CURRENTS WITH THE 3-EQUATION MODEL
This time we consider flows that are steady in
time but developing in the downstream direction.
The calculations are performed with the
3-equatoin model The upstream boundary
conditions are A necessary condition for
solving the flow downstream is that the initial
flow be supercritical, i.e. Rio (Rgqs/U3)o lt 1.
(This does not mean that subcritical flows
cannot be treated. They require, however,
integration in the upstream rather than
downstream direction.
22
SAMPLE CALCULATION SELF-ACCELERATION
The 3-equation model does not have a fixed point
(equilibrium corresponding to normal flow). This
notwithstanding, both self-accelerating and
self-decelerating regimes can be defined. The
example below shows the calculation conditions
for an example generated with TurbCurrSpatIg3equ.x
ls
23
RESULTS SELF-ACCELERATION
Note that both U and qs increase downstream. U
reaches a very high value of 8 m/s 4000 m
downstream of the starting point. The bed slope
is 0.03.
24
RESULTS SELF-DECELERATION
The calculation conditions are exactly the same
as those of the previous slide, except that the
bed slope is lowered to 0.02.
25
TURBIDITY CURRENTS WITH THE 4-EQUATION MODEL
The governing equations are
and the upstream initial conditions are specified
as
26
SAMPLE CALCULATION WITH THE 4-EQUATION MODEL
The calculation conditions correspond to the
example of self-acceleration fiver for the 3-
equation model. The calculation is performed
using TurbCurrSpatIg4equ.xls.
27
STRATIFICATION EFFECTS CHANGE THE NATURE OF THE
SOLUTION
The effect of damping of the turbulence due to
stratification effects in the 4-equation model
pushes this case into the regime of
self-deceleration.
28
SELF-ACCELERATION WITH THE 4-EQUATION MODEL
If the slope S is increased to 0.075, the flow
does go into self-acceleration. But the velocity
realized 4 km downstream is much less than the
case using the 3-equation model presented
earlier. The inclusion of stratification effects
still allow for self-acceleration, but the
conditions are more stringent.
29
EXCAVATION OF SUBMARINE CANYONS
Self-acceleration is a way by which a relatively
small flow, generated by any condition such as
breaching, hyperpycnal flow, retrogressive delta
failure, or wave action, can generate a current
strong enough to excavate submarine canyons.
Xu, Noble, Rosenfeld, Paull (USGS, NPS, MBARI)
30
WHEN THE CALCULATION FAILS
Under some conditions, the calculation fails at
some point downstream. There are various reasons
for this (e.g. time step too large), but among
them is an important physical reason. Consider
the example input shown below for the 3-equation
model.
The calculation is supposed to be performed out
to x 0.25 x 1000 250 m.
31
WHEN THE CALCULATION FAILS contd.
The numerical model fails at x 110 m, well
short of the intended distance of x 250 m
32
WHAT HAPPENED?
Note that the Richardson number Ri rises to 1 at
x 110 m. This implies that there must be a
hydraulic jump to subcritical flow upstream of
this point, because the bed slope is too low to
support the flow. The precise location of the
hydraulic jump is determined by dowstream
conditions. Its location must be determined by
upstream integration of the backwater equations
from a control point downstream.
hydraulic jump
H
33
IN A SUBMARINE CANYON-FAN SYSTEM, THE DECLINE IN
SLOPE DOWNSTREAM CAN CAUSE A HYDRAULIC JUMP
This jump can often be associated with the
transition from the canyon to the fan. Recall
that subcritical flow has a very low entrainment
entrainment rate of ambient water. This can
allow the flow to stay more or less confined
within a leveed channel.
Hydraulic jump?
34
EVIDENCE FOR THE HYDRAULIC JUMP
The 4-equation model applied to an erodible bed
indicates the formation of sediment waves in the
vicinity of the hydraulic jump. (More about this
will be presented in a subsequent chapter.) A
sample calculation is shown below.
Kostic and Parker (2006)
35
CANYON-FAN TRANSITION ON THE NIGER MARGIN
Lets look here
From Prather and Pirmez (2003)
36
THE SEDIMENT WAVES
sediment waves
37
LIMITS TO SELF-ACCELERATION

• Self-acceleration cannot occur indefinitely. The
  following factors limit it.
• Declining slope downstream must eventually move
  the flow out of the self-acceleration regime.
• Under some conditions, it appears that strong
  stratification effects near slope breaks can turn
  off the turbulence and kill the flow.
• Even in a submarine canyon of sufficient
  steepness to support continued self-acceleration,
  at some point the bed is eroded to bedrock
  (stiff, consolidated clay in many submarine
  canyons), and so no more sediment can be
  entrained (except for that supplied by a slow
  rate of bed incision).
• Such conditions are called bypass
  conditions. The sediment load is simply
  transferred downstream without erosion or
  deposition, so that the suspended sediment load
  becomes constant. In a canyon of constant width,
  then, whatever settles is immediately entrained
  without further bed erosion, and

38
BYPASS CONDITIONS AND EXCAVATION OF SUBMARINE
CANYONS
incision
The passage of repeated turbidity currents that
are powerful enough to reach bypass conditions
can (one flow at a time) slowly cause channel
incision, leading to the formation of the canyon.
Canyon incision cannot occur until the bed is
swept clean of easily erodible sediment.
39
NORMAL FLOW
40
SELF-ACCELERATION DOES NOT APPLY ONLY TO FLOWS
THAT ARE STEADY IN TIME AND DEVELOPING IN SPACE
The calculation uses the 4-equation model
retaining the time terms. It shows how a small
pulse-like flow can self-accelerate into a long,
quasi-continuous flow on a steep slope, and then
decelerate on a lower slope (0 in this case).
Pratson et al. (2001)
41
REFERENCES
Fukushima, Y., Parker, G. and Pantin, H., 1985,
Prediction of igniting turbidity currents in
Scripps Submarine Canyon. Marine Geology, 67,
55-81. Fukushima, Y. and Parker, G., 1990,
Numerical simulation of powder snow avalanches.
Y. Fukushima and G. Parker, Journal of
Glaciology, 36(2), 229 237. Kostic, S. and
Parker, G., 2006, The response of turbidity
currents to a canyon-fan transition internal
hydraulic jumps and depositional signatures,
Journal of Hydraulic Research, 44(5)
631653. Pantin, H. M., 1979, Interaction between
velocity and effective density in turbidity flow
phase-plane analysis, with criteria for
autosuspension, Marine Geology, 31,
59-99. Parker, G., 1982, Conditions for the
ignition of catastrophically erosive turbidity
currents. Marine Geology, 46, pp. 307-327,
1982. Parker, G., Y. Fukushima, and H. M. Pantin,
1986, Self-accelerating turbidity currents.
Journal of Fluid Mechanics, 171, 45-181. Parker,
G., M. H. Garcia, Y. Fukushima, and W. Yu, 1987,
Experiments on turbidity currents over an
erodible bed., Journal of Hydraulic Research,
25(1), 123-147. Pirmez, C., 1994, Growth of a
submarine meandering channellevee system on
Amazon Fan, PhD Thesis, Columbia University, New
York, 587 p. Pirmez, C. and Imran, J., 2003,
Reconstruction of turbidity currents in a
meandering submarine channel, Marine and
Petroleum Geology 20(6-8), 823-849. Prather, B.
E. and Pirmez, C., 2003 Evolution of a shallow
ponded basin, Niger Delta slope, Annual Meeting
Expanded Abstracts, American Association of
Petroleum Geologists, 12, 140-141. Pratson, L. F,
Imran, J, Hutton, E.W.H., Parker, G. and
Syvitski, J.P.M., 2001, Debris Flows vs.
Turbidity Currents a Modeling Comparison of
Their Dynamics and Deposit, Computers
Geosciences 27, 701716.