DIMENSIONLESS BANKFULL HYDRAULIC RELATIONS FOR EARTH AND TITAN

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DIMENSIONLESS BANKFULL HYDRAULIC RELATIONS FOR
EARTH AND TITAN
European Space Agency
Gary Parker Dept. of Civil Environmental
Engineering and Dept. of Geology University of
Illinois
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UNTIL RECENTLY TITAN WAS SHROUDED IN MYSTERY

• What we knew or could reasonably infer
• Larger than Mercury
• Atmospheric pressure 1.5 Earth atmospheres near
  surface
• 95? K near surface
• Atmosphere of nitrogen (mostly), methane, ethane
• Crustal material of water/ice
• Near triple point of methane/ethane possibility
  of
• a. methane/ethane oceans
• b. methane/ethane precipitation as liquid/solid
• 7. Possibility of rivers of liquid methane
  carrying sediment of solid water ice!

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AND THEN JANUARY 14, 2005 ARRIVED!
I was glued to the internet! I had waited for
years!
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MARS VERSUS TITAN
Mars shows evidence of ancient rivers of flowing
water that carried sediment similar to that of
the Earths crust.
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MARS VERSUS TITAN contd.
But the era of flowing rivers was a long time
ago, as evidenced by the fairly intense impact
cratering of Mars, and may not has lasted very
long as compared to Earth.
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MARS VERSUS TITAN contd.
Tectonic ridges?
Titan shows evidence of active tectonics,
vulcanism, aeolian and fluvial reworking, and has
very few impact craters so its surface is likely
active in modern geological time!
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MARS VERSUS TITAN contd.
Volcano?
Titan shows evidence of active tectonics,
vulcanism, aeolian and fluvial reworking, and has
very few impact craters so its surface is likely
active in modern geological time!
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MARS VERSUS TITAN contd.
Aeolian dunes?
Titan shows evidence of active tectonics,
vulcanism, aeolian and fluvial reworking, and has
very few impact craters so its surface is likely
active in modern geological time!
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MARS VERSUS TITAN contd.
River drainage basin?
Titan shows evidence of active tectonics,
vulcanism, aeolian and fluvial reworking, and has
very few impact craters so its surface is likely
active in modern geological time!
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MARS VERSUS TITAN contd.
Impact crater
Titan shows evidence of active tectonics,
vulcanism, aeolian and fluvial reworking, and has
very few impact craters so its surface is likely
active in modern geological time!
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ALLUVIAL GRAVEL-BED RIVERS ON TITAN?
The evidence suggests that at least near where
Huygens touched down, there is a plethora of
alluvium in the gravel and sand sizes. The
gravel presumably consists of water ice and
appears to be fluvially rounded.
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CAN OUR KNOWLEDGE OF ALLUVIAL GRAVEL-BED RIVERS
ON EARTH HELP US MAKE INFERENCE ABOUT TITAN?
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IF WE KNEW THE PHYSICS BEHIND RELATIONS FOR
BANKFULL GEOMETRY HERE ON EARTH

• Bankfull Depth Hbf (Qbf)0.4
• Bankfull Width Bbf (Qbf)0.5
• Bed Slope S (Qbf)-0.3
• where Qbf bankfull discharge

we might be able to extend the relations to Titan.
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WE BEGIN WITH EARTH
The Parameters Qbf bankfull discharge
(m3/s) QbT,bf volume bedload transport rate at
bankfull discharge (m3/s) Bbf bankfull width
(m) Hbf bankfull depth (m) S bed slope
(1) D surface geometric mean or median grain
size (m) ? density of water (kg/m3) ?s density
of sediment (kg/m3) R (?s/ ?) 1 submerged
specific gravity of sediment 1.65
(1) g gravitational acceleration
(m/s2) ? kinematic viscosity of water
(m2/s) The forms sought dimensionless versions
of
Why dimensionless? In order to allow scaling
between Earth and Titan!
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Meet my friends the DIMENSIONLESS PARAMETERS
Particle Reynolds number
Dimensionless bankfull discharge
Dimensionless bankfull depth
Dimensionless bankfull width
Down-channel bed slope
Dimensionless bankfull bedload transport rate
Bankfull Shields number
Shields number at threshold of motion
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DATA SETS FOR GRAVEL-BED RIVERS ON EARTH

• Alberta streams, Canada1
• Britain streams (mostly Wales)2
• Idaho streams, USA3
• Colorado River, USA (reach averages)
• 1 Kellerhals, R., Neill, C. R. and Bray, D. I.,
  1972, Hydraulic and
• geomorphic characteristics of rivers in Alberta,
  River Engineering
• and Surface Hydrology Report, Research Council of
  Alberta, Canada,
• No. 72-1.
• 2 Charlton, F. G., Brown, P. M. and Benson, R.
  W., 1978, The
• hydraulic geometry of some gravel rivers in
  Britain, Report INT 180,
• Hydraulics Research Station, Wallingford,
  England, 48 p.
• 3 Parker, G., Toro-Escobar, C. M., Ramey, M. and
  Beck S., 2003,
• The effect of floodwater extraction on the
  morphology
• of mountain streams, Journal of Hydraulic
  Engineering, 129(11),
• 2003.
• 4 Pitlick, J. and Cress, R., 2002, Downstream
  changes in the channel of a
• large gravel bed river, Water Resources Research
  38(10), 1216,
• doi10.1029/2001WR000898, 2002.

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WHAT THE DATA SAY WIDTH, DEPTH, SLOPE
The four independent sets of data form a coherent
set!
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REGRESSION RELATIONS BASED ON THE DATA
To a high degree of approximation,
Remarkable, no?
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WHAT DOES THIS MEAN?
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WHAT THE DATA SAY BANKFULL SHIELDS NUMBER
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THE PHYSICS BEHIND IT ALL
Assume the following relations. Manning-Strickler
resistance relation Parker-Einstein bedload
relation Relation for bankfull Shields number
Channel form relation of type of Parker
(1978) Gravel yield relation
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THE RELATIONS OF THE PREVIOUS SLIDE YIELD
PRECISELY THE OBSERVED DIMENSIONLESS RELATIONS!
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GENERALIZATION FOR OTHER PLANETS/SATELLITES
Manning-Strickler resistance relation Parker-Ei
nstein bedload relation Relation for bankfull
Shields number Channel form relation of type
of Parker (1978) Gravel yield relation
(volume to mass)

The presence of g and R allow us to go from
to
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BACK-CALCULATED DIMENSIONALLY HOMOGENEOUS
BANKFULL HYDRAULIC RELATIONS FOR ALLUVIAL GRAVEL
RIVERS ON

ARBITRARY HEAVENLY BODIES
The presence of g and R allow us to go from
to
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FROM
TO

Parameter Earth Titan
Pressure E-atmo p 1 1.5
Temperature ?K T 293 95
Grav. accel. m/s2 g 9.81 1.40
Fluid dens. kg/m3 ? 1000 446
Sed. Dens. kg/m3 ?s 2650 931
(?s/?) - 1 R 1.65 1.09
Kin. Viscosity m2/s ? 1.00x10-6 4.04x10-7
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CONSIDER A STREAM WITH THE SAME BANKFULL
DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE
D HOW SHOULD TITAN COMPARE WITH EARTH?

E Earth, T Titan
From
to
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CONSIDER A STREAM WITH THE SAME BANKFULL
DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE
D HOW SHOULD TITAN COMPARE WITH EARTH?

E Earth, T Titan
1.48 x 0.83 1.23
1.57 x 1.56 2.46
0.72 x 0.80 0.57
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SO FOR THE SAME BANKFULL DISCHARGE Qbf AND
CHARACTERISTIC GRAIN SIZE D

A gravel-bed river on might be 1.23 x the
bankfull depth, 2.46 x the bankfull width
and 0.57 x the down-channel slope of a
gravel-bed river on
Could braiding be more common on Titan?
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BUT WAIT A MINUTE! IS GRAVEL ON TITAN GRAVEL
ON EARTH?

For dynamic similarity in grain Reynolds
number or or So the answer is yes
to a reasonable approximation!
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GRAIN REYNOLDS INVARIANCE

Besides, the dynamics of sediment transport
becomes approximately invariant to particle
Reynolds number for or D gt 8.8 mm on
Earth or D gt 10.6 mm on Titan based on the
condition ?c/?c,asymp ? 0.90 using
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WHAT ABOUT AEOLIAN PROCESSES ON TITAN?
Let Ua wind velocity, ?a atmospheric density,
Cf drag coefficient, ?s sediment density, D
grain size. Scaling for mobility of grain size
D Atmospheric density Earth 293?K 1 E-atmo,
?a 1.21 kg/m3 Titan (nitrogen) 95?K 1.5
E-atmo, ?a 5.39 kg/m3 Assuming Reynolds
invariance (Cf ? constant), critical velocity Uac
to blow around size D scales as Much easier to
blow sediment around on Titan! But much less
solar heating to drive meteorology!
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QUESTIONS OR COMMENTS?