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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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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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But a thick shroud of smog produced by the breakdown of methane under ultraviolet light prevented any surface visualization.

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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This and other images of Titan courtesy European Space Agency and NASA

Cassini/Huygens Mission:

very strong evidence for

rivers of liquid methane carrying sediment of water ice

AND THEN JANUARY 14, 2005 ARRIVED!

I was glued to the internet! I had waited for years!


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MARS VERSUS TITAN Agency and NASA

Mars shows evidence of ancient rivers of flowing water that carried sediment similar to that of the Earth’s crust.


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MARS VERSUS TITAN contd. Agency and NASA

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. Agency and NASA

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. Agency and NASA

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. Agency and NASA

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. Agency and NASA

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. Agency and NASA

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? Agency and NASA

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 Agency and NASAEARTH 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 GEOMETRY HERE ON EARTHEARTH

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 GEOMETRY HERE ON EARTH 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 GEOMETRY HERE 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,

  • doi:10.1029/2001WR000898, 2002.


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WHAT THE DATA SAY: WIDTH, DEPTH, SLOPE GEOMETRY HERE ON EARTH

The four independent sets of data form a coherent set!


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REGRESSION RELATIONS BASED ON THE DATA GEOMETRY HERE ON EARTH

To a high degree of approximation,

Remarkable, no?


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WHAT DOES THIS MEAN? GEOMETRY HERE ON EARTH



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THE PHYSICS BEHIND IT ALL GEOMETRY HERE ON EARTH

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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GENERALIZATION FOR OTHER PLANETS/SATELLITES OBSERVED DIMENSIONLESS 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 (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 RELATIONS FOR

TO


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CONSIDER A STREAM WITH THE SAME BANKFULL DISCHARGE Q RELATIONS FORbf 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 Q RELATIONS FORbf 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 Q RELATIONS FORbf 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! RELATIONS FOR

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 RELATIONS FOR

Besides, the dynamics of sediment transport becomes approximately invariant to particle Reynolds number for

or

D >~ 8.8 mm on Earth

or

D >~ 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? RELATIONS FOR

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? RELATIONS FOR


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