dimensionless bankfull hydraulic relations for earth and titan
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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DIMENSIONLESS BANKFULL HYDRAULIC RELATIONS FOR EARTH AND TITAN
Gary ParkerDept. of Civil & Environmental Engineering and Dept. of Geology
University of Illinois
European Space Agency
2
UNTIL RECENTLY TITAN WAS SHROUDED IN MYSTERYWhat we knew or could reasonably infer:1. Larger than Mercury2. Atmospheric pressure ~ 1.5 Earth atmospheres near surface3. ~ 95 K near surface4. Atmosphere of nitrogen (mostly), methane, ethane5. Crustal material of water/ice6. Near triple point of methane/ethane: possibility of
a. methane/ethane oceansb. methane/ethane precipitation as liquid/solid
7. Possibility of rivers of liquid methane carrying sediment of solid water ice!
But a thick shroud of smog produced by the breakdown of methane under ultraviolet light prevented any surface visualization.
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AND THEN JANUARY 14, 2005 ARRIVED!
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
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 Earth’s 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.
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!
Tectonic ridges?
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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 EARTHThe 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 ofbTsbh n
bfbf,bTnbf
nbfbf
nbfbf Q~Q,Q~S,Q~B,Q~H
Why dimensionless?In order to allow scaling between Earth and Titan!
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Meet my friends the DIMENSIONLESS PARAMETERS
5/2bf
bf5/1
QHgH~
Particle Reynolds number
2bf,bT
T DgDQ
Q̂
5/2bf
bf5/1
QBgB~
2bf
DgDQQ̂
DRgDpRe
RDSHbf
bf
Dimensionless bankfull discharge
Dimensionless bankfull depth
Dimensionless bankfull width
Dimensionless bankfull bedload transport rate
Bankfull Shields number
Shields number at threshold of motion]1006.022.0[5.0 )7.7(6.0
pc
6.0p ReRe
S Down-channel bed slope
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DATA SETS FOR GRAVEL-BED RIVERS ON EARTH
1. Alberta streams, Canada1
2. Britain streams (mostly Wales)2
3. Idaho streams, USA3
4. 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 morphologyof mountain streams, Journal of Hydraulic Engineering, 129(11), 2003.4 Pitlick, J. and Cress, R., 2002, Downstream changes in the channel of alarge gravel bed river, Water Resources Research 38(10), 1216,doi:10.1029/2001WR000898, 2002.
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WHAT THE DATA SAY: WIDTH, DEPTH, SLOPEThe four independent sets of data form a coherent set!
0.0001
0.001
0.01
0.1
1
10
100
1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07
Qhat
Btil
de, H
tilde
, S
Britain widthAlberta widthIdaho widthColorado widthBritain depthAlberta depthIdaho depthColorado depthBritain slopeAlberta slopeIdaho slopeColorado slope
H~
B~
S
S,H~,B~
Q̂
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y = 0.3785x4E-05
y = 4.6977x0.0661
y = 0.1003x-0.3438
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
Qdim
Bdi
mtil
de, H
dim
tilde
, S
BdimtildeHdimtildeSPower (Hdimtilde)Power (Bdimtilde)Power (S)
REGRESSION RELATIONS BASED ON THE DATA3438.00661.000004.0 Q̂1003.0S,Q̂698.4B~,Q̂3785.0H~
To a high degree of approximation,
3785.0H~H~ c Remarkable, no?
Q̂
S,H~,B~
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WHAT DOES THIS MEAN?
4.0bf5/1bf
4.0bfbf
Qg3785.0H
orQ~H
4661.0bf
5/10661.02/5bf
4661.0bfbf
QgDg698.4B
orQ~B
3438.0bf
3438.02/5
3438.0bf
QDg1001.0S
orQ~S
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WHAT THE DATA SAY: BANKFULL SHIELDS NUMBER
62.1r,03.0c
bfc
)average(0486.0~bf
Shields Diagram with Threshold for Motion, Threshold for Significant Suspension and Bankfull Shields Number for Gravel-bed Streams
0.001
0.01
0.1
1
10
1 10 100 1000 10000 100000 1000000
Rep
*
suspensionmotionAltaBritIdaColoAverage
threshold of motion (modified Shields curve)
threshold for significant suspension
]1006.022.0[5.0 )7.7(6.0pc
6.0p ReRe
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THE PHYSICS BEHIND IT ALLAssume 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
2645.0bf
bfbfbf
bf
DH732.3
gHHBQCz
5.4
bf
c2/3bf
bf
bf,bT 12.11DRgDB
Q
0562.0bf Q̂02301.0
62.1rc
bf
5504.0T Q̂003176.0Q̂
1
10
100
1 10 100 1000
Hhat
Cz
CzFitCz
D/Hbf
0.001
0.01
0.1
1
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
Qhat
taus
bf tausbfFitQ
Q̂
bf
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THE RELATIONS OF THE PREVIOUS SLIDE YIELD PRECISELY THE OBSERVED DIMENSIONLESS
RELATIONS!
3438.0
0661.0
o
Q̂1003.0S
Q̂698.4B~
3785.0H~H~
y = 0.3785x4E-05
y = 4.6977x0.0661
y = 0.1003x-0.3438
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
Qdim
Bdi
mtil
de, H
dim
tilde
, S
S,H~,B~
Q̂
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GENERALIZATION FOR OTHER PLANETS/SATELLITESManning-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)
2645.0bf
bfbfbf
bf
DH732.3
gHHBQCz
5.4
bf
c2/3bf
bf
bf,bT 12.11DRgDB
Q
0562.0
2bfbf
DgDQ02301.0
RDSH
623.1rc
bf
5504.0
2bf
2bf,bT
DgDQ
R100841.0
DgDQ
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
The presence of g and R allow us to go from
to
4.0bf5/1
7908.0
bf Qg
)R1(1751.0H
1653.0
4661.0bf
2331.0bf DQ
g)R1(R992.15B
3438.0bf
8595.0
7908.0
1719.0
QD
)R1(gR1314.0S
ARBITRARY HEAVENLY BODIES
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FROM TO
Parameter Earth TitanPressure E-atmo p 1 1.5Temperature K T ~ 293 ~ 95Grav. accel. m/s2 g 9.81 1.40Fluid dens. kg/m3 1000 446Sed. 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?
From
to
7908.0
E
T
5/1
E
T
E,bf
T,bf
)R1()R1(
gg
HH
1
E
T
2/1
E
T
2331.0
E
T
E,bf
T,bf
)R1()R1(
RR
gg
BB
7908.0
E
T
E
T
1719.0
E
T
E
T
)R1()R1(
RR
gg
SS
E = Earth, T = Titan
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CONSIDER A STREAM WITH THE SAME BANKFULL DISCHARGE Qbf AND CHARACTERISTIC GRAIN SIZE D
HOW SHOULD TITAN COMPARE WITH EARTH?
7908.0
E
T
5/1
E
T
E,bf
T,bf
)R1()R1(
gg
HH
1
E
T
2/1
E
T
2331.0
E
T
E,bf
T,bf
)R1()R1(
RR
gg
BB
7908.0
E
T
E
T
1719.0
E
T
E
T
)R1()R1(
RR
gg
SS
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 be1.23 x the bankfull depth,2.46 x the bankfull width and0.57 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!
E,pT,p ReRe
E
EEEE
T
TTTT DDgRDDgR
20.1gg
RR
DD
3/1
E
T
3/1
E
T
3/2
E
T
E
T
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GRAIN REYNOLDS INVARIANCE
3320~p Re
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
]1006.022.0[5.0 )7.7(6.0pc
6.0p ReRe
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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 293K 1 E-atmo, a = 1.21 kg/m3
Titan (nitrogen) 95K 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!
WHAT ABOUT AEOLIAN PROCESSES ON TITAN?
Es
2af
Ts
2af
gDUC
gDUC
aa
16.0gg
UU
2/1
E
T
2/1
E,s
T,s
2/1
E,a
T,a
E,ac
T,ac
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QUESTIONS OR COMMENTS?