frontal dynamics of powder snow avalanches
DESCRIPTION
Frontal Dynamics of Powder Snow Avalanches. Cian Carroll, Barbara Turnbull and Michel Louge. EGU General Assembly, Vienna, April 27, 2012. Thanks to Christophe Ancey, Perry Bartelt, Othmar Buser, Jim McElwaine, Florence & Mohamed Naiim, Matthew Scase, Betty Sovilla. - PowerPoint PPT PresentationTRANSCRIPT
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Frontal Dynamics of Powder Snow Avalanches
Cian Carroll, Barbara Turnbull and Michel Louge
EGU General Assembly, Vienna, April 27, 2012
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Sponsored by ACS Petroleum Research Fund
Thanks to Christophe Ancey, Perry Bartelt, Othmar Buser, Jim McElwaine,Florence & Mohamed Naiim, Matthew Scase,Betty Sovilla
Sovilla, et al, JGR (2010)
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Field datarapid eruption
Issler (2002) Sovilla et al (2006)
time (s)
heig
ht (m
)
time (s)
stat
ic p
ress
ure
(Pa)
McElwaine & Turnbull JGR (2005)
depression
Sovilla, et al JGR (2006)
slope
width
distance (m) distance (m)
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Consider avalanche head
rapid eruption
Issler (2002)Sovilla et al (2006)
source
avalanche rest frame
avalanche head
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Principal assumptions in the cloud
source
avalanche head • Negligible basal shear stress• Negligible air entrainment• Inviscid• Uniform mixture density
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Rankine half-body potential flow
€
Ri = 2′ ρ − ρ( )
′ ρ g ′ H
′ U 2
€
ζ ≡1− ρ / ′ ρ
€
H → ′ H = H /δSwelling
Rankine, Proc. Roy. Soc. (1864)
€
p + ρu 2
2+ ρgz = ′ p + ′ ρ
′ u 2
2+ ′ ρ gz
€
′ U = δUSlowing
U’
U
€
δ =1−ζ1+ Ri
€
p = ′ p
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Experiments and simulations on eruption currents
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Static pressure in the cloud
€
p − pa
(1/2) ′ ρ ′ U 2 = 2(x / ′ b ) −1(x / ′ b )2 + ( ′ h / ′ b )2
pressure p, air density , cloud density ’ stagnation-source distance b’
fluidized depth h’
€
x / ′ b €
p − pa
(1/2) ′ ρ ′ U 2
€
⇒ surface pressure time - history
prediction
data: McElwaine and Turnbull
JGR (2005)
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Porous snow pack interface
€
∇2 p = 0
pore pressure p
€
′ h
€
′ b
Pore pressure gradients defeat cohesion
rapid eruption Issler (2002)
time (s)
heig
ht (m
)
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Porous snow pack
€
R ≡ 2ρ cg ′ b μ e
′ ρ ′ U 2
snowpack density c, friction e
interface
€
∇2 p = 0
pore pressure p
€
′ h
€
′ b
Pore pressure gradients defeat cohesion
2
y
x
s
12
Mohr-Coulomb failure
€
′ h ′ b
€
R€
h'b'
≈ 1Ra1
€
a1 ≈ 0.42
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Frontal Dynamics
€
∂p∂s
= 0
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Mass balance
€
˙ m e = ′ ρ ′ H W( ) ′ U
€
˙ m s = ρ c λ ′ h cosα W( ) U
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Mass balance
€
˙ m s = ˙ m e ⇔′ h ′ b
= πρρ c cosα
⎛ ⎝ ⎜
⎞ ⎠ ⎟
1λ 1+ Ri( ) 1−ζ( )
snowpack density c, friction e, inclination , entrained fraction of fluidized depth h’
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Stability
€
˙ m s = ˙ m e ⇔′ h ′ b
= πρρ c cosα
⎛ ⎝ ⎜
⎞ ⎠ ⎟
1λ 1+ Ri( ) 1−ζ( )
snowpack density c, friction e, inclination , entrained fraction of fluidized depth h’
€
′ h ′ b ⇒ (Ri,ζ )Snowpack eruption feeds the cloud:
Cloud pressure fluidizes snowpack:
€
(Ri,ζ )⇒ ′ h ′ b
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Stability diagram
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€
ζ ≡1− ρ′ ρ
€
Ri = 2′ ρ − ρ( )
′ ρ g ′ H
′ U 2
unstable Ri stable ζ unstable
stableRi unstable ζ stable
cloud height
density
€
′ H = (1− 2a1)U 2
2g
€
′ h = ρU 2(1− 2a1)2gρ c cosα
€
′ =
11− 2a1
entrained depth€
=1/χ 0
€
=1.05 /χ 0
€
χ0 = a0 cosαμ e
a1
ρ c
πρ ⎛ ⎝ ⎜
⎞ ⎠ ⎟
1−a1
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Frontal Dynamics
€
∂∂t
′ ρ ′ b 2W aVU − aM ′ U ( ) + ρ ′ b 2W aμU[ ] = aV ′ b 2Wρgsinα
acceleration momentum added mass weight + buoyancy
€
aV ≈ 3
€
aM ≈ 3.3
€
aμ ≈ 3.3
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Acceleration
€
∂U∂t
= 1− 2a1
1+ δaM /aV
⎛ ⎝ ⎜
⎞ ⎠ ⎟gsinα − δaM /aV
1+ δaM /aV
⎛ ⎝ ⎜
⎞ ⎠ ⎟U 2 d lnW
dx
gravity channel width W
distance (m) distance (m)
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Other predictions
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Height vs distance
cloud height
€
′ H = (1− 2a1)U 2
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Froude number vs distance
cloud Froude number
€
2g ′ H U 2 = (1− 2a1)
Vallet, et al, CRST (2004)Sovilla, Burlando & Bartelt JGR (2006)
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Volume growth
volume growth
€
V = H 'WUdt∫
Measurements: Vallet, et al, CRST (2004)
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total volume
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Impact pressure ≠ static pressure
Cloud arrest
€
pI = ′ p + 12
′ ρ urel2
€
pI
ρ2
U 2= 2 −ζ
1−ζ ⎛ ⎝ ⎜
⎞ ⎠ ⎟+ 2
1−ζ ⎛ ⎝ ⎜
⎞ ⎠ ⎟
ˆ x ˆ x 2 + ˆ y 2
−δ ⎡ ⎣ ⎢
⎤ ⎦ ⎥− 2 ˆ y β
(1−ζ )Impact
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pI /(ρ /2)U 2
€
x /b
increasing heightAn impact pressure
decreasing with heightdoes not necessarily
imply densitystratification.
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Air entrainment
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Air entrainment into the head
€
˙ m air
˙ m source
≈ 18
(1−ζ )2 1− exp −b /rc( )[ ] source radius rc
€
ζ =1− ρ / ′ ρ €
˙ m air
˙ m source
€
˙ m air
˙ m source
< 31/ 2
πδ(1−ζ ) f v, with fv =
1− 2Ria2 for Ria <1
0.2 /Ria otherwise, Ria ≡ Riδ 2 cosα
2(1−ζ )
Ancey, JGR (2004)
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Conclusions• Our model of eruption currents is closed without
material input from surface erosion or interface air entrainment.
• Porous snowpacks synergistically eject massive amounts of snow into the head of powder clouds.
• Suspension density swells the cloud and weakens its internal velocity field.
• Mass balance stability sets cloud growth.• Changes in channel width affect acceleration.• Experiments should record cloud density and pore
pressure.
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Thank you
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Cian Carroll
Barbara Turnbull
Betty Sovilla
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