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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−ζ
1+ 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
1
2
Mohr-Coulomb failure
€
′ h ′ b
€
R€
h'
b'≈
1
Ra1
€
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
=πρ
ρ c cosα
⎛
⎝ ⎜
⎞
⎠ ⎟
1
λ 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
=πρ
ρ c cosα
⎛
⎝ ⎜
⎞
⎠ ⎟
1
λ 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
€
′ =
1
1− 2a1
entrained depth€
=1/χ 0
€
=1.05 /χ 0
€
χ0 =a0 cosα
μ ea1
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
2g
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Vallet, et al, CRST (2004) QuickTime™ and a decompressor
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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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air entrainment in the tail
total volume
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Impact pressure ≠ static pressure
Cloud arrest
€
pI = ′ p +1
2′ ρ urel
2
€
pI
ρ
2U 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
≈1
8(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 ) fv, 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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