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

avalanche headQuickTime™ and a

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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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