Download - Frontal Dynamics of Powder Snow Avalanches

Frontal Dynamics of Powder Snow Avalanches

Cian Carroll, Barbara Turnbull and Michel Louge

EGU General Assembly, Vienna, April 27, 2012

QuickTime™ and a decompressor

are needed to see this picture.

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)



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)



Consider avalanche head

rapid eruption

Issler (2002)Sovilla et al (2006)

source

avalanche rest frame

avalanche head



Principal assumptions in the cloud

source

avalanche head• Negligible basal shear stress• Negligible air entrainment• Inviscid• Uniform mixture density



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



Experiments and simulations on eruption currents





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)



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)



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



Frontal Dynamics

€

∂p

∂s= 0

avalanche headQuickTime™ and a

decompressorare needed to see this picture.

Mass balance

€

˙ m e = ′ ρ ′ H W( ) ′ U

€

˙ m s = ρ c λ ′ h cosα W( ) U



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’



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



Stability diagram



€

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



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



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)



Other predictions



Height vs distance

cloud height

€

′ H = (1− 2a1)U 2

2g



Vallet, et al, CRST (2004) QuickTime™ and a decompressor


Froude number vs distance

cloud Froude number

€

2g ′ H

U 2= (1− 2a1)

Vallet, et al, CRST (2004)Sovilla, Burlando & Bartelt JGR (2006)





Volume growth

volume growth

€

V = H 'WUdt∫

Measurements: Vallet, et al, CRST (2004)



air entrainment in the tail

total volume



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


are needed to see this picture.€

pI /(ρ /2)U 2

€

x /b

increasing heightAn impact pressure

decreasing with heightdoes not necessarily

imply densitystratification.



Air entrainment



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)





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.



Thank you









Cian Carroll

Barbara Turnbull

Betty Sovilla

