A v a l a n c h e

D y n a m i c s

a n d

D e f e n ce

slope length,L.Using these parameters, Wederive a simi-
larity law fora simple model of a granular avalanche.
The equation of motion of a group of granular material
is as follows;
mdv/dt=mg(sinq-m cosq) -av(1)
dx/dt=v(2)
wheremandqare themassand inclinationangle of
chute respectively, andmandaare the coefficients of the
resisting force.
To make Eqs.(1) and (2) dimensionless,let velocity,v,
distance,x,and time,t,be

Thismeansthatthe
steady motion is domi-
nant under the scaling
conditionrepresented
by Eq.(12),which also
canbedescribedby
dimensionless number
as:

V_e² / Lg^*<< 1.(14)

Therefore,Eq.(14)or
Eq.(12) governs thefor-
mation of the head-tail
structure.V² /Lg^*is a
kind of Froude^e
number
(wecallthissystem
sizeFroudenumber)
thatV and Larethe ter-
minalvelocityand
slopelength,respec-
tively.

v=Vv,x=Lx ,t=Tt .

(3)

whereVandTare the scales of velocityand time respec-
tively.
Then Eqs.(1) and (2) are

dv/d t=Tg* / V- ( Tg* / Ve)v=Tg* /V(1 - (V/ V)v)(4) dx/dt= (TV/ L)ve (5)

where
mg^* / a=V
e ^{,g*=g(sinq=mcosq).}

If
T=V_e/ g^*=T
L=V_e² / g^*=L^{c ,}
c ^,
V=V_e ,

then Eqs.(4) and (5) are

(6)

(7)
(8)
(9)

IfV<<V_e,

then

dv/dt= 1.

(15)

(16)

dv/dt= 1 -v(10)
dx/dt=v.(11)
This means that both accelerating motion and steady
motion appear under the scales represented by
Eqs.(7)-(9).

If

In this case,only acceleration is dominant,therefore, the
head-tail structure cannot be formed.
In table 3, typicalcases in naturalavalanchesandgranu-
laravalanches are summarized forthesame systemsize
Froude number, 0.1.TTB avalanches on 25 to 100 m slope
and SFP avalanches on 1to4mare similarwithnatural
powdersnow avalancheson a few kiro meters slope, and

V=V_eandL>> L _c,that is,T>>T_c ,

(12)

Fig.8 Schematic diagram of head-tail structure formation.

206