Transcript 低粘性マグマの噴火様式 - University of New

Conduit models (WK-1, YK-1)
to investigate transition between
explosive and effusive eruptions
T. Koyaguchi
University of Tokyo
Collaboration with:
Andy Woods, Shigeo Yoshida, Helene Massol, Noriko Mitani
etc.
Explosive or Effusive
Unzen, 1991
Pinatubo, 1991
Key observations
What is the minimal model
to explain these extreme eruption styles?
Basic Equation for WK-1
(Woods and Koyaguchi, 1993)
Mass conservation
d  (1  n)v
0
dz
liquid：
gas：

[  v{n  sP1/ 2 (1  n)}]  Qw
dz
Qw  2 rn K ( P  Ph   ) / h
 : overpressure parameter
Momentum conservation
Equation of State
v
dv
1 dP

gF
dz
 dz
 8 v
  r 2
F 
2
0.0025 v

r
1

1 n

nRT
P

l
n  n0  sP1/ 2
T  constant
Normalized depth
A method to systematically investigate
the features of solutions (Shooting method)
lithostatic
hydrostatic
for different mass flux
Normalized pressure
Determine exit pressure by systematically changing mass flux, Q
Obatain the relationship between DP and Q
Multiple steady solutions and
“negative friction”
Exit pressure
Overpressure para meter（α）～1 atm
Mass flux
General features of results
Geological implication of
the presence of multiple solutions
Exit pressure
出
口
の
圧
力
Increase of
chamber pressure
Atmospheric
pressure + load
Sonic
solution
Sub-sonic solution
Dome eruption
Mass flux
Dome collapse
Explosive eruption
 Mass flux
Decrease in total friction
due to descending
fragmentation surface
Mass flux
Exit pressure
Pressure drop due to
viscous friction
Exit pressure
Origin of the multiple solutions
Pressure drop due to
turbulent friction

Exit pressure
Mass flux
Mass flux
Mass flux
Purpose of YK-1
(Yoshida and Koyaguchi, 1999)
Gas-loss
through
conduit wall
WK-1
Gas may escape
vertically.
YK-1
What is
the minimal model to
express the effects
of relative velocity?
Basic equations for YK-1
・2-velocity model
・presence of fractured turbulent flow regime
Mass conservation
Ql  l ul 1     const.
Qg   g ug  const.
Momentum conservation
d
dP
Qlul    1     l (1   ) g  Flg  Fl w
dz
dz
d
dP
Q
u



  g g  Flg  Fgw

g g
dz
dz
Equation of state
P   g RT
 : Gas volume fraction
Ql : Liquid mass flux (kg/m2・s)
Qg : Gas mass flux (kg/m2・s)
Constitutive equation describing
wall friction
Before fragmentation
Flw 
8
u , Fgw  0
2 1
rc
← Poiseuille flow
After fragmentation
Flw  0, Fgw 
w
4rc
 g u g2 sgn(u g )
←Trubulent flow
Tentatively critical void fraction (f=0.8) was chosen as a
fragmentation criterion.
Constitutive equations describing
gas-liquid friction
Bubbly flow
  0.6 : Flg 
3
 (1   )(u g  ul )
2
rb
1t
←Stokes’ terminal veolcity
 3
  lg

 g (1   )(ug  ul )2  sgn(ug  ul )
0.6    0.7 : Flg   2  (1   )(ug  ul )  
 rb
  4rb

t
t
  0.6
0.7  0.6
Fractured turbulent flow
0.7    0.8 : Flg 
lg
4rc
 g (1   )(u g  ul ) 2 sgn(u g  ul ) ← turbulent pipe flow
1t
0.8    0.85
    3C 
: Flg   lg   D   g (1   )(ug  ul )2 sgn(ug  ul )
 4rb   8ra 
t
t
  0.8
0.85  0.8
Gas-particle flow
  0.85 : Flg 
3CD
 g (1   )(u g  ul ) 2 sgn(u g  ul ) ←high Re terminal velocity
8ra
Essense of YK-1
Whatever the details of the constitutive equations may be…
Wall friction ～friction between liquid and gas
Both are determined by gas viscosity.
Wall friction >>friction between liquid and gas
Determined by gas viscosity
Determined by liquid viscosity.
Wall friction ～ friction between liquid and gas
Both are determined by liquid viscosity.
velocity
pressure
depth
Void fraction
General features of results
pressure
pressure
End