Geoscience Reference
In-Depth Information
in the flask,
n
represents molar number of it and
α
is constant. Also an
equation of a rate of gas supply in the flask is written as,
d
n
d
t
=
β,
(3)
where
β
is a constant.
Using Eqs. (1)-(3) and so on, we can write a spouting period
T
as
V
0
αβ
+
f
k
S
ρgαβ
(
f
k
+
P
0
+
ρgH
)
,
T
=
(4)
where
S
represents an area of a cross section of the pipe.
4. A Dynamical Model of a Periodic Bubbling Spring
A derivation of a dynamical model and a modified dynamical model of a
geyser (a periodic bubbling spring) is stated in Ref. 4 in detail. Finally, an
evolution equation of temporal variations of height of top of a water pole
packed in the spouting pipe of a periodic bubbling spring is written as
(
n
0
+
βt
)(
V
0
+
Sx
)
ρH
d
3
x
d
t
3
(
n
0
+
βt
)(
V
0
+
Sx
)
d
2
x
d
t
2
+
8
πηH
S
+(
n
0
+
βt
)
PS
d
x
d
t
=(
V
0
+
Sx
)
Pβ,
(5)
where
n
0
represents molar number of gas in a underground space just before
the water pole's beginning to move up to the upper entrance of the spouting
pipe and
η
represents viscosity coecient. And
x
is regarded as a position
of the lower interface between water and gas of the water pole and an upper
direction of a vertical line is regarded as a plus direction of the
x
-axis.
5. A Combined Model Combining the Mathematical Model
(a Static Model) and the Improved Dynamical Model of
Periodic Bubbling Spring
A combined model consists of the mathematical model (the static model)
and the improved dynamical model of a periodic bubbling spring. Con-
cretely, temporal variations of height of top of the water pole of a periodic
bubbling spring as shown Fig. 1 is dealt with by the improved dynamical
model. And a spouting period, that is, time from a beginning of a pause
mode to next one is dealt with by the static model.
