CONVECTION CURRENTS I N GEYSERS.
A. L. Hales, B.A.
(Received 1936 September 9)
I . Rayleigh * and Jeffreys t have shown that when a liquid of infinite
horizontal extent is heated from below viscosity prevents convection until
a critical temperature gradient is reached. Jeffreys 1 suggested that the
behaviour of geysers may be explained in this way. The suggestion is that
“the water may be unable to conduct the new heat away except at such
a temperature gradient that boiling must occur at a certain depth, while
viscosity prevents convection.” The present paper is an attempt to investigate this idea. T h e method used is similar to that employed by Rayleigh
and Jeffreys. For the purpose of the paper a geyser is regarded as a circular
cylindrical tube.
2. The equations of motion of a fluid are, in vector form,
iV
- - v x curl v = F - I gradp -grad
at
P
&v2 + 4v
- grad (div v) - v curl curl V,
(2.1)
3
where F and v are the force and velocity vectors, v is the kinematic viscosity,
and a x b denotes the vector product of a and b.
T h e equation of continuity is
*
dt
+ div (pv) = 0.
We now make use of the principle of the exchange of stabilities and
a
write - =o, and, in addition, assume that the departures from the undisturbed
at
state are small quantities, the squares and products of which may be neglected.
We substitute
P =Po +P‘,
where the subscript zero denotes the value of the variable in the undisturbed
state.
T h e equation of continuity then reduces to
* Phil. Mag., 32, 529, 1916.
-f Phil. Mag., 2 , 833, 1926.
1 H.Jeffreys, Earthquakes and Mountains, p. 116,1935.
Convection Currents in Geysers
1937 Jan.
v , and w being the r,
co-ordinates.
u,
+,
and z components of v in cylindrical polar
If [pol denotes the change in
of
ry)E,
po
in the system,-1 aP0
-w
Po ax
Wmax
whereas -aw
- is of the order
ax
h '
~
is of the order
It follows that if the per-
1 aP0
centage change in density in the system is small -w
Po ax
and therefore
.
I a(ru) I av aw
dlvv=-+--+-=o.
r a+ ax
r ar
On the same assumption
123
may be neglected
(2.5)
(2.1)is
F p - grad p - vp
curl curl v = 0,
(2.6)
F =(o,
0,g), z being measured downwards.
T h e component equations of (2.6)are
since p , is not a function of r and +.
I n the undisturbed state (2.9)is
aP0
gpo-- az =o,
and therefore
(2.9)may
(2.10)
be rewritten as
gp - aptfVpoV2W =o.
az
f
(2.11)
Forming the divergence of the equations (2.6)
p
div F + F .grad p - VZp - v grad p .curl curl v - upo div curl curl v =o,
i.e. with the former assumption and the use of
aP'
- VZp'+g-.
ax
Eliminating p' between
(2.1I)
and (2.13)
(2.10)
(2.12)
Mr. A. L. Hales,
124
4,
1
The equation of heat conduction is
dT
at
- = KV'T,
where T is the absolute temperature.
Assuming that in the undisturbed state aT0 is constant and equal to
~
ax
p,
(2.16)
W p = KV'T',
where
T = Toi- T'.
Further, p is related to T by the equation
(2.17)
- aT),
p =&(I
where pc is constant and a is the coefficient of thermal expansion. Expressing
p' in terms of T'
p' = -p,aT'.
(2.18)
Equation
(2.15)becomes
on substituting for w and p'
We now suppose that
kx
T' = T I cos n$ sin -,
a
kx
u' = u l cos n$ cos -,
a
kz
7.1'
= v l sin n$ cos -,
a
w'
= w 1cos
(2.20)
kx
n$ sin -,
a
amrr
where TI,
ul, vl, and w 1 are functions of r only, and k = -, a being the
h
radius, h the depth, and m, n integers.
At x = o and x = h T' and w are zero. u' and w' are not, however, zero.
By analogy with the principle of Saint-Venant it is assumed that when the
length of the tube is large compared with its radius the conditions in its
interior are not seriously affected by the distribution of velocities at the ends.
Equation (2.19)then is
(2.21)
where
d2
v2=-++-
dr2
I
d n2
-----.
r dr
k2
r2 a2
1937 Jan.
Convection Currents in Geysers
3. Solution of the Equation.-If
and (2.21) becomes
(c2+3)3k4
+ k2)3- 4(c2
- I)2
(
c
where
=--(c2
2 a3
-0,
+ 3132k4 --.
_gpca13a4
4(c2 - 1)
VPoK
The solutions of (3.2) for a2 are
+
(c - I)2
(c I)2
k2,
2(c - I )
4k2
a 1 2 =c2-
a22 z -~
I’
k2.
2(c + I)
a32 =-
The solution for T1is therefore
j=3
T ~ j=1
= ~ A ,a J, ~ ( ~ ) .
The other possible solution is infinite for r =o.
K
(3.4)
It follows from (2.16) that
j=3
wl= - - z A j ( a , 2 + k 2 ) J n
Pa2,=1
(3.5)
On elimination of p’ between (2.7) and (2.11) we find
u1 2nv1 adV2w, gp,aadT,
V 2 U l - - - -r2 r2 k dr
vpok dr = 0.
(3.6)
From (2.5) it follows that
I d(ru,) nvl kw,
(3.7)
r d r f-++--0.
r
a
When the value of v1 given by (3.7) is substituted in (3.6) it is found that
-~
adV2wl
zdu,
- + ul
- =-gp,aadT,
__ - - ~
- 2kw,
- .
r dr r2 vpok dr k dr
ar
V2U,+_
(3.8)
We now substitute for T , and w1 and the right side of (3.8) becomes
j=3
K
z
j=3
C A , a jJ i ( ay ) - kPa4
__ j=lA , ai( aj2 + k 2 ) 2 J ; ( y )
V P & j=1
2kKj = 3
+,-C4ai2
Pa rj,l
+k2)Jn(Y),
or
since from (3.2) and (3.3)
gp,a
KC
k&P
K
(aj2+k2)3
?------=-=-
vp&
* Jn(x)
kpa4
at
’
is the Bessel function of the first kind, of order n.
(3.9)
Mr. A. L. Hales,
I 26
Jn’(x) may be expressed in terms of Jn and JnPl by the equations
*
n
- ,.7n(~),
Jn’(X) = J n - l ( X )
n
-J n + l ( X ) *
Jn‘(X) =;Jn (X)
The equation for u1 is therefore
The particular integral of this equation is
and from (3.7)
Kkn jG3A3(
a32+ R 2 )
V 1 = iga23,1
- Z
Jn(y)*
a32
The complementary function that is finite at r =o is
A
u1= -In(
r
and from (3.7)
):>n(
v1 =A{
)kr;
7
- ;In-.(:)}.
T h e complete solutions for u1 and v 1 are therefore
-J n - l i ? ) } ,
4 . The Symmetric Case.-In this case u, v , w, and T’are independent
of 4. The derivation of the equation (2.19) and the solution (3.4) with
n = o are still valid. The equation can also be derived directly. T h e
solutions for u1 and v 1 must, however, be discussed again.
T h e equation (3.7) is now
~d--(rul) + kw, = 0.
r dr
a
-
~
T h e particular integral of this equation is
* Gray and Mathews, Bessel Functions, p. 1 3 , 1895.
t I&) is the modified Bessel function of the first kind.
(4.1)
1937 Jan.
127
Convection Currents in Geysers
and the complementary function satisfies
d
(4.3)
-(rul)
dr
=o.
The solution of (4.3) is infinite at r =o, and the complete solution is therefore given by (4.2).
It follows from (2.8) that
and the solution that is finite at r =o is
v = .II($).
(4.5)
The boundary condition v = o at r = a can be satisfied only by A = 0.
5. Symmetric Case.-Boundary conditions : TI=o, w 1=o, u1 =o at r =a.
These conditions result in the following equations for the constants
A,, A , and A , :
AIJo(a1)
+
AI(%' $- k 2 ) J o ( a ~ )
+Az(a22
I
,
aI2+k2 ,
a12
+_
k2 _
JI(%)
__
_
a1
Jo(a1)'
+ A3J0(
u3)
(5.1)
+ k2)Jo(az) fA3(a32+k2)Jo(%)=o,
I
,
a,'+k2 ,
a
z
2
+_
k2 J1(
~ 2 )
__
__
a2
=O>
Jo(a2)'
I
+ k2
(5.2)
= 0,
a32
~3~
+ k2
a3
J1( a3)
(5.4)
J0(a3)
or
where
This equation determines c, and therefore C, in terms of k. It is found
that the ranges c = 3 to CQ, c = 3 to I , c =o to I , c =o to - I , c = - 3 to - I ,
and c = - 3 to - cg give identical roots for C, and so it is necessary only
to discuss the roots in the range c =3 to 00.
The least roots for c and C are given below.
Mr. A. L. Hales,
I 28
TABLE
I
Limiting
k
value
k-to
0.I
452.0
4256
452.9
C
c
6. Symmetric Case.-Boundary
0.3
0’4
476.4
459.6
269.7
465.5
0-2
1067
455.4
conditions :
0.5
174.0
473.2
dT1 =0,w1=o,
__
. dr
u,
= o a t r =a.
Instead of (5.1)we have
A I ~ L , J I+(A~ zI )~Ji(a2)
, + A3a3Jl(a3) =o,
and the consistency equation is
1
a12
(6.4
a32
7
or
4c- aJo(a1)
Ji(4
+ (c + I)(C - 3)- P z I o ( P 2 )
- (c -
I)(C
Il(P2)
+ 3)--
a3J0(a3)
=0.
(6.4)
Jl(a3)
The least roots are given in Table 11.
TABLE
I1
Limiting
value
k
k-to
0.I
0.2
0.3
452.0
4255
452.6
1066
454.5
475.4
457.7
C
C
0.4
268.7
462.2
0.5
173.0
468.0
7. Case n = I .-Boundary conditions : T, = 0,w 1= 0,u1 = 0,v , = o at r =a.
The equations for A,, A2, A, and A are
C Aj( +k2)Jl(
aj2
j=1
uj)
= 0,
1937 Jan.
Convection Currents in Geysers
129
Eliminating A between (7.3) and (7.4)
The consistency equation for (7.1), (7.2), and (7.5) is
= 0,
1
,
1
,
I
al2i-k2
,
a22+k2
,
a32+k2
(7.6)
where
or
The least roots are given below.
TABLE
I11
Limiting
value
k
k-to
C
c
216.0
0 .I
2940
216.1
0.2
737.6
217.7
0.3
329.8
220.3
0.4
0.5
187.1
121.0
224.0
228.9
For n = z the least roots are, when K=o.I, c=5278 and C=696.5.
8. Although the solution has been discussed for another set of boundary
conditions the ones which are of interest are T = 0,u =o, v =o, w =o at r =a.
The consistency equation gives, for any value of k, a least value of C,
and therefore of /3, which will make the equations for A,, A, and A, consistent. In other words, it gives a least value of /I for which convection
currents are possible. It follows that the currents most easily excited are
those for which n = I , discussed in paragraph 7.
C is an increasing function of k, so that the first current to arise will be
that for which k is least, and therefore that for which m = I . As C is a
slowly increasing function of k the currents for which m = 2 , . . . will be
excited by temperature gradients not much greater than that which will
excite m = I. The currents are spirals, and the resultant motion will be
similar to that observed when a test-tube is heated.
Jeffreys * has shown that for a compressible fluid /3 must be replaced
* Proc. Camb. Phil. SOC.,26, 270, 1930.
G 9
130
Convection Currents in Geysers
4, 1
by /3-/3,,
Po being the adiabatic temperature gradient. Po is roughly
10-7"A. per cm., and therefore the correction is not important in this
case.
It was not found possible to test the results by actual observations, as
the only geyser for which the necessary observations of temperature gradient,
radius and depth could be found was the Great Geyser in Iceland. Bunsen *
made observations on this geyser in 1846. The radius is approximately
3 metres and the depth 21 metres. The observations of temperature differ
widely except at the bottom, and therefore the estimate of I O - ~"C. per cm.
can only be regarded as giving the order of magnitude.
For these values of u and h, and taking m = I , k =o-2 approximately,
we find from Table I11 C =zzo. Calculating the maximum temperature
gradient which is possible if convection does not exist, we find /3 = 3 x 1 0 - l ~
"C. per cm. T h e values of the constants used were a =7 x I O - ~ , p = I ,
v = 3 x I O - ~ , K = 1.5 x I O - ~c. g. S. u.
It follows that convection should be possible for this geyser and therefore that eruptions should not take place. The result is not unexpected
since v and K above are the molecular coefficients, whereas it has been found
necessary in other cases of large scale motions (e.g. air currents, ocean
currents and the flow of fluids through channels) to replace these by the
corresponding "eddy" coefficients. The values of the eddy coefficients vary
with the scale of the motion, and therefore it is not possible to assign values
to the eddy viscosity E and the eddy conductivity K,. On the basis of the
eddy current theory proposed by Taylor t E = K,. On this assumption we
find that in order that the calculated temperature gradient should exceed
that observed E and K , must be greater than 130. The values of E found in
other fields range from I to 400.
The only point at which the depth enters into the discussion is in the
determination of k, and as C varies comparatively slowly with k we can
say that u4/3 is approximately constant for the critical case. This means
that if u4/3is greater than a constant found from the tables for C and equation
(3.3) (v and K replaced by E and K , respectively), convection currents will
arise. Thus the geyser will be in a constant state of ebullition and will
not erupt periodically. In geysers for which a4/3is less than the constant
convection currents are prevented by viscosity and the geyser will therefore
erupt periodically provided that the temperature gradient is sufficient for
the temperature at some depth to exceed the boiling-point at that depth.
T h e rate of increase of the boiling-point with pressure is 2-76x 10-5 "C. per
dyne, and of the pressure with depth 941dynes per
( p =0.96). It follows
that the rate of increase of the boiling-point with depth is 2.60 x 10-2 "C.
per cm. If the temperature gradient is less than this the boiling-point
will not be reached and eruption will not occur. For the Great Geyser /3
was less than the above value, but, as pointed out earlier, the estimate was
rough.
* Poggen. Annul., 72,1847. I was unable to refer to this journal, and the information used was taken from the Encyclopcedia Britannica, 11th ed., 1910.
t Phil. Trans., A, 215, I , 1915.
1937 Jan.
Note on Mr. Hales's Paper
131
In conclusion, my thanks are due to Dr. H. Jeffreys for suggesting the
problem.
Summary.
Convection currents in a circular tube heated at the bottom are investigated. It is found that, as in the case of a liquid of infinite horizontal
extent, viscosity prevents convection until a certain critical temperature
gradient is reached. The possibility that this gives a criterion as to whether
a geyser will erupt, or not, is discussed.
University of the Witwatersrand,
Johannesburg.
NOTE ON MR. HALES'S PAPER.
Harold Jeflreys, D S c . , F.R.S.
(Received 1936 September 9)
With the values of v, K and a adopted by Hales the value of pa4 for
the critical case is about 0.0014. The viscosity fits water at 95' C. The
following are a few critical values of j? for various radii of the column,
together with the depths needed for an increase of temperature of 100~:a (cm.)
p("C/cm.)
Depth (cm.)
0.05
220
0.4
0. I
14
7
0.5
-022
4.500
1.0
.oo14
70,000
It appears that if stability is to be maintained by viscosity and conduction
while the boiling-point is reached at some depth, either the tube must be
very narrow or the column of water very deep. The values for the smaller
radii should be within the range of experimental test. The depths needed
to reach the boiling-point in a geyser should possibly be divided by 10 or
so, since the water at the top is already hot. On the whole it appears
possible that if the column of water is 2 cm. or less in diameter stability
can be preserved, and that the only way of disposing of the heat entering
at the bottom will be by intermittent boiling, but the data available do not
permit a thorough test.
I doubt the suggestion towards the end of the paper that eddy conduction
and viscosity can play any important part in controlling geysers. They
arise in a fluid already in fairly rapid motion, but they do not exist in
equilibrium and should have nothing to do with the initiation of departures
from it.