Jökull - 01.12.1973, Page 30
r 1 „ 4 A , b,-,
H =p!«g {^-Cba-btanh4)]
2ML
cb
sinh b' [tanh —
tanh
F’}-
Apg AD
(43)
b =
2 _
T cL
12Ky L
2 . _ 3
.p CgA^L
rHi ^ s fi 4
'1 4K Lb .2
tanh-^
4
8KM
2
pA L cb
a- b , b'
sinh b itanh ” -tanh —
Apg AD j-
(44)
kCí,btenh|)- sinh b'(tanh| - tanh^-)
b 1
4 Ap AD_ (46)
Pl“ T1L
and factoring the expression T^gLp^ a/4 from
the right hand side of (44) we can write
b=(^)k(b>b') 45)
where N' is given by (28) and k (b, b') is given
by (46). Equation (46) can be expressed in the
form (47)
k (b, b') = g (b) + m (b, b') + n (b, b') (47)
The function g (b) is defined by (26'). Tlie
functions m (b, b') and n (b, b') are due to the
phase transition. These correspond to the latent
heat effect and the density effect of the phase
transition respectively. We expand m (b, b') and
n (b, b') for small b and b' to determine the
modification of the phase transition on the
critical Rayleigh number. Keeping only terms
of Iowest order, and setting L = 2h, the ex-
pression for m (b, b') reduces to
m(b,b') A)b2 (48)
“ cTih \ h /
In order to determine n (b, b') for small b
and b', an estimate of AD, the phase level
difference, is required. Assuming tlrat the slope
of the Clapeyron curve is constant, equation
(34) may be integrated with the result that
Tc = yP + 0 (49)
where 0 is an empirical constant. Since only a
small range of the equilibrium curve is con-
sidered in the analysis, the error involved in
this procedure is negligible. Writing the pres-
sure as P = px gz, where z is the depth, (49)
becomes
Tc = PiSyz + 0 (5°)
At the phase boundary, the temperature (40)
and (42) must coincide with the transition tem-
perature. Therefore,
Ti(di) = Pi£Ydi + 0 (51)
Tiii(d2) = plgY(L-d2) + 0 (52)
28 JÖKULL 23. ÁR