Jökull - 01.12.1966, Síða 24

In the present context the most important
solutions o£ (4) involve a fluid current heated
from below and starting at x = 0 with a temp-
erature 0 = 0. The initial temperature and
the temperature at the surface z = h can be
assumed to be zero. A class of interesting solu-
tions of this type can be derived on the basis
of following type of solution.
Corresponding solutions for the case of a
stratified fluid layer of a finite thickness can
be obtained rather easily on the basis of the
general solution (5). Vertical variations of u
as well as az can be taken into account. But
in view of the available observational material,
these refinements appear somewhat irrelevant
at this juncture.
0(x, z, t) = H(t —x/u)0i(x, z) +
(1—H(t —x/u))02(t, z), (5)
where H(t) is the unit step function and 0i
and 02 are functions satisfying the following
parabolic equations 301 320! - b , 30 2 0202 (6)
3x 3z2 3t “ dz2
where b = az/u, and
0i(O, z) = 0 02(O, z) = 0 (7)
©i(x, h) = 0 02(t, h) = 0
The boundary conditions at the lower bound-
ary z = 0 have to be prescribed in accordance
with the temperature or the vertical heat flow
there.
Of special interest is the stationary solution
in the case of a uniform and constant vertical
heat flow from below. This solution is given
by ©i (x, y) with the boundary condition
30!
z = 0 — oca_ —— = q, (8)
z ðz
where q is the density and c the heat capacity
of the fluicl. For an infinitely thick layer, i.e.
h — co, the solution 0i satisfying (6) to (8) is
given bv Carslaw and Jaeger (1959).
0i (x,z ) =
2q \/bx
gcaz
ierfc (z/2\/bx),
(9)
and the temperature at the lower boundary
z = 0 is given by
0i (x, 0)
1. 12q Vbx
oca
1. 12q
QC
V—
v a„u
(10)
Written in terms of the corresponding vertical
lapse rate y0 = q/Qcaz, equation (10) takes the
form
01 (x, 0) = 1. 12y0VbV (11)
178 JÖKULL
(4) THE EDDY DIFFUSIVITY AND
SUPERADIABATIC TEMPERATURE
LAPSE RATES IN THE BOTTOM
BOUNDARY LAYER
Since the terrestrial heat flow is quite small
and varies only from about 0.02 to 0.2 watts/m2,
with a global average of 0.06 watts/m2, the
question arises whether it has any noticeable
influence on the dynamics ancl the temperature
field in the bottom boundary layer. In the
present context we are more interested in the
local structure of the field, since the global in-
fluence of the terrestrial heat flow has been
discussed elsewhere. For example, the data
given by Knauss (1962) indicate that the tem-
perature of the deep waters in the Pacific is
affected by the flow of heat from the earth’s
interior. The Pacific deep water, which fills
the Pacific basin from the bottom up to a
level of 2,000 to 2,500 m, originates in the
Antartic south of 60° S and moves far into the
North Pacific. The total distance covered by
this current is more than 13,000 km and the
average velocity of flow appears to be of the
order of 10~3 m/sec. On the passage to the
North, the water temperature is raised by a
total of about 0.6° C over a distance of 12,000
km giving an average northward gradient of
5 x 10~8° C/m. A simple calculation indicates
that the flow of heat from the earth’s interior
accounts for the temperature rise in the lowest
300 m of this current.
One of the first questions to arise is whether
significant superadiabatic temperature lapse
rates are possible in the bottom boundary laver,
and whether such lapse rates could extencl tens
of meters above the floor. Superadiabatic condi-
tions on tliis scale could lead to instabilities
and noticeable temperature fluctuations.
The continuity of the heat flow through the
solid-water interface requires that superadiaba-
tic lapse rates be present in a thin layer above
the ocean floor. The vertical extension of this