Jökull - 01.12.1966, Side 22

The present paper will discuss the problems
of the superadiabatic lapse rates and convec-
tive motions from a point of view which differs
slightly from that of Lee and Cox (1966). Since
very little is at present known about turbulence
in the oceans, the following discussion will
necessarily be of a very incomplete and tenta-
tive character.
(2) FREE AND FORCED CONVECTION
The criterion (Jeffreys, 1926) for the onset
of free convection in a stationary horizontal
layer of a homogeneous fluid, which is being
heated from below, is given by the value of the
Rayleigh number R. The layer is unstable if
R = agyh4/ua > C. (1)
where a is the coefficient of thermal expansion,
g the acceleration of gravity, y the vertical tem-
perature lapse rate at the onset of convection,
h the thickness of the layer, v the kinematic
viscosity, a the thermal diffusivity and C a
crtitical number depending on the boundary
conditions. For a layer with a free surface and
a rigid bottom the critical number is 1100. The
model is based on a constant temperature lapse
rate at the onset of convection, that is, on the
assumption of a very slow heating of the layer.
In the case of a compressible fluid, y represents
the effective rate, that is the lapse rate of the
potential temperature.
The Rayleigh number may also be expressed
in terms of a given upward heat flow q,
R = «g(q-q0)h4/rak’ (2)
where q0 is the heat flow caused by the adia-
batic lapse rate and k is the thermal conduc-
tivity.
The global average of the terrestrial heat
flow of 0.06 watts/m2 gives a lapse rate of 0.1
°C/m in stationary sea water. If values for sea
water are inserted for the other parameters in
(1) we find that this gradient and C = 1100
give a marginal thickness of around 0.05 m.
In view of the fourth power of the thickness
in (1) it is obvious that any stationary layer
of water which is permeated by the terrestrial
heat flow will be highly unstable, even if the
thickness is only a few meters. For example,
the critical temperature lapse rate in a layer
of thickness of 5 m will be 10~9 °C/m, and
under the influence of the terrestrial heat flow
its Rayleigh number will exceed the critical
value by a factor of 108. Such highly unstable
layers develop convective motion on a very fine
scale. The temperature at all levels deviates
only little from the adiabatic temperature.
This result holds for stationary layers where
a cell-type convection develops when R > C.
Jeffreys (1928) has shown that the condition of
stationarity can be relaxed, and that the above
criterion (1) also holds in the case where a
steady laminar current passes through the layer,
provided all quantities involved are indepen-
dent of the coordinate in the direction of the
current. The free convection will then occur
in strips parallel to the steady current. In the
case of oceanic currents, the condition of in-
dependence of the coordinate in the direction
of flow will in general not be satisfied. But
the deviations are probably so small that Jef-
frey's result will be applicable in most cases
of interest.
The condition of laminarity is essential. The
eddy viscosity and eddy conductivity of turbu-
lent currents generally exceed the correspond-
ing molecular quantities by several orders of
magnitude. Moreover, the eddy transport is
strongly anisotropic. Whereas the molecular dif-
fusivity of heat in sea water is 0.14 x 10~6
m2/sec, Sverdrup (1957) gives for the vertical
eddy diffusivity values of 2 X 10-6 to 10-2
m2/sec and for the horizontal eddy cliffusivity
an even wider range of 10-1 to 104 m2/sec.
The effective Rayleigh number therefore ap-
pears to be radically affected by the turbulence,
resulting in an enhanced stability against free
convection. However it must be emphasized
that this conclusion is necessarily a qualitative
one. The eddy transport phenomena are only
poorly known and the eddy transport coeffic-
ients are not well defined quantities since they
depend even on the scale of the turbulnce. It
is therefore not possible to arrive at a stability
criterion for turbulent fluid layers by applying
eddy transport coefficients to the criterion given
in (!)•
The eddy transport of heat is of a dynamical
origin and is referred to as forced convection
in contrast to the above discussed case of free
convection where the convection motion is
caused by the buoyant forces. The distinction
between the regimes of forced and free convec-
tion is important. A fluid layer where forced
176 JÖKULL