Jökull - 01.12.1966, Side 23

convection is the prevailing agent of heat trans-
port can be stable against free convection. The
transition between the two regimes is probably
not sharp, and there is a certain overlap.
In a stratified fluid a measure of the ratio
between the buoyant and the dynamic forces
is given by the Richardson number
r. _ g((d0/dz)-|3) _ _ «gy
o(dú/dz)2 (dú/dz)2 ’
where z is the vertical coordinate, o the density,
fi the density lapse rate at adiabatic conditions
and ú(z) the mean horizontal velocity. The
second expression for Ri is derived on the
basis of the assumption that the density de-
pends only on the temperature. In the case of
sea water this would imply constant salinity.
Since the bottom waters of the deep oceans are
locally homogeneous, the following considera-
tions will be based on this assumption The
Richardson number may possibly provide the
basis for a criterion for the transition from
fully forced to dominantly free convection.
It is well known that both regimes of forced
and free convection of heat can be observed
in the lowest layers of the atmosphere where
the winds are generally turbulent. At relatively
high winds, forced convection conditions may
prevail in spite of superadiabatic temperature
lapse rates. The enhanced stability is due to
the eddy transport of momentum and heat.
On the other hand, high lapse rates and
strong buoyant forces are built up on clear hot
days with moderate or low winds. The lower
layers of the atmosphere become unstable with
the onset of free convection as the result. This
situation is especially conspicuous in desert
areas where very high ground temperatures are
reached during the day. Large bubbles of hot
air break away from the ground and penetrate
up to great heights. These bubbles form the
well known thermal winds, or thermals as they
are generally referred to. The space near the
ground, vacated by the thermal, becomes oc-
cupied by cooler air coming from above, result-
ing in a rather sudden local temperature drop
at the ground. Conditions of free convection in
the lower atmosphere are theref'ore characteriz-
ed by relatively large temperature fluctuations
near the ground.
Priestley (1955) has investigated the transi-
tion between the regimes of fully forced and
dominantly free convection in the atmosphere
near the ground. Normally, there will be a
regime of forced convection in a boundary layer
above the ground, whereas free convection may
prevail at higher levels. Using the Richardson
number at an elevation of 1.5 m, he finds that
at this level the transition occurs when 0.02 <
— Ri < 0.05. The same critical number will
probably not apply to other levels.
(3) DIFFUSION TYPE MODELS
In modern meteorology (see Sutton, 1953)
eddy transport processes are frequently describ-
ed and studied on the basis of diffusion type
models. In the present case of a horizontal layer
of fluid this leads to the following model. Let
the coordinate system be placed at the bottom
of the layer with the z-axis pointing upward.
Moreover, let the fluid flow in the direction
of the x-axis and the velocity u(z) be indepen-
dent of x and y. If @(P) is the potential tem-
perature at a point P, the diffusion model leads
to the following equation
3 / 30 \ 3 ( 30 \ (3)
cv" V‘v 0y / +'?z Y'* (7 )
where ax, ay and az are the coefficients of eddy
diffusion which are functions of space and
time.
Molecular conduction and sources within the
fluid are being neglected. Initial and boundary
conditions have to be prescribed in accordance
with the given situation.
For the present purpose where only order-of-
magnitude considerations are involved, equa-
tion (3) can be simplified considerably. It
should be possible to obtain adequate approx-
imations by assuming that the velocitv u and
the eddy coefficients are constants and equal
to average values. Moreover, in the case of ex-
tensive horizontal layers, the vertical diffusion
predominates and the horizontal component
can be neglected. These simplifications lead to
the following equation
30 30 S20
+ u -— = a, ——, (4)
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