SLJRFACI? LAYER IN UNSTABLE CONDITIONS
137
The eddy conductivity K h may be defined by the equation q = Khpr and
since 4 is a constant, the variation of Fz with height - is determined by the
variation of K , . To find this we notice that q - wp implies that q - o,, ow C
where C is the correlation coefficient. We know, however, that warm parcels
rise and cold parcels descend so that C should be of order one. This assumption is strongly supported by observations (Deardorff and Willis. 1967) in
which C is 0.5-0.6 over a wide range of Rayleigh numbers. Both theories
then should yield q - o,cr,. The quantity cr,, may be estimated by assuming
op - pz I where I is a length over which the total buoyancy is conserved. It is
reasonable that I - z as is the case in turbulent shear flow above a surface.
Indeed, I z implies I - H for the container as a whole; this implies that the
energy-containing eddies fill the entire container and this is observed in
experiments. The three estimates q - K,, p,, q - G,, ow, op - p , z lead to
K, - TO,
and the problem of finding K, and, therefore, 3, is reduced to finding o w .
This is the point of departure for the present theory and the similarity
theory. The latter, as shown by Ktaichnan (1962) assumes o i 7 op I which is
a seemingly reasonable assumption that the kinetic and potential energies
are of the same order, or that the vertical acceleration of a parcel is of the
order of the buoyancy force. This estimate, and the earlier estimates, leads to
P* - Y .
In the present theory the last order of magnitude estimate is not made.
Instead experimental observations are used (Malkus, 1954b: Deardorff and
Willis, 1967) that the kinetic energy averaged over the entire container is of
order H Ap. This result holds with considerable accuracy over a range of
Rayleigh numbers from lo5 to lo9. This has a physical interpretation that
the vertical velocity of a parcel is of an order obtained by imagining that it
conserves its density and rises freely in the unstable environment from an
origin in the thermal boundary layer. It suggests, therefore, that o i - z A p
at any given level z. If we adopt this estimate, we obtain
-
2/3 -413
p, = ~ ( ~ ~ ) 5 / 6 ~ 1 / 3
J/Z
(22 1
I 2
whcre B is a constant for zero shear but in general may be considered a
function of V . This contrasts with the z - * ’ ~ dependence of the similarity
thcory. In essence, it is easy to see that the basic difference in the two theories
can be reduced to a difference in the estimate of the time scale T of the eddy
motion. The present theory assumes T depends on z and Ap only and the
similarity theory assumes that T depends on 4 and z only. Actually. if one
acknowledges that Ap is a more fundamental parameter than q, the present
assumption is preferable if one also feels that vanishingly small molecular
coefficients should not directly affect the time scale of the eddies. In any case,
137
The eddy conductivity K h may be defined by the equation q = Khpr and
since 4 is a constant, the variation of Fz with height - is determined by the
variation of K , . To find this we notice that q - wp implies that q - o,, ow C
where C is the correlation coefficient. We know, however, that warm parcels
rise and cold parcels descend so that C should be of order one. This assumption is strongly supported by observations (Deardorff and Willis. 1967) in
which C is 0.5-0.6 over a wide range of Rayleigh numbers. Both theories
then should yield q - o,cr,. The quantity cr,, may be estimated by assuming
op - pz I where I is a length over which the total buoyancy is conserved. It is
reasonable that I - z as is the case in turbulent shear flow above a surface.
Indeed, I z implies I - H for the container as a whole; this implies that the
energy-containing eddies fill the entire container and this is observed in
experiments. The three estimates q - K,, p,, q - G,, ow, op - p , z lead to
K, - TO,
and the problem of finding K, and, therefore, 3, is reduced to finding o w .
This is the point of departure for the present theory and the similarity
theory. The latter, as shown by Ktaichnan (1962) assumes o i 7 op I which is
a seemingly reasonable assumption that the kinetic and potential energies
are of the same order, or that the vertical acceleration of a parcel is of the
order of the buoyancy force. This estimate, and the earlier estimates, leads to
P* - Y .
In the present theory the last order of magnitude estimate is not made.
Instead experimental observations are used (Malkus, 1954b: Deardorff and
Willis, 1967) that the kinetic energy averaged over the entire container is of
order H Ap. This result holds with considerable accuracy over a range of
Rayleigh numbers from lo5 to lo9. This has a physical interpretation that
the vertical velocity of a parcel is of an order obtained by imagining that it
conserves its density and rises freely in the unstable environment from an
origin in the thermal boundary layer. It suggests, therefore, that o i - z A p
at any given level z. If we adopt this estimate, we obtain
-
2/3 -413
p, = ~ ( ~ ~ ) 5 / 6 ~ 1 / 3
J/Z
(22 1
I 2
whcre B is a constant for zero shear but in general may be considered a
function of V . This contrasts with the z - * ’ ~ dependence of the similarity
thcory. In essence, it is easy to see that the basic difference in the two theories
can be reduced to a difference in the estimate of the time scale T of the eddy
motion. The present theory assumes T depends on z and Ap only and the
similarity theory assumes that T depends on 4 and z only. Actually. if one
acknowledges that Ap is a more fundamental parameter than q, the present
assumption is preferable if one also feels that vanishingly small molecular
coefficients should not directly affect the time scale of the eddies. In any case,