obtained quite simply for V = 0 by assuming. by analogy to shearing flow,
thiit j), should involve only q and 2 in a region above the thermal boundary
layer. We see, therefore, that the two similarity arguments, with and without
shear. are the same. If either is to be believed, they should predict the
behavior of mean quantities both in the atmosphere and laboratory. We
have already indicated one of the major disagreements in atmospheric measurements. In the laboratory there is controversy about the behavior of p z .
Townsend (1959) proposed & cx z - " where 1.3 < n < 2.5. Data of Croft
(1958) yield z - ~ ; ' .
The similarity arguments can be used to predict other quantities, for
example, rms p or n,, r. z - 1'3 compared with measurements in which exponents range from -0.48 to -0.70 (Rossby, 1969; Somerswles and Gazda.
1969). Finally Townsend (1959) gives an interesting result for the "dissipation function" for buoyancy fluctuations 6. He finds (5 cx z-'" compared
with z - from similarity theory.
We may conclude that evidence in atmospheric and laboratory investigations does not inspire confidence in the similarity theories when heating
effects are strong. We have seen that all theories reduce to the argument that
mean quantities should depend only on q and E when the shear is zero and
that the argument was inspired by the success of the theory that only rand z
are important in shearing flow above a surface. In the latter case, however. T
is proportional to ( A U ) ~ and is either very weakly dependent on v (smooth
wall) or independent of v (rough wall). In the case of heating, however, q and,
therefore, either nu or o, , directly involve the molecular coefficients.
4. A NEW THEORY
A new theory, differing from the similarity theory, has been proposed by
the author (Long, 1974) and involves the behavior of a large number of
mean quantities including mean buoyancy gradient, mean density gradient.
rms velocities and buoyancies, length and time scales, and eddy viscosity and
conductivity. We are concentrating in this paper on the properties of the
mean buoyancy gradient, and we will therefore confine attention to a few
aspects of the new theory relevant to finding pZ in a region well above the
thermal boundary layer. For simplicity, we will again assume v = K and
H = cc. For the time being, we may also consider the shear to be zero.
The similarity theory for Au = 0 may be based on simple dimensional
analysis once it has been decided that p , , for example, should depend only
on 4 and z. On the other hand, as shown by Kraichnan (1962). it may also be
hased on cstiinatcs of orders of magnitude of certain quantities as z -+ x
and this approach reveals the essential difference between the present theory
and the similarity theory.
thiit j), should involve only q and 2 in a region above the thermal boundary
layer. We see, therefore, that the two similarity arguments, with and without
shear. are the same. If either is to be believed, they should predict the
behavior of mean quantities both in the atmosphere and laboratory. We
have already indicated one of the major disagreements in atmospheric measurements. In the laboratory there is controversy about the behavior of p z .
Townsend (1959) proposed & cx z - " where 1.3 < n < 2.5. Data of Croft
(1958) yield z - ~ ; ' .
The similarity arguments can be used to predict other quantities, for
example, rms p or n,, r. z - 1'3 compared with measurements in which exponents range from -0.48 to -0.70 (Rossby, 1969; Somerswles and Gazda.
1969). Finally Townsend (1959) gives an interesting result for the "dissipation function" for buoyancy fluctuations 6. He finds (5 cx z-'" compared
with z - from similarity theory.
We may conclude that evidence in atmospheric and laboratory investigations does not inspire confidence in the similarity theories when heating
effects are strong. We have seen that all theories reduce to the argument that
mean quantities should depend only on q and E when the shear is zero and
that the argument was inspired by the success of the theory that only rand z
are important in shearing flow above a surface. In the latter case, however. T
is proportional to ( A U ) ~ and is either very weakly dependent on v (smooth
wall) or independent of v (rough wall). In the case of heating, however, q and,
therefore, either nu or o, , directly involve the molecular coefficients.
4. A NEW THEORY
A new theory, differing from the similarity theory, has been proposed by
the author (Long, 1974) and involves the behavior of a large number of
mean quantities including mean buoyancy gradient, mean density gradient.
rms velocities and buoyancies, length and time scales, and eddy viscosity and
conductivity. We are concentrating in this paper on the properties of the
mean buoyancy gradient, and we will therefore confine attention to a few
aspects of the new theory relevant to finding pZ in a region well above the
thermal boundary layer. For simplicity, we will again assume v = K and
H = cc. For the time being, we may also consider the shear to be zero.
The similarity theory for Au = 0 may be based on simple dimensional
analysis once it has been decided that p , , for example, should depend only
on 4 and z. On the other hand, as shown by Kraichnan (1962). it may also be
hased on cstiinatcs of orders of magnitude of certain quantities as z -+ x
and this approach reveals the essential difference between the present theory
and the similarity theory.