I32
RORI'.HI R. 1.0F;G
smiiller than those in the atmosphere. Thc second reason is that all experiments. except the recent one by Townsend, are without shear whereas the
utmosphcre is usually in appreciable mean motion. We illustrate the importancc of the latter remark by considering an idealized experiment involving
statistically steady, turbulent flow of a Boussinesq liquid betwceri horizontal
smooth plates ;it z = 0 and z = H. The lower plate is heated and nioves
dong the saxis at a speed -Au; the upper is cooled and moves along
the . Y axis at speed Au. Thus, the ensemble mean velocity ii ar z = H j 2 is
zero. The temperature difference corresponds to an increment of buoyancy
2 Ap where buoyancy is defined as
( 1 )
Here p , is density, pO is the mean value of the density at : = H / 2 , and g is
gravity. Thus, p = 0 at z = H / 2 . Two extreme cases correspond to stationary, heated plates and to moving plates with Ap = 0.
P = [ ( P I - PO)/VOIB
2. DISCUSSION OF TliP T W O CASES
I f Au = O,A/) p 0, dimensional analysis leads to forms of the mean quantities in terms of several nondimensional parameters. The mean buoyancy
gradient. for example, is
(4
P: = (AP,':)f'(P* c'o 1 5 )
where z is the vertical coordinate and
(3)
P = \llK,
= H(Ap)' ' / K 2 ', C = 4Ap)l;3,'Kz,3
In Eqs. (3), P is the Prandtl number, to is proportional to the cube root of
the Rayleigh number,
Ra = H' AylvK
and 4: may be regarded as the ratio of the height above the lower surface to
the thickness d7. - K*/'/(Ajj)' ' of the thermal boundary layer. We confine
;ittention, of course, to situations and regions in which To and 5 are large. . _
Many mcasuremcnts haw been made of the buoyancy flux q = KF, - wp.
wlierc we now denotc fluctuating velocities by u, L', w, and fluctuating buoyancy by p. The flux 4 is constant with height in a steady state and therefore
can be expressed as
(4)
All experiments indicate a very weak dependence on to when this number is
large (Turner, 1973) and it seems likely thatf, may be considered independent of to at large Rayleigh numbers. I t is then inescapable that y is directly
RORI'.HI R. 1.0F;G
smiiller than those in the atmosphere. Thc second reason is that all experiments. except the recent one by Townsend, are without shear whereas the
utmosphcre is usually in appreciable mean motion. We illustrate the importancc of the latter remark by considering an idealized experiment involving
statistically steady, turbulent flow of a Boussinesq liquid betwceri horizontal
smooth plates ;it z = 0 and z = H. The lower plate is heated and nioves
dong the saxis at a speed -Au; the upper is cooled and moves along
the . Y axis at speed Au. Thus, the ensemble mean velocity ii ar z = H j 2 is
zero. The temperature difference corresponds to an increment of buoyancy
2 Ap where buoyancy is defined as
( 1 )
Here p , is density, pO is the mean value of the density at : = H / 2 , and g is
gravity. Thus, p = 0 at z = H / 2 . Two extreme cases correspond to stationary, heated plates and to moving plates with Ap = 0.
P = [ ( P I - PO)/VOIB
2. DISCUSSION OF TliP T W O CASES
I f Au = O,A/) p 0, dimensional analysis leads to forms of the mean quantities in terms of several nondimensional parameters. The mean buoyancy
gradient. for example, is
(4
P: = (AP,':)f'(P* c'o 1 5 )
where z is the vertical coordinate and
(3)
P = \llK,
= H(Ap)' ' / K 2 ', C = 4Ap)l;3,'Kz,3
In Eqs. (3), P is the Prandtl number, to is proportional to the cube root of
the Rayleigh number,
Ra = H' AylvK
and 4: may be regarded as the ratio of the height above the lower surface to
the thickness d7. - K*/'/(Ajj)' ' of the thermal boundary layer. We confine
;ittention, of course, to situations and regions in which To and 5 are large. . _
Many mcasuremcnts haw been made of the buoyancy flux q = KF, - wp.
wlierc we now denotc fluctuating velocities by u, L', w, and fluctuating buoyancy by p. The flux 4 is constant with height in a steady state and therefore
can be expressed as
(4)
All experiments indicate a very weak dependence on to when this number is
large (Turner, 1973) and it seems likely thatf, may be considered independent of to at large Rayleigh numbers. I t is then inescapable that y is directly