city scales which are then prescribed. Althouzh this has worked well for the
neutral case. the validity o f the various assumptions involved is highly
yuestionable for stable conditions.
From the observations in the stable atmosphere, there is a strong indication that the log-linear profile extends well into the height range where the
flux variation must be considered (see Carl et ul., 1973). In neutral conditions also the logarithmic law holds up to 10-15 7; of the boundary layer
height even though the u? flux may vary by more than 40 7; in this layer
(see, e.g., Thuillier and Lapp, 1964; Panofsky. 1973; Wyngtlard et al., this
volume). Kccxntly, Tennekes (1973) has given an explanation for this.
Based on the above observational evidence an expression for the eddy
viscosity function can be derived for the lower part of neutral and stable
boundary layers if we can approximate the flux variation in this layer by a
simple function. Equation (13) may be considered to be the lowest-order
approximation, some of the limitation of which we have pointed out earlier.
A better choice seems to be the exponential function
which has the same initial slope as implied by Eq. (13) (this, of course, is
required by the equation of motion) and at the same time has some finite
curvature. Equation (25) fits the calculated profiles for the neutral case by
Deardorff (1972) and Wyngaard ct ul. (this volume) quite well up to a value
of < 2 0.15. The eddy viscosity distribution which is consistent with
Eqs. (3a) and (25) is given as
w herc
is a stiibility parameter.
The prescription of K, through Eq. (26) [a similar form was proposed
earlier by Arya (1973). for p, = 01 is not explicit, since ce/u+ is not known
until after the solution of Eqs. (19) and (20). To begin, some reasonable
value of uu/u* is assumed for a given p,, , and Eqs. (19). (20). and (26) are
solved numerically using a version of the “shooting method” (for more
details, wc Arya, 1973). The new calculated value of v,,/u* replaces the old
one and the equations are solved iteratively until theassumed and calculated
values of this parameter agree within a specified tolerance. Although our
assumptions or hypotheses implicit in the derivation of Eq. (26)are not valid
in the upper layer. the above eddy viscosity distribution will be assumed for
1 tic whole boundary layer since the solutions are considered not too sensitive
to the details of K, distribution in the upper layer (see, e.g., Arya, 1973).
neutral case. the validity o f the various assumptions involved is highly
yuestionable for stable conditions.
From the observations in the stable atmosphere, there is a strong indication that the log-linear profile extends well into the height range where the
flux variation must be considered (see Carl et ul., 1973). In neutral conditions also the logarithmic law holds up to 10-15 7; of the boundary layer
height even though the u? flux may vary by more than 40 7; in this layer
(see, e.g., Thuillier and Lapp, 1964; Panofsky. 1973; Wyngtlard et al., this
volume). Kccxntly, Tennekes (1973) has given an explanation for this.
Based on the above observational evidence an expression for the eddy
viscosity function can be derived for the lower part of neutral and stable
boundary layers if we can approximate the flux variation in this layer by a
simple function. Equation (13) may be considered to be the lowest-order
approximation, some of the limitation of which we have pointed out earlier.
A better choice seems to be the exponential function
which has the same initial slope as implied by Eq. (13) (this, of course, is
required by the equation of motion) and at the same time has some finite
curvature. Equation (25) fits the calculated profiles for the neutral case by
Deardorff (1972) and Wyngaard ct ul. (this volume) quite well up to a value
of < 2 0.15. The eddy viscosity distribution which is consistent with
Eqs. (3a) and (25) is given as
w herc
is a stiibility parameter.
The prescription of K, through Eq. (26) [a similar form was proposed
earlier by Arya (1973). for p, = 01 is not explicit, since ce/u+ is not known
until after the solution of Eqs. (19) and (20). To begin, some reasonable
value of uu/u* is assumed for a given p,, , and Eqs. (19). (20). and (26) are
solved numerically using a version of the “shooting method” (for more
details, wc Arya, 1973). The new calculated value of v,,/u* replaces the old
one and the equations are solved iteratively until theassumed and calculated
values of this parameter agree within a specified tolerance. Although our
assumptions or hypotheses implicit in the derivation of Eq. (26)are not valid
in the upper layer. the above eddy viscosity distribution will be assumed for
1 tic whole boundary layer since the solutions are considered not too sensitive
to the details of K, distribution in the upper layer (see, e.g., Arya, 1973).