for second moments exactly, and model third order terms (Donaldson.
1972a,b; Mellor. 1974; Daly and Harlow. 1970; Jones and Launder. 1972:
Ng and Spnlding. 1072). There is some justitication for this approach (which
we will follow): as we shall show later. whilc it is not possible to construct n
riitionill mcdel at secorid order. it is iIt third order. However, the model
constructed still rests on a fallacy: kinetic theory concepts are embodied,
implying that length and time scales of the transporting mechanism (the
turbulence) are small relative to length and time scales of the mean motion.
This is, of course, known not to be the case for turbulence. There is thus an
article of faith involved: if a crude assumption for second moments predicts
tirst moments adequately, perhaps a crude assumption for third moments
will predict second moments adequately.
Some of tliesc third order closure schemes are incomplete in the sense that
they providc no prediction for one of the scales (which may be taken to be
equivalent to ;I lciigth scale) (Donaldson, 1972a,b; Mellor. 1974). Others
(Daly and Harlow, 1970; Ng and Spalding. 1072; Jones and Launder, 1972)
do providc it supplementary equation equivalent to one for a length scale.
All of them, however. suffer from a basic flaw: they do not present any
method for generating the models used for the third order terms. Since the
models are constructed on an ad hoc basis, usually being required to have
only the same general tensor character as the terms modeled, models are
occasionally constructed that behave incorrectly with Reynolds number
(C'orrsin. 1972). or as we shall see, the right term is included for the wrong
reason. or important terms are omitted.
We will present here two related techniques which make it possible to
generate. in ii consistent and straightforward manner, models of all orders of
the third moments, and of all order in Reynolds number. The technique is
equi1lly applicable to stratification, to pollution dispersal, to chemical reactions, etc. Many of the terms generated are essentially those suggested by
othcr authors on an i d hoc basis. However, in the case of the third order
translwrt tcrnis. we will find that it is inconsistent within the model not to
iillow thc llux of onc second order quantity to be produced by gradients of
d l thc others, much as a molecular llux of salt can be produced in a liquid by
il tenipcrature gradient, and vice vcrsa. This opens the possibility of upgradient diffusion, an important process in atmospheric modeling. Unlike
thc situation in kinetic theory, where the crossdiffusion coefficients are
ordinarily small. the turbulent crossdiffusion coefficients may be substantial. However. in the (artificial) situation of constant eddy viscosity and
CoIlstiint structure, the forms obtained reduce to the classical forms assumed
by other authors on an ad hoc basis.
I t is possible that we will conclude ultimately that these third order closures, though within our reach computationally. are, for some purposes.
1972a,b; Mellor. 1974; Daly and Harlow. 1970; Jones and Launder. 1972:
Ng and Spnlding. 1072). There is some justitication for this approach (which
we will follow): as we shall show later. whilc it is not possible to construct n
riitionill mcdel at secorid order. it is iIt third order. However, the model
constructed still rests on a fallacy: kinetic theory concepts are embodied,
implying that length and time scales of the transporting mechanism (the
turbulence) are small relative to length and time scales of the mean motion.
This is, of course, known not to be the case for turbulence. There is thus an
article of faith involved: if a crude assumption for second moments predicts
tirst moments adequately, perhaps a crude assumption for third moments
will predict second moments adequately.
Some of tliesc third order closure schemes are incomplete in the sense that
they providc no prediction for one of the scales (which may be taken to be
equivalent to ;I lciigth scale) (Donaldson, 1972a,b; Mellor. 1974). Others
(Daly and Harlow, 1970; Ng and Spalding. 1072; Jones and Launder, 1972)
do providc it supplementary equation equivalent to one for a length scale.
All of them, however. suffer from a basic flaw: they do not present any
method for generating the models used for the third order terms. Since the
models are constructed on an ad hoc basis, usually being required to have
only the same general tensor character as the terms modeled, models are
occasionally constructed that behave incorrectly with Reynolds number
(C'orrsin. 1972). or as we shall see, the right term is included for the wrong
reason. or important terms are omitted.
We will present here two related techniques which make it possible to
generate. in ii consistent and straightforward manner, models of all orders of
the third moments, and of all order in Reynolds number. The technique is
equi1lly applicable to stratification, to pollution dispersal, to chemical reactions, etc. Many of the terms generated are essentially those suggested by
othcr authors on an i d hoc basis. However, in the case of the third order
translwrt tcrnis. we will find that it is inconsistent within the model not to
iillow thc llux of onc second order quantity to be produced by gradients of
d l thc others, much as a molecular llux of salt can be produced in a liquid by
il tenipcrature gradient, and vice vcrsa. This opens the possibility of upgradient diffusion, an important process in atmospheric modeling. Unlike
thc situation in kinetic theory, where the crossdiffusion coefficients are
ordinarily small. the turbulent crossdiffusion coefficients may be substantial. However. in the (artificial) situation of constant eddy viscosity and
CoIlstiint structure, the forms obtained reduce to the classical forms assumed
by other authors on an ad hoc basis.
I t is possible that we will conclude ultimately that these third order closures, though within our reach computationally. are, for some purposes.