Geoscience Reference
In-Depth Information
Since a variance is positive, the sign of a production term in a variance budget such
as Eq. (5.7) is clear: when placed on the right side of the equation it is positive. But
this does not hold for fluxes, which can be of either sign.
The mean-gradient production term is a contraction of kinematic Reynolds stress
and the mean scalar gradient. The turbulent scalar flux so produced need not be
aligned with the gradient of C producing it. In the vertical ( i
3) flu x budg et in the
horizontally homogeneous surface layer, for example, this term is
=
u 3 u 3 ∂C/∂x 3 ,
and the flux and mean gradient are aligned. But in the horizontal ( i
=
1) budget
the term is
u 1 u 3 ∂C/∂x 3 , which is the rate of production of horizontal scalar flux
by interaction of the turbulence with the vertical gradient of C .
The tilting production term changes the magnitude and direction of the scalar
flux through tilting by the mean velocity gradient. It is analogous to the stretching
and tilting term in Eq. (1.28) for vorticity. Each of these two production terms is of
order su 2 / .
The pressure-covariance term in Eq. (5.39) has long been considered difficult, if
not impossible, to measure reliably, and it was neglected in some early studies. The
other terms in the budget have been measured in field programs in the atmospheric
surface layer, allowing this pressure covariance to be inferred from their imbalance;
it is now known to be important ( Subsection 5.5.3 ). It has also been measured
directly ( Wilczak and Bedard , 2004 ).
As we explain in the Appendix, the molecular term in the scalar flux budget
(5.39) is conventionally neglected on the grounds of local isotropy so it reduces to
c ∂p
∂x i
.
∂cu i
∂t
U j ∂cu i
u j u i ∂C
cu j ∂U i
∂cu i u j
∂x j
1
ρ
=−
∂x j
∂x j
∂x j
(5.40)
In steady state this expresses a balance among, in order, mean advection; mean
production through the interaction of Reynolds stress and the mean scalar gradient,
and the interaction of scalar flux and mean velocity gradient; turbulent transport;
and destruction through pressure effects. Its leading terms are of order su 2 / .
5.4.2 The u i u k budget
Using the same scaling and loc al-iso tropy arguments to simplify the molecular
terms (Appendix), the budget of u i u k reduces to
∂u i u k
∂t
U j ∂u i u k
=−
( mean advection )
∂x j
u j u k ∂U i
u j u i ∂U k
∂x j
∂x j
( mean-gradient production )
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