Geoscience Reference
In-Depth Information
“model parameterization science” to ensure continued progress in meteorological
model development (
NRC
,
2005
).
Appendix
Scaling the molecular-diffusivity terms in the budgets
of scalar flux and Reynolds stress
The molecular-diffusivity terms in the scalar flux budget
(5.39)
involve one large-
scale and one small-scale quantity, and so by Guideline 7 we try to avoid scaling
them. Instead we rewrite them:
γ
cu
i,j
,j
−
γ u
i
c
,jj
=
γ(u
i
c)
,jj
−
γ u
i,j
c
,j
,
(5.64)
=
ν(u
i
c)
,jj
−
ν
u
i
c
,j
,j
−
νc
,j
u
i,j
.
νcu
i,jj
The first terms on the right side of
Eqs. (5.64)
represent molecular diffusion. They
scale as
su
2
/
(the order of the leading terms in
Eq. (5.39)
) times
R
−
1
γ/ν
and
t
R
−
1
, respectively, so they are negligible.
Velocity and scalar derivatives scale as
t
∼
u
3
/ν
1
/
2
,
,j
∼
s
2
u/γ
1
/
2
.
(/ν)
1
/
2
u
i,j
∼
υ/η
∼
∼
s
d
/η
(5.65)
Thus the covariances in the second terms on the right side of
Eqs. (5.64)
scale as
cu
i,j
<s
u
3
/ν
1
/
2
=
s
2
u
3
/ν
1
/
2
,
u
i
c
,j
<
s
2
u
3
/γ
1
/
2
,
(5.66)
the “
<
” indicating upper bounds since we are scaling covariances of large- and
small-scale quantities, which are not well correlated
(Problem 5.17)
. Thus, the
second terms on the right side of
Eqs. (5.64)
scale as
γ
cu
i,j
,j
,ν
u
i
c
,j
,j
<su
2
/ (γ/ν) R
−
1
/
2
<su
2
/ (γ/ν)
1
/
2
R
−
1
/
2
.
(5.67)
t
t
Thus this pair of terms in
Eqs. (5.64)
is also negligible a
t lar
ge
R
t
.
We conclude that the molecular term in the budget of
u
i
c
reduces to
γ u
i
c
,jj
+
νcu
i,jj
−
(γ
+
ν)u
i,j
c
,j
≡−
χ
u
i
c
.
(5.68)
