Assuming j h = v and combining Eqs. (3.82) and (3.83), yields
@ u 0
i h
0
@t
þ u j
@ u 0
i h
0
@x j
¼ Àh
0 u 0
j
@U i
@x j
À u 0
i u 0
j
@h
@x j
þ d i3
h
0 h v
h v
!
gÀ
I
II
III
IV
V
À
@ h
0 u 0
i u 0
j
@x j
þ f c e ij3 u j h
À Á À
1
q
@ p 0 h
0
@x i
À p 0 @h
0
@x i
2
4
3
5 þ v
@ 2 u 0
i h
0
ð Þ
@x 2
j
VI
VII
VIII
IX
X
À2v0
@u 0
i
@x j
@h0
@x j
À
u 0
i
q c p
@R 0
nj
@x j
XI
XII
ð3:84Þ
Terms I and II relate to storage and to advection correlation between the
fluctuations of velocity and temperature; terms III, IV, and V reflect production or
consumption by average flow, tangential tensions due to turbulence and buoyancy;
term VI deals with transport by turbulent motions; term VII represents the Coriolis
forces; terms VIII and IX represent a redistribution due to the return to isotropy and
the transport by the pressure correlation term; terms X and XI represent the
molecular diffusion and viscous dissipation, respectively; term XI can be represented by 2e u i h , following the definition of at Eq. (3.68). Lastly, the term XII is
related to the correlations between velocity and the radiative budget fluctuations.
For the surface boundary layer, the equation can be simplified by eliminating the
Coriolis forces as well as term XII (including radiative balance fluctuations) and
molecular diffusion:
@ u 0
i h
0
À Á
@t
þ u j
@ u 0
i h
0
À Á
@x j
¼ Àh
0 u 0
j
@u i
@x j
À u 0
i u 0
j
@h
@x j
þ d i3
h
0 h v
h v
g À
@ h
0 u 0
i u 0
j
À
Á
@x j
À
1
q
@ p 0 h
0
À Á
@x i
À p 0 @h
0
@x i
!
À 2e u i h
ð3:85Þ
In the case of vertical flow of sensible heat, (i = 3), given the conditions of
homogeneity in the horizontal plane and absence of advective effects (Stull 1994),
we have
@ u 0
3
h0
ð Þ
@t
¼ Àu 0
3
2 @h
@x 3
À
@ h 0u 02
3
À Á
@x 3
þ
þ
gh0h
0
v
h m
À
1
q
@ p 0 h
0
À Á
@x 3
À p 0 @h
0
@x 3
!
À 2e u 3 h
ð3:86Þ
3.5 Introduction to Turbulent Motion Equations
53