V+ = V / G
θ2
wθ2
Θ
(44b)
= -2wθ
-
- Nθ .
t
z
z
(39)
Ug + = Ug / G
(44c)
I
II
III
IV
Vg + = Vg / G
In eq 39, term I is the time rate of change of temper-
(44d)
ature variance. Term II is the production of temperature
t+ = f t
variance through the interaction of the vertical temper-
(44e)
ature flux ( wθ ) with the mean temperature gradient.
z+ = z / z0 .
Term III is the turbulent transport of temperature vari-
(44f)
ance by the vertical velocity fluctuations. Term IV, which
by definition is
Here
(
)
1/ 2
θθ
G = Ug + Vg2
2
(45)
Nθ ≡ 2 D
(40)
xj xj
is always positive. Since it appears with a minus sign in
aerodynamic roughness length of the surface. In addi-
eq 39, in analogy with ε, it represents the rate at which
tion, I will make uw and vw nondimensional by divid-
molecular processes dissipate the temperature variance.
2
From the conservation equation for specific humid-
will say much more about later.
ity, eq 8, we can follow exactly the same procedure as
Following Tennekes and Lumley (1972, p. 168 ff.),
above for potential temperature to derive conservation
I make eq 16 nondimensional
equations for the mean specific humidity
(uw / u* ) +
2
(
)
U+
1
1
2
=-
V+ - Vg +
Q
wq
Q
+
= Dw 2
(41)
Cg Ro* t+
z+
Cg Ro*
t
z
z
(46a)
and for the humidity variance
(vw / u* ) -
2
(
)
V+
1
1
=-
U+ - Vg+ .
2
2
q
Q
wq
= -2wq
-
- Nq
z+
Cg Ro* t+
Cg Ro*
(42)
t
z
z
(46b)
Here
where
u*
Cg ≡
(47)
G
q q
Nq ≡ 2 Dw
(43)
xj xj
is the geostrophic drag coefficient, which, as I will show
later, is typically 0.03. And
is the dissipation rate of specific humidity variance.
u
Ro* ≡
As in the TKE equation, eq 33, it is common in eq 39
*
(48)
f z0
and 42 to assume steady state and that the turbulent trans-
port terms are small. Production then again equals dis-
called the friction Rossby number (Tennekes and Lum-
ley 1972, p. 170), is typically of order 106 over sea ice.
sipation (Large and Pond 1982; Panofsky and Dutton
1984, p. 94).
If our scaling is accurate in eq 46, the terms contain-
ing the nondimensional variables should all be of order
one. When the first and last terms in each of these two
MONIN-OBUKHOV SIMILARITY
equations, however, are divided by Cg Ro* , terms of
order 104 result. Consequently, in the ASL, all that
The equations for the mean velocity components, eq
16, the mean potential temperature, eq 38, and the mean
remains of eq 46 are
specific humidity, eq 41, provide some important insights
(uw / u*2 ) ≅ 0
into processes in the atmospheric surface layer.
Let me define nondimensional versions (denoted by
(49a)
z+
+) of the variables in eq 16 that are appropriate in an
ASL
(vw / u* ) ≅ 0.
2
(49b)
U+ = U / G
(44a)
z+
6
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