THE NEAR-SURFACE LAYER OF THE OCEAN
representing the shear turbulence production and the dissipation, the
assumption of a constant stress layer, and the surface boundary conditions
produce the relationship:
1/ 4
/ M
q u B S
.
(3.37)
Substituting q and l in (3.32) with (3.33) and (3.37) respectively gives the
classic formula (3.4) for the dissipation rate in the logarithmic layer:
3/ 4
3
3
0
0
/ m
sh
u B S
u
B z z
z z
H
N
N
.
(3.38)
Note that constants in the Mellor and Yamada (1982) closure scheme are
chosen so that
3
1
M
S B { to ensure that for z >> z 0 the logarithmic layer
asymptote (3.4) is achieved; the logarithmic layer (or constant stress layer)
asymptote requires that the Coriolis terms in (3.28) and (3.29) are neglected.
For the asymptotic steady state regime, when TKE is produced at the
surface according to (3.35) and the shear production term is eliminated from
(3.31), the TKE equation represents a balance between the downward
diffusion of energy injected at the surface and dissipation wv
H . Thus (3.31)
reduces to
3
3
2
0
0
3
q
q
q
S
B
z z
z
z
z z
N
ª
º
w
w
«
»
w
w
«
»
¬
¼
.
(3.39)
The solution to (3.39) is as follows:
3
0
0
n
n
q c z z
c z z
,
(3.40)
where
4
.
2
)]
/(
3
[
2
/
1
2 B
S
n
q N
, and constants c and c are to be
determined from (3.36) and the condition of turbulence decay at large depth:
b = 0 at z o f . The c term in (3.40) represents decay away from the
surface while the c term should be equal to zero to satisfy the boundary
condition at z o f . The turbulence velocity scale is then described as
follows:
/ 3
1/ 3
0
n
q c
z z
,
(3.41)
178