THE NEAR-SURFACE LAYER OF THE OCEAN
2
3
3
0
( )/
( )
( , )
B
x
B
x
x
x
E k
B B
k kd
k F k h
c
k
S
D
P
P
<
|
|
³
G
,
(5.13)
where F is a universal function of its nondimensional arguments kh and P.
For kh <<1, we assume that
( , ) B
F kh
c
P
P
|
, since kh can formally be
dropped out of the number of determining parameters. The explicit form of
F as function of kh can in principle be obtained by merging turbulent
boundary layer spectra (Kamal et al., 1972; Wijesekera et al., 2001) with the
spectrum of two-dimensional turbulence.
After replacing buoyancy B in (5.13) with its expression via density Ú ̓ ̓
the one-dimensional horizontal wavenumber spectrum of density is as
follows:
3
( )/ var(
)
x
x
B
x
E k
c
k
U
U
P
w
|
(5.14)
where B
c is another universal function of its nondimensional argument P.
As we can see from Figure 5-8, in the wavenumber range,
4
4 10
x
k
!
m
-1 , which corresponds to the wavelength range
16
O
km, the
experimental spectrum follows the
3
x
k
law predicted by equation (5.14).
Function
B
c P used to show the k
-3 spectral law in Figure 5-8 depends on
parameter P, which is related to the vertical mixing process and horizontal
buoyancy gradients via equation (5.12).
5.3.6 Numerical diagnostics of nonlinear diffusion equation
Numerical diagnostics of the nonlinear diffusion equation (5.9) can help
to understand the essential physics beyond the universal spectrum (5.14). We
consider the following simplified form of equation (5.9):
3
t
x
x
B
B
B
J ª
º
w
w w
)
¬
¼
,
(5.15)
where
0
, /
B
g
x t
U
U
'
and J h W h is the mixed layer depth, and
Wis the vertical mixing time scale. Boundary conditions are set as follows:
0
x B
w
at x = -'L/2 and x = 'L/2, where 'L is the domain size.
For an axisymmetric case, equation (5.9) can be written in polar
coordinates (r, D), where r is the radial distance, and D is the polar angle.
Being extended evenly for r < 0, the axisymmetric version of equation has
the same form and the same boundary conditions as equation (5.15) but with
x replaced by r.
304
2
3
3
0
( )/
( )
( , )
B
x
B
x
x
x
E k
B B
k kd
k F k h
c
k
S
D
P
P
<
|
|
³
G
,
(5.13)
where F is a universal function of its nondimensional arguments kh and P.
For kh <<1, we assume that
( , ) B
F kh
c
P
P
|
, since kh can formally be
dropped out of the number of determining parameters. The explicit form of
F as function of kh can in principle be obtained by merging turbulent
boundary layer spectra (Kamal et al., 1972; Wijesekera et al., 2001) with the
spectrum of two-dimensional turbulence.
After replacing buoyancy B in (5.13) with its expression via density Ú ̓ ̓
the one-dimensional horizontal wavenumber spectrum of density is as
follows:
3
( )/ var(
)
x
x
B
x
E k
c
k
U
U
P
w
|
(5.14)
where B
c is another universal function of its nondimensional argument P.
As we can see from Figure 5-8, in the wavenumber range,
4
4 10
x
k
!
m
-1 , which corresponds to the wavelength range
16
O
km, the
experimental spectrum follows the
3
x
k
law predicted by equation (5.14).
Function
B
c P used to show the k
-3 spectral law in Figure 5-8 depends on
parameter P, which is related to the vertical mixing process and horizontal
buoyancy gradients via equation (5.12).
5.3.6 Numerical diagnostics of nonlinear diffusion equation
Numerical diagnostics of the nonlinear diffusion equation (5.9) can help
to understand the essential physics beyond the universal spectrum (5.14). We
consider the following simplified form of equation (5.9):
3
t
x
x
B
B
B
J ª
º
w
w w
)
¬
¼
,
(5.15)
where
0
, /
B
g
x t
U
U
'
and J h W h is the mixed layer depth, and
Wis the vertical mixing time scale. Boundary conditions are set as follows:
0
x B
w
at x = -'L/2 and x = 'L/2, where 'L is the domain size.
For an axisymmetric case, equation (5.9) can be written in polar
coordinates (r, D), where r is the radial distance, and D is the polar angle.
Being extended evenly for r < 0, the axisymmetric version of equation has
the same form and the same boundary conditions as equation (5.15) but with
x replaced by r.
304