Chapter 3: NEAR- SURFACE TURBULENCE
the wave field.” This allows one to analyze the near-surface processes
between wave troughs and crests.
The amplitude of wave oscillations decreases as
exp kz with depth,
where k is the wavenumber and z is the vertical coordinate (conventionally
directed upward). Csanady’s (1984) coordinate system can, therefore, only
be used close to the air-sea interface (i.e., at
1
z
k
). Soloviev (1992)
proposed an extension of the Csanady (1984) approach by introducing a
Lagrangian coordinate system that accounts for the depth attenuation of the
surface wave-induced perturbation. This co-ordinate system is described by
the following transformation:
x
x c
; y y
c
;
)
,
( t
z
z
z
K
c
,
(3.1)
where x and y are the horizontal axes, z is the vertical coordinate fixed to the
still water level of the ocean. Wave displacement, K , is given as a FourierStieltjes integral (Iyanaga and Kawada, 1980),
( , , , )
exp(
) exp[ (
)] ( , )
x y z t
k z
i k l
t dZ k
K
Z
Z
³³
G G
G
,
(3.2)
where
)
,
( Z
k
dZ
G
is the Fourier-Stieltjes amplitude introduced in such a way
that
( , ) ( , )
( , )
dZ k
dZ k
k
dkd
K
Z
Z
Z
Z
)
G
G
G
G
,
( , )
k
K
Z
)
G
is the surface wave
spectrum, k
G
is the wavenumber vector with components
y
x k
k ,
, and l
G
is
the vector with components
y
x, .
An important property of transformation (3.1)-(3.2) is that kinematic
effects of waves on the near-surface flow in this Lagrangian coordinate
system are largely eliminated. Transformation (3.1)-(3.2) projects the surface
layer disturbed by the linear potential waves into a flat near-wall layer.
Turbulence properties of this near-wall layer may differ from the classical
wall layer because of slip boundary conditions. Nevertheless, it is sometimes
convenient to compare the near-surface turbulence to well-known properties
of the turbulent boundary layer near a wall.
3.1.2 Wall layer analogy
A classic wall layer consists of an inner and outer part. Following Nowell
(1983), the inner part of the wall layer is defined as 0
0.2
z
h
d
; the outer
part, as 0.2h z h
d d (where z is the distance to wall and h is the depth of
the mixed layer of the thickness of the turbulent boundary layer). Some
145
the wave field.” This allows one to analyze the near-surface processes
between wave troughs and crests.
The amplitude of wave oscillations decreases as
exp kz with depth,
where k is the wavenumber and z is the vertical coordinate (conventionally
directed upward). Csanady’s (1984) coordinate system can, therefore, only
be used close to the air-sea interface (i.e., at
1
z
k
). Soloviev (1992)
proposed an extension of the Csanady (1984) approach by introducing a
Lagrangian coordinate system that accounts for the depth attenuation of the
surface wave-induced perturbation. This co-ordinate system is described by
the following transformation:
x
x c
; y y
c
;
)
,
( t
z
z
z
K
c
,
(3.1)
where x and y are the horizontal axes, z is the vertical coordinate fixed to the
still water level of the ocean. Wave displacement, K , is given as a FourierStieltjes integral (Iyanaga and Kawada, 1980),
( , , , )
exp(
) exp[ (
)] ( , )
x y z t
k z
i k l
t dZ k
K
Z
Z
³³
G G
G
,
(3.2)
where
)
,
( Z
k
dZ
G
is the Fourier-Stieltjes amplitude introduced in such a way
that
( , ) ( , )
( , )
dZ k
dZ k
k
dkd
K
Z
Z
Z
Z
)
G
G
G
G
,
( , )
k
K
Z
)
G
is the surface wave
spectrum, k
G
is the wavenumber vector with components
y
x k
k ,
, and l
G
is
the vector with components
y
x, .
An important property of transformation (3.1)-(3.2) is that kinematic
effects of waves on the near-surface flow in this Lagrangian coordinate
system are largely eliminated. Transformation (3.1)-(3.2) projects the surface
layer disturbed by the linear potential waves into a flat near-wall layer.
Turbulence properties of this near-wall layer may differ from the classical
wall layer because of slip boundary conditions. Nevertheless, it is sometimes
convenient to compare the near-surface turbulence to well-known properties
of the turbulent boundary layer near a wall.
3.1.2 Wall layer analogy
A classic wall layer consists of an inner and outer part. Following Nowell
(1983), the inner part of the wall layer is defined as 0
0.2
z
h
d
; the outer
part, as 0.2h z h
d d (where z is the distance to wall and h is the depth of
the mixed layer of the thickness of the turbulent boundary layer). Some
145