THE NEAR-SURFACE LAYER OF THE OCEAN
1.2.2 Boundary-layer approximation
Adjacent to the air-sea interface there are boundary layers in both the
atmosphere and the ocean. Their thickness is much less than their horizontal
dimension, which leads to the so-called boundary-layer approximation.
Under this approximation, the motion and continuity equations (1.1)-(1.4)
are as follows (Mellor, 1996):
1
1 zx
u
u
u
u
p
u
v
w
fv
t
x
y
z
x
z
W
U
U
w
w
w
w
w
w
w
w
w
w
w
w
,
(1.6)
1
1 zy
v
v
v
v
p
u
v
w
fu
t
x
y
z
y
z
W
U
U
w
w
w
w
w
w
w
w
w
w
w
w
,
(1.7)
p
g
z
U
w
w
,
(1.8)
0
u
v
w
x
y
z
w
w
w
w
w
w
.
(1.9)
Under the boundary-layer approximation, the conservation equation for
vertical momentum (1.3) reduces to the hydrostatic equation (1.8). This
equation provides a substantial simplification of the governing system of
equations because the vertical distribution of pressure can be determined
from the density field, and it does not directly depend on the fluid motion.
The heat, salinity, and other material transport equations in the upper
ocean under the boundary layer approximation are as follows:
rV
R
p
Q
I
Q
c
u
v
w
t
x
y
z
z
z
z
U
§
·
w
w
w4
w4
w4
w4
w
¨
¸
w
w
w
w
w
w
w
©
¹
,
(1.10)
rV
J
S
S
S
S
J
u
v
w
t
x
y
z
z
z
U
§
·
w
w
w
w
w
w
¨
¸
w
w
w
w
w
w
©
¹
.
(1.11)
6