Chapter 1: INTRODUCTION
where U is the water density, and
D u
v
w
Dt
x
y
z
t
w
w
w w
w
w
w
w
. The
rectangular coordinate system has its origin at the sea surface, with x
directed eastward, y directed northward, z directed upward, and u, v, and w
represent the corresponding velocity components; t is time;
2 sin
f
M
:
and
2 cos
y
f
M
:
, where : is the magnitude of the Earth’s rotation vector
and M is the latitude; p is pressure; g is the acceleration due to gravity.
Symbol W denotes different components of the stress tensor, due to viscosity
(and to fluctuating velocities if u, v, and w refer to ensemble mean
quantities.)
In the Boussinesq approximation, which is applicable to many oceanic
conditions, the density changes in equations (1.1)-(1.3) can be neglected
except in the gravity term entering (1.3) where U is multiplied by g . The
Boussinesq approximation is not valid for acoustic applications and for twophase environments like the surface wave breaker.
The system of equations (1.1)-(1.3) is complemented with the continuity
equation for incompressible fluids,
0
u
v
w
x
y
z
w
w
w
w
w
w
.
(1.4)
In the general case of an isotropic Newtonian fluid, the different
components of the viscous stress tensor depend linearly on strain
components (Mellor, 1996):
2
2
2
xx
yx
zx
xy
yy
zy
M
xz
yz
zz
u
u
v
u
w
x
y
x
z
x
v
u
v
v
w
K
x
y
y
z
y
w
u
w
v
w
x
z
y
z
z
W
W
W
W
W
W
U
W
W
W
ª
º
w
w
w
w
w
«
»
w
w
w
w
w
«
»
ª
º
«
»
w
w
w
w
w
«
»
«
»
«
»
w
w
w
w
w
«
»
«
»
¬
¼
«
»
w
w
w
w
w
«
»
w
w
w
w
w
¬
¼
,
(1.5)
where U is the water density that under the Boussinesq approximation can
be treated as a constant, and M
K is the kinematic viscosity. (Newtonian
fluids are fluids that have linear dependence of the stress tensor on strain
components.)
5