Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 50

3

Response to forcing: the governing equations
and some basic solutions

In responding to the forcing discussed in the previous chapter, the motions of the
shelf seas are governed by fundamental physical laws. The principles involved,
namely those of the dynamics expressed in Newton's laws of motion, and those
of kinematics set by the geometrical rules of motion, are fundamentally the same
as those which solid body movement must obey. However, to express these laws
in an appropriate form for fluids is a little more difficult than for the solid body
case. Starting from the basic principles we shall show in this chapter, without
detailed derivations, how the equations of motion arise and illustrate the role
played by each of the main terms in the equations. Along the way, we shall
examine some simple force balances which describe particular forms of motion
that are of special importance in the ocean generally and in the shelf seas in
particular.

3.1 Kinematics: the rules of continuity

...................................................................................
When water moves in the ocean, it has to obey some fundamental constraints which
are independent of the forces driving the motion. These rules, termed kinematics, are
mainly concerned with maintaining the continuity of the fluid and, in some cases,
conserving its properties. Most fundamental and important is the rule that mass must
be conserved. Since seawater is almost, but not quite, incompressible, we can often
assert that volume as well as mass is conserved. When this is the case, we can readily
derive a relation (see Box 3.1 ) between the velocity components u, v, w:

@ u

x þ @ y þ @

v w

@

z ¼

0 : ð 3 : 1 Þ

@ @

This statement is the same for any Cartesian coordinate system, but we shall follow
a general convention in oceanography and choose the z axis positive upwards,
with z 0 at the surface, and the x and y axes positive eastward and northward

respectively. An alternative, and sometimes more convenient, form of the volume
continuity statement comes from integrating Equation (3.1) between the bottom
(z

¼

¼ h) and the surface to give: