Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 52

Setting this rate of increase equal to the cuboid volume the rate of density increase,

we have, after dividing both sides by dx dy dz, a statement of mass continuity:

@ð u Þ

þ @ð

v Þ

þ @ð

w Þ

¼ @
@

t
If we can regard density as constant, then this relation simplifies to:

@ x @ y @ z

@ u

x þ @ y þ @

v w

@

z ¼

0 ð 3 : 3 Þ

@ @

which is a widely used statement of the continuity of volume. Equation (3.3) can be
conveniently abbreviated to the statement that the divergence of the velocity is zero,
i.e. r:

¼

u 0 where u

¼ð

u ; v ; w Þ is the vector representing the velocity components.

As well as insisting on the continuity of the fluid, we can often apply conservation
rules to its properties. For example, changes in the total mass of salt in a region of the
shelf seas must be matched by the net input or output at the boundaries. Such
statements impose simple but powerful constraints which can greatly assist analysis.
A further simplification which we can frequently exploit is that flow in the sea is
mainly in the horizontal plane. This is because, in most parts of the ocean, density
increases with depth so that a particle displaced upwards or downwards experiences
a restoring force. Vertical movement is thus suppressed by stratification so that
vertical velocities are very much less than those in the horizontal plane and the
motion can often be regarded as two dimensional (2D). There are exceptions to this
horizontal 2D pattern. For example, Fig. 3.1 shows two situations in which substan-
tial vertical motions can occur in the ocean. In the first ( Fig. 3.1a ), local cooling of the
sea surface can lead to the sinking of denser water in convective chimneys with surface
flow converging at the top of the chimney and divergent flow away from the base of
the chimney below. In the second example ( Fig. 3.1b ), horizontal flow at the surface
converges along the line of a front, with vertical flows along the sloping density
surfaces and a line divergence further down the water column. An example of such
a line convergence, occurring along the plume front at the mouth of an estuary, can
be seen in Fig. 3.1c . As we shall see later in this chapter, important exceptions to 2D
flow also occur in the upwelling and downwelling motions at coastal boundaries.

3.2 Dynamics: applying Newton's Laws

...................................................................................
The central principle of dynamics is Newton's second law of motion. This law is
concerned with changes in the momentum of a body and can be expressed concisely
in the statement that the acceleration of a body a is determined by the net force on the
body F divided by its mass m, i.e. a

¼ F = m. To express this relation for a fluid, we