Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 51

h @^

x þ @^

u v

þ @
@ t ¼

0 ð 3 : 2 Þ

@ @ y

where v are the velocities averaged over the depth h and the vertical velocity w

at the surface has been replaced by the rate of change of surface level

u and ^ ^

@ / @ t.

Box 3.1 Continuity statements

Consider the transport of fluid mass through a small fixed cuboid with dimensions
dx, dy, dz which will act as a 'control volume', as illustrated below. We want to
express the fact that what flows into the cuboid must come out, assuming that fluid is
not destroyed or created within the volume of the cuboid. The fluid has density r and
velocity components u, v, w (in Cartesian coordinates x, y, z) which generally will
vary in space and time.

y

u

u x

x

u

u

x 2 x

2

u

z

x

x

Figure B3.1

We start with flow in the x direction, with the mass flux, ru, in the centre of the
cuboid. The mass flow into the cuboid through the left face is the face area

2 , where the second term in the bracket
accounts for any change in u and/or r in the x direction. For the right face,
the mass flow will again be different due to the changes in u and r,andisgiven
by

@ð u Þ

@ x x

velocity density i.e. ð y z Þ u

2 . The net gain of fluid mass due to flow in the x direction is
the difference between what enters the box minus what leaves the box, which is just

þ @ð u Þ

@ x x

ð y z Þ u

z @ð

u Þ

@ð

u Þ

x y

x ¼

cuboid volume :

@ @ x

Combining with equivalent expressions for flow in the y and z directions, we see that
the net rate of gain of mass in the control volume is:

@ð

x þ @ð

u Þ

y þ @ð

v Þ w Þ

x y z :

@ @ @ z