Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 59

not distinguish between these forms of stress but represent the combined stress by
t x (z) with the convention that this is the stress exerted by the fluid on the lower side of a
plane on the fluid above it. With this definition, the bottom face of the cuboid
in Fig. 3.3b experiences a force (t x

z)/2)dxdy in the positive x direction.
At the same time, the upper face of the cuboid exerts a force on the layer above
it given by (t x þ

dz( @ t x / @

¼
reaction), the cuboid will experience an equal and opposite force in the negative x
direction. The net force on the cuboid in the x direction will then be:

dz( @ t x / @ z)/2)dxdy. According to Newton's third law (action

t x @ t x þ @ ¼ @

t x
@

t x
@

t x
@

z

z = 2

z

z = 2 x y

z

z x y

ð 3 : 12 Þ

@

t x
@

per unit volume :

z

Similarly, the force on a particle for flow in the y direction will be given by the
analogous term

@ t y / @ z.

3.2.4 The equations of motion (and hydrostatics)

We can now include both pressure and frictional stresses to state the equations of
horizontal motion in x and y as:

Du
Dt ¼

1 @ p

x þ @

t x
@

F x
;

fv þ

@ z

ð 3 : 13 Þ

Dv
Dt ¼

1 @ p

y þ @

t y
@

F y

fu þ

@ z

where F x and F y now represent any additional forces which may be acting (e.g. tidal
forces). We have included only the frictional stresses associated with vertical changes
in the horizontal flow (termed vertical shear) and therefore neglected stresses due to
horizontal shear in the flow. This is a reasonable simplification for much of the
ocean, where conditions are laterally uniform, or nearly so, and vertical shear stresses
dominate. In regions of rapid horizontal changes, additional stress terms will need to
be included.
A dynamical statement analogous to the x and y equations is also available for the
vertical dimension z, i.e.:

Dw
Dt ¼

1 @ p

z

g : ð 3 : 14 Þ

@

It has a simpler form because, as explained above, the vertical component of the
Coriolis force can be neglected and frictional forces acting in the vertical are usually
small compared with g. In many situations, we can further simplify this equation by
appealing to the fact that vertical velocities are small and consequently the vertical
acceleration term Dw/Dt can be neglected in comparison with g. The only force in the