Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 57

by u

x , i.e. the distance travelled in the x direction multiplied by the rate of change
of u in the same direction. Adding the equivalent contributions for movement in the y
and z directions, the total change du following a particle is just:

t @ u
@

¼ @

u

t @

u

t @

u

t @

u

u

t

t þ u

x þ

v

y þ

w

z :

ð 3 : 8 Þ

@ @ @ @

Dividing by dt and then allowing time increment dt to tend to zero, we have for the
total rate of acceleration following a particle:

Du

Dt ¼ @

u

u @

u

v @

u

w @

u

t þ x þ y þ z :

ð 3 : 9 Þ

@ @ @ @

We can use the operator D/Dt, termed the total or material derivative, to express the
rate of change for any property of a particle as it moves through the fluid. The last
three terms on the right of Equation (3.9) are the non-linear terms. In many cases,
when the spatial gradients are not too large, we can make the linearising

approximation that D/Dt t and neglect the non-linear terms.

With this form for the acceleration terms, we can now write the dynamical state-
ments (Equation 3.5) for motion of a unit volume (we replace mass m with density r)in
the x and y direction as:

¼ @ / @

Du
Dt ¼

F x
;

Dv
Dt ¼

F y

fv þ fu þ ð 3 : 10 Þ

where we have included the Coriolis force components ( Equation 3.7) and the terms
F x and F y represent the net force (per unit volume) acting in the x and y directions
respectively.

3.2.3 Internal forces: how do we include pressure and frictional forces?

In addition to the externally imposed forces such as wind stress acting at the sea
surface, there are two important classes of force which operate in the interior of the
fluid. Fluid particles exert normal forces on each other, where we are using 'normal'
in the mathematical sense as acting perpendicular to a plane. All of the individual
normal forces between fluid particles combine to make up the pressure which is
defined as the normal force per unit area. At the same time, particles influence each
other via frictional forces which act tangentially to exert stresses (force per unit
area). 2 In the following paragraphs, we shall examine the way these two internal
forces contribute to the dynamics, starting with the pressure field.
Pressure acts the same in all directions, i.e. the normal force on a small, solid plane
located in the fluid would be the same whatever the plane's orientation. To determine

2 For a pressure force, think of pushing down on the table. For the tangential shear force, think of the
force you exert if you slide your hand along the table.