Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 60

vertical of comparable magnitude to g is then the pressure gradient term @ p/ @ z so

that, to a good approximation, the force balance in the vertical is simply:

1 @ p

g or @

p

0 ¼

z z ¼

g : ð 3 : 15 Þ

@ @

This is the hydrostatic approximation in which the force of gravity on a particle is
balanced by the Archimedean upthrust exerted on it by the pressure field. The
equation is readily integrated to determine the pressure field of the ocean from the
density field which is obtained from measurements of temperature and salinity
profiles. In order to compute the pressure field accurately from density, we must
take account of the smallest variations of density which we can resolve from tem-
perature and salinity measurements. In practice, this means an accuracy in density of
=

10 5 , which requires temperature accurate to better than 0.0015 C and salinity
to better than 0.012 (PSS). This level of accuracy is not, however, necessary when we
are dealing with density as part of the inertia term (i.e. mass acceleration), in which

case, it is usually good enough to use a fixed, reference density value 3 which we will
denote by the symbol r 0 . These different ways of treating density appear, at first
sight, to be inconsistent but are readily justified and are widely used in what is
referred to as the Boussinesq approximation (see Phillips, 1966 ) . We shall frequently
make use of this simplifying assumption in what follows.
The hydrostatic force balance ( 3.15 ), together with the x and y dynamical state-
ments (Equation 3.13) and the continuity constraint (Equation 3.3) makes up the full
set of four equations governing motion, termed the equations of motion. They have
been derived under a particular set of assumptions (2D horizontal, incompressible,
hydrostatic flow) and provide a suitable theoretical basis for most of the topics we
shall discuss. However, you will appreciate that more general forms of the equations
of motion may be needed when our assumptions are not applicable.
In general, the equations of motion give us four equations for the four unknowns
(pressure and the three components of velocity) which can, in principle, be solved
when friction can be neglected. If frictional stresses are important, then, to close the
problem, we also need to know how to relate the turbulent stress components t x and
t y to other properties of the flow. We shall return to the question of how to achieve
this turbulence closure in Chapter 4 but, for now, we will proceed to examine some
relatively simple solutions of the equations of motion in which the frictional stresses
are neglected or specified in a very simple form.

3.3 Geostrophic flow

...................................................................................
We start with the case of steady flow in which only the pressure gradient and the
Coriolis force are involved and all other forces are excluded. The resulting balance
leads to important relations between the velocity and density fields which are applic-
able to a wide range of situations in the ocean and the atmosphere.

3 A commonly used value in shelf sea studies is r 0 ¼

1026 kg m 3 .