Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 62

pressure is constant on an isobar, i.e. along an isobar changes in pressure in the
x direction are balanced by changes in the y direction:

¼ @

p

þ @

p

p

x

x

y

y ¼ 0

@ @

p ¼ @

¼

ð 3 : 20 Þ

=@

dy
dx

p x

so that tan c :

@ p =@ y

Geostrophic flow is, therefore, parallel to the isobars, or perpendicular to the
pressure gradient. The result is initially somewhat counterintuitive and in contrast
to the situation in flows at the laboratory scale where fluid usually moves down the
pressure gradient. Geostrophic flow, or a close approximation to it, occurs widely in
the atmosphere and in the ocean. Equations (3.16) are extensively used in meteorology
and oceanography to determine the velocity field from the pressure distribution. Before
the advent of recording current meters, knowledge of the ocean currents was largely
based on estimates of the geostrophic flow derived in this way from surveys of
temperature and salinity.

3.3.2 The gradient equation

The pressure at a depth z, p(z), arising from the vertical distribution of density through
the overlying water column, is derived by integrating the hydrostatic relation:

ð
z

ð ð

0

0
z

p ð z Þ p 0 ¼ gdz ¼ gdz þ gdz

ð 3 : 21 Þ

ð

0
z

¼ þ s g

gdz

where p 0 is the atmospheric pressure acting at the surface, is the surface elevation

above mean sea level and r s is the density at the surface which is assumed to be
constant between z .

We next take the derivative to find the horizontal pressure gradient for two cases:

¼ 0andz ¼

(i) The density r is constant or a function of z only
In this simple but important case, the density structure is the same everywhere in
x and y and so the pressure gradient becomes:

ð

0
z ð

@ p

n ¼ @

s @
@

s @
@

z Þ gdz þ g

n ¼

g

n :

ð 3 : 22 Þ

@ @ n

Note that the integral term is constant and has no variability in the horizontal, so it
has a zero horizontal derivative. If we ignore any difference (typically

< 1%) between