Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 65

to obtain relative currents between any two isobars p 1 and p 2 from their relative
slope y r as:

g
f

g
f tan

C 2 C 1 ¼

ð 2 1 Þ r ð : Þ

tan tan 3 28

where the final approximation takes advantage of isobaric slope angles being generally
<< 1 . In order to complete knowledge of the geostrophic currents relative to the Earth,

we need to know the slope of one isobar relative to the horizontal or the velocity on
one isobar. In the past (before 1960), when direct measurements of current were rarely
available, oceanographers relied heavily on geostrophic calculations to estimate
ocean currents. In efforts to make geostrophic current estimates absolute rather than
relative, it was often assumed, without proof, that there was a level of no motion at
which the isobars were horizontal. In the shelf seas, comparison with current meter
measurements indicates that assuming that there is no flow at the seabed is usually a
reasonable approximation in determining geostrophic currents from the density field.

3.3.3 Thermal wind

An alternative (and somewhat simpler) approach to the application of the geo-
strophic balance is to calculate the so called thermal wind, a term which clearly comes
from meteorology and indicates the origin of the method. The geostrophic balance
Equations (3.16) are combined with the hydrostatic relation (Equation 3.15) to
eliminate the pressure by cross-differentiating the equations. For example, using
the x equation and the hydrostatic relation, we have:

2 p

@ p

! @

x ¼ 0 f @

v

x ¼ 0 fv

@ @ @ @

z z

2 p

@ p

! @ g @

@ z ¼

g

x ¼

@ @ z @ x

so that:

0 f @

v

g @
@

z ¼ x :

ð 3 : 29 Þ

@

Repeating the same procedure for the geostrophic balance in the y direction, the two
components of the thermal wind shear can be written as:

@ v

0 f @

g @ u

0 f @

g

z ¼ x ; z ¼ y :

ð 3 : 30 Þ

@ @ @ @

In order to deduce the velocity profile, these equations can be integrated upwards
from the bed using the observed density gradients and the condition that u

¼ v ¼ 0 at

the bottom (z ¼ h). As we shall see in Chapter 8 , the thermal wind method provides