These allow the calculation of the
vertical variation of velocity from the density field.
The name thermal is an artifact from the original
meteorological use where the temperature field was used
as a proxy for the density field. In oceanography, the fact
that salinity can also significantly contribute to variations
in the density field leads to the use of density rather than
temperature. The thermal wind equations are derived from
the horizontal equations of motion and the hydrostatic equation,
beginning with the equations of motion reduced to the geostrophic
equations, i.e.
where
is the Coriolis parameter, u and
v the horizontal velocity components, p the pressure,
and
the density.
The vertical derivative of each equation is taken, the order
of differentiation switched for the pressure, and the
hydrostatic equation (
) substituted to
obtain
These equations only give the variation of the velocity with
depth. Further information must be supplied to obtain absolute
velocities.