**Pressure
and Height Gradient
**

Conceptually,
the pressure gradient is the rate at which pressure decreases over
distance. The partial derivative
notation is consistent with our discussions of its interpretation from
Wednesday. The equivalence sign (instead of the
equals sign) is used because the right hand side represents a finite difference
approximation of the infinitesmal derivative on the left. (This allows us to estimate the
derivative on maps).

in
the Cartesian or rectangular coordinate system, and

in
the natural coodinate sytem

As
you will see, the horizontal wind depends strongly on the horizontal pressure
gradients. In the example used in
Metr 201, if the air parcel has dimensions of 100 km along the x-axis, the
pressure gradient would be evaluated

There
is no pressure gradient along the y-axis for this case.

As
we will see in future lectures, instead of looking at level surfaces,
meteorologists and oceanographers often look at constant pressure
surfaces. The chart corresponding to the 18000 foot pressure map in x, y, p coordinates, for example, is the 500 mb chart. The variation of heights on that pressure surface is comparable to the variation of pressure on a level surface.

On
a level surface, isobars can be used to express the pressure gradient. As we will see, on a constant pressure
surface, ÒheightÓ contours can be used to express the same mathematical
concept. ÒHeightÓ refers to the
elevation (usually in meters or decameters) above sealevel at which the given
pressure is found.

On
height maps, the expressions
analogous to those above are:

where
z is the height of the given pressure surface above sealevel (meteorology) and
above mean ocean surface levels (oceanography). Oceanographers commonly use these latter expressions to
develop their ideas about motion in the ocean. In meteorology (and oceanography), for example, one is required to do a finite difference estimate of the height gradient normal to the flow to calculate the geostrophic flow magnitude.