Introduction - An Introduction to Dynamic Meteorology - page 23

Thus in the isobaric coordinate system the horizontal pressure gradient force is
measured by the gradient of geopotential at constant pressure. Density no longer
appears explicitly in the pressure gradient force; this is a distinct advantage of the
isobaric system.

1.6.3 A Generalized Vertical Coordinate

Any single-valued monotonic function of pressure or height may be used as the
independent vertical coordinate. For example, in many numerical weather predic-
tion models, pressure normalized by the pressure at the ground

[ ≡

σ p(x, y, z, t)/

p s (x,y,t) is used as a vertical coordinate. This choice guarantees that the ground

is a coordinate surface (σ

]

1) even in the presence of spatial and temporal surface
pressure variations. Thus, this so-called σ coordinate system is particularly useful
in regions of strong topographic variations.
We now obtain a general expression for the horizontal pressure gradient, which
is applicable to any vertical coordinate s

≡

s(x, y, z, t) that is a single-valued
monotonic function of height. Referring to Fig. 1.12 we see that for a horizontal
distance δx the pressure difference evaluated along a surface of constant s is related
to that evaluated at constant z by the relationship

=

p C − p A p C − p B δz

δx +

p B − p A

=

δx δz δx

Taking the limits as δx, δz → 0 we obtain

∂p
∂x

∂z
∂x

∂p
∂x

∂p
∂z

s = s +

(1.27)

z

Fig. 1.12 Transformation of the pressure gradient force to s coordinates.