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The Change of Variables in Meteorological Equations

Meteorology has developed a special facility in the techniques of
changing variables. This developed from the practical matter of wanting to
use pressure as the vertical coordinate rather than geometric height. Hydrostatic
balance gives pressure as a monotonic function of height. But the change
of variables is not quite so simple as it might seem. First, the
monotonic function for pressure as function of height may be different at
different locations. Secondly, and more importantly, the partial derivatives
with respect to horizontal distances with pressure held constant are different
from those quantitities with height held constant. The fundamental
relationship, which probably should be a named lemma, is:

Let s(x,y,z,t) be any function that is monotonic in z if the other variables
are held fixed and let q be any variable. Then:

In explaining the above result it is convenient to leave y and t entirely out of
the picture. Suppose s=s(x,z) and q=q(x,z). In constructing Δx and
Δq when Δs=0 it must be that z changes. The value of
Δq depends upon not only the Δx but also the corresponding
Δz that results in Δs being zero. Thus

Δq =(∂q/∂x)_{z}Δx+(∂q/∂z)_{x}Δz

Division by Δx and taking the limit as Δx goes to zero gives the
required result.

More can be and needs to be said about (∂z/∂x)_{s}).
In constructing the increments in x and z, Δz must be such that
Δs = (∂s/∂x)Δx + (∂s/∂z)Δz = 0

This latter condition requires that:

Δz = -[(∂s/∂x)_{z}/(∂s/∂z)_{x}]Δx

When this value for Δz is substituted into the equation for
Δq the result is:

Some interesting things happen for specific cases. Suppose q is the same
variable as s. Then the first formula above becomes:

(∂s/∂x)_{s} = (∂s/∂x)_{z} + (∂s/∂z)_{x}(∂z/∂x)_{s} but, by definition,
(∂s/∂x)_{s} = 0
and thus
(∂s/∂x)_{z} = - (∂s/∂z)_{x}(∂z/∂x)_{s}

This is an equation that has puzzled students for generations because the
the usual intuition dois not expect the minus sign. J.R. Holton in his
An Introduction to Dynamic Meteorology displays a version of this
relation when s=pressure=p on pages 21 and 22 and notes the importance of the minus
but gives the wrong explanation. He says that the minus sign arises because
δz<0 for δp>0 but this shows up in terms of the sign of
∂p/δz not as a factor of -1 for that expression.