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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:

(∂q/∂x)s = (∂q/∂x)z + (∂q/∂z)x(∂z/∂x)s

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)zΔx+(∂q/∂z)xΔz

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:

Δq = (∂q/∂x)zΔx
- (∂q/∂z)x[(∂sss//∂x)z/(∂s/∂z)x]Δx

Division by Δx and allowing it to go to zero gives

(∂q/∂x)s = (∂q/∂x)z
- (∂q/∂z)x[(∂s/∂x)z/(∂s/∂z)x]

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.

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