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and Divergence |
The geopotential tendency equation is:
where χ=∂Φ/∂t and
where η_{g}=ζ_{g}+f.
Consider the case where the geopotential field is
where the ψ functions are linear combinations of the sine and cosine functions; i.e., any functions such that ψ"(z)=-ψ(z).
The B term of the geopotential tendency equation is computed as follows:
Thus
where K^{2}=k^{2}+l^{2}.
Therefore the geostrophic approximation
of relative vorticity is
The components of the gradient of this relative vorticity are:
The advection of the gradient of relative vorticity is given by:
The last two terms cancel each other so that if the planetary vorticity f is taken as constant
and thus the B term of the geopotential tendency equation is
The computation of the C term of the geopotential tendency equation is as follows:
The components of the gradient of this quanitity are:
Therefore,
Thus the C term of the geopotential tendency equation is given by
Therefore the RHS of the geopotential tendency equation is:
The expression (f_{0}π/p_{0}) re-occurs in the analysis so it is convenient to denote it as υ. This means that the condition for the geopotential tendency to be zero is
As noted by C.A. Riegel or his editor A.F.C. Bridger in Fundamentals of Atmospheric Dynamics and Thermodynamics,
It is well known that the vorticity of the geostrophic wind is a rather good approximation of the vorticity of the actual wind, but the divergence of the geostrophic wind is an exceptionally poor approximation to the divergence of the real wind. This is true for mass divergence and for velocity divergence, and the geostrophic wind should never be used for divergence calculations.
One approach to an alternate determination of divergence δ is to note that the continuity equation in isobaric coordinates is
For adiabatic flow Holton (p. 166) gives the omega equation in the form
The components of (∂V_{g}/∂p) are:
Thus
Therefore
This means that ω is proportional to ψ_{1}'(πp/p_{0})ψ_{2}'(kx)ψ_{3}(ly) and hence
Therefore,
where
If
then the denominator of D reduces to 2K^{2} and the numerator is K^{4}/(f_{0}π/p_{0}) and hence D reduces to
This means that
This is in comparison with geopotential height of
Thus since δ is proportional to ψ_{2}'(kx)ψ_{3}(ly) but Z depends upon ψ_{2}(kx)ψ_{3}(ly) divergence δ and geopotential height Z are 90° out of phase in the x direction while being in phase in the y direction.
Below are given the contour plots for the data given in JR Holton's problem M6.5. The first two are for p=750 mb and the third combines those two graphs. The horizontal and vertical scales are in units of 100 km.
The following are for p=250 mb.
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