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 Geopotential Tendencyand Divergence

## The Geopotential Tendency Equation

The geopotential tendency equation is:

#### [∇2+∂/∂p((f02/σ)(∂/∂p)]χ = B - C

where χ=∂Φ/∂t and

#### B = -f0Vg·∇((1/f0)∇2+f) = -f0Vg·∇ηg C = ∂/∂[(f02/σ)Vg·∇(∂Φ/∂p)]

where ηgg+f.

Consider the case where the geopotential field is

#### Φ = Φ0(p) - f0U0yψ1(πp/p0) + (f0V/k)ψ2(kx)ψ3(ly)

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:

• The components of the gradient of the geopotential field are:

#### ∂Φ/∂x = f0Vψ2'(kx)ψ3(ly) ∂Φ/∂y = -f0U0ψ1(πp/p0) + (f0Vl/k)ψ2(kx)ψ3'(ly)

• The geostrophic wind is Vg = (1/f0)k×∇Φ. This evaluates to:

Thus

#### ∇2Φ = -(k+l2/k)ψ2(kx)ψ3(ly) -(K2/k)ψ2(kx)ψ3(ly)

where K2=k2+l2.

Therefore the geostrophic approximation
of relative vorticity is

#### ζg = (1/f0)∇2Φ = -(K2V/k)ψ2(kx)ψ3(ly)

The components of the gradient of this relative vorticity are:

#### Vg·∇ζg = -K2U0Vψ1(πp/p0) + (K2Vl/k)ψ2(kx)ψ2'(kx)ψ3(ly)ψ3'(ly) - (K2Vl/k)ψ2(kx)ψ2'(kx)ψ3(ly)ψ3'(ly)

The last two terms cancel each other so that if the planetary vorticity f is taken as constant

#### Vg·∇ηg = -K2U0Vψ11<(πp/p0)

and thus the B term of the geopotential tendency equation is

#### B = -f0(Vg·∇ηg) = f0K2U0Vψ1(πp/p0)

The computation of the C term of the geopotential tendency equation is as follows:

#### ∂Φ/∂p = Φ0'(p) - (f0Uyπ/p0)ψ1'(πp/p0)

The components of the gradient of this quanitity are:

Therefore,

#### Vg·∇(∂Φ/∂p) = - (f0U0Vπ/p0)ψ1'(πp/p0)ψ2'(kx)ψ3(ly) and ∂/∂p(Vg·∇(∂Φ/∂p)) = - (f0U0Vπ2/p02)ψ1(πp/p0)ψ2'(kx)ψ3(ly)

Thus the C term of the geopotential tendency equation is given by

#### C = (f02/σ)∂/∂p(Vg·∇(∂Φ/∂p)) = (f03U0Vπ2/σp02)ψ1(πp/p0)ψ2'(kx)ψ3(ly)

Therefore the RHS of the geopotential tendency equation is:

#### B-C = [f0K2U0V - (f03U0Vπ2/σp02)]ψ1(πp/p0)ψ2'(kx)ψ3(ly) and hence B-C=0 if f0K2U0V - (f03U0Vπ2/σp02 = 0 which reduces to K2 = (1/σ)(f0π/p0)2

The expression (f0π/p0) 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

## The Estimation of Divergence

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

#### ∇·V + ∂ω/∂p = 0 and so δ = -∂ω/∂p

For adiabatic flow Holton (p. 166) gives the omega equation in the form

#### [∇2+∂/∂p((f02/σ)(∂/∂p)]ω = (f0/σ)(∂Vg/∂p)·∇ηg

The components of (∂Vg/∂p) are:

Thus

Therefore

#### [∇2+∂/∂p((f02/σ)(∂/∂p)]ω = -(f0K2U0Vπ/σp0)ψ1'(πp/p0)ψ2'(kx)ψ3(ly)

This means that ω is proportional to ψ1'(πp/p02'(kx)ψ3(ly) and hence

Therefore,

where

If

#### K2 = (1/σ)(f0π/p0)2

then the denominator of D reduces to 2K2 and the numerator is K4/(f0π/p0) and hence D reduces to

This means that

#### ∂ω/∂p = -D(π/p0)ψ1'((πp/p0)ψ2'(kx)ψ3(ly) and hence δ = ∇·V = -(∂ω/∂p) = D(π/p0)ψ1'(πp/p0)ψ2'(kx)ψ3(ly) or, equivalently, δ = (K2U0V/2f0)ψ1'(πp/p0)ψ2'(kx)ψ3(ly)

This is in comparison with geopotential height of

#### Z = Φ/g = Φ0(p)/g - (f0U0y/g)ψ1(πp/p0) + (f0V/gk)ψ2(kx)ψ3(ly)

Thus since δ is proportional to ψ2'(kx)ψ3(ly) but Z depends upon ψ2(kx)ψ3(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.  