The Standard Derivation

In the following the variables in red represent vectors or the gradient operator which produces a vector. The vector of geostrophic wind velocity V_gcan be expressed as

V_g = (1/ρf)k×∇_H(p)

where k is the unit vertical vector, ∇_H(p) is the horizontal gradient of air pressure, ρ is the mass density of air, f is the Coriolus parameter and × is the vector cross product.

When this is differentiated with respect to height z the result is:

∂V_g/∂z =
(-1/ρ²f)(∂ρ/∂z)k× ∇_H(p) + (1/ρf)k×∂(∇_H(p))/∂z
= (-1/ρ²f)(∂ρ/∂z)k×∇_H(p) + (1/ρf)k×∇_H(∂p/∂z)

The first term on the right can be rearranged and -ρg can be substituted for ∂p/∂z on the basis of the hydrostatic equation to get:

∂V_g/∂z =
(-1/ρ)(∂ρ/∂z)[(1/ρf)k×∇_H(p)] + (1/ρf)k×∇_H(-ρg)

Since (1/ρf)k×∇_H(p) is just V_g and g is a constant the above equation reduces to:

∂V_g/∂z = (-1/ρ)(∂ρ/∂z)V_g −(g/ρf)k×∇_H(ρ)

Since (1/ρ)(∂ρ/∂z) = ∂(ln ρ)/∂z and (1/ρ)∇(ρ) = ∇(ln ρ) the above equation can be further simplified to:

∂V_g/∂z = − (∂(ln ρ)/∂z)V_g − (g/f)k×∇_H(ln ρ)

The equation of state for an ideal gas, p=ρRT, is equivalent to

ln(ρ) = ln(p) - ln(R) - ln(T)
and hence
∂(ln(ρ))/∂z = ∂(ln(p))/∂z - ∂(ln(T))/∂z
and likewise
∇_H(ln ρ) = ∇_H(ln p) - ∇_H(ln T)

But from the hydrostatic equation

∂(ln p)/∂z = -g/RT.

Therefore

∂V_g/∂z =
(g/RT + ∂ln T/∂z)V_g - (g/f)k×∇_H(ln p)
+ (g/f)k×∇_H(ln T)

But k×∇_H(ln p) is none other than (f/RT)V_g so:

∂V_g/∂z =
(-g/RT)V_g + ∂(ln T)/∂z)V_g + (g/f)(f/RT)V_g
+ (g/f)k×∇_H(ln T)

Thus the two terms on the right involving V_g cancel leaving

∂V_g/∂z = (∂(ln T)/∂z)V_g + (g/f)k×∇_H(ln T)

The first term on the right is neglectible compared to the second so the relation between the vertical shear in wind direction and the thermal gradient is:

Since k×∇_H(ln T) is perpendicular to ∇_H(ln T) in the horizontal plane the direction of ∂V_g/∂z is parallel to the isotherms.

A Shorter Derivation

Using the Gas Law the geostrophic wind can be expressed as

V_g = (RT/pf)k×∇_H(p) = (RT/f)k×∇_H(ln p)

Note that the hydrostatic equation is equivalent to:

∂(ln p)/∂z = - g/RT

Therefore when the geostrophic wind equation is differentiated with respect to z the result

∂V_g/∂z =
(R/f)(∂T/∂z)k×∇_H(ln p) + (RT/f)k×∇_H(∂(ln p)/∂z))

which can be expressed as

∂V_g/∂z =
((1/T)∂T/∂z)((RT/f)k×∇_H(ln p)
+ (RT/f)k×∇_H(-g/RT)
or equivalently
= ∂(ln T)/∂z)V_g - (gT/f)k×∇_H(1/T)

But

T∇_H(1/T) = T[-(1/T²)∇_H(T)]
= - (1/T)∇_H(T) = - ∇_H(ln T)

This result substituted into the equation for vertical wind shear gives

∂V_g/∂z = (∂(ln T)/∂z)V_g + (g/f)k×∇_H(ln T)

Again the first term on the RHS is negligible compared to the second term.