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 Thayer Watkins
Silicon Valley
& Tornado Alley
USA

The Thermal Wind Equation:

The Relation Between
the Horizontal Temperature Gradient and
the Vertical Wind Shear

The Standard Derivation

The geostrophic wind can be expressed in vector form as (where variables in red represent vectors):


Vg = (1/ρf)k×∇Hp
 

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


∂Vg/∂z =
(-1/ρ2f)(∂ρ/∂z)k× ∇Hp + (1/ρf)k×∂(∇Hp)/∂z
= (-1/ρ2f)(∂ρ/∂z)k×∇Hp + (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:


∂Vg/∂z =
(-1/ρ)(∂ρ/∂z)[(1/ρf)k×∇Hp] + (1/ρf)k×∇H(-ρg)
 

Since 1/ρf)k×∇Hp is just Vg and g is a constant the above equation reduces to:


∂Vg/∂z =
(-1/ρ)(∂ρ/∂z)Vg - (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:


∂Vg/∂z =
- ∂(ln ρ)/∂z)Vg - (g/f)k×∇H(ln ρ)
 

The equation of state 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


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

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


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

Thus the two terms on the right involving Vg cancel leaving


∂Vg/∂z = (∂(ln T)/∂z)Vg + (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:



∂Vg/∂z = + (g/f)k×∇H(ln T)
 

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

A Shorter Derivation

Using the Gas Law the geostrophic wind can be expressed as


Vg = (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


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

can be expressed as


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


∂Vg/∂z = (∂(ln T)/∂z)Vg + (g/f)k×∇H(ln T)
 

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