San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Thermal Wind Equation:

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

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 Vgcan be expressed as

Vg = (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:

Vg/∂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:

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

Since (1/ρf)k×H(p) 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 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.


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))

which 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)


TH(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)

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

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