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

In the following the variables in red represent vectors or the gradient operator which produces a vector. The vector of geostrophic wind velocity V_{g}can be expressed as
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:
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:
Since (1/ρf)k×∇_{H}(p) is just V_{g} and g is a constant the above equation reduces to:
Since (1/ρ)(∂ρ/∂z) = ∂(ln ρ)/∂z and (1/ρ)∇(ρ) = ∇(ln ρ) the above equation can be further simplified to:
The equation of state for an ideal gas, p=ρRT, is equivalent to
But from the hydrostatic equation
Therefore
But k×∇_{H}(ln p) is none other than (f/RT)V_{g} so:
Thus the two terms on the right involving V_{g} cancel leaving
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.
Using the Gas Law the geostrophic wind can be expressed as
Note that the hydrostatic equation is equivalent to:
Therefore when the geostrophic wind equation is differentiated with respect to z the result
which can be expressed as
But
This result substituted into the equation for vertical wind shear gives
Again the first term on the RHS is negligible compared to the second term.
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