|San José State University|
& Tornado Alley
The purpose of this material is to derive the equation for the
time rate of change of vorticity in such a way as to point up the role
that the intersection of the pressure levels and density levels plays
in the development of vorticity. For an inertial frame of reference
the equations of motion for a
parcel of air are, in vector form,:
dv/dt = -(1/ρ)∇p -
gk + f
where v is the velocity vector, ρ the density, p pressure, g the acceleration due to gravity, k the unit vertical vector and f the vector of friction forces. The pressure gradient term
is especially important.
This term can be put into an interesting form by noting that from the definition of potential temperature θ:
and when the gradient operator ∇ is applied to this equation the result is
From the defintion of entropy s it follows that
Multiplying through by T and noting that cp∇T is the same as ∇h, where h stands for enthalpy, results in
Thus if the pressure gradient term in the equations of motion is replaced with T∇s - ∇h the result is
Since k is the same as ∇z the above equation is equivalent to
The motion-following derivative dv/dt is composed of an instaneous rate of change at a point and an advection term; i.e.,
The advection (inertial) term v·∇v can be expressed1 as
Thus the equations of motion for the atmosphere can be expressed in vector form as
The curl operator ∇× can be applied to this equation. The curl of any gradient of a scalar field vanishes; i.e., ∇×∇γ=0 for any scalar field γ because of the equality of cross derivatives. Therefore under the curl operation ∇(v2/2 + h + gz) vanishes.
Also, because the curl of a curl vanishes,
The result of applying the curl operator to the left-hand side of the above equation of motion (1) and taking into account the interchangeability of the time and space derivatives is
Equating this to the result of applying the curl operation to the right-hand side of the equation (1) gives
This form of the vorticity equation points out the role of the intersection or non-intersection of the isothermal surface and the isoentropic surface through the term ∇T×∇s, which has a magnitude equal to |∇T||∇s|sin(φ) where φ is the angle between the two vectors.
Note that since ∇s = cp∇T/T - R∇p/p
Since by the ideal gas law
Thus the ∇T×∇s term in the vorticity equation can be replaced by a term involving ∇p×∇ρ. Generally all of these cross product terms, called solenoid terms, are proportional and they all vanish when the atmosphere is barotropic; i.e.; when ∇ρ always has the same direction as ∇p.
1This follows from the vector identity
with A=B=v; i.e.,
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