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A Derivation of the Vorticity Equation

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

-(1/ρ)p

is especially important.

This term can be put into an interesting form by noting that from the definition of potential temperature θ:

ln(θ) = ln(T) + κln(p0) - κln(p)

and when the gradient operator is applied to this equation the result is

θ/θ = T/T - κp/p

From the defintion of entropy s it follows that

s = cpθ/θ
and hence from the previous equation
 
s = cpT/T - Rp/p
since κ = R/cp.

Multiplying through by T and noting that cpT is the same as h, where h stands for enthalpy, results in

Ts - h = -(RT/p)p = -(1/ρ)p

Thus if the pressure gradient term in the equations of motion is replaced with Ts - h the result is

dv/dt = Ts - h −gk + f

Since k is the same as z the above equation is equivalent to

dv/dt = Ts − h −gz + f
or, equivalently
dv/dt = Ts − (h + gz) + f

The motion-following derivative dv/dt is composed of an instaneous rate of change at a point and an advection term; i.e.,

dv/dt = ∂v/∂t + v·∇v

The advection (inertial) term v·v can be expressed1 as

(v2/2) - v×(×v)
but ×v is just the vorticity vector q
so
v·v = (v2/2) - v×q

Thus the equations of motion for the atmosphere can be expressed in vector form as

(1)   ∂v/∂t = v×q + Ts − (v2/2 + h + gz) + f

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,

×(T×s) = s.

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

×(∂v/∂t) = ∂(×v)/∂t = ∂q/∂t
 

Equating this to the result of applying the curl operation to the right-hand side of the equation (1) gives

q/∂t = −×(v×q) + s + ×f

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 s, which has a magnitude equal to |T||s|sin(φ) where φ is the angle between the two vectors.

Note that since s = cpT/T - Rp/p

s = T×(cpT/T - Rp/p)
= -T×(Rp/p)
= -(R/p)p)

Since by the ideal gas law

ln(T) = ln(p) - ln(ρ) - ln(R)
T/T = p/p - ρ/ρ
and hence
p = - (T/ρ)ρ

Thus the s term in the vorticity equation can be replaced by a term involving 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

(A·B) = (A·)B + (B·)A + A×(×B) + B×(×A)

with A=B=v; i.e.,

(v·v) = 2(v·)v + 2v×(×v)
and hence
(v·)v = (1/2)(v·v) - v×(∇×s;v).



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