The blue vector perpendicular to the blue-outlined plane is the gradient vector for the f field. The red vector perpendicular to the red-outlined plane is the gradient vector for the g field. The green vector in the line of intersection of the two planes is their vector cross product. If the two constant surfaces coincide then the angle between the gradients is zero and the solenoid term vanishes. This condition prevails when the atmosphere is barotropic; i.e., when the atmospheric density is a function only of the pressure.

Derivation of the Equation for the Time Rate of Change 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 - κ∇p/p
 

From the defintion of entropy s it follows that

∇s = c_p∇θ/θ
 
and hence from the previous equation
 
∇s = c_p∇T/T − R∇p/p
since κ = R/c_p.
 

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

T∇s − ∇h = −(RT/p)∇p = −(1/ρ)∇p
 

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

dv/dt = T∇s − ∇h −gk + f
 

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

dv/dt = T∇s − ∇h −g∇z + f
or, equivalently
dv/dt = T∇s − ∇(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 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 + T∇s − ∇(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) = ∇T×∇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) + ∇T×∇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 ∇T×∇s.

Note that since ∇s = c_p∇T/T - R∇p/p

∇T×∇s = ∇T×(c_p∇T/T - R∇p/p)
= −∇T×(R∇p/p)
= −(R/p)(∇T×∇p)
 

Since by the ideal gas law

ln(T) = ln(p) − ln(ρ) − ln(R)
∇T/T = ∇p/p − ∇ρ/ρ
and hence
∇T×∇p = −(T/ρ)(∇p×∇ρ)
 

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.

Combining the last two derivations gives

∇T×∇s = (RT/pρ)(∇p×∇ρ)
= (1/ρ²)∇p×∇ρ
 

The −∇×(v×q) term in the vorticity equation reduces² to

q(∇·v) + [(v·∇)q − (q·∇)v]

These two terms are known as the divergence term and the twisting term, respectively.

Not much can be done analytically with the friction term except note that it is directed opposite to the velocity vector. It is of a low order of magnitude than the other terms and is usually neglected in the analysis.

The term vorticity in meterology usually refers to the vertical component of the vorticity vector. The vertical component of a vector may be obtained analytically by take the dot product of the vector with the unit vector in the vertical direction k; i.e., ζ = k·q. Thus the time rate of change of vorticity ζ is given by

dζ/dt = ζ(∇·v) + k·[(v·∇)q −(q·∇)v] + (1/ρ²)k·(∇p×∇ρ)
 

The terms on the right-hand side of the equation are known, respectively, as the divergence term, the twisting or tilting term and the solenoid term.

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×(∇×v).
 

² The vector identity

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

when A=v and B=q=∇×v reduces to

∇×(v×q) = −q(∇·v) − (v·∇)q + (q·∇)v
 

because the divergence of the curl of a vector vanishes.