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

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.

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).