Fundamentals of Plasma Physics

16.3 Isomorphism to incompressible 2D hydrodynamics 463

16.3 Isomorphism to incompressible 2D hydrodynamics

Consider now the equations governing an incompressible two-dimensionalfluid with ve-
locityu=ur(r,θ)ˆr+uθ(r,θ)ˆθ.Incompressibility means that the mass densityρis constant
and uniform so that the continuity equation reduces to

∇·u=0. (16.13)

Using the vector identityu·∇u=∇u^2 / 2 −u×∇×uthefluid equation of motion can be
written as

ρ

(

∂u

∂t

+

1

2

∇u^2 −u×∇×u

)

=−∇P. (16.14)

Taking the curl of this equation and defining the vorticity vectorΩ=∇×ugives

∂Ω

∂t

=∇×(u×Ω) (16.15)

which has the same form as the ideal MHD induction equation, Eq.(2.81), and socan be
interpreted as indicating that the vorticity being frozen into the convectingfluid (Kelvin
vorticity theorem). Because

∇×u=ˆz

(

1

r

∂

∂r

(ruθ)−

1

r

∂ur

∂θ

)

(16.16)

the vorticity vector lies in thezdirection and may be written as

Ω=Ωˆz. (16.17)

Thezcomponent of Eq.(16.15) is

∂Ω

∂t

= ˆz·∇×(u×Ωˆz)

= ∇·((u×Ωˆz)׈z)

= −∇·(uΩ)

= −u·∇Ω. (16.18)

The incompressibility condition Eq.(16.13) means that the velocity can be written in terms
of a stream-functionψ,
u=−∇ψ×zˆ (16.19)
so Eq.(16.18) can be written as

∂Ω

∂t

=∇ψ׈z·∇Ω. (16.20)

Furthermore, thezcomponent of the curl of Eq.(16.19) can also be expressed in terms of
ψsince

Ω = ˆz·∇×u

= −zˆ·∇×(∇ψ׈z)

= −∇·((∇ψ×zˆ)×zˆ)

= ∇^2 ψ. (16.21)