The Lamb–Chaplygin dipole model is a mathematical description for a particular inviscid and steady dipolar vortex flow. It is a non-trivial solution to the two-dimensional Euler equations. The model is named after Horace Lamb and Sergey Alexeyevich Chaplygin, who independently discovered this flow structure. This dipole is the two-dimensional analogue of Hill's spherical vortex.
The model A two-dimensional (2D), solenoidal vector field u {\displaystyle \mathbf {u} } may be described by a scalar stream function ψ {\displaystyle \psi } , via u = − e z × ∇ ψ {\displaystyle \mathbf {u} =-\mathbf {e_{z}} \times \mathbf {\nabla } \psi } , where e z {\displaystyle \mathbf {e_{z}} } is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream function is related to the vorticity ω {\displaystyle \omega } via a Poisson equation: − ∇ 2 ψ = ω {\displaystyle -\nabla ^{2}\psi =\omega } . The Lamb–Chaplygin model follows from demanding the following characteristics:
The dipole has a circular atmosphere/separatrix with radius R {\displaystyle R} : ψ ( r = R ) = 0 {\displaystyle \psi \left(r=R\right)=0} . The dipole propages through an otherwise irrotational fluid ( ω ( r > R ) = 0 ) {\displaystyle \omega (r>R)=0)} at translation velocity U {\displaystyle U} . The flow is steady in the co-moving frame of reference: ω ( r < R ) = f ( ψ ) {\displaystyle \omega (r<R)=f\left(\psi \right)} . Inside the atmosphere, there is a linear relation between the vorticity and the stream function ω = k 2 ψ {\displaystyle \omega =k^{2}\psi }
The solution ψ {\displaystyle \psi } in cylindrical coordinates ( r , θ {\displaystyle r,\theta } ), in the co-moving frame of reference reads:
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