title: Lamb–Chaplygin dipole
text: # Lamb–Chaplygin dipole
  
  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 $\mathbf {u}$ may be described by a scalar stream function $\psi$, via $\mathbf {u} =-\mathbf {e_{z}} \times \mathbf {\nabla } \psi$, where $\mathbf {e_{z}}$ is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream function is related to the vorticity $\omega$ via a Poisson equation: $-\nabla ^{2}\psi =\omega$. The Lamb–Chaplygin model follows from demanding the following characteristics:
  
  * The dipole has a circular atmosphere/separatrix with radius $R$: $\psi \left(r=R\right)=0$.
  * The dipole propages through an otherwise irrorational fluid ($\omega (r>R)=0)$ at translation velocity $U$.
  * The flow is steady in the co-moving frame of reference: $\omega (r<R)=f\left(\psi \right)$.
  * Inside the atmosphere, there is a linear relation between the vorticity and the stream function $\omega =k^{2}\psi$
  
  The solution $\psi$ in cylindrical coordinates ($r,\theta$), in the co-moving frame of reference reads:
  
  ${\begin{aligned}\psi ={\begin{cases}{\frac {-2UJ_{1}(kr)}{kJ_{0}(kR)}}\mathrm {sin} (\theta ),&{\text{for }}r<R,\\U\left({\frac {R^{2}}{r}}-r\right)\mathrm {sin} (\theta ),&{\text{for }}r\geq R,\end{cases}}\end{aligned}}$
  
  where $J_{0}{\text{ and }}J_{1}$ are the zeroth and first Bessel functions of the first kind, respectively. Further, the value of $k$ is such that $kR=3.8317...$, the first non-trivial zero of the first Bessel function of the first kind.
  
  ## Usage and considerations
  
  Since the seminal work of P. Orlandi, the Lamb–Chaplygin vortex model has been a popular choice for numerical studies on vortex-environment interactions. The fact that it does not deform make it a prime candidate for consistent flow initialization. A less favorable property is that the second derivative of the flow field at the dipole's edge is not continuous. Further, it serves a framework for stability analysis on dipolar-vortex structures.
  
  