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:

$$ \begin{align} \psi = \begin{cases} \frac{-2 U J_{1}(kr)}{kJ_{0}(kR)}\mathrm{sin}(\theta), & \text{for } r < R, \\ U\left(\frac{R^2}{r}-r\right)\mathrm{sin}(\theta), & \text{for } r \geq R, \end {cases} \end{align} $$

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.