Lamb–Chaplygin dipole

The flow structure of the 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 $𝐮$ may be described by a scalar stream function $\psi$, via $𝐮=-{𝐞}_{𝐳}\times \mathrm{\nabla }\psi$, where ${𝐞}_{𝐳}$ 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: $-{\mathrm{\nabla }}^2\psi =\omega$. The Lamb–Chaplygin model follows from demanding the following characteristics: Template:Citation needed

• The dipole has a circular atmosphere/separatrix with radius $R$: $\psi (r=R)=0$.
• 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:

$\begin{array}{c}\psi =\{\begin{array}{cc}\frac{-2U{J}_1(kr)}{k{J}_0(kR)}\mathrm{s}\mathrm{i}\mathrm{n}(\theta ),& \text{for }r<R,\\ U(\frac{R^2}{r}-r)\mathrm{s}\mathrm{i}\mathrm{n}(\theta ),& \text{for }r\ge R,\end{array}\end{array}$

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.Template:Citation needed

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.