Physics: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 [math]\displaystyle{ \mathbf{u} }[/math] may be described by a scalar stream function [math]\displaystyle{ \psi }[/math], via [math]\displaystyle{ \mathbf{u} = -\mathbf{e_z} \times \mathbf{\nabla} \psi }[/math], where [math]\displaystyle{ \mathbf{e_z} }[/math] is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream function is related to the vorticity [math]\displaystyle{ \omega }[/math] via a Poisson equation: [math]\displaystyle{ -\nabla^2\psi = \omega }[/math]. The Lamb–Chaplygin model follows from demanding the following characteristics:

• The dipole has a circular atmosphere/separatrix with radius [math]\displaystyle{ R }[/math]: [math]\displaystyle{ \psi\left(r = R\right) = 0 }[/math].
• The dipole propages through an otherwise irrorational fluid ([math]\displaystyle{ \omega(r \gt R) = 0) }[/math] at translation velocity [math]\displaystyle{ U }[/math].
• The flow is steady in the co-moving frame of reference: [math]\displaystyle{ \omega (r \lt R) = f\left(\psi\right) }[/math].
• Inside the atmosphere, there is a linear relation between the vorticity and the stream function [math]\displaystyle{ \omega = k^2 \psi }[/math]

The solution [math]\displaystyle{ \psi }[/math] in cylindrical coordinates ([math]\displaystyle{ r, \theta }[/math]), in the co-moving frame of reference reads:

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

where [math]\displaystyle{ J_0 \text{ and } J_1 }[/math] are the zeroth and first Bessel functions of the first kind, respectively. Further, the value of [math]\displaystyle{ k }[/math] is such that [math]\displaystyle{ kR = 3.8317... }[/math], 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.