A two-dimensional (2D), solenoidal vector field may be described by a scalar stream function , via , where is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream function is related to the vorticity via a Poisson equation: . The Lamb–Chaplygin model follows from demanding the following characteristics: [citation needed]
- The dipole has a circular atmosphere/separatrix with radius : .
- The dipole propages through an otherwise irrorational fluid ( at translation velocity .
- The flow is steady in the co-moving frame of reference: .
- Inside the atmosphere, there is a linear relation between the vorticity and the stream function
The solution in cylindrical coordinates (), in the co-moving frame of reference reads:
where are the zeroth and first Bessel functions of the first kind, respectively. Further, the value of is such that , the first non-trivial zero of the first Bessel function of the first kind.[citation needed]