J.H.G.M. van Geffen and G.J.F. van Heijst

Fluid Dynamics Research22, 191--213 (1998)

The Lamb dipole initially has a linear relation between vorticity
*omega* and streamfunction *psi*, namely
*omega=k²psi*, where the value of *k*
follows from *ak=*3.8317. As the dipole moves and grows, the
*omega,psi*-relation becomes nonlinear, at first near the edge
and the axis of the dipole, but as time goes on the nonlinearity spreads
towards the vorticity extrema. As long as there is still a linear part in
the *omega,psi*-relation around the extrema of vorticity, a
*k*-value can
be found. It appears that *k* decreases as function of time, but the
product *a*k* increases with time.
During this evolution the dipole retains its major characteristics: a more
or less circular form and a more or less linear *omega,psi*-relation.
These characteristics are named "Lamb-like".

As the dipole evolves further, a larger and larger part of the
*omega,psi*-relation becomes nonlinear, until finally the
entire *omega,psi*-relation can be described by
*omega=C**sinh*(2*psi)* for a certain constant *C*,
which depends only on
time. The dipole has then lost its main Lamb-like characteristics, although
it is still circular to within a few per cent.
The time scale at which this process takes
place is determined mainly by the strength of the viscous effects; it does
not so much depend on the strength of the initial dipole (a stronger initial
dipole only moves faster).

Since the Lamb dipole appears to retain its characteristics for a long time, tests have been done with other 2D dipolar vorticity structures as initial vorticity distribution. The alternative dipoles studied have the same symmetry property as the Lamb dipole: two patches of oppositely signed vorticity which lie symmetrically about the line of motion. These dipoles evolve into a Lamb-like dipole, followed by a decay like that of the Lamb dipole itself. The time it takes to form a Lamb-like dipole depends on the initial vorticity distribution: the further it is away from a Lamb dipole, the more time it takes.

The main conclusion therefore is that a dipolar vortex with Lamb-like characteristics is a very stable vorticity structure in a viscous finite fluid, and initial vorticity distributions that are not too different from a Lamb dipole evolve to a dipolar structure with Lamb-like characteristics.

**Note on notation here**

*
Greek characters and equations cannot (yet) be represented properly
on HTML-pages, hence they are given here in italics, however poor this
representation is.
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Jos van Geffen** --
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