  A complete model of the dynamics of scrape-off layer filaments will be rather
complex, including temperature evolution, three dimensional geometry and finite
Larmor radius effects. However, the basic mechanism of
$\boldsymbol{E}\times\boldsymbol{B}$ advection due to electrostatic potential
driven by the diamagnetic current can be captured in a much simpler model; a
complete understanding of the physics in the simpler model will then aid
interpretation of more complex simulations, by allowing the new effects to be
disentangled. Here we consider such a simple model, which assumes cold ions and
isothermal electrons and is reduced to two dimensions. We derive the scaling
with width and amplitude of the velocity of isolated scrape-off layer
filaments, allowing for arbitrary elliptical cross-sections, where previously
only circular cross-sections have been considered analytically. We also put the
scaling with amplitude in a new and more satisfactory form. The analytical
results are extensively validated with two dimensional simulations and also
compared, with reasonable agreement, to three dimensional simulations having
minimal variation parallel to the magnetic field.
