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 $boldsymbol{E}timesboldsymbol{B}$ 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.
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