Copyright (2000) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Physics of Fluid, Vol. 12, No. 5, 2000 and may be found at X. Y. Yin, D. J. Sun, M. J. Wei, and J. Z. Wu, Absolute and convective instability character of slender viscous vortices, Physics of Fluid, Vol. 12, No. 5, 2000.