EQUATION-OF-MOTION OF A DIFFUSING VORTEX SHEET

Moore's (1978) equation for following the evolution of a thin layer of uniform vorticity in two dimensions is extended to the case of a non- uniform, instantaneously known, vorticity distribution, using the meth od of matched asymptotic expansions. In general, the vorticity distrib ution satisfies a boundary-layer equation. This has a similarity solut ion in the case of a vortex layer of small thickness in a viscous flui d. Using this solution, an equation of motion of a diffusing vortex sh eet is obtained. The equation retains the simplicity of Birkhoff's int egro-differential equation for a vortex sheet, while incorporating the effect of viscous diffusion approximately. The equation is used to st udy the growth of long waves on a Rayleigh layer.