4.2 Physical meaning of the equations

The equations (4.8) through (4.13) derived in the previous section are the core of the redistribution method. While they were derived using mathematical arguments, some have a clear physical meaning. For example, the lowest order equation (4.8) conserves circulation for each vortex.

Next, (4.9) conserves the center of vorticity; and (4.10) implies the correct expansion of the mean diameter. These conservation laws are expressions of the physical laws of conservation of linear and angular momentum, [122].

The positivity condition (4.13) expresses the physical fact that reverse vorticity cannot form spontaneously in the middle of a flow field.

The size (4.2) of the redistribution region corresponds to the typical distance of order $O(\sqrt{\nu\Delta t})$ over which the vorticity of a vortex diffuses during a time-step. It ensures that numerically the vorticity diffuses out over a distance of the same order.

Together, these properties imply that even if numerical resolution is poor, the possible effects of the errors remain quite limited. No false circulation, linear or angular momentum, or reversed vorticity can be created by the numerical errors. The center of vorticity is unaffected and the root mean square size of the vortex system expands at the correct rate. The vorticity will not expand over a region much larger than the physical one. The long range errors in velocity, which are determined by the vorticity moments, vanish. Disjoint sets of vortices much more than $O(\sqrt{\nu\Delta t})$ apart satisfy the conservation laws individually.