Overshooting: Mixing Length Yields Divergent Results

Overshooting (OV) is the signature of the nonlocal nature of convection. To describe the latter, one needs five nonlocal, coupled differential equations to describe turbulent kinetic energies (total K and in the z -direction K z ), potential energy, convective flux, and rate of dissipation ε. We show analytically that if ε is assumed to be given by the local expression, ε = K 3/2 l -1 (mixing length l = α H p or l = z / a , since the region is small in extent), the remaining differential equations exhibit singularities (divergences) for specific values of a within the range of values usually employed. No solution can be found. Thus, OV results from such an approach are quite accidental, as they stem from an arbitrary fine tuning of a .