Analytic 1D Approximation of the Divertor Broadening S in the Divertor   Region for Conductive Heat Transport
  Topic is the divertor broadening $S$, being a result of perpendicular transport in the scrape-off layer and resulting in a better distribution of the power load onto the divertor target. Recent studies show a scaling of the divertor broadening with an inverse power law to the target temperature $T_t$, promising its reduction to be a way of distributing the power entering the divertor volume onto a large surface area.   It is shown that for pure conductive transport in the divertor region the suggested inverse power law scaling to $T_t$ is only valid for high target electron temperatures. For decreasing target temperatures ($T_t < 20\,$eV) the increase of $S$ stagnates and the conductive model results in a finite value of $S$ even for zero target temperature. It is concluded that the target temperature is no valid parameter for a power law scaling, as it is not representative for the entire divertor volume. This is shown in simulations solving the 2D heat diffusion equation, which is used as reference for an analytic 1D model describing the divertor broadening along a field line solely by the ratio of the perpendicular to the parallel diffusivity.   By assuming the temperature dependence of these two quantities an integral form of $S$ is derived, relying only on the temperature distribution along the separatrix between X-point and target. Integration along the separatrix results in an approximation for $S$, being in agreement with the 2D simulations. This model is also applicable to scenarios including heat losses, e.g. due to radiation.   Convective transport can not be neglected for high recycling conditions, hence the derived expression for $S$ is not expected to hold. However, based on the non-vanishing parallel transport, respective the finite parallel transport time, the divertor broadening is expected to reach a finite value. 
Dirk Nille and Bernhard Sieglin and Thomas Eich
physics.plasm-ph