The substantial drop of the plasma temperature along magnetic field lines with increasing plasma density is one of the main features in tokamak divertors. As a result the temperature gradient at the divertor plates becomes very steep and the boundary condition normally applied for the parallel Mach number M at the target, M t = 1, cannot be satisfied. In this case the value of M t based on the general form of the Bohm criterion, M t P 1, has to be determined from the continuity of plasma parameters.
In the present paper a new approach to resolve the Mach number at the target for such a situation is proposed. This method avoids the singularity problem that arises by treating the particle balance and parallel motion equations in a differential form. Instead, the integral representation of the equations is formulated for an arbitrary form of particle and momentum sources. The approach can also take into account transport perpendicular to the magnetic field lines.
The proposed method is demonstrated on the example of a one-dimensional stationary model for the scrape-off layer (SOL) plasma and includes the continuity-, parallel momentum-and heat transfer equations. The recycled neutrals are described in the diffusion approximation. In the case of low density the normal condition M t = 1 is satisfied and the results are in agreement with the two-point model. At high enough plasma density solutions with the supersonic flow at the divertor plates, M t > 1, are found. These states correspond to a partially detached plasma with a temperature of a few eV.