3. Consideration of detachment as a dissipative structure

The most visible approach to understand the detachment mechanisms is to analyze the power balance at the plasma edge by applying the concept of dissipative structures [36]. The power transported from the plasma core by the plasma heat conduction is lost from the edge region mostly through two channels: (i) the plasma particle outflow through the separatrix and (ii) the radiation of light impurities such as carbon sputtered from the divertor target plates.

On the one hand, the density of the former channel qcon is normally monotonously increasing with the plasma temperature T at the plasma edge. On the other hand, it is well known [7] that the impurity radiation density qrad has a maximum as a function of T: by a too low temperature, electrons do not have enough energy to excite impurity species, by a too high temperature, impurities are strongly ionized and have a very large excitation energy. In particular, for carbon, the best "radiators" are the Li-like ions C3+. Qualitatively, qcon, qrad, and their sum qloss are displayed as functions of T in Figure 17.

#### Figure 17.

Temperature dependence for the energy loss channels from the plasma edge with the plasma conduction and convection through the separatrix, qcon, with impurity radiation, qrad, and their sum qloss.

#### Figure 18.

Temperature dependence for the density of the energy loss from the plasma edge, qloss, for three magnitudes of the plasma density n<sup>1</sup> <n<sup>2</sup> <n<sup>3</sup> (solid curves) and of the heat flux to the edge from the plasma core, qheat: In a steady state, these balance each other, qloss ¼ qheat:

Both qcon and qrad increase with plasma density n [7]. Indeed, the larger the content of plasma particles, the higher their loss from the device. Likewise, qrad increases both with the density of exciting electrons n and that of the exciting impurity species nI ¼ cIn. Typically, the relative concentration cI of carbon impurityin LHD plasmas is of order of 1%.

In a steady state, the energy loss from the plasma edge has to be balanced by the heat transfer from the plasma core, with the density qheat, which in many cases is weakly dependent both on the edge temperature and on the plasma density. Figure 18 shows qloss and qheat versus T for three magnitudes of n. One can see that a moderate increase of the plasma density from n<sup>1</sup> to n3, by less than 40%, results in a very strong drop in the stationary edge temperature, from its level Tat of several tens of eVs in an attached plasma state A to Tdet of 1 eV in a detached state D. In the latter case, electrons and ions in the plasma effectively recombine one with another as it is indicated by spectroscopic measurements.

For the intermediary level of the density, n2,there are three steady states. It is straightforward to comprehend that the one with the in-between temperature is unstable. Indeed, an infinitesimal spontaneous deviation from this state, e.g., with diminishing T, leads to an increase of the energy losses and a further decrease of the edge temperature. Finally, the plasma will get the one of two stable states with a low temperature.
