A unified model of density limit in fusion plasmas

Abstract

A limit for the edge density, ruled by radiation losses from light impurities, is established by a minimal cylindrical magneto-thermal equilibrium model. For ohmic tokamak and reversed field pinch the limit scales linearly with the plasma current, as the empirical Greenwald limit. The auxiliary heating adds a further dependence, scaling with the 0.4 power, in agreement with Lmode tokamak experiments. For a pure externally heated configuration the limit takes on a Sudolike form, depending mainly on the input power, and is compatible with recent Stellarator scalings.A discharge-terminating density limit (DL) is found in all the magnetic confinement fusion devices [1]. One of the main interpretative branches invokes impurity radiation losses, which scale with the square of the density. The consequent cooling of the plasma can become critical at high density, giving rise to a variety of instabilities, both thermal [2][3][4][5][6][7][8], and MHD [9][10][11][12][13]. Given the rich phenomenology, DL seems elusive of an explanation based on a single mechanism. This letter presents a complementary approach to the problem, analysing, in cylindrical geometry, the feasibility of a magneto-thermal equilibrium with realistic temperature profile, rather than addressing specific instabilities. Such a study provides a unified interpretation of the phenomenon, given that a DL ruled by light impurities radiation (experiments show that any significant contamination by heavy impurities is just detrimental towards the achievement of high densities [1]), quantitatively consistent with experimental scalings, emerges naturally for all the magnetic configurations. In particular, we found a Greenwald-like scaling [1] for tokamak and reversed field pinch (RFP), and a Sudo-like scaling [14,15] for a pure externally heated configuration, taken as approximation of the stellarator. We are aware that this analysis cannot 2 exhaust the topic, since some instability mechanism is necessary to describe the dynamical route to the plasma termination. Consequently, we speak of an 'equilibrium DL' and not of the DL in the ultimate sense. This work has been inspired by analyses of the ohmic tokamak presented in[16] (section 7.8) and in [17] (section 8). The differences rely in a more general approach, besides a more accurate treatment of the profile dependent terms.We introduce a minimal cylindrical equilibrium model (each quantity depending on the radial coordinate r only), analytically treated with a formalism able to unify the configurations with an applied electric field, namely the tokamak, both ohmic and with additional heating, and the RFP, considered as purely ohmic. Basically, we will take integral relations from the heat transport equation, in some way similar to those carried out in [18] apart for the simpler geometry, and combine them with Ohm's law at r=0 (on-axis). Ohm's law is replaced by a suitable scaling for the energy confinement time in the case of pure auxiliary heating. [ ]Here, T is the electron temperature and K an effec...

Plasma-wall self-organization in magnetic fusion

“…The new (IP/a 4 ) 4/9 scaling is a part of those derived in [9,8,10], which are in much better agreement with the tokamak and RFP databases. They describe the density limit as a radiative one, and therefore include naturally a clear explicit dependence on P .…”

“…They describe the density limit as a radiative one, and therefore include naturally a clear explicit dependence on P . The theory and good agreement with experiments extends to the stellarator [9,11,12,10], which provides a unified view of density limits in magnetic confinement.…”

“…Therefore, the simplest version of our iterative model for radiation opens the prospect of a radiative density limit occurring for a fraction of radiated power as low as 1/2. After the outcome of [9] recalled in the introduction, this is another prediction of a radiative density limit without necessarily a very high fraction of radiated power.…”

Increasing the density in Wendelstein 7-X: benefits and limitations

“…In Refs. [15,16] the heat diffusivity was estimated by using scaling laws for the energy confinement time τ E together with the definition of an effective heat diffusivity χ eff = a 2 /τ E . Such an approach to estimate the heat transport will of course only result in meaningful predictions for parameter ranges where the τ E -scaling is valid.…”

“…In Ref. [16] it is shown that the Sudo-limit is a special case of Eq. ( 5) and, hence, the two models introduced above are essentially identical.…”