A Unified theory of transport barriers (TBs) in magnetically confined systems

Abstract

A thermodynamic model of a plasma boundary layer, characterized by enhanced temperature contrasts is proposed. The theory is constructed to determine the inner boundary temperature $T_1$ for a specified outer (colder) boundary temperature $T_0$, the heat flux $F$ entering the inner boundary, and the parameters defining the layer. The system shows bifurcation and switches to a stable high gradient state if the heat flux $F$ entering through the inner boundary exceeds a critical value $F_c$. However there is an additional stringent condition for the transition to occur; the edge temperature $T_0$ must exceed a critical value $T_c$- no transition is possible if $T_0<T_c$ even for arbitrary large $F$. Equally important is the finding that $F_c$ is not a monotonic function of $T_0$ but has a minimum at $T_{optimum}$ (= $4T_c$ )in the model calculation. The confinement peaks at $T_{optimum}$. The basic conceptual physics is obviously simple: The high contrast state becomes the preferred state when the incoming power into the layer is preferentially converted into coherent motions like the fluid flows and currents (undermining the standard diffusive processes that keep the lower temperature contrast). The purely macroscopic thermodynamic model bears excellent comparison with experimental and detailed microscopic investigations of the H-mode. Deeper plausibility reasons for the workability of this heat engine, creating the simultaneous existence of an ordered state and large entropy production, are suggested.

Figures (5)

• Figure 1: Equivalent diagram of a heat engine in a boundary layer. For ${\Delta T}> 0$,the heat engine can work to drive Flows (P is the power driving the flow which equals the dissipative power in steady state.). The flow produces an additional “nonlinear impedance” $Z = \eta(P) P$ that sustains the temperature contrast ${\Delta T}$ yielding a free energy to drive the flow itself.
• Figure 2: ITPA from Ref. Kotschenreuther_2024.
• Figure 3: This plot shows parameterizations of data from Alcator C-Mod published in Ref. Cmod. The LH threshold power is shown in Fig. A as a function of line-averaged density (compare with Fig. 4 from Ref. Cmod). The parameterization, $P_{LH} = 1.5/\bar{n}^2 + 0.2\bar{n}^{3.5}$, is taken directly from Ref. Cmod. Fig. B shows the relation between the separatrix temperature and the line-averaged density (compare with Fig. 7 D from Ref. Cmod), and Fig. C converts the data into the relation between the threshold power and separatrix temperature, making it directly relevant for the theory presented here.
• Figure 4: A. shows the parameterization of the electron electron temperature at $\Psi_N = 0.95$ at the LH-transition (parameterized to match Fig. 7 A in Ref. Cmod). Along with the separatrix temperature and threshold power, this allows for the calculation of the threshold resistivity, $\eta_0$ (recall Eq. \ref{['resistivity']}), shown in B.
• Figure 5: . A comparison of the experimental dependence (black) of heat flux at the LH transition and the theoretical predictions from Eq. \ref{['threshold2']} (solid colors) and Eq. \ref{['Fparabola']} (dashed color). The theoretical predictions are shown for a range of values of $\eta_0$ justified by the intrinsic uncertainties in the experimental gradients. The theoretical prediction of $F_{min}$ matches the experimental value for a $\approx15\%$ reduction in $\eta_0$. However, the curvature is substantially under-predicted by the theory.