For this value of θ the intersection occurs in the regime without stable stationary states at the edge of the plasma. Without those the system will oscillate according to a stable limit cycle, as is created by the Hopf bifurcation.
The -parameter space for fixed of (a) -flow shear model of this paper, and (b) Zohm's model. 10 The cusp-shaped short-dashed lines indicate the fold bifurcations ( and ). The Hopf bifurcations consist of two parts, the dashed lines ( and ) lead to the sharp L-H and H-L transitions. At and the Hopf bifurcations are in reversed order, such that in the region surrounded by the solid curve only limit cycle solutions are possible.
(a) Maximal Hysteresis ( ): L-H transition. (b) Maximal hysteresis ( ): H-L transition. (c) Maxwell's equal area rule ( ). (d) Generalised equal area rule ( , ), where the relation between K(X) and F(X) is given in Eq. (18) .
A contour plot of the control parameter in space and time, in which the special contours and are indicated. The asymmetric thick contour corresponds to the generalised equal area rule, (Eq. (18) ), bounding the region in that exhibits H-mode transport. depends on the local slope and has special values: , , and .
(a) The edge steady state solutions of Zohm's model for different values of the control parameter, θ. The profiles corresponding to these edge solutions are indicated in the surrounding graphs together with the jump trajectory during the L-H transition. The profiles are drawn from (i.e., ) with diffusivity Dcore to theedge at r = 0 (i.e., ) with diffusivity De . (b) , L-mode: The profile of the diffusivity squared which is almost constant over the entire spatial region. (c) , L-mode: Approaching the L-H transition De will slightly drop. (d) , The L-H transition: At the threshold value the Hopf bifurcation makes the L-mode unstable and the system jumps to the stable H-mode branch following the indicated trajectory; G = constant. (e) , “thin-barrier H-mode:” The edge state did not yet exceed the GEA-threshold, Eq. (25) . The barrier width is set by the viscosity, i.e., of the order of an ion gyro-radius. (f) , “thick-barrier H-mode:” profile with an enlarged pedestal, whose width is set by the GEA rule.
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