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Transport bifurcation induced by sheared toroidal flow in tokamak plasmasa)
a)Paper XI3 3, Bull. Am. Phys. Soc. 55, 374 (2010).
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Image of FIG. 1.
FIG. 1.

(Color online) ITG instability and turbulence at zero flow shear: (a) linear growth rate; (b) turbulent heat flux vs the normalised temperature gradient κ = R 0/L T . The case of κ = 11, further explored in Fig. 3, is marked by *.

Image of FIG. 2.
FIG. 2.

(Color online) Turbulent heat (a) and toroidal angular momentum (b) fluxes (normalised to gyro-Bohm values) as functions of flow shear for different values of the ion temperature gradient κ ; (c) turbulent Prandtl number vs flow shear (for cases where the heat flux is non-zero).

Image of FIG. 3.
FIG. 3.

(Color online) Transiently growing linear modes at κ = 11. (a) The heat flux as a function of time, normalised to its value at t 0, where , so chosen to skip the short initial transient associated with a particular choice of initial condition. All modes grow transiently for γ E  > 0. (b) Duration of transient growth τγ from t = t 0 to the peak value of Q t vs flow shear.

Image of FIG. 4.
FIG. 4.

(Color online) Two measures of the strength of the transiently growing linear modes at κ = 11: (a) the effective growth rate of the mode and (b) the number of exponentiations of the heat flux during the growing phase, both vs. the flow shear. The apparent discontinuity at γ E  = 0 in (a) is a result of taking the average growth rate of the heat flux over 0 < t < τγ rather than the initial or peak growth rate; for all γ E  > 0 the average will include a period where the growth rate tends to zero.

Image of FIG. 5.
FIG. 5.

(Color online) The spectrum of turbulent fluctuations for normal and subcritical turbulence: (arbitrary units) vs k y and for (a) γ E  = 0.0, κ = 12 (ITG turbulence with no flow shear) and (b) γ E  = 2.2, κ = 12 (subcritical, strong PVG-driven turbulence). It should be noted that as the sign of k y is opposite in GS2 to the present work, these graphs were plotted via a parity transformation. Also shown in (b) is the line of maximum transient amplification (27), as calculated in Ref. 14.

Image of FIG. 6.
FIG. 6.

(Color online) Momentum flux divided by the heat flux (each normalised by the corresponding gyro-Bohm estimate) vs the flow gradient at a constant value of the heat flux Q = 2.6 Q gB plotted using (a) interpolation from the data explained in Sec. V C and (b) the parameterisation from Sec. VI. Also shown are the neoclassical contribution to the momentum flux (dotted line) and the momentum flux at constant heat flux without the neoclassical contribution (dashed line).

Image of FIG. 7.
FIG. 7.

(Color online) Turbulent heat flux vs the temperature gradient for different values of the flow shear, showing (a) low-shear values γ E  < 1 (b) a close up of the low Q t region in (a), and (c) high-shear values γ E  ≥ 1.

Image of FIG. 8.
FIG. 8.

(Color online) The dependence of the heat transport stiffness dQ/dκ on the flow shear in the low-Q t and intermediate-Q t regions, measured using simulations close to both thresholds (points, see Fig. 7) and using the parameterised model of Sec. ??? (lines).

Image of FIG. 9.
FIG. 9.

(Color online) The first and second critical thresholds, κ c 1 and κ c 2, measured using simulations close to both thresholds (points, see Fig. 7) and using the parameterised model of Sec. ??? (lines).

Image of FIG. 10.
FIG. 10.

(Color online) The modelled heat flux (lines) shown along with the simulated data points for (a) low flow shear and (b) high flow shear. Legends as in Figs. 7(a) and 7(c).

Image of FIG. 11.
FIG. 11.

(Color online) Heat flux Q vs the temperature gradient κ at a constant ratio of the momentum flux to the heat flux Π/Q (in units of ) plotted using (a) interpolation from the data (Sec. V C) and (b) the parameterised model (Sec. VI). Also shown is the neoclassical contribution to the heat flux. Interpolation in (a) is impossible near this neoclassical line, where the contours are closely spaced and Π/Q is multivalued. (b) also shows the maximum possible temperature gradient at low Q for a given Π/Q (labelled “max κ”). (c) Plots the same curves as (b), showing both Q against γ E and κ, and the curves projected on the γ E -κ plane, illustrating the increase in the flow shear along each curve of constant Π/Q. Points A and B in (a) correspond to points A and B in Fig. 6(a).

Image of FIG. 12.
FIG. 12.

(Color online) The region in which bifurcations can occur, calculated using the analysis of Ref. 22 and the parameterisation of the fluxes described in Sec. VI. The cross indicates the location of the bifurcation described in Sec. V E. The dashed line represents the optimum value of Q for a given value of Π/Q, Eq. (50).

Image of FIG. 13.
FIG. 13.

(Color online) The regions (shaded) in which transitions can happen, as a function of the beam power and the beam particle energy, for three different devices. The dashed lines indicate, for each device, the value of P NBI for a given E NBI which would lead to the optimum temperature gradient, as described in Sec. VII E. The points indicate typical values of P NBI and E NBI for each device. The projected P NBI and E NBI for MAST Upgrade were taken from Ref. 61.


Generic image for table
Table I.

Typical plasma properties from the ITER profile database (Ref. 60). The symbol a denotes the minor radius of the device. The temperature was calculated as the mean of the core (TI0) and edge (TI95) temperatures.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Transport bifurcation induced by sheared toroidal flow in tokamak plasmasa)