Steady-state tokamak research: Core physics

Tokamaks have demonstrated excellent plasma confinement capability because of their symmetry but has an intrinsic drawback because of their pulsed inductive operation. Efforts have been made in the past 20 years to realize steady-state operation, the most successful utilizing a bootstrap current. In this review, progress in understanding tokamak physics related to steady-state operation is described to investigate the scientific feasibility of a steady-state tokamak fusion power system.

Figure 1

Poloidal cross section of a tokamak with divertor. The plasma is vertically elongated and is also triangular.

Figure 2

Layout of steady-state tokamak reactor.

Figure 3

(${\beta }_t$, ${\beta }_p$) diagram and operating points of ITER, SSTR, and ARIES-I. ITER-Q10 stands for an inductive operation, and ITER-ss stands for a steady-state operation.

Figure 4

Energy flow diagram of steady-state tokamak reactor. The plant efficiency ${\eta }_{\mathrm{plant}}=P_{\mathrm{net}}/P_{\mathrm{th}}$, and the ratio of net electric power output to thermal power output is given by ${\eta }_{\mathrm{plant}}={\eta }_{\mathrm{th}}-{\eta }_{\mathrm{aux}}-1/{\gamma }_{CD}{\gamma }_{BD}(1+Q)$.

Figure 5

The plant efficiency (${\eta }_{\mathrm{plant}}$) as a function of plasma energy gain ($Q$) for He-cooled and water-cooled systems in the steady-state tokamak reactor.

Figure 6

WS profiles using central NBCD and $n=1$, 2, and 3 ideal MHD stability as a function of the central safety factor $q_0$ for the SSTR. From 224.

Figure 7

WS profile and ideal ${\beta }_N$ limits with and without wall stabilization. From 269.

Figure 8

Negative-shear scenario for the SSTR, where off-axis NBCD is used. From 299.

Figure 9

Negative-shear plasma profile and ideal MHD stability limit with and without wall stabilization. From 269.

Figure 10

Current hole equilibrium.

Figure 11

Distortion of velocity distribution function in collisionless tokamak. From 230.

Figure 12

Comparison of experimental and simulated surface voltages with and without trapped-particle correction. $\times$, ○, and ● correspond to different analysis methods and plasma species. From 446.

Figure 13

Comparison of experimental and simulated local poloidal fields with and without trapped-particle correction in TFTR. Better agreement with experiment is obtained using the neoclassical resistivity. From 20.

Figure 14

(a) Time evolution of surface voltage, (b) bootstrap fraction, and (c) internal inductance from measurement and calculation. High-power NB is injected perpendicularly so that the NB does not drive significant current. The measured surface loop voltage is consistent with the 80% bootstrap current fraction. From 230.

Figure 15

Comparison of experimental and numerical bootstrap current fractions as a function of poloidal beta. From 230.

Figure 16

Comparison of measured and calculated current-profiles in bootstrap-current-dominated JT-60U discharge. From 341.

Figure 17

Composition of current profile in full bootstrap discharge in JT-60U. From 391.

Figure 18

(a) Comparison of shielding factors between the Start-Cordey theory and magnetic island (MI) methods with ${\nu }_a^{*}=0$. From 230. (b) Collisionality dependence of the shielding factor from Eq. (41) and Sauter’s $L_{31}$ compared with the Lin-Liu model. From 173.

Figure 19

Comparison of experimentally measured NBCD current profile with calculations with and without MSI. From 130.

Figure 20

Neutral-beam-current drive efficiency ${\eta }_{\mathrm{CD}}={\overline{n}}_{e}R{I}_p^{\mathrm{CD}}/P_{\mathrm{CD}}$ at central electron temperature $T_e(0)$. From 295.

Figure 21

Schematic explanation of Fisch-Boozer and Ohkawa mechanisms. From 315.

Figure 22

Comparison of measured and calculated EC-driven current. From 309.

Figure 23

(a) Effect of inboard and outboard injection on ECCD efficiency. From 61. (b) Experimental result. From 386.

Figure 24

Observation of neoclassical toroidal viscosity (NTV) at NSTX. From 448.

Figure 25

Observation of offset toroidal rotation due to NTV at DIII-D. From 118.

Figure 26

Gradient-flux relation in toroidal momentum showing a clear offset in flux as evidence of nondiffusive (off-diagonal) transport of toroidal momentum. From 184.

Figure 27

Intrinsic rotation by Reynolds stress in ITG turbulence. From 113.

Figure 28

Excitation of toroidal flow by different divertor geometries. From 248.

Figure 29

Comparison of ERATO-J and MARG2D simulations for identifying the stability boundary. From 1.

Figure 30

Observation of $m/n=3/2$ NTM at TFTR. (a) Measured evolution of island width, (b) calculated island width using modified Rutherford equation, and (c) calculated island width with fixed parameters ($\eta ,w_c\equiv {ϵ}^{1/2}{L}_q{\beta }_p/L_p,{\Delta }^{\prime }$). From 48.

Figure 31

Stabilization of NTM for fundamental $O$-mode ECCD. From 193.

Figure 32

Comparison of experimental measurement and numerical calculation based on the modified Rutherford equation. From 194.

Figure 33

Distortion of the tearing island due to viscous drag. From 250.

Figure 34

Comparison of linear radial eigenmode structures among small (strong coupling), intermediate, and large (weak coupling) $\Delta r$ (left panel) and nonlinear time evolution of kinetic energies of $m/n=3/1,6/2,9/3,\dots$ (right panel). From 198.

Figure 35

(a) Time evolution of disruptive TM in NS discharge; (b) $T_e$, ${\tilde{T}}_e$, and $q$ profiles just before collapse. From 395.

Figure 36

Time evolution of nonlinear DTM for intermediate $\Delta r$ for different resistivity $\eta$. ME and KE are the magnetic and kinetic energy, respectively. From 198.

Figure 37

(a) Time evolution of the $\frac{3}{1}$ magnetic energy in nonlinearly destabilized DTM for various plasma resistivities $\eta$; (b) current distributions for various plasma resistivities. From 199.

Figure 38

Contour plot of current distribution for (a) nonlinearly destabilized DTM (point current) and (b)  standard DTM (sheet current). From 199.

Figure 39

(a) Density profile, (b) shear Alfvén wave, and (c) shear Alfvén wave modified to KAW.

Figure 40

Relation among rotating plasma, resistive wall mode, and wall.

Figure 41

Discharge trajectories of DIII-D and JT-60U in the ($V_t{|}_{q=2}$, $C_{\beta }$) plane. The X’s and boxes show onsets of RWM. From 117 and 393.

Figure 42

$n$ dependence of the beta limit showing agreement between WKB theory (dashed line) and PEST (solid line) in high-$n$ regime and unique oscillatory behavior at low to intermediate $n$ obtained by treating $n$ as a continuous variable. From 71.

Figure 43

Stability boundaries in $(ϵ{\beta }_p,g)$ for various pressure profiles [$\mathrm{A}\mathrm{\text{:}}dp/d\psi \sim 0.3(1-\psi {)}^{0.5}+0.7$, to $\mathrm{C}\mathrm{\text{:}}dp/d\psi \sim (1-\psi {)}^2$, $\mathrm{D}\mathrm{\text{:}}dp/d\psi \sim (1-\psi {)}^5$], where $g$ is the normalized beta ${\beta }_N$. The circle shows the region of observed ${\beta }_p$ collapses in high-${\beta }_p$ experiments in JT-60U. From 301.

Figure 44

(a) Pressure profiles (A) and (B) in weak-negative-shear plasma and (b) $ϵ{\beta }_p$ stability improvement from (A) to (B) in ($q_{\min }$, $ϵ{\beta }_p$). From 300.

Figure 45

Stability diagram of the high-$q$ ($q_{95}=7.3$) equilibria in the ($j_{\mathrm{ped}}$, ${\alpha }_{96}$) plane. Toroidal rotation weakly destabilizes the peeling-ballooning modes. From 2.

Figure 46

The effect of triangularity and high-$q$ operation for ELM stability. From 396.

Figure 47

Comparison of peeling-balooning eigenmodes for small and large ELMs. From 80.

Figure 48

ELM frequency variation with toroidal rotation at the pedestal top. From 298.

Figure 49

Edge pedestal and edge bootstrap current in collisionless plasma. From 371.

Figure 50

ELM energy loss as a function of pedestal collisionality and time evolution of ELM for different powers. From 158.

Figure 51

Suppression of type-I ELM by $n=3$ RMP. From 85.

Figure 52

Time evolution of high-${\beta }_p$ discharge showing the formation of ITB and subsequent BLM (observed by ECE) leading to $H$-mode edge pedestal formation. (1)–(4) correspond to different times between 5.6 and 6 s. From 238.

Figure 53

SAR and gap structure due to toroidal coupling of $m=1$ and $m=2$ modes in the $(q,\omega )$ plane.

Figure 54

Shear Alfvén resonance and gap spectrum due to toroidal coupling of $m=1$ and $m=2$ modes. From 53.

Figure 55

Amplification of AE amplitude by RSAE-TAE interaction and the RSAE-TAE diagram. From 392.

Figure 56

Fast frequency sweeping (FS) mode observed at JT-60U. From 363.

Figure 57

Comparison of fast FS mode in JT-60U and simulation using the BB model. From 255.

Figure 58

Axisymmetric trimagnetic island equilibrium. From 397.

Figure 59

Time evolution of central current density with and without resistive $m/n=1/0$ modes. From 181.

Figure 60

Bump and void avalanches near marginal stability.

Figure 61

Symmetry breaking of void and bump avalanches by $d{E}_r$ shear.

Figure 62

Gyrokinetic full-$f$ simulation of toroidal ITG turbulence, showing outward propagation of the bump where $d{E}_r/dr>0$ and inward propagation of void where $d{E}_r/dr<0$. From 192.

Figure 63

Normalized ion heat flux as a function of $R/L_{Ti}$ for different toroidal rotations at JET. From 272.

Figure 64

(a) Eigenmode structure in poloidal cross section. (b) Radial harmonics of toroidal ITG with $n=15$ and $a/{\rho }_i=324$. From 189.

Figure 65

(a) Radial variation of normalized zonal flow, GAM, and sound wave frequencies [$f[v_i/a]$, $f_{\mathrm{GAM}}=\sqrt{2(\Gamma +\tau )T_{\mathrm{eq}}}(a/2\pi R)$, and $f_{\mathrm{sound}}=\sqrt{(\Gamma +\tau )T_{\mathrm{eq}}}(a/2\pi qR)$]. (b) Radial variation of time-averaged zonal flow drives. From 282.

Figure 66

Equivalent ion thermal diffusivity vs $R/L_T$ from the Lawrence Livermore National Laboratory (LLNL) gyrokinetic simulation, showing the Dimits shift, as compared with the 1994 Institute of Fusion Study/Princeton Plasma Physics Laboratory (IFS/PPPL) gyrofluid simulation. From 75.

Figure 67

Transient GAM–zonal-flow interplay in the Dimits shift regime $R/L_T=3.83$ and above $R/L_T=4.47$. From 280.

Figure 68

Electron heat flux as a function of $R/L_T$ showing the existence of critical electron temperature gradient. The open symbols correspond to $r/a=0.2$, 0.4, and 0.6, respectively, and the full symbols correspond to another series of discharges. From 169.

Figure 69

Comparison of radial profile of $d{T}_e/dr{|}_c$ between theory and experiment. From 211.

Figure 70

(a) Poloidal cross section and (b) radial harmonics of the linear toroidal ETG eigenmode in positive shear. From 190.

Figure 71

ETG turbulence structures of zonal flow and streamers in NS plasma. From 190.

Figure 72

Stability diagram of ITG and TEM modes for $T_e=T_i$, trapped-electron fraction of 0.3, and collisionless limit (${\nu }_{*}=0$). From 116.

Figure 73

Normalized electron thermal diffusivity vs $a/L_{T_e}$ from gyrokinetic simulation compared with experiment. From 137.

Figure 74

Schematic diagram of positive feedback effect on $E_r$ from $d{P}_i/dr$, $d{T}_i/dr$, and $v_{\zeta }$.

Figure 75

ITB profiles in NS shear plasma and $E_r$, ${\chi }_i$, and ${\chi }_e$ profiles. From 364.

Figure 76

Difference in ${\chi }_i$ of ITBs in balanced, co- and counter-NB injection in NS plasma. From 339.

Figure 77

Density peaking factor as a function of effective collisionality. From 8.

Figure 78

(a) Extended high bootstrap current fraction discharge duration obtained by rotation control. From 341. (b) Attainment of reactor-relevant moderate-$q_{95}$ high bootstrap current fraction discharge with wall stabilization. From 342.

Figure 79

Control flow for steady-state tokamak operation. Plasma profile, field, flow, macrofluctuations, and microfluctuations are interlinked and complex, while control knobs are not plentiful.