Abstract
The nonlinear interplay between the resistive wall mode (RWM) and the toroidal plasma flow is numerically investigated in a full toroidal geometry, by simultaneously solving the initial value problems for the n = 1 RWM and the n = 0 toroidal force balance equation. Here, n is the toroidal mode number. The neoclassical toroidal viscous torque is identified as the major momentum sink that brakes the toroidal plasma flow during the nonlinear evolution of the RWM. This holds for a mode that is initially either unstable or stable. For an initially stable RWM, the braking of the flow, and hence the eventual growth of the mode, depends critically on the initial perturbation amplitude.
This work was partfunded by The RCUK Energy Programme under Grant No. EP/I501045 and The European Communities under the contract of Association between EURATOM and CCFE. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
Youwen Sun would like to acknowledge the support from the National Magnetic Confinement Fusion Science Program of China under Grant Nos. 2013GB102000 and 2012GB105000 and the National Natural Science Foundation of China under Grant Nos. 11205199 and 10725523.
I. INTRODUCTION
II. TOROIDAL FORMULATION OF NONLINEAR COUPLING BETWEEN RWM AND PLASMA FLOW
III. NUMERICAL RESULTS
A. Equilibrium
B. Linear stability
C. Interaction between initially unstable RWM and flow
D. Interaction between initially stable RWM and flow
IV. CONCLUSION AND DISCUSSION
Key Topics
 Resistive wall mode
 91.0
 Plasma flows
 50.0
 Toroidal plasma confinement
 45.0
 Torque
 39.0
 Magnetohydrodynamics
 14.0
H05H1/02
Figures
The plasma boundary shape of the toroidal equilibrium.
The plasma boundary shape of the toroidal equilibrium.
Radial profiles of the equilibrium quantities for (a) the safety factor q, (b) the plasma pressure normalized by , (c) the plasma density normalized to unity at the magnetic axis, and (d) the toroidal rotation frequency of the plasma, normalized by the onaxis Alfvén frequency.
Radial profiles of the equilibrium quantities for (a) the safety factor q, (b) the plasma pressure normalized by , (c) the plasma density normalized to unity at the magnetic axis, and (d) the toroidal rotation frequency of the plasma, normalized by the onaxis Alfvén frequency.
Linear stability of (a) the ideal external, pressuredriven kink mode and (b) the resistive wall mode in the presence of the plasma flow.
Linear stability of (a) the ideal external, pressuredriven kink mode and (b) the resistive wall mode in the presence of the plasma flow.
Evolution of an initially unstable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond to time: 1– , 2– , 3– , and 4– . Both the electromagnetic and the NTV torques are included in this simulation.
Evolution of an initially unstable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond to time: 1– , 2– , 3– , and 4– . Both the electromagnetic and the NTV torques are included in this simulation.
Time trace of the net toroidal electromagnetic (solid line) and NTV (dashed line) torques acting on the plasma during the nonlinear evolution of an initially unstable RWM as described in Fig. 4 .
Time trace of the net toroidal electromagnetic (solid line) and NTV (dashed line) torques acting on the plasma during the nonlinear evolution of an initially unstable RWM as described in Fig. 4 .
Radial profiles of the initial (i.e., before closing the nonlinear coupling loop) E × B rotation frequency , the precessional drift frequency of deeply trapped thermal ions at thermal velocity, and the ionion collision frequency . ɛ is approximately the inverse aspect ratio.
Radial profiles of the initial (i.e., before closing the nonlinear coupling loop) E × B rotation frequency , the precessional drift frequency of deeply trapped thermal ions at thermal velocity, and the ionion collision frequency . ɛ is approximately the inverse aspect ratio.
Evolution of an initially unstable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond to time: 1– , 2– , and 3– . Only the electromagnetic is included in this simulation.
Evolution of an initially unstable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond to time: 1– , 2– , and 3– . Only the electromagnetic is included in this simulation.
Time traces of the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, with and without inclusion of the NTV torque in the simulation. The dashed curve corresponds to the exponential growth of the initially unstable linear mode. The dashed vertical line indicates the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on.
Time traces of the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, with and without inclusion of the NTV torque in the simulation. The dashed curve corresponds to the exponential growth of the initially unstable linear mode. The dashed vertical line indicates the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on.
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the plasma rotation frequency at the q = 2 and q = 3 surfaces, and (d) the net toroidal electromagnetic and NTV torques acting on the plasma. The dashed vertical lines in (a), (b), and (c) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The initial mode amplitude, normalized by B _{0}, is .
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the plasma rotation frequency at the q = 2 and q = 3 surfaces, and (d) the net toroidal electromagnetic and NTV torques acting on the plasma. The dashed vertical lines in (a), (b), and (c) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The initial mode amplitude, normalized by B _{0}, is .
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond totime: 1– , 2– , and 3– . The initial mode amplitude, normalized by B _{0}, is . Both the electromagnetic and the NTV torques are included in this simulation.
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond totime: 1– , 2– , and 3– . The initial mode amplitude, normalized by B _{0}, is . Both the electromagnetic and the NTV torques are included in this simulation.
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the change of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The initial mode amplitude, normalized by B _{0}, is . Only the electromagnetic torque is included in this simulation.
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the change of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The initial mode amplitude, normalized by B _{0}, is . Only the electromagnetic torque is included in this simulation.
Time traces of the net toroidal electromagnetic (solid line) and NTV (dashed line) torques acting on the plasma, during the nonlinear evolution of an initially stable RWM, with the initial mode amplitude : (a) both electromagnetic and NTV torques are included in the toroidal torque balance and (b) only the electromagnetic torque is included.
Time traces of the net toroidal electromagnetic (solid line) and NTV (dashed line) torques acting on the plasma, during the nonlinear evolution of an initially stable RWM, with the initial mode amplitude : (a) both electromagnetic and NTV torques are included in the toroidal torque balance and (b) only the electromagnetic torque is included.
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond to time: 1– , 2– , 3– , and 4– . The initial mode amplitude, normalized by B _{0}, is . Both the electromagnetic and the NTV torques are included in this simulation.
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond to time: 1– , 2– , 3– , and 4– . The initial mode amplitude, normalized by B _{0}, is . Both the electromagnetic and the NTV torques are included in this simulation.
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond to time: 1– , 2– , and 3– . The initial mode amplitude, normalized by B _{0}, is . Only, the electromagnetic torque is included in this simulation.
Evolution of an initially stable RWM: (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, (b) the real and imaginary parts of the perturbed radial field b ^{1} at the q = 2 surface, (c) the radial profile of the plasma rotation frequency, and (d) the plasma rotation frequency at the q = 2 and q = 3 surfaces. The dashed vertical lines in (a), (b), and (d) indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. The dashed vertical lines in (c) indicate the location of the q = 2 and 3 rational surfaces, respectively. The numbered lines in (c) correspond to time: 1– , 2– , and 3– . The initial mode amplitude, normalized by B _{0}, is . Only, the electromagnetic torque is included in this simulation.
Time traces of the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, with and without inclusion of the NTV torque in the simulations for an initially stable RWM, with the initial mode amplitude at (a) and (b) . The dashed curves correspond to the exponential decay of the initially stable linear mode. The dashed vertical lines indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on.
Time traces of the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, with and without inclusion of the NTV torque in the simulations for an initially stable RWM, with the initial mode amplitude at (a) and (b) . The dashed curves correspond to the exponential decay of the initially stable linear mode. The dashed vertical lines indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on.
Time traces of (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, and (b) the plasma rotation frequency at the q = 2 and q = 3 surfaces, with various choices of the initial mode amplitude. The dashed curves in (a) correspond to the exponential decay of the initially stable linear mode. The dashed vertical lines indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. Both the electromagnetic and the NTV torques are included in the simulations.
Time traces of (a) the amplitude of the perturbed radial field b ^{1} at the q = 2 surface, and (b) the plasma rotation frequency at the q = 2 and q = 3 surfaces, with various choices of the initial mode amplitude. The dashed curves in (a) correspond to the exponential decay of the initially stable linear mode. The dashed vertical lines indicate the moment of time when the nonlinear coupling between the mode and the plasma flow is switched on. Both the electromagnetic and the NTV torques are included in the simulations.
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