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ABSTRACT: A theory for the superbanana plateau resonance that occurs at the phase space boundary is presented to refine the comprehensive theory for neoclassical toroidal plasma viscosity in tokamaks. The results of the theory reproduce those of the standard superbanana plateau resonance theory when the resonance occurs away from the boundary. It shows that the strength of the superbanana plateau resonance weakens in the vicinity of the phase space boundary. It also indicates that it is important to know the resonance pitch angle parameter when comparing the results of the superbanana plateau resonance with those of the experiments. The theory is used to refine the kernel for the superbanana plateau resonance in the approximate expression for the neoclassical toroidal plasma viscosity.
Nuclear Fusion 06/2011; 51(7):073043. · 4.09 Impact Factor
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ABSTRACT: Neoclassical toroidal plasma viscosity (NTV) torque induced by non-axisymmetric magnetic perturbation in the collisionless regimes in tokamaks is modelled by solving the bounce-averaged drift kinetic equation numerically. The detailed comparison between the numerical and the analytic solutions of NTV is discussed in this paper. In different asymptotic limits of the collisionless regimes, the numerical solutions are in good agreement with the analytic results. The numerical results are different from the analytic results calculated from the smoothly connected formula in the transit regimes. The pitch angle scattering is especially important in the regime. The final difference between the numerical and the analytic results can be up to a factor of 2 near the transition between the non-resonant and resonant regimes. This reveals the importance of the boundary condition of the pitch angle space. The sign of the electric field is found to be important in the calculation of the NTV torque. It shows that the effect of the resonant particles makes the NTV torque more important for the lower collisionality and lower rotation cases, which are the International Thermonuclear Experimental Reactor relevant conditions. It also shows that the electron NTV torque is important in the low collisionality case. This numerical method can be applied for modelling the NTV torque in different collisionality regimes and their transitions in tokamaks without additional approximations.
Nuclear Fusion 04/2011; 51(5):053015. · 4.09 Impact Factor
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ABSTRACT: Error fields and resistive magnetohydrodynamic modes are ubiquitous in real tokamaks. They break the toroidal symmetry in |B| in tokamaks. Here, B is the magnetic field. There are two mechanisms that break the symmetry on the perturbed magnetic surface: one is the perturbed field itself and the other results from the distortion of the magnetic surface due to the perturbed field. The broken toroidal symmetry leads to enhanced neoclassical toroidal plasma viscosity and consequently the rate of the toroidal flow damping. The neoclassical toroidal plasma viscosity also results in a steady-state toroidal plasma flow. In addition, the neoclassical toroidal plasma viscosity in the vicinity of the magnetic islands provides a mechanism to determine the island rotation frequency, which is an important quantity for the island stability. Here, the theory for neoclassical toroidal plasma viscosity in the vicinity of the magnetic island is extended to include the effects of the collisional boundary layer that lead to scaling in the transport fluxes, where ν is the collision frequency.
Nuclear Fusion 03/2011; 51(4):043013. · 4.09 Impact Factor
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ABSTRACT: The usual ripple transport calculations lead to a ν−1 scaling of the transport coefficients with collision frequency ν. The paper extends and clarifies this scaling by taking into account the fact that the dominant contributions to transport come from particles in the high-energy tail of the distribution function. This means that particles with energy E ~(4–6)T dominate, and this restricts the ν−1 scaling range to ν(T) > (E/T)5/2 δωd(T) ~ 100 δωd(T), where T is the thermal energy, δ is the ripple-well depth, and is the time to drift over the region where the localized ripple well exists. In addition, transport coefficients are derived in weak-ripple (α ≡ /Nqδ 1, with the inverse aspect ratio, N the number of toroidal coils, q the safety factor), low-collisionality regimes where the radial step size is determined by the distance a particle drifts before it 'collisionlessly detraps' from a ripple well. The maximum transport rate is found to be about two orders of magnitude smaller than usually assumed, primarily because of the restriction of the ν−1 scaling regime to the much higher collision frequencies.
Nuclear Fusion 01/2011; 22(8):1061. · 4.09 Impact Factor
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ABSTRACT: Flux–force relation, a fundamental relation that relates transport fluxes to forces, for non-axisymmetric tori in general magnetic flux coordinates that are not Hamada coordinates, is derived. The derivation is based on kinetic theory instead of fluid theory. It is shown that pressure force also contributes to the relation in non-Hamada coordinates in general to make the relation compatible with kinetic theory and to make it coordinates invariant. The results are applied to the theory for the neoclassical toroidal viscosity in tokamaks that have error fields or resistive magnetohydrodynamic (MHD) modes.
Nuclear Fusion 11/2010; 50(12):125012. · 4.09 Impact Factor
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ABSTRACT: The effects of finite gradient B drift are included in the collisional boundary layer analysis to improve the accuracy of the neoclassical toroidal plasma viscosity in tokamaks that have error fields or magnetohydrodynamic activities present. Depending on the sign of the electric charge of the species and that of the radial electric field, the effects of finite gradient B drift can either reduce, if the E × B drift is in the same direction of the gradient B drift, or enhance, if these two drifts are in the opposite direction, the magnitude of the neoclassical toroidal plasma viscosity. Here, E is the electric field and B is the magnetic field. However, because the gradient B drift depends on the effective pitch angle, the net effects have to be properly weighted by integrating over the particle energy.
Nuclear Fusion 11/2010; 50(12):125008. · 4.09 Impact Factor
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ABSTRACT: Bumpiness in a magnetic field enhances the magnitude of the plasma viscosity and increases the rate of the plasma flow damping. A general solution of the neoclassical toroidal plasma viscosity (NTV) torque induced by nonaxisymmetric magnetic perturbation (NAMP) in the collisionless regimes in tokamaks is obtained in this Letter. The plasma angular momentum can be strongly changed, when there is a small deviation of the toroidal symmetry caused by a NAMP of the order of 0.1% of the toroidal field strength.
Physical Review Letters 10/2010; 105(14):145002. · 7.37 Impact Factor
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ABSTRACT: An approximate analytic expression for neoclassical toroidal plasma viscosity in tokamaks that have error fields or magnetohydrodynamic activities is presented. The expression smoothly joins transport fluxes or plasma viscosity in all the known collisionality regimes derived from the solution of the bounce averaged drift kinetic equation and should be useful in modelling results of existing and future tokamak experiments. It also incorporates some of the extensions of the known expressions to include the effects of finite ∇B drift in the non-resonant transport processes. Here, B is the magnitude of the magnetic field. The toroidal momentum balance equation is a nonlinear function of the radial electric field when the neoclassical plasma viscosity is dominant. It can have bifurcated solutions for the radial electric field and may lead to better plasma confinement as a result.
Nuclear Fusion 01/2010; 50(2):025022. · 4.09 Impact Factor
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ABSTRACT: Bootstrap current and transport fluxes associated with bounce–transit and drift resonance in tokamak plasmas that have error fields or magnetohydrodynamic activity are calculated by solving the parallel and the toroidal components of the momentum and the heat flux balance equation. Because error fields are not localized around the surfaces where the safety factor q is a rational number, the bootstrap current densities induced by error fields on either side of the rational surface do not cancel. Thus, there is a net contribution to the equilibrium bootstrap current. This may offer a possibility to control bootstrap current density using error fields. Other transport fluxes, such as parallel mass flow, radial electric field, particle and heat flux, are also presented. The complete flow vector calculated can be compared with the experimental measurements.
Plasma Physics and Controlled Fusion 01/2010; 52(2):025005. · 2.42 Impact Factor
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ABSTRACT: An Eulerian approach to treat the bounce–transit and precession drift resonance in tokamaks is developed by expanding the poloidal angle dependence in terms of a Jacobian elliptic function in the low collisionality regime instead of integrating along the unperturbed particle trajectories. One of the advantages is that a complex Coulomb collision operator can be adopted in the approach. The full thermodynamic forces are kept to conserve momentum in the collision processes. To illustrate the method, the approach is applied to calculate the neoclassical toroidal plasma viscosity in both the resonant plateau regime and the Pfirsch–Schluter regime. Both trapped particles and circulating particles contribute to the resonant plateau regime through the bounce and drift resonance and the transit and drift resonance, respectively. The dependences on plasma parameters for both regimes are found to be the same as that of the nonlinear plasma viscosity. The resonant plateau regime naturally connects to the Pfirsch–Schluter regime in the collisional limit.
Plasma Physics and Controlled Fusion 06/2009; 51(7):075015. · 2.42 Impact Factor
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ABSTRACT: Neoclassical toroidal plasma viscosity in the superbanana regime resulting from the symmetry breaking magnetohydrodynamic activities or error fields in tokamaks is calculated. The superbanana regime becomes important when the E × B drift frequency is comparable to or smaller than the ∇B drift frequency and when the effective collision frequency is less than the bounce frequency of the superbananas. Here, E is the radial electric field, B is the magnetic field and B = |B|. The width of the superbananas depends on the relative strength of the magnetic field perturbations to the inverse aspect ratio of tokamaks. It is independent of the particle energy and the magnitude of the equilibrium magnetic field. The viscosity and the transport fluxes are found to be proportional to the collision frequency. The toroidal flow damping rate is greatly enhanced in this regime.
Plasma Physics and Controlled Fusion 02/2009; 51(5):055003. · 2.42 Impact Factor
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ABSTRACT: A theoretical model for the neoclassical toroidal plasma viscosity for tokamaks resulting from the symmetry breaking magnetic field perturbation caused by the collisionless detrapping/retrapping mechanism is presented. The results of the model are valid when the width of the collisional boundary layer is narrower than the width of the layer of the detrapping/retrapping process in the pitch angle space, and when the normalized magnitude of the perturbed magnetic field is much smaller than the inverse aspect ratio. The neoclassical toroidal plasma viscosity derived from this model scales with the collision frequency. The results can be used in modeling plasma rotation when the toroidal symmetry of |B| of the tokamaks is broken. Here, B is the magnetic field.
Plasma Physics and Controlled Fusion 12/2008; 51(3):035004. · 2.42 Impact Factor
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ABSTRACT: Neoclassical toroidal plasma viscosity in the superbanana plateau regime induced by the broken toroidal symmetry in |B| resulting from the error fields or magnetohydrodynamic activities in tokamaks is calculated. The transport processes in this regime are dominated by particles that have vanishing bounce averaged precession frequency, i.e. by the resonant particles. The superbanana plateau regime becomes important when the E × B drift speed is comparable to or smaller than the ∇B drift speed. Here, E is the radial electric field, B is the magnetic field and B = |B|. The resonance occurs regardless of the sign of the radial electric field or the electric charge a species carries. The plasma viscosity and the transport fluxes in this regime are found to be independent of the collision frequency.
Plasma Physics and Controlled Fusion 12/2008; 51(3):035009. · 2.42 Impact Factor
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ABSTRACT: It is demonstrated that the pitch angle integrals in the transport fluxes in the ν regime calculated in
K. C. Shang [Phys. Plasmas 10, 1443 (2003)
] are divergent as the trapped-circulating boundary is approached. Here, ν is the collision frequency. The origin of this divergence results from the logarithmic dependence in the bounce averaged radial drift velocity. A collisional boundary layer analysis is developed to remove the singularity. The resultant pitch angle integrals now include not only the original physics of the ν regime but also the boundary layer physics. The transport fluxes, caused by the particles inside the boundary layer, scale as .
Physics of Plasmas 08/2008; 15(8):082506-082506-7. · 2.15 Impact Factor
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ABSTRACT: The effects of orbit squeezing are important to neoclassical and anomalous transport fluxes in the region where plasma confinement is improved. This occurs in the edge region after the transition from the low confinement mode (L-mode) to the high confinement mode (H-mode) or in the vicinity of low-order rational surfaces. Neoclassical toroidal viscosity in tokamaks induced by the broken toroidal symmetry resulting from the activity of magnetohydrodynamic instabilities or error fields is calculated to include orbit squeezing effects. It is found that in the 1/ν regime, the magnitude of the neoclassical toroidal viscosity is enhanced by a factor of ∣S∣3/2, where ν is the collision frequency and S is the orbit squeezing factor; while in the ν regime, it is reduced by a factor of ∣S∣1/2. A boundary layer analysis is performed to remove the singularity in the vicinity of the trapped-circulating boundary in the ν regime. As a result, the well-known regime is recovered. Orbit squeezing has little effect in the regime. The implications on the physics of H-mode confinement are also discussed.
Physics of Plasmas 08/2008; 15(8):082505-082505-8. · 2.15 Impact Factor
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A C Sontag,
S A Sabbagh,
W Zhu,
J E Menard,
R E Bell,
J M Bialek,
M G Bell,
D A Gates,
A H Glasser,
B P Leblanc, K C Shaing,
D Stutman,
K L Tritz
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ABSTRACT: The National Spherical Torus Experiment (NSTX) offers an operational space characterized by high-beta (β t = 39%, β N > 7, β N /β no-wall N > 1.5) and low aspect ratio (A > 1.27) to leverage the plasma parameter dependences of RWM stabilization and plasma rotation damping physics giving greater confidence for extrapolation to ITER. Significant new capability for RWM research has been added to the device with the commissioning of a set of six non-axisymmetric magnetic field coils, allowing generation of fields with dominant toroidal mode number, n, of 1–3. These coils have been used to study the dependence of resonant field amplification on applied field frequency and RWM stabilization physics by reducing the toroidal rotation profile below its steady-state value through non-resonant magnetic braking. Modification of plasma rotation profiles shows that rotation outside q = 2.5 is not required for passive RWM stability and there is large variation in the RWM critical rotation at the q = 2 surface, both of which are consistent with distributed dissipation models.
Nucl. Fusion. 01/2007; 47.
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W Zhu,
S A Sabbagh,
R E Bell,
J M Bialek,
M G Bell,
B P LeBlanc,
S M Kaye,
F M Levinton,
J E Menard, K C Shaing,
A C Sontag,
H Yuh
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ABSTRACT: Dissipation of plasma toroidal angular momentum is observed in the National Spherical Torus Experiment due to applied nonaxisymmetric magnetic fields and their plasma-induced increase by resonant field amplification and resistive wall mode destabilization. The measured decrease of the plasma toroidal angular momentum profile is compared to calculations of nonresonant drag torque based on the theory of neoclassical toroidal viscosity. Quantitative agreement between experiment and theory is found when the effect of toroidally trapped particles is included.
Physical Review Letters 07/2006; 96(22):225002. · 7.37 Impact Factor
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S A Sabbagh,
A C Sontag,
J M Bialek,
D A Gates,
A H Glasser,
J E Menard,
W Zhu,
M G Bell,
R E Bell,
A Bondeson, [......],
S M Kaye,
L L Lao,
B P Leblanc,
Y Q Liu,
R Maingi,
D Mueller, K C Shaing,
D Stutman,
K Tritz,
C Zhang
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ABSTRACT: The National Spherical Torus Experiment (NSTX) has demonstrated the advantages of low aspect ratio geometry in accessing high toroidal and normalized plasma beta, β t ≡ 2µ 0 p/B 2 0 and β N ≡ 10 8 β t aB 0 /I p . Experiments have reached β t = 39% and β N = 7.2 through boundary and profile optimization. High β N plasmas can exceed the ideal no-wall stability limit, β Nno-wall , for periods much greater than the wall eddy current decay time. Resistive wall mode (RWM) physics is studied to understand mode stabilization in these plasmas. The toroidal mode spectrum of unstable RWMs has been measured with mode number n up to 3. The critical rotation frequency of Bondeson–Chu, crit = ω A /(4q 2), describes well the RWM stability of NSTX plasmas when applied over the entire rotation profile and in conjunction with the ideal stability criterion. Rotation damping and global rotation collapse observed in plasmas exceeding β Nno-wall differs from the damping observed during tearing mode activity and can be described qualitatively by drag due to neoclassical toroidal viscosity in the helically perturbed field of an ideal displacement. Resonant field amplification of an applied n = 1 field perturbation has been measured and increases with increasing β N . Equilibria are reconstructed including measured ion and electron pressure, toroidal rotation and flux isotherm constraint in plasmas with core rotation ω φ /ω A up to 0.48. Peak pressure shifts of 18% of the minor radius from the magnetic axis have been reconstructed.
Nucl. Fusion. 01/2006; 4635(52).
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Aaron C. Sontag,
S. A. Sabbagh,
W. Zhu,
J. M. Bialek,
J. E. Menard,
D. A. Gates,
A. H. Glasser,
R. E. Bell,
B. P. LeBlanc,
M. G. Bell, [......],
M. S. Chu,
C. C. Hegna,
S. M. Kaye,
L. L. Lao,
Y. Liu,
R. Maingi,
D. Mueller, K. C. Shaing,
D. Stutman,
K. Tritz
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ABSTRACT: The resistive wall mode (RWM) poses a limit to the maximum β that can be sustained in magnetic fusion experiments. RWM stabilization physics at low aspect ratio is studied in high-β National Spherical Torus Experiment (NSTX) [
M. Ono, S. M. Kaye, Y.-K. M. Peng et al., Nucl. Fusion 40, 557 (2000)
] plasmas (βt up to 39%; βN up to 6.8) to understand and alleviate this constraint. Plasmas with increased q in NSTX have been maintained with β above the computed ideal no-wall β limit for more than 20 wall times with no signs of RWM growth in cases where toroidal rotation ωϕ>ωA/4q2 across the entire plasma cross section. Plasmas that violate this stability criterion can suffer a RWM induced collapse within a few wall times. This critical rotation profile for stabilization is in agreement with drift-kinetic theory applied to low frequency magnetohydrodynamics modes [
A. Bondeson and M. S. Chu, Phys. Plasmas 3, 3013 (1996)
]. A toroidally symmetric array of internal sensors has been used to observe n = 1–3 RWMs in NSTX. This array consists of Bp and Br sensors both above and below the midplane at 12 toroidal locations instrumented to detect toroidal mode numbers of n = 1–3. RWM perturbations exceeding 30 G have been measured with mode growth rates on the order of 5 ms. Small modes (δB<10 G) which cause minor drops in β, with growth rates ∼ 1500 s−1 have been observed when βN exceeds 6. Resonant field amplification of an externally applied error field by the stable RWM has been observed.
Physics of Plasmas 04/2005; 12(5):056112-056112-7. · 2.15 Impact Factor
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K C Shaing
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ABSTRACT: It is shown that in the vicinity of a magnetic island, the symmetry of the equilibrium magnetic-field strength B is broken due to the finite width of the islands. The magnitude of this broken symmetry is of the order of (deltaB/B)(1/2), where deltaB is the perturbed magnetic-field strength. This leads to enhanced plasma transport. The symmetry-breaking induced-transport flux in tokamaks with islands is calculated.
Physical Review Letters 01/2002; 87(24):245003. · 7.37 Impact Factor
