Resistive Wall Effects Research Articles

The impact of the resistive-wall impedance on the ILSF storage ring

The resistive contribution of the vacuum chamber represents a significant source of the collective effects and should be analysed in several aspects. Due to the NEG-coated re-designed ILSF vacuum chamber, the resistive-wall effects were carefully studied. The measurement setup of the RW test was simulated and compared with the analytical equations. Finally, the longitudinal threshold was presented here to estimate the RW effects on the single bunch instability.

Effect of resistive wall on thermal quench in JET disruptions

Experimental data, simulations, and theory are presented for a JET tokamak thermal quench. The emphasis is on the timescale of the bulk plasma thermal energy loss. The simulations suggest that the thermal energy loss is caused by a resistive wall tearing mode, and experimental data are consistent with this conclusion. The timescale of the thermal quench is the inverse of the mode growth rate.

Resolution of the paradox discovered by Freidberg in the Shafranov's theory on plasma equilibrium

Predictions of the Shafranov's analytical theory are so highly respected that some of them are reprinted without any check for consistency. One example of such a kind is considered here: the equation for the plasma shift in a tokamak first introduced in Shafranov, J. Nucl. Energy C 5, 521 (1963). Since then, it has entered a number of review papers and textbooks, though, when tested by Freidberg [Freidberg, Rev. Mod. Phys. 54, 801 (1982)], it revealed some peculiar features. The main (long unnoticed) discovery was that the external vertical magnetic field required by that equation at zero shift was significantly different from the value given by the other famous Shafranov's formula for the same field in the same configuration. It is the tremendous scale of disagreement in the very core of the otherwise perfect theory that attracts attention. Here, a wrong element in the construction is found and repaired. It is established that it must be the incorrectly imposed boundary conditions for the magnetic field at the vacuum vessel wall treated as an ideal conductor. The proposed replacement eliminates the contradiction and extends the model by incorporating the resistive-wall effects.

Diamagnetic diagnostics of rapid events in the T-15MD tokamak

The resistive wall effect on the diamagnetic signal is analysed. The calculations are performed by the DINA code for the T-15MD tokamak. The accompanying analytics reveals the main dependencies allowing easy extrapolation of the results to any other tokamak. It is explicitly shown that the key parameter in the task is the decay time of the poloidal current induced in the wall. This current can be neglected for the events with transient times longer by factor of ten. If the plasma evolves faster, the difference between the measured signal and the plasma-produced variation of the toroidal magnetic flux will be larger than 10%, up to 100% for typically rapid thermal quenches. Incorporation of the induction equation for the wall current greatly improves the diagnostic capabilities.

Drift kinetic effects on plasma response in high beta spherical tokamak experiments

The high β plasma response to rotating external magnetic perturbations is numerically studied and compared with the National Spherical Torus Experiment (NSTX). The hybrid magnetohydrodynamic(MHD)-kinetic modeling shows that drift kinetic effects are important in resolving the disagreement of plasma response between the ideal MHD prediction and the NSTX experimental observation when plasma pressure reaches and exceeds the no-wall limit (Troyon et al 1984 Plasma Phys. Control. Fusion 26 209). Since the external rotating fields and high plasma rotation are presented in the NSTX experiments, the importance of the resistive wall effect and plasma rotation in determining the plasma response is also identified, where the resistive wall suppresses the plasma response through the wall eddy current. The inertial energy due to plasma rotation destabilizes the plasma. The complexity of the plasma response in this study indicates that MHD modeling, including comprehensive physics, e.g. the drift kinetic effects, resistive wall and plasma rotation, are essential in order to reliably predict the plasma behavior in a high beta spherical tokamak device.

Pfirsch-Tasso versus standard approaches in the plasma stability theory including the resistive wall effects

The study is devoted to theoretical description of plasma stability in toroidal fusion systems with a resistive wall. Its aim is elimination of contradictions between the models recently developed for the resistive wall mode analysis and the Pfirsch-Tasso approach originated from the paper published in 1971 [D. Pfirsch and H. Tasso, Nucl. Fusion 11, 259 (1971)]. The main relations have been given there without detailed proofs. Here, a missing chain of derivations is restored and earlier unknown limitations that restrict the applicability of the Pfirsch-Tasso energy principle are established. Its replacement valid in a wider area is proposed. The new result is free from the constraints implicitly imposed in the Pfirsch-Tasso procedure and can be used with any plasma model (not necessarily ideal) and for arbitrary perturbations. The proposed extensions allow applications for analysis of the rotational stabilization and optimization of the ITER scenarios.

Sideways force due to coupled kink modes in tokamaks

The paper is devoted to calculation of the sideways force on the vacuum vessel wall in tokamaks at the early stage of disruptions before possible appearance of the halo currents. The theory developed so far predicts quite a large force due to a single-helicity kink mode. This concept is revisited here. It is demonstrated that the existing single-mode models are incompatible with the natural requirement that the sideways force on the plasma must be practically zero. The latter can be satisfied by incorporating a secondary coupled mode, which is the main new element here compared to the earlier analytical approaches. With this difference, the derivations are performed in the standard cylindrical model that is widely used in the resistive wall mode studies. The resistive wall effects are accounted for as determining the wall reaction on the plasma-driven perturbations. The derived expressions explicitly reveal the sideways force dependence on γτw with a maximum at γτw=O(1), where γ is the kink growth rate and τw is the resistive wall time. It is proved that the amplitude of the sideways force produced by the kink modes must be much smaller than expected from the existing scalings.

Multiscale Modeling of a Short Relativistic Electron Bunch Interacting With Long Resistive Structures at Cryogenic Temperatures

This paper presents multiscale modeling of electromagnetic interaction between a short relativistic electron bunch and long resistive structures at cryogenic temperatures. The considered physical problems include multiple spatial scales such as structure, bunch size, and skin depth. In order to simulate such multiscale problems, a directionally implicit finite-difference time-domain (FDTD) scheme based on Maxwell's equations in the presence of a relativistic bunch current is developed by using a magnetically mixed Newmark-leapfrog formulation. With the so-called magic time step, the resultant FDTD scheme has no accumulation of numerical dispersion errors for all frequencies in the direction of the motion. In order to model the anomalous skin effect (ASE) of metals in time domain, the developed scheme is combined with a rational function approximation of the first-order Leontovich surface impedance boundary condition. The scheme is verified in comparison with analytical approaches. ASE resistive wall effects are assessed for beam pipes and a step collimator.

Plasma stability theory including the resistive wall effects

Plasma stabilization due to a nearby conducting wall can provide access to better performance in some scenarios in tokamaks. This was proved by experiments with an essential gain in${\it\beta}$and demonstrated as a long-lasting effect at sufficiently fast plasma rotation in the DIII-D tokamak (see, for example, Straitet al.,Nucl. Fusion, vol. 43, 2003, pp. 430–440). The rotational stabilization is the central topic of this review, though eventually the mode rotation gains significance. The analysis is based on the first-principle equations describing the energy balance with dissipation in the resistive wall. The method emphasizes derivation of the dispersion relations for the modes which are faster than the conventional resistive wall modes, but slower than the ideal magnetohydrodynamics modes. Both the standard thin wall and ideal-wall approximations are not valid in this range. Here, these are replaced by an approach incorporating the skin effect in the wall. This new element in the stability theory makes the energy sink a nonlinear function of the complex growth rate. An important consequence is that a mode rotating above a critical level can provide a damping effect sufficient for instability suppression. Estimates are given and applications are discussed.

Linear stability studies including resistive wall effects with the CASTOR/STARWALL code

CASTOR/STARWALL is a code package for linear stability studies consisting of the CASTROR_3DW code, the STARWALL code, and a hybrid version of both codes named CASTOR3D that is currently under development. The latter solves an extended eigenvalue problem consisting of the plasma part defined by the weak form of the perturbed single-fluid MagnetoHydroDynamic (MHD) equations, and a vacuum part which is derived from an energy functional. This new code can describe wall resistivity and plasma inertia simultaneously. Furthermore, besides the so far used straight field line coordinates, additionally, general flux coordinates are implemented which are more appropriate for the description of instabilities located close to the separatrix (e.g. Edge Localized Modes (ELMs)). The formulation of the eigenvalue problem in the new coordinates is not limited to axisymmetric equilibria, so that, finally, stability studies of resistive and rotating, 3D equilibria should be possible. In this paper we will sketch the theory, report on the progress of the code development, and present first results.

Resistive wall wakefields in the extreme anomalous skin effect regime

Usual treatments of resistive wall effects in accelerators are limited to the normal skin effect regime of electrical conductivity in metals. Therefore they do not generally apply to the situations when beam-exposed metallic surfaces of the vacuum chamber are held at cryogenic temperatures, where simple metals exhibit anomalous skin effect behavior. These situations occasionally occur in accelerators with cold-bore devices, such as small-gap superconducting undulators. The amount of anomalous resistivity material can be substantial to significantly influence beam dynamics. To accurately estimate these effects, we expand the conventional treatment of resistive wall in accelerators into the extreme anomalous skin effect region. Starting with the surface impedance expressions, we derive resistive wall related quantities commonly used in accelerator physics, such as wake functions, wake potentials, loss factor, etc. in the extreme anomalous skin effect region. We follow with examples for resistive wall generated heat and transverse mode-coupling instability.

Magnetic calculations of a superconducting undulator at the ESRF

2D and 3D models of a superconducting undulator suitable for the ESRF storage ring have been developed. The models are based on the performance of superconducting NbTi wire available from the cabling industry. Iron is used in the poles to enhance the peak field. The 2D model shows that it is possible to obtain a peak field of 1.18 T, corresponding to a K-value of 1.65, for a period length of 15 mm, a magnetic gap of 6.5 mm, and a current density of 1050 A/mm 2 in the superconducting coils. 1050 A/mm 2 corresponds to 80% of the critical current density in the coils. A 3D model of the central parts of the undulator close to the storage ring shows that it is possible to obtain a negligible first and second field integral at the nominal current density of 1050 A/mm 2 and moderate field integrals at lower current densities. A full 3D model including the current leads and the path of the current carrying wires shows that an accurate arrangement of the current leads and path of the current carrying wires is essential but difficult to obtain. A precise design of the current leads and the path for the current carrying wire has not yet been found and this requires further empirical knowledge of the winding technique. The heat load to the cold mass of a 100 period superconducting undulator with a vertical accelerator vacuum aperture of 6.0 mm and a period length of 15 mm length has been estimated. The heat load is in the range 3.4–5.8 W, depending on the filling mode of the ESRF storage ring. The dominating heat load is the heating due to the resistive wall effect. Four local cooling machines, each with a cooling capacity of 1.5 W at about 4 K, would give enough cooling capacity for the foreseen filling modes of the ESRF storage ring.

Ferromagnetic and resistive wall effects on the beta limit in a tokamak

The ferromagnetic and resistive wall effects on the beta limit in a high aspect ratio tokamak are investigated. It is shown that the beta limit is reduced to 90% of that without ferromagnetic effect for a high aspect ratio tokamak, when the relative permeability of the ferromagnetic wall is 2. In these analyses, parabolic profiles for both plasma current and pressure are employed and the radius and thickness of the resistive wall are rw = 1.43a and d = 0.07a (a is plasma minor radius), respectively. The stability window with respect to the external kink modes is shown to be reduced by a high permeability effect in the case of a uniform current tokamak plasma, which is different from the case where only the finite resistivity effect is considered. The effect of toroidal plasma flow is also investigated, and it is shown that the toroidal background flow velocity of 0.3vpa, vpa is poloidal Alfvén velocity, is sufficient for the resistive wall to have the stability effect of an ideal wall. The ferromagnetic effect of the wall destabilizes both resistive wall and ideal kink modes.

Stacking of a Low Current Electron Beam in a Harmonic Potential Trap by RF-Repeller

Particles of a low current beam can be accumulated in a harmonic potential well in a high vacuum environment when they are repelled back by a local rf electric field of the frequency close to the bounce motion in the well. Here, proper damping mechanisms for the repelled particle blobs are necessary to suppress their bounce motions. This stacking method was experimentally proved for electron beams of 1.1 µA using a Multi-Ring-Electrode trap and the results were compared with numerical estimations based on a single particle model. The observed damping was much larger than the estimated one from the resistive wall effect. The stacking efficiency was nearly the same as the estimated one at the stacked number N less than 1×107 but it decreased with N. Experimentally obtained relationship amongst the stacked number, the incident beam energy, the rf frequency and its amplitude behaved qualitatively in the same way as the numerical results. The accumulation proceeded until the well was filled up with electrons.

Numerical studies of confinement scaling in the conventional reversed field pinch

Scaling laws for reversed field pinch (RFP) confinement parameters versus plasma current and density are found from computer simulations. The RFP dynamics at high Lundquist numbers approaching 106 is studied using a high resolution, 3-D, resistive MHD numerical code. Optimum plasma conditions are attained by assuming that the transport coefficients are classical, and by ignoring radiation losses and resistive wall effects. Anomalous global transport results from classical parallel heat conduction along stochastic field lines in the plasma core. The pinch parameter is Θ = 1.8 and the aspect ratio is R/a = 1.25. Poloidal beta is found to scale as βθ ∝ (I/N)-0.40I-0.40 and energy confinement time as τE ∝ (I/N)0.34I0.34. On-axis temperature scales as T(0) ∝ (I/N)0.56I0.56. Experimental results from T2, RFX and MST agree well with the above numerical results and also with the obtained magnetic fluctuation scaling ∝ S-0.14, where S is the Lundquist number. Thus stochastic core field lines appear to persist also at higher, reactor relevant currents and temperatures in the conventional RFP, indicating the need to further pursue confinement enhancement techniques.

External Field Threshold for the Unlocking of Magnetic Islands in the Presence of Resistive Wall Effects and Toroidal Mode Coupling

The unlocking of static resistive tearing modes by rotating external magnetic perturbations such as those which may arise from the natural tokamak error-field is discussed. For a single mode the balance between the accelerating torque imparted by the error-field and the braking torque due to the interaction of the mode with the wall sets a threshold for the error-field current (H P Furth et al 1963 Phys. Fluids 6 459), below which the mode frequency will not lock to the error-field frequency. If the mode resonant with the error-field is coupled to another mode with a rational surface located elsewhere in the plasma, the unlocking process is more elaborated and substantial modifications to the current threshold are expected. The present analysis may contribute to the explanation of some mode unlocking events in tokamak discharges with a non negligible error-field.

Confinement scaling laws for the conventional reversed-field pinch

A series of high resolution, 3D, resistive MHD numerical simulations of the reversed-field pinch are performed to obtain scaling laws for poloidal beta and energy confinement at Lundquist numbers approaching 10(6). Optimum plasma conditions are attained by taking the transport coefficients to be classical, and ignoring radiation losses and resistive wall effects. We find that poloidal beta scales as beta(straight theta) approximately I-0.40 and that the energy confinement time scales as tau(E) approximately I0.34 for fixed I/N, with aspect ratio R/a = 1.25.

A self-consistent nonlinear theory of resistive-wall instability in a relativistic electron beam

A self-consistent nonlinear theory of resistive-wall instability is developed for a relativistic electron beam propagating through a grounded cylindrical resistive tube. The theory is based on the assumption that the frequency of the resistive-wall instability is lower than the cutoff frequency of the waveguide. The theory is concentrated on study of the beam current modulation directly related to the resistive-wall klystron, in which a relativistic electron beam is modulated at the first cavity and propagates downstream through the resistive wall. Because of the self-excitation of the space charge waves by the resistive-wall instability, a highly nonlinear current modulation of the electron beam is accomplished as the beam propagates downstream. A partial integrodifferential equation is obtained in terms of the initial energy modulation (ε), the self-field effects (h), and the resistive-wall effects (κ). Analytically investigating the partial integrodifferential equation, a scaling law of the propagation distance zm at which the maximum current modulation occurs is obtained. It is found in general that the self-field effects dominate over the resistive-wall effects at the beginning of the propagation. As the beam propagates farther downstream, the resistive-wall effects dominate. Because of a relatively large growth rate of the instability, the required tube length of the klystron is short for most applications.

Transient state of a bunched electron beam subject to resistive-wall instability.

This paper analyzes the transient state of a bunched electron beam traversing a narrow beam pipe under the influence of transverse resistive-wall effects. Because the electron beam in the wigglers of future free-electron lasers (FEL's) is required to travel in pipes with small diameters, this study is relevant in determining the beam stability in these FEL's. The analysis is restricted to the case in which the first mode dominates. First, an integral representation of the single-mode solution is obtained. Second, the transient-state solution is obtained for the case in which the focusing force dominates the resistive-wall force. The maximum transient amplitude is derived for the case in which a single bunch is off-axis. Third, this result is used to compute the maximum transient amplitudes when the bunches enter the system with random and with equal initial displacements, respectively. These analytic expressions of maximum transient amplitudes are useful in predicting when the transient could become significant. They are compared with earlier derivations and with numerical simulations. These expressions are also applied to various proposed FEL's. Results showed that the transverse resistive-wall instability could not pose a problem in the operation of these FEL's.

Transverse resistive wall effects on the dynamics of a bunched electron beam.

In the wigglers of future free-electron lasers, the electron beam will be required to travel over a length of 10 m or more in pipes with small diameters. Transverse resistive wall effects could lead to beam breakup during this transport. To investigate this possibility, the equation of motion for a bunched beam is solved analytically. Results show that a steady-state solution is reached for times larger than the diffusion time. This solution can either oscillate or grow exponentially with the length of the pipe, depending on the relative magnitudes of the resistive wall effect and the focusing force in the wiggler. The magnitude of the resistive wall effect depends on the pipe radius b (it increases as 1/b/sup 2/) but is independent of the thickness and conductivity of the pipe. The thickness and conductivity affect only the time required to reach the steady-state solution. The possibility of a significant transient is also discussed.