The external kink related sideways wall force is investigated on ultralow-q RFX-mod plasma discharge. Several theoretical predictions are experimentally verified. The presence of coupled kink modes is observed when the wall is subject to a sideways force. The absence of an integral electromagnetic force on the plasma is analyzed in such conditions. The dependence of sideways force on the mode growth rate is also revealed. A comparison between different theoretical predictions of the sideways force magnitude is performed for both flat-top and ramp-down phases. The results are compared with the direct calculation of sideways force provided by the ex-vessel magnetic measurements. These analyses show that the sideways force related to kink-like mode is much smaller than predicted by models that approximate the plasma as a rigid current-carrying ring or that neglect the presence of the wall.

Using the SPIDER code, three series of calculations of plasma equilibrium in a tokamak are performed—with low, medium, and high pressure. In each series, the cross section is purely elliptical with elongation varying from K=1 (circular plasma) to K=2.4. For these configurations, the integrals S1÷S3 defining the right-hand sides of the virial relations in Lao et al. [Nucl. Fusion 25, 1421 (1985)] are calculated. Their dependences on plasma parameters, elongation K of magnetic surfaces, and radial derivatives of K and Shafranov shift Δ are analyzed. The accuracy of analytical expressions for S1÷S3 proposed in Pustovitov [Phys. Plasmas 29, 092507 (2022)] is assessed.

The problem of the plasma stability against the resistive wall modes is considered from two sides, theoretical and experimental. The main subject is the dispersion relation and its verification, which is commonly understood as a comparison of the predicted and observed stability thresholds. As in the conventional magnetohydrodynamics, the growth rate γ and the angular rotation frequency ω of the mode are found from the energy balance with account of some dissipation in the plasma, additional to the resistive losses in the vacuum vessel wall. The resulting relations are integral, which allows the same γ and ω with different integrands. It is shown explicitly that only two fitting parameters are needed for getting a perfect agreement of such results with measured γ and ω. This explains why all attempts with so-called kinetic relations have been good in that. This also reveals the reason for the earlier finding [A. M. Garofalo, Fusion Sci. Technol. 48, 918 (2005)] that a number of models provided the stability regardless of the type of dissipation as long as the dissipation was sufficiently large. It is shown here that such “degeneracy” is a general property. One consequence is that a similar success with any model cannot guarantee its validity, and none of them can be recommended to ITER immediately. It is also explained that the edge harmonic oscillations can be a promising candidate for testing the dissipation channels missing in the kinetic dispersion relations.

The study is focused on the formula proposed by M. Ferrara et al. [Nucl. Fusion 48, 065002 (2008)] as an upper bound on the value of the internal inductance ℓ_i in tokamaks. That was postulated to be true under the assumption that the toroidal current density does not change sign. It is shown here that the latter condition allows the results for ℓ_i above the mentioned “upper bound.” This is demonstrated explicitly with easily integrable distributions and the outcome exceeding the bound by 30 ℓ_i because the result must also depend on the plasma pressure.

The dynamic problem of plasma equilibrium in a tokamak is considered taking into account the electromagnetic reaction of the vacuum vessel resistive wall. The currents induced in the wall during transient events contribute to the external magnetic field that determines the plasma shape and position. Accordingly, the plasma geometry must evolve so that the inductive excitation of the wall current would properly compensate for the resistive losses. Simultaneous consideration of these factors presents the main difficulty of the description. It is performed in a general form using the Green's function method that guarantees the mathematical accuracy of expressions for the magnetic fields from each source. At the same time, it is desirable to minimize the related complications, which is one of the goals here. The starting point is the standard solution of the external equilibrium problem given by integral relating the poloidal magnetic flux to the magnetic field at the plasma boundary. In the evolutionary problem, the additional equations for the plasma-wall electromagnetic coupling are transformed to an equation with a similar integral over the wall, but with either the time derivative of the poloidal magnetic flux or the wall current density in the integrand. The mentioned similarity allows to use the already developed techniques, which makes this formulation compact and convenient. It provides the basis for extension of the existing analytical theory of equilibrium to the case with non-circular plasma and wall.

The problem of inferring the internal inductance ℓ i from external magnetic measurements in tokamaks is considered. It was practically resolved in JET (Barana O et al 2002 Plasma Phys. Control. Fusion 44 2271), but several questions remain to be addressed before extrapolation of that method on other tokamaks. These naturally arise because of the integral nature of ℓ i determined by unknown distribution of the magnetic field inside the plasma. Without universal solutions, the quality of approximations should be examined along with potential experimental enforcements of the existing procedures. This is the main goal of the study that is fully analytical. Several relevant quantities are calculated for the large-aspect-ratio plasma with elliptical shifted magnetic surfaces. Usually the plasma elongation K is mentioned as a key factor in the task. Here it is explicitly shown that not only K , but its radial derivative too plays a role. The latter parameter did not appear in the JET scaling and in the accompanying discussions. It is also shown that the K -dependent term representing there the first iteration actually brings a systematic error. Therefore, several improvements to that scaling can be proposed. The study is based on the use of virial relations for the toroidal axisymmetric configurations.

The estimation of integral wall force using solely magnetic measurement in RFX-mod experiment is presented. The vertical and sideways forces are directly obtained from the magnetic field measured outside the vacuum vessel. Several theoretical predictions related to tokamak are also verified for the reversed field pinch configuration. The contribution of different modes to the force is also considered and analyzed. This method of calculation would be relevant for future nuclear fusion reactors where magnetic measurements will be located only outside the vacuum vessel.

The dispersion relation for the resistive wall modes (RWMs) is derived without the use of the trial function bHF proposed in S. W. Haney and J. P. Freidberg [Phys. Fluids B 1, 1637 (1989)] for the magnetic perturbation b outside the plasma. Another difference from the Haney–Freidberg (HF) approach is the incorporation of non-ideal effects in the plasma description. These enter the final result through the energy functional and affect the external solution for b through the boundary conditions only. This allows to perform the derivations in a general form without constraints on the dissipation mechanisms in the plasma. Then, the main mathematical difficulties are related to the description of the energy flow outside the plasma. This part of the task is presented with details allowing easy comparisons with the reference HF case. Being universally applicable, the resulting dispersion relation covers the existing variants, including those based on the so-called kinetic approaches. It shows that, because of its integral nature, the same predictions can be expected from various models for the plasma. Another conclusion is that, with a non-ideal contribution, just one or two free parameters would be enough to get agreement with experimental data on the plasma stability boundary. This, however, does not guarantee that the same choice of the fitting coefficients will be similarly efficient on other devices. The proposed relations provide a unified approach to the problem of plasma stability against RWMs.

In the plasma equilibrium theory, Gajewski's analytical expression [Gajewski, Phys. Fluids 15, 70 (1972)] for the poloidal magnetic flux ψ outside the plasma is known. It was obtained as a solution of the two-dimensional Laplace equation outside an infinite straight cylinder with an elliptical cross section and a uniform current density j ζ. An example of its use for analysis of static configurations is given in the study by Porcelli and Yolbarsop [Phys. Plasmas 26, 054501 (2019)]. Here, we consider the question of its applicability in dynamic problems including, for example, the current quench (CQ) or vertical displacement event (VDE), when the electromagnetic response of the vacuum vessel to the plasma magnetic field evolution has to be accounted for. It is shown that the mentioned cylindrical model does not provide enough information for calculation of the current induced in the wall. Mathematically, this manifests itself in the fact that Gajewski's expression contains an indefinite constant of integration ψ b (hereinafter it is ψ at the plasma boundary), which, in analytical applications, is replaced either by zero or by a value that makes ψ = 0 on the magnetic axis. This does not affect the magnitude of the magnetic field B, but it would incorrectly give the electric field at ∂ B / ∂ t ≠ 0. To eliminate this shortcoming, an additional block of calculations in the toroidal geometry is needed. Here, the problem is solved analytically. The resulting final expression with ψ b well-defined in the toroidal configuration also includes the effects of the Shafranov's shift and inhomogeneity of j ζ. The proposed extensions allow generalization of the earlier results to a wider area and cover such events as CQ or VDE.

The solution of the sideways force problem is finally needed for the ITER project. The task became urgent when the extreme danger of such a force was perceived. The predictions were based on the so-called Noll's formula derived under some simplifications. One of them was the prescription of the plasma motion without testing its compatibility with the force balance condition. Later, an alternative approach has been proposed [D. V. Mironov and V. D. Pustovitov, Phys. Plasmas 24, 092508 (2017)], where the key element was the absence of an integral electromagnetic force on the plasma. Another important improvement was a proper treatment of the vacuum vessel wall. Now the extensions of the previously developed models leading to or supporting Noll's formula are proposed with the resistive wall reaction similarly incorporated. The main attributes of those approaches, the plasma displacements, are kept the same as in the original versions. Precisely, these are the plasma tilt or the (1,1) kink mode. Two forces are calculated with such displacements: on the plasma and on the vacuum vessel wall. The former is shown to be far from zero in the analyzed cases, violating thereby the force-free condition. This does not happen when this constraint goes first. It becomes a selection rule for allowable perturbations. These roughly resemble the tilt and (1,1) mode but differ from them, which changes the result dramatically. The maximal force that can be produced by such kink-like modes compatible with the force balance cannot reach even one tenth of Noll's force. The quantitative comparisons of the competing models are provided.

In the quarter-century since the danger of sideways force on tokamak walls during disruptions was first recognized, substantial progress has been made in understanding the connection between plasma kink and the force. Less is known, however, about the effect of the wall asymmetry on force generation. Here, we explore how irregularly situated ports lead to large sideways forces even for a symmetric driver such as a typical current quench. Further, we compare the effectiveness of both mechanisms.

Virial relations are traditionally considered as providing the diamagnetic parameter, poloidal beta βp, and internal inductance ℓi through the integrals determined by the plasma shape and poloidal magnetic field at the plasma–vacuum interface. This gives rise to discussions of their potential applications for diagnostic purposes. Recently, this concept was analyzed in the numerical study of Bongard et al. [Phys. Plasmas 23, 072508 (2016)]. Here, we analytically calculate three main virial integrals (traditionally denoted as S1, S2, and S3) for the plasma with elliptical cross section. The results are expressed through the plasma elongation, its radial derivative, and a similar derivative Δ′ of the Shafranov shift, all taken at the plasma boundary. The geometry of magnetic surfaces inside is not constrained, which guaranties the applicability of the results in a wide area. It is shown that S1 must be a constant, S3 weakly depends on ℓi, and only S2 is a sensitive function of the plasma state through Δ′. This makes S2 the quantity most suitable for diagnostics, while independence of S1 on the plasma shape, βp, and ℓi can be good for calibrations. The difficulties of inferring ℓi from the measured S3 are now shown explicitly.

The study is devoted to theoretical analysis of some models for calculating the disruption forces in tokamaks and scalings for these forces which can be proposed for ITER. It is motivated by the necessity of reliable predictions for ITER. The task includes the evaluation of the existing models, resolution of the conflicts between them, elimination of contradictions by proper improvements, elaboration of recommendations for dedicated studies. Better qualities of the modelling and higher accuracy are the ultimate theoretical goals. In recent years, there was a steady progress in developing a physics basis for calculating the forces, which gave rise to new trends and ideas. It was discovered, in particular, that the wall resistivity, penetration of the magnetic perturbation through the wall, the poloidal current induced in the wall, the kink-mode coupling, plasma position in the vacuum vessel must be the elements essentially affecting the disruption forces. These and related predictions along with earlier less sophisticated concepts and results are analysed here.

First, it is shown that during electromagnetic transients in COMPASS-U the poloidal field coils must drain sizeably the current from the vessel and, therefore, reduce disruption forces and their duration. Next, the role of poloidal eddy current (which is absent in some approaches) in the dynamics of vertical and radial forces is found to be essential. Finally, to verify the CarMa0NL modelling for COMPASS-U, the numerical results are cross-validated with general analytical predictions (Pustovitov 2015 Nucl. Fusion 55 113032): the computed vertical force on the tokamak wall is found to be almost zero during rapid (jump-like) transients, as it should be because of strong skin-effect. This test proves the credibility of the simulation model and computational realization.

The possibility of generation of the rotating sideways force on the wall by the kink modes is analytically investigated. The approach is basically the same as that developed earlier in (Mironov and Pustovitov 2017 Phys. Plasmas 24 092508) for the locked modes, but now their rotation is allowed. Its main elements are ∂ b /∂ t ≠ 0 (described by the growth rate γ and angular rotation frequency ω of the magnetic perturbation b ), resistive dissipation in the wall, and the requirement of zero sideways force on the plasma. These make the approach greatly different from those resulting in the so-called Noll’s formula. The result is also different; it predicts a force an order of magnitude smaller. Nevertheless, such a force can be dangerous at the resonance frequency of the vacuum vessel. The derived relations show that the rotating force must be maximal at ωτ w = O (1), where τ w is the resistive wall time. For the faster modes it decreases roughly as ∼1/ ω .

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.

The earlier analytical model (Pustovitov and Kiramov 2018 Plasma Phys. Controlled Fusion 60, 045011) for plasma-wall electromagnetic interaction in tokamaks is revised and extended. In that model, the magnetic field B was treated as tangential to the vacuum vessel (VV) wall, which implied some pre-selected plasma position inside the VV. Here, in contrast, the plasma shift Delta b<i with respect to VV is a free parameter, and the wall is not a magnetic surface. The derived expressions explicitly show the effect of Delta b<i, or the normal component of B on the poloidal distribution of the disruption force on the wall, while the integral radial force is found to be insensitive to Delta bCarMa0NL simulations (Isernia et al 2019 Plasma Phys. Control. Fusion 61 115003), which are now explained and confirmed analytically, including the role and interplay of the normal and tangential forces as functions of Delta b<i. The standard large-aspect-ratio model of an axisymmetric tokamak with circular plasma and an almost coaxial wall is used. Additionally, to facilitate comparison with previous studies, the ideal-wall reaction is assumed.

Electromagnetic forces generated during hard-to-predict transient events, represent a serious constraint for the operation and design of tokamaks A sudden loss of plasma stability, triggering plasma thermal and current quenches, leads to the induction of eddy currents in the conducting structures surrounding the plasma column. Interaction of these currents with the magnetic field is responsible for a j x B local force that might compromise the integrity of the device. Here we evaluate the effect of poloidal currents induced in the wall on the local and global forces. To test the earlier analytical predictions (Pustovitov and Kiramov 2018 Plasma Phys. Controlled Fusion 60, 045011), we consider a circular tokamak by the numerical tool CarMa0NL (Villone et al 2013 Plasma Phys. Controlled Fusion 55, 095008). The results confirm the necessity of incorporating the poloidal currents into the task, as these strongly affect the local stress distribution and the global radial force. The overall agreement between analytical and numerical computation is an additional evidence that CarMa0NL is a sound tool for the prediction of disruption forces in tokamaks At the same time, the simulation conditions in which the agreement is less satisfactory allow to identify which are the most restrictive assumptions of the analytical model, pointing out the way for future theoretical work.