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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 S 1 ÷ S 3 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 S 1 ÷ S 3 proposed in Pustovitov [Phys. Plasmas 29, 092507 (2022)] is assessed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The dispersion relation for the resistive wall modes (RWMs) is derived without the use of the trial function b H F 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Virial relations are traditionally considered as providing the diamagnetic parameter, poloidal beta β p , and internal inductance ℓ i 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 S 1 , S 2 , and S 3 ) 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 S 1 must be a constant, S 3 weakly depends on ℓ i , and only S 2 is a sensitive function of the plasma state through Δ ′. This makes S 2 the quantity most suitable for diagnostics, while independence of S 1 on the plasma shape, β p , and ℓ i can be good for calibrations. The difficulties of inferring ℓ i from the measured S 3 are now shown explicitly. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 with respect to VV is a free parameter, and the wall is not a magnetic surface. The derived expressions explicitly show the effect of , 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. Similar results have been obtained recently in CarMa0NL 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. 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. [ABSTRACT FROM AUTHOR]

The study is devoted to the problem of the magnetic field diffusion through a toroidal resistive shell (wall with respect to the interior). This is the same task as that considered in Dialetis et al. [J. Appl. Phys. 69, 1813 (1991)], but with a new element: current-carrying plasma inside the vessel. This extends the study on tokamaks with a resistive wall. The shape and position of the magnetically confined plasma must react on the field variations which brings considerable complications in the plasma electromagnetic description as compared to that of the rigid wall and external conductors. The proposed algorithm is devised so that the plasma properties are fully accounted for. It is based on the Green's function technique providing correct asymptotic behavior of the solutions that determine, through Maxwell equations and Ohm's law, the current induced in the wall during transient events. For tokamaks, this gives a closure in the analytical approaches incorporating the plasma-wall electromagnetic interaction with non-ideal wall reaction. This is needed for disruption modeling and, in particular, for evaluation of the disruption forces on the wall in large tokamaks like ITER with expected plasma current quench from 15 MA to zero in 35 ms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 csw 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. [ABSTRACT FROM AUTHOR]