Response to “Comment on ‘Continuum modes in rotating plasmas: General equations and continuous spectra for large aspect ratio tokamaks’ ” [Phys. Plasmas 19, 064701 (2012)]

We are very pleased with the interest caused by our paper and are grateful to Professor Goedbloed for his comment.¹ We are really sorry that we have not noticed the work presented in Ref. 2 (henceforth referred to as I), although we have cited other relevant papers by the same authors. Partly, it can be excused by the clear trend of I to trace some common features of “transonic” flows in a rather general form suitable for different objects while we have discussed the MHD modes of large-aspect-ratio tokamak plasma only. (Note that concrete results have been obtained in I for more specific equilibria as we did in our paper³ below referred to as II.)

We accept the remark in the Comment concerning a possible reduction of our equations to the matrix form used in I. It is not a surprise at all that the general set of differential equations describing the coupling of Alfvén and sound modes (30) and (31) (hereafter we follow the equation numbering and notations used in II) is equivalent to Eqs. (83)–(85) of I. One can see this using Eqs. (32)–(34). Excluding the terms containing $Ω$ and $Ω2$ by the use of these equalities one can easily rearrange the corresponding terms in Eqs. (30) and (31) and rewrite them in the form,

In the limiting case of purely toroidal plasma rotation the above equations (and the original Eqs. (30) and (31) as well) definitely describe all possible continuous modes for tokamak equilibrium with isentropic magnetic surfaces—as it has been underlined in Sec. IV B 2 of II and has been fairly mentioned in the Comment. For such an equilibrium the entropy perturbation is equal to zero, $p'=(Γp/ρ)ρ'$⁠, and, therefore, the convective modes can not exist. However, there is a place for convective effects in the case of other equilibria with poloidal variation of entropy (see, e.g., Refs. 2 and 4–6). Due to tokamak axial symmetry, the dependencies of equilibrium mass density, temperature, or entropy on poloidal flux and poloidal angle cannot be specified by stationary MHD equations. It means that tokamak equilibria with purely toroidal flows are degenerated. It was just such a degeneration, which allowed us to find analytically the unstable zonal flows in Ref. 6.

A large piece of the Comment is devoted to the proposition to go out of use in MHD the conventional sonic Mach number used in II and in a lot of papers by the other authors (see, e.g., Refs. 4–7). Although this subject is rather general and has no specific relation to the commented paper II, there is a sense to note the following. Indeed, the poloidal Alfvén Mach number M (⁠$M2=κ2/ρ$⁠) plays the key role for MHD equilibrium and stability analysis. Nevertheless, the sonic Mach numbers are used not only due to a tradition but also due to very transparent physical reasons. There are 3 types of terms in the MHD force balance equation related to inertia, pressure and Ampere’s force, and, generally speaking, each two may be scaled with respect to the third; it is simply a matter of convenience. In the case of low-beta tokamaks it is convenient to keep the sonic Mach numbers when:

1. the angular velocities of poloidal and toroidal plasma rotation are relatively small;

2. the low-frequency perturbations like the geodesic acoustic modes and its modifications are studied.

The frequencies of such modes are of the order of sound frequency $cs/R0$ or less. The frequency of slow magnetosonic wave in the low-pressure case is of the same order, $cs/qR0$ (⁠$R0$ is the major tokamak radius, $cs$ is the sound speed). As it is indicated in II, we have restricted ourselves to the case of slow poloidal plasma rotation and of relatively slow toroidal rotation such that the poloidal and toroidal sonic Mach numbers are of the order of unity,

For such flows $M2≡κ2/ρ=(Γp/B2)MP2≪1$⁠.

The next critical issue of the Comment concerns the problem of “transonic” transitions of equilibria that was not discussed in our paper II. Indeed, it was not due to a very simple reason. We worked with low-beta, large-aspect-ratio tokamak plasmas with relatively slow rotation $(MP,MT)≃1$⁠. In addition, we assumed that $|1-MP2|≃1$⁠. These assumptions evidently guarantee the existence of equilibrium with nested magnetic surfaces of circular cross-section, hence there is no point for a discussion. This fact easily follows also from Eqs. (60) and (61) of I, which formally give estimates $ρ1/ρ0≃a/R0$ and $Δ'≃a/R0$ (⁠$ρ0$ and $ρ1$ are poloidally uniform and poloidally oscillating parts of the mass density, correspondingly; $Δ'$ is the radial derivative of the Shafranov’s shift).

Moreover, these equations of paper I show that under assumptions made the equilibrium with nested magnetic surfaces exists even in the case of faster poloidal rotations, but such that $2≤MP2≪B2/Γp≡2/Γβ$⁠. In the case of low-pressure plasma the first region of hyperbolic flow regime $Mc2<M2<Ms2$ (in notations of I) is very narrow because $Mc2≃Ms2≃Γp/B2$⁠.

We would like to emphasize again that the area of working parameters we have chosen does not invoke any problem with equilibrium, and from our point of view, there is nothing to comment here. Nevertheless, being inspired with the claim that “equilibrium and stability are no longer separate issues…” specially marked in the Comment, we would like to make our vision of the problem more clear.

First of all, we think that the equilibrium and stability problems should always be separated just by the logic of the theory of perturbations. Linear stability analysis has a sense only in the case when the initial equilibrium state exists. It means that equilibrium problem should always be solved beforehand—independently of the type of equilibrium equation. If a solution of the equilibrium equations satisfying the specific boundary conditions cannot be designed, there is no place for spectral stability analysis.

It is appropriate to mention here that a problem of equilibrium cannot be reduced only to the problem of possible violation of ellipticity condition that is specially stressed in the Comment. Indeed, $Mc,Ms$⁠, and $Mf$⁠, whose combination determines the type of the equilibrium (Grad-Shafranov) equation, are not constant and depend on their turn on the solution of equilibrium equations that is a priori unknown. Ipso facto, the equilibrium equation can be attributed to the certain (elliptic or hyperbolic) class only locally, while the equilibrium conditions have to be satisfied everywhere in the domain occupied by plasma. From this point of view, the requirement to have the equation of the elliptic class is neither necessary nor sufficient for existence of smooth global equilibrium solution. Since the equilibrium plasma parameters obey the nonlinear equation and have to satisfy the boundary conditions, there can be some singularities even when the equilibrium equation is elliptic everywhere within plasma domain. For instance, there are well known Soloviev’s solutions⁸ for a static equilibrium problem, which demonstrate a necessity to adjust very precisely the flux functions entering the Grad-Shafranov equation to find an equilibrium configuration with nested magnetic surfaces appropriate for tokamak plasma confinement and to avoid X-points and separatrices within plasma domain. Definitely, the most of existing equilibrium codes are not able to resolve such complicated equilibria and can be applied to rather simple configurations with nested magnetic surfaces. Otherwise, there are examples of MHD flows, which have a different magnetic structure (consistent with corresponding boundary conditions), but can be a subject for stability analysis. Summarizing the above mentioned, we conclude that the existence of equilibrium is the only crucial point for a subsequent stability analysis.

Final remark of the Comment concerns the particular case of gravitating “fat” accretion disk considered in I but not in II. Indeed, the paper II (as it follows from its title) concerns the large aspect ratio tokamak case only. However, analyzing this particular case, we have found some new results absent in I. Dispersion relation (45)–(47) of our paper II defining the continuous spectrum in low-beta, large aspect ratio tokamaks is a general equation in three-mode approximation. Despite a statement of general treatment of the problem, such a general analytical dispersion relation is absent in I. Even in the case of modes localized near the rational magnetic surfaces, when $k∥≪1/qR$⁠, dispersion relation (117) of I is just a particular case of Eq. (54). It strictly follows from Eq. (54) under two additional assumptions made in I:

1. equilibrium plasma motion is directed strictly parallel to the magnetic field, so that $Ω=0$⁠;

2. both poloidal and toroidal angular velocities are large compared to $cs/R0$⁠, so that $MT=MP≫1$⁠.

The last assumption is equivalent to Eq. (64) of I. In this limiting case of perturbations localized at the rational magnetic surfaces according to Eq. (119) of I there are no unstable modes in tokamak case. This result easily follows from the general stability criterion (53). Indeed, assuming $MT=MP≫1$ we obtain $A=(1+q2/2)MP4>0$⁠. At the same time, in the case when $MP2>1,|1-MP2|≃1$ and $MP2$ is of the order of unity an unstable mode of perturbations with $k∥=0$ does exist if criterion (53) is satisfied. In particular, in the case of predominant poloidal plasma rotation, such that $ΩP≫ΩT$⁠, an instability takes place when criterion (58) is satisfied and this criterion is very realistic. Moreover, in the case of perturbations with $(m,n)≠0$ localized away from the rational surfaces another, lower-frequency unstable mode arises when $k∥2cA2≪cs2/R02$ and criterion (58) is satisfied. This mode is described by Eq. (60) of II. We have numerically studied the coupling of Alfvén and sound modes in II in the case, when the Alfvén frequency is comparable to the sound frequency, $k∥2vA2≃cs2/R02$⁠, and have found a stabilization of the above instabilities due to this coupling. Thus, in our opinion we have obtained the new interesting and important for tokamak plasma physics results, which are not covered by I.

The work was partly supported by the Russian Foundation for Basic Research (Grant No. 10-02-01302).