The Brunt–Väisälä frequency of rotating tokamak plasmas
Introduction
One of the very few viable ways to confine a plasma at the high temperatures required for nuclear fusion, is using magnetic fields. One of the most promising designs suitable for the commercial production of energy is the tokamak, in which the plasma is confined in the shape of a torus. A tokamak force equilibrium can be described quite well by a balance between the pressure gradient, the Lorentz force, and centrifugal forces. This force balance can be described by the fluid model of magnetohydrodynamics (MHD). Assuming axisymmetry, the static MHD equations can be reduced to a single partial differential equation, called the Grad–Shafranov equation [1], [2]. The solution ψ is a stream function for the poloidal magnetic field, and can be used as a coordinate that labels the magnetic surfaces.
Given the dependence of the pressure and the toroidal magnetic field on ψ, the solution of the Grad–Shafranov equation gives ψ as a function of the spatial coordinates, completing the description of the equilibrium. Assuming this dependence to be linear, a simple polynomial solution was obtained by Solov’ev [3]. Incidentally, this solution is identical to Hill’s solution for the Stokes stream function of a spherical vortex [4], which long preceded the conception of the idea of a tokamak in the 1950’s. Solutions expressed in special mathematical functions were also found for a quadratic dependence [5], [6], [7] and a mixed linear and quadratic dependence [8], [9], [10], [11]. The most general linear solution is obtained in Refs. [12], [13]. More exotic dependencies yielding analytical solutions are considered in Refs. [14], [15]. Convenient homogeneous solutions are provided in Refs. [16], [17].
In the presence of toroidal rotation, one additionally has to specify how the angular rotation frequency depends on ψ. Assuming that the ratio between the angular frequency squared and the static temperature is constant and that the magnetic surfaces have constant temperature or entropy, analytical solutions were found for a linear [18] or a mixed linear and quadratic dependence [19] of the pressure and the toroidal magnetic field on ψ. For isothermal magnetic surfaces, solutions for a quadratic or mixed dependence were already obtained in Refs. [20], [21], or in spherical coordinates in Ref. [22]. For more general flows, solutions have been found assuming plasma incompressibility, see e.g. Refs. [23], [24].
All existing analytical equilibrium solutions including toroidal flow, make an assumption on which quantity is constant on the magnetic surfaces. In the present work, we assume a general parametrization of this quantity from the outset. The freedom of this choice is retained in the equilibrium equation and the subsequently derived class of analytical solutions. These analytical solutions are generalizations of those of Ref. [18] and without rotation reduce to the polynomial Solov’ev solution [3]. For numerical stability calculations, an accurate numerical representation of the equilibrium is required. In order to test the accuracy of the equilibrium solver, the Solov’ev solution is frequently used [25], [26], [27], [28], [29].
Various equilibrium codes that include toroidal rotation exist. Most of these codes assume isothermal magnetic surfaces [30], [31], [32], [33], [34], [35], [36] while some assume flux surfaces of constant density [37], [38]. The equilibrium code FINESSE [39] allows in case of purely toroidal flow the freedom to choose either the temperature, the density, or the entropy to be constant within the magnetic surfaces [40]. Only rarely, analytical solutions have been used to test an equilibrium code including toroidal flow [35]. In the present work we will use the derived analytical solutions, to test the convergence behavior of FINESSE.
The Solov’ev equilibrium solution [3] is also frequently used for stability calculations as a standardized reference equilibrium [27], [28], [41], [42], [43]. For stability calculations including toroidal flow, less well-defined reference test cases are generally used to benchmark numerical codes. Again, the derived analytical solutions would be ideally suited to serve as a standardized reference equilibrium including rotation.
Also in the analytical stability calculations in this work, the freedom to specify which quantity is constant on the magnetic surfaces of the equilibrium is retained. An analytical expression is derived for the low-frequency continuous Alfvén spectrum of a large aspect ratio tokamak plasma, including the effects of compressibility and toroidal rotation. This expression still allows for the choice of which quantity is constant on the magnetic surfaces of the equilibrium, and as such generalizes the results of Refs. [44], [45], [46], [47] for isothermal flux surfaces and the recent result of Ref. [48] for magnetic surfaces of constant density.
The lowest-frequency branch of the derived expression is shown to be equal to the Brunt–Väisälä-frequency associated with the centrifugal convective effect. For a tokamak plasma with magnetic surfaces of constant density, the convective instability destabilizes the continuous spectrum. Simulations show how instability spreads from a localized part of the tokamak to almost the entire plasma for sonic rotational velocities. For a tokamak with isothermal magnetic surfaces, rotation induces a gap in the continuous spectrum. Good correspondence between the continuum frequencies from numerical simulations and the derived analytical expression is obtained, showing its usefulness for the validation of stability codes.
Section snippets
Magnetohydrodynamic equilibria
In this section we will introduce the considered geometry and basic equations that are used in the analysis of the subsequent sections. Although this work pertains to toroidal rotation, in order to provide some context and introduce concepts that will arise in the stability analyses, Section 2.3 is devoted to more general equilibrium flows.
Analytical equilibria with toroidal flow
The extended Grad–Shafranov equation introduced in the previous section will be solved analytically under simplifying assumptions. To increase the versatility of the thus obtained analytical solutions, various homogeneous solutions that may be added are discussed. Expressions for equilibrium properties like the Mach number, plasma beta, and the safety factor are subsequently derived, allowing control over the characteristics of the solutions. Finally, alternative analytical solutions are
The Brunt–Väisälä frequency
The gravitational stability of an adiabatically displaced air parcel in the atmosphere depends on the vertical stratification of pressure and density. Something similar holds for plasma under the influence of the centrifugal force. By analogy, in Section 4.2.2 expressions are derived for the oscillation frequency of plasma confined to the circular magnetic surfaces of a large aspect ratio tokamak.
Waves and instabilities
Up to now we have been discussing plasma configurations in which the Lorentz force, the pressure force, and the centrifugal force are balanced. We do not yet know if these configurations are stable against slight deviations from equilibrium. A small perturbation may create a restoring force, resulting in a wave within the plasma. The resulting force inequilibrium caused by the perturbation may however also amplify the original perturbation, leading to instability.
First, the Eulerian approach to
Numerical calculations
In this section we numerically analyse the waves and instabilities arising in the analytical equilibria of Eq. (21). The discrete spectrum of global modes and instabilities arising in these equilibria have been discussed in Refs. [70] and [64], respectively. Here, we will focus on the low-frequency continuous MHD spectrum. In particular we will look at the continuum frequencies given by Eq. (77a). We consider the cases ζ = γ/(γ − 1), ζ → ∞, and ζ → 1 corresponding to isentropic (S(ψ)),
Conclusion
The axisymmetric stationary ideal MHD equations are reduced to an extended Grad–Shafranov equation Eq. (18) that allows one to specify which thermodynamic quantity is constant on the magnetic surfaces. Analytical equilibrium solutions (21) have been derived that still contain this freedom. An arbitrary number of vacuum solutions may be added, allowing control over characteristics like the aspect ratio, ellipticity, and triangularity of the equilibrium. Expressions for the Mach number, average
Acknowledgments
We gratefully acknowledge the valuable comments received from J. W. S. Blokland on a draft version of the manuscript. This work, supported by NWO and the European Communities under the contract of the Association EURATOM/FOM, was carried out within the framework of the European Fusion Program. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
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