Numerical simulation and optimal control in plasma physics. With applications to Tokamaks.

*(English)*Zbl 0717.76009
Wiley/Gauthier-Villars Series in Modern Applied Mathematics. Chichester etc.: Wiley; Paris: Gauthier-Villars. xvi, 363 p. £49.95 (1989).

The subject of the monography is grouped into seven chapters. In the first five chapters the stationary axisymmetric equilibrium of the tokamak plasma is studied. Namely: the boundary value problem (BVP) pertaining to the tokamak configuration is based upon the so called Grad- Shafranov equations of magnetohydrostatics (Chapter I), the importance of the problem of free boundary of the plasma ring and the interface conditions is discussed. The BVP is then formulated in terms of differential operator equations which in general are nonlinear ones. The linearization procedure is studied and the ways of the numerical solution procedures are discussed. Results are shown for some important tokamak prototypes: TFR, JET, TORE Supra and INTOR.

Then (Ch. II) the static control of the plasma boundary by external current is studied, which aims at the causal influence on the plasma radial position, height and shape by means of the variation of additional current distribution for which the possibility is built in the apparatus. A general approach is formulated in terms of optimal control with control parameters, equation of state and cost-function. Some results are shown for the prototypes. Next (Ch. III) a simplified model study is shown (tokamak without iron, and plasma current density being a linear function) to study the mathematical existence of the solution. Determination of the equilibrium solution is the aim of the next investigation (Ch. IV) for which the current distribution in the magnetizing coils is modelled by a given function. The equilibrium solution branches and the stability of the horizontal displacements are considered for some machines. The next step (Ch. V) is devoted to some plasma diagnostic procedures through magnetic measurements which is reduced to an ultrarapid determination of the plasma boundary. Some details of the application possibilities of this diagnostic tool are discussed for the prototypes.

Then the evolution of the equilibrium is studied (Ch. VI) at the diffusion time scale. On the basis of the transport equations a system of diffusion equations is derived to obtain a law of time evolution. Their relation to ideas of classical and neo-classical diffusion model is discussed and other possibilities are mentioned. The relative BVPs are formulated and the numerical methods of solution are discussed with applications to the prototypes. The last item (Ch. VII) is devoted to high aspect-ratio circular plasmas of arbitrary cross section, their stability and control with respect to horizontal displacements. The virial theorem method of V. D. Shafranov is the base of this research.

The monography is essentially a mathematical study of the technical and physical state of art, an essay of synthesis as regards the formulation and practical solution methods of the tokamak problems. Besides the own results of the author, references to an extensive literature are given.

Then (Ch. II) the static control of the plasma boundary by external current is studied, which aims at the causal influence on the plasma radial position, height and shape by means of the variation of additional current distribution for which the possibility is built in the apparatus. A general approach is formulated in terms of optimal control with control parameters, equation of state and cost-function. Some results are shown for the prototypes. Next (Ch. III) a simplified model study is shown (tokamak without iron, and plasma current density being a linear function) to study the mathematical existence of the solution. Determination of the equilibrium solution is the aim of the next investigation (Ch. IV) for which the current distribution in the magnetizing coils is modelled by a given function. The equilibrium solution branches and the stability of the horizontal displacements are considered for some machines. The next step (Ch. V) is devoted to some plasma diagnostic procedures through magnetic measurements which is reduced to an ultrarapid determination of the plasma boundary. Some details of the application possibilities of this diagnostic tool are discussed for the prototypes.

Then the evolution of the equilibrium is studied (Ch. VI) at the diffusion time scale. On the basis of the transport equations a system of diffusion equations is derived to obtain a law of time evolution. Their relation to ideas of classical and neo-classical diffusion model is discussed and other possibilities are mentioned. The relative BVPs are formulated and the numerical methods of solution are discussed with applications to the prototypes. The last item (Ch. VII) is devoted to high aspect-ratio circular plasmas of arbitrary cross section, their stability and control with respect to horizontal displacements. The virial theorem method of V. D. Shafranov is the base of this research.

The monography is essentially a mathematical study of the technical and physical state of art, an essay of synthesis as regards the formulation and practical solution methods of the tokamak problems. Besides the own results of the author, references to an extensive literature are given.

Reviewer: I.Abonyi

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

65K10 | Numerical optimization and variational techniques |