Optimal control of tokamak and stellarator plasma behaviour

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Abstract

The control of plasma transport, laminar and turbulent, is investigated, using the methods of scaling, optimal control and adaptive Monte Carlo simulations. For this purpose, the asymptotic behaviour of kinetic equation is considered in order to obtain finite-dimensional invariant manifolds, and in this way the finite-dimensional theory of control can be applied. We imagine the labyrinth of open doors and after applying self-similarity, the motion moved through all the desired doors in repeatable ways as Brownian motions. We take local actions for each piece of contractive ergodic motion, and, after self-organization in adaptive invariant measures, the optimum movement of particles is obtained according to the principle of maximum entropy. This is true for deterministic and stochastic cases that serve as models for plasma dynamics.

Introduction

Turbulent transport in tokamak is very challenging scientific problem. It is an intrinsically nonlinear problem, involving a wide range of space and time scales. Computer simulations are a valuable tool in tracking these problems, and are used in combinations with experiments. Our goal is to develop quantitative predictions of tokamak turbulence, and to use them in design of future fusion reactors.

The temperature is defined only for a system in equilibrium; this does not prevent us, of course, from extending the concept to nonequilibrium states. This can be done by either assuming a local equilibrium or, using some basic relation that includes the equilibrium, to define temperatures of nonequilibrium states as well.

Singularly perturbed systems are those whose order is reduced when the parasitic parameter is neglected. The main purpose is ill-conditioning resulting from interaction of slow and fast dynamic modes. This kind of system, for any fixed ε, can be used to represent many important physical systems subject to random failures and structure changes. Heat is lost from fusion devices via turbulent transport caused by convective instabilities. These instabilities are fine scale (centimeter eddy sizes) and fast time-scale (microsecond).

There is an urgent need to take the step from microscopic turbulence to the macroscopic transport. To achieve fusion in magnetically confined plasma, it is necessary to hold plasma of tens of meters cubed in a stable configuration for many seconds. Inside this plasma are physical processes in a vast range of space and time scales.

Section snippets

Transport theory and scaling

The kinetic equations contain both wavelike and diffusion like characteristics in various time scales. The following trick is used—modification of the faster time-scales phenomena, making use of the assumption that the slower time-scale motion depends only on the asymptotic response of the faster time-scale motion. Long time-scale kinetic modelling of the interaction requires techniques to reduce noise and average over short times. The sampling resolution is chosen adaptively. A probability

Complexity function

The scheme of calculation of complexity function [3] is similar to one used for calculation of the Lyapunov exponents, except for some details. We consider the complexity function and the exit time distribution. We obtain the fractal space–time properties of trajectories. It can be noticed that specific features of Hamiltonian systems are to have chaotic dynamics strongly nonuniform in phase space and strongly intermittent in time. Within such situations the polynomial complexity and anomalous

Existence of invariant manifolds

For equations in the infinite dimension there are finite-dimensional locally invariant submanifolds. They have to map domain into itself and generate local flows. We can calculate the tangent spaces on the canonical basis, and we obtain the derivative that admits local flows or admit local semiflow. A preliminary study of the microinstabilities, which may be responsible for the turbulent transport, is provided using theory—based scalings for the scrape-off layer width. From this the design

Finite dimensional optimal control

We are interested to having controllability within a neighbourhood of an equilibrium point. The domain of attraction of a control set is defined. We can construct the union of the basins of attraction of all the equilibrium points. Modern computers have begun to grasp this complexity, but the available spatial resolution is still too low for a full model. We need to resolve the scale on which smoothing occurs—just a few millimetres and we also need to determine the initial conditions.

We have

Approach to ergodicity in Monte Carlo simulations

Alternative method for simulation of transport is the Monte Carlo method, which can be used together with scaling process. A goal of Monte Carlo simulations in statistical mechanics is the calculations of ensemble mean values of thermodynamic quantities. Ensemble mean values are multidimensional integrals over configuration space, where f(x, v) is the probability of finding system in the state defined by (x, v). Monte Carlo simulations generate a sampling of configuration space (xk, vk) by the use

Statistics of poincare recurrences

For a uniformly hyperbolic map on the torus, the decay low is exponential. Instead of a system, we consider a skew integrable map of the cylinder, which is not ergodic and has zero entropy. This map describes a shear flow, and has a local mixing property. We rigorously prove that the statistics of first return is of polynomial type around the fixed points. This result can be extended to quasi-integrable area preserving maps, such as the standard map for small coupling. For all aperiodic ergodic

Conclusion

In this paper, the problem how to describe optimal work of a future tokamak and a stellarator is solved. For this purpose, we imagine that the trajectories of plasmic flows must go toward several open doors, and when in given situation they have uniform shape after self-organization we can expect the optimisation of the results. Also, design of future tokamaks demands the possibilities of work in regime of adaptively recurrence, and in such a way is obtained circle description of future

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