Development of a genetic algorithm for transport studies in high-temperature laboratory plasmas

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Università di Pisa

Dipartimento di Fisica Enrico Fermi

Corso di Laurea Specialistica in Scienze Fisiche

Development of a genetic algorithm for

transport studies in high-temperature

laboratory plasmas

Candidato: Relatori:

Manuel Scotto d'Abusco Prof. Francesco Pegoraro

Dr. Fabio Sattin Correlatore: Dr. Lorella Carraro

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Joy in looking and comprehending is nature's most beautiful gift

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Chapter 1 Introduction

Nuclear fusion is a fundamental physical process which takes place in the stars. In the hot core of a star the atoms are completely ionized, i.e they are in a plasma state, and at temperature of the order of 10 million Kelvin degrees, lighter nuclei overcome Coulomb repulsion yielding nuclear fusion reactions. Thanks to the released energy, the stars heat up counteracting their gravitational collapse and a stationary phase is reached.

On the Earth, in order to exploit fusion reactions as a controlled source of energy, we attempt to conne the plasma by magnetic elds and heat it up until a high enough temperature is reached. Several years of attempts have highlighted numerous problems and a very fascinating physics.

In this chapter I will give a general introduction about nuclear fusion, explaining what is a plasma and which are the features of a magnetic connement machine. Then I will underline some problems relative to fusion machine, especially focusing on the trouble of impurity transport and their formation. Lastly I will give some experimental achievements for an RFP fusion device.

1.1 The termonuclear fusion

The nuclear fusion, as it takes place in the star core, starts with four protons which originate an alfa particle turning the mass dierence between reagents and nal products into kinetic energy of the reaction products. In laboratories on the Earth, it is preferred to use dierently reagents, because proton-proton reaction is characterized by a small cross section that makes these reactions possible only on the star, where the large abundance of hydrogen and the possibility to maintain the plasma conned over log time scales, compensates the low reaction probability.

The choice of the most suitable reagents is determined by some aspects such as the Coulomb repulsion, which depends on the product of the charge of the reagent nuclei, and the trend of the cross section as a function of the reagent energies. These characteristics promote the use of lighter nuclei which have small charge. Furthermore fusion of light nuclei is an exothermic process so it produces energy converting a part of their mass in energy as given by the Einstein relation: E = mc2_{; conversely fusion of heavy nuclei}

(elements with mass above56_{F e}_{) is an endothermic process.}

It can be concluded that the most convenient reaction is that between deuterium and tritium:

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2_{D +}3_{T →}4 _{He(3.56M ev) + n(14.03M ev)} _(1.1)

that for temperature between 20 and 100 keV ,those at whom a nuclear reactor will work, has a cross section two order of magnitude higher than other possible reactions. This is displayed in gure 1.1.

Figure 1.1 Cross section for typical fusion reaction: deuterium-deuterium(D-D),deuterium-tritium (D-T), deuterium-helium(D-3_He). 1 barn = 10−28_m2. Image from [19]

Unfortunately tritium is a radioactive element with an lifetime of 12 years so it is not found in nature. However it can be produced by the following reaction:

6_{Li + n →}4 _{He(2.10M eV ) +}3_{T (2.70M eV )} _(1.2)

These resources are potentially inexhaustible, indeed deuterium can be extracted from water where it is present in the quantity of 33g for ton, while the estimation of lithium reserves are sucient to satisfy humanity's energy demand for millions of years.

1.1.1 Plasma parameters

At temperature values near those required for nuclear fusion the matter is completely ion-ized, i.e. it is in a plasma state. The dominant aspect of a plasma is the collective behavior of its particles. Each one of them interacts with the other particles through long range forces. The length scale that characterizes the range of action of charge inhomogeneities is given by the Debye length that is dened as:

λD =

r ₀kBTe

nee2

(1.3) where Te and ne are the temperature and numeric density of electrons respectively,

while kB is the Boltzmann constant. Over scale lengths larger than the Debye length the

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The time scale over which the plasma neutralizes local charge unbalance is determined by the inverse of the plasma frequency

ωp =

s n2

e

me0 (1.4)

Over this time scale, the electric forces rearrange the spatial charge density in order to return to an equilibrium state. In fusion plasmas the typical values for the Debye length stay in an interval between the micrometer and the millimeter, while typical values for the plasma frequency stay between the gigahertz and the terahertz.

For time scale and spatial scales above the two quantities dened above, the plasma can be described as a neutral uid giving rise to a magnetohydrodynamics model, of which I will use some equations in the next chapters.

1.2 The magnetic connement concept

Fusion reactions take place only for specic conditions of density and temperature that allow to overcome the Coulomb barrier. Magnetic connement is based on the fact that, in the presence of a stationary magnetic eld, the motion of the particles can be decomposed in a motion parallel to the magnetic eld and in a circular motion in perpendicular direction respect to magnetic eld, the latter characterized by the cyclotron frequency and the Larmor radius dened as:

Ωc =

q|B|

m rL= mv⊥

q|B| (1.5)

where v⊥ is the velocity of the particle in direction perpendicular to the magnetic eld.

These quantities depend on the mass and on the charge of the particles thus protons and heavier ions will make orbits with radius bigger than those of electrons, but with lower frequencies.

The global motion of the particle is a helical curve with it axis that coincides with the eld line. The fact that particles follow a motion in direction parallel to the eld line has led to the conclusion of using closed magnetic eld line conguration . For this reason, toroidal conguration systems have been adopted. However the presence in torus of non uniform magnetic elds or the gradients due to the curvature, give rise to drift motion in radial direction that increases the plasma interactions with the wall of the chamber. The solution consists in the use of helical magnetic elds so that the drift motion points once in direction and once towards the other compensating the eects.

1.2.1 The ignition criterion

The aim of the nuclear fusion is that of obtaining from fusion reaction more energy than that used to conne the plasma. Unfortunately the connement doesn't last forever be-cause there are mechanisms of loss of energy, due to for example to the bremsstrahlung radiation of accelerated particles, and mechanisms of transport that decrease the charac-teristic connement time of the particles and energy. It would be helpful that a part of the produced energy be invested in the reduction of this losses. For example the α-particles produced from the reaction described in equation (1.2),which are charged particles, could

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Figure 1.2: Representation of the helical motion of the particles in a toroidal system. Image from [1]

be conned and used in order to transfer a part of the fusion energy to plasma; while the energy stored in kinetic motion of neutrons will inevitably come out of the magnetic chamber because they can't be conned, but can be used from an external system to convert thermal energy into electric energy.

The ignition criterion gives a mathematical relation that combines temperature T, density n and connement time τE expressing that the fusion reactor is self-sustained

with the energy produced from the α-particle. This is called also as the Lawson criterion and is given by:

nT τE > 3 · 1021m3keV s (1.6)

This means that for a typical fusion machine with n ∼ 1020_m−3_{, T ∼ 20keV , we}

should have a connement time of τE ∼ 1s. The density is conditioned from the material

of the rst wall that doesn't support high power density, while temperature is forced by the trend of the cross section which for the used fusion reagents has a maximum for T = 20keV (see gure 1.1).

In the gure 1.3 are displayed some value for the triple products of the equation (1.6) reached by fusion devices in the past years. The aim of the fusion research remains the ignition of the plasma. This goal could be approached with ITER, whose technical parameters indicate that the energy produced will be 10 times higher than the one used.

1.2.2 The Reversed Field Pinch (RFP) conguration

In this subsection I will concentrate on a particular type of magnetic conguration, called RFP, which is the one on which I have developed my thesis work. In particular I will underline some aspects that distinguish it from the most common tokamak conguration. Both congurations have an axisymmetric toroidal geometry, and are characterized from magnetic elds that have both a toroidal component and a poloidal component. The former is produced by the current which ows in external coils that wrap in poloidal direction around the torus, while the second is produced by a plasma current that ows in toroidal direction. The sum of the two components lead to helical eld lines that twist on nested magnetic surfaces. We can dene a parameter, called safety factor, that quanties the number of toroidal and poloidal turns which a magnetic eld line completes before rejoining with itself. It expression is given in cylindrical geometry by:

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Figure 1.3: Typical nT τE values joined by past experiments. Image from [1]

q(r) = rBφ(r) RBθ

(1.7) If q is a rational number of the type m/n, with m and n two integers, the eld line rejoins itself after m toroidal turns and n poloidal turns, while if q assumes irrational value the eld line will ergodically cover the entire toroidal surface.

The rst dierence between tokamak and RFP appears in the magnitude of the toroidal and poloidal magnetic component. Tokamak has a toroidal eld one order greater than the poloidal one, while in RFP the two components have comparable magnitude; furthermore in the outer region of the plasma, the RFP toroidal eld reverses its sign. In gure 1.4 it is displayed the axisymmetric conguration of an RFP and the radial prole of the magnetic eld components.

The poloidal eld is produced from the plasma current IP, so in order to produce

poloidal elds of strength comparable with the toroidal one, in a RFP this current must be very high. This is another dierence with tokamak conguration which needs only a small value of IP. Such high value of IP allows reaching high temperature by using ohmic

heating only, without the need for additional heating methods (radio frequency, neutral injection), which are instead required in the tokamaks.

Tokamak and RFP have dierent proles of the safety factor given by the equation (1.7), caused from the reversal of the torodial eld at the edge of the plasma. This is displayed in gure 1.5.

The magnetic eld reversal is observed only for some value of the Pinch parameter dened as:

Θ = Bθ(a) hBφi

= µ0aIP

2Φ (1.8)

where a, Φ are respectively the minor radius of the torus and the ux of the magnetic eld, Bθ, Bφ the poloidal and the toroidal eld components, and h i represents the average

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Figure 1.4: radial prole of the toroidal and poloidal magnetic eld components. Image from [1]

Figure 1.5 Radial prole of the safety factor in tokamak and RFP conguration. m and n are the toroidal and poloidal mode number respectively. This gure shows how the safety factor changes as a function of the mode number n with m xed to 1. Image from [1] over a poloidal section. Its typical values for the RFP are greater then one while in tokamak, where we have a small poloidal component, it assumes values lower than one.

1.3 Multiple helicity state (MH) and quasi-single

he-licity state (QSH)

The most important feature of an RFP conguration is the presence of Multiple Helicity states (MH) and Quasi Single Helicity states (QSH). These states are due to small per-turbations of the magnetic eld which can be described in cylindrical coordinate as the sum of spectral modes of the type:

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Bj,m,n(r)expi(mθ +

n Rz)

(1.9) where θ is the poloidal angle, z the axial coordinate, R the length of the cylinder axis and j is a subscript that indicates one of the component of the magnetic eld. m, n are the mode numbers that take in consideration the poloidal and toroidal periodicity.

MH states are characterized by a wide spectra of helical perturbations each one with of dierent values of m and n. These modes interact among them inducing a turbulent behavior characterizing the system. For example these modes inuence the spatial distri-bution of the eld lines allowing some to move freely from the center regions of the plasma to the outermost regions mapping the whole space. This causes rapid particle transport degrading very quickly the connement properties. The magnetic conguration is chaotic and no magnetic surfaces exist, the plasma energy connement is very low. These states are experimentally observed for low value of IP

QSH states are characterized by a reduction of the magnetic chaos and the predomi-nance of a single mode above the others. From the recent experiment on RFX-mod (an RFP fusion device situated in Padova), has been observed that these states are regulated from the plasma current IP. It was also noted that increasing IP the system spends ever

much time in the QSH transient state, and the secondary modes reduce further their amplitude. In this state are also observed strong transport barriers characterized by high values of the gradient of the electron temperature that enhances the connement proper-ties in the plasma core. In the inner region of the plasma it has been experimentally found an electron temperature two times greater than that in MH state. In the gures 1.6 a and 1.6 b are respectively shown typical RFX-mod mode spectrum and temperature proles for a MH and a QSH state.

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Figure 1.6: In gure (a) are shown the spectrum of modes in MH and QSH states, in (b) is shown the radial prole of the electron temperature in MH(white dots) and QSH (black dots) states. Image from [1]

1.4 The RFX-mod machine

In this subsection I would give some technical features of the RFX-mod machine. It is the biggest experimental fusion machine that exploits a RFP magnetic conguration, and it

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is situated in Padova (Italy).

It fundamental geometrical features and technical specications are described in the list below:

• it is constituted by a torus with major radius equals to R = 2 m, and minor radius equals to a = 0.5 m, so it is characterized by an aspect ratio equal to R/a = 4; • the inner surface of the connement chamber is covered by with 18 mm thick

graphite tiles. The material is chosen for its low atomic number and good me-chanical properties, in order to reduce high radiation losses due to impurities in the plasma,

• the connement chamber is surrounded with a copper shell, that assumes an impor-tant role to control the plasma stability;

• it is surrounded by 48 poloidal coils that generate a toroidal magnetic eld with maximum intensity equal to 0.8 T ;

• it has a system of 192 coils for feedback control of the magnetic eld, which have the task of controlling the outermost surface of the plasma in order to reduce interaction of the plasma with the wall;

• typical values for the electron temperature, density and plasma current are: n = 5 · 1019 m−3, Te = 1.3 keV, IP ≤ 2 M A

1.5 Why impurity transport

The plasma contained in a fusion machine is not completely constituted by ions of the basic fuel (generally deuterium) and electrons but is often contaminated by impurity ions produced by interaction with the wall of the chamber. The observed impurity elements are classied using criteria of dierent type 1_{. For example the are usually divided in}

light impurities and in heavy impurities on the basis of their atomic number. The rst are completely ionized in the core of the plasma, where the temperature is high, therefore they don't emit central line radiation. The second remain partially ionized also in the innermost plasma regions because the ionization potential of their last hydrogen-like ion is much higher than the central electron temperature.

Another criterion of classication is according the mechanism of production of impu-rity elements, that usually consists in dividing desorbed and eroded impurities. Typical desorbed impurities are oxygen, carbon, nitrogen, and chlorine that are usually produced from thermal desorption, desorption due to particle and photon bombardment and chem-ical reaction that release gaseous contaminants. The concentration (ratio with respect to the main gas ions) of desorbed impurities usually are around 1-3 %.

Eroded impurities, instead, are the elements that form the metallic component of the plasma wall such as Ti, Fe, Cr and Ni. They are usually less abundant, and are more easily produced through plasma ions sputtering, that striking the wall ejects particles from it.

Once these impurities have been released they start their path into the plasma and are progressively ionized to higher charge states by electron impacts, as soon as they

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penetrate in inner regions of the plasma where the electron temperature increases. Radial velocity of this movement perpendicular to the magnetic eld lines is much smaller than the thermal velocity of the parallel motion, thus a uniform spread over magnetic surfaces is rapidly reached.

Impurity ions with low degree of ionization are mainly found in the periphery because they need only low temperature, on the other hand in inner region we nd ions with high degree of ionization. The nal result is a shell structure of the ionized charge states, the position in this shell being determined mainly by the electron temperature and density prole. In stationary condition an equilibrium is reached where the shell structure is maintained by continuous recycling impurity ux.

Impurities have both benecial and harmful eects on the plasma that justify exten-sive experimental and theoretical works in order to better understand their transport, production, and removal techniques. One of the most important eects are the radiation losses,that are an important component to consider in the energy balance. Heavy impu-rities are the most dangerous ones because, since are incompletely stripped also in the plasma core, radiate strongly, and above a certain concentration all the input power will be radiated away, so it is important to keep heavy impurities away from the central region of the plasma.

On the other hand light impurities have also benecial eect, having low ionization potential, they are situated mostly in the periphery of the plasma, where their radiation help the plasma energy transfer, avoiding damage to the wall of the chamber, and also the release of other heavy impurities. Furthermore in the plasma core they are completely stripped, so they radiate not much, only bremmstrahlung radiation.

Lastly impurity ions have direct eect on transport perpendicular to the magnetic surface, determining the changes in the transport coecients, but this is an argument of which I will widely discuss in the next chapters of this thesis.

In general it is experimentally observed that increasing the impurity concentration the plasma stability decreases and instabilities develop at lower vale of IP.

1.5.1 Impurity removal

Impurity accumulation is dangerous for the plasma stability, so some techniques are de-veloped to reduce and to remove impurity content 2_.

Before to beginning a new experimental session,the wall is pretreated to eliminate any remaining impurity residues. The procedure is called baking: the surface of the wall is heated at high temperature where impurity residues leave the wall by thermal desorption. There exist also some methods that work when the machine is in operation. One of these is the use of a limiter, i.e. a piece of material xed on the wall that extends inwards into the outermost regions of the plasma. In this part the particles have overcome the last closed magnetic surface,so they diuse slowly, but the fast parallel motion leads them to crash with this piece so as to avoid ending up on the wall. Unfortunately its not very ecient because impurities from the limiter are released.

To solve this inconvenience a completely dierent concept to reduce the incoming impurity ux has been developed, this is the magnetic divertor. The divertor concept is based on the rearranging of the magnetic eld in order to produce a null point in the

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poloidal eld near the edge of the plasma. In this point the boundary plasma is diverted in a separate chamber where it is neutralized and result products are pumped away, avoiding their reintroduction in the main chamber.A divertor is usually implemented using conductors concentric with the plasma current, preserving the axisymmetry of the fusion machine.

Divertors are doubly advantageous because they directly remove the impurity and at the same time they avoid further inuxes due to bombardment of the walls by recycling plasma.

The limiter and the divertor are schematically illustrated in the gure below.

Figure 1.7: schematic representation of (a) a limiter and (b) a divertor. Image from [19]

1.6 Thesis Outline

The presence of impurities within the plasma has always represented a problem for a fusion machine, but their presence is inevitable. In every fusion device there is a rst wall located on the inside of the connement chamber to shield it from the hot plasma. In RFX-mod the rst wall is made of graphite and interactions with plasma release carbon atoms. For this reason the impurities studies have become increasingly important.

What is proposed in this thesis is a model for studying the transport of impurities. Treating them like a uid, their dynamics can be described in good approximation by a diusive motion. In cylindrical geometry, appropriate for the shape of magnetically conned machines, it is formalized through the equation:

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∂nz ∂t = − 1 r ∂ ∂r r − D(r)∂nz ∂r + v(r)nz + Sz(r) (1.10)

where nz represents the density of a generic ion of charge z, and Sz is a source term

that take into account recombination and ionization processes.

The coecients D(r) and v(r), incorporate the interactions between impurities, the main gas and the magnetic eld, and assume an important role for the understanding of the transport dynamics. Dierent theories give estimates of D and v, but none reproduces satisfactorily the experimental observations. For this reason in this thesis it was preferred to use an indirect approach, i.e. to infer the most suitable prole of D and v, from exper-imental measurements. From the mathematical point of view the procedure considered is an optimization problem, in which we look for a set of parameters that minimize the distance between a set of experimental data and the simulated signals.

A genetic algorithm is developed which has the task of nding the best spatial prole for D, v that reproduces experimental measures of x-ray brightness, line-emission bright-ness, electron density and total radiated power of carbon and oxygen impurities. D and v proles are modeled through a set of real numbers that represents the values that they assume in the radial point of the mesh used by the transport code. D, v represents the parameters of the system and, once generated by the genetic algorithm, are fed to the transport code which computes the simulated counterpart of the experimental data. Lastly a subroutine is built which has the task of computing the agreement between experimental and simulated data. This thesis proposes a method to connect each other three elements: (1) a set of experimental data of RFX, (2) an one dimensional transport code for carbon and oxygen impurities, (3) an optimization algorithm, i.e the genetic algorithm.

The results of the application of this method to some reference cases are presented through the thesis. A series of simulation are performed with the purpose of optimizing dierent kind of physical quantities and the results of comparison between the D,v proles obtained from the reconstructions and their theoretical predictions will be presented.

In some cases the reconstructions will be enough robust to be used as a starting point for an elaboration of a transport theory.

The thesis has been structured in this way:

Chapter 2 : this chapter is devoted to the presentation of the fundamental features of an optimization algorithm. Then the development and testing of a genetic algorithm is presented step by step;

Chapter 3 : in this chapter the simulation model is presented giving some information about its features, then it is explained how the genetic algorithm interfaces with it. Furthermore some information about the subroutines that calculate the accordance with experimental data are given;

Chapter 4 : a summary of theoretical transport models is presented, and for each one the calculation of diusive and convective coecients is carried out;

Chapter 5 : the results of the analysis performed optimizing dierent kind of physical quantities are presented. Then the obtained transport coecients will be compared with the theoretical predictions of previous chapter;

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Chapter 2 The Genetic Algorithm

Genetic Algorithms (GAs) are heuristic approaches to solve problems of global optimiza-tion over. In general, the task is to optimize certain properties of a system by choosing the system parameters. These algorithms are of stochastic type and have been developed by analogy with natural phenomena. They are based on an initial population of individuals that can interact and modify their characteristics in order to better survive in the envi-ronment, this is the simple rule of 'natural selection'. During my master thesis I developed a genetic algorithm to optimize transport reconstruction. In this chapter I will explain some fundamental characters of GA and how to use them in a proper way.

2.1 The problem of Global Optimization

The standard approach to an optimization problem begins by designing an objective function that can model the problem's features while incorporating any constraints. The aim of the optimization is to nd the point in the model parameters space which gives rise to the optimum of the assigned objective function, usually called 'tness'. Global optimization algorithms have a wide range of applicability, for example they can used in searching the global maximum or minimum of a mathematical function, or for determining some quantities of a physical system which are left as free parameters. Just this one will be the kind of problem on which we well concern in this work, thus throughout this thesis with tness function we will mean the dierence between the measured value for the analyzed problem in a point of the parameter space and that predicted by the model. Lower is this gap and better is the used model, so the problem will be reduced to a minimization of the tness function.

Before explaining how a genetic algorithm works I list some of the properties that should be presented in a good algorithm of optimization 1

1. ability to handle non-dierentiable, non linear and multimodal tness function; 2. parallelizability, to overcome computational expensive tness function;

3. few control variables to steer the minimization, robust and easy to use;

4. consistent convergence to the global extremum in consecutive independent trials;

1_{The listed properties are the same given in [2], pag. 342}

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2.1.1 The structure of a Genetic Algorithm

As already said GA is a powerful tool for the search of global maxima or minima of a real function. It starts from an initial population of individuals that are possible candidate solutions, and evolves these randomly exploring the solution space. Finally it selects only the better solutions, namely those with lower values of tness function. Precisely it follows the scheme shown in gure 2.1:

Figure 2.1 Flowchart of GA Initialization

GA starts from a xed number of individuals NP, each one consisting of an array of real numbers Xi,0 = {x1,i,0, ...., xj,i,0}. Every individual is made of as many components as the

number of parameters n to be determined.

In this rst step GA randomly generates each component j of the NP individuals within a range Min, Max given as input from users:

xj,i,0 ∈ Minj , M axj

j = 1, ...., n i = 1, ...., N P

where the sub-scripts are the respectively component, population and generation in-dices.

Mutation

In this step a vector, that we will call target vector, is chosen from the population of individuals and every component of this vector is modied with the addition of a ran-dom perturbation. In this way the initial candidate solutions expand as a cloud and ex-plore other possible better solutions. The vector Vi,G+1 = {v1,i,G+1, ...., vj,i,G+1} is usually

generated adding the dierence between two or more vectors of the existing population Xi,G= {x1,i,G, ...., xj,i,G}, but there are also other formulae that I list below:

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2. Vi,G+1 = XBest,G+ F × (Xr1,G− Xr2,G)

3. Vi,G+1 = Xi,G+ F × (XBest,G− Xr3,G) + F × (Xr1,G− Xr2,G)

4. Vi,G+1 = XBest,G+ F × (Xr1,G− Xr2,G) + F × (Xr3,G− Xr4,G)

5. Vi,G+1 = Xr1,G+ F × (Xr2,G− Xr3,G) + F × (Xr4,G− Xr5,G)

This operation is repeated for each individual i; r1, r2, r3, r4, r5 ∈ {1, ..., N P } are

indices randomly chosen so that r1 6= r2 6= r3 6= r4 6= r5 6= i, it is in this sense that

GA is a stochastic method. F is a control parameter that x the mutation strength; the higher, the more changes we will have in the j component of the i individual. It is a scalar number chosen of the order of unity because if it is zero the vector components remain unchanged while if it assumes very high values the new components will be completely dierent. For my purposes F usually stays within the interval (0, 2]. XBestis the individual

with the best tness found in the G'th generation. In the gure 2.2 there is a representation of the rst mutation strategy.

Figure 2.2 An example of two dimensional tness function with its minimum, contour lines and generated vector Vi,G+1, from [2]

Crossover

In order to improve the diversity of the perturbed vectors a crossover operation is per-formed. In this step a trial vector Ui,G+1 = {u1,i,G+1, u2,i,G+1, ...., un,i,G+1} is generated

as follows:

uj,i,G+1 = vj,i,G+1 if r ≤ Cr or j = k

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where r is a real number randomly chosen in the interval [0, 1], Cr is the crossover rate with Cr ∈ [0, 1] and k ∈ [1, n] is a random component's index chosen for each individual in order to change at least one component. This operation is repeated for each component j and individual i. The generation of a trial vector is represented in gure 2.3.

Figure 2.3 Illustration of crossover for j = 8 parameters Evaluation and Parent Selection

I assemble these two steps because they are unavoidably connected. The trial vectors Ui,G+1generated during a crossover process are evaluated by the tness function f(Ui,G+1),

then these values are compared with those of the previous generation f(Xi,G). The

indi-viduals for the next generation are selected following this rule: Xi,G+1 = Ui,G+1 if f (Ui,G+1) ≤ f (Xi,G)

Xi,G+1 = Xi,G otherwise

So only the individuals with lower tness value will survive, with the result that the next generation is composed of better individuals then the previous one.

Exit Condition

In this step the validation of a condition given as input is veried. If it's satised the algorithm stops otherwise a new iteration starts. This condition is usually set on the maximum number of iterations or if some threshold on the tness value is reached.

With this features GAs satisfy all the proprieties needed to be a good optimization algorithm. In eect they fulll requirement (1) because they are direct search method, so they search set of points around the current point looking for the one where the value of

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the tness function is lower then the value at the current point, this means that it's not important if the tness function is dierentiable or continuous.

Requirement (2) is important because many times the evaluation of the tness function can demand from minutes to hours. GAs fulll this requirement by using independently perturbed population vectors. Every population vector is perturbed by the dierence of two randomly chosen population vectors, so GAs require few input in agreement with the requirement (3).

The requirement (4) is absolutely the most important. GAs are independent from the problem to being optimized, they only search a solution in large space of candidate solu-tions, this doesn't guarantees that an optimal solution is ever found. They are constructed to nd the optimal or near optimal solution but sometimes it can happen that they could stop in local minima , especially in the case of many parameters and constraints. John Henry Holland was the rst to theorize the convergence of GA. In his book 'Adaptation in Natural and Articial Systems' in 1975 he enunciated and demonstrated the schema theorem. The latter, that in literature is often called The Holland Theorem, is the funda-mental result in the theory of genetic algorithms. For the structure of the algorithm itself, it is dicult to demonstrate its convergence from a mathematical point of view, however the Holland theorem claries the implicit principles of operation of genetic algorithms, and for this reason I report its enunciate and its demonstration in Appendix A.

2.2 Implementation and testing of a genetic algorithm

In the previous section I widely discussed fundamental features and properties of GA. In this section I will explain the reasons that lead me to choose one type of evolution strategy rather than another one. Then I will explain some issues I have faced during the development of GA and how I solved them. In a second moment I will analyze speed, preci-sion, reliability comparing two dierent evolution formulas and parent selection methods. In order to dierentiate dierent variants of GA I will use the notation introduced by Storn and Price in "Dierential Evolution A Simple and Ecient Heuristic for Global Optimization over Continuous Spaces". The notation consists of a sequence of this type: x /y / z where :

• x species the vector currently mutated that could be the vector of lowest tness function of the current population "best", or a randomly chosen population vec-tor"rand";

• y the number of vectors used for the mutation; • z the crossover scheme;

2.2.1 Specication of control parameters, troubles and

precau-tions of a GA implementation

In this section I will give more detailed informations about the GA that I developed. It is a versatile algorithm because it can easily switch between two dierent evolution strategies or even use both at the same time. Using the notation just introduced it uses a "Rand/1/Bin " and a "Best/1/Bin" evolution strategies. "Bin" is to be understood

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Figure 2.4: contour plot of the function in (2.5) for n=2. In green are displayed the evolution in the parameter space of the 50 individuals used for the optimization.

as Binomial Crossover, namely the exchange takes place only between components with the same j-index of the current vector and its mutation. This operation is repeated for each component of the target vector and it involves only two components at a time. There are other crossover schemes, another one for example is the cut point crossover "cpt", where for each vector, the index of one component is chosen in a random way, and all the components from the beginning of the vector to this index are exchanged with the respective component of another randomly chosen vector. All the others components remain unchanged.

An important aspect to stress is the choice of control parameters. In the general GA's description I introduced some parameters such as the Mutation Strength F and the Crossover Rate Cr. I have faced the problem to quantify these parameters. In my specic case I chose:

F = 0.5 + rnd · 0.4 with rnd a unif orm random number ∈ [0, 1] (2.1)

Cr = 0.9 (2.2)

With such a high crossover rate the exploration of new regions of the solution space is encouraged, while mutation force from 0.5 to 0.9 is a right compromise between evolution of the initial candidate population and convergence of the algorithm. The next problem to investigate is that of the range [Lj, Uj]where each component is generated. In the rst

generation the max and the min available values are given as input from the users. In this case the GA produces random numbers within the extremes of the interval of each

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component. The extremes are chosen on reasonable assumptions (for example physical constraints) about which is the most suitable search interval. At the beginning it could not even be known therefore it could be useful to do analysis on dierent intervals. A problem arises when during a mutation process the mutated component could overcame the interval boundaries. To prevent this problem, I adopted the following strategy:

(

min(Uj, 2Lj− ui,j,G) ui,j,G ≤ Lj

max(Lj, 2Uj− ui,j,G) ui,j,G ≥ Uj

(2.3) in this way I preserved a uniform search within the interval and I avoided accumulation at some points. With these simple but essential precautions, I successfully developed a GA which we can see in action on a bidimensional test function in gure 2.4 and 2.5.

Figure 2.5 step evolution of GA. The dots in red represent the individuals of the popula-tion. The j-components are the x and y coordinates.

2.2.2 Comparison of two evolution strategies

I have already mentioned that the algorithm I built is capable of working with two dierent mutation formulas. I briey summarize their notation to give more clarity:

(a) Vi,G+1 = Xr1,G+ F × (Xr2,G− Xr3,G) Rand/1/Bin

(b) Vi,G+1= XBest,G+ F × (Xr1,G− Xr2,G) Best/1/Bin

(2.4) In this section I will show some tests to understand which one is the most suitable strategy for my specic problem. My purpose is to understand which is faster and more precise. I evaluated the eciency of my algorithm in the global maximum research of a specic

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mathematical function. It is the same function displayed in gure 2.5 but now I analyze it in multidimensional case. f = n X i=0 cos2(2x) · exp(−(0.4 · x)2) (2.5) This function has a global maximum in zero where f = n. In multidimensional space this function has a maximum for all components equal to 0, so the function to optimize is: f itness = n − n X i=0 cos2(2x) · exp(−(0.4 · x)2) (2.6)

(a) Best/1/Bin (b) Rand/1/Bin

Figure 2.6 Comparison between GAs. Black line in the top graphics is the best solution foundend by GA. In red solutions within a range of 2-times the lowest one. Red dots of the bottom plots are the lowest tness values.

I used the tness value to decide which of the new candidate solutions will survive and will be added to the next generation.

First I tried out the precision and the speed of convergence. In gure 2.6 it is shown a comparison between Rand/1/Bin and Best/1/Bin for a population with NP = 100 and n = 15. In the graphics on the top of the gure are displayed in black the vector with lowest tness value and in red all the solutions with tness values within an interval of 2-times the lowest one, while the plots in the bottom of the gure display a report of the tness values of each vector for all the iterations.

Comparing subplots (a) and (b) it's evident that, since the time for one simulation is the same for both the algorithms, Best/1/Bin is faster than Rand/1/Bin while the latter is extremely precise. It's worth to note that in (a) all solutions converge to the same

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tness value while in (b) there is a signicative spread. This is not a problem because the worst vector, in terms of tness value, of the nal generation is however more precise of the best vector produced from a Best/1/Bin strategy .

At this point we have a fast but not precise algorithm and another very precise but extremely slow one. I did some tests following a dierent way of selecting the individuals of the new generation. Until now these are chosen by comparing the i-th trail vector with the i-th vector of the previous generation and selecting the vector of lowest tness value. In this way, even if the trial vector has a lower tness value than any other vector of the previous generation, it is however discarded. So I have decided to change the way of selecting the individuals for the new generation so as to keep all the vectors with the lower tness values chosen among the trial vectors and the vectors of the previous generation. Then I apply this type of parent selection, that I called Better Selection, to a GA that follows a Rand/1/Bin strategy. The idea is that random mutation prevents the possibility of locking in a local minimum, but completely destroys the j-th component of the current vector, and this slows down the search for the optimal solution, on the other hand if each mutated component arises from better individuals, the convergence is accelerated. I considered this way a right compromise between speed and precision, and I test it on the function in equation (2.5). The results are shown in gure 2.7

Figure 2.7 The same plots of 2.6 but with another parent selection strategy. NP = 100, n = 15

Using this upgrade, Rand/1/Bin converges quickly furthermore all individuals merge at the same tness value without spread. The lowest tness value improves and the con-vergence speed is comparable with a Best/1/Bin strategy. The strip of red lines is a bit thinner, that means that the algorithm produces nearest solutions to the optimal solution and is more stable. Unfortunately we can also note that a point is still out of the optimal value, it is only one point out of fteen but this attests to a limitation of the algorithm.

Now I have attempted to increase the interval range for each component from {Lj, Uj} =

[−5, 5] to {Lj, Uj} = [−10, 10]. The result is presented in gure 2.8

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(a) Best/1/Bin (b) Rand/1/Bin, Better Selection

Figure 2.8 Comparison between Rand/1/Bin with better selection strategy and Best/1/Bin in a wider range. NP = 100, n = 15.

is getting larger and the number of points out of their optimal values increase. Instead the eciency of Rand/1/Bin with parent selection of the type better selection, remains constant. I obtain another result testing Rand/1/Bin (Better Selection) with more indi-viduals; this is shown in gure 2.9

This gure shows that increasing the individuals of the population the algorithm meets its goal and it nds the best possible solution. The number of iterations slightly increase but this is due to the fact that I am using more vector.

The purpose of these tests was not only to verify the eciency of the algorithm but also to understand how to make it work at its best. What emerges clearly from the tests is that we need to provide a sucient number of individuals in order to have the possibility of to nd the optimal solution, and that this number increases as parameters increase. With other tests I noticed that a rule of thumb is to use a number of individuals about 10 times the number of parameters.

Using these expedients I can assert that Rand/1/Bin with parent selection of the type Better Selection is precise, fast and very stable and it is the best candidate for the reconstruction of transport coecient proles.

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Chapter 3 The Collisional Radiative Model

GA is a powerful tool that can be used to optimize physics problems. One could construct an appropriate model and minimize the dierence between theoretical results and the respective experimental measures, to determine some free parameters not specied in the model. This makes it possible to use the most general possible models and to avoid some physical constraints on unknown physical parameters. It's important remember that the dierence in the results must be minimal so we need to be careful in choosing the model. In this chapter I will present the impurity transport equation used to reconstruct convective V(r) and diusive D(r) coecient proles, the latter produced and selected by a genetic algorithm that work in parallel with the transport code. These coecients in general are functions that depend on time t, space ~x, gradients of temperature T, electric and magnetic elds. This can be point out by deriving the single-uid MHD transport equations in 1-D cylindrical geometry. One could manage MHD uid equations to obtain relation of this form:

∂Q ∂t = 1 r · ∂ ∂r rD∂Q ∂r + V Q + S(Q, r, t) (3.1) for each physical variables that are mass, energy and magnetic ux in a plasma. In S are contained sources and sink terms. The system of equations can be numerically solved but this isn't the way I followed in my work for some reasons that I will explain in later section.

I prefer to focus on a predictive model that solves the particle continuity equation with a given set of boundary conditions. The latter can be used to reproduce the impu-rity density evolution inferred from spectroscopic measurements by using the D and V coecients as free parameters in a least squares t.

3.1 Continuity Equation

Transport studies usually characterize the evolution of the impurity ux in terms of radial diusion,D, and convection, V, proles. The transport of impurity ions is described by 1-D continuity equation in cylindrical coordinates of this type:

∂ni,z(r)

∂t = −∇ · Γi,z(r) + Si,z (3.2) 29

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where Γi,zis the radial ux density of the atomic species i of charge state z={0, 1, ..., ZN}

with ZN the atomic nuclear charge. Si,r is the source representing ionization,

recombina-tion and charge-exchange processes. The impurity ux density Γi,z is expressed as the sum

of a diusive and convective term:

Γi,z = −Di∇ni,z+ vini,z (3.3)

where Di and viare the radially dependent diusion coecient and convective velocity

of the i-th atomic species and the assumption is made that they are the same for all charge states of the impurity. A negative ux implies that the impurity ions are owing towards the core of the plasma, on the other hand a positive ux implies an impurity accumulation near the edge. Di is always positive while vi can be positive or negative depending on the

physical driving transport mechanism.

Using equation (3.3), the zero-ux (Γi,z = 0) peaking factor, dened as _L1_i,z, can be

written as: 1 Li,z = |∇ni,z| ni,z = |vi| Di (3.4) where |∇ni,z| ∼ ∂ni,z

∂r and vi = v(r). The magnitude of the peaking factor informs

of the amount of accumulation. Di is always positive so the signs of the peaking factor

reveals the direction of particle convection. It's worthy to note that for zero ux, namely in stationary condition, only the ratio of vi

Di can be determined. Transient event, such as

pellet injections, gas pu or neutral beam, disturbs the impurity equilibrium (Γi,z 6= 0)

and allows Di and vi to be evaluated separately, rather then as a ratio. In later section (see

subsection 3.3) a transport code that use equation (3.2) for carbon and oxygen impurities will be explained. It is used both in stationary and non-stationary conditions where a transient phase is induced by a carbon pellet injection.

3.2 Trasport in 1-D cylindrical plasma

(forse superuo)

The D and V coecients in general are functions of time, space, gradients of tempera-ture, electric and magnetic eld. As example this is clearly shown in the derivation of uid equations for transport of density, energy and magnetic ux 1_{. For simplicity the model}

will be presented in 1-D cylindrical geometry and the viscosity will be neglected because it usually is not a dominant eect. The analysis is carried out by single-uid resistive MHD model with some simplications list below:

a) the inertial terms in the momentum equation are neglected because the characteristic time scale for transport is long compared to the ideal MHD time scale. The system slowly evolves in time through a continuing sequence of MHD equilibrium states each satisfying J × B = ∇p

b) The resistivity in the Oms's law is separated into perpendicular and parallel compo-nents: ηJ → η⊥J⊥+ηkJk to distinguish particle diusion related to η⊥and magnetic

diusion related to ηk

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c) Thermal conduction, sources and sinks term are added to the energy equation. In fusion plasma a reasonable assumption is Ti ≈ Te = T so a single energy equation

is used.

d) The physical variables in exam are n, T, v=ver, B = Bθeθ+ Bzez, E = Eθeθ+ Ezez

and are considered all functions of (r,t).

With this simplication the starting equation system describing transport model can be written as: ∂n ∂t + 1 r ∂ ∂r(rnv) = 0 ∂ ∂r p + B 2 z 2µ0 + Bθ µ0r ∂ ∂r(rBθ) = 0 E + v × B = η⊥J⊥+ ηk Jk BB 3n ∂T ∂t + v ∂T ∂r +2nT r ∂ ∂r(rv) = −∇ ·q + S ∂Bθ ∂t = ∂Ez ∂r ∂Bz ∂t = − 1 r ∂ ∂r(rEθ) µ0Jθ = − ∂Bz ∂r µ0Jz = 1 r ∂ ∂r(rBθ) (3.5)

where p = 2nT and the current components that are in Ohm's law are given by J⊥ = JθBz − JzBθ B2 (Bzeθ− Bθez) = 1 B2 ∂p ∂r(Bzeθ− Bθez) Jk = JθBθ+ JzBz B (3.6) S is a source term to account for ohmic heating, external heating, fusion alpha particle and radiation losses. The heat ux vector is expressed in terms of thermal diusion and temperature gradients in this way:

q = −nχ∂T

∂rer (3.7)

The exact value of thermal diusion is not important for the aim of this dissertation, it is assumed as a known quantity. Adding the quas-static pressure balance equation J × B = ∇p we have ve equations for ve unknown quantities n, T, Bθ, Bz, v and the

system can be mathematically solved although it is quite dicult. A simple example of how to reduce the model to a closed set of transport equations in tokamak approximation will be given in the next section.

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3.2.1 Transport equation in Tokamak approximation

The equation system (3.5) is dicult to solve both analytically and numerically, but it can be reduced to simple set of equations with tokamak approximation. This approximation assumes that the dominant magnetic eld component points in the axial direction and is independent of r and t. One can write Bz(r, t) = B0+ δBz(r, t), where B0 = const. and

δBz B0. The rst step to reduce the model is to eliminate the electric eld in Faraday's

law by means of Ohm's law. In this way the following equations for the time evolution of the magnetic components are obtained:

∂Bθ ∂t + ∂ ∂r(Bθv) = − ∂ ∂r η⊥Bθ B2 ∂p ∂r − ηk Bz B Jk ∂Bz ∂t + 1 r ∂ ∂r(rBzv) = − 1 r ∂ ∂r r η⊥Bz B2 ∂p ∂r + ηk Bθ B Jk (3.8)

Now we can apply the tokamak approximation to the left hand side of the Bz evolution

in this manner: ∂Bz ∂t + 1 r ∂ ∂r(rBzv) ≈ ∂ ∂t+ v ∂ ∂r δBz+ B0 r ∂ ∂r(rv) ≈ B0 r ∂ ∂r(rv) (3.9) using this approximation we can integrate over r to obtain an explicit expression for v: v ≈ − ηk µ0B02 Bθ r ∂ ∂r(rBθ) − η⊥ B2 0 ∂p ∂r (3.10)

where the approximation Jk ≈ Jz is used. This expression of v can be substituted in

the time evolution equations for n, T, Bθ and one nally arrives to the following set of

simplied transport equations: ∂n ∂t = 1 r ∂ ∂r r2nT η⊥ B2 0 ∂n ∂r + n T ∂T ∂r + 2ηk βpη⊥ n rBθ ∂rBθ ∂r , 3n∂T ∂T = 1 r ∂ ∂r rnχ∂T ∂r + S, ∂rBθ ∂t = r ∂ ∂r ηk µ0r ∂rBθ ∂r . (3.11) with, βp = 4µ_B0nT2

θ . For simplicity the compression and convective terms in energy

equation are neglected. In classical transport theory these terms are small compared with thermal diusivity: χ Dn. At this moment the classical description is more than enough.

Dening particle diusion and convective coecients Dn, vn, DT, DB as shown below,

one can manages these equations to obtain relation of the form of generic transport equation (3.1).

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Dn = 2nT η⊥ B2 0 vn = Dn 1 T ∂T ∂r + 2ηk βpη⊥rBθ ∂rBθ ∂r DT = nχ DB = ηk µ0 (3.12)

The mathematically expressions in the system (3.12) clearly point out the delicate issue of the dependence of transport coecients from other quantities such as density, temperature and them gradients. In this way one obtains three coupled non-linear partial dierential equations simpler than the starting equation but it is still quite dicult to be solved.

3.2.2 Beyond the tokamak approximation

In previous section I analyzed a classical transport model in order to show the non trivial dependence of transport coecients from the physical quantities of the system and their gradients. In tokamak approximation a relative simple closed system of transport equa-tions is obtained. Although it is a fairly common situation I will argue some dierences with a generic case, which takes account of the dependence of the axial magnetic eld Bz(r)from the radial coordinate, and of the fact that the components Bθ(r)and Bz(r)are

of comparable magnitude. For example this could be the RFP-case where toroidal and poloidal magnetic components are of comparable strength and the sign of the toroidal component reverses at edge of the chamber. Let's start from the equation (3.8). Assuming stationary conditions for the axial magnetic eld we can integrate over the radial coordi-nate the Bz component without using tokamak approximation. This is possible because

one has radial derivative from both sides but in this case Bz = Bz(r) is a function of r.

1 r ∂ ∂r(rBzv) = − 1 r ∂ ∂r r η⊥Bz B2 ∂p ∂r + ηk Bθ B Jk v = − η⊥ B2 ∂p ∂r + ηk Bθ Bz(r)B Jk (3.13)

Now the approximation Jk ≈ Jz is not still valid so one can substitute the more general

relation of equation (3.6). Using p = 2nT one obtains the following equation for the time evolution of density: ∂n ∂t = 1 r ∂ ∂r r nη⊥T B2_(r) ∂n ∂r+ η⊥n2 B2_(r) ∂T ∂r − nηkB_θ2 Bz(r)B2(r) 1 µ0 ∂Bz ∂r + nηkBθ B2_(r)µ 0 1 r ∂ ∂r(rBθ) (3.14) This derivation shows that density evolution holds the form of a diusive-convective equation as that of (3.1), furthermore the convective coecient depends on the radial gradient of the Bz component. One can formally derive equations of the same type for

energy too but this is beyond the aim of my dissertation, the results will be an equation more complicate than energy transport equation in the system (3.5).

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This little discussion points out that in general magnetic conguration, transport equations become more complicated, because the dependence from the gradient of axial magnetic component can no longer be neglected. The model is very dicult to solve numerically, furthermore the description is limited by a classical view of the particle transport. The physical mechanism that rule particle transport is often unknown so it's not right to constrain the transport coecients to have a specic form. It's for this reason and in order to avoid very computationally demanding calculations that I prefer to use a simpler and more general model as that of equation (3.2). The latter works in parallel with GA that selects the most suitable transport coecients in order to reproduce the best agreement with experimental data.

3.3 The simulation model

In RFX-mod machine the main impurity species are carbon and oxygen. 1-D collisional-radiative (CR) transport code is used to reproduce experimental line-emission, continuum radiation and total radiated power associated with impurities. The model solves the con-tinuity equations for ion states of the dierent species:

∂nz ∂t = − 1 r ∂ ∂r r − D(r)∂nz ∂r + v(r)nz + Sz(r) Sz(r) = neIz−1−[Iz+ Rz]ne+ RzCXnn nz +Rz+1ne+ RCXz+1nnnz+1 z = 0, 1, ..., ZN (3.15) ZN is the nuclear charge of the atomic species, nz the ion density, D and v are the

radially dependent diusion coecient and the convective velocity respectively. They are assumed independent of the charge of the ions. S is the source term that takes into account the recombination, charge exchange and ionization mechanism. Iz is the

ion-ization rate from the ion of charge Z, Rz is the recombination rate (radiative+three

body+dielectronic),RCX

z is the charge exchange rate and nn is the hydrogen neutral

den-sity. Impurity ionization is mainly due to electron collisions, while recombination processes take place through radiative, dielectronic and three-body recombination 2_{. The rst}

cor-responds to the spontaneous emission of a photon with wavelength related to the released energy. In the second an incident electron is resonantly captured by the recombined ion in a level with large principal quantum number,and the excess energy is spent in the excitation of a bound electron, then in a second step the ion can be radiatively stabilized below the ionization potential by emission of a photon. Lastly three-body recombination involves two electrons one of which has the task of carry away the excess energy and momentum. It is important only for high density plasmas.

The continuity equations (3.15) are initially solved for the ground states of the ions; then, for each radial position, the collisional radiative model is applied to calculate the steady-state excited level populations of the He-like and H-like ions. The code computes population of each ion states and from these calculate the total radiated power, the free-free bremsstrahlung and the free-bound recombination emission in the soft x-rays SXR spectral range. For each impurity there are dierent charge states, so the spectrum emission will be given by the following sum (in cm−3_{· s}−1_{· eV}−1_{) [6]}

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dW dE = n 2 e9.6 × 10 −14 e −E Te E√Te X z nz ne z2gf f+ z2gf bβ(z) (3.16) where β(z) = ξ µ3 χµ Te exp χµ Te + ∞ X ν=1 2χH Te z2 (µ + ν)3 exp z2_χ H (µ + ν)2_T e (3.17) In these formulas dW is the power radiated per unit volume, z is the ionic charge, E and Te are the x-ray energy and temperature respectively and are expressed in units of

electronvolts as the ionization potential for hydrogen, χH, and the ionization potential for

the atomic ground state, χµ. gf f and gf bare the free-free and the free-bound Gaunt factors

evaluated at each step taking into account their dependence on the electron temperature and on the eective charge. They are multiplicative factors that take into account quantum corrections to the classical calculation of the emitted radiation. In the equation (3.17) the rst term represents recombination to the lower states, with ground-conguration µ and all them assumed with the same ionization potential χµ. ξ is the number of vacancies in

the µ-shell before recombination. While the second term represents recombination eects to states with higher principal quantum number.

Temperature Te(r, t) and electron density ne(r, t) proles are given as inputs to the

code.

Te(r = 0, t)is measured by a double-foil technique SXR system [8] , Te(r, t)is inferred

from a 10-chord SXR diagnostic [9] with a time resolution of 1 ms and the edge tem-perature Te(a) is deduced from the measurements with the thermal He-beam diagnostic

[10] . The central line-averaged electron density ne(r = 0, t) is measured by a 16-chord

interferometer [11] while density prole, ne(r, t), is obtained by inversion of interferometer

data using the following analytic expression: ne(r) = y00− y00− y11− y01 r a α − y11 r a β (3.18) y00, y11, y01, α, β are constant parameters determined by the inversion algorithm to

reproduce the experimental measurements. In gure 3.1 are shown some temperature and electron density proles in the RFX-mod experiments.

Other inputs to give to the code are the hydrogen neutral prole, boundary conditions for the impurity inuxes and the main ion temperature. The rst is calculated using a Monte Carlo code [12] that models the neutral density prole nn(r)through the measured

Hα line emission from the edge of the plasma. The nn(r) prole is xed in shape and its

time evolution is given by the product of the normalized density value with ne(0, t)at any

time. Neutral impurity atoms are treated as entering from the wall of the chamber with a constant inux velocity v0 and their density n0 are described by a continuity equation

of this type:

hΓ0i = −v0n0(r)

hΓ0i(a) = Γwall

(3.19) where Γwall is the total impurity inuxes from the wall r = a that are measured with

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(a) (b)

gure 3.1 Typical density and temperature prole, in (a) are displayed electron density proles at dierent time, the data refer to the shot 24597. In gure b is displayed tem-perature prole measured using Tthomson scattering, the data refer to the shot 30305 Ti ≈ 0.7 · Te; although direct measurements of ion temperature are not available the latter

can be inferred from Doppler broadening measurements on several impurity lines (O7+

, O6+, O4+, C3+).

As already mentioned the transport coecients D(r) and v(r) are chosen by GA in order to achieve a satisfactory agreement with experimental data. Typical experimental signals used to optimize transport coecients are line brightness prole, central electron density and total radiated power. The simulated ones are computed by post-process sub-routines at whom I will dedicate a specic subsection.

The separately determination of the transport coecients is only possible by simula-tions in non stationary condisimula-tions. A transient phase is induced by carbon pellet injection and in this case the code computes the pellet ablation and the consequent release of par-ticles due to an additional source term. This tool allows better investigation of the central regions of plasma where the impurity concentration is usually too low to record a clean signal. Some fundamental features about pellet injection simulations will be given in a later subsection.

3.3.1 Collisional radiative model explanation

The calculation of emitted radiation by plasmas requires the knowledge of both the dis-tribution of ions among dierent ionization states and the populations of excited levels of each ions. For this purpose one should resolve a complex system of rate equations that describe population balancing due to processes of ionization, recombination, collisional excitation and de-excitation, radiative decay and absorption. The full problem is quite dicult to resolve so approximations are generally used. Corona model (CM) and colli-sional radiative model (CRM) 3 _{are two approximations frequently used. In the case of}

low electron density ne value, the corona model is the most suitable approximation. This

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is based on balance between collisional ionization (and excitation) and recombination(and spontaneous decay). In this model free electrons are assumed to have Maxwellian velocity distribution and the number of ions in the excited levels are negligible in comparison to the ground level. Three-body recombinations are irrelevant at this electron density so, population between two ionization states of charge z and z+1 are connected by:

nzneIz(Te) = nz+1neRz+1(Te, ne) (3.20)

where the charge-exchange recombination is neglected. In this approximation the pop-ulation densities of excited level are determined by balance between the rate of collisional excitation from the ground level and the rate of spontaneous radiative decay. For electron density suciently low, collision don't interfere with radiative emission. This condition isn't always satised in RFP machine where electron density of 3.5 × e19 _m−3 _{is reached.}

At this ne value the rate of collision increases until depopulation of excited level is

com-pletely determined by collisional de-excitation so CM fails. There is also the possibility that radiative transitions are not allowed because the excited state is a metastable state so it has a long lifetime and de-population can take place only by collisions. For these situations CRM is more suitable. It take into account not only balancing between ex-cited levels and ground level but also between the exex-cited states themselves. For each ionic state the temporal evolution of population densities (excited plus ground state) can be divided according to the relaxation time in long lifetime states, such as the ground state and metastable states, and short lifetime states, such as excited states. The popu-lation densities of ground and metastable states evolve in temporal scale comparable to the time for plasma ion transport across the temperature or density scale lengths, while excited states are considered quasi-static because the time of population redistribution among them is very short comparing to plasma temporal scale. Using these simplica-tions one can describe the continuity equation for the population densities considering the collisional-radiative matrix ,C(k,j)

i,z , where o-diagonal elements contain collisional and

ra-diative coecients linking states k and j for an ion of i-th specie and charge state z, while diagonal elements,Ck,k

i,z , contain the total loss rate coecient from stat k. Dening ri,z+1k

as the recombination rate from the z + 1 ionization state into the level k, the temporal evolution of population densities is given in equation (3.21):

" dn(ρ)_i,z dt 0 # = " C_i,z(ρ,) C_i,z(ρ,j) C_i,z(k,) C_i,z(k,j) # " n(ρ)_i,z n(j)_i,z # + neni,z+1 " r_i,z+1(ρ) r_i,z+1(k) # (3.21) where ρ and represents long-lived time dependent states, while k and j represent short-lived states. An implicit sum over repeated superscript indices is implied. From this equation one can nd the following expression for the population of quasi-static excited levels:

nj_i,z = −neni,z+1 C (j,k) i,z

−1

r_i,z+1(k) − C_i,zj,k)−1C_i,z(k,ρ)n(ρ)_i,z (3.22) Substituting n(j)

i,z into equation (3.21) for n (ρ)

i,z one obtains a time dependent equation

for the long-lived states that takes into account of the redistribution eects of population densities of excited states, allowing the evaluation of eective ionization and recombination rate coecients:

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dn(ρ)_i,z dt =C (ρ,) i,z − C (ρ,j) i,z C (j,k) i,z −1

C_i,z(k,ρ)n(ρ)_i,z +r(ρ)_i,z+1− C_i,z(ρ,j) C_i,zj,k−1r(k)_i,z+1neni,z+1

dn(ρ)_i,z dt = I (ρ) i,zn (ρ) i,z + R (ρ) i,z+1neni,z+1

I_i,z(ρ) =C_i,z(ρ,)− C_i,z(ρ,j) C_i,z(j,k)−1C_i,z(k,ρ) R(ρ)_i,z+1 =r(ρ)_i,z+1− C_i,z(ρ,j) C_i,zj,k−1r(k)_i,z+1 (3.23) The matrix elements C(j,k)

i,z contain all the relevant process that determine population

and depopulation of the atomic levels and the summation over all ρ-states gives the ionization and recombination coecients of equation (3.15). Adding a ux term that takes into account the variation of charge state distribution due to transport of particles, the Collisional radiative model is now complete.

3.3.2 The pellet ablation

Simulations with the pellet carbon injection are performed to separately determine convec-tive and diusive transport coecients. In this case the code computes the pellet ablation considering an additional neutral source term due to the carbon atoms released from the pellet.

(a) (b)

gure 3.2: representation of the pellet ablation, in gure 3.2a is shown the ablation cloud while in gure 3.2b is represented the electrons stick to the eld line that impact with the ablation cloud. rg rp, thus q1(r) ≈ q2(r), where rg is the electron gyro-radius

When the pellet enters in the plasma 4_{, a lot of electrons, ions or neutral atoms strike}

it. The ux of energy transported by these particles causes the vaporization of pellet

Development of a genetic algorithm for transport studies in high-temperature laboratory plasmas

Università di Pisa

Dipartimento di Fisica Enrico Fermi

Development of a genetic algorithm for

transport studies in high-temperature

laboratory plasmas

Contents

Chapter 1

Introduction

1.1 The termonuclear fusion

1.1.1 Plasma parameters

1.2 The magnetic connement concept

1.2.1 The ignition criterion

1.2.2 The Reversed Field Pinch (RFP) conguration

1.3 Multiple helicity state (MH) and quasi-single

he-licity state (QSH)

1.4 The RFX-mod machine

1.5 Why impurity transport

1.5.1 Impurity removal

1.6 Thesis Outline

Chapter 2

The Genetic Algorithm

2.1 The problem of Global Optimization

2.1.1 The structure of a Genetic Algorithm

2.2 Implementation and testing of a genetic algorithm

2.2.1 Specication of control parameters, troubles and

precau-tions of a GA implementation

2.2.2 Comparison of two evolution strategies

Chapter 3

The Collisional Radiative Model

3.1 Continuity Equation

3.2 Trasport in 1-D cylindrical plasma

3.2.1 Transport equation in Tokamak approximation

3.2.2 Beyond the tokamak approximation

3.3 The simulation model

3.3.1 Collisional radiative model explanation

3.3.2 The pellet ablation

Dipartimento di Fisica Enrico Fermi

1.2 The magnetic connement concept

1.2.2 The Reversed Field Pinch (RFP) conguration

2.2.1 Specication of control parameters, troubles and